## Going The Distance

The main way to get a handle on your ship definition is to decide what kinds of missions it will be capable of. Let's decide that the Solar Guard cruiser Polaris will be capable of taking off from Terra, travelling to Mars, landing on Mars, taking off from Mars, travelling to Terra, and landing on Terra. All without re-fuelling.

Keep in mind that this is an incredibly silly sort of ship to design. Any real spacecraft designer would design two craft: one surface to orbit shuttle, and one orbit to orbit vehicle.

## Delta-V

The main number of interest is deltaV. This means "change of velocity" and is usually measured in meters per second (m/s) or kilometers per second (km/s). A spacecraft's maximum deltaV can be though of as how fast it will wind up traveling at if it keeps thrusting until the propellant tanks are dry.

If that means nothing to you, don't worry. The important thing is that a "mission" can be rated according to how much deltaV is required. For instance: lift off from Terra, Hohmann orbit to Mars, and Mars landing, is a mission which would take a deltaV of about 18,290 m/s. If the spacecraft has equal or more deltaV capacity than the mission, it is capable of performing that mission. The sum of all the deltaV requirements in a mission is called the deltaV budget.

This is why it makes sense to describe a ship's performance in terms of its total deltaV capacity, instead of its "range" or some other factor equally silly and meaningless. In Michael McCollum's classic Antares Dawn, when the captain asks the helmsman how much propellant they have, the helmsman replies that they have only 2200 kps (kilometers per second) left in the tanks.

The basic deltaV cost for liftoff and landing is what is needed to achieve orbital (or circular) velocity.

For a back-of-the-envelope calculation, figure boosting from Terra's surface into LEO will require about 9,400 m/s of deltaV. For other planets use the equation:

Δvo = sqrt[ (G * Pm) / Pr ]

where:

• Δvo = deltaV to lift off into orbit or land on a planet from orbit (m/s)
• G = 0.00000000006673 or 6.673e-11 (gravitational constant, don't ask)
• Pm = planet's mass (kg)
• Pr = planet's radius (m)
• sqrt[x] = square root of x

Δvo is what you will use for missions like the Space Shuttle, where you just climb into orbit, deliver or pick up something, then land from orbit. However, if the mission involved travelling to other planets, you will have to use Δesc instead. This is "escape velocity", and is also the delta V required to land from deep space instead of landing from orbit.

Δesc = sqrt[ (2 * G * Pm) / Pr ]

Δesc = sqrt[ (1.3346e-10 * Pm) / Pr ]

where:

• Δesc = deltaV for escape velocity from a planet (m/s)
• G = 0.00000000006673 or 6.673e-11 (gravitational constant)
• 1.3346e-10 = 2 * G
• Pm = planet's mass (kg)
• Pr = planet's radius (m)

So for our Polaris mission, basic deltaV for Terra escape or capture: 11,180 m/s, basic deltaV for Mars escape or capture: 5030 m/s

Please note that Δesc already includes the deltaV for Δvo. In other words, when figuring the total deltaV for a given mission, you will add in either Δesc or Δvo, but not both. Use Δvo for surface-to-orbit missions and use Δesc for planet-to-planet missions

## Drag

The above equation does not take into account gravitational drag or atmospheric drag. Both are very difficult to estimate.

For a back-of-the-envelope calculation, figure boosting from Terra's surface into LEO will require an extra 1,500 m/s to 2,000 m/s to compensate for the combined effects of atmospheric drag and gravity drag.

Gravitational drag (aka "gravity tax") depends on the planet's gravity, the angle of the flight path, and the acceleration of the spacecraft. For Terra, the first approximation is 762 m/s (acceleration of ten gees). You won't use this equation, but the actual first approximation is

Δvd = gp * tL

where:

• Δvd = deltaV to counteract gravitational drag (m/s)
• gp = acceleration due to gravity on planet's surface (m/s2)(this assumes that the majority of the burn is close to the ground)
• tL = duration of liftoff or duration of liftoff burn (seconds)

Arthur Harrill has made a nifty Excel Spreadsheet that calculates the liftoff deltaV for any given planet.

Gravitational drag grows worse with each second of burn, so one wants to reduce the burn time. Unfortunately reducing the burn time is the same as increasing the acceleration, and there is a limit to what the human frame can stand. Thorarinn Gunnarsson noted that the eyes are very vulnerable to high-gravity acceleration, second only to bad hearts and full bladders.

You won't use this equation either but

tL = Δvo / A

where:

• A = spacecraft's acceleration (m/s2)

The spacecraft's acceleration will be discussed on the page about blast-off.

The equation you will use is this:

Apg = A / gp

Δvd = Δesc / Apg

where:

• Apg = acceleration of spacecraft in terms of planetary gravities

For our Polaris mission, with an acceleration of 10 g, gravitational drag during Terra lift off will be 11,180 m/s / 10 = 1,118 m/s.

Atmospheric drag only occurs on planets with atmospheres (Δva). There ain't many planets in the solar system with atmospheres. At least none that you'd care to land on. Landing on Jupiter is a quick way to convert your spacecraft into a tiny ball of crumpled metal. The same holds true for Venus, except that the tiny ball will be acid-etched. So for a planet with no atmosphere, Δva will be zero.

For Terra, the first approximation is 610 m/s. It is not possible to give a general equation for atmospheric drag due to the large number of factors and variables. You can probably get away with proportional scaling, comparing atmospheric density, assuming you can find data on planetary atmospheric density (translation: I don't know how to do it).

## Total Delta-V

The total lift-off or landing deltaV is the basic deltaV plus the extra deltaV due to atmospheric drag (if any) and gravitational drag.

Δtvo = Δvo + Δvd + Δva

Δtesc = Δesc + Δvd + Δva

where:

• Δtvo = total orbital deltaV (m/s)
• Δtesc = total escape deltaV (m/s)
• Δvo = basic deltaV cost for liftoff and orbital landing (m/s)
• Δesc = basic deltaV cost for escape and deep space landing (m/s)
• Δvd = deltaV to counteract gravitational drag (m/s)
• Δva = deltaV to counteract atmospheric drag (m/s)

So the total deltaV to lift off from Terra for our Polaris mission is 11,180 + 1118 + 610 = 12,908 m/s. Maybe 13,058 if you add in about 150 m/s for course corrections and as a safety margin.

## Lift-off Acceleration Profile

So you want to keep the acceleration at a maximum of 4g or so otherwise the astronauts cannot manipulate the controls (max of 30g to avoid causing serious injury). But you want to spend as little time as possible getting into orbit in order to minimize gravitational drag. Therefore you want to maintain a steady 4g (throttling back the thrust as the mass of the propellant drops) until you get into orbit, right?

Well, I found that it was not that simple. You see, if you are lifting off from a planet with an atmosphere, you have to have to keep your spacecraft's speed such that the maximum dynamic pressure (or "Max Q") is too low to shred the ship into titanium confetti. The Space Shuttle's acceleration profile keeps Max Q below about 700 pounds per square foot, but a more sturdy spacecraft could probably survive 800 pounds per square foot.

On the NASA Spaceflight forum I asked what the optimal "acceleration profile" would be for an atomic rocket with a thrust-to-weight ratio above 1, an unreasonable specific impulse of 20,000 (a NSWR), single-stage surface to orbit.

A gentleman who goes by the Internet handle of "Strangequark" was kind enough to answer me.

The acceleration profile says that the spacecraft takes off and accelerates at 4g for about five seconds. From second 5 to second 8 it drastically throttles back to an acceleration of about 0.25g. From second 8 to second 50 it gradually increases acceleration until it is back to 4g. It then stays at 4g until second 215, where it achieves orbit and the engine is shut off.

## Hohmann Transfer Orbits

Now we need to figure the deltaV for Terra-Mars transits.

Hohmann Transfers

Current spacecraft propulsion systems are so feeble that they cannot manage much more than the lowest deltaV missions. So they tend to use a lot of "Hohmann transfer orbits".

A Hohmann orbit between two planets is guaranteed to take the smallest amount of deltaV possible. For the Terra-Mars Hohmann, deltaV is 5,596 m/s.

Notice that the deltaV required to get into orbit is 11,180 m/s while the Terra-Mars deltaV is only 5,596 m/s. As Robert Heinlein noted, once one gets into Earth orbit, you are "halfway to anywhere."

And note that it is not strictly necessary that the destination be a physical planet. It can be a virtual point in space, like a reserved slot in geostationary orbit for your communication satellite obtained at great expense and prolonged negotiation with the International Telecommunication Union. Communication satellites are generally delivered via Hohmann transfer, the equations still work even though there is not a planet at the destination. The virtual point still mathematically moves and acts like a planet, even though there ain't nuttin' there.

Drawbacks of Hohmann Transfers

Unfortunately a Hohmann orbit also takes the maximum amount of transit time. For the Terra-Mars Hohmann mission, transit time is about 8.6 months.

The other drawback is that there are only certain times that one can depart for a given mission, the so-called "Synodic period" or Hohmann launch window. The start and destination planets have to be in the correct positions. For the Terra-Mars Hohmann mission, the Hohmann launch windows occur only every 26 months! If you do not launch at the proper time, when you get to the destination planet's orbit the planet won't be there. And then your life span is the same as your rapidly dwindling oxygen supply.

Hop David has computed the cosmic train schedule for Hohmann railroad towns in the Asteroid belt.

### CALCULATING HOHMANN TRANSFERS

If you are in a hurry and just want the transfer parameters between solar system major planets, you can use Erik Max Francis' Mission Tables. These provide the Hohmann delta-V requirements, the transit time, and the delay between Hohmann launch windows.

If the planets you want are not in the tables (because you've made your own solar system or something), the equations are below:

Hohmann Components

A Hohmann transfer consists of three phases:

1. Insertion Burn: A large burn to leave circular orbit around starting planet and enter the Hohmann transfer
2. A long Coasting Phase where the spacecraft travels on an elliptical orbit with engines off
3. Arrival Burn: A large burn to leave the Hohmann transfer and enter into a circular orbit around the destination planet (otherwise you are doing a flyby mission)

So the total delta V required is the Insertion Burn plus the Arrival Burn.

Note that when launching only an idiot or somebody absolutely desperate will have their Hohmann going contrary to the planet's native orbital motion. Launching in the same direction as the orbital motion means your spacecraft starts out will that motion as free delta V. The Terra-Mars insertion burn requires 32,731 m/s of delta V. Launching with Terra's orbital motion means the ship starts out with 29,785 m/s for free, and only has to burn for an additional 2,946 m/s. And in the same way the Mars arrival burn in theory requires 21,476 m/s but by using Mars orbital velocity the ship only needs 2,650 m/s. The total delta V required is only 5,596 m/s, not the outrageous 54,207 m/s it needs in theory.

Also note that with a Hohmann, the starting point and the ending point will be 180° from each other. That is, if you draw a line from the start point, the center point, and the end point, you will make a straight line.

Calculating Hohmann Delta V

Warning: the following technique is a simplification. It assumes that the planet orbits are perfectly circular, and the two orbits are coplanar. Neither of these are true in reality, but they are close enough for goverment work. The following technique will give you figures that are in the ballpark, but please do not use them for real astrogation. The perfect technique that gives perfect results is a nightmare of mathematical calculation. If you really want to know, find a copy of Fundamentals of Astrodynamics or Introduction to Space Flight and I salute you.

At the start, you have to chose the starting planet and destination planet (or moon, or asteroids, or whatever). Both have to be orbiting the same primary object, the sun or central planet.

First you need "μprimary" ("mu") the gravitational parameter for the sun or planet at the center. If you are calculating Hohmann transfers between planets orbiting Sol, I've precalculated the value of μ for you:

μSolPrimary = 1.32715×1020 m3/s2

If you are doing something fancy like transfers between the moons of Jupiter, you have to calculate μprimary for yourself, using the mass of the central body:

μprimary = 6.674×10-11 * Mprimary

where Mprimary = mass of central planet or moon (in kilograms). 6.674×10-11 is Newton's gravitational constant expressed in units such that the resulting delta V will be in meters per second, instead of something worthless like furlongs per fortnight. So for Jupiter, Planetary Fact Sheets tell you it has a mass of 1,898.3×1024 kilograms, therefore its μprimary is 1.2669×1017

For both the starting and destination planets you'll need:

• The mean orbital radius in meters, i.e., the distance between the planet and the primary. Remember 1 AU = 1.496×1011 meters, since very few astronomical books are silly enough to give orbital radii in meters.
• The planet's mass in kilograms
• The planet's mean radius in meters, i.e., distance from the center of the planet and the surface
• The altitude of the parking orbit in meters, i.e., the distance between the planet's surface and the orbiting spacecraft. The orbital altitude at the start planet and destination planet can be totally different from each other. To make life easier on you the parking orbits are assumed to be circular.

Now for the Hohmann delta V calculation. This will give the delta V required to leave low orbit around the starting planet and brake into low orbit around the destination planet. For a crewed mission presumably the crew want to return home again, so you'll have to do the calculations over again with the start and destination data swapped. This will give the delta V for the homeward trip. Add these together to find the minimal delta V rating for the spacecraft.

Yes, there certainly are a lot of equations. That's why they call it rocket science. You probably should make a spreadsheet or something to do the work for you. I tried to encode the following into a spreadsheet (download Microsoft Excel 97-2003 XLS, download Libre Office Calc ODS). It may contains mistakes, use at your own risk.

The "s" subscript means "starting planet" and the "d" subscript means "destination planet". Note that this symbol "∞" should be an infinity symbol, a figure 8 lying on its side. Apologies if your browser cannot render it. In some textbooks they use instead the subscript "inf".

μprimary = 6.674×10-11 * Mprimary

OrbitVelocitys = sqrt[ μprimary / OrbitRadiuss ]

Velocitys = sqrt[ μprimary * ((2 / OrbitRadiuss) - (1 / SemimajorAxis)) ]

Velocity∞s = abs[ Velocitys - OrbitVelocitys ]

μs = 6.674×10-11 * Ms

ParkingOrbitCircularVels = sqrt[ μs / ParkingOrbitRadiuss ]

VelocityescS = sqrt[ (2 * μs) / ParkingOrbitRadiuss ]

VelocityhyperS = sqrt[ Velocity∞s2 + VelocityescS2 ]

DeltaVs = VelocityhyperS - ParkingOrbitCircularVels

OrbitVelocityd = sqrt[ μprimary / OrbitRadiusd ]

Velocityd = sqrt[ μprimary * ((2 / OrbitRadiusd) - (1 / SemimajorAxis)) ]

Velocity∞d = abs[ Velocityd - OrbitVelocityd ]

μd = 6.674×10-11 * Md

ParkingOrbitCircularVeld = sqrt[ μd / ParkingOrbitRadiusd ]

VelocityescD = sqrt[(2 * μd) / ParkingOrbitRadiusd]

VelocityhyperD = sqrt[Velocity∞d2 + VelocityescD2]

DeltaVd = VelocityhyperD - ParkingOrbitCircularVeld

DeltaV = abs[DeltaVs] + abs[DeltaVd]

where:

• x2 = square of x
• sqrt[x] = square root of x
• abs[x] = absolute value of x, that is, remove any negative sign
• SemimajorAxis = Semi-major axis of Hohmann Transfer orbit (meters)
• μprimary = mass of primary star (kg) (or whatever) that starting and destination planets are orbiting, multiplied by gravitational constant
• μs = mass of starting planet (kg), multiplied by gravitational constant
• OrbitVelocitys = orbital velocity of the starting planet (m/s), i.e., free delta V
• Velocitys = velocity of Insertion Burn (m/s)
• Velocity∞s = actual velocity needed for Insertion Burn after taking advantage of the free delta V. Called "hyperbolic velocity at infinity" (m/s)
• ParkingOrbitAltitudes = altitude of ship's parking orbit above starting planet's surface (m)
• ParkingOrbitRadiuss = radius of ship's parking orbit at starting planet (m)
• VelocityescS = local escape velocity from starting planet (m/s)
• DeltaVs = delta V required to insert spacecraft in parking orbit around starting planet into Hohmann transfer (m/s)
• μd = mass of destination planet (kg), multiplied by gravitational constant
• OrbitVelocityd = orbital velocity of the destination planet (m/s), i.e., free delta V
• Velocityd = velocity of Arrival Burn (m/s)
• Velocity∞d = actual deta V needed for Arrival Burn after taking advantage of the free delta V. Called "hyperbolic velocity at infinity" (m/s)
• ParkingOrbitAltituded = altitude of ship's parking orbit above destination planet's surface (m)
• ParkingOrbitRadiusd = radius of ship's parking orbit at destination planet (m)
• VelocityescD = local escape velocity from destination planet (m/s)
• DeltaVd = delta V required for spacecraft to leave Hohmann transfer and enter parking orbit around destination (m/s)
• DeltaV = actual total delta V needed for the entire Hohmann transfer, which is what you were doing all these calculations for in the first place

NOMENCLATURE NOTE:

Depending upon which NASA document you are reading, Velocity∞s is also called Departure V-infinity or C3. In missions it is sometimes called Trans-{destination planet}-Injection, e.g., TMI = Trans-Mars Injection.

Velocity∞d is also called Arrival V-infinity or V. In missions it is sometimes called {destination planet}-Orbit Insertion, e.g., MOI = Mars Orbit Insertion.

The Vis Viva Equation

How does the above mess of equations work? By the power of the Vis Viva Equation aka "orbital-energy-invariance law". It is used multiple times.

If you don't give a rat's heinie about how this works, please skip ahead to the next section.

If a planet, moon, spacecraft, or whatever is in an elliptical (non-circular) orbit around a primary object (sun or moon), the Vis Viva equation is:

μprimary = G * Mprimary

V = sqrt[ μprimary * ((2/r) - (1/a)) ]

where

Mprimary = mass of primary object (kg)
G = Newton's constant of gravitation = 6.674×10-11 N⋅kg-1⋅m2
μprimary = standard gravitational parameter of the primary object
V = orbital velocity at a given point along the elliptical orbit (m/s)
r = distance from primary of the given point along the elliptical orbit (m)
a = semi-major axis of elliptical orbit (m)
sqrt[x] = square root of x

According to Kepler's Third Law, a planet in an elliptical orbit around a primary has a different orbital velocity at different points in the orbit. The closer that orbital point is to the primary, the faster the orbital velocity is.

If you have a circular orbit, r = a so the equation reduces to:

V = sqrt[ μprimary / r ]

and the orbital velocity is the same at all points in the circular orbit.

The elliptical Vis Viva equation is used to calculate Velocitys and Velocityd.

The circular Vis Viva equation is used to calculate OrbitalVelocitys, OrbitalVelocityd, ParkingOrbitCircularVels, and ParkingOrbitCircularVeld

Calculating Hohmann Travel Time

Th = 0.5 * sqrt[ (4 * π2 * SemimajorAxis3) / μprimary ]

where:

• Th = Hohmann travel time (seconds)
• SemimajorAxis = Semi-major axis of Hohmann Transfer orbit (meters) from above calculation
• μprimary = given above, depends on mass of central body
• sqrt[x] = square root of x
• x2 = square of x
• x3 = raise x to the third power
• π = 3.14159...

Remember:

• seconds / 2,592,000 = months
• seconds / 31,536,000 = years

The "0.5" factor is because in a Hohmann, the spacecraft only travels over half the Hohmann orbit before it reaches the destination.

Calculating Hohmann Launch Windows

Hohmann launch windows occur at each synodic period between the two planets.

OrbitPeriodi = 2 * π * sqrt[OrbitRadiusi3 / μprimary]

OrbitPeriods = 2 * π * sqrt[OrbitRadiuss3 / μprimary]

SynodicPeriod = 1 / ( (1/OrbitPeriodi) - (1/OrbitPeriods))

where:

• SynodicPeriod = time delay between Hohmann launch windows (seconds)
• OrbitRadiusi = orbital radius of planet closer to central object (meters)
• OrbitRadiuss = orbital radius of planet further away from central object (meters)
• OrbitPeriodi = one planetary year for the inferior planet (seconds)
• OrbitPeriods = one planetary year for the superior planet (seconds)
• μprimary = given above, depends on mass of central body
• x3 = raise x to the third power
• π ≅ 3.14159...

Remember:

• seconds / 2,592,000 = months
• seconds / 31,536,000 = years

Calculating Launch Timing

This is for calculating two things:

1. What is the configuration of the two planets indicating it is time to launch?
2. If you do a Hohmann from planet A to planet B, how long do you have to wait on planet B before the launch window to planet A opens?

For the first question, the best I can do is indicate the angular separation between the two planets when the Hohmann window opens. For example: with the Terra-Mars Hohmann, when the launch window opens, what is angle Terra-Sol-Mars? Note that 0° is where the start planet is at. And at the end of the Hohmann journey, both the spacecraft and the destination planet will be at 180° from the the location of the start planet at the beginning of the journey.

α = π * (1 - ( (1/(2*sqrt[2])) * sqrt[(r1/r2 + 1)3]))
or
α = π * (1 - ( 0.35355 * sqrt[(r1/r2 + 1)3]))

where:

• α = Phase Angle, or angle StartPlanet-CenterObject-DestPlanet (radians). If negative, DestPlanet is behind StarPlanet, otherwise it is ahead.
• x3 = raise x to the third power
• π ≅ 3.14159...
• 0.35355 ≅ 1 / (2 * sqrt[2])

Convert radians into decimal degrees by muliplying by (180/π), which is approximately 57.29578...

Calculating Stayover Time Before Return Trip

For details about how long the ship will have to delay at Mars before the return trip Hohmann window opens, refer here

### Hohmanns In More Detail

Planetary Transfer Calculator is an on-line calculator for various types of transfers (including Hohmanns and torchship brachistochrone transfers). It can calculate ballistic transfers between planets and moons, and powered (constant acceleration) transfers between stars (including effects of relativity). It can also calculate propagation delay due to the absolute speed of light between planets and moons.

Back-of-the-envelope Orbital Transfer Calculator is an on-line calculator for Hohmann trajectories (only) created by Pete Wildsmith. It is basically a wrapper around Erik Max Francis' BOTE Python library. There are some simplifications which reduce the accuracy a bit, read the docs at the "BOTE Python library" link under "Limitations" for details.

For a more in-depth look at the equations for the deltaV of a given Hohmann mission, go here

There is a more in-depth example of calculating both Hohmann and more energetic orbits using Fundamentals of Astrodynamics at the incomparable Voyage to Arcturus. The entries in question are here, here, and here. The discussion is about the superiority of Nuclear-Ion propulsion as compared to Nuclear-Thermal propulsion.

There are good basic tutorials on orbital mechanics and trajectory here, here and here.

There is a simple listing of the appropriate equations at http://scienceworld.wolfram.com/physics/HohmannTransferOrbit.html and at http://en.wikipedia.org/wiki/Hohmann_transfer_orbit

Here is an Excel spreadsheet called "Pesky Belter" which will calculate Hohmann deltaV, transit times, and synodic periods.

Erik Max Francis has written a freeware Hohmann orbit calculator in Python, available here. Be warned that the documentation is rudimentary, and operating the calculator requires a beginners knowledge of the Python language.

Information about the mass and semi-major axes of various planets can be found here: http://nssdc.gsfc.nasa.gov/planetary/planetfact.html

The Windows utility program Swing-by calculator can be found at http://www.jaqar.com

There is a freeware Windows program called Orbiter that allows one to fly around the solar system using real physics. A gentleman named Steven Ouellette has created an Orbiter add-on that re-creates the Rolling Stone from the Heinlein novel of the same name, along with the mission it flew (follow the above link).

You are probably using Hohmann transfer orbits because your rocket ain't a torchship. That is the spacecraft has such a pathetically small amount of delta-V that it is forced to use bargain-basement bin cheap Hohmanns instead of fast but hideously expensive Brachistochrone tranfers.

Since the ship is on such a tight delta-V budget it cannot afford to leave the pre-plotted Hohmann trajectory. If you do, you'll run out of the propellant you need to reach your destination, the ship will sail off into the Big Dark, and everybody will die when the oxygen runs out. This was highlighted in a famous story called The Cold Equations by Tom Godwin.

The net result is that when it comes to side trips, rockets are about as capable of that as is a railroad locomotive. The rocket has to stick to its planned Hohmann like it was a choo-choo train on solid steel girders. Much like a locomotive, leaving the tracks for an off-road excursion is a disaster (yes I know that some cargo spacecraft are arranged like train with the engines in the front dragging the cargo behind, but that's another matter).

## Mid-Course corrections

When a probe or spacecraft performs a maneuver, the idea is to enter into a pre-calculated trajectory (hopefully arriving at your destination). But nobody and nothing is perfect. The performance of the maneuver might be a hair off, though not enough to be immediately noticeable. Mission Control or the spacecraft's astrogator has the job of monitoring the spacecraft's current position and vector at this specific point in time, to see if the spacecraft is still on track for the specified trajectory. If it is not in the groove, the astrogator will calculate a mid-course corrections (Trajectory Correction Maneuver or TCM). This is a tiny maneuver to put the spacecraft back on track.

Currently I have no idea how to calculate such a thing. In Proceeding of the Symposium on Manned Planetary Missions 1963/1964 they suggested that with then-current navigation gear the delta V required for TCM on the Terra-Mars trajectory was about 105 m/s and 92 m/s for the Mars-Terra trajectory.

## Other Transfer Orbits

And if you have a Torchship with an outrageous amount of delta V, you can do a Brachistochrone transfer. This is kind of the opposite of a Hohmann, it is a maximum delta V cost / minimum transit time trajectory.

You launch whenever you want, none of this "launch window" nonsense. Point the nose of your spacecraft at Mars, burn the engine for 1 gee of acceleration for 1.75 days, do a skew flip to aim your tail at Mars, and burn for 1.75 days of 1 gee deceleration. You get to Mars in 3.5 days flat...

...provided your spacecraft is a torchship that can manage a whopping 2,990,000 meters per second of delta V!

## Pork-Chop Plots

Expending more deltaV than a Hohmann requires can also allow a ship to depart more often than the Hohmann's limit of one per synodic period, but this is hideously complicated to calculate (no, I don't know how to do this either, it is called Lambert's problem).

Instead of calculating this, you can look it up in graphs called a "pork-chop" plot for a given Hohmann trajectory (so-called because some rocket engineer with an odd sense of humor thought the contour lines looked vaguely like a pork chop).

There is an example of how to use a pork-chop plot here.

If you are lucky you can find them in various NASA documents, though almost all of them are for the Terra-Mars mission. Failing that, there are some on-line calculatiors and stand alone applications that will plot them for you.

Luckily (for Windows users at least) there is a Windows program called Swing-by Calculator by http://www.jaqar.com which can calculate all the orbits over a series of dates and export a datafile, which can be imported into Excel, which can then draw the pork chop plot. Full instructions on how to do this is included with the software, which currently is free so long as it is not used for commercial purposes. Unfortunately Swing-By calculator seems to have vanished.

For missions to asteroids you can use NASA's Jet Propulsion Laboratory's online Small-Body Mission-Design Tool.

There is an online calculator called EasyPorkshop. It draws two separate types of plot, Trajectory injection and Orbit Insertion. It cannot draw Total delta-v plots.

Windows users can use Trajectory Optimization Tool by Adam Harden. It also draws two separate types of plot, Trajectory injection and Orbit Insertion. It cannot draw Total delta-v plots.

There is an old-school Windows command-line program with no graphic user interface that only does Earth-Mars porkchop plots here. Go to section A Computer Program for Creating Pork Chop Plots of Ballistic Earth-to-Mars Trajectories, and download PDF document, Zipped file of executable program, and JPL DE421 ephemeris binary data file. It is actually written in Fortran, so good luck with that.

Planetary Transfer Calculator is an on-line calculator that can create Total delta-v plots plots. Alas, they are somewhat tiny and the scales are not labeled.

Now you have to understand that there are four types of pork-chop plots. Remember that every Hohmann trajectory has two propulsion burns:

1. TRAJECTORY INJECTION BURN: burn that pays the delta-V cost to inject the spacecraft into the Hohmann trajectory to the destination planet. The amount of delta-V is called Velocity∞s, Departure V-infinity or C3. In missions it is sometimes called Trans-{destination planet}-Injection, e.g., TMI = Trans-Mars Injection.

2. ORBIT INSERTION BURN: burn that pays the delta-V cost to take the spacecraft out of the Hohmann trajectory and insert it into a circular orbit around the destination planet. The amount of delta-V is called Velocity∞d, Arrival V-infinity or V. In missions it is sometimes called {destination planet}-Orbit Insertion, e.g., MOI = Mars Orbit Insertion.

The total mission delta-V is the sum of the trajectory injection and orbit insertion delta-Vs. This is sometimes called Vtotal.

So the four types of pork-chop plots are:

1. Plot shows Trajectory Injection delta-V contour lines
2. Plot shows Orbit Insertion delta-V contour lines
3. Plot shows both Trajectory Injection and Orbit Insertion as two sets of overlapping contour lines
4. Plot shows Total Mission delta-V contour lines (i.e, the sum of Trajectory Injection and Orbit Insertion)

A pork-chop plot shows the delta-V for a given Hohmann mission: starting in an orbit around Planet A and ending up in an orbit around Planet B. Usually the x-axis shows departure date of the spacecraft. The y-axis is either the arrival date of the spacecraft OR it is the duration of the trajectory (the transit time). If the y-axis is arrival date, there are sometimes diagonal scale lines displaying transit time.

The delta-V (C3, V, or Vtotal) is displayed as a series of contour lines, often colored. This shows as two bullseyes, with the center of one of the bullseyes being the lowest delta-V and the most economical Hohmann transfer.

For every Departure Date and Arrival Date, you find the intersection of the corresponding x and y axis values, and see what delta-V contour it lies in. This is how much delta-V your spacecraft will need. The center of the bullseye with the lowest delta-V defines the allowed departure and arrival dates if you are on a tight budget. By varying your departure time you can see the deltaV cost of launching at other than the proper synodic period. By varying your arrival time you can see the deltaV cost of shortening the duration of the trip.

For an easy to read explanation of pork-chop plots, check out Hollister Davis' Deboning the Porkchop Plot.

For more than you want to know about pork-chop plots, read On the nature of Earth-Mars porkchop plots.

### Using Pork-Chop Plots

The captain of the spacecraft will ask the astrogator for a mission plan to travel from planet A to planet B in trip time T. The astrogator will determine a family of mission plans, with the current ship's delta-V capacity as the upper limit (or the ship will not be capable of performing that mission) and with the captain's specfied trip duration time as the lower limit (or the captain will be unhappy). You see, a Hohmann trajectory generally uses the least delta-V, but also has the longest possible mission time, and the mission can only start on specific dates ("launch windows") as well. By increasing the delta-V used the launch window can be altered and the mission time can be reduced.

What the astrogator will do is have the navigation computer draw a pork-chop plot, which is a graph with departure times on one axis, arrival times on the other axis, and total delta-V requirements drawn as contour lines in the graph. Cross out the areas of delta-V that are too high for the spacecraft to manage, cross out the part of the graph with a mission duration that is too long to suit the captain, and what remains are the possible missions.

If it turns out there is no possible mission inside the stated parameters, the astrogator will have to confer with the captain over what is possible.

## Interplanetary Transport Network

Actually there is a type of transfer orbit that requires even less deltaV than a Hohmann, the so-called "Interplanetary Transport Network" However, this transfer's practicality is questionable, for a manned mission at least. On the plus side it requires exceedingly small amounts of deltaV. On the minus side, as one would expect, it is so slow it makes a Hohmann look like a hypersonic bullet train. A Hohmann can travel from Earth orbit to Lunar orbit in a few days, the Interplanetary Transport Network takes two months.

This was developed for uncrewed space probes who didn't have to worry about dragging along months of life support supplies.

## Tourist Season

Due to spacecraft taking advantage of Hohmann transfers, they will tend to arrive all at the same time at the destination planet, stay until the launch window for Hohmann transfer back to Terra, and be absent for the many months before the next Hohmann timed arrival. In other words, Mars will have a "tourist season" and an "off season". I use the word "tourist" but this actually means "anybody traveling or shipping anything to Mars who wants to avail themselves of the reduced delta V cost of Hohmann transfer."

The ships in transit will tend to be in a relatively compact group. Clever operators will have special ships in the group: not to travel to Mars but to do business with the other ships in the group (with an eye to making lots of money). Things like being an interplanetary 7-11 all night convenience store, selling those vital little necessities (that you forgot to pack) at inflated prices. A fancy restaurant spaceship for when you are truly fed up with eating those nasty freeze-dried rations. A space-going showboat for outer space riverboat gambling. An expensive health clinic. A flying bar with a wide variety of vacuum-distilled liquors (anybody for a Pan Galactic Gargle Blaster?). Not to mention a orbital brothel. Fans of TOS Battlestar Galactica will be reminded of the Rising Star, luxury liner and casino in space.

It might be possible to make an Aldrin Cycler into such an enterprise, but the timing would be tricky.

For the Martian tourist season:

• At Terra, Hohmann launch window to Mars happens every 2.17 years (26 months). Tourists ship launch into Hohmann trajectory.
• Tourist ships spend 0.70873 years (8.5 months) in transit to Mars. Convenience ships do a booming business.
• Tourist ships arrive at Mars. Start of the Martian tourist season
• 1.25 years (15.3 months or 459 days) after tourist ships arrive, Hohmann window to Terra opens. Departure of tourist ships and end of Martian tourist season.
• Tourist ships spend 0.70873 years (8.5 months) in transit to Terra. Convenience ships do a booming business.

Due to the way the Hohmann windows overlap, the Martian tourist season will be 1 year 3 months and 7 days long, and the Martian off season will be 8 months and 15 days long. As with any seasonal place, during tourist season the prices of anything tourist related will be inflated.

YearEvent
0.00Terra ⇒ Mars launch window opens. Tourist fleet Alfa departs Terra.
0.71Tourist fleet Alfa arrives at Mars. End of Martian off season, start of Martian tourist season.
1.98Mars ⇒ Terra launch window opens. Tourist fleet Alfa departs Mars. End of Martian touist season, start of Martian off season.
2.17Terra ⇒ Mars launch window opens. Tourist fleet Bravo departs Terra.
2.69Tourist fleet Alfa arrives at Terra.
2.88Tourist fleet Bravo at Mars. End of Martian off season, start of Martian tourist season.
4.15Mars ⇒ Terra launch window opens. Tourist fleet Bravo departs Mars. End of Martian touist season, start of Martian off season.
4.33Terra ⇒ Mars launch window opens. Tourist fleet Charlie departs Terra.
4.86Tourist fleet Bravo arrives at Terra.

## A Working Example

Table 2: DeltaV budget using Hohmann transfers
StageDelta-v
(m/s)
Terra liftoff and
insertion into
Hohmann transfer
to Mars
14,070
Mars landing5030
Mars liftoff and
insertion into
Hohmann transfer
to Terra
7520
Terra landing12,908
Total39,528
Table 1: DeltaV budget for our Polaris mission.
StageDelta-v
(m/s)
Terra liftoff12,908
Hohmann to Mars5590
Mars landing5030
Mars liftoff5030
Hohmann to Terra5590
Terra landing12,908
Total47,056

Solar Guard cruiser Polaris needs a deltaV of at least 47,056 m/s in order to perform the mission specified in Table 1.

If the propulsion system has enough acceleration to achieve the Hohmann deltaV while still close to the planet it lifted off from, the total deltaV requirements can be reduced. Doing the liftoff and the Hohmann insertion as one long burn does this. Ordinarily the totals of the liftoff and Hohmann deltaVs are simply added together. If done as one long burn, it will be:

ΔvTotal = sqrt( ΔvLiftoff2 + ΔvHohmann2)

For instance, instead of Mars Liftoff and Hohmann being 5030 + 5590 = 10620 it will be sqrt( 50302 + 55902 ) = 7520.

How does this work? Well, it is an example of the Oberth effect (see below). Doing one long burn ensures that more of your propellant is expended low in the gravity well. And in case you are wondering, multi-stage rockets count as "one long burn," even though there is a small interrupting between stages.

Therefore, from Table 2, the Polaris needs a deltaV of at least 39,528 m/s in order to perform the mission specified.

8.6 months one way is pretty pathetic. Of course spending more deltaV can decrease the time.

Much easier of course is to examine a Pork Chop plot from Swing By Calculator. You can see from the left plot below how it reaches the point of diminishing returns quite quickly.

If you want to cheat, you can look up some of the missions in Jon Roger's Mission Table.

## Sample Delta-V Budgets

From the Wikipedia article Delta-v Budget.
• Launch from Terra's surface to LEO—this not only requires an increase of velocity from 0 to 7.8 km/s, but also typically 1.5–2 km/s for atmospheric drag and gravity drag
• Re-entry from LEO—the delta-v required is the orbital maneuvering burn to lower perigee into the atmosphere, atmospheric drag takes care of the rest.

### Stationkeeping

ManeuverAverage delta-v per year [m/s]Maximum per year [m/s]
Drag compensation in 400–500 km LEO< 25< 100
Drag compensation in 500–600 km LEO< 5< 25
Drag compensation in > 600 km LEO< 7.5
Station-keeping in geostationary orbit50–55
Station-keeping in L1/L230–100
Station-keeping in lunar orbit0–400
Attitude control (3-axis)2–6
Spin-up or despin5–10
Stage booster separation5–10

### Terra–Luna space

Delta-v needed to move inside Terra–Luna system (speeds lower than escape velocity) are given in km/s. This table assumes that the Oberth effect is being used—this is possible with high thrust chemical propulsion but not with current (As of 2011) electrical propulsion.

The return to LEO figures assume that a heat shield and aerobraking/aerocapture is used to reduce the speed by up to 3.2 km/s. The heat shield increases the mass, possibly by 15%. Where a heat shield is not used the higher from LEO Delta-v figure applies, the extra propellant is likely to be heavier than a heat shield. LEO-Ken refers to a low earth orbit with an inclination to the equator of 28 degrees, corresponding to a launch from Kennedy Space Center. LEO-Eq is an equatorial orbit.

∆V km/s from/toLEO-KenLEO-EqGEOEML-1EML-2EML-4/5LLOLunaC3=0
Terra9.3–10
Low Earth orbit (LEO-Ken)4.244.333.773.433.974.045.933.22
Low Earth orbit (LEO-Eq)4.243.903.773.433.994.045.933.22
Geostationary orbit (GEO)2.061.631.381.471.712.053.921.30
Lagrangian point 1 (EML-1)0.770.771.380.140.330.642.520.14
Lagrangian point 2 (EML-2)0.330.331.470.140.340.642.520.14
Lagrangian point 4/5 (EML-4/5)0.840.981.710.330.340.982.580.43
Low lunar orbit (LLO)1.311.312.050.640.650.981.871.40
Luna2.742.743.922.522.532.581.872.80
Terra escape velocity (C3=0)0.000.001.300.140.140.431.402.80

### Terra–Luna space—low thrust

Current electric ion thrusters produce a very low thrust (milli-newtons, yielding a small fraction of a g), so the Oberth effect cannot normally be used. This results in the journey requiring a higher delta-v and frequently a large increase in time compared to a high thrust chemical rocket. Nonetheless, the high specific impulse of electrical thrusters may significantly reduce the cost of the flight. For missions in the Terra–Luna system, an increase in journey time from days to months could be unacceptable for human space flight, but differences in flight time for interplanetary flights are less significant and could be favorable.

The table below presents delta-v's in km/s, normally accurate to 2 significant figures and will be the same in both directions, unless aerobreaking is used as described in the high thrust section above.

FromTodelta-v (km/s)
Low Earth orbit (LEO)Earth–Moon Lagrangian 1 (EML-1)7.0
Low Earth orbit (LEO)Geostationary Earth orbit (GEO)6.0
Low Earth orbit (LEO)Low Lunar orbit (LLO)8.0
Low Earth orbit (LEO)Sun–Earth Lagrangian 1 (SEL-1)7.4
Low Earth orbit (LEO)Sun–Earth Lagrangian 2 (SEL-2)7.4
Earth–Moon Lagrangian 1 (EML-1)Low Lunar orbit (LLO)0.60–0.80
Earth–Moon Lagrangian 1 (EML-1)Geostationary Earth orbit (GEO)1.4–1.75
Earth–Moon Lagrangian 1 (EML-1)Sun-Earth Lagrangian 2 (SEL-2)0.30–0.40

### Interplanetary

The spacecraft is assumed to be using chemical propulsion and the Oberth effect.

FromToDelta-v (km/s)
LEOMars transfer orbit4.3
Terra escape velocity (C3=0)Mars transfer orbit0.6
Mars transfer orbitMars capture orbit0.9
Mars Capture orbitDeimos transfer orbit0.2
Deimos transfer orbitDeimos surface0.7
Deimos transfer orbitPhobos transfer orbit0.3
Phobos transfer orbitPhobos surface0.5
Mars capture orbitLow Mars orbit1.4
Low Mars orbitMars surface4.1
EML-2Mars transfer orbit<1.0
Mars transfer orbitLow Mars Orbit2.7
Terra escape velocity (C3=0)Closest NEO0.8–2.0

According to Marsden and Ross, "The energy levels of the Sun–Earth L1 and L2 points differ from those of the Earth–Moon system by only 50 m/s (as measured by maneuver velocity)."

### Near-Earth objects

Near-Earth objects are asteroids that are within the orbit of Mars. The delta-v to return from them are usually quite small, sometimes as low as 60 m/s, using aerobraking in Earth's atmosphere. However, heat shields are required for this, which add mass and constrain spacecraft geometry. The orbital phasing can be problematic; once rendezvous has been achieved, low delta-v return windows can be fairly far apart (more than a year, often many years), depending on the body.

However, the delta-v to reach near-Earth objects is usually over 3.8 km/s, which is still less than the delta-v to reach the Moon's surface. In general bodies that are much further away or closer to the Sun than Earth have more frequent windows for travel, but usually require larger delta-vs.

## Oberth Effect

The Oberth Effect is a clever way for a spacecraft to steal some extra delta V from a nearby planet (). The spacecraft travels in a parabolic orbit that comes exceedingly close to a planet (or sun), and does a delta V burn at the closest approach (apogee). The spacecraft leaves the planet with much more delta V than it actually burned, apparently from nowhere. Actually the extra delta V comes from the potential energy from the mass of the propellant expended.

No, the Oberth Effect is not the same as a gravitational slingshot. Gravitational slingshots give you free delta V for velocity and vector changes without you having to burn any fuel at all. It also happens with close approaches to planets, but the free delta V can only be in certain directions. Yes, you can use both the Oberth Effect and Gravitational Slingshots in the same maneuver.

The closer you graze the planet or sun, the better, that is, the lower the periapsis or perihelion (there are all sorts of cute names for periapsis depending upon the astronomical object you are approaching, you can read about them in the link). Remember that these are measured from the center of the planet or sun, not their surface. This means that if your ship's parabolic orbit has a periapsis of 4000 kilometers from Terra's center, the fact that the radius of the Terra is about 6378 kilometers means you are about to convert you and your ship into a smoking crater. Do not forget that some planets have atmospheres which raise the danger zone even higher. And approaching too close to the Sun will incinerate your ship.

The first thing you will need to calculate is the escape velocity at periapsis. It is:

Vesc = sqrt((2 * G * M) / r)

r = (2 * G * M) / (Vesc2)

where:

• Vesc = escape velocity at periapsis (m/s)
• G = Gravitational Constant = 6.67428e-11 (m3 kg-1
• s-2)
• M = mass of planet or sun (kg)
• r = periapsis (m)

To actually calculate the bonus delta V you will get from the Oberth Maneuver:

Vf = sqrt((Δv + sqrt(Vh2 + Vesc2))2 - Vesc2)

Δv = sqrt(Vf2 + Vesc2) - sqrt(Vh2 + Vesc2)

where:

• Vf = final velocity (m/s)
• Vh = initial velocity before Oberth Maneuver(m/s)
• Δv = amount of delta V burn at periapsis (m/s)
• Vesc = escape velocity at periapsis (m/s)

From The Rolling Stones by Robert Heinlein (1952). The ship has departed from the Moon, and is about to perform the Oberth Maneuver around Earth en route to Mars.

## Gravitational Slingshot

A Gravitational Slingshot is a clever way for a spacecraft to use the relative motion and gravity of a planet to alter the direction and velocity spacecraft, with said spacecraft burning no propellant at all. There are limits to the directions the ship's vector can be altered to. NASA and other space agencies are quite fond of such slingshots because their ships always have a pathetically low delta-V capability. Nothing better than free delta-V.

No, a Gravitational Slingshot is not the same as the Oberth Effect. The Oberth Effect allows a spacecraft to get bonus delta-V when burning propellant. It also happens with close approaches to planets, but the free delta V can be in any desired direction. Yes, you can use both the Oberth Effect and Gravitational Slingshots in the same maneuver.

It appears like you are getting something for nothing, but you ain't. The laws of physics always balances their books (eventually). What happens is that the spacecraft is stealing energy from the planet. It is just that the planet is so huge and the spacecraft is so tiny, that the craft could steal energy trillions of times before the change in the planet's orbit became detectable by our current scientific instruments. It is like stealing drops of water from the Pacific ocean, the Sun would grow old and die before you noticed any lowering of sea level.

NASA was excited back last century when they spotted an alignment of planets in the solar system occurring in the late 1970s that would allow a space probe to do a series of gravitational slingshots and visit most of the planets. This alignment only happens every 175 years. NASA called it the Grand Tour. Sadly pressure from both the congressional holders of NASA's budget and from the new Space Shuttle program forced the cancelling of the Grand Tour. It was replaced by the drastically down-scaled Voyager program. Meanwhile the Shuttle program suffered costs overruns that devoured NASA budget while utterly failing its design goal of reducing the cost of space access.

### Smuggler's Turn

I've seen this a few times in science fiction but I cannot seem to find any accepted name for it. Perhaps one of you readers can. For now I'll call it The Phssthpok Maneuver. TV Tropes talks about the Spaceship Slingshot Stunt which is not quite the same thing, more like just a gravitational slingsot.

Anyway our heroes are in a spacecraft being hotly pursued by the bad guys, and the heroes cannot see to shake the baddies off their tail. So the heroes dive their ship on a close pass to a planet / gas giant / sun / white dwarf / neutron star / black hole and use either the Oberth effect, gravitational slingshot, or both, to do a bootlegger's turn and escape by shooting off at a wild tangent. The bad guys either are too cowardly to try it, cannot match the velocity, or cannot anticipate the unexpected vector change.

The key is to get as close as possible to something with lots of gravity in order to magnify your efforts to escape.

## Delta-V Maps

This section has been moved here.

## General Orbits

This section has been moved here.

## Atomic Rockets notices

This week's featured addition is SPIN POLARIZATION FOR FUSION PROPULSION

This week's featured addition is INsTAR

This week's featured addition is NTR ALTERNATIVES TO LIQUID HYDROGEN