The Mathematics of a Trip to Mars? 373
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 (Score:4, Funny)
Mathematics is for Mathematicians (Score:5, Funny)
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Re:Extemely Complex Calculations (Score:3, Funny)
You'd think it was rocket science.
Re:Extemely Complex Calculations (Score:3, Funny)
Pointy end goes up
Re:Extemely Complex Calculations (Score:4, Funny)
How do you know if an astrodynamicist is an extrovert?
He looks at your feet when he talks to you!
Trip to mars dont seem that "simple" (Score:2, Interesting)
I'm willing to bet what ever they use has a TON of factors build into it.
Re:Trip to mars dont seem that "simple" (Score:5, Informative)
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.
Porkchop Plots (Score:4, Insightful)
-everphilski-
Re:Trip to mars dont seem that "simple" (Score:5, Funny)
<cheap shot> // TODO: INSERT ENGLISH-METRIC CONVERSION
Here's a line of the code:
</cheap shot>
Re:Trip to mars dont seem that "simple" (Score:3, Informative)
For an amatuer you could get by with the Earth and Moon (even exclude Sun...although it is large it is much further away) for initial trajectory, then consolidate Earth and Moon and add the Sun, drop the Earth o
Why would you expect us to? (Score:4, Funny)
Hmmm. (Score:4, Funny)
Re:Hmmm. (Score:3, Funny)
Re:Hmmm. (Score:3)
method... (Score:2, Funny)
Earth(+moon), Mars, Sun...I think that will get you there, as long as you dodge the moon on your way out
Re:method... (Score:5, Insightful)
Re:method... (Score:4, Funny)
y = (o*u) / (ar^e + an - (i*(d-i)+(o-t)))
Keep dreaming. It's a complicated thing. You have to factor in the gravity of all the planets, moons, other large bodies and the ship itself. You have to make sure the thing doesn't crash into any asteriods, and it's going to have to make course corrections en route, to avoid things and to stay on track. Maybe the best route is to loop around a planet, and get a speed boost from its gravity. Maybe there isn't enough fuel to do it the fastest way, maybe the timeframe for one route is too small, etc etc etc.
Sorry to break it to you, but there isn't an equation that spits on a directional vector for you to shoot your rocket at. If you want to play pretend, download some solar system simulator, I'd assume at least one of them has some dinky flight planner thingy.
Re:method... (Score:4, Funny)
Re:method... (Score:3, Funny)
I think it would be pretty damn difficult to hit a directional vector with a wad of spit.
Numerical integration (Score:3, Funny)
Come on, this ain't rocket science, people. Oh, wait...
Re:Numerical integration (Score:2)
That and the rediculously long lead times between projects. The project they were interviewing for launches in 2012! That's a long time to slip away before anyone notices your mistakes.
Re:Numerical integration (Score:3, Funny)
I asked a rocket scientist what he said to indicate that something isn't so difficult. He said: I say, "this isn't nuclear physics."
I asked a nuclear physicist what he said to incicate that something isn't so difficult. He said: I say, "this isn't brain surgery."
I asked a brain surgeon what he said to indicate that something isn't so difficult. He said: I say, "this isn't rocket science."
What about (Score:3, Funny)
I am sure that can get you to Mars.
Re:What about (Score:2)
Re:What about (Score:3, Interesting)
Bonus question: calculate the length of the queue for the bathrooms.
Anyway, making a bunch of assumptions (like everyone jumps a half meter high, and weighs 75 Kg) the earth's recoil is a tenth of the diameter of a proton.
Re:What about (Score:2)
Did he take into account that all the people aren't in one place?
Re:What about (Score:3, Insightful)
No change it orbit of the center of mass of the earth-jumpers system, sure. But the earth itself would most certainly change its orbit. Of course, the earth's gravity would soon pull the jumpers back just as the jumpers' gravity would pull the earth back, and the earth's orbit would return to its initial orbit.
The Slashdot "common" (Score:3, Funny)
Since trips to Mars seems commonplace (NASA has sent one every 26 months)
Was I the only one to think... Slashdot... commonplace... once every 2 years....
"Having Sex is commonplace for me"... the new Slashdot definition of commonplace.
Re:The Slashdot "common" (Score:2, Funny)
only if you're talking dupes and its every 2 hrs
Re:The Slashdot "common" (Score:2, Funny)
"Having Sex is commonplace for me"... the new Slashdot definition of commonplace.
Lucky bastard! My sex life can't even be measured in earth time.
Re:The Slashdot "common" (Score:5, Informative)
easy (Score:4, Funny)
It's like a message in a bottle, but so much cooler.
Re:easy: 4 step program (Score:4, Funny)
1) Leave Earth
2) ???
3) Arrive Mars
4) PROFIT!!!
Simplifying interplanetary control software (Score:3, Funny)
Ok ok...I understand some of you will be rolling your eyes at this stage, struggling to understand how on earth a piece of command line software designed for the installation and maintenance of Debian packages could even be remotely applicable to designing a robust mission control interface for missions to the Mars. I will explain. Basically, think of the Earth as a large Debian mirror, equipped with many astronaut 'files'. Imagine the space ship as a .deb package, safely protecting all the astronauts from the harsh vacuum of space. The Mars (or Mars...this solution is cross-platform after all) is your local host. The Sun is...well...that creaky old Sun Ultra 5 from yesterday's OSnews article that no one wants to go close to lest they get burned or flamed by Sun zealots. OK...now how does the system work?
Basically, a mission controller wants to 'install' a 'package' of astronauts from the Earth 'mirror' onto the Mars 'host'. It's 5am, the mission controller hasn't slept for 3 days, and every command sent from Houston is critical. Enter apt-get. The initial launch command would be something like:
apt-get install astronauts
Great! The launch vehicle is on its way! Since the 'link' between the 'mirror' and the 'host' is quite slow (imagine an old school 9600 baud leased line), the 'package' 'download' may take a few days to complete. This is where the mission control staff go to work on getting their Gentoo boxes compiling KDE. When the 'package' is 'downloaded', it's important to check that no astronauts were hurt along the way. The mission controller enters the following command:
apt-get check
This wil check for 'broken dependencies'. So far, so good! The '.deb package' will now successfully 'install' onto the 'host', meaning the astronauts can land on the Mars, and perform their critical experiments. However, all good things must come to and end, and the 'package' will need to be removed from the host. Mission control to the rescue.
apt-get remove astronauts
Excellent! Tom Hanks, Gary Sinese and that other guy are now on their way home. Again, this is a slow link, so our 'host' may take a few days to remove it from it's 'hard disk'. Once the capsule has landed back on Earth, it will be ready for the next group of astronauts to make their journey. But no-one would want to spend 10 days locked up in a small space filled with cast-off cans of Jolt Cola and empty Penguin Mint containers. The capsule will need to be tidied up! Mission control enters one last command to complete the mission:
apt-get autoclean
Done! Another successful Mars shot. Mission control is a breeze with the new apt-get mission control system. No more complicated GUIs, voice recognition or toggle switches. apt-get to infinity and beyond!
Re:Simplifying interplanetary control software (Score:2)
Applying open source software to the safety of Astronauts.
Infact, I wonder if an apt-get implementation could have solved the foam insulation problems of the shuttle in much the same way.
I look forward to the community's response!
Security and Open and Available Software (Score:5, Informative)
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 (Score:5, Insightful)
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.
According to one guy... (Score:3, Informative)
Truck bombs are actually far more frightening than most people think. Think about it for a moment - the IRA's attack on Manchester in
You're not kidding! (Score:3, Informative)
Here's a link http://deseretnews.com/dn/view/0,1249,600155076,00
Re:You're not kidding! (Score:3, Funny)
But the most impressive thing to me is, did they really rebuild that entire stretch of highway, including the railroad tracks, in THIRTY-SIX HOURS?!!? What the fuck! You've gotta send UDOT to New
Re:Missing the point (Score:3)
Nope. Not really. (Score:5, Insightful)
"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
-everphilski-
Much more than a 2-body problem ... (Score:5, Informative)
Cheers,
IT
Re:Much more than a 2-body problem ... (Score:2)
Fortunately, the latest mars launch is nice and simple [wikipedia.org]. Though the launch, correction maneuvers, and mars orbits aren't clear at all. I'd still wet my pants if I could get some data on the precise 3D coordinates/velocities/maneuvers used for the actual orbiter. Does NASA release this kind of data?
Re:Much more than a 2-body problem ... (Score:4, Informative)
SPICE [nasa.gov]-formated files are used for the MRO. Some SPICE kernels are/will be available here [nasa.gov].
As always, Wikipedia to the rescue (Score:3, Informative)
http://en.wikipedia.org/wiki/Gravitational_slingsh ot [wikipedia.org]
http://en.wikipedia.org/wiki/Interplanetary_Superh ighway [wikipedia.org]
http://en.wikipedia.org/wiki/Category:Astrodynamic s [wikipedia.org]
JPL has a good intro (Score:5, Interesting)
http://marsprogram.jpl.nasa.gov/spotlight/porkcho
Re:JPL has a good intro (Score:5, Informative)
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:
Re:JPL has a good intro (Score:3, Informative)
Simple Newtonian (Score:5, Informative)
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 (Score:5, Insightful)
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
Re:Simple Newtonian (Score:4, Funny)
As long as you don't get your units mixed up. :-)
Re:Simple Newtonian (Score:2)
Re:Simple Newtonian (Score:2)
I'm only speculating here, but I think you'd get more precise results (than using GR) if mars' atmosphere is modelled correctly and all the things which can affect its density (temperature etc.) get proper attention.
Re:Simple Newtonian (Score:2, Informative)
At the end of the day, at the scales that are involved, you might have improved the accuracy by a foot or two but it's
Re:Simple Newtonian (Score:3, Interesting)
Let's consider an Earth-moon tranfer, for the launch, the gravitational effect
Orbiter (Score:5, Informative)
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 (Score:5, Informative)
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 (Score:5, Informative)
Re:Trajectory Math (Score:2)
Yea. Maybe that was my only typing/brain-use mistake in the post, but I doubt it.
Re:Trajectory Math (Score:2, Funny)
(But you realize if it'd been a feet/meters error, you never would've heard the end of it....)
Re:Trajectory Math (Score:3, Funny)
Re:Trajectory Math (Score:2)
Two more things... (Score:5, Informative)
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 (Score:5, Informative)
Low Thrust and Space Lanes (Score:3, Interesting)
I find recent work on low thrust trajectories the most fascinating. I was made aware of it in Science News a few months ago. Although the combined influence of the Sun and all the planets form a chaotic system (in principle not predictable beyond certain time limits),
Re:Trajectory Math (Score:2)
While this is usually the case, it wouldn't be for a manned Mars mission. For relatively modest increases in available fuel (delta-v) (or equiavlently, decrease in payload) you can get quite large decreases in travel time.
I don't have numbers, but you might be able to reduce 26 months to 15 months for 20% more delta-V. This is, however, a dimishing-returns situation, e.g. a 3 month trip would be hugely expen
Re:Trajectory Math (Score:5, Informative)
Re:Trajectory Math (Score:4, Interesting)
Hohmann transfers are never the minimum energy orbital transfer. IPS (interplanetary superhighway) orbits are lower energy for all cases, although they take much longer. (To be fair, IPS [wikipedia.org] orbits are new - 1997-ish - and before that, Hohmann transfers were the minimum energy orbital transfer). IPS orbits are so low energy that it basically takes the same delta-V to get almost anywhere in the solar system - the delta-V to get to a Lagrange point.
For manned missions, however, you don't really care about lowest-energy, because orbits are always tradeoffs between transit time and energy, and manned missions want the shortest transit time feasible.
Basics of Space Flight (Score:2)
Automatic Differentiation (Score:2)
get someone else to foot the bill.. (Score:3, Funny)
sit back and watch all the funds get diverted to a new space program.
Some info (Score:4, Informative)
The best info I've found so far is actually a do-it-yourself [orbitermars.co.uk] exercise... there's a space-travel simulator [ucl.ac.uk] that you can use to try to figure out how to get to mars, along with some helper apps that do some math for you.
In terms of starting, basic data... you can ignore the effects of the MRO on the two planets, since it's so small. But the positions of the two planets can be gotten from here [navy.mil]. To understand the coordinates used, study here [wikipedia.org].
I'd like to find some decent open-source apps to visualize the orbits in 3D... at least a static diagram, if not an animation.
Worth a prize (Score:5, Interesting)
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.
One Semester Course in Orbital Mechanics (Score:2, Interesting)
TransX! (Score:5, Informative)
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.
Re:TransX! (Score:2)
G from other bodies convergent (Score:2)
Mainly there's just four significant factors.
1. How much to get off this heap of dirt we call Earth.
2. How fast we want to get there, which basically directly related to the next question...
3. How much does it take to slow down once you get there. (Since there's no friction in outer space...)
4. All the previous in reverse order, knowing that you don't need to get back t
Re:G from other bodies convergent (Score:2)
D.E.'s aside. You transform the D.E.'s into a solvable problem in the Linear Programming world.
"Fundamentals of Astrodynamics" (Score:3, Informative)
A fun physics exercise is to model a slingshot maneuver and then try to figure out *why* your rocket burn is more effective if you dip inside gravity well when you do it.
Re:"Fundamentals of Astrodynamics" (Score:2)
Re:"Fundamentals of Astrodynamics" (Score:3, Informative)
I too was going to suggest Bate, Mueller and White, but I see somebody beat me to it.
The sections on Hohmann Transfer Orbits and Patched Conics would see to answer the OP's question. Not good enough to actually fly a mission, but more than good enough to get the orders of magnitude (delta v, elapsed time, etc.) and figure out what else you need to figure out.
...laura
Re:"Fundamentals of Astrodynamics" (Score:3, Informative)
Pretty Simple first-pass calculation (Score:2, Informative)
Consider an elliptical orbit around the sun (aren't they all...) with a major axis where perihelion is earth's distance, aphelion is Mars' distance from the sun. I don't know the formula, but you should be able to find it on the net.
Now calculate the orbit time. You start your trip tangent to the earth, and blast away faster than the earth is circling the sun (but in the same direction). You catch up to Mars as the top of your orbit is tangent to (grazes the inside of) Mars' orbit. Therefor
Genetic Algorithms (Score:2)
the Simple Answer and the Complex Answer (Score:5, Informative)
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-
Re:the Simple Answer and the Complex Answer (Score:2, Informative)
As for the procedure, for most missions to Mars, the launch vehicle takes care of ste
Why go to Mars? (Score:3, Funny)
If you would just increase my funding and personnel there would be fewer accidents.
--Dr Bertregur.
Satellite Tool Kit (Score:2, Informative)
Want to plan a trip to Mars? no problem using the Astrogator plug-in you're in buisness. However it will set you back several thousands of dollars....
Re:Satellite Tool Kit (Score:3, Interesting)
"Fundamentals of Astrodynamics" (Score:5, Informative)
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.
A good book which touches on the topic..... (Score:4, Informative)
The Basics of Spaceflight (Score:3, Informative)
The concept is easy. (Score:3, Informative)
Atomic Rocket (Score:3, Informative)
Atomic Rocket [projectrho.com] has some interesting reading. It's a nice mix of (as far as I can tell) good physics and some science fiction theory.
Basically, the whole site was designed to help new sci-fi authors make their stories closer to scientific reality. So there's a lot of info not only on the various requirement for a mars trip with different types of engines (everything from chemical thrusters to ion drives to nuclear rockets to the sci-fi only torch ships) to what the requirements would be for a crew living on such a ship and what sort of person defense would actually be reasonable.
It's a fun read, and quite educational as well, if not as hard-core science-y as some of the replies.
Who's selling MATLAB... (Score:3, Interesting)
... Earth-to-Mars calculation packages and, more importantly, who's buying them? I mean, that got to be a niche market if there ever was one.
In a former life, I worked for a group that did some work for the Naval Research Laboratories. Some of the work involved LEO satellites. When asked what software package they were using (knowing that there were several available through COSMIC) to do the calculations, I still recall the answer: ``Oh, we just write our own.'' (As though they do it whenever they need such a program, probably while eating their corn flakes in the morning. Heck, they probably did just that. :-) Being mere mortals, we bought the sources for one of the nicer COSMIC packages. Name of it escapes me.)
So is there really a market for doing interplanetary orbital calculations that someone's actually able to sell a package for $150 a pop? The folks that are actually able to send something from Earth to Mars I suspect are already able to whip out this code in short order. (Dang, I wish I'd watched `The Day the Earth Stood Still' over the weekend like I wanted to. Then I'd be able to include that nifty quote that Klaatu uttered about ``being good enough to get me from planet to planet''.)
Re:Who's selling MATLAB... (Score:3, Informative)
Klatu: I thought you would have solved it by now. You substitute this term here---and then the answer follows by variation of parameters.
Physicist: That gives the first-order terms. But what about the higher orders?
Klatu: Negligible.
Physicist: You've tested this theory?
Klatu [shrugs]
Other Software. (Score:3, Informative)
Von Braun's Short & Sweet Rendition (Score:3, Interesting)
After the war, while helping the US Army launch liberated V2s in New Mexico, Wernher continued to screw off, and eventually scribbled enough material for a small book, Das Marsprojekt. It was quickly offered in English translation as The Mars Project [uillinois.edu], and is still available in paperback. It's only 90 pages cover to cover, and covers all of the basic math, engineering concepts, and logistics of loading up the wagon for a trip.
In particular, the orbital calculations are laid out and illustrated in such a way that anyone with any faculty in math can come to grips with it. THEN, you can go apeshit with tomes such as Introduction To Space Dynamics [amazon.com].
The solar-system map they use is public at JPL (Score:4, Informative)
They use high-precision numerical integration for the trajectory of the spacecraft, using one of the standard high-precision general ephemerides as background data. (Textbooks mentioned by posters elsewhere in this thread decribe in general terms the astronav. techniques used for mission planning, but as soon as they get down to mapping the trajectory as precisely as possible, they need the background ephemeris as well.)
For the recent Mars missions, the background ephemeris is a very highly refined ephemeris "DE410" produced by the JPL, this appears to be a local improvement intended especially to reduce errors in the neighborhood of Mars and Saturn, relative to the DE405 ephemeris which remains the world standard for official ephemeris publications. It seems they got an accuracy in the region of Mars as close as only "a few meters"!!!
See details of DE410 on the public JPL site here [nasa.gov], and especially you might want to look at the background report on DE410 [nasa.gov].
-wb-