What's The Mission?

• 
  The Polaris from Danger In Deep Space by Cary Rockwell, 1953

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.

• 
  Artwork by John Polgreen

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:

Δ_vo = sqrt[ (G * P_m) / P_r ]

where:

• Δ_vo = 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)
• P_m = planet's mass (kg)
• P_r = planet's radius (m)
• sqrt[x] = square root of x

Δ_vo 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 Δ_esc instead. This is "escape velocity", and is also the delta V required to land from deep space instead of landing from orbit.

Δ_esc = sqrt[ (2 * G * P_m) / P_r ]

Δ_esc = sqrt[ (1.3346e-10 * P_m) / P_r ]

where:

• Δ_esc = deltaV for escape velocity from a planet (m/s)
• G = 0.00000000006673 or 6.673e-11 (gravitational constant)
• 1.3346e-10 = 2 * G
• P_m = planet's mass (kg)
• P_r = 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 Δ_esc already includes the deltaV for Δ_vo. In other words, when figuring the total deltaV for a given mission, you will add in either Δ_esc or Δ_vo, but not both. Use Δ_vo for surface-to-orbit missions and use Δ_esc for planet-to-planet missions

• 
  Artwork by Ed Emshwiller, 1961
• 
  Dynamic Science Fiction, October.

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

Gravitational drag 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

Δ_vd = g_p * t_L

where:

• Δ_vd = deltaV to counteract gravitational drag (m/s)
• g_p = acceleration due to gravity on planet's surface (m/s²)(this assumes that the majority of the burn is close to the ground)
• t_L = 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

t_L = Δ_vo / 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:

A_pg = A / g_p

Δ_vd = Δ_esc / A_pg

where:

• A_pg = 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 (Δ_va). 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, Δ_va 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).

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

Δ_tvo = Δ_vo + Δ_vd + Δ_va

Δ_tesc = Δ_esc + Δ_vd + Δ_va

where:

• Δ_tvo = total orbital deltaV (m/s²)
• Δ_tesc = total escape deltaV (m/s²)
• Δ_vo = basic deltaV cost for liftoff and orbital landing (m/s²)
• Δ_esc = basic deltaV cost for escape and deep space landing (m/s²)
• Δ_vd = deltaV to counteract gravitational drag (m/s²)
• Δ_va = deltaV to counteract atmospheric drag (m/s²)

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.

In the following, "positive longitudinal" means sitting in a chair with your feet on the ground and your head above your heart. "Transverse forces prone" means lying face down. "Transverse forces supine" means lying on one's back.

So you want to keep the acceleration at a maximum of 4g or so otherwise the astronauts cannot manipulate the controls. 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.

• 
  Chart created by Strangequark

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.

• 
  From We Claim These Stars by Poul Anderson, 1959

• 
  Hohmann orbit to a superior planet (Terra to Mars)
• 
  Hohmann orbit to an inferior planet (Terra to Venus)
• 
  "Interplanetary Transport Network"
• 
  "Pork-chop" plot, by Winchell Chung jr.

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

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 5590 m/s.

Notice that the deltaV required to get into orbit is 11180 m/s while the Terra-Mars deltaV is only 5590 m/s. As Robert Heinlein noted, once one gets into Earth orbit, you are "halfway to anywhere."

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.

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 years.

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). This is displayed in something called a C3 or "pork-chop" plot.

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. As a replacement, Windows users can use Trajectory Optimization Tool by Adam Harden.

Alternatively 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.

On the plot, departure times are on one axis, and arrival times are on another. Every intersection corresponds to an orbit. The color of the intersection tells the amount of deltaV required. The spot at the center of the big bulls-eye is the Hohmann orbit. 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.

On this plot I also drew diagonal white lines for trip duration. For instance, all the intersections between the "9 month" and "10 month" lines have a duration between nine months and ten months.

Yes, I know that this chart shows the Hohmann mission taking about 16,000 m/s of deltaV and ten and a half months. instead of 5590 m/s and 8.6 months. My only conclusion is that the Swing-by Calculator is taking many more variables into account, or I made some rather drastic mistakes while calculating the numbers by hand. But, as Tom Lehrer explained in his song New Math, "But the idea is the important thing."

For step by step instructions on calculating 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

There is a java applet at http://www.exodusproject.com/NavComp/index.html

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).

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:

Δ_vTotal = sqrt( Δ_vLiftoff² + Δ_vHohmann²)

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. The http://www.exodusproject.com/NavComp/index.html applet can calculate that for you. What you do is alter the value of the "semi-latus rectum" hit "calculate" and see the new time and deltaV. (For Terra-Mars Hohmann, the semi-latus rectum is 1.207607)

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.

• 
  Pork chop by Winchell Chung jr.
• 
  Actual Pork chop from NASA. The important parts to an SF author are C3l[blue] = delta-V for the launch, and TTIME[red] = mission duration. If anybody cares, SEP[green] = Sun-Earth-Probe angle (if too small solar static drowns out probe signal) and Ls[magenta] = Earth-Sun-Mars angle (angle made by drawing a line from Earth to the Sun then to Mars). Click for larger image.

These are "maps" of the delta-V cost to move from one "location" to another. Keep in mind that some of the locations are actually orbits. If there is a planet with an atmosphere involved, "aerobraking" may be used (i.e., diving through the planet's atmosphere to use friction to burn off delta-V for free in lieu of expending expensive propellant).

• LEO: Low Earth Orbit. Earth orbit from 160 kilometers to 2,000 kilometers from the Earth's surface (below 200 kilometers Earth's atmosphere will cause the orbit to decay). The International Space Station is in an orbit that varies from 320 km to 400 km.
• GEO: Geosynchronous Earth Orbit. Earth orbit at 42,164 km from the Earth's center (35,786 kilometres from Earth's surface). Where the orbital period is one sidereal day. A satellite in GEO where the orbit is over the Earth's equator is in geostationary orbit. Such a satellite as viewed from Earth is in a fixed location in the sky, which is intensely desirable real-estate for telecommunications satellites. These are called "Clarke orbits" after Sir. Arthur C. Clarke. Competition is fierce for slots in geostationary orbit, slots are allocation by the International Telecommunication Union.
• EML1: Earth-Moon Lagrangian point 1. On the line connecting the centers of the Earth and the Moon, the L1 point is where the gravity of the two bodies cancels out. It allows easy access to both Earth and Lunar orbits, and would be a good place for an orbital propellant depot and/or space station. It has many other uses. It is about 344,000 km from Earth's center.

• 
  Chart is from Rockets and Space Transportation. Delta Vs are in kilometers per second. "AB" means "aerobraking", that is, the planet's atmosphere may be used to change delta V instead of expending thrust.
• 
  Chart by Wolfkeeper diagraming delta V requirements in cis-Lunar and Martian space. Topologically this is almost identical to the previous map, but there are some differences.
• 
  In terms of Delta-V, Earth-Moon-Lagrange-1 (EML1) is very close to LEO, GEO and lunar volatiles (moon ice/propellant). By Hop David.
• 
  Earth-Moon-Lagrange-1 (EML1) is only 1.2 kilometers/second from grazing Mars' atmosphere. From there the remaining velocity changed needed can be accomplished with aerobraking. By Hop David.
• 
  In terms of delta-V, Earth-Moon-Lagrange-1 (EML1) is only 2.5 km/sec from the moon and 3.8 km/sec from LEO. If aerobraking drag passes are used, it would only take .7 km/sec to get from EML1 to LEO (red lines indicate one-way delta-V saving aerobraking paths). By Hop David.
• 
  It takes about .65 km/sec to drop from Earth-Moon-Lagrange-1 (EML1) to a 300 km altitude perigee.
  At perigee the cargo is moving nearly escape speed, 3.1 km/sec faster than a circular orbit at that altitude.
  3.1 - .65 is about 2.4. EML1 has about a 2.4 km/sec advantage over LEO.
  From a high apogee, plane changes are inexpensive. So it's easier to pick your inclinitation from EML1. EML1 moves 360 degrees about the earth each month, so you can choose your longitude of perigee when a launch window occurs.
  This 2.4 km/sec advantage not only applies to trans Mars insertions, but any beyond earth orbit destination (near earth asteroids, Venus, Ceres, etc.)
  By Hop David.

The Oberth Effect is a clever way for a spacecraft to steal some extra delta V from a nearby planet (No, this is not the same as a gravitational slingshot). 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. 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.

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:

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

r = (2 * G * M) / (V_esc²)

where:

• V_esc = 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:

V_f = sqrt((Δ_v + sqrt(V_h² + V_esc²))² - V_esc²)

Δ_v = sqrt(V_f² + V_esc²) - sqrt(V_h² + V_esc²)

where:

• V_f = final velocity (m/s)
• V_h = initial velocity before Oberth Maneuver(m/s)
• Δ_v = amount of delta V burn at periapsis (m/s)
• V_esc = 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.