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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?"
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.
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 about this in Science News, and you can find more info (in as much detail as you'd like!) on Shane Ross' homepage.
Cheers,
IT
Power corrupts. PowerPoint corrupts absolutely.
Easier: Orbiter.
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.
Kneel Before Christ!
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...
Argh. The laws of science be a harsh mistress.
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.
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. 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.
Also good is this site.
For explanation of gravity assists see this site.
Also see, Science World at Wolrram
The best info I've found so far is actually a do-it-yourself exercise... there's a space-travel simulator 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. To understand the coordinates used, study here.
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.
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, is an excellent way to learn this stuff. It comes on in reruns from time to time.
Duncan Sharpe's TransX
C'mon Orbiter 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.
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.
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 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-
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 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 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.
Take a look at the book "Mining the Sky" by John S. Lewis. Without getting into a deep mathmatical treatment, he does lay out what goes into calculating sending missions to and from Mars, Earth orbit, the moon, and the asteroid belt. If I am not mistaken, somewhere in there he even explains the significance of the oft heard NASA term "launch window". (It's basically when your launch site (Florida, for instance) and your target (Mars or the ISS) share a favored geometerical relationship in space-time.) While it is lite on the equations, I think this will have most of what you are looking for.... Now if I can just find my copy. BTW.. Lewis' books are a must read for anyone interested in what's up there, whether it's the moon, Mars, or beyond.
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:
"It takes considerable knowledge just to realize the extent of your own ignorance." - Thomas Sowell
The orbital mechanics that the Hohlmann transfer to Mars takes advantage of allow a "cheap" (low-energy) shot at Mars about every 2 years.
You see? You see? Your stupid minds! Stupid! Stupid!
In the land of the blind, the one-eyed man is king.
"Those large trucks on the Interstate that you see every day have a weight limit of about 65,000 lbs."
0 .html to a story of an explosion caused when a semi overturned and caught fire in a canyon about 35 miles from my home. It occurred last Wednesday. The semi was hauling 38,000 lbs. of explosives. Not one person died! That stretch of highway is highly-travelled and pretty dangerous on its own without exploding vehicles. If you look at the images of the road, you'll likely agree that it's quite an amazing thing that no one died. Nearly the entire semi and trailer were gone. The explosion left a crater about 20 to 35 feet deep and 60 to 80 feet wide.
h tml to a Salt Lake TV station that received a video taken by someone travelling on the highway during the explosion. (The streaming video worked quite well on my Mac - Tiger & Safari - , so I'm pretty sure it'll work for most anyone)
Here's a link http://deseretnews.com/dn/view/0,1249,600155076,0
Here is another link http://kutv.com/topstories/local_story_226191800.
I plan to give those truckers an even wider berth from now on.
They don't use two-body approximations for the NASA missions to Mars!
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, and especially you might want to look at the background report on DE410.
-wb-