1. Rockets don't got a "range" like a car running out of gasoline. Instead they have "maximum velocity" called "Delta V".
2. A rocket's propellant tanks are like a "wallet" full of delta-V "money"
3. Each rocket maneuver has a delta-V "cost" which has to be paid with delta-V money. Don't have enough in your wallet? Then you can't afford to do that manuever.
4. A space mission is composed of several manuevers, each with a delta-V cost. The total is the delta-V mission cost.
5. Refilling the propellant tank is like putting more money in the wallet. This is why rocket engineers are trying to figure out how to put gasoline stations in space. Because it sucks trying to do a long trip in your car on one tank.
6. Most rockets are poor, with very little money in their wallet. So they can only afford the cheapest maneuvers
7. One of the cheapest maneuvers is called a "Hohmann transfer." Using one to go from Terra to Mars takes about 5,700 meters per second of delta-V money and 8.6 months of travel time. And about the same to return home.
8. Also you can't just take off on a Mars Hohmann any time you want. Terra and Mars have to be in the proper spots, which happens every 26 months. This is called a "Hohmann Launch Window." And once you reach Mars you have to wait 15.3 months before the return launch window happens.

All those cute starship spec sheets you see with moronic entries like "range" or "maximum distance" betray a dire lack of spaceflight knowledge. Spacecraft ain't automobiles, if they run out of gas they don't drift to a halt. DeltaV is the key.

Mercury's mass is 3.302e23 kg and radius is 2.439e6 m. sqrt[ (6.673e-11 * 3.302e23) / 2.439e6] = 3006 m/s liftoff deltaV.

Mercury's mass is 3.302e23 kg and radius is 2.439e6 m. sqrt[ (1.3346e-10 * 3.302e23) / 2.439e6] = 4251 m/s escape velocity deltaV.

Mercury's surface gravity is 3.70 m/s. Say that the duration of liftoff burn is 30 seconds. Then the gravitational drag would be 3.70 * 30 = 110 m/s Gravitational Drag.

Mercury's basic deltaV cost for liftoff is 3006 m/s. If the spacecraft has an acceleration of 10 g, or 98.1 m/s, then 3006 / 98.1 = 30 seconds Liftoff Burn Duration.

Say the spacecraft can accelerate at 10 g (Terra gravities), or 98.1 m/s. Mercury's surface gravity is 3.70 m/s. So the spacecraft can accelerate at 98.1 / 3.70 = 26.5 Mercury Gravities. Since the deltaV for escape velocity is 4251 m/s, then 4251 / 26.5 = 160 m/s Gravitational Drag (which is close enough for government work to 110 m/s).

Well, for the very unreasonable case where you have an infinitely throttleable rocket, and a 4 g upper limit, you would do something like what’s in the attached. Basically, you keep your accel as high as you can, while staying under a maximum dynamic pressure limit (0.5 * air density * velocity²). Once you’re through the thick part of the atmosphere, density drops off, and you can punch it again. I chose 800 psf for the attached graph, because it is a reasonable upper limit on maximum dynamic pressure (or "Max Q").

In 1925 Walter Hohmann made your life incredibly easier when he discovered Hohmann transfer orbits. Be grateful.

For a Terra-Mars Hohmann with central body being Sol, Terra is starting planet, Mars is destination planet.

• Mass of Sol = 1.9885×10³⁰ kg = M_primary
• Terra's orbital radius = 1.000 AU = 1.496×10¹¹ meters = OrbitRadius_s
• Mars orbital radius = 1.524 AU = 2.280×10¹¹ meters = OrbitRadius_d
• Mass of Terra = 5.9720×10²⁴ kg = M_s
• Mass of Mars = 6.4171×10²³ kg = M_d
• Mean Radius of Terra = 6.3710×10⁶ m = PlanetRadius_s
• Mean Radius of Mars = 3.3895×10⁶ m = PlanetRadius_d
• Terra Parking Orbit Altitude = 300 km = 300,000 m = ParkingOrbitAltitude_s
• Mars Parking Orbit Altitude = 300 km = 300,000 m = ParkingOrbitAltitude_d

Doing the math:

• SemimajorAxis = (OrbitRadius_s + OrbitRadius_d) / 2
• SemimajorAxis = (1.496×10¹¹ + 2.280×10¹¹) / 2
• SemimajorAxis = 1.888×10¹¹ meters
  
  
• μ_primary = 6.674×10^-11 * M_primary
• μ_primary = 6.674×10^-11 * 1.9885×10³⁰ kg
• μ_primary = 1.32715×10²⁰ m³/s²
  
  
• OrbitVelocity_s = Sqrt[ μ_primary / OrbitRadius_s]
• OrbitVelocity_s = Sqrt[1.32715×10²⁰ / 1.496×10¹¹]
• OrbitVelocity_s = 29,785 meters/sec
  
  
• Velocity_s = sqrt[ μ_primary * ((2 / OrbitRadius_s) - (1 / SemimajorAxis))]
• Velocity_s = sqrt[1.32715×10²⁰ * ((2 / 1.496×10¹¹) - (1 / 1.888×10¹¹))]
• Velocity_s = 32,731 meters/sec
  
  
• Velocity_∞s = abs[ Velocity_s - OrbitVelocity_s ]
• Velocity_∞s = abs[ 32,731 - 29,785 ]
• Velocity_∞s = 2,946 meters/sec
  
  
• μ_s = 6.674×10^-11 * M_s
• μ_s = 6.674×10^-11 * 5.9720×10²⁴
• μ_s = 3.9857×10¹⁴
  
  
• ParkingOrbitRadius_s = PlanetRadius_s + ParkingOrbitAltitude_s
• ParkingOrbitRadius_s = 6.3710×10⁶ + 300,000
• ParkingOrbitRadius_s = 6.671×10⁶ m
  
  
• ParkingOrbitCircularVel_s = sqrt[ μ_s / ParkingOrbitRadius_s ]
• ParkingOrbitCircularVel_s = sqrt[ 3.9857×10¹⁴ / 6.671×10⁶ ]
• ParkingOrbitCircularVel_s = 7,730 m/s
  
  
• Velocity_escS = sqrt[ (2 * μ_s) / ParkingOrbitRadius_s ]
• Velocity_escS = sqrt[ (2 * 3.9857×10¹⁴) / 6.671×10⁶ ]
• Velocity_escS =10,931 m/s
  
  
• Velocity_hyperS = sqrt[ Velocity_s∞² + Velocity_sesc² ]
• Velocity_hyperS = sqrt[ 2,946² + 10,931² ]
• Velocity_hyperS = 11,321 m/s
  
  
• DeltaV_s = Velocity_hyperS - ParkingOrbitCircularVel_s
• DeltaV_s = 11,321 - 7,730
• DeltaV_s = 3,592 m/s
  
  
  
  
• OrbitVelocity_d = sqrt[ μ_primary / OrbitRadius_d]
• OrbitVelocity_d = sqrt[1.32715×10²⁰ / 2.280×10¹¹]
• OrbitVelocity_d = 24,126 meters/sec
  
  
• Velocity_d = sqrt[ μ_primary * ((2 / OrbitRadius_d) - (1 / SemimajorAxis))]
• Velocity_d = sqrt[1.32715×10²⁰ * ((2 / 2.280×10¹¹) - (1 / 1.888×10¹¹))]
• Velocity_d = 21,476 meters/sec
  
  
• Velocity_∞d = abs[ Velocity_d - OrbitVelocity_d ]
• Velocity_∞d = abs[ 21,476 - 24,126 ]
• Velocity_∞d = 2,650 m/s
  
  
• μ_d = 6.674×10^-11 * M_d
• μ_d = 6.674×10^-11 * 6.4171×10²³
• μ_d = 4.2828×10¹³
  
  
• ParkingOrbitRadius_d = PlanetRadius_d + ParkingOrbitAltitude_d
• ParkingOrbitRadius_d = 3.3895×10⁶ + 300,000
• ParkingOrbitRadius_d = 3.6895×10⁶ m
  
  
• ParkingOrbitCircularVel_d = sqrt[ μ_d / ParkingOrbitRadius_d ]
• ParkingOrbitCircularVel_d = sqrt[ 4.2828×10¹³ / 3.6895×10⁶ ]
• ParkingOrbitCircularVel_d = 3,407 m/s
  
  
• Velocity_escD = sqrt[(2 * μ_d) / ParkingOrbitRadius_d]
• Velocity_escD = sqrt[(2 * 4.2828×10¹³) / 3.6895×10⁶]
• Velocity_escD = 4,818 m/s
  
  
• Velocity_hyperD = sqrt[Velocity_∞d² + Velocity_escD²]
• Velocity_hyperD = sqrt[2,650² + 4,818²]
• Velocity_hyperD = 5,499 m/s
  
  
• DeltaV_d = Velocity_hyperD - ParkingOrbitCircularVel_d
• DeltaV_d = 5,499 - 3,407
• DeltaV_d = 2,092
  
  
• DeltaV = abs[DeltaV_s] + abs[DeltaV_d]
• DeltaV = abs[2,946] + abs[2,092]
• DeltaV = 3,592 + 2,650
• DeltaV = 5,684 meters/sec

So the Polaris has to be capable of 5,684 m/s of delta V in order to do the Terra-Mars Hohmann transfer from Low Earth Orbit to Low Mars Orbit.

For travel time of a Terra-Mars Hohmann with central body being Sol:

• μ_primary = 1.32715×10²⁰
• SemimajorAxis = 1.888×10¹¹ meters

Doing the math:

• T_h = 0.5 * sqrt[(4 * π² * SemimajorAxis³) / μ_primary]
• T_h = 0.5 * sqrt[(4 * 3.14159² * (1.888×10¹¹)³) / 1.32715×10²⁰]
• T_h = 0.5 * sqrt[2.65684×10³⁵ / 1.32715×10²⁰]
• T_h = 0.5 * sqrt[2,001,915,283,599,362]
• T_h = 0.5 * 44,742,768
• T_h = 22,371,384 seconds = 8.6 months

Delay between Hohmann launch windows for Terra-Mars Hohmann, central body is Sol, Terra is the inferior planet.

• μ_primary = 1.32715×10²⁰
• OrbitRadius_i = 1.496×10¹¹ meters
• OrbitRadius_s = 2.280×10¹¹ meters

Doing the math:

• OrbitPeriod_i = 2 * π * sqrt[OrbitRadius_i³ / μ_primary]
• OrbitPeriod_i = 2 * 3.14159 * sqrt[(1.496×10¹¹)³ / 1.32715×10²⁰]
• OrbitPeriod_i = 2 * 3.14159 * sqrt[3.348071936×10³³ / 1.32715×10²⁰]
• OrbitPeriod_i = 2 * 3.14159 * sqrt[25,227,532,200,580]
• OrbitPeriod_i = 2 * 3.14159 * 5,022,702
• OrbitPeriod_i = 31,558,565 seconds
  
  
• OrbitPeriod_s = 2 * π * sqrt[OrbitRadius_s³ / μ_primary]
• OrbitPeriod_s = 2 * 3.14159 * sqrt[(2.280×10¹¹)³ / 1.32715×10²⁰]
• OrbitPeriod_s = 2 * 3.14159 * sqrt[1.1852352×10³⁴ / 1.32715×10²⁰]
• OrbitPeriod_s = 2 * 3.14159 * sqrt[89,306,800,286,328]
• OrbitPeriod_s = 2 * 3.14159 * 9,450,228
• OrbitPeriod_s = 59,377,531 seconds
  
  
• SynodicPeriod = 1 / ( (1 / OrbitPeriod_i) - (1 / OrbitPeriod_s))
• SynodicPeriod = 1 / ( (1 / 31,558,565) - (1 / 59,377,531))
• SynodicPeriod = 67,359,430 seconds = 26 months = 2.14 years

• 

Angle between planets for Terra-Mars Hohmann.

• r1 = 1.496×10¹¹ meters
• r2 = 2.280×10¹¹ meters

Doing the math:

• α = π * (1 - ( (1/(2*sqrt[2])) * sqrt[(r1/r2 + 1)³]))
• α = 3.14159 * (1 - (0.35355 * sqrt[(1.496×10¹¹/2.280×10¹¹ + 1)³]))
• α = 3.14159 * (1 - (0.35355 * sqrt[(0.65617 + 1)³]))
• α = 3.14159 * (1 - (0.35355 * sqrt[1.65617³]))
• α = 3.14159 * (1 - (0.35355 * sqrt[4.54271]))
• α = 3.14159 * (1 - (0.35355 * 2.13136))
• α = 3.14159 * (1 - 0.75355)
• α = 3.14159 * 0.24645
• α = +0.77424 radians = +44.36°

Since Phase Angle α is positive, Mars is ahead of Terra.

For the Mars-Terra Hohmann, Phase Angle α = -1.31229 radians, or -75.19° behind Mars.

In a typical round trip, you start at a planet (say, Earth), then execute a Hohmann transfer to another planet (say, Mars). However, you cannot return immediately, since Earth and Mars are then not in the right places. You must wait a certain amount of time before taking another Hohmann transfer back.

How long will that be?

Working out the problem turns out to be relatively straightforward. (For maintaining intuition, I'll continue with the example of visiting Mars, but note that the analysis remains generally applicable.)

First, we define four times:

• t₀: the time of departure from Earth.
• t₁: the time of arrival at Mars.
• t₂: the time of departure from Mars.
• t₃: the time of arrival at Earth.

Next, we define the (heliocentric) angular position of these planets at given times. As time progresses over the planet's orbital period, these angles sweep out 2π radians of passage over a single planetary year (a period of time P_E and P_M for Earth and Mars, respectively). We are only interested in their values at the four points above, though. E.g. θ_M,2 is the angle of Mars at time t₂.

We need to define how these angles relate to each other. Let's call t_H := t₁-t₀ the duration of the Hohmann transfer; notice that it is the same going out as coming back. For the time period t₀ to t₁, the spacecraft is on a Hohmann transfer from Earth to Mars. The angles relate as:

In English, this just says that the (angular) positions of both Earth and Mars advance over the time of the outgoing transfer.

At Mars, we'll wait for some unknown time t_W (waiting time). Again, the planets' angular positions advance:

Finally, coming back, we do another Hohmann:

We still need some more information to solve this problem, though. The first two pieces are that a Hohmann transfer always takes you an angle π around the center:

The first equation says that the spacecraft departing Earth must arrive at Mars after going halfway around the orbit, while the second says that the spacecraft leaving Mars must arrive back at Earth, again after a half-orbit. We need to be more-careful, though. Although the angles for the Earth and Mars can be taken to increase monotonically, one of them might do a full orbit while the other has not. We therefore need to be able to add/subtract multiples of 2π of the angles to get them to agree:

We need just one more equation. The entire problem can be shifted by a constant amount around the Sun. To constrain it, without loss of generality we can simply set:

(I'll also rename the remaining k₁ to just k.)

We can rewrite this huge mess of equations as a matrix equation to impose some semblance of sanity on the complexity:

Solving this is not too difficult (I did it by hand first before checking my result with sympy). The only part of it we care about is the value for t_W, which works out to be:

Because k is just some integer, we can remove that -1. Generalizing the notation a little, so that the home planet has period P₀ and the visited planet has period P₁, we get:

For k, any value can be chosen so long as the resulting t_W is nonnegative.

---

This was the result I gave here (the Google+ question which nerd-sniped me into doing this whole thing). I'm fairly convinced it's correct, but, it's still a little unsatisfactory. Choosing k from the integers is obnoxious. It would be nice to choose it from the natural numbers (i.e., ℕ := {1,2,3,…}), and thereby get an increasing sequence of possible departure dates, starting from a value that is the earliest.

Requiring that t_W ≥ 0, some algebra shows that, if P₀ > P₁, then we must have k ≥ (-2t_H)/(P₀). Similarly, if P₀ < P₁, then k ≤ (-2t_H)/(P₀). And, of course, if P₀ = P₁, then the planets are co-orbital and you cannot travel between them with Hohmann transfers (so we shall assume this is not the case forthwith).

We now want to generate a sequence of k values from natural-valued n values that produces the correct result either way:

Some ad-hoc finagling with the intersection of these lines in the middle produces:

By substituting back into our answer, we can get an equation that tells you every possible waiting time, starting from the first, indexed by n ∈ ℕ = {1,2,3,…}:

I'm less-confident of this result than the previous but, as we'll see below, it seems to work—at-least for the P₀ < P₁ case I tested.

---

Let's check and demonstrate our work by returning to the Earth/Mars example we started with.

The actual values are (note sidereal value for P_i whereas Gregorian calendar used to convert from months):

P₀ = 365.256363004
P₁ = 686.980
t_H =“8.5 months” ≈ 259 days
t_W =“14.9 months” ≈ 454 days

According to our first formula, we get:

k	tW
-3	≈ 1235 days
-2	≈ 455 days
-1	≈ -325 days
0	≈ -1105 days
1	≈ -1885 days
2	≈ -2665 days
3	≈ -3445 days

Therefore, k must be -2 or less. Notice also the separation between the possible waiting times. They should come in multiples of the transfer window times (because if you miss your departure date for the aligned planets, you'll have to wait until the next transfer window). For Earth/Mars, this is “26 months” ≈ 791 days. Indeed, we see that successive launch times differ by roughly this many days. The agreement is to within 1.5% or so, which is pretty good given the imprecision of t_H and the expected t_W, as well as the astronomical fact that the orbits are imperfect.

For our second formula, we have:

n	tW
1	≈ 455 days
2	≈ 1235 days
3	≈ 2015 days

This confirms that the formula works when P₀ < P₁, but the case for P₀ > P₁ remains untested.

Terra Space Station and the school ship Randolph are in a circular orbit 22,300 miles above the surface of the Earth, where they circle the Earth in exactly twenty-four hours, the natural period of a body at that distance.

Since the Earth's rotation exactly matches their period, they face always one side of the Earth — the ninetieth western meridian, to be exact. Their orbit lies in the ecliptic, the plane of the Earth's orbit around the Sun, rather than in the plane of the Earth's equator. This results in them swinging north and south each day as seen from the earth. When it is noon in the Middle West, Terra Station and the Randolph lie over the Gulf of Mexico; at midnight they lie over the South Pacific.

The state of Colorado moves eastward about 830 miles per hour. Terra Station and the Randolph also move eastward nearly 7000 miles per hour — 1.93 miles per second, to be finicky. The pilot of the Bolivar had to arrive at the Randolph precisely matched in course and speed. To do this he must break his ship away from our heavy planet, throw her into an elliptical orbit just tangent to the circular orbit of the Randolph and with that tangency so exactly placed that, when he matched speeds, the two ships would lie relatively motionless although plunging ahead at two miles per second. This last maneuver was no easy matter like jockeying a copter over a landing platform, as the two speeds, unadjusted, would differ by 3000 miles an hour.

Getting the Bolivar from Colorado to the Randolph, and all other problems of journeying between the planets, are subject to precise and elegant mathematical solution under four laws formulated by the saintly, absent-minded Sir Isaac Newton nearly four centuries earlier than this flight of the Bolivar — the three Laws of Motion and the Law of Gravitation. These laws are simple; their application in space to get from where you are to where you want to be, at the correct time with the correct course and speed, is a nightmare of complicated, fussy computation.

• 

CONJUNCTION CLASS

For high-thrust rockets, the most fuel-efficient way to get to Mars is called a Hohmann transfer. It is an ellipse that just grazes the orbits of both Earth and Mars, thereby making the most use of the planets’ own orbital motion. The spacecraft blasts off when Mars is ahead of Earth by an angle of about 45 degrees (which happens every 26 months). It glides outward and catches up with Mars on exactly the opposite side of the sun from Earth’s original position. Such a planetary configuration is known to astronomers as a conjunction. To return, the astronauts wait until Mars is about 75 degrees ahead of Earth, launch onto an inward arc and let Earth catch up with them.

Each leg requires two bursts of acceleration. From Earth’s surface, a velocity boost of about 11.5 kilometers per second breaks free of the planet’s pull and enters the transfer orbit. Alternatively, starting from low Earth orbit, where the ship is already moving rapidly, the engines must impart about 3.5 kilometers per second. (From lunar orbit the impulse would be even smaller, which is one reason that the moon featured in earlier mission plans. But most current proposals skip it as an unnecessary and costly detour.) At Mars, retrorockets or aerobraking must slow the ship by about 2 kilometers per second to enter orbit or 5.5 kilometers per second to land. The return leg reverses the sequence. The whole trip typically takes just over two and a half years: 260 days for each leg (increasing astronaut exposure to galactic cosmic radiation) and 460 days on Mars. In practice, because the planetary orbits are elliptical and inclined, the optimal trajectory can be somewhat shorter or longer. Leading plans, such as Mars Direct and NASA’s reference mission, favor conjunction-class missions but quicken the journey by burning modest amounts of extra fuel. Careful planning can also ensure that the ship will circle back to Earth naturally if the engines fail (a strategy similar to that used by Apollo 13).

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• 

OPPOSITION CLASS

To keep the trip short (reducing astronaut exposure to galactic cosmic radiation), NASA planners traditionally considered opposition-class trajectories, so called because Earth makes its closest approach to Mars—a configuration known to astronomers as an opposition—at some point in the mission choreography. These trajectories involve an extra burst of acceleration, administered en route. A typical trip takes one and a half years: 220 days getting there, 30 days on Mars and 290 days coming back. The return swoops toward the sun, perhaps swinging by Venus, and approaches Earth from behind. The sequence can be flipped so that the outbound leg is the longer one. Although such trajectories have fallen into disfavor—it seems a long trip for such a short stay—they could be adapted for ultrapowerful nuclear rockets or “cycler” schemes in which the ship shuttles back and forth between the planets without stopping.

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• 

LOW THRUST

Low-thrust rockets such as ion drive save fuel but are too weak to pull free of Earth’s gravity in one go (high specfic impulse but very low thrust). They must slowly expand their orbits, spiraling outward like a car switchbacking up a mountain. Reaching escape velocity could take up to a year, which is a long time to expose the crew to the Van Allen radiation belts that surround Earth. One idea is to use low-thrust rockets only for hauling freight. Another is to move a vacant ship to the point of escape, ferry astronauts up on a “space taxi” akin to the shuttle and then fire another rocket for the final push to Mars. The second rocket could either be high or low thrust. In one analysis of the latter possibility, a pulsed inductive thruster fires for 40 days, coasts for 85 days and fires for another 20 days or so on arrival at the Red Planet.

A VASIMR engine opens up other options. Staying in low gear (moderate thrust but low efficiency), it can spiral low efficiency), it can spiral out of Earth orbit in 30 days. Spare propellant shields the astronauts from radiation. The interplanetary cruise takes another 85 days. For the first half, the rocket upshifts; at the midpoint it begins to brake by downshifting. On arrival at Mars, part of the ship detaches and lands while the rest—including the module for the return flight—flies past the planet, continues braking and enters orbit 131 days later.

Abstract

     You have a rocket in a high circular orbit around a massive central body (a planet, or the Sun) and wish to escape with the fastest possible speed at infinity for a given amount of fuel. In 1929 Hermann Oberth showed that firing two separate impulses (one retrograde, one prograde) could be more effective than a direct transfer that expends all the fuel at once. This is due to the Oberth effect, whereby a small impulse applied at periapsis can produce a large change in the rocket’s orbital mechanical energy, without violating energy conservation. In 1959 Theodore Edelbaum showed that this effect could be exploited further by using up to three separate impulses: prograde, retrograde, then prograde. The use of more than one impulse to escape can produce a final speed even faster than that of a fictional spacecraft that is unaffected by gravity. We compare the three escape strategies in terms of their final speeds attainable, and the time required to reach a given distance from the central body. To do so, in an Appendix we use conservation laws to derive a “radial Kepler equation” for hyperbolic trajectories, which provides a direct relationship between travel time and distance from the central body. The 3-impulse Edelbaum maneuver can be applied to interplanetary transfers, exploration of the outer solar system and beyond, and (in time reverse) efficient arrival and orbital capture. The physics principles employed are appropriate for an undergraduate mechanics course.

I. INTRODUCTION

     Newton’s laws of motion and universal gravitation allow us to relate the geometrical properties of an orbit to conserved quantities such as angular momentum and energy. Changing the velocity of an orbiting object changes these quantities, and therefore the orbital path. One of the earliest descriptions of such orbital maneuvers was Newton’s cannon, a thought experiment that demonstrates the effects of changing a projectile’s launch speed on its resulting orbit — increasing this speed increases the size and period of the orbit. For launch speeds greater than the local escape speed, the projectile does not return, but follows a hyperbolic escape path and approaches an asymptotic final speed v_∞ far from the central body.

     In the 20th century, pioneers of astrodynamics began to use these principles to chart the courses of hypothetical spacecraft that propel themselves by the directed expulsion of matter carried as fuel. The resulting impulses change their orbital paths. Even for a simple system consisting of a rocket and a massive central body, a variety of maneuvers are possible, such as Hohmann and bi-elliptic transfers between orbits, with differing numbers of impulses employed to use the minimal amount of fuel.

     In this paper we discuss three strategies for a spacecraft in a circular orbit to achieve a high-speed escape with v_∞ > 0. We compare their fuel requirements to achieve a given v_∞, and their travel times to reach a given distance from the central body.

     In Sec. II we start with conservation laws to develop equations that relate orbital speed and distance. In Sec. III we use these to compare three escape strategies in terms of fuel usage. In Sec. IV, with the aid of an Appendix, we derive expressions for the travel time from the original orbit out to a given distance. In Sec. V we conclude by discussing applications and providing suggestions for student investigations.

II. BASIC EQUATIONS FOR ORBITAL MANEUVERS

     We assume that the mass of the spacecraft is tiny relative to the central body, such that any change in the latter’s motion can be ignored. The central body can then be used to define an inertial frame in which we analyze the spacecraft’s motions.

     To simplify our analysis, we consider only impulses of duration much less than the orbital period around the central body. Equivalently, for such idealized impulsive maneuvers the rocket can change its velocity without changing its position, so that the pre- and post-impulse Keplerian orbits intersect. For such impulses to have the greatest effect on a spacecraft’s orbital energy and angular momentum, they must be applied either prograde (in the direction of motion, by expelling exhaust backwards) or retrograde relative to the spacecraft velocity vector. This results in all paths being coplanar, reducing our analysis to two dimensions. (We discuss orbital plane changes briefly in Sec V.)

1. Speed as a function of distance

     Between impulses, an object in freefall in the gravitational field of an isolated spherical body of mass M will have a constant specific energy (where we use “specific” to mean “per unit mass”),

     Here, v is the magnitude of its velocity vector, r is its distance from the central body’s center, and G is the gravitational constant. (This energy properly belongs to the system of the spacecraft and central object, but since the massive central body is assumed stationary, we assign it to the spacecraft for brevity.)

     From Eq. (1), in order to escape with an asymptotic final speed v_∞ at infinity, an unpowered spacecraft at a distance r_dep must have a departure speed v_dep there given by

     For the maneuvers described in this paper, we also need to relate the orbital speed of a spacecraft to its position in a bound elliptical orbit. Let its speed be v_A at apoapsis (farthest from the central body, at distance r_A) and v_P at periapsis (the closest point, distance r_P). Due to the central nature of the gravitational field, the specific angular momentum is conserved and given by

where v_θ is the azimuthal component of the spacecraft’s velocity. Therefore, at the extremes of the ellipse where there is no radial motion,

Combining Eqs. (1) and (4) we can solve for apoapsis and periapsis speeds in terms of the corresponding distances

The first square-root factor in each equation is the circular

2. Relating impulse to fuel usage and energy input

     Typically, a rocket engine is modeled as one that expels its fuel continuously, even for short-duration impulses that represent the idealized limit of minimizing fuel usage and transfer times. Conservation of momentum leads to the rocket equation, attributed to Konstantin Tsiolkovsky, which relates the speed change Δv to the initial and remaining mass of the rocket,

where m_i and m_f are the masses of the rocket (including fuel) before and after the impulse, respectively, and v_ex is the effective exhaust speed. (The gravitational attraction between the rocket and exhaust mass can be ignored.) Equation (6) does not depend on the rate of fuel consumption, and applies to multiple intermittent firings of the rocket engine. For this reason, we shall follow the standard convention of using total Δv as a proxy for fuel cost when comparing strategies to achieve an orbital transfer.

     When a mass of fuel (m_i - m_f) is expelled to obtain a given Δv, the chemical energy converted to kinetic energy is ΔE_chem = ½ (m_i - m_f)v_ex². A fraction of this input energy changes the mechanical energy of the rocket, while the rest is carried away in the exhaust, as we shall discuss in Sec. III.5.

III. ESCAPE FROM A CIRCULAR ORBIT

     Initially, we have a spacecraft in a circular orbit of radius r₀ with orbital speed v₀ = √GM/r0 (This relation can be proven by centripetal arguments, or by setting r_A = r_P = r₀ in Eq. 5.) We desire to achieve a target final speed v_∞ for the smallest total Δv, so that the smallest amount of fuel is required.

1. Single impulse for direct escape

     The simplest escape strategy is for the rocket to fire a single, prograde impulse from the circular orbit, to increase its speed past the local escape speed √2v₀, as shown in Fig. 1(a). To find the specific impulse Δv_D for such a direct escape to produce a final speed v_∞, use Eq. (2) with v_dep = v₀ + Δv_D, and r_dep = r₀, i.e.

This function is plotted as a green dotted line in Fig. 2.

2. Two-impulse escape - the “Oberth maneuver”

     In 1929, Hermann Oberth described a more fuel-efficient strategy to obtain the same v_∞ from a high circular orbit (of radius many times that of the central body). First use a retrograde impulse to drop closer to the body, then apply a prograde impulse at the periapsis of that elliptical orbit, as shown in Fig. 1(b). Author Robert Heinlein (who had worked with Oberth on the pioneering science fiction film Destination Moon) described this maneuver in chapter 7 of his novel The Rolling Stones:

“A ship leaving the Moon or a space station for some distant planet can go faster on less fuel by dropping first toward the Earth, then performing her principal acceleration while as close to the Earth as possible.”

     Fig. 1(a), 1(b), 1(c). Three escape strategies from a counter-clockwise circular orbit, all with a total Δv = 1.1 v₀. Dots show the spacecraft position at intervals of 1/40th of the original circular orbital period. The rocket images mark the position and direction of each impulsive thrust.
     (a) Direct (single-impulse) escape.
     (b) Oberth (2-impulse) escape with r_in = 0.15 r₀.
     (c) Edelbaum (3-impulse) escape with r_out = 1.50 r₀ and r_in = 0.15 r₀.
     (These values were chosen for clarity of plotting, and are not the same as those adopted for Figs. 2 and 3.)

     Reference 7 explains the physics of this Oberth effect in detail. If a rocket body of mass m moves at speed v, a forward specific impulse Δv increases its kinetic energy by ½m(v+Δv)² - ½mv² which increases linearly with v. Since an orbiting rocket travels faster as it moves deeper into the potential well, firing an impulse as close as possible to the central body allows it to maximize the energy gained, in principle without limit as r_P→0. (In practice r_P is limited by the size of the central body).

     The increase in the mechanical energy of the system of rocket+exhaust comes from the chemical energy converted in the impulse ΔE_chem, which is constant regardless of where the impulse was fired. Therefore, any additional energy gained by a fast-moving rocket must be obtained at the expense of the mechanical energy carried by the expelled fuel, as we discuss in Sec. III.5.

     The two-impulse Oberth maneuver is defined by the periapsis distance r_in of the elliptical orbit produced by the initial, retrograde impulse at radius r₀. From Eq. (5) with r_A = r₀ and r_P = r_in, the required velocity change for this first impulse Δv_Ob1 is

where the leading negative sign denotes the retrograde direction. Once the spacecraft reaches periapsis, it will be moving at speed v_P given by Eq. (5) with r_A = r₀ and r_P = r_in. To achieve a final speed v_∞, use Eqs. (2) and (5) with r_dep = r_in to find the second prograde velocity boost Δv_Ob2 from v_P to v_dep,

The total specific impulse for the Oberth escape maneuver is the sum of the absolute values of Eqs. (8) and (9), i.e. Δv_Ob = |Δv_Ob1|+ Δv_Ob2, which simplifies to

Equation (10) is plotted in Fig. 2 as a dashed purple curve for r_in = 0.05r₀. This curve crosses that described by Eq. (7) at Δv_Ob = Δv_D = v₀, independent of r_in, for which the resulting v_∞ = √2v₀, coincidentally equal to the local escape speed from the original orbit. For desired values of v_∞ larger than this, Δv_Ob < Δv_D, so the two-impulse Oberth maneuver will require less fuel than the single-impulse direct escape (cf. Fig. 2).

     As r_in is decreased, the “Oberth advantage” (the difference Δv_D - Δv_Ob) gets larger for v_∞ > √2v₀, since the fuel expelled at periapsis is placed into a lower energy orbit. In the theoretical limit as r_in → 0, the dashed purple curve in Fig. 2 becomes flat, and Eq. (8) shows that Δv_Ob1 → -v₀ for any v_∞. This speed change stops the spacecraft in its original orbit and causes it to fall in radially towards the center of attraction. Then, the tiniest boost Δv_Ob2 at r_in = 0 could produce any value of v_∞ desired.

3. Three-impulse escape - the “Edelbaum maneuver”

     In 1959 Theodore Edelbaum described a coplanar escape maneuver that employs three impulses: a prograde boost to raise apoapsis, followed by a retrograde impulse to cause the spacecraft to fall in to a low periapsis, and thence a final boost to escape. An example is shown in Fig. 1(c). By falling in from a larger radius than the original circular orbit, the resulting faster periapsis speed enhances the Oberth effect, by transferring more of the expelled fuel’s chemical and mechanical energy to the spacecraft.

     The transfer from circular orbit to the intermediate ellipse is defined by the apoapsis radius of that ellipse, r_A = r_out, with periapsis at the original orbit radius, r_P = r₀. Equation (5) gives the required post-impulse velocity v_P, from which we find the necessary boost Δv_Ed1,

The resulting apoapsis speed is also found from Eq. (5). Once the spacecraft reaches this position, it fires a second, retrograde impulse to lower its periapsis distance to a new r_P = r_in < r₀. The required velocity change Δv_Ed2 is then

The speed v_P of the rocket after it falls from apoapsis at rout to periapsis at r_in is given by Eq. (5), and the required velocity for escape from there to a specified v_∞ is found from Eq. (2) with r_dep = r_in. The difference is the final velocity boost Δv_Ed3, i.e.

The total speed change for an Edelbaum escape is the sum of the absolute values of Eqs. (11), (12), and (13), Δv_Ed = Δv_Ed1 + |Δv_Ed2| + Δv_Ed3, and can be simplified to

Equation (14) is plotted in Fig. 2 as a blue dot-dashed curve for r_in = 0.05r₀ and r_out = 2.5r₀. It is also valid for computing Δv_Ob if r_out = r₀, and for Δv_D if r_out = r_in = r₀.

     Compared to the two-impulse Oberth escape maneuver, the Edelbaum escape always has a lower overall Δv for a given v_∞ by an amount

independent of v_∞, with a corresponding savings in fuel expenditure. (Readers can show that Eq. 15 is positive for any r_in < r₀ and r_out > r₀.) This difference can be seen as the constant vertical distance between the dashed purple (Oberth) and dot-dashed blue (Edelbaum) curves in Fig. 2.

     In Fig. 2, the dot-dashed blue curve for Δv_Ed crosses the dotted green curve for Δv_D when

     For larger desired values of v_∞, the Edelbaum maneuver requires a lower total Δv than the direct or Oberth escapes, and so is fuel-optimal for such high-speed escapes.
     In the dual limit r_out → ∞ and r_in → 0, Δv_Ed → (√2 - 1)v₀. This corresponds to a transfer which first sends the spacecraft very far from the central body, then (much later) applies a very small retrograde impulse to nullify its angular momentum, causing it to fall radially inwards to r_in = 0, where a tiny impulse can produce any value of v∞ desired. While this bi-parabolic trajectory has the smallest possible Δv (≈ 0.41v₀) to achieve any desired v_∞, it requires an infinite travel time.

     We can invert and simplify Eq. (14) to give an expression for the mission’s final v_∞ for an available overall Δv provided by the fuel,

This relation also applies to the 2-impulse Oberth escape if one sets r_out = r₀, and to the direct escape if one sets r_out = r_in = r₀.

4. Four or more impulses?

     A valid question is whether additional fuel savings can be achieved using more than three impulses. It has been shown that any fuel-optimal coplanar transfer will consist of no more than three impulses. This also holds true for coplanar bound orbit-to-orbit transfers, which are either Hohmann (two impulses) or bi-elliptic (three). Coplanar escape strategies that produce fuel-optimal transfers with more than three impulses can always be reduced to an equivalent with three or fewer.

     As an example, consider the following 4-impulse strategy: (1) first slow down from circular orbit to an intermediate inner radius r_in, (2) use the Oberth effect there to boost to r_out, (3) slow down again to lower periapsis to r_in, prior to (4) boosting to escape. We have shown above that more efficient escapes are realized as r_out → ∞ and r_in → 0 (to increase periapsis speed and hence the Oberth advantage of the final impulse). Therefore, any intermediate ellipse can be made more fuel-efficient by increasing r_out and decreasing r_in to their mission-constrained extremes. After doing so, the optimal 4-impulse escape degenerates to the same path as the 3-impulse escape and is no more efficient.

     For idealized impulsive maneuvers, a rocket in a closed orbit will always return to the position at which the impulse was applied (unless another impulse is made). A prograde or retrograde impulse at one apsis of an orbit raises or lowers the other apsis. It further follows from Eq. (6) that multiple impulses fired at the same position in orbit are fuel-equivalent to a single, combined impulse.

     For example, if the rocket engine cannot apply the retrograde impulse for the Oberth or Edelbaum maneuver all at once, it can lower its periapsis successively to r_in by multiple firings each time it passes through apoapsis, without any loss of overall fuel efficiency. Students can use the equations in Sec. II to design and evaluate their own multi-impulse maneuvers.

5. Energetics of impulsive escape maneuvers

     How can the same fuel expenditure (and hence energy converted ΔE_chem) produce different final rocket speeds v_∞ for the three escape strategies? The initial mechanical energy of the rocket+fuel system of mass m_i in a circular orbit is E_sys,i = -½m_iv₀² from Eq. (1). After all the fuel is expelled, the mechanical energy added to the system is ΔE_chem from Sec. II.2, which provides the final rocket mass m_f with a speed change Δv given by Eq. (6). The final mechanical energy E_sys,f of the system of rocket+exhaust is then

which is constant for a given Δv and fuel mass expended, and independent of the number of impulses. Therefore, if one wants the rocket to end up with the highest possible mechanical energy m_fε = ½m_fv_∞², one should choose an impulse strategy that leaves the combined exhaust masses with the lowest.

     As an exercise, by expressing v_ex as a multiple of v₀, students can calculate ΔE_chem and the change in kinetic energy of the rocket (due to changes in its speed and mass) after each impulse. The difference gives the mechanical energy change (positive, negative, or zero) of the fuel mass expelled. This will reveal for each strategy how the same final system energy E_sys,f is distributed between the escaping rocket and the total mass of expelled fuel.

6. Comparison with a “no gravity” rocket

     A remarkable result is that both Oberth and Edelbaum escapes can attain a higher v_∞ than a fictional rocket that is unaffected by the central body’s gravity, which boosts tangentially from orbit at r₀ to a constant speed v_∞ = v₀ + Δv, shown as a dotted gray line in Fig. 2. For example, use Eq. (17) to calculate v_∞ for each escape strategy using a mission total Δv = 1.25v₀. With r_in = 0.05r₀, the Oberth escape gives v_∞ = 2.30v₀; additionally setting r_out = 2.5r₀ gives v_∞ = 2.92v₀ for the Edelbaum escape — both faster than v_∞ = 2.25v₀ for the “no gravity” rocket.

     The Oberth and Edelbaum trajectories can end up with larger values of v_∞ because the fuel on board the “no gravity” rocket carries no gravitational potential energy relative to the central body. With no potential energy to “steal” from the expelled fuel, that spacecraft’s kinetic energy gain is limited to a fraction of the fuel’s chemical and kinetic energy only.

IV. TRAVEL TIME COMPARISON

     Compared to a single-impulse direct escape, both the Oberth and Edelbaum escapes can produce faster final speeds for a given total Δv (and thus fuel). However, these maneuvers require additional elliptical orbit segments where the spacecraft moves slowly. For very distant destinations (r >> r₀), the extra time on these segments may not be important, but for intermediate distances there will be a trade-off in total time to destination. In this section we calculate the travel times from the initial circular orbit to a specified distance r from the central body, to determine which of the three strategies will reach that distance in the shortest time.

1. Hyperbolic segment

     All three escape strategies culminate in a hyperbolic escape from a prograde impulse applied at periapsis. For the direct escape, the departure radius r_dep = r₀; for the Oberth and Edelbaum escapes, r_dep = r_in.

     In the Appendix, we derive a general relationship for a hyperbolic orbit between the time since periapsis and the distance r from the central body. Equation (A6) gives the time of flight from a periapsis distance r_P out to a destination distance r . We can rewrite that equation in terms of the original circular speed v₀ and period T₀ = 2πr₀/v₀,

2. Elliptical segments (2- and 3-impulse escapes)

     For the direct impulse, there are no intermediate orbits so the total travel time is just the hyperbolic segment time t_hyp(r₀,r). For the Oberth and Edelbaum transfers, Kepler’s third law provides the time spent on the elliptical segments. The transfer time t_ell(r_A, r_P) between apoapsis and periapsis is half the period of an orbit of semi-major axis (r_A + r_P)/2, as seen in Fig. 1,

     To obtain the total travel time for the two-impulse Oberth maneuver, we must include the time for the infalling segment t_ell(r₀, r_in). For the three-impulse Edelbaum maneuver, we must include two elliptical segment transfer times t_ell(r₀, r_out) + t_ell(r_out, r_in).

3. Shortest travel time for Edelbaum escape

     Expressions for the total travel times for the three escape strategies are summarized in Table I. These are plotted in Fig. 3 as a function of the Edelbaum maneuver “swing out” radius r_out, using the same parameters as for Fig. 2 and a destination distance r = 200r₀. Clearly for this case there is an optimal swing-out radius r_out ≈ 2.5r₀ for the Edelbaum transfer to minimize the travel time. For larger values of r_out, the additional travel time in the first elliptical segment of Fig. 1(c) offsets the Oberth effect’s advantage of gaining a larger v_∞ for the same overall Δv.

     There is no tractable closed-form solution for the time-minimizing r_out as a function of destination distance r and mission constraints Δv_Ed and r_in. Instead, students can use Eq. (17), followed by Eqs. (19) and (20), to evaluate travel times for a range of values of r_out given the values of the other parameters. As the destination distance r increases, the time-optimal value of rout also increases, since the elliptical segments take up a smaller fraction of the overall trip duration, while the larger v_∞ reduces the remaining time spent on the hyperbolic segment.

4. Travel times compared

     Despite their slow starts, spacecraft executing Oberth and Edelbaum escapes can overtake a spacecraft on a direct escape and arrive at a distant destination sooner, as shown in Fig. 3. As discussed in Sec. III.6, a spacecraft on either of the 2- or 3-impulse trajectories can even overtake a fictional rocket that experiences no gravitational attraction to the central body. The travel time t_nog for this putative “no gravity” rocket is simply the length of the linear segment from r₀ to r divided by its constant v_∞ = v₀ + Δv,

and is shown as the dotted gray line in Fig. 3.

V. DISCUSSION

     For a spacecraft that can barely escape the central body (v_∞ ≈ 0), Fig. 2 shows that a single-impulse direct transfer is most efficient in terms of Δv. This is also the case for a spacecraft in a low orbit that cannot approach the central body any closer than its original orbital radius r₀. For a high-speed escape from a high circular orbit, the Oberth and Edelbaum maneuvers can produce a larger v_∞ for a given Δv, as shown in Fig. 2. However, to obtain the benefit of reduced travel times, the destination distance must be much greater than the original orbital radius, since the Oberth and Edelbaum transfers require slow elliptical segments prior to the hyperbolic escape. A mission that calls for a high-speed escape from a lunar-like orbit around Earth (r₀ ≈ 60 Earth radii and v₀ ≈ 1 km/s), or from a high orbit around any planet or asteroid, can benefit from employing the Oberth or Edelbaum maneuver.

     Most mission concepts are constrained by the total Δv available from the fuel on board. If employing either the Oberth or Edelbaum transfer, the inner radius r_in of the intermediate orbit is limited by the radius of the central body (including its atmosphere), and in the case of the Sun, by heating and radiation considerations. (The Parker Solar Probe makes perihelion passes as close as 10 solar radii ≈ 0.05 AU, but an escaping spacecraft would only have to do so once.)

     To date, no mission has used an Oberth or Edelbaum transfer to send a spacecraft to the outer solar system. This is mainly because the “Oberth advantage” over direct escape works best for values of total Δv ≈ v₀ (≈ 30 km/s for a heliocentric orbit at r₀ = 1 AU), which is not currently attainable by chemical rockets. Some mission concepts have proposed hybrid gravity-assist/Oberth trajectories that use a carefully timed fly-by of Jupiter (r_out = 5.2 AU) to provide some of the retrograde Δv_Ed2 necessary to cause the spacecraft to fall in close to the Sun.

     Thus far we have restricted our analysis to coplanar orbits and impulses. However, if one desires to change the plane of the escape hyperbola, the Δv (and fuel cost) to do so is greatly reduced by applying the plane-change impulse simultaneously with the second, retrograde impulse of the Edelbaum transfer at r_out, where the spacecraft is moving slowest.

     An important application of the 3-impulse Edelbaum maneuver is in time reverse, for efficient arrival and capture into a chosen circular orbit when a spacecraft approaches a planetary body at high relative speed. For approaches with v_∞ > v₀, the optimal strategy is shown in Fig 1(c) by reversing the direction of the arrows, and consists of a braking impulse at periapsis (which could be partially achieved by repeated aerobraking passes) into an eccentric orbit, followed by a periapsis-raising prograde impulse, and a third, “circularization burn”.

     For student discussions, Table II summarizes pros and cons of the Edelbaum transfer compared to direct escape and gravity-assist (“slingshot”) trajectories. Students can explore how the functions in Figs. 2 and 3 change as they vary r_in, r_out, and total Δv to plan their own escape and approach maneuvers. They can also extend the analysis presented here to incorporate noncircular initial orbits, which can alter the choice of fuel-optimal transfer strategy. Trajectories can be simulated using software such as Systems Tool Kit (STK), NASA’s General Mission Analysis Tool, Orbiter, or Kerbal Space Program.

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      Space manifolds form the boundaries of dynamic channels to provide fast transport to the innermost and outermost reaches of the solar system. Such features are an important element in spacecraft navigation and mission design, providing a window to the apparently erratic nature of comets and their trajectories. In a new report now published on Science Advances, Nataša Todorović and a team of researchers in Serbia and the U.S. revealed a notable and unexpected ornamental structure of manifolds in the solar system. This architecture was connected in a series of arches spreading from the asteroid belt to Uranus and beyond. The strongest manifolds were found linked to Jupiter with profound control on small bodies across a wide and previously unknown range of three-body energies. The orbits of these manifolds encountered Jupiter on rapid time-scales to transform into collisional or escaping trajectories to reach Neptune's distance merely within a decade. In this way, much like a celestial highway, all planets generate similar manifolds across the solar system for fast transport throughout.

Navigating chaos in the solar system

     In this work, Todorović et al. used fast Lyapunov indicator (FLI); a dynamic quantity used to detect chaos, to detect the presence and global structure of space manifolds. They captured the instabilities acting on orbital time scales with the sensitive and well-established numerical tool to define regions of fast transport in the solar system. Chaos in the solar system is inextricably linked to the stability or instability of manifolds forming intricate structures whose mutual interaction can enable chaotic transport. The general properties can be described relative to the planar, circular and restricted three-body problem (PCR3BP) approximating the motion of natural and artificial celestial bodies. While this concept is far from being fully understood, modern geometric insights have revolutionized spacecraft design trajectories and helped build new space-based astronomical observatories to transform our understanding of the cosmos.

The dynamics of space manifolds that allow the grand tour of the solar system via an interplanetary transport network has also contributed to the transit mechanisms of the Jupiter-family comets (JFCs). The JFCs are the evolutionary products of trans-Neptunian objects that continue to evolve through the giant planet region as Centaurs and into the inner solar system. Cometary and asteroidal bodies occupying orbits in the region between Jupiter and Neptune and Centaurs are dynamic and unstable with lifetimes of only a few million years. Astrophysicists usually use vastly diverse time scales to model detailed dynamic pathways that connect different time zones of the outer solar system.

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Greeks and Trojans—the global structure of space manifolds

     Todorović et al. considered the short-term (100-year) evolution of massless test particles (TPs) located on orbitals between the main asteroid belt and Uranus. They presented the data in dynamic maps based on two widely used orbit integration packages ORBIT9 and REBOUND while developing a force model containing seven major planets from Venus to Neptune as perturbers alongside the Sun/Jupiter/test particle three-body system. Co-orbital asteroids known as "Greeks" and "Trojans" followed the same orbit as Jupiter but led or trailed the planet by an angular distance.

     The team computed the FLI (fast Lyapunov indicator) across 100 years for a large grid, where lighter regions represented orbits located on stable manifolds and darker regions represented those away from them. The researchers noted an emergence of a large "V-shaped" chaotic structure connected to a series of arches at increasing heliocentric distances and nearly following the Perihelion line of Jupiter. The stable manifolds led to chaotic motion due to complex interactions with the corresponding unstable manifolds. These manifolds were analytically highly complex. Furthermore, as expected, Jupiter was the dominant perturber of the system and responsible for the majority of the rich chaotic architecture—tracked all the way beyond Neptune.

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Rapid scattering and collisions, followed by the Centaur: Jupiter-family comets orbital gateway

     To understand the dynamics of manifold- and close-encounter physics in the system, Todorović et al. used software packages to accurately track evolutions through close approaches with Jupiter. Using Jovian-minimum-distance maps for the Greek and Trojan orbital configurations, the team showed how all orbits along the chaotic structures entered Jupiter's Hill sphere during the course of their evolution. To understand close encounter dynamics, the team investigated Lagrange equilibrium points (L₁ and L₂), which define positions in space where the gravitational pull of two large masses precisely equalled the centripetal force required for a small object to move with them. All close-encounter trajectories visited the neighborhood of either L₁ or L₂ Lagrange points, casting light on the poorly understood Greek-Trojan dichotomy of escaped Jupiter Trojan asteroids.

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     Among the test particles (TPs) approaching Jupiter, a few dozen directly collided and their Jovicentric distances became less than Jupiter's radius. Nearly 2000 TPs transitioned from bound elliptical orbits to unbound hyperbolic escape orbits as a result of manifold-induced encounters. The transitioning orbits then reached Uranus and Neptune within 38 and 46 years; the fastest test particles arrived in the Neptunian region under a decade. Scattering or collision with Jupiter was at least several orders of magnitude shorter than those previously reported. Todorović et al. next observed the path of comet 39P/Oterma based on previous work conducted more than two decades ago, where the comet closely followed the invariant manifold structures associated with L₁ and L₂. The work showed how the invariant manifolds were the true orbital gateway that seemed to influence the low-inclination orbits closer to the Lagrange points of outer planets.

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Outlook on chaotic transport

In this way, Nataša Todorović and colleagues reported manifolds that act across orbital timescales of several decades in this work, in contrast to the tens to thousands of millions orbital revolutions that are traditionally considered. Additional information through quantitative studies will provide deeper insights into the transport between the two belts of minor bodies and the terrestrial planet region. The team expect to combine these observations with theory and simulations to improve the existing understanding of celestial transport. The observed effect of Jupiter-induced, large-scale transport on a decadal time scale is no surprise, since space missions have been historically designed for Jupiter-assisted transport, including flybys of Voyager 1 and Voyager 2.

More information: Nataša Todorović et al. The arches of chaos in the Solar System, Science Advances (2020). DOI: 10.1126/sciadv.abd1313

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A free-return trajectory is a trajectory of a spacecraft traveling away from a primary body (for example, the Earth) where gravity due to a secondary body (for example, the Moon) causes the spacecraft to return to the primary body without propulsion (hence the term free).

(ed note: Translation, if an Apollo lunar mission had an engine malfunction in the first half of the journey, the spacecraft would go sailing off into an eccentric Terran orbit and all the astronauts would die. But if it was using a free-return trajectory, the Lunar gravity would automatically sling the spacecraft back towards Terra with no engines needed and the astronauts could land on Terra Firma.)

Earth–Moon

The first spacecraft traveled using free-return trajectory on October 4, 1959 was Russian «Луна-3» (Moon-3). Then, free-return trajectories were introduced by Arthur Schwaniger of NASA in 1963 with reference to the Earth–Moon system. Limiting the discussion to the case of the Earth and the Moon, if the trajectory at some point crosses the line going through the centre of the Earth and the centre of the moon, then we can distinguish between:

• A circumlunar free-return trajectory around the Moon. The spacecraft passes behind the Moon. It moves there in a direction opposite to that of the Moon. If the craft's orbit begins in a normal (west to east) direction near Earth, then it makes a figure 8 around the Earth and Moon.
• A cislunar free-return trajectory. The spacecraft goes beyond the orbit of the Moon, returns to inside the Moon's orbit, moves in front of the Moon while being diverted by the Moon's gravity to a path away from the Earth to beyond the orbit of the Moon again, and is drawn back to Earth by Earth's gravity. (There is no real distinction between these trajectories and similar ones that never go beyond the Moon's orbit, but the latter may not get very close to the Moon, so are not considered as relevant.)

For trajectories in the plane of the Moon's orbit with small periselenum radius (close approach of the Moon), the flight time for a cislunar free-return trajectory is longer than for the circumlunar free-return trajectory with the same periselenum radius. Flight time for a cislunar free-return trajectory decreases with increasing periselenum radius, while flight time for a circumlunar free-return trajectory increases with periselenum radius.

Using the simplified model where the orbit of the Moon around the Earth is circular, Schwaniger found that there exists a free-return trajectory in the plane of the orbit of the Moon which is periodic: after returning to low altitude above the Earth (the perigee radius is a parameter, typically 6555 km) the spacecraft would return to the Moon, etc. This periodic trajectory is counter-rotational (it goes from east to west when near the Earth). It has a period of about 650 hours (compare with a sidereal month, which is 655.7 hours, or 27.3 days). Considering the trajectory in an inertial (non-rotating) frame of reference, the perigee occurs directly under the Moon when the Moon is on one side of the Earth. Speed at perigee is about 10.91 km/s. After 3 days it reaches the Moon's orbit, but now more or less on the opposite side of the Earth from the Moon. After a few more days, the craft reaches its (first) apogee and begins to fall back toward the Earth, but as it approaches the Moon's orbit, the Moon arrives, and there is a gravitational interaction. The craft passes on the near side of the Moon at a radius of 2150 km (410 km above the surface) and is thrown back outwards, where it reaches a second apogee. It then falls back toward the Earth, goes around to the other side, and goes through another perigee close to where the first perigee had taken place. By this time the Moon has moved almost half an orbit and is again directly over the craft at perigee.

There will of course be similar trajectories with periods of about two sidereal months, three sidereal months, and so on. In each case, the two apogees will be further and further away from Earth. These were not considered by Schwaniger.

This kind of trajectory can occur of course for similar three-body problems; this problem is an example of a circular restricted three-body problem.

While in a true free-return trajectory no propulsion is applied, in practice there may be small mid-course corrections or other maneuvers.

A free-return trajectory may be the initial trajectory to allow a safe return in the event of a systems failure; this was applied in the Apollo 8, Apollo 10, and Apollo 11 lunar missions. In such a case a free return to a suitable reentry situation is more useful than returning to near the Earth, but then needing propulsion anyway to prevent moving away from it again. Since all went well, these Apollo missions did not have to take advantage of the free return and inserted into orbit upon arrival at the Moon.

Due to the landing-site restrictions that resulted from constraining the launch to a free return that flew by the Moon, subsequent Apollo missions, starting with Apollo 12 and including the ill-fated Apollo 13, used a hybrid trajectory that launched to a highly elliptical Earth orbit that fell short of the Moon with effectively a free return to the atmospheric entry corridor. They then performed a mid-course maneuver to change to a trans-Lunar trajectory that was not a free return. This retained the safety characteristics of being on a free return upon launch and only departed from free return once the systems were checked out and the lunar module was docked with the command module, providing back-up maneuver capabilities. In fact, within hours after the accident, Apollo 13 used the lunar module to maneuver from its planned trajectory to a free-return trajectory. Apollo 13 was the only Apollo mission to actually turn around the Moon in a free-return trajectory (however, two hours after perilune, propulsion was applied to speed the return to Earth by 10 hours and move the landing spot from the Indian Ocean to the Pacific Ocean).

Earth–Mars

A free-return transfer orbit to Mars is also possible. As with the Moon, this option is mostly considered for manned missions. Robert Zubrin, in his book The Case for Mars, discusses various trajectories to Mars for his mission design Mars Direct. The Hohmann transfer orbit can be made free-return. It takes 250 days in the transit to Mars, and in the case of a free-return style abort without the use of propulsion at Mars, 1.5 years to get back to Earth, at a total delta-v requirement of 3.34 km/s. Zubrin advocates a slightly faster transfer, that takes only 180 days to Mars, but 2 years back to Earth in case of an abort. This route comes also at the cost of a higher delta-v of 5.08 km/s. Zubrin claims that even faster routes have a significantly higher delta-v cost and free-return duration (e.g. transfer to Mars in 130 days takes 7.93 km/s delta-v and 4 years on the free return), and thus advocates for the 180-day transfer even if more efficient propulsion systems, that are claimed to enable faster transfers, should materialize. A free return is also the part of various other mission designs, such as Mars Semi-Direct and Inspiration Mars.

However it should be noted that, travel duration (to Mars or back to Earth) and delta-v requirement depend on the departure year (eg. 2020 or 2022 or so on). 2-year-free-return means from Earth to Mars (aborted there) and then back to earth all combine total is 2 years (0.5 yrs + 1.5 yrs). If entry corridor to Mars is limited (eg. +/- 0.5 deg entry with <9km/s speed as in the reference), 2-year-return is not possible for some years and for some years, delta-v kick of 0.6km/s to 2.7km/s at Mars may be needed to reach back to Earth.

NASA published the Design Reference Architecture 5.0 for Mars in 2009, advocating a 174-day transfer to Mars, which is close to Zubrin's proposed trajectory. It cites a delta-v requirement of approximately 4 km/s for the trans-Mars injection, but does not mention to the duration of a free return to Earth.

Abstract

     The use of electric propulsion for Mars has been explored since the 1970’s when men looked to travel beyond the moon. The use of electric propulsion has been recommended in several studies as a low-risk, lower cost approach to the robotic Mars sample return missions. Electric propulsion has been evaluated for delivery of Mars cargo using power systems order of magnitude beyond state-of-the-art. Electric propulsion has also been considered for fast transits to Mars supporting manned exploration activities. Results of generalized electric propulsion transits form Earth to Mars are presented. Trades are presented as a generalized assessment based on spacecraft mass-to-power ratio, trip time, and propulsion system performance including variable and constant specific impulse, and efficiency.

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II. Fast Transits

     For the purpose of these trades, a fast transit is any transit faster than the equivalent impulsive low energy transit. NASA has flown several missions to Mars using an opposition trajectory; roughly that of a heliocentric Hohmann transfer to Mars. These low energy transits take approximately 250 days and 180 degrees transit around the sun. Therefore all trades are for only Type I trajectories; less than 180o of heliocentric transit. Example plots of Type I impulsive trajectories are shown in figure 1. In practice, there is a very small penalty, for Mars direct entry missions, to reduce the transit time closer to 200 days. The transfers for Mars Odyssey, Spirit and Opportunity trajectories were 200, 208, and 202 days respectively. The penalty to arrive at Mars much faster than 200 days, however; quickly becomes impractical using impulsive propulsion options.

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     The optimal ΔV for a given time of flight fluctuates over the 15 year Earth-Mars cycle due to the eccentricity and inclination of Mars’ orbit. Previous studies have evaluated the ΔV sensitivities and various concepts for alternative architectures with reduced transfer times using impulsive trajectories. While these are beyond the scope of the study, non-electric solutions are not considered viable below 120 days.

     NASA completed several propulsion trades for human Mars missions in the development of the Design Reference Architecture (DRA). The NASA baseline departure is from a 407 km circular orbit using a two-burn Earth escape, to reduce gravity losses, and has transfer times from 175 to 225 days dependent on the launch opportunity. At Mars, the vehicles are inserted in a 1-sol orbit (250 km x 22,793 km) with propulsive insertion for crewed vehicles and both aerocapture and propulsive capture for the cargo vehicles. The crewed Mars Transfer Vehicle (MTV) has an initial mass to low Earth orbit of 356.4 t with a “payload” of 62.8 t and total dry mass fraction of 43%. A high elliptical parking orbit for departure is a higher performance alternative for a reduced departure ΔV. Because chemical propulsion transits are not the focus of these analyses, only reference, impulsive departures were assumed for optimistic results and simplified analysis. The architecture team baselined a nuclear thermal rocket (NTR) with an I_sp capability of 875 - 950s. An optimistic assumption of 1,000 s specific impulse was used for the impulsive comparison analyses. Figure 2 provides the dry mass fraction for an Earth-to-Mars transit using the baseline LEO departure, a high elliptical departure, and an aero Mars Orbit Insertion (MOI) entry from high elliptical departure as a function of transit time.

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The total delivered dray mass fraction is simply calculated by Tsiolkovsky’s “rocket” equation, equation (1). Even with optimistic assumptions, an NTR does not have practical delivered mass fractions for transfer times shorter than 2 - 3 months. The high fidelity DRA mission analysis results limit the NTR capability to a minimum potential transfer time of approximately 140 days for the optimal single launch opportunity of 2035.

A. Variable vs. Constant Specific Impulse

     Extensive Copernicus calculations were conducted to quantify the effect of the variable specific impulse for the Earth-Mars transits. Parametric trades included varying the transfer times from Earth to Mars from 30 – 250 days and jet power values between 50 kW to 500 MW. For each case, the initial mass and departure date were optimized to produce the maximum delivered mass. The trades were constrained to a specific impulse range from 2,000 s to 10,000 s. To simplify the parametric trades, the spacecraft was assumed to depart the Earth and arrive at Mars with a V_∞ of 0 km/s.

     The first results, shown in figure 3, are the dry mass fractions delivered as a function of transfer time. For the variable specific impulse, the delivered dry mass fraction is only dependent on transfer time, and is valid for all power levels. The constant specific impulse cases tended to optimized to the lower I_SP boundary of 2,000 s, especially for the very short transfer times. The variable I_SP will also operate at the minimum constrained I_SP near both Earth and Mars, but can operate at a higher I_SP when the spacecraft is transitioning from accelerating away from the Earth and performing the rendezvous with Mars. The ability to leverage both high thrust when necessary and higher specific impulse when there are minimal maneuver inefficiencies results in propellant savings for the variable specific impulse option. In all cases, the variable I_SP outperforms the constant I_SP. Also, the advantage of variable specific impulse increases for the most stressing cases; the very fast transfer times. For example, a 40 day transit with constant specific impulse can only deliver a 10% dry mass fraction while a variable specific impulse transit can deliver a 22% dry mass fraction. This is compared to a 250 day transit that can deliver a 66% versus a 68% dry mass fractions for constant and variable specific impulse cases respectively. Figure 3 illustrates that the transfer time increase required to match performance is significant for the very short transfers; a 70 day variable I_SP transit is comparable to a 100 day constant I_SP transit with respect to mass fraction.

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     Increasing power only has a negligible effect on the relative performance because, as will be shown later, the ΔV is primarily dependent on the transfer time with a secondary effort due to the thrust-to-power ratio of the propulsion system. The trades are optimized to produce the maximum final delivered mass for a given transfer time; adding additional power only linearly increases the absolute delivered mass capability. Therefore, the critical comparison is the advantage that can be observed in the delivered mass-to-power ratio, system alpha, for the transits. The delivered mass-topower ratio is defined as the total dry mass that can be delivered to the final orbit for a given power level. Figure 4 provides a comparison of delivered mass-topower ratios for the variable and constant specific impulse. Again, the variable specific impulse always outperforms the constant I_SP solution, but the advantage is relatively small until the transit times become very short. As the transit times fall below 40 days, the comparison begins to diverge; increasing the benefit of a variable specific impulse system. For a 30 day transit, the variable I_SP delivered mass-to-power ratio is more than double the constant I_SP ratio.

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B. System Trades

     The majority of the trades are from the baseline LEO departure, using the electric propulsion to escape, performing the transit to Mars, arriving with a V_∞ of 0 km/s, and then spiraling down to Phobos. Data is presented with respect to total system mass-to-power ratios from these four mission milestones: Initial Mass at LEO (IMLEO), Earth escape, Mars Arrival, and Phobos arrival. Note that for the accelerations and mass fractions associated with the fast transits, there is negligible performance difference between a low Mars circular orbit and a Phobos rendezvous at altitude of 5,981 km.

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Figure 5. Trajectories to from Earth to Mars for 200 through 25 days.

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Rendezvous

     The rendezvous mission from LEO to Phobos for various transfer times is shown in figure 5. The transfer time refers to the time between the Earth escape and the arrival at Mars with a V_∞ of 0 km/s. There is additional time associated with the spiral phase from LEO to escape and from Mars arrival to Phobos. The trades are conducted without a duty cycle constraint. From figure 5, the thruster is operating for the majority of the transfer time until the mission durations get very short. For the lower specific impulse and lowest transfer times, the thruster operating time is at its lowest. The thruster operating time as a function of specific impulse for several mission times is shown in figure 6. Figures 5 and 6 show for longer duration missions, the thruster is operating nearly the entire mission. For the shorter duration missions, the thruster accelerates the spacecraft and then has a significant coast period prior to thrusting to rendezvous with Mars.

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     It is also important to note that the ΔV changes significantly with trip time, but also to a lesser extent with thrust-to-power ratio. Figure 7 provides the ΔV versus specific impulse and transfer time. The specific impulse dependency is based on a constant efficiency resulting in a lower thrust for the same power. The ΔV and I_SP can be used to calculate the mass fraction through Tsiolkovsky’s equation, but is calculated directly from the trajectory performance output; shown in figure 8.

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     The two columns from figure 8 are results assuming either staging of the spacecraft at escape or departure from LEO without staging. Because of the large mass fraction required to raise the vehicle from a low-Earth-orbit to escape, one may prefer to stage the spacecraft. For a crewed mission, it is likely that the vehicle would be raised near escape, potentially into a high elliptical orbit and wait for a crew module to dock with the spacecraft prior to escape. For this scenario, the dry mass fraction delivered from escape is improved by the ΔV savings over the LEO departure. This is shown in figure 9 for all specific impulse and transfer times evaluated. At 2,000 s specific impulse, just over 40% of the departure mass is delivered to Phobos, but 57% of the mass from escape is delivered to Phobos on a 200 days transit.

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     The key parameters for mission design for chemical missions are typically the ΔV and specific impulse to characterize the spacecraft in terms of delivered mass performance. For electric propulsion missions, the key parameters are often the thrust-topower ratio of the propulsion system and the spacecraft power-to-mass ratio. The thrust-to-power ratio has a secondary influence on the mission ΔV and the power-to-mass ratio, i.e. P/m ≈ F/m = a, has a strong influence on the achievable mission trip time. Because the trades are conducted with fixed trip times and propulsion system thrust-to-power ratios, the key performance parameter that can be derived from the analyses is the delivered mass-to-power ratio, alpha. Again, because we are fixing trip time and thruster performance, the resulting parameter is the delivered mass-to-power ratio. The delivered alpha is illustrated in figure 10. Figure 10 is based on a 2,000 s I_SP propulsion system and the power is jet power.

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     Figure 10 shows that for every kW of spacecraft power available, a 2,000 s propulsion system can deliver 111 kg to Phobos given a 200 days transfer, 54 kg for a 150 day transfer, 18 kg for a 100 day transfer, etc. Figure 10 also provide the start mass alpha required to deliver those masses. Therefore a 1 kW, 2000 s propulsion system would require a start mass of 273 kg in LEO, it would have a mass of 197 kg at escape, and 120 kg when it arrives at Mars to eventually deliver 111 kg to Phobos. Figure 10 highlights the challenge of very fast transits to Mars; as the trip time decreases, the mass that can be delivered for a given power also decreases exponentially. This is expected because of the ΔV increase as was shown in figure 7. Figure 11 provides the delivered alpha capability for various specific impulse capabilities to Phobos. As will be shown later, figure 9 and 11 can be used for performance trades and feasibility assessments for any payload and trip time to Mars.

     Up until now, all data is based on jet power; assuming each kW of power is available for either thrust or specific impulse following equation (2) assuming 100% efficiency. This is convenient for analysis, but it does not lend itself for evaluating the power and propulsion system requirements based on realistic efficiencies.

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     When propulsion system efficiency is addressed, the delivered mass capability for a given power will decrease; linearly if thrust is held constant. The efficiency is the total percentage of electric power generated that is converted into jet power. Losses include everything from power conditioning, ionization losses, beam divergence, etc. Figures 12 - 15 provides the deliver mass performance for a 2,000 s, 3,000 s, 4,000 s, and 5,000 s propulsion system at various efficiencies.

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     The data clearly shows two competing constraints, mass fraction and the delivered mass-to-power ratio. If we want to have any appreciable payload at very short transfer times, we desire lower specific impulse option. If we want to have practical dry mass fractions, we desire a higher specific impulse option.

Direct or Constrained Entry Velocities

     One method to greatly reduce the mission ΔV is to allow for a direct entry into the Mars atmosphere. This would allow the propulsion system to accelerate the vehicle to a very high velocity without an arrival ΔV penalty. However, the spacecraft must have some method to reduce velocity at arrival. Crewed missions during the Apollo program experienced entry velocities on the order of 11 km/s at Earth; unmanned entry vehicles for science missions have been designed for up to 15 km/s. Analysis for crewed missions to Mars recommended constraining the entry velocity to 10 km/s; corresponding to a V_∞ of 8.65 km/s at a 300 km entry altitude. Unfortunately, as shown in figure 16, the direct entry velocities greatly exceed the human rated limit for transfer times less than 100 days. Therefore, the fast transits to Mars must leverage the propulsion system to slow down before entering the atmosphere. Nevertheless, the potential for mass savings should not be ignored entirely. For a 40 day transfer, the rendezvous transit ΔV is approximately 50 km/s, but approaches only 26 km/s if direct entry was tolerable. Figure 17 provides the delivered alpha using a direct entry and a comparison to a rendezvous. If a direct entry were tolerable, the performance for transfer times under 50 days appears far more practical than the rendezvous solutions.

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     Methods used by scientific spacecraft include entering into a very high elliptical orbit and use aerobraking to reduce the energy of the orbit without onboard propulsion. Unfortunately, the time of the aerobraking sequence would exceed the transfer time to Mars and eliminate any safety benefit achieved by the rapid transit. An aerocapture system is also impractical for such high entry velocities. The human body’s tolerance to vibration is even lower than a directed g-force. Staying within the atmosphere long enough to bleed off 20 - 30 km/s is not something that can be achieved within the tolerances of the human body. It is possible to use the electric propulsion system to constrain the arrival velocity and save approximately 8.5 km/s. The payoff for a constrained entry velocity must be traded with the mass increase imposed on the entry capsule and also the risk increase from a direct entry to Mars without an opportunity to assess the landing site and conditions before entry.

C. Solar vs. Constant Power

     The analysis shown previously is based on having a constant power available to the thrusters. This could be done using a nuclear power source, or by sizing a solar array such that the power fall-off never drops below the propulsion system input power. For missions to Mars, the power drop-off is relatively reasonable. Mars aphelion is only 1.67 AU, and therefore the power drop-off is estimated at approximately 1/2.78. As was shown in figure 5, the majority of the mission ΔV is during the initial departure from the Earth, when the spacecraft is closest to the sun. There is a penalty for the power degradation, but a solar solution may still provide a higher near-term performance than a nuclear alternative. Figure 18 illustrates a trajectory using SEP for a 50 day transit to Mars. The thruster does operate longer during the arrival arc because less power is available. Because the spiral time scales with power, it was initially expected that the spiral to low Mars would be a significant detriment for SEP on crewed missions, however, if the lower delivered alpha is tolerable, the spiral time is not significantly longer than the nuclear option because the arrival mass is lower. The delivered alpha is approximately 75% of the constant power solution; so depending on the alpha of the solar power system, it may not be practical to oversize the solar array. Either the power system will be oversized at 1 AU or the propulsion system with be undersized at 1 AU. The power and propulsion system alphas can be traded with the delivered alpha to assess the benefit of over sizing the solar array or tolerating the performance decrement.

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IV. Results

     Mass fraction and system limitations have been traded to illustrate near-term and far-term viable solutions for fast transits to Mars. Analyses are “spreadsheet” level using generalized system assumptions and not full spacecraft design. Several feasibility cases have been run, and those results are provided in the appendix. A few cases are shown using tables 1 and 2. The values in the tables represent the remaining mass-to-power ratio available for payload; calculated by equation (3). The results are based on various system allocations that are dependent on either the power level or the mass fraction of the launched mass. For example, the power, propulsion, and potentially thermal subsystems can be scaled with power, while structures and propellant tanks are subsystems that may scale as a fraction of the launched mass.

Example Case:

     Assume a 60% efficient propulsion system that operates at 5,000 s specific impulse and has an alpha of 0.5 kg/kW and a power generation system alpha of 0.5 kg/kw, a tankage fraction of 10%, and structure, communications, thermal control, etc. are another 10% of the dry mass. The totals for non-payload subsystems are 1 kg/kW and 20% of the dry mass allocation. If we want to evaluate a 100 day transfer staged at escape to Phobos, it has a delivered alpha of 6.5 kg/kW and a delivered dry mass fraction of 62%. The payload capability is:

Then for a payload mass requirement, MPayload, the spacecraft allocation can be simply calculated as:

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     Therefore, if we wanted to send a payload to Mars of 50,000 kg with this spacecraft, we would need a power system capable of providing 14.66 MW to a spacecraft with a launch mass of 153,350 kg including 58,020 kg for propellant, and 45,330 kg for the remaining dry mass: 14,660 kg for the power and propulsion subsystems, and 30,670 kg for propellant tanks and remaining subsystems.

     If we try to send the same 50,000 kg payload using a variable Isp system; α_Delivered = 11.16 kg/kW and MFDelivered = 41%, and follow equation 3 we calculate a value for α_Payload of 4.72 kg/kW. This translates to a power and launch mass requirement of 10.6 MW and 288,470 kg respectively. This highlights an important note for using variable specific impulse results based on this methodology; because the trades are based on maximizing the delivered alpha, the variable specific impulse case will always use less power to deliver the same payload mass for a fixed transfer time, but that may come at a higher launch mass. One could employ a methodology to minimize IMLEO, but the purpose of the chosen methodology was to determine the maximum delivered mass-to-power ratio to understand the propulsion and power limiting case. In any methodology the variable specific impulse outperforms the constant specific impulse.

     Table 2 provides the 60% efficient propulsion system solution with no dry mass dependence on either power or launched mass. Table two provides the limiting cases for a 60% efficient system. The values in red signify that the delivered mass fraction is less than 30%. Figure two illustrates that there is an exponential decrease in performance as trip time is decrease, that the highest delivered mass-to-power ratio is achieved at low specific impulse, but that performance comes at the cost of a high propellant mass fraction.

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V. Technology Status and Predictions

     While beyond the scope of these analyses, a brief summary on technology status to meet the performance requirments to enable fast transits to Mars is provided. The summary below is not intended to be a comprehensive assessment of the available and required technologies, rather provide a reference point for scaling a few of the key technologies that will be required to enable transfer times to Mars in less than 100 days.

A. Propulsion

     There are several options for electric propulsion systems as the available power scales from 10s of kW to 10s of MW. In many cases the same propulsion system will not be ideal over the entire range of input powers. For example, Hall thrusters may be suitable up to a few hundred kilowatts, but the system may become cumbersome at several megawatts. Also, a magnetoplasmadynamic (MPD) thruster may not make sense for a 100 kW system, but an MPD, ELF, or VASIMR may offers system benefits at > 100 KW, with VASIMR having an added advantage of variable I_SP.

     Though it is beyond the scope of this analysis, a survey of propulsion technologies is relevant to near-term feasibility. Therefore, a few propulsion options are presented in table 3 from an Air Force Research Laboratory status of high power electric propulsion activities.7 For relative comparison, a 200 kW thruster system is shown based on available scaling data. The tables has been updated as new data has become available. These specific mass estimates do not include auxiliary hardware, thermal management, etc. that would increase the system specific mass. Aside from thermal management, there is an expectation that a propulsion system scaled at the megawatt level could be less than 1 kg/kW. Efficiencies in table 1 are thruster efficiencies and does not account for power conditioning and any propulsion related subsystems. Optimistically, total system efficiencies of 40% - 80% may be possible for high power EP systems.

     While the advantage of variable to constant specific impulse may not be a significant driver, there is a distinct advantage in technology development for an evolutionary path of power and propulsion system implementation. Near-term missions are likely to be in the gravity well of the Earth with power levels below 100kW. These systems typically optimize to lower specific impulse. As the power system increases; so will the optimal specific impulse for the mission. While an individual mission may not see a benefit from the variable specific impulse, working out the technology challenges of a system that can vary specific impulse with significant design changes does provide a larger return on investment for intermediate steps. For example, at 50 kW, investments may focus on Hall technology ideal for 2,000 - 3,000 s specific impulses. An intermediate 300 kW system may employ gridded-ion engine technology for 4,000 - 6,000 s operation. When multi-MW power systems become available, the thruster technology may shift again to MPDs, PITs, ELFs, VASIMR, etc. Certainly system level integration and scaling challenges will advance the state-of-the-art, but the thruster challenges will likely require parallel investment of independent technologies. Solving a challenge such as thruster lifetime on a Hall thruster may not be directly applicable to the future systems.

     Also, if one does consider the possibility of multi-hundred megawatt power systems with near zero mass, the divergence of performance for trip times is likely to hold for missions beyond Mars. As the mission performance pushes the boundary of a practical ΔV that can ever be achieved with any system, a variable specific impulse system may offer double the performance capability.

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B. Nuclear Power

     As power systems increase to hundreds of kilowatts and multi-megawatts, nuclear power must be considered. The highest fidelity near-term predictions based on NASA studies from the Prometheus project predicted nuclear power system alpha 30 – 40 kg/kW. The baseline power system produced 200 kW with a core mass of 1,569 kg for a core alpha of 7.8 kg/kw before you include the shielding, control system, and power conversion system. The complete reactor module had a system alpha of approximately 16.5 kg/kW. This reactor module is for the power production, but does not include the mass of the heat rejection system; another 12.8 kg/kW. This brings the JIMO total power system alpha to approximately 30 kg/kw. This system alpha is before margin and is still more than an order of magnitude higher than required for transfer times <90 days.

     It is not expected that turbo-Brayton reactor systems will fall below 10 kg/kW, even for larger power output systems. However, alterative lower-TRL options exist including a Magnetohydrodynamic (MHD) system with a projected alpha approaching 1 kg/kW. Even lower system alpha systems have been proposed using Fissioning Plasma Core Reactor (FPCR) with MHD power generation. The high operating temperatures of an MHD generator (efficiencies near 30%) can be combined with heat recovery steam cycles for overall system efficiencies approaching 60%. These studies highlight the potential for significant improvement in system alphas, but provide little insight into the often over looked power management and distribution (PMAD) system and thermal management for the waste heat generated by the power and propulsion systems.

     A broad NASA study predicted the 200 kW class nuclear power specific masses would approach 30 kg/kW, consistent with the higher fidelity Prometheus results, and also predicts that 100 MW class systems could approach 8 kg/kW including the often over looked power PMAD, radiators, and power conversion redundancy for a 10 year mission. Using more aggressive assumptions and shorter lifetime, the study predicts 6 kg/kW may be achievable. The study predicted the power conditioning and control system to have an alpha of 2 kg/kW and the distribution lines with another 0.5 kg/kW. Data from several studies starting with human Mars mission studies in 1996 through the present from throughout NASA has been compiled and is shown in figure 19. The data is consistent with expectations that Brayton technology will be limited near 10 kg/kW while farther term technology may approach power system alphas near 1 kg/kW for 10s – 100s of MW power.

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C. Solar Power

     The potential does exist to use solar power for exploration within the inner solar system. Solar array technology continues to improve in cell efficiencies and in system alpha. The expanded use of electric prolusion, and the synergistic benefit of electric propulsion and increasing power availability is driving commercial and government towards higher power systems. The SOA solar arrays are currently 15 - 20 kg/kW. The DARPA Fast Access Spacecraft Testbed (FAST) program has progressed the SOA has with near-term projected goals of 8 kg/kW at 1 AU. NASA studies have far-term predictions approaching 3.3 kg/kW including the array, based on Stretched Lens Array Square Rigger (SLASR) technology with advanced cells, gimbals, booms, power cabling, etc. A point design for a 232kW system had a current best estimate (CBE) system mass of 781 kg. Using an advanced cell, a 100 kW End-of-Life, after 10 years at GEO, stretched lens array design has a predicted mass performance of 1.85 kg/kW.

     Figure 20 illustrates the trend of spacecraft power available over time. The International Space Station (ISS) represents the largest space power asset at over 100 kW. The trends of available power illustrate a doubling of power approximately every four years. Base on this trend, a 1 MW, 10 MW and 100 MW solar power system would be available in 13, 27, and 41 years respectively. This does not preclude the possibility of revolutionary improvements in technology, but the evolutionary advancements will not have 100 MW class solar power systems available until after 2050.

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D. Heat Rejection

     Another challenge of very high power generation and propulsion is the waste heat generated by the systems. For use on the interplanetary transfer, assuming a view to deep space, the radiators have an area requirement of approximately 2.2 m²/kW of heat dissipation. The radiators for the International Space Station (ISS) have an area density of 14.64 kg/m². The ISS radiators include mass for deployment support structure. The fast transit electric propulsion vehicle would likely also require deployable radiators. During the NASA Capabilities, Requirements, Analysis, and Integration (CRAI) study, NASA estimated NASA radiators could obtain 10 kg/m² in the near-term with investment. The CRAI team also provided a long-term goal of approaching 2.5 kg/m² using advanced radiator materials and doubling the SOA heat transport capabilities. Therefore the near-term performance would be approximately 22 kg/kW and in the long-term reach 5.5 kg/kW. The majority of the heat waste heat is likely to come from the power generation. If we use the most optimistic assumptions, a 60% efficient power generator and an 80% efficient propulsion system, our total efficiency from thermal heat generated to jet power is only 48%. Therefore, for each kW of jet power, the spacecraft must dissipate 1.08 kW of thermal waste heat. In other words, the long range heat rejection system alpha goal almost 6 kg/kW of electric propulsion jet power. Relative to the performance goals of fast transit, even if the power and propulsion system were without mass, the mass of the heat rejection system alone would prevent the possibility of a transfer to Mars in less than two months. Note this challenge is exacerbated for the nuclear power option. A solar powered option may only need to dissipate 1 – 0.25 kW of waste heat for every 1 kW of jet power produced for 50% - 80% efficient propulsion systems. This would translate into an approximate heat rejection alpha as low as 5.5 – 1.4 kg/kW in the near-term and far-term respectively.

VI. Conclusion

     In 2002, it was reported that 39 transits to Mars may be possible with 200MW NEP system with variable specific impulse using a power system alpha below 1 kg/kW. Results indicate that this may be possible if the power, propulsion, PMAD, thermal control, etc. can approach a combined system alpha approaching 0.5 kg/kW. Variable I_SP may double the performance over constant ISP for trip times less than 40 days; alleviating the requirement closer to 1 kg/kW. In all cases, variable specific impulse can increase both delivered mass fraction and delivered alpha compared to fixed I_SP. State-of-the-art and near-term technology predictions indicate options may exist for a path to achieve very low alpha propulsion systems. However, the power system alpha is far from that necessary to achieve transfers less than 50 days, and would take a revolutionary technology advancement by an order of magnitude; not a evolutionary advancement in Brayton technology. Also, the PMAD and thermal control challenges associated with the tremendous power transmission and conditioning, in addition to waste heat dissipation orders of magnitude beyond those demonstrated, are not likely to be addressed in the near-term. The waste heat rejection system has a total alpha beyond those required for transfers less than 60 days, and there is no plan to achieve the alpha reduction required. Solar array technology may offer the lowest alpha achievable without significant investment in space based nuclear reactors. With advanced solar array technology, combined with high power propulsion system development, crewed missions to Mars may become practical for transfer times of a few months. Therefore, electric propulsion does the have the potential to perform transits to Mars faster than practical with the baselined NTR system. Crewed transits to Mars in a few weeks are not possible using any foreseeable technology available.

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Appendix

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Question

      NICK T: On a porkchop plot for a given departure date (X axis), why are there two local energy minimums along the arrival date (Y-axis) for a transfer between Earth and Mars? The Hohmann transfer orbit seems to have one optimal solution, but the plot's two minimums seem fairly distinct in time but similar in energy; 15.0 to 15.5 and 15.5 to 16.0 in terms of C3L (something I only vaguely understand; KSP mostly deals in delta-V) (C3L is the delta-V of the Trajectory Injection)
     A NASA page about sending spacecraft to Mars refers to "Type 1" and "Type 2" transfer orbits, which clashes with a perhaps simple view that there should be a singular optimal Hohmann transfer orbit; the smallest ellipse tangent to two orbits. Granted, the planets' orbits are not perfectly circular, but the different transit times (red contours) for local minimums differ by almost a factor of two! (about 200 days and about 400 days) I'm not sure if I'm reading the plot correctly though, as there is a disconnect between the given Earth-Mars plot, and the prose on the page as well, which talks about 7 month (~200 day) or 10 month (~300 day) transfers.

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Answer

What are porkchop plots?

     When the goal is to send a spacecraft from the Earth to another planet, it's not enough to reach the target planet's orbit. The vehicle has to meet the target planet itself. The amount of energy needed to accomplish this varies widely depending on departure and arrival dates. A porkchop plot is a graphical interplanetary mission planning tool that depicts as a contour plot the required energy as a function of departure date and arrival date. The energy needed by the launch vehicle is key in determining the feasibility of such a mission. A mission plan that requires more energy than a launch vehicle could possibly provide is not feasible. In addition to feasibility, a porkchop plot aids in planning key mission operations and in planning the optimal trajectory between the two planets. The plot shown in the question and replicated below show launch energy. It does not show the change in energy needed at arrival. Other porkchop plots do show this as a second set of contour lines.
     The original post asks three key questions:

• Why is there a gap in porkchop plots?
• Why don’t they use an optimal Hohmann transfer?
• Why are there two local energy minima?

     Before answering the above, it will help to explain how a porkchop plot is constructed.

How are porkchop plots constructed?

     As a mission planning tool, a porkchop plot makes certain simplifying assumptions with regard to reaching the target planet. Later on, more detailed analyses address those simplifying assumptions. The key simplifying assumptions used in making a porkchop plot are the patched conic approximation and impulsive maneuvers.
     These assumptions reduce the problem to one of finding Keplerian orbits about the Sun that take the spacecraft from the vicinity of the Earth to the vicinity of the target planet in the requisite amount of time. Finding such transfer orbits is the subject of Lambert's problem. A number of such transfer orbits might exist. I’ll denote the angle subtended by the departure point, the Sun, and arrival point as θ, with . The principal value of this angle will be between 0° and 180°, inclusive. For now I’ll ignore cases where θ is 0° or 180°. This means that the transfer plane is well-defined and that the number of solutions is finite.
     Lambert's problem does not have closed form solutions; a number of iterative techniques have been developed to find solutions. One solution, “the short way”, or “Type 1” transfers, has the change in true anomaly equal to θ as described above. Another solution, the “long way”, or “Type 2” transfers, has the change in true anomaly equal to 360°-θ. Other solutions may exist as well. For example, one way to transfer from Earth to Mars in 2.5 years is to make more than a complete orbit during the transfer. Porkchop plots typically only show the Type 1 and Type 2 solutions, and typically only show at most one of these two solutions for a given pair of departure and arrival dates. If one of the two solutions is close to optimal, the other solution will inevitably follow a retrograde path and thus will involve huge expenditures of energy. There’s no reason to show these highly sub-optimal solutions.

Why is there a gap in porkchop plots?

     The plot can be cleaned up further by removing cases where the better of the two solutions still involves huge energy expenditures. Huge energy expenditures are obviously going to result when the transfer time is very short or very long. A not so obvious place where this happens is when the angle subtended between the line from the Earth and Sun at departure and the target planet and Sun at arrival is nearly 180°. That the Earth and target planet have slightly different orbital planes means that the transfer plane will be nearly orthogonal to the planetary orbital planes when the transfer angle is close to but not equal to 180°. This makes the approach of having a maneuver at departure and a maneuver at arrival extremely expensive for those transfers that are close to 180°. Removing those very expensive transfers from view is what creates the gap in the porkchop plot.
     This excessive cost for near 180° transfers is to some extent an artifact of the approach used to create a porkchop plot. Adding a third maneuver enables the use of much smaller in-plane maneuvers at the start and end, with a small plane change somewhere along route. There’s a problem, however. This mid-course plane change would necessarily mean thrust from the spacecraft itself. This is undesirable. The spacecraft itself provides very little of the energy with the two burn approach. The energy for the Earth departure comes from the launch vehicle, and in the case of Mars, most of the energy for Mars arrival comes from aerobraking. It’s better to fold that plane change into the maneuvers at departure and arrival so as to keep the thrust needed by the spacecraft down to a minimum.

Why don’t they use an optimal Hohmann transfer?

     An optimal Hohmann transfer doesn’t exist. Hohmann transfers are in-plane maneuvers that transfer from one circular orbit to another that share a common orbital plane. Planetary orbits are slightly inclined with respect to one another and are elliptical rather than circular. Generalizing the concept of a Hohmann transfer to that of a 180° transfer, in most cases that 180° transfer doesn’t exist. When it does, that the orbits are not coplanar and that the orbits are not circular means that this 180° transfer is no longer optimal.

Why are there two local energy minima?

     There aren’t just two local energy minima. There are a countable infinite number of local minima. A porkchop plot only shows the first two. The other solutions take even longer than Type 1 and Type 2 transfers and are more sensitive to errors.

      Changing direction causes ΔV (change in velocity), often more than a change in speed. Compare the velocity vectors below. When going the same direction, the difference is 1 km/s. When at right angles the difference is 5 km/s. We know this from driving in traffic. Two cars going almost the same speed hit each other. If they’re in the same lane going the same direction, it’s a mild bump. If one car runs a red light and T-bones a car in cross traffic, the impact is serious:


     This is the strength of a Hohmann transfer orbit. Velocity vectors are pointing the same direction at departure as well as destination. No direction change is needed, only a speed change:


     Note the Hohmann transfer path moves 180 degrees about the sun:


     A Hohmann transfer assumes the departure and destination orbits are co-planar. But what if the destination orbit is inclined?

Orbit Planes and Spherical Trigonometry

     A plane passing through a sphere’s center cuts the sphere along a great circle. A group of planes all sharing a common point can be represented as great circles on a sphere. Since every orbit about the sun is a conic section having the sun as a focus, each orbital plane shares the sun as a common point. Representing the orbital planes as great circles is convenient. There are already a lot of theorems in spherical trigonometry which gives us a suite of tools for looking at angles between orbital planes.
     The shortest path (or geodesic) along a spherical surface between two points is an arc of a great circle. If we set the sphere’s radius to be 1, the arc length is also the angular separation in radians.
     A familar group of great circles are the longitude lines on a globe. The equator is the only great circle among the latitude lines. All the longitude lines are great circles passing through the poles.
     Let’s use the equatorial great circle to represent the departure plane. Recall the Hohmann transfer moves 180 degrees about the center. In this illustration, latitude and longitude for departure and destination is (0°, 0°) and (7°, 180°). The only great circle connecting these points is a polar orbit nearly 90° from the departure and destination planes! Big plane changes at departure and destination destroys the virtue of a Hohmann orbit.


     I’ve also tried to demonstrate this in this video:

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     The big delta V needed for large plane changes makes the ridge (gap between the two blobs) in a porkchop plot:


     Porkchop plots are drawn by doing iterations of various Lambert Space Triangles. Lambert iterations give polar transfer orbits when departure and destination longitudes differ by 180°.
     Does this mean Hohmann transfers are no good if the destination orbit’s inclined? No, the big plane changes can be avoided with a mid course plane change. Here is a broken plane transfer where a plane change burn is done at the ascending node:


     The line where the destination and departure planes intersect form the ascending and descending nodes. Starting in the departure plane and doing a plane change at the node avoids the two major plane changes. The departure and destination planes differ by an angle called i, for inclination.
     Changing a vector by an angle i takes dv of v * 2 * sin(i/2).


     The Vis Viva Equation tells us v = sqrt(μ(2/r - 1/a)). So v ranges from sqrt (μ((1-e)/(a(1+e)))) at aphelion to sqrt (μ((1+e)/(a(1-e)))) at perihelion. Let's look at a Ceres transfer orbit. An ellipse with a 1.88 a.u. semi major axis and eccentricity 0.47 will have speeds ranging from 36 km/s (at perihelion) to 13 km/s (at apohelion). Inclination's about 10.6 degrees. So plane change ranges from 36 km/s   * 2 * sin(10°/2) to 13 km/s   * 2 * sin(10°/2) or from 6.7 to 2.4 km/s. Is the a 2.4 km/s plane change at aphelion the best we can do?  No, it's possible to have less plane change expense.
     Launch is at the perihelion of an outbound Hohmann orbit. If the launch coincides with a node, the entire plane change can be done during earth departure or at arrival. Then the delta V entails a speed change as well as a direction change. Doing a single plane change/speed change burn saves delta V as shown by this diagram:


     Law of cosines tells us for a triangle a, b, c, a² + b² - 2ab cos(i) = c². In this case, i is the angle between a and b and c is the delta V needed from the combined plane change and speed change.
     At aphelion, a combined speed change/plane change only costs 0.76 km/s more than the speed change alone.
     When launching deep in earth’s gravity well, we enjoy an Oberth benefit. Ceres' gravity well lends a little Oberth benefit at the destination. If the line of nodes coincides with transfer orbit's line of apsides, plane change can cost as little as 0.52 km/s extra.
     This indicates as much plane change as possible should be made at departure and arrival. What sort of plane changes should we make to minimize the angle of the midcourse plane change?
     The fattest part of an orange slice is right in the middle:


     The angular separation at launch has to be some part of the orange slice. To minimize the angle between transfer plane and destination plane, the angular separation at launch should be in the middle. Having the transfer plane intersect the destination plane 90° from launch minimizes plane change angle.


     An object on an elliptical path moves slower as it moves further from the sun, so doing plane changes further out are cheaper. The 90º from launch is a minimum. There will be a larger plane change angle 100 degrees from launch, but velocity will be slower. Also plane change lessens as flight path angle increases. I hope to talk about this more when I have time.
     But for now I believe this shows that the Lambert iterations greatly exaggerates plane change expense for a Hohmann path where departure and destination points are 180° degrees apart. Most of that plane change expense can be eliminated by choosing a good place to do a midcourse plane change

     A PDF on Broken Plane Maneuvers Fernando Abilleria of NASA Jet Propulsion Laboratory

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The Interplanetary Transport Network (ITN) is a collection of gravitationally determined pathways through the Solar System that require very little energy for an object to follow. The ITN makes particular use of Lagrange points as locations where trajectories through space are redirected using little or no energy. These points have the peculiar property of allowing objects to orbit around them, despite lacking an object to orbit. While they use little energy, the transport can take a very long time. Shane Ross has said "Due to the long time needed to achieve the low energy transfers between planets, the Interplanetary Superhighway is impractical for transfers such as from Earth to Mars at present

History

Interplanetary transfer orbits are solutions to the gravitational "restricted three-body problem", which, for the general case, does not have exact solutions, and is addressed by numerical analysis approximations. However, a small number of exact solutions exist, most notably the five orbits referred to as "Lagrange points", which are orbital solutions for circular orbits in the case when one body is significantly more massive.

The key to discovering the Interplanetary Transport Network was the investigation of the nature of the winding paths near the Earth-Sun and Earth-Moon Lagrange points. They were first investigated by Jules-Henri Poincaré in the 1890s. He noticed that the paths leading to and from any of those points would almost always settle, for a time, on an orbit about that point. There are in fact an infinite number of paths taking one to the point and away from it, and all of which require nearly zero change in energy to reach. When plotted, they form a tube with the orbit about the Lagrange point at one end.

The derivation of these paths traces back to mathematicians Charles C. Conley and Richard P. McGehee in 1968. Hiten, Japan's first lunar probe, was moved into lunar orbit using similar insight into the nature of paths between the Earth and the Moon. Beginning in 1997, Martin Lo, Shane D. Ross, and others wrote a series of papers identifying the mathematical basis that applied the technique to the Genesis solar wind sample return, and to Lunar and Jovian missions. They referred to it as an Interplanetary Superhighway (IPS).

Paths

As it turns out, it is very easy to transit from a path leading to the point to one leading back out. This makes sense, since the orbit is unstable, which implies one will eventually end up on one of the outbound paths after spending no energy at all. Edward Belbruno coined the term "weak stability boundary" or "fuzzy boundary" for this effect.

With careful calculation, one can pick which outbound path one wants. This turned out to be useful, as many of these paths lead to some interesting points in space, such as the Earth's Moon or between the Galilean moons of Jupiter. As a result, for the cost of reaching the Earth–Sun L₂ point, which is rather low energy value, one can travel to a number of very interesting points for a little or no additional fuel cost. But the trip from Earth to Mars or other distant location would likely take thousands of years.

The transfers are so low-energy that they make travel to almost any point in the Solar System possible. On the downside, these transfers are very slow. For trips from Earth to other planets, they are not useful for manned or unmanned probes, as the trip would take many generations. Nevertheless, they have already been used to transfer spacecraft to the Earth–Sun L₁ point, a useful point for studying the Sun that was employed in a number of recent missions, including the Genesis mission, the first to return solar wind samples to Earth. The network is also relevant to understanding Solar System dynamics; Comet Shoemaker–Levy 9 followed such a trajectory on its collision path with Jupiter.

Further explanation>

The ITN is based around a series of orbital paths predicted by chaos theory and the restricted three-body problem leading to and from the unstable orbits around the Lagrange points – points in space where the gravity between various bodies balances with the centrifugal force of an object there. For any two bodies in which one body orbits around the other, such as a star/planet or planet/moon system, there are three such points, denoted L₁ through L₃. For instance, the Earth–Moon L₁ point lies on a line between the two, where gravitational forces between them exactly balance with the centrifugal force of an object placed in orbit there. For two bodies whose ratio of masses exceeds 24.96, there are two additional stable points denoted as L₄ and L₅. These five points have particularly low delta-v requirements, and appear to be the lowest-energy transfers possible, even lower than the common Hohmann transfer orbit that has dominated orbital navigation in the past.

Although the forces balance at these points, the first three points (the ones on the line between a certain large mass, e.g. a star, and a smaller, orbiting mass, e.g. a planet) are not stable equilibrium points. If a spacecraft placed at the Earth–Moon L₁ point is given even a slight nudge towards the Moon, for instance, the Moon's gravity will now be greater and the spacecraft will be pulled away from the L₁ point. The entire system is in motion, so the spacecraft will not actually hit the Moon, but will travel in a winding path, off into space. There is, however, a semi-stable orbit around each of these points, called a halo orbit. The orbits for two of the points, L₄ and L₅, are stable, but the halo orbits for L₁ through L₃ are stable only on the order of months.

In addition to orbits around Lagrange points, the rich dynamics that arise from the gravitational pull of more than one mass yield interesting trajectories, also known as low energy transfers. For example, the gravity environment of the Sun–Earth–Moon system allows spacecraft to travel great distances on very little fuel, albeit on an often circuitous route.

Missions

Launched in 1978, the ISEE-3 spacecraft was sent on a mission to orbit around one of the Lagrange points. The spacecraft was able to maneuver around the Earth's neighborhood using little fuel by taking advantage of the unique gravity environment. After the primary mission was completed, ISEE-3 went on to accomplish other goals, including a flight through the geomagnetic tail and a comet flyby. The mission was subsequently renamed the International Cometary Explorer (ICE).

The first low energy transfer using what would later be called the ITN was the rescue of Japan's Hiten lunar mission in 1991. Another example of the use of the ITN was NASA's 2001–2003 Genesis mission, which orbited the Sun–Earth L₁ point for over two years collecting material, before being redirected to the L₂ Lagrange point, and finally redirected from there back to Earth. The 2003–2006 SMART-1 of the European Space Agency used another low energy transfer from the ITN. In a more recent example, the Chinese spacecraft Chang'e 2 used the ITN to travel from lunar orbit to the Earth-Sun L₂ point, then on to fly by the asteroid 4179 Toutatis.

      Wikipedia describes the Interplanetary Transport Network as "… pathways through the Solar System that require very little energy for an object to follow." See this Wikipedia article. They also say "While they use very little energy, the transport can take a very long time."

     Low energy paths that take a very long time? I often hear this parroted in space exploration forums and it always leaves me scratching my head.

     The lowest energy path I know of to bodies in the inner solar system is the Hohmann orbit. Or if the destination is noticeably elliptical, a transfer orbit that is tangent to both the departure and destination orbit. Although I think of bitangential transfer orbits as a more general version of the Hohmann orbit.

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     In the case of Mars, a bitangential orbit is 8.5 months give or take a month or two. Is there a path that takes a lot longer and uses almost no energy? I know of no such path.

L1 and L2

     The interplanetary Superhighway supposedly relies on weak stability or weak instability boundaries between L1 and/or L2 regions. Here is an online text on 3 body Mechanics and their use in space mission design. The authors are Koon, Lo, Marsden and Ross. Shane Ross is one of the more prominent evangelists spreading the gospel of the Interplanetary Super Highway.

     The focus of this online textbook is the L1 and L2 regions. From page 10:

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     L1 and L2 are necks between realms. In the above illustration the central body is the sun, and orbiting body Jupiter. L1 and L2 are necks or gateways between three realms: the Sun realm, the Jupiter Realm and the exterior realm.

     Travel between these realms can be accomplished by weak stability or weak instability boundaries that emanate from L1 or L2. From page 11 of the same textbook:

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My terms for various Lagrange necks

     First letter is the central body, the second letter is the orbiting body.

Earth Moon L1: EML1
Earth Moon L2: EML2

Sun Earth L1: SEL1
Sun Earth L2: SEL2

Sun Mars L1: SML1
Sun Mars L2: SML2

     Since I'm a lazy typist that is what I'll use for the rest of this post.

EML1 and 2

     I am very excited about the earth-moon Lagrange necks. They've been prominent in many of my blog posts. Here's a post entirely devoted to EML2.

     EML1 and 2 are about 5/6 and 7/6 of a lunar distance from earth:

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     Both necks move at the same angular velocity as the moon. So EML1 moves substantially slower than an ordinary earth orbit would at that altitude. EML2 moves substantially faster.

     It takes only a tiny nudge and send objects in these regions rolling about the slopes of the effective potential hills. Outside of the moon's influence they tend to fall into ordinary two body ellipses (for a short time).

     Here's the ellipse an object moving at EML1 velocity and altitude would follow if the moon weren't there:

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     In practice an EML1 object nudged earthward will near the moon on the fifth apogee. If coming from behind, the moon's gravitational tug can slow the object which lowers perigee.

     Here is an orbital sim where the moon's influence lowered perigee four times:

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     I've run sims where repeated lunar tugs have lowered perigees to atmosphere grazing perigees. Once perigee passes through the upper atmosphere, we can use aerobraking to circularize the orbit.

     Orbits are time reversible. Could we use the lunar gravity assists to get from LEO to higher orbits? Unfortunately, aerobraking isn't time reversible. The atmosphere can't increase orbital speed to achieve a higher apogee. And low earth orbit has a substantially different Jacobi constant than those orbits dwelling closer to the borders of a Hill Sphere.

     So to get to the lunar realm, we're stuck with the 3.1 km/s LEO burn needed to raise apogee. But once apogee is raised, many doors open.

     There are low energy paths that lead from EML1 to EML2. EML2 is an exciting location.

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     In the above illustration I have an apogee beyond SEL2. But by timing the release from EML2, we could aim for other regions of the Hill Sphere, including SEL1.

     Here is a sim where slightly different nudges send payloads from EML2:

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     See how the sun bends the path as apogee nears the Hill Sphere? From EML2 there are a multitude of wildly different paths we can choose. In this illustration I like pellet #3 (orange). It has a very low perigee that is moving about 10.8 km/s. And it got to this perigee with just a tiny nudge from EML2. Pellet # 4 is on it's way to a retrograde earth orbit. Most of the other pellets are saying good bye to earth's Hill Sphere.

     I am enthusiastic about using EML1 and EML2 as hubs for travel about the earth-moon neighborhood. But a little less excited about travel about the solar system.

We've left Earth's Hill Sphere. Now what?

     Recall that EML1 and 2 are ~5/6 and 7/6 of a lunar distance from the earth. SEL1 and 2 are much less dramatic: 99% and 101% of an A.U. from the sun. Objects released from these locations don't vary much from earth's orbit:

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     Running orbital sims gets pretty much the same result pictured above.

     Mars is even worse:

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     Are there weak instability boundary transfers leading from SEL2 to SML1? I don't think this particular highway exists.

     To get a 1.52 A.U. aphelion, we need a departure Vinfinity of 3 km/s. To be sure EML2 can help us out in achieving this Vinfinity. In other words we could use lunar assists to depart on a Hohmann orbit. But a Hohmann orbit is different from the tube of weak instability boundaries we're led to imagine.

     And once we arrive at a 1.52 aphelion. we have an arrival 2.7 km/s Vinfinity we need to get rid of.

     Pass through SML1 at 2.7 km/s and you'll be waving Mars goodbye. The Lagrange necks work their mojo on near parabolic orbits. And an earth to Mars Hohmann is decidedly hyperbolic with regard to Mars.

     What about Phobos and Deimos? The Martian moons are too small to lend a helpful gravity assist. We need to get rid of the 2.7 km/s Vinf and neither SML1 nor the moons are going to do it for us.

Mars ballistic capture by Belbruno & Toppotu

     Edward Belbruno is another well known evangelist for the Interplanetary Superhighway (though he likes to call them ballistic captures). Belbruno cowrote this pdf on ballistic Mars capture.

     Here is a screen capture from the pdf:

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     The path from Earth@Departure to Xc is pretty much a Hohmann transfer. In fact they assume the usual departure for Mars burn. Arrival is a little different. They do a 2 km/s heliocentric circularization burn at Xc (which is above Mars' perihelion). This particular path takes an extra year or so to reach Mars.

     So they accomplish Mars capture with a 2 km/s arrival burn. At first glance this seems like a 0.7 km/s improvement over the 2.7 km/s arrival Vinf.

     Or it seems like an advantage to those unaware of the Oberth benefit. If making the burn deep in Mars' gravity well, capture can be achieved for as little as 0.7 km/s.

     Comparing capture burns it's 2 km/s vs 0.7 km/s. So what do we get for an extra year of travel time? 1.3 km/s flushed down the toilet!

What About Ion Engines?

     "What about ion engines?" a Belbruno defender might object. "They don't have the thrust to enjoy an Oberth benefit. So Belbruno's 0.7 km/s benefit is legit if your space craft is low thrust."

     Belbruno & friends are looking at a trip from a nearly zero earth C3 to a nearly zero Mars C3. In other words from the edge of one Hill Sphere to another.

     So to compare apples to apples I'll look at a Hohmann from SEL2 to SML1. I want to point out I'm not using Lagrange necks as key holes down some mysterious tube. They're simply the closest parts of neighboring Hill Spheres.

     "Wait a minute..." says Belbruno's defender, "We're talking Hall thrusters. So no Hohmann ellipse, but a spiral."

     Low earth orbit moves about 4° per minute. So a low-thrust burn lasting days does indeed result in a spiral. But Earth's heliocentric orbit moves about a degree per day while Mars' heliocentric orbit moves about half a degree per day. At this more leisurely pace, a 4 or 5 day burn looks more an impulsive burn. The transfer between Hill Spheres is more Hohmann-like than the spiral out of earth's gravity well.

     Instead of a 1 x 1.524 AU orbit, the new Hohmann is a 1.01 x 1.517 AU ellipse. The new Hohmann's perihelion is a little slower, the new aphelion a little faster.

     Moreover, SEL2 moves at the same angular velocity as earth. So it's speed is about 101% earth's speed. Likewise SML1 moves at about 99.6% Mars' speed.

     With this revised scenario, aphelion rendezvous delta V is now more like 2.4 km/s. Still, Belbruno's 2 km/s capture burn saves 0.4 km/s.

     0.4 km/s is better than chopped liver, right? Well, recall ion engines with very good I_SP. I'll look at an exhaust velocity of of 30 km/s.

e^2.4/30 - 1 = 0.083
e^2/30 - 1 = 0.069

     So given a 100 tonne payload, rendezvous xenon is 8.3 tonnes for Hohmann vs 6.9 tonnes for Belbruno's ballistic capture.

108.3/106.9 = 1.0134

     We're adding a year to our trip time for a one percent mass improvement? Sorry, I don't see this a great trade-off.

Summary

     The virtually zero energy looooong trips between planets are an urban legend.

     I'll be pleasantly surprised if I'm wrong. To convince me otherwise, show me the beef. Show me the zero energy trajectory from an earth Lagrange neck to a Mars Lagrange neck.

     Until then I'll think of this post as a dose of Snopes for space cadets.

     I'd like to thank Mike Loucks and John P. Corrico Jr. I've held these opinions for awhile but didn't have the confidence to voice them. Who am I but an amateur with no formal training? But talking with these guys I was pleasantly surprised to find some of my heretical views were shared by pros. Without their input I would not have had the guts to publish this post.

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In the meantime he had another worry; strung out behind him were several more ships, all headed for Mars. For the next several days there would be frequent departures from the Moon, all ships taking advantage of the one favorable period in every twenty-six months when the passage to Mars was relatively 'cheap', i.e., when the minimum-fuel ellipse tangent to both planet's orbits would actually make rendezvous with Mars rather than arrive foolishly at some totally untenanted part of Mars' orbit. Except for military vessels and super expensive passenger-ships, all traffic for Mars left at this one time.

During the four-day period bracketing the ideal instant of departure ships leaving Leyport paid a fancy premium for the privilege over and above the standard service fee. Only a large ship could afford such a fee; the saving in cost of single-H reactive mass had to be greater than the fee. The Rolling Stone had departed just before the premium charge went into effect; consequently she had trailing her like beads on a string a round dozen of ships, all headed down to Earth, to tack around her toward Mars.

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Hazel looked them over. 'Mr d'Avril, don't you have something a bit larger?'

'Well, yes, ma'am, I do — but I hate to rent larger ones to such a small family with the tourist season just opening up: I'll bring in a cot for the youngster.'

(ed note: The "tourist season" on Mars starts with the earliest arrival time of a spacecraft on a Terra-Mars Hohmann trajectory)

'See here, I don't want to buy this du — this place. I just want to use it for a while.'

Mr d'Avril looked hurt. 'You needn't do either one, ma'am. With ships arriving every day now I'll have my pick of tenants. My prices are considered very reasonable. The Property Owner's Association has tried to get me to up 'em — and that's a fact'

Hazel dug into her memory to recall how to compare a hotel price with a monthly rental — add a zero to the daily rate; that was it Why, the man must be telling the truth! — if the hotel rates she had gotten were any guide. She shook her head. 'I'm just a country girl, Mr d'Avril. How much did this place cost to build?'

Again he looked hurt 'You're not looking at it properly, ma'am. Every so often we have a big load of tourists dumped on us. They stay awhile, then they go away and we have no rent coming in at all. And you'd be surprised how these cold nights nibble away at a house. We can't build the way the Martians could.'

Hazel gave up. 'Is that season discount you mentioned good from now to Venus departure?'

'Sorry, ma'am. It has to be the whole season.' The next favorable time to shape an orbit for Venus was ninety-six Earth-standard days away — ninety-four Mars days — whereas the 'whole season' ran for the next fifteen months, more than half a Martian year before Earth and Mars would again be in a position to permit a minimum-fuel orbit.

It was true. Even in the bars that catered to inner planet types, the mix was rarely better than one Earther or Martian in ten (Belters). Squinting out at the crowd, Miller saw that the short, stocky men and women were nearer a third.

"Ship come in?" he asked.

"Yeah."

"EMCN?" he asked. The Earth-Mars Coalition Navy often passed through Ceres on its way to Saturn, Jupiter, and the stations of the Belt, but Miller hadn't been paying enough attention to the relative position of the planets to know where the orbits all stood.

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      This essay was written in 1952, long before the Mariner space probes gave us our close-up glimpses of the tantalizing red planet. Nevertheless, most of the concepts presented here are still quite valid, though we now know that Mars is even more rugged than anticipated. In particular, the atmospheric pressure is so low (about one-hundredth of Earth’s) (actually more like 0.00628 of Earth) that simple breathing masks will not give sufficient protection; we will have to wear spacesuits.

     So you’re going to Mars? That’s still quite an adventure —though I suppose that in another ten years no one will think twice about it. Sometimes it’s hard to remember that the first ships reached Mars scarcely more than half a century ago and that our colony on the planet is less than thirty years old. (By the way, don’t use that word when you get there. Base, settlement, or whatever you like—but not colony, unless you want to hear the ice tinkling all around you.)

     I suppose you’ve read all the forms and tourist literature they gave you at the Department of Extraterrestrial Affairs. But there’s a lot you won’t learn just by reading, so here are some pointers and background information that may make your trip more enjoyable. I won’t say it’s right up to date—things change so rapidly, and it’s a year since I got back from Mars myself—but on the whole you’ll find it pretty reliable.

     Presumably you’re going just for curiosity and excitement—because you want to see what life is like out on the new frontier. It’s only fair, therefore, to point out that most of your fellow passengers will be engineers, scientists, or administrators traveling to Mars—some of them not for the first time—because they’ve got a job of work to do. So whatever your achievements here on Earth, it’s advisable not to talk too much about them, as you’ll be among people who’ve had to tackle much tougher propositions. I won’t say that you’ll find them boastful: it’s simply that they’ve got a lot to be proud of, and they don’t mind who knows it.

     If you haven’t booked your passage yet, remember that the cost of the ticket varies considerably according to the relative positions of Mars and Earth. That’s a complication we don’t have to worry about when we’re traveling from country to country on our own globe, but Mars can be six times farther away at one time than at another. Oddly enough, the shortest trips are the most expensive, since they involve the greatest changes of speed as you hop from one orbit to the other. And in space, speed, not distance, is what costs money.

     Incidentally, I’d like to know how you've managed it. I believe the cheapest round trip comes to about $30,000, and unless the firm is backing you or you’ve got a very elastic expense account—Oh, all right, if you don’t want to talk about it…

     I take it you’re O.K. on the medical side. That examination isn’t for fun, nor is it intended to scare anyone off. The physical strain involved in space flight is negligible— but you’ll be spending at least two months on the trip, and it would be a pity if your teeth or your appendix started to misbehave. See what I mean?

     You’re probably wondering how you can possibly manage on the weight allowance you’ve got. Well, it can be done. The first thing to remember is that you don’t need to take any suits. There’s no weather inside a spaceship; the temperature never varies more than a couple of degrees over the whole trip, and it’s held at a fairly high value so that all you’ll want is an ultra-lightweight tropical kit. When you get to Mars you’ll buy What you need there and dump it when you return. The great thing to remember is only carry the stuff you actually need on the trip. I strongly advise you to buy one of the complete travel kits—a store like Abercrombie & Finch can supply the approved outfits. They’re expensive, but will save you money on excess baggage charges.

     Take a camera by all means—there’s a chance of some unforgettable shots as you leave Earth and when you approach Mars. But there’s nothing to photograph on the voyage itself, and I’d advise you to take all your pictures on the outward trip. You can sell a good camera on Mars for five times its price here—and save yourself the cost of freighting it home. They don’t mention that in the official handouts.

     Now that we’ve brought up the subject of money, I’d better remind you that the Martian economy is quite different from anything you’ll meet on Earth. Down here, it doesn’t cost you anything to breathe, even though you’ve got to pay to eat. But on Mars the very air has to be synthesized—they break down the oxides in the ground to do this—so every time you fill your lungs someone has to foot the bill. Food production is planned in the same way —each of the cities, remember, is a carefully balanced ecological system, like a well-organized aquarium. No parasites can be allowed, so everyone has to pay a basic tax which entitles him to air, food, and the shelter of the domes. The tax varies from city to city, but averages about $10 a day. Since everyone earns at least ten times as much as this, they can all afford to go on breathing.

     You’ll have to pay this tax, of course, and you’ll find it rather hard to spend much more money than this. Once the basic needs for life are taken care of, there aren’t many luxuries on Mars. When they’ve got used to the idea of having tourists around, no doubt they’ll get organized, but as things are now you’ll find that most reasonable requests won’t cost you anything. However, I should make arrangements to transfer a substantial credit balance to the Bank of Mars—if you’ve still got anything left. You can do that by radio, of course, before you leave Earth.

     So much for the preliminaries; now some points about the trip itself. The ferry rocket will probably leave from the New Guinea field, which is about two miles above sea level on the top of the Orange Range. People sometimes wonder why they chose such an out-of-the-way spot. That’s simple: it’s on the equator, so a ship gets the full thousand-mile-an-hour boost of the Earth’s spin as it takes off—and there’s the whole width of the Pacific for jettisoned fuel tanks to fall into. And if you’ve ever heard a spaceship taking off, you’ll understand why the launching sites have to be a few hundred miles from civilization. (gee, I guess it is just too bad for all those uncivilized natives living there who are rapidly going deaf)

     Don’t be alarmed by anything you’ve been told about the strain of blast-off. There’s really nothing to it if you’re in good health—and you won’t be allowed inside a spaceship unless you are. You just lie down on the acceleration couch, put in your earplugs, and relax. It takes over a minute for the full thrust to build up, and by that time you’re quite accustomed to it. You’ll have some difficulty in breathing, perhaps—it’s never bothered me—but if you don’t attempt to move you’ll hardly feel the increase of weight. What you will notice is the noise, which is slightly unbelievable. Still, it lasts only five minutes, and by the end of that time you’ll be up in the orbit and the motors will cut out. Don’t worry about your hearing; it will get back to normal in a couple of hours.

     You won’t see a great deal until you get aboard the space station, because there are no viewing ports on the ferry rockets and passengers aren’t encouraged to wander around. It usually takes about thirty minutes to make the necessary steering corrections and to match speed with the station; you’ll know when that's happened from the rather alarming “clang” as the air locks make contact. Then you can undo your safety belt, and of course you’ll want to see what it’s like being weightless.

     Now, take your time, and do exactly what you’re told. Hang on to the guide rope through the air lock and don’t try to go flying around like a bird. There’ll be plenty of time for that later: there’s not enough room in the ferry, and if you attempt any of the usual tricks you’ll not only injure yourself but may damage the equipment as well.

     Space Station One, which is where the ferries and the liners meet to transfer their cargoes, takes just two hours to make one circuit of the Earth. You’ll spend all your time in the observation lounge: everyone does, no matter how many times they’ve been out into space. I won’t attempt to describe that incredible view; I’ll merely remind you that in the hundred and twenty minutes it takes the station to complete its orbit you’ll see the Earth wax from a thin crescent to a gigantic, multicolored disk, and then shrink again to a black shield eclipsing the stars. As you pass over the night side you’ll see the lights of cities down there in the darkness, like patches of phosphorescence. And the stars! You’ll realize that you’ve never really seen them before in your life.

     But enough of these purple passages; let’s stick to business. You’ll probably remain on Space Station One for about twelve hours, which will give you plenty of opportunity to see how you like weightlessness. It doesn’t take long to learn how to move around; the main secret is to avoid all violent motions—otherwise you may crack your head on the ceiling. Except, of course, that there isn’t a ceiling since there’s no up or down any more. At first you’ll find that confusing: you’ll have to stop and decide which direction you want to move in, and then adjust your personal reference system to fit. After a few days in space it will be second nature to you.

     Don’t forget that the station is your last link with Earth. If you want to make any final purchases, or leave something to be sent home—do it then. You won’t have another chance for a good many million miles. But beware of buying items that the station shop assures you are “just the thing on Mars.”

     You’ll go aboard the liner when you’ve had your final medical check, and the steward will show you to the little cabin that will be your home for the next few months. Don’t be upset because you can touch all the walls without moving from one spot. You’ll only have to sleep there, after all, and you’ve got the rest of the ship to stretch your legs in.

     If you’re on one of the larger liners, there’ll be about a hundred other passengers and a crew of perhaps twenty. You’ll get to know them all by the end of the voyage. There’s nothing on Earth quite like the atmosphere in a spaceship. You’re a little, self-contained community floating in vacuum millions of miles from anywhere, kept alive in a bubble of plastic and metal. If you’re a good mixer, you’1l find the experience very stimulating. But it has its disadvantages. The one great danger of space flight is that some prize bore may get on the passenger list—and short of pushing him out of the air lock there’s nothing anyone can do about it.

     It won’t take you long to find your way around the ship and to get used to its gadgets. Handling liquids is the main skill you’ll have to acquire: your first attempts at drinking are apt to be messy. Oddly enough, taking a shower is quite simple. You do it in sort of a plastic cocoon, and a circulating air current carries the water out at the bottom.

     At first the absence of gravity may make sleeping difficult—you’ll miss your accustomed weight. That’s why the sheets over the bunks have spring tensioning. They’ll keep you from drifting out while you sleep, and their pressure will give you a spurious sensation of weight.

     But learning to live under zero gravity is something one can’t be taught in advance: you have to find out by experience and practical demonstration. I believe you’ll enjoy it, and when the novelty’s worn off you’ll take it completely for granted. Then the problem will be getting used to gravity again when you reach Mars!

     Unlike the take-off of the ferry rocket from Earth, the breakaway of the liner from its satellite orbit is so gentle and protracted that it lacks all drama. When the loading and instrument checks have been completed, the ship will uncouple from the Space Station and drift a few miles away. You’ll hardly notice it when the atomic drive goes on; there will be the faintest of vibrations and a feeble sensation of weight. The ship’s acceleration is so small, in fact, that you’l1 weigh only a few ounces, which will scarcely interfere with your freedom of movement at all. Its only effect will be to make things drift slowly to one end of the cabin if they’re left lying around.

     Although the liner’s acceleration is so small that it will take hours to break away from Earth and head out into space, after a week of continuous drive the ship will have built up a colossal speed. Then the motors will be cut out and you’ll carry on under your own momentum until you reach the orbit of Mars and have to start thinking about slowing down.

     Whether your weeks in space are boring or not depends very much on you and your fellow passengers. Quite a number of entertainments get organized on the voyage, and a good deal of money is liable to change hands before the end of the trip. (It’s a curious fact, but the crew usually seems to come out on top.) You’ll have plenty of time for reading, and the ship will have a good library of micro-books. There will be radio and TV contact with Earth and Mars for the whole voyage, so you’ll be able to keep in touch with things—if you want to.

     On my first trip, I spent a lot of my time learning my way around the stars and looking at clusters and nebulae through a small telescope I borrowed from the navigation officer. Even if you’ve never felt the slightest interest in astronomy before, you’ll probably be a keen observer before the end of the voyage. Having the stars all around you—not merely overhead—is an experience you’ll never forget.

     As far as outside events are concerned, you realize, of course, that absolutely nothing can happen during the voyage. Once the drive has cut out, you’ll seem to be hanging motionless in space: you’ll be no more conscious of your speed than you are of Earth’s seventy thousand miles an hour around the Sun right now. The only evidence of your velocity will be the slow movement of the nearer planets against the background of the stars—and you’ll have to watch carefully for a good many hours before you can detect even this.

     By the way, I hope you aren’t one of those foolish people who are still frightened about meteors. They see that enormous chunk of nickel-steel in New York’s American Museum of Natural History and imagine that’s the sort of thing you’ll run smack into as soon as you leave the atmosphere—forgetting that there’s rather a lot of room in space and that even the biggest ship is a mighty small target. You’d have to sit out there and wait a good many centuries before a meteor big enough to puncture the hull came along. It hasn’t happened to a spaceship yet.

     One of the big moments of the trip will come when you realize that Mars has begun to show a visible disk. The first feature you’ll be able to see with the naked eye will be one of the polar caps, glittering like a tiny star on the edge of the planet. A few days later the dark areas—the so-called seas—will begin to appear, and presently you’ll glimpse the prominent triangle of the Syrtis Major. In the week before landing, as the planet swims nearer and nearer, you’ll get to know its geography pretty thoroughly.

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     The braking period doesn’t last very long, as the ship has lost a good deal of its speed in the climb outward from the Sun. When it’s. over you’ll be dropping down onto Phobos, the inner moon of Mars, which acts as a natural space station about four thousand miles above the surface of the planet. Though Phobos is only a jagged lump of rock not much bigger than some terrestrial mountains, it’s reassuring to be in contact with something solid again after so many weeks in space.

     When the ship has settled down into the landing cradle, the air lock will be coupled up and you’ll go through a connecting tube into the port. Since Phobos is much too small to have an appreciable gravity, you’ll still be effectively weightless. While the ship’s being unloaded the immigration officials will check your papers. I don’t know the point of this; I’ve never heard of anyone being sent all the way back to Earth after having got this far!

     There are two things you mustn’t miss at Port Phobos. The restaurant there is quite good, even though the food is largely synthetic; it’s very small, and only goes into action When a liner docks, but it does its best to give you a fine welcome to Mars. And after a couple of months you’ll have got rather tired of the shipboard menu.

     The other item is the centrifuge; I believe that’s compulsory now. You go inside and it will spin you up to half a gravity, or rather more than the weight Mars will give you when you land. It’s simply a little cabin on a rotating arm, and there's room to walk around inside so that you can practice using your legs again. You probably won't like the feeling; life in a spaceship can make you lazy.

     The ferry rockets that will take you down to Mars will be waiting when the ship docks. If you’re unlucky you’ll hang around at the port for some hours, because they can’t carry more than twenty passengers and there are only two ferries in service. The actual descent to the planet takes about three hours, and it’s the only time on the whole trip when you’ll get any impression of speed. Those ferries enter the atmosphere at over five thousand miles an hour and go halfway around Mars before they lose enough speed through air resistance to land like ordinary aircraft.

     You’ll land, of course, at Port Lowell: besides being the largest settlement on Mars it’s still the only place that has the facilities for handling spaceships. From the air the plastic pressure domes look like a cluster of bubbles—a very pretty sight when the Sun catches them. Don’t be alarmed if one of them is deflated. That doesn’t mean that there’s been an accident. The domes are let down at fairly frequent intervals so that the envelopes can be checked for leaks. If you’re lucky you may see one being pumped up—it’s quite impressive.

     After two months in a spaceship, even Port Lowell will seem a mighty metropolis. (Actually, I believe its population is now well over twenty thousand.) You’ll find the people energetic, inquisitive, forthright—and very friendly, unless they think you’re trying to be superior.

     It’s a good working rule never to criticize anything you see on Mars. As I said before, they’re very proud of their achievements—and after all you are a guest, even if a paying one.

     Port Lowell has practically everything you’ll find in a city on Earth, though of course on a smaller scale. You’ll come across many reminders of “home.” For example, the main street in the city is Fifth Avenue—but surprisingly enough you’ll find Piccadilly Circus where it crosses Broadway.

     The port, like all the major settlements, lies in the dark belt of vegetation that roughly follows the Equator and occupies about half the southern hemisphere. The northern hemisphere is almost all desert—the red oxides that give the planet its ruddy color. Some of these desert regions are very beautiful; they’re far older than anything on the surface of our Barth, because there’s been little weathering on Mars to wear down the rocks—at least since the seas dried up, more than 500 million years ago.

     You shouldn’t attempt to leave the city until you’ve become quite accustomed to living in an oxygen-rich, low-pressure atmosphere. You'll have grown fairly well acclimated on the trip, because the air in the spaceship will have been slowly adjusted to conditions on Mars. Outside the domes, the pressure of the natural Martian atmosphere is about equal to that on the top of Mount Everest—and it contains practically no oxygen. So when you go out you’ll have to wear a helmet, or travel in one of those pressurized jeeps they call “sand fleas.”

     Wearing a helmet, by the way, is nothing like the nuisance you’d expect it to be. The equipment is very light and compact and, as long as you don’t do anything silly, is quite foolproof. As it’s very unlikely that you’ll ever go out without an experienced guide, you’ll have no need to worry. Thanks to the low gravity, enough oxygen for twelve hours’ normal working can be carried quite easily —and you’ll never be away from shelter as long as that.

     Don’t attempt to imitate any of the locals you may see walking around without oxygen gear. They’re second-generation colonists and are used to the low pressure. They can’t breathe the Martian atmosphere any more than you can, but like the old-time native pearl divers they can make one lungful last for several minutes when necessary. Even so, it’s a silly sort of trick and they’re not supposed to do it.

     As you know, the other great obstacle to life on Mars is the low temperature. The highest thermometer reading ever recorded is somewhere in the eighties, but that’s quite exceptional. In the long winters, and during the night in summer or winter, it never rises above freezing. And I believe the record low is minus one hundred and ninety!

     Well, you won’t be outdoors at night, and for the sort of excursions you’ll be doing, all that’s needed is a simple thermosuit. It’s very light, and traps the body heat so effectively that no other source of warmth is needed.

     No doubt you’ll want to see as much of Mars as you can during your stay. There are only two methods of transport outside the cities—sand fleas for short ranges and aircraft for longer distances. Don’t misunderstand me when I say “short ranges”—a sand flea with a full charge of power cells is good for a couple of thousand miles, and it can do eighty miles an hour over good ground. Mars could never have been explored without them. You can survey a planet from space, but in the end someone with a pick and shovel has to do the dirty work filling in the map.

     One thing that few visitors realize is just how big Mars is. Although it seems small beside the Earth, its land area is almost as great because so much of our planet is covered with oceans. So it’s hardly surprising that there are vast regions that have never been properly explored, particularly around the poles. Those stubborn people who still believe that there was once an indigenous Martian civilization pin their hopes on these great blanks. Every so often you hear rumors of some wonderful archaeological discovery in the wastelands, but nothing ever comes of it.

     Personally, I don’t believe there ever were any Martians —but the planet is interesting enough for its own sake. You’ll be fascinated by the plant life and the queer animals that manage to live without oxygen, migrating each year from hemisphere to hemisphere, across the ancient sea beds, to avoid the ferocious winter. The fight for survival on Mars has been fierce, and evolution has produced some pretty odd results. Don’t go investigating any Martian life forms unless you have a guide, or you may get some unpleasant surprises. Some plants are so hungry for heat that they may try to wrap themselves around you.

     Well, that’s all I’ve got to say, except to wish you a pleasant trip. Oh, there is one other thing. My boy collects stamps, and I rather let him down when I was on Mars. If you could drop me a few letters while you’re there— there’s no need to put anything in them if you’re too busy —I’d be much obliged. He’s trying to collect a set of spacemail covers portmarked from each of the principal Martian cities, and if you could help—thanks a lot!

      For any given vehicle design,  what one assumes for mission delta-vees,  vehicle weight statements,  course corrections,  and landing burn requirements greatly affects the payload that can be carried.  The effect is exponential:  variation in required mass ratio with changes in delta-vee and exhaust velocity.
     This analysis looks at trips from low Earth orbit to direct entry at Mars,  and for the return,  a direct launch from Mars to a direct entry at Earth.  The scope is min-energy Hohmann transfer plus 3 faster trajectories (see ref. 1). 
     The vehicle under analysis is the 2019 version of the Spacex “Starship” design,  as described in ref. 2.  The most significant items about that vehicle model are the inert mass and the maximum propellant load.  For this study,  the vehicle is presumed fully loaded with propellant at Earth departure,  and at Mars departure.  See also Figure 1.  Evaporative losses are ignored.

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     Since a prototype has yet to fly,  the design target inert mass of 120 metric tons is presumed as baseline.  Uncertainty demands that inert mass growth be investigated.  To that end,  the average of that design target and the 200 metric ton inert mass of the so-called “Mark 1 prototype” (that average is some 160 metric tons) is used to explore that effect.
     As currently proposed,  the vehicle has six engines.  Three are the sea level version of the “Raptor” engine design,  and the other three are vacuum versions of the same engine design (basically just a larger expansion bell).  I have already reverse-engineered fairly-realistic performance for these in ref. 3.  Because of the smaller bells,  the sea level engines gimbal significantly,  while the vacuum engines cannot.  Thus it is the sea level engines that must be used to land on Mars as well as Earth:  gimballing is required for vehicle attitude control.

Analysis Process

     As shown in Figure 2,  the analysis process is not a simple single-operation calculation.  The vehicle model provides a weight statement and engine performance.  The mission has delta-vee requirements for departure,  course correction,  and landing,  which must be appropriated factored (in order to get mass ratio-effective values).   There are two sets of analysis:  the outbound leg from Earth to Mars,  and the return leg from Mars to Earth. 
     Each leg analyzes 3 burns.  Earth departure,  and course correction are done with the vacuum “Raptor” engines,  while the landing on Mars is done with the sea level “Raptors” to obtain the necessary gimballing.  Mars departure and course correction are done with the vacuum “Raptor” engines (Mars atmospheric pressure is essentially vacuum).  The Earth landing is done with the sea level “Raptors” to get the gimballing and to get the atmospheric backpressure capability.  

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     This analysis is best done in a spreadsheet,  which then responds instantly to changes in one of the constants (like an inert mass or a delta-vee).  That is what I did here. 
     Referring again to Figure 2,  for each burn,  there is an appropriate vehicle ignition mass.  At departure,  it is the ignition mass from the weight statement.  For each subsequent burn,  it is the previous burn’s burnout mass.  Each burn’s burnout mass is its ignition mass divided by the required mass ratio for that burn,  in turn figured from that burn’s delta vee and the appropriate exhaust velocity.
     For each burn,  the change in vehicle mass from ignition to burnout is the propellant mass used for that burn.  For the first burn,  the propellant remaining (after the burn) is the initial propellant load minus the propellant mass used for that burn.  For the subsequent burns,  propellant remaining is the previous value of propellant remaining,  minus the propellant used for that burn. 
     After the final burn,  the propellant remaining cannot be a negative number!  If it is,  one reduces the payload number originally input,  and does all the calculations again.  If this done in a spreadsheet,  this update is automatic!  Ideally,  the propellant remaining should be exactly zero,  but for estimating purposes here,  a small positive fraction of a ton (out of 1200 tons) is “close enough”.
     Thus it is payload that is determined in this analysis.   This particular input (payload mass) is revised iteratively until the final burn’s remaining-propellant estimate is essentially zero.  That is the maximum payload value feasible for the mission case.

Orbits and the Associated Delta Vees

     As indicated in ref. 2,  I have looked at a Hohmann min energy transfer orbit,  and 3 faster transfers with shorter flight times.  All of these are transfer ellipses with their perihelions located at Earth’s orbit.  For Hohmann transfer,  the apohelion is at Mars’s orbit.  For the faster transfers,  apohelion is increasingly far beyond Mars’s orbit.  Why this is so is explained in the reference.  See Figures 3 and 4.
     Note that the overall period of the transfer orbit is important for abort purposes.  If the period is an exact integer multiple of one Earth year,  then Earth will be at the orbit perihelion point simultaneously with anything traveling along that entire transfer orbit.  This offers the possibility of aborting the direct entry and descent at Mars,  if conditions happen to be bad when the encounter happens.  Otherwise,  the spacecraft is committed to entry and descent,  no matter what.  

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     The cases examined in ref. 1 were all computed for Earth and Mars at their average distances from the sun.  The larger transfer ellipse with the longer period occurs when both Earth and Mars are at their farthest distances from the sun.  This leads to larger delta vees to reach transfer perihelion velocity for the trip to Mars,  and larger velocity on the transfer orbit for the trip back to Earth.
     Ref 1 has the required velocities and delta-vees,  but the most pertinent data are repeated here:

Transfer	E.depdV, km/s	trip time, days	M. Vint, km/s
Hohmann	3.659	259	5.69
2-yr abort	4.347	128	7.40
No abort	4.859	110	7.36
3-yr abort	5.223	102	6.53
Transfer	M.depdV, km/s	trip time, days	E.Vint, km/s
Hohmann	5.800	259	11.57
2-yr abort	7.548	128	12.26
No abort	7.509	110	12.77
3-yr abort	6.653	102	13.14

     I did not examine the worst cases for all the transfer orbits in ref. 1,  but I do have the  increase in perihelion velocity for the worst case Earth departure on a Hohmann transfer for Mars:  0.20 km/s higher than average.  I also have the increase in apohelion velocity for the worst case Mars departure on a Hohmann transfer for Earth:  0.16 km/s higher than average.
     I cheated here:  I used those worst-case Hohmann increases for all the faster trajectories as well.  That’s not “right”,  but it should be close enough to see the relative size of the effect of worst case over average conditions.  I also used the same additive changes on the entry velocities.
     Because of the precision trajectory requirements for direct entry while moving above planetary escape speed,  some sort of course correction burn or burns will simply be required.  With this kind of analysis,  I have no way to evaluate that need.  So I just guessed:  0.5 km/s delta-vee capability in terms of propellant reserves. 
     Because this is just a guess,  I did not run any sensitivity analysis on it.  However,  the delta-vee budget proposed here is factor 2.5 larger than the difference average-to-worst-case for the trip to Mars,  which suggests it is “plenty”.   It is about factor 3 larger than the difference average-to-worst-case for the return trip to Earth.  You can get a qualitative sense of this effect from examining that average-vs-worst case effect.

Propellant Budgets for Direct Landings

     With this vehicle (or just about any other vehicle),  entry must be made at a shallow angle relative to local horizontal.  Down lift is required to avoid bouncing off the atmosphere,  since entry interface speed V_int exceeds planetary escape speed.  This is true at both Mars and at Earth.  Once speed has dropped to about orbit speed,  the vehicle must roll to up lift,  to keep the trajectory from too-quickly steepening downward. 
     The hypersonics end at roughly local Mach 3 speeds,  which is around 0.7-1 km/s velocity,  near 5 km altitude on Mars,  and near 45 km altitude on Earth  which has about the same air pressure.  Up to that point,  entry at Mars and Earth look very much alike,  excepting the altitude.  After that point they diverge sharply,  as illustrated in Figure 5.

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     The descent and landing at Earth require the ship to decelerate to transonic speed,  then pull up to a 90-degree angle of attack (AOA,  measured relative to the wind vector).  Thus,  as the trajectory rapidly steepens to vertical,  the ship executes a broadside “belly-flop” rather like a skydiver. 
     At low altitude where the air is much denser,  the terminal speed in the “belly-flop” will be well subsonic.  I assumed 0.5 Mach,  but that might be a little conservative.  This is the point where AOA increases to 180 degrees (tail-first),  and the landing engines get ignited.  From there,  touchdown is retropropulsive.
     The landing on Mars is quite different.  The ship comes out of hypersonics very close to the surface,  still at high AOA and still very supersonic.  From there,  the ship must pitch to higher AOA and pull up,  actually ascending back toward 5 km altitude.  This ascent is energy management:  speed drops rapidly as altitude increases.  It’s not quite a “tail slide” maneuver,  but it is similar to one. 
     At the local peak altitude,  the ship is moving at about local Mach 1,  and pitches to tail first attitude,  igniting the landing engines.  From there,  touchdown is retropropulsive.  The Martian “air” at the surface is very thin indeed,  as the figure indicates.  It may be that thrust is required to assist lift toward bending the trajectory upward:  the engines would have to be ignited earlier,  and at higher speed,  as indicated in the figure.  Whether this is necessary is just not yet known.
     The low point preceding the local pull-up is at some supersonic speed;  I just assumed about local Mach 1.5,  as indicated in the figure.  That would correspond to a factor 1.5 larger landing delta-vee requirement,  implying a larger landing propellant budget. 
     In either case,  I also use an “eyeball” factor of 1.5 upon the kinematic landing delta-vee,  to cover gravity loss effects,  maneuver requirements,  and any hover or near-hover to divert laterally to avoid obstacles. 
     So,  for purposes of this sensitivity analysis,  the Earth landing is not of much interest,  but the Mars landing is.  The sensitivity analysis looks at the effects of Mach 1.5-sized vs Mach 1-sized touchdown delta-vee.

Analysis Results

     The scope of the sensitivity analysis is illustrated in Figure 6.  As indicated earlier,  the orbital delta-vee increases worst-case-vs-average,  for Hohmann transfer,  were applied additively to the departure delta-vees for the faster trajectories.  No attempt was made to vary the course correction budgets.   Growth in vehicle design inert mass was examined.  An increase in the Mars touchdown delta-vee was examined.  Nothing else was considered.

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     The results start with the worst vs average orbital delta-vee sensitivity.  These results are given in Figure 7.  These are the plots from the spreadsheet,  copied and pasted into the figure.  There are 4 such plots in the figure:  the top two are for the outbound journey Earth to Mars.  The bottom two are for the return journey Mars to Earth.  Results for all 4 transfer orbit cases are shown simultaneously by using trip time as the abscissa. 
     Each has 4 data points:  these are for the Hohmann transfer at 259 days flight time,  the 2-year abort orbit at 128 days,  the non-abort orbit at 110 days,  and the 3-year abort at 102 days.  Be aware that the curves are probably not really straight between the Hohmann orbit and the 2-year abort orbit.  I did not run enough fast transfer cases in ref. 1 to get a smooth curve here.
     The most significant thing in the left hand figure for the outbound trip is the about-40 ton loss of max payload between average and worst case for the Hohmann transfer.  This is a lot less than the about-130 ton payload loss using the 2-year abort orbit instead of Hohmann transfer,  or the about-210 ton payload loss for using the 3-year abort orbit. 
     The average-vs-worst-case deficits are somewhat similar on the faster orbits.  The Mars entry interface velocity trend in the right-hand figure is obviously very nonlinear.  Yet,  all the calculated values fall below the entry velocity from low Earth orbit (LEO).  Any heat shield capable of serving for return from LEO will serve this Mars entry purpose,  which would be the governing case if the trip were one-way only.  There’s only a small change in entry speeds for average-vs-worst orbit case in this estimated analysis. 
     The return voyage has trends shaped quite differently.  For Hohmann transfer,  the worst-vs-average payload loss is about 20 tons.  The deficits on the faster orbits should be similar.  The deficit for using the 2-year abort orbit instead of Hohmann is far larger at about 110 tons,  and that’s from a small return payload to begin with. 
     In the right hand Earth entry interface speed plot,  the blue and orange curves in the entry interface plot fall only slightly apart.  Note that all the entry velocities are much higher than the just-below-escape speed seen with Apollo returning from the moon.  The faster transfer orbits,  and even the Hohmann transfer,  are substantially more demanding than a lunar return entry.  It is clearly the direct-entry Earth return that will size the heat shield design!  

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     Results for the effects of inert mass growth sensitivity are given in Figure 8.  This is the same 4-plot format as Figure 7.  For the outbound trip to Mars,  the Hohmann mass penalty for inert mass growth is about the same 40 ton deficit as for worst-case orbit distances.  It is similar for the faster trajectories.  It is the return trip that most suffers from vehicle inert mass growth.  We lose about 40 tons from an already small return payload on the Hohmann transfer.  However the 2-year abort trajectory and the no-abort trajectory are entirely infeasible,  with their max payloads calculated as negative.  Note that both the Mars and Earth entry interface velocities are unaffected by this sensitivity.  The orange and blue curves fall right on top of each other.

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     The sensitivities to the need for a thrusted pull-up on Mars are given in Figure 9.  This follows the same format as Figures 7 and 8.  Bear in mind that the nominal design lights the engines for touchdown at about Mach 1 speed.  For this analysis,  the engines are ignited earlier,  at about Mach 1.5 flight speed,  to assist lift in pulling up to the Mach 1 “flip”,  to tail-first attitude.  That makes the landing delta vee about 1.5 times larger.  (Note that each case is also factored up by 1.5 further,  to cover any maneuver / hover needs for the touchdown.)
     What the figure shows is about the same 40-ton payload loss on the voyage to Mars to cover the increased landing propellant requirement for the Hohmann transfer.  Effects on the faster transfers are similar.  This trend is comparable to the worst-case orbit losses.  The return payload is entirely unaffected,  as the landing occurs prior to refueling and loading for the trip home.
     Both the Earth and Mars entry interface velocities are unaffected by this Mars thrusted pull-up scenario.  The orange and blue curves fall right on top of each other. 

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Final Remarks

     #1.  These results are only approximate!  Real 3-body orbital analysis,  and real entry-trajectory lifting flight dynamics models,  must be used to get better answers.  Nevertheless,  the trends are quite clear from this approximate analysis.
     #2.  Flying on faster transfer orbits will cost a lot of payload capability,  on both the outbound voyage,  and the return voyage.  This effect is much worse on the return voyage,  where the allowable payload is just inherently smaller.
     #3.  The effects of worst-case orbital positions-relative-to-average,  of Mars and Earth,  have a significant effect on payload,  but it is only half or less the effect of choosing faster transfer orbits.
     #4.  The effect of vehicle inert mass growth from the design target of 120 metric tons to an arbitrary but realistic 160 metric tons is comparable to the effect of worst-case vs average orbits on the outbound voyage.  However it has catastrophic effects on the return voyage!  This is enough to prevent faster-than-Hohmann transfers on the voyage home,  for this vehicle model.
     #5.  The effects of needing a thrusted pull-up for the Mars landing is comparable to the effects of worst-case orbit distances on the outbound voyage.  This has no effects upon the return voyage.
     #6.  It is the direct Earth entry velocity that will design the vehicle heat shield for any vehicle capable of making the return.  This is substantially more challenging than was the return from the moon.  For deliberately-designed one-way vehicles to Mars,  the heat shield design requirements are comparable to entry from low Earth orbit.
     #7.  My personal opinions are that thrusted pull-up will be needed,  along with the need to fly when Earth and Mars orbital distances are worst-case,  plus there will be a little inert mass growth (say by 20 metric tons to 140 metric tons vehicle inert mass).  That kind of thing is the proper design point for this vehicle,  not the most rosy projections!  Estimated performance data for this design case (at 140 metric ton inert mass) are in Figure 10 (same basic format as Figures 7,  8,  and 9).   Note that two of the faster transfers home are precluded.  The feasible one has a very small max payload value compared to Hohmann transfer. 

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     #8.  Bear in mind that the rather high max allowable payload figures feasible to Mars for Hohmann transfer are incompatible with what can be aboard “Starship” for launch to low Earth orbit.  The payloads for the faster transfers to Mars look more like what can be ferried up to LEO.  That suggests that a faster transfer to Mars is most compatible with the projected “Starship” / “Super Heavy” system design characteristics,  as these were evaluated in references 2 and 3.
     #9.  Bear also in mind that a faster transfer orbit to Mars ought to include abort capability,  in case conditions at arrival prove too bad to attempt the landing.  There is simply not the propellant available to enter orbit and wait for better conditions.  Thus life support supplies must be carried to last the entire period of the transfer orbit,  and a full-capability heat shield for direct Earth entry must be used.
     #10.  The fast transfer home need not be limited by abort capability.  It can be a different transfer orbit than the outbound trip.  Surprisingly,  the shapes of the plotted curves suggest that something faster than the “3-year abort” orbit could be used for the return home.
     #11.  Given a way to combine two payloads to LEO into one “Starship” by cargo transfer operations on orbit,  then (and only then) the very large payloads to Mars indicated for Hohmann transfer become feasible.  Like on-orbit cryogenic refueling,  this on-orbit cargo transfer capability does not yet exist,  not even as a concept (on-orbit refueling at least exists as a concept).

References:

     #1. G. W. Johnson,  “Interplanetary Trajectories and Requirements”,  posted 21 November 2019, 
     #2. G. W. Johnson,  “Reverse-Engineering the 2019 Version of the Spacex “Starship”/ ”Super Heavy” Design”,  posted 22 October 2019, 
     #3. G. W. Johnson,  “Reverse-Engineered “Raptor” Engine Performance”,  posted 26 September 2019, 

My text for this sermon is the set of delta v maps, especially the second of them, at the still ever-growing Atomic Rockets site. These maps show the combined speed changes, delta v in the biz, that you need to carry out common missions in Earth and Mars orbital space, such as going from low Earth orbit to lunar orbit and back.

Here is a table showing some of the missions from the delta v maps, plus a few others that I have guesstimated myself:

Patrol Missions

Mission	Delta V
Low earth orbit (LEO) to geosynch and return	5700 m/s powered (plus 2500 m/s aerobraking)
LEO to lunar surface (one way)	5500 m/s (all powered)
LEO to lunar L4/L5 and return (estimated)	4800 m/s powered (plus 3200 m/s aerobraking)
LEO to low lunar orbit and return	4600 m/s powered (plus 3200 m/s aerobraking)
Geosynch to low lunar orbit and return (estimated)	4200 m/s (all powered)
Lunar orbit to lunar surface and return	3200 m/s (all powered)
LEO inclination change by 40 deg (estimated)	5400 m/s (all powered)
LEO to circle the Moon and return retrograde (estimated)	3200 m/s powered (plus 3200 m/s aerobraking)
Mars surface to Deimos (one way)	6000 m/s (all powered)
LEO to low Mars orbit (LMO) and return	6100 m/s powered (plus 5500 m/s aerobraking)

Entries marked "(estimated)" are not in source table; delta v estimates are mine. ("Plus x m/s aerobraking" means ordinarily the engine would be responsible for that delta V as well, but it can be obtained for free via aerobraking. E.g., LEO to geosynch and return costs 8,200 m/s with no aerobraking)

Two things stand out in this list. One is how helpful aerobraking can be if you are inbound toward Earth, or any world with a substantial atmosphere. Many craft in orbital space will be true aerospace vehicles, built to burn off excess speed by streaking through the upper atmosphere at Mach 25 up to Mach 35.

But what really stands out is how easily within the reach of chemical fuels these missions are. Chemfuel has a poor reputation among space geeks because it barely manages the most important mission of all, from Earth to low orbit. Once in orbit, however, chemfuel has acceptable fuel economy for speeds of a few kilometers per second, and rocket engines put out enormous thrust for their weight.

(ed note: with 4,400 m/s exhaust velocity oxygen-hydrogen chemical rockets:

3100 m/s ΔV requires a very reasonable mass ratio of 2 {50% of wet mass is fuel}

6100 m/s ΔV requires a mass ratio of 4 {75% fuel} which is right at the upper limit of economical mass ratios )

In fact, transport class rocket ships working routes in orbital space can have mass proportions not far different from transport aircraft flying the longest nonstop global routes.

A jetliner taking off on a maximum-range flight may carry 40 percent of its total weight in fuel, with 45 percent for the plane itself and 15 percent in payload. A moonship, the one that gets you to lunar orbit, might be 60 percent propellant on departure from low Earth orbit, with 25 percent for the spacecraft and the same 15 percent payload. The lander that takes you to the lunar surface and back gets away with 55 percent propellant, 25 percent for the spacecraft, and 20 percent payload.

(These figures are for hydrogen and oxygen as propellants, currently somewhat out of favor because liquid hydrogen is bulky, hard to work with, and boils away so readily. But H2-O2 is the best performer, and may be available on the Moon if lunar ice appears in concentrations that can be shoveled into a hopper. Increase propellant load by about half for kerosene and oxygen, or 'storable' propellants.)

(ed note: so the point is that chemical rockets are perfectly adequate for missions to Mars or cis-Lunar space provided there is a network of orbital propellant depots suppled by in-situ resource allocation. An orbital propellant depot in LEO supplied by Lunar ice would do the trick. An orbital depot in Low Mars Orbit supplied by Deimos ice would also be very useful.)

      One of our favorite SF themes is the "Belter Civilization," which usually seeks—and gets—independence from the colonial masters on Earth. Belters make their livings as asteroid miners, and they flit from asteroid to asteroid, slicing up planetoids for the rich veins of metal we'll presumably find in them.

     In the usual story, the miners go off on long prospecting tours, leaving their families on a "settled" rock. The Belt Capital is usually located on Ceres or some other central place which may or may not have been extensively transformed; and when Belters get together, it's always in an asteroid city.

     The Belters don't ever come to Earth or any other planet. Indeed, they regard planets as "holes," deep gravity wells which can trap them and use up their precious fuels. The assumption here is that it's far less costly to flit from asteroid to asteroid than it is to land on a planet or get into close orbit around one.

     Another assumption, generally not stated in the stories, is that fuels are expensive and scarce, and the Belters have to conserve reaction mass; this is why, in the usual Belter story, you conserve both time and energy by never going outside the Belt. Scarce as fuel is, though, I suppose the Belters have a source of it locally or they couldn't contemplate independence. They must have fuel for their ships and energy for their artificial environments. Without those, there'd be no Belter Civilization. Even if we discover something of fabulous value in the Belt we can't operate without energy and fuel.

     Those are not, by the way, the same thing. Nuclear fission reactors and large solar panels could provide enough power for a permanent Belt station, and if there were something valuable enough out there we could put a reactor onto an asteroid now. Rocket fuels are something else again. To make a rocket work, you must have reaction mass: something to get moving fast backwards and dump overboard. Unfortunately, asteroids are rock, and rocks don't make very good rocket fuel. We'll come back to what the Belters might do about that later.

     For the moment, let's see how difficult travel to and in the Belt is. We’ll use the same measure as last time, the total change in ship velocity required to perform the mission. This is called delta v, and you should recall that a ship with a given fuel efficiency and ratio of fuel to non-fuel weight will have a unique calculatable delta v. It doesn't matter whether the pilot uses that delta v in little increments or in one big burn: the sum of velocity changes remains the same.

     Similarly, various mission delta-v requirements can be calculated from the laws of orbital mechanics independent of the ship used. Figure 14 gives the delta-v requirements for getting around the Earth-Jupiter portion of the solar system. We're assuming that getting to Earth orbit is free, whether with the laser launching system I described previously, or with shuttles, or whatever, so all missions to or from Earth begin and end in orbit.

     The first thing we see is that landing on an asteroid isn't much easier than going to Mars; in fact, Ceres is harder to get to than Mars. This is because not only are the asteroids a long way out, but they don't help you catch up to them; they've so little mass that you have to chase them down. Thus, once among the asteroids, you may well want to stay there and not use up all that delta v coming back to Earth.

     Then, too, travel between Ceres and a theoretical asteroid 2 AU out is a lot cheaper than getting to Earth from either of them. (One AU, or astronomical unit, is the distance from Earth to the Sun and is 93,000,000 miles, or 149,500,000 = 1.5 × 10⁸ kilometers.) It takes 8 km/sec to get to the 2 AU rock, but only 3.2 more to get from there to Ceres.

     So far so good, and we're well on the way to developing a Belt Civilization. There's already a small nit to pick, though: although travel to Mars itself is costly, it's as easy to get to Mars orbit as it is to go from asteroid to asteroid. Thus, if a laser-launch system could be built on Mars, making travel to and from Mars orbit cheap, Mars might well become the Belt Capital.

     Politics being what they are, though, perhaps the Martians (well, Mars colonists?) won't like having all those crude asteroid miners on their planet, and the Belters will have to build their own capital at some convenient place such as Ceres or the 2 AU rock, saving both their feelings and some energy. However, we've so far said nothing about how long it takes to get from one place to another. The delta v's in Figure 14 are for minimum energy trips, Hohmann transfer orbits, and to use a Hohmann orbit you must start and finish with origin and destination precisely opposite the Sun. You can't just boom out when you feel like it; you must wait for the precise geometry, otherwise the delta-v requirements go up to ridiculous values.

     You get a launch window once each synodic period. A synodic period is the time it takes two planets, or planetoids, to go around the sun and come back to precisely the same positions relative to each other: from, say, being on opposite sides of the Sun until they’re in opposition again, which is what we need for a Hohmann journey.

     The synodic periods and travel times are given in Figure 15, and our Belt Civilization is in trouble again. Not only does it take 1.57 years to get from Ceres to 2 AU (or vice versa), but you can only do it once each 7 years! Travel to and from Mars isn't a lot better, either. The Belters aren't going to visit their Capital very often, and one wonders if a civilization can be built among colonies that can only visit each other every seven to nine years, spending years in travel times to do it.

     By contrast, you can get from Earth to the flying rocks every year and a half, spending another year-and-a-half in transit. That's no short time either, but it beats the nine years of the Ceres-asteroid visitations.

     Perhaps, though, we haven't been quite fair to the Belters. Asteroids aren't as widely separated as Ceres and our 2 AU rock Most textbooks claim the asteroids are concentrated between 2.1 and 3.3 AU out from the Sun. We'll assume they're all in the same plane (they aren't), so the Belt area works out to 4.6 × 10²⁷ square centimeters. The books say there are about 100,000 asteroids visible with the Palomar Eye, but we want to be fair (and make things simple) so we'll assume there are 460,000 asteroids interesting enough to want to visit, or one every 10²² cm² within the Belt. That means the asteroids lie on an average of 10¹¹ cm apart, which happens to be 10⁶ km or one million kilometers, about three times the distance from Earth to the Moon.

It is incredibly rare when the laws of the universe let you get something for nothing. Give your heartfelt gratitude to Hermann Oberth for uncovering one for you. It will be a lifesaver.

What is the escape velocity 300 kilometers above the surface of Mars?

300 km = 300,000 meters. Mars has a radius of about 3,396,000 meters. So r = 3,396,000 + 300,000 = 3,696,000 meters. Mars also has a mass of 6.4185e23 kg, you can find this in NASA's incredibly useful Planetary Fact Sheets.

• V_esc = sqrt((2 * G * M) / r)
• V_esc = sqrt((2 * 6.67428e-11 * 6.4185e23) / 3,696,000)
• V_esc = sqrt(85,678,000,000,000 / 3,696,000)
• V_esc = sqrt(23,181,000)
• V_esc = 4814 m/s = 4.81 km/s

What is periapsis around the Sun that will give an escape velocity of 200 km/sec?

200 km/sec = 200,000 m/sec. The mass of the Sun is about 1.9891e30 kg.

• r = (2 * G * M) / (V_esc²)
• r = (2 * 6.67428e-11 * 1.9891e30) / (200,000²)
• r = 265,436,115,600,000,000,000 / 40,000,000,000
• r = 6,635,902,890 meters = 6,636,000 kilometers

Given that you are going to travel a parabolic orbit around the Sun that has an escape velocity of 200 km/s at periapsis, you have an initial velocity of 3.2 km/s, and you wish to exit the Oberth Maneuver with a final velocity of 50 km/s, calculate the required Δ_v burn at periapsis.

V_esc = 200 km/s = 200,000 m/s. V_h = 3.2 km/s = 3200 m/s.V_f = 50 km/s = 50,000 m/s.

• Δ_v = sqrt(V_f² + V_esc²) - sqrt(V_h² + V_esc²)
• Δ_v = sqrt(50,000² + 200,000²) - sqrt(3200² + 200,000²)
• Δ_v = sqrt(2,500,000,000 + 40,000,000,000) - sqrt(10,240,000 + 40,000,000,000)
• Δ_v = sqrt(42,500,000,000) - sqrt(40,010,240,000)
• Δ_v = 206,000 - 200,000
• Δ_v = 6,000 m/s = 6 km/s

So by burning 6 km/s of Δ_v, you get an actual Δ_v increase of 46.8 km/s. That's 40.8 km/s for free. Sweet!

      If onboard fuel is available to produce a velocity change, another type of swingby can do even better. This involves a close approach to the Sun, rather than to one of the planets. The trick is to swoop in close to the solar surface and apply all available thrust near perihelion, the point of closest approach.

     Suppose that your ship has a small velocity far from the Sun. Allow it to drop toward the Sun, so that it comes close enough almost to graze the solar surface. When it is at its closest, use your onboard fuel to give a 10 kms/second kick in speed; then your ship will move away and leave the solar system completely, with a terminal velocity far from the Sun of 110 kms/second.

     The question that inevitably arises with such a boost at perihelion is, where did that "extra" energy come from? If the velocity boost had been given without swooping in close to the Sun, the ship would have left the solar system at 10 kms/second. Simply by arranging that the same boost be given near the Sun, the ship leaves at 110 kms/second. And yet the Sun seems to have done no work. The solar energy has not decreased at all. It sounds impossible, something for nothing.

     The answer to this puzzle is a simple one, but it leaves many people worried. It is based on the fact that kinetic energy changes as the square of velocity, and the argument runs as follows: The Sun increases the speed of the spacecraft during its run towards the solar surface, so that our ship, at rest far from Sol, will be moving at 600 kms/second as it sweeps past the solar photosphere. The kinetic energy of a body with velocity V is V²/2 per unit mass, so for an object moving at 600 kms/second, a 10 kms/second velocity boost increases the kinetic energy per unit mass by (610²-600²)/2 = 6,050 units. If the same velocity boost had been used to change the speed from 0 to 10 kms/second, the change in kinetic energy per unit mass would have been only 50 units. Thus by applying our speed boost at the right moment, when the velocity is already high, we increase the energy change by a factor of 6,050/50 = 121, which is equivalent to a factor of 11 (the square root of 121) in final speed. Our 10 kms/second boost has been transformed to a 110 kms/second boost.

     All that the Sun has done to the spaceship is to change the speed relative to the Sun at which the velocity boost is applied. The fact that kinetic energy goes as the square of velocity does the rest.

     If this still seems to be getting something for nothing, in a way it is. Certainly, no penalty is paid for the increased velocity—except for the possible danger of sweeping in so close to the Sun's surface. And the closer that one can come to the center of gravitational attraction when applying a velocity boost, the more gratifying the result.

     Let us push the limits. One cannot go close to the Sun's center without hitting the solar surface, but an approach to within 20 kilometers of the center of a neutron star of solar mass would convert a 10 kms/second velocity boost provided at the right moment to a final departure speed from the neutron star of over 1,500 kms/second. An impressive gain, though the tidal forces derived from a gravitational field of over 10,000,000 gees might leave the ship's passengers a little the worse for wear.

     Suppose one were to perform the swingby with a speed much greater than that obtained by falling from rest? Would the gain in velocity be greater? Unfortunately, it works the other way round. The gain in speed is maximum if you fall in with zero velocity from a long way away. In the case of Sol, the biggest boost you can obtain from your 10 kms/second velocity kick is an extra 100 kms/second. That's not fast enough to take us to Alpha Centauri in a hurry. A speed of 110 kms/second implies a travel time of 11,800 years.

One way to look at the Oberth effect is in terms of gravitational potential energy. In the reference frame of the planet, the sum of kinetic energy and potential energy is conserved.

So, consider that when you do a rocket thrust, your rocket thruster pumps some kinetic energy into the system and then the result is your rocket ship going off in one path and the exhaust going off in another path. The total energy will be equal to your initial energy plus the energy provided by the rocket thruster.

But that total energy is split between the rocket ship and the exhaust. The Oberth effect is an observation that your rocket ship ends up with more energy if the exhaust ends up with less energy. By "dumping" the exhaust when you're lower in the gravity well, it ends up in a lower orbit with less energy. Therefore, your rocket ship ends up with more energy.

Furthermore, it's quite easy to calculate from first principles the benefit of a general Oberth maneuver, and helps to make it understandable.

Let's say we're in circular orbit at a distance r around a planet of mass M, such that our orbital speed around the planet is v_cir. Let's say we want to execute a burn that will give us a hyperbolic excess of v_inf — that is, we want to burn now such that we ultimately end up with a speed at infinity of v_inf. (If this were to commit to a Hohmann transfer orbit, then v_inf would be the Hohmann orbit transfer insertion deltavee.)

So we need to make some burn deltav that will give us a total initial speed of v_ini = v_cir + deltav. deltav is what we want to solve for. Well, after our burn leaves us with a speed of v_ini, we make no other burns, and so we're strictly under the influence of gravity. That means that the total energy immediately after our burn is complete E will be equal to our total energy after we've escaped the planet entirely and have ended up with our proper hypberbolic speed, E':

E = E'

Since total energy is the sum of the kinetic and potential energies, then

K + U = K' + U'

The kinetic energies should be obvious; they're just (1/2) m v² for the circular and hyperbolic excess speeds, respectively. For potential energy, this is also relatively straightforward. The potential energies are similarly easy to find since U(r) = -G m M/r. Initially we're at distance r; finally we're at distance r → ∞. So:

(1/2) m v_ini² - G m M/r = (1/2) m v_inf² + 0

We can simplify this by noting that G m M/r is also just the escape speed from the planet at our initial distance, which we'll call v_esc. Substituting and canceling the (1/2) m terms:

v_ini² - v_esc² = v_inf²

Now just substitute the expanded value for v_ini and solve for deltavee:

deltavee = √(v_inf² + v_esc²) - v_cir

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A gravity-well maneuver involves what appears to be a contradiction in the law of conservation of energy. A ship leaving the Moon or a space station for some distant planet can go faster on less fuel by dropping first toward Earth, then performing her principal acceleration while as close to Earth as possible. To be sure, a ship gains kinetic energy (speed) in falling towards Earth, but one would expect that she would lose exactly the same amount of kinetic energy as she coasted away from Earth.

The trick lies in the fact that the reactive mass or 'fuel' is itself mass and as such has potential energy of position when the ship leaves the Moon. The reactive mass used in accelerating near Earth (that is to say, at the bottom of the gravity well) has lost its energy of position by falling down the gravity well. That energy has to go somewhere, and so it does — into the ship, as kinetic energy. The ship ends up going faster for the same force and duration of thrust than she possibly could by departing directly from the Moon or from a space station. The mathematics of this is somewhat baffling — but it works.

8.12 Gravity swingbys.

There is one form of velocity increase that needs neither onboard rockets nor an external propulsion source. In fact, it can hardly be called a propulsion system in the usual sense of the word. If a spacecraft flies close to a planet it can, under the right circumstances, obtain a velocity boost from the planet's gravitational field. This technique is used routinely in interplanetary missions. It was used to get the Galileo spacecraft to Jupiter, and to permit Pioneer 10 and 11 and Voyager 1 and 2 to escape the solar system. Jupiter, with a mass 318 times that of Earth, can give a velocity kick of up to 30 kms/second to a passing spacecraft. So far as the spaceship is concerned, there will be no feeling of onboard acceleration as the speed increases. An observer on the ship experiences free fall, even while accelerating relative to the Sun.

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In orbital mechanics and aerospace engineering, a gravitational slingshot, gravity assist maneuver, or swing-by is the use of the relative movement (e.g. orbit around the Sun) and gravity of a planet or other astronomical object to alter the path and speed of a spacecraft, typically to save propellant and reduce expense.

Gravity assistance can be used to accelerate a spacecraft, that is, to increase or decrease its speed or redirect its path. The "assist" is provided by the motion of the gravitating body as it pulls on the spacecraft. The gravity assist maneuver was first used in 1959 when the Soviet probe Luna 3 photographed the far side of Earth's Moon and it was used by interplanetary probes from Mariner 10 onwards, including the two Voyager probes' notable flybys of Jupiter and Saturn.

Explanation

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A gravity assist around a planet changes a spacecraft's velocity (relative to the Sun) by entering and leaving the gravitational sphere of influence of a planet. The spacecraft's speed increases as it approaches the planet and decreases while escaping its gravitational pull (which is approximately the same), but because the planet orbits the Sun the spacecraft is affected by this motion during the maneuver. To increase speed, the spacecraft flies with the movement of the planet (taking a small amount of the planet's orbital energy); to decrease speed, the spacecraft flies against the movement of the planet. The sum of the kinetic energies of both bodies remains constant (see elastic collision). A slingshot maneuver can therefore be used to change the spaceship's trajectory and speed relative to the Sun.

A close terrestrial analogy is provided by a tennis ball bouncing off the front of a moving train. Imagine standing on a train platform, and throwing a ball at 30 km/h toward a train approaching at 50 km/h. The driver of the train sees the ball approaching at 80 km/h and then departing at 80 km/h after the ball bounces elastically off the front of the train. Because of the train's motion, however, that departure is at 130 km/h relative to the train platform; the ball has added twice the train's velocity to its own.

Translating this analogy into space: in the planet reference frame, the spaceship has a vertical velocity of v relative to the planet. After the slingshot occurs the spaceship is leaving on a course 90 degrees to that which it arrived on. It will still have a velocity of v, but in the horizontal direction. In the Sun reference frame, the planet has a horizontal velocity of v, and by using the Pythagorean Theorem, the spaceship initially has a total velocity of √2v. After the spaceship leaves the planet, it will have a velocity of v + v = 2v, gaining around 0.6v.

This oversimplified example is impossible to refine without additional details regarding the orbit, but if the spaceship travels in a path which forms a hyperbola, it can leave the planet in the opposite direction without firing its engine. This example is also one of many trajectories and gains of speed the spaceship can have.

This explanation might seem to violate the conservation of energy and momentum, apparently adding velocity to the spacecraft out of nothing, but the spacecraft's effects on the planet must also be taken into consideration to provide a complete picture of the mechanics involved. The linear momentum gained by the spaceship is equal in magnitude to that lost by the planet, so the spacecraft gains velocity and the planet loses velocity. However, the planet's enormous mass compared to the spacecraft makes the resulting change in its speed negligibly small. These effects on the planet are so slight (because planets are so much more massive than spacecraft) that they can be ignored in the calculation.

Realistic portrayals of encounters in space require the consideration of three dimensions. The same principles apply, only adding the planet's velocity to that of the spacecraft requires vector addition, as shown below.

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Due to the reversibility of orbits, gravitational slingshots can also be used to reduce the speed of a spacecraft. Both Mariner 10 and MESSENGER performed this maneuver to reach Mercury.

If even more speed is needed than available from gravity assist alone, the most economical way to utilize a rocket burn is to do it near the periapsis (closest approach). A given rocket burn always provides the same change in velocity (Δv), but the change in kinetic energy is proportional to the vehicle's velocity at the time of the burn. So to get the most kinetic energy from the burn, the burn must occur at the vehicle's maximum velocity, at periapsis. Oberth effect describes this technique in more detail.

Purpose

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A spacecraft traveling from Earth to an inner planet will increase its relative speed because it is falling toward the Sun, and a spacecraft traveling from Earth to an outer planet will decrease its speed because it is leaving the vicinity of the Sun.

Although the orbital speed of an inner planet is greater than that of the Earth, a spacecraft traveling to an inner planet, even at the minimum speed needed to reach it, is still accelerated by the Sun's gravity to a speed notably greater than the orbital speed of that destination planet. If the spacecraft's purpose is only to fly by the inner planet, then there is typically no need to slow the spacecraft. However, if the spacecraft is to be inserted into orbit about that inner planet, then there must be some way to slow it down.

Similarly, while the orbital speed of an outer planet is less than that of the Earth, a spacecraft leaving the Earth at the minimum speed needed to travel to some outer planet is slowed by the Sun's gravity to a speed far less than the orbital speed of that outer planet. Thus, there must be some way to accelerate the spacecraft when it reaches that outer planet if it is to enter orbit about it. However, if the spacecraft is accelerated to more than the minimum required, less total propellant will be needed to enter orbit about the target planet. In addition, accelerating the spacecraft early in the flight reduces the travel time.

Rocket engines can certainly be used to increase and decrease the speed of the spacecraft. However, rocket thrust takes propellant, propellant has mass, and even a small change in velocity (known as Δv, or "delta-v", the delta symbol being used to represent a change and "v" signifying velocity) translates to a far larger requirement for propellant needed to escape Earth's gravity well. This is because not only must the primary-stage engines lift the extra propellant, they must also lift the extra propellant beyond that, which is needed to lift that additional propellant. Thus the liftoff mass requirement increases exponentially with an increase in the required delta-v of the spacecraft.

Because additional fuel is needed to lift fuel into space, space missions are designed with a tight propellant "budget", known as the "delta-v budget". The delta-v budget is in effect the total propellant that will be available after leaving the earth, for speeding up, slowing down, stabilization against external buffeting (by particles or other external effects), or direction changes, if it cannot acquire more propellant. The entire mission must be planned within that capability. Therefore, methods of speed and direction change that do not require fuel to be burned are advantageous, because they allow extra maneuvering capability and course enhancement, without spending fuel from the limited amount which has been carried into space. Gravity assist maneuvers can greatly change the speed of a spacecraft without expending propellant, and can save significant amounts of propellant, so they are a very common technique to save fuel.

Limits

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The main practical limit to the use of a gravity assist maneuver is that planets and other large masses are seldom in the right places to enable a voyage to a particular destination. For example, the Voyager missions which started in the late 1970s were made possible by the "Grand Tour" alignment of Jupiter, Saturn, Uranus and Neptune. A similar alignment will not occur again until the middle of the 22nd century. That is an extreme case, but even for less ambitious missions there are years when the planets are scattered in unsuitable parts of their orbits.

Another limitation is the atmosphere, if any, of the available planet. The closer the spacecraft can approach, the faster its periapsis speed as gravity accelerates the spacecraft, allowing for more kinetic energy to be gained from a rocket burn. However, if a spacecraft gets too deep into the atmosphere, the energy lost to drag can exceed that gained from the planet's gravity. On the other hand, the atmosphere can be used to accomplish aerobraking. There have also been theoretical proposals to use aerodynamic lift as the spacecraft flies through the atmosphere. This maneuver, called an aerogravity assist, could bend the trajectory through a larger angle than gravity alone, and hence increase the gain in energy.

Even in the case of an airless body, there is a limit to how close a spacecraft may approach. The magnitude of the achievable change in velocity depends on the spacecraft's approach velocity and the planet's escape velocity at the point of closest approach (limited by either the surface or the atmosphere.)

Interplanetary slingshots using the Sun itself are not possible because the Sun is at rest relative to the Solar System as a whole. However, thrusting when near the Sun has the same effect as the powered slingshot described as the Oberth effect. This has the potential to magnify a spacecraft's thrusting power enormously, but is limited by the spacecraft's ability to resist the heat.

An interstellar slingshot using the Sun is conceivable, involving for example an object coming from elsewhere in our galaxy and swinging past the Sun to boost its galactic travel. The energy and angular momentum would then come from the Sun's orbit around the Milky Way. This concept features prominently in Arthur C. Clarke's 1972 award-winning novel Rendezvous With Rama; his story concerns an interstellar spacecraft that uses the Sun to perform this sort of maneuver, and in the process alarms many nervous humans.

A rotating black hole might provide additional assistance, if its spin axis is aligned the right way. General relativity predicts that a large spinning mass produces frame-dragging—close to the object, space itself is dragged around in the direction of the spin. Any ordinary rotating object produces this effect. Although attempts to measure frame dragging about the Sun have produced no clear evidence, experiments performed by Gravity Probe B have detected frame-dragging effects caused by Earth. General relativity predicts that a spinning black hole is surrounded by a region of space, called the ergosphere, within which standing still (with respect to the black hole's spin) is impossible, because space itself is dragged at the speed of light in the same direction as the black hole's spin. The Penrose process may offer a way to gain energy from the ergosphere, although it would require the spaceship to dump some "ballast" into the black hole, and the spaceship would have had to expend energy to carry the "ballast" to the black hole.

The Tisserand parameter and gravity assists

The use of gravity assists is constrained by a conserved quantity called the Tisserand parameter (or invariant). This is an approximation to the Jacobi constant of the restricted three-body problem. Considering the case a comet orbiting the Sun and the effects a Jupiter encounter would have, Tisserand showed that

will remain constant (where a is the comet's semi-major axis, e its eccentricity, i its inclination, and a_J is the semi-major axis of Jupiter). This applies when the comet is sufficiently far from Jupiter to have well-defined orbital elements, and to the extent that Jupiter is much less massive than the Sun and on a circular orbit.

This quantity is conserved for any system of three objects, one of which has negligible mass, and another of which is of intermediate mass and on a circular orbit. For example, the Sun, Earth and a spacecraft, or Saturn, Titan and the Cassini spacecraft (using the semi-major axis of the perturbing body instead of a_J.) This imposes a constraint on how a gravity assist may be used to alter a spacecraft's orbit.

The Tisserand parameter will change if the spacecraft makes a propulsive maneuver or a gravity assist of some fourth object. This is one reason why many spacecraft frequently combine Earth and Venus (or Mars) gravity assists or also perform large deep space maneuvers.