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?"

Comments

Trip to mars dont seem that "simple"

[ResQuad]

Nasa has probably built a nifty model that will tell them the best launch times and dates. And I highly doubt that this model is as simple as 2 bodies. Everything out there has gravity, hell even the moon could be a problem.

I'm willing to bet what ever they use has a TON of factors build into it.

JPL has a good intro

[cunniff]

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

Worth a prize

[radtea]

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

[LabRat]

The web page the OP found looks pretty cool..though I agree it's a little too condensed to be useful for a complete beginner. While I don't want to imply that orbital mechanics is out of the reach of intelligent, math-oriented folks without some sort of formal instruction....a course in the subject matter can definitely help. I took a class with the author of this book http://www.amazon.com/exec/obidos/tg/detail/-/0292 751052/qid=1124147579/sr=8-2/ref=sr_8_xs_ap_i1_xgl 14/102-9094747-8542529?v=glance&s=books&n=507846

With a decent mathematical background, the book could be followed fairly well to get an idea of what it takes to calculate the trajectory for a Martian mission. There are other books out there too...but I am familiar with this one since I used it in college. Dr. Szebehely was an awesome prof, by the way...everyone should have the privilege of learning from someone like him at least once in their lives.

Of course, in the "real Solar System", the gravity of Jupiter can be a real factor, in addition to the other planets (depending on how close you need your calculations to be)...and unfortunately only the 2-body problem can be easily solved in a general closed form. For other scenarios, numerical methods that calculate the trajectory "step-by-step" must be employed.

Good Luck!

Low Thrust and Space Lanes

[CustomDesigned]

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 too slow but is useful for unmanned spacecraft sometimes.

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), there exist stable transfer lanes which function like chaotic attractors (thanks mainly to the stabilizing influence of Jupiter). Once you get your unmanned craft into the lane, it needs only to apply corrections now and then to stay in the lane - and gravity will take it all the way to its destination! The time required is measured in years rather than months, but it makes unmanned missions much more economical.

The "slingshot" trajectories of earlier out planet explorers were similar, but had to be calculated in full for each mission. This new technique creates a 3D (orbital plane plus time) map of the space lanes - which looks like a maze of twisting tubes of varying diameters. To plan your trip, you find a lane near earth that goes to your destination. You need fuel for getting to the lane, course corrections while travelling, and exiting the lane.

As described in the Novel Oxygen , we could send unmanned supply ships to Mars via the economical low thrust space lane routes. The manned mission would come later, when the supplies have and/or will have arrived, and will be lighter and cheaper since it will only need food, water, etc for the trip, and not for the extended stay required by Holman transfer trajectories for the speedier manned trip. Fuel for the return trip would also be sent ahead. (Clearly, I would want some reduncancy, and robot surveillance to ensure that said supplies have truly arrived safely.)

Re the novel: of *course* something goes wrong. Think Apollo 13, but on a *much* longer trip. That's all I'll say.

Re:Satellite Tool Kit

[GraWil]

AGI (makers of STK) was started by two former General Electric (space division) employees and their software has become industry standard. It is used by most space agencies including ESA & NASA. Note that the price point is high and roughly equal to the engineering time they envison their software replaces. A relatively base model will set you back about $30K USD if you want something with opengl graphics visualization. If you want to plan a mission to Mars you'll need astrogator and probably the visualization so your looking at $50K USD. There are academic discounts of about 20%. For perspective, I'm using STK right now for a Mars mission trade study.

I use the software daily and while I cough at the price and maintenance, it does what it is supposed to do most of the time. Sadly, it does crash a fair bit under windows and they stopped developing for unix a few versions back...

Re:Trajectory Math

[barawn]

Hohmann transfers are not always the minimum orbit energy orbit.

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 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.

Re:Simple Newtonian

[Keebler71]

Yup, we got to the moon (and will go further) using simple two-body equations... most of which are not-too-distantly related to the equations you used in high school physics. This is done by breaking the mission into smaller segments, the dynamics of which are governed using the assumption that only the body that has the greatest gravitational affect on the vehicle is currently active. This is called a patched conic approach

Let's consider an Earth-moon tranfer, for the launch, the gravitational effects of the moon are miniscule and ignored. The vehicle is propagated outward from the Earth along an ellipse (or parabola) with the Earth at one focus. The laucnh dates and launch headings are adjusted such that this outbound orbit gets "close" to the moon. Now if you follow this orbit outwards, at some point it will get so close to the moon that the moon's gravity will dominate the Earth's effects. At this point you resolve the vehicle's state into the moon's (non-inertial) coordinate system in which frame that arrival orbit probably looks like a hyperbola. Now you follow that conic in to the periapsis (closest point of approach) and subtract just enough energy to result in a closed orbit about the moon. Voila! You are now in lunar orbit and never solved a three-body problem (at least not analytically). Of course, the devil is in the details, in this case, splinig the conics together.

Who's selling MATLAB...

[rnturn]

... 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''.)

Von Braun's Short & Sweet Rendition

[cmholm]

One of the things the Wehrmacht's V2 program manager periodically whined about was the tendency of Von Braun and the other engineers to spend their time daydreaming about spaceflight, rather than the practical matter of lofting a ton of high explosives.

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, 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.

Re:According to one guy...

[SgtChaireBourne]

True. The truck bomb which destroyed the Marine barracks in Beirut in 1983 was one of the largest non-nuclear explosions in history.

And for that one, like a good number of similar attacks, the brass had received intel in advance warning specifying darn near every key detail of who, what, when, where, why and how. And like other similar attacks, they ordered the guards to a less ready state and left a drive, clear of obstacles, through from the gate to the lobby.

It was probably in the interest of someone's domestic political agenda to let that one happen. Just like on the 10th of September 2001 in Europe, flights and flight paths were locked down tight though not in the US. Hey if nothing happens then fine. However, if the attack goes through, then you have the perfect excuse to ram through the Patriot Act and other anti-American treachery.

Re:What about

[OwnedByTwoCats]

I remember doing the calculation in College. The really amazing thing is that you could fit all four billion people, each one having a square meter for a chair, in an area only 64 kilometers on a side. With the population increase since then, we're up to a square 77.5 km on a side.

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.