The Mathematics of a Trip to Mars?

hakonhaugnes wonders: "Since trips to Mars seems commonplace (NASA has sent one every 26 months), I thought it made sense to try to understand how the interplanetary trajectory is calculated. NASA's page is deploringly void of intricate details. I found this excellent page, but it still left me feeling that I was missing something. Surely the calculus must go beyond two bodies (mars/earth)? (It seems there are commercial MATLAB scripts available but at $150 it went beyond the defensible to satisfy my curiosity). Are there any curious Slashdot readers with the usual great insight into how to calculate a trip to Mars?"

Extemely Complex Calculations

What with all the epicycles and all;-)

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Why would you expect us to?

I see more budget cuts have caused NASA to outsource to the open-source

Security and Open and Available Software

There has been a very long tradition of making source code developed by Government projects available to the general computing public. This is the true "public domain" software that has existed since the beginning of computing. I believe many bits of code from NASA made it into the public domain over the years.

I would bet that the information you desire is now considered to be highly classified and thus not available. You could produce trajectory information for ballistic missiles and who knows how it might be mis-construed as useful to those "terrorists" of whom the US is so fearful these days.

Besides... you might find a units of measure error or two if you got to see this code.

Fear and Fear Itself

The mathematical models for ballistic missiles isn't what's stopping "terrorists" from making them. What stops terrorists is that it's so much cheaper, faster, more reliable and easier to load a truck full of fertilizer and fuel oil, then blow up a skyscraper or maybe a bridge. Or just release a $25 video "around election time", which is about 18 months every 2 years (75% of the time). Both of which create terror, which is the entire point of terrorism.

There was a time when such math was secret, and strategic. But we caught up to the Soviets shortly after they tested that ballistic missile math on Sputnik, in the late 1950s. A half century later, our open society has proven more than a match for such "proprietary" losers. If we can stay that way, despite the exaggerated bugbears that people throw around to justify the secrecy that kills both science and liberty.

Nope. Not really.

Any orbital mechanics textbook will give you more than enough information to calculate this for yourself. One of my final exam questions in spacecraft design was to design a moon mission, in about 15 minutes. Mars isn't much harder, just further away, it's the same problem.

"Elements of Spacecraft Design" by Charles D. Brown has a few good chapters on orbital mechanics. Check a local university library, cause the book cost me nigh unto $100 :P

-everphilski-

Much more than a 2-body problem ...

Several of the people I work with in Caltech's Control and Dynamical Systems department work on celestial mechanics and calculating space flight trajectories -- and I can assure you, it's some pretty complicated stuff, involving invariant manifolds and (IIRC) patching together different three-body systems. There's a good popular article [sciencenews.org] about this in Science News, and you can find more info (in as much detail as you'd like!) on Shane Ross' homepage [caltech.edu].

Cheers,
IT

JPL has a good intro

I was an intern at JPL a couple of decades ago, and they always started with a "porkchop plot" (or "butterfly plot") of possible trajectories and their energy requirements. Here is a webpage that documents that to some extent:

http://marsprogram.jpl.nasa.gov/spotlight/porkchop All.html [nasa.gov]

Re:JPL has a good intro

I did some graduate research/internship at JPL. They still use the porkchop plots which are published in volumes spanning the next decade or two. The "bible" at JPL (as far as I could tell) was The Interplanetary Mission Design Handbook Vol 1, Part 2 [cdeagle.com]. This is the document that I carry around with me to work or on travel if I think I am going to do a little research on the side (unfortunately my paying job has nothing to do with astrodynamics.) It covers pretty much all of the relevant equations for the various phases of an interplanetary mission (launch, transfer, arrival) as well as some other stuff. This is 35 pages of raw meat - little explanation, no derivations. Just the facts. I think the actual pork chop plots are in either other volumes or other parts of this volume (my paper copy had them right after this section).

Anyway, without at least some education in orbital mechanics/astrodynamics, the above ref will probably be a little overwhelming. To get up to speed I recommend the following:

• Fundamentals of Astrodynamics [amazon.com] by Bate, Mueller & While. Undergrad text, should be no problem if you have had calculus.
• Fundamentals of Astrodynamics and Applications [amazon.com] by Vallado. (This is usually referred to as "Vallado" - in fact I never even knew its title until I just looked it up!) This one is much more in-depth and is certainly found on the desk of anyone who does research in this field. Most of the stuff from the JPL handbook is in here, plus lots and lots of other stuff

Simple Newtonian

I recently had a NASA guy come to speak to my research group at my medical school in Houston. We were talking about the long term effect of micro-gravity on human physiology (round trip to Mars). Anyway he told us that most of the mathematical calculations that the Space Flight Center here in Houston use are the "simple" Newtonian laws of motion. He claimed they were suitable for calculating trajectories to the Moon, Mars, etc...

Re:Simple Newtonian

Anyway he told us that most of the mathematical calculations that the Space Flight Center here in Houston use are the "simple" Newtonian laws of motion.
Sure. To use Einstein's general relativity would be overkill as the changes are too small.

But Newtons laws can get arbitrarily complex with the number of bodies that go into the equation.

One is newton's axiom.
Two is still easy and taught in school. Kepler ellipses etc. Together with the rocket equation (also only newton), it gives everything needed to go to earth orbit.

But.. three is not analytically solvable. From there, numerics takes over and this is still a very active field of research, still far from perfect. But they're surely good enough :-)

Orbiter

Orbiter is a great way to learn how those trips are done. It is a free simulator for windows and is available at www.orbitersim.com.
It has tools for calculating all sorts of interplanetary transfers and you can actually perform the flight from launch to landing on mars with all kinds of spacecraft.

Trajectory Math

I recently wrote some trajectory software for NASA. What I worked on is an approximation used for mission planning, not actual trajectories. I work with people who live and breathe this stuff and have worked on high-thrust and low-thrust trajectories for missions to the outer planets. I am mostly a software engineer, but I learned a lot from them while working on this project.

The key here is the energy required. Space travel is still dominated by propulsion. That is, the engines and the fuel they need, and the fuel needed to launch that fuel to orbit, etc., is where most of the cost is.

It is important to travel on a trajectory, called the transfer orbit, that requires the least energy. For a high thrust spacecraft, the minimum energy trajectory is called a Holman transfer [nasa.gov]. Simply, it is an orbit that just touches the orbits of both planets. The periapsis, the closest point to the sun, touches the orbit of the one planet and the apoapsis, the furtherest point, touches the other planet. For this to work, the destination planet needs to be half an orbit away when the spacecraft arrives. This is a lot easier to see in a picture.

For Earth to Mars, the spacecraft launches and then the thrusters fire to change the spacecraft's orbit of the sun from Earth's orbit to the transfer orbit. It then travels half of the transfer orbit and fires its thrusters to change its orbit to match Mars. This can be done by aerocapture, aerobraking or propulsion. The opportunity for a Holman transfer to Mars occurs every 26 years. It is based on the length of the orbit for the bodies being transferred between. The return trip also needs to be a Holman transfer to save fuel. The opportunity does not occur until many months after arrival. I forget the actual number. That is why Mars trips will have a long stay on Mars before returning.

Low thrust is different. Low thrust spacecraft thrust all or most of the time during the trip and the trajectory is more complicated. It is not usable for manned flight because it is to slow but is useful for unmanned spacecraft sometimes.

This is called Celestial Mechanics. When you add propulsion, it becomes Orbital Mechanics.

The best site I have found is NASA's Spacefligh Basics [nasa.gov].

Also good is this site [braeunig.us].

For explanation of gravity assists see this site [ednet.ns.ca].

Also see, Science World at Wolrram [wolfram.com]

Re:Trajectory Math

I think you mean 26 months, rather than years.

Two more things...

I forgot to answer part of the question. It is mostly a two-body problem--two bodies at a time. Launch is Earth and the spacecraft. Then it is the Sun and the spacecraft. Then it is Mars and the spacecraft. However, to go straight to a transfer orbit without orbiting the earth first, there is only one time of day to launch (local time), different for each destination. In this way the destination planet is considered at Earth launch.

For my mission planning software we never considered more than two bodes at a time. For the real stuff, they probably consider more than two bodies at a time, but the other bodies are just correction factors.

The Mechanical Universe [caltech.edu], is an excellent way to learn this stuff. It comes on in reruns from time to time.

Re:Trajectory Math

It's a Hohmann transfer orbit, not "Holman". And you get a window for an Earth-Mars transfer orbit every 25+ months, not 26 years. And constant-thrust spacecraft are generally faster on interplanetary missions than high-thrust, brief impulse craft, with the exceptions being gravity assist manouvers and really low acceleration designs such as old-school ion thrusters, (which NASA only uses because they are scared to death of actually flying anything really innovative such as a VaSIMR or a magsail). Constant-acceleration designs are not used for manned missions yet because we havn't sent any manned missions far enough to take advantage of the method, but any manned interplanetary mission would be crazy not to use a magsail because it provides an artificial magnetosphere for charged-particle shielding while at the same time providing continuous thrust with minimal power consumption and no reaction mass by deflecting the solar wind.

Re:Trajectory Math

Overall - nice post. Unfortunately, IAARS and need to clarify a few minor details:

• It's a Hohmann Transfer, not a Holman
• Hohmann transfers are not always the minimum orbit energy orbit. If the ratio of the circular orbit radii is greater than 12 or so, then a bi-elliptic trajectory (three-burn) is optimal.
• Actually, Celestial Mechanics is the study of the orbits of celestial bodies (naturally occurring) - man-made vehicles bring forth the distinction of Astrodynamics.
• Mars' minimum energy transfers occur slightly more than every two years, not 26.
• Oh, and I probably should be much more careful when using the work "optimal". Hohmann tranfers are only "optimal" if they are co-planar, the burns are considered impulses, and the effects of gravity assists, continuous thrust or any other non-conservative forces are ignored. In reality, we need to take advantage at least one of the above to even get to the outer planets on any type of reasonable time-line.

Worth a prize

I helped judge the Canada-Wide Science Fair a few years ago, and the person my judging team ranked the highest had set himself precisely this problem: how do you really calculate the trajectory of a spacecraft from Earth to Mars? His solution was a wonderful exploration of the gory details of the problem--he had parts of the orbit that could be approximated reasonably in closed form (basically when the spacecraft was far away from everything, especially Jupiter) and other bits where there were three-body and more calculations.

He understood error estimation and the importance of computing the same quantity several different ways so that they act as a check on each other. He also had modeled aspects of the spacecraft itself, the rotational moments, effects of changing fuel mass, etc, etc, etc. In short, he understood that science is more of an art than a science. It was really nice work.

TransX!

Duncan Sharpe's TransX [orbitermars.co.uk]

C'mon Orbiter [ucl.ac.uk] fans, you were thinking the exact same thing when you read this article... Planning a trip to Mars? Just hit Shift-J and start plotting your Hohmann transfer orbit insertion burn. [wikipedia.org]

For those who are lost:

ORBITER is a free flight simulator that goes beyond the confines of Earth's atmosphere. Launch the Space Shuttle from Kennedy Space Center to deploy a satellite, rendezvous with the International Space Station or take the futuristic Delta-glider for a tour through the solar system - the choice is yours.
But make no mistake - ORBITER is not a space shooter. The emphasis is firmly on realism, and the learning curve can be steep. Be prepared to invest some time and effort to brush up on your orbital mechanics background. A good starting point is JPL's Space Flight Learners' Workbook. [nasa.gov]

also...

TransX is [Duncan Sharpe's] eXtended Transfer MFD. It's designed for planning trips across the solar system, or even just to the moon. It's full-featured, with support for complex flight plans, including slingshot trajectories. And naturally, there's a manual that comes with it.

the Simple Answer and the Complex Answer

The saying is "From low earth orbit, you are halfway to anywhere in the solar system." The delta-V (change in velocity) required to get to low earth orbit is about 7.6 m/s neglecting gravity and drag losses. The velocity to escape is about 13 m/s. Add in a little bit of velocity to correct your orbit to make it to Mars and it's about right, 14 m/s. (actually it'll be a bit more if you're launching from Kennedy, you have to get rid of that pesky inclination and that's an expensive maneuver, even combining it with the trans-martian injection it's expensive.

Here's the actual procedure.
1. surface to low earth orbit.
2. circularize low earth orbit. [hohmann transfer]
3. correct orbital parameters (longitude of ascending node, argument of periapsis, orbital inclination)
4. low earth orbit to trans-martian-injection [hohmann transfer]
(3 and 4 can be combined, to a point, in order to save delta-V.)
5. burn to circularize martian orbit [hohmann transfer]
6. correct orbital parameters (Same as 3)
7. Burn to descend to surface
The actual math is too much for a slashdot post. Sorry. If you are truly curious check out "Elements of Spacecraft Design" by Charles D. Brown.

-everphilski-

"Fundamentals of Astrodynamics"

is the book you want. It's by Bate, Mueller, and White, and it works from first principles up to "how we designed the Apollo lunar trajectories".

The easiest way to conceive of interplanetary orbits is to first pretend that they lie in a single plane (the plane of the ecliptic) and then pretend that the planets themselves are insignificant for most of the trip -- so you consider only the gravitational field of the Sun. Then your orbit is an ellipse. It's pretty easy to show that, if you're going at Earth's orbital velocity, the ellipse that gets you from Earth's orbit to any other nearly circular orbit with the least change in velocity (ie rocket fuel) is an ellipse that is tangent to both orbits.

Once you've figured that out, you have to figure out when to launch to get to Mars's orbit in the same place that Mars happens to be. Those times happen at a particular phase of Mars's and Earth's orbit.

You can do pretty well by pretending that you can neglect the Sun entirely until you get far enough from the Earth, then you can neglect Earth and Mars entirely until you get close enough to Mars. That is the technique that was used for Apollo trajectories -- the "method of spliced conics". You can hear some evidence of it in the Apollo 13 movie, when they talk about "entering the Moon's gravitational field" or something like that -- the Moon's gravitational field extends throughout the Universe, of course, but to simplify the calculations they neglected everything but the mass with the strongest gravitational force on the capsule.

Nowadays you can get really, really good orbital elements [nasa.gov] for each of the planets online, which lets you calculate exactly where each planet is at any given time. You can just code up an insanely cheesy inverse-square-law integrator in PDL [perl.org] or one of the other free languages -- or even a spreadsheet -- and find a good orbit by trial and error using the gravitational fields of all the large bodies in the solar system.

Re:Trip to mars dont seem that "simple"

Easier: Orbiter [ucl.ac.uk].

I once worked on a more complex version (after writing a simpler version), but got distracted to other projects somewhere between the finished code to implement Kirchoff's laws for the electrical system and the unfinished code to calculate the volume and mass of a fuel tank.

Re:Trip to mars dont seem that "simple"

Nasa has probably built a nifty model...

<cheap shot>
Here's a line of the code: // TODO: INSERT ENGLISH-METRIC CONVERSION
</cheap shot>

Re:method...

Um, okay, great. Gravity from lets say Jupiter stops at the asteriod belt right? Every thing can make a tiny difference. Also not the poster is asking how to plot a course, and you're giving the equation to calculate the newtonian gravity between two objects. Related, yes. An answer, no. Knowing how many newtons of force your getting from all these bodies doesn't solve the problem. You're fired.

Re:The Slashdot "common"

The orbital mechanics that the Hohlmann transfer to Mars takes advantage of allow a "cheap" (low-energy) shot at Mars about every 2 years.