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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?"
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
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.
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.
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make install -not war
Yup. You have to do trade studies with lots of iterations. On one axis you have launch date, on the other you have arrival date, and you start plotting. You can then vary your trans-martian-injection burn, and what your get are called Porkchop plots, cause they look like porkchops.
-everphilski-
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.
:P
"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-
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.
Because that approach would take too much fuel. I believe that NASA tries to calculate a launch pattern that more or less "slings" the object in an arc that will meet up with Mars after the necessary months of travel.
Everyone who's up in arms over the idea of simplifying the problem needs to calm down. As in most cases, taking into account anything and everything that could effect the trajectory of the spacecraft midflight in mathematical terms creates an overwhelming problem. Science, and good science at that, is constantly conducted using mathematical simplifications (or conducted accepting some form of error... even the most basic measurements, for example, are not accurate in the truest sense). The trick is knowing when and where to make those simplifications and understanding both what you are including the equations and what you are leaving out.
Trying to understand multi-body, multi-plane interplanetary transfers taking into account the effect of radiation pressure, atmospheric drag in LEO, blah blah blah is really not necessary (and becomes extremely complex, as has already been discussed numerous times in previous posts) if all you're trying to do is understand the basic mechanics of a trip to Mars or any other planet, for that matter. Even the mathematics related to bodies that would have an effect on a Mars mission can be simplified through the use of ideas like sphere of influence (SOI; which celestial body has the prevailing gravitational influence on a spacecraft at which points in its trajectory) and the like.
If you're just trying to begin to understand how interplanetary travel works, start with the basics. Then work your way out into Lagrange points, the effects of dark matter on deep space missions, and gravitational assist trajectories.