Quote:
Originally Posted by Anthony
it's worth remembering that this stuff is rocket science, all but the simplest problems are impractical to do by hand, or with the level of math it's practical to put into a spreadsheet.
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Yup, this is the heart of the matter. As a somewhat avid player of Kerbal Space Program, I can say transfer orbits are very much beyond a simple spreadsheet. Just to give you a hint:
In addition to the orbital/escape velocities of Mars and Earth, you need to take into consideration their own orbital velocities around the sun. (14.9 and 18.5 mps respectively). A spacecraft on a minimum energy Hohmann transfer orbit from Earth to Mars will actually arrive at Mars with a slower orbital velocity (around the Sun) than Mars. You start off with a greater velocity than Earth, but as you climb out of the Suns gravity well, you slow down. Doing a quick Google search claims that to get to a transfer orbit to Mars from Earth escape is about 0.3 mps, which will put you at 18.8 mps. You'll then need 0.6 mps to make a capture orbit at Mars, so you'll be going 14.3 mps when you get there. The climb to Mar's orbit cost you 4.5 mps, which you'll have to pay no matter how directly or indirectly you get there. It doesn't come from your Delta-v in low energy transfers, since you already had it from Earth's orbit. But if you wait for minimum distance and burn straight "up" to Mars, you'll notice it.
The ultimate (for a Reactionless system) would be to thrust towards Mars until you reach your suicide burn distance, then doing a suicide burn on approach. (A suicide burn is a deceleration burn at maximum that brings you just to zero velocity at the target). (Generally speaking, in order to just burn at the target, then suicide burn to slow down, you must be able to thrust to overcome the gravitational attraction of the Sun at your distance. At 1 AU this is 0.0006 G or so, so it doesn't take much, but low thrust designs won't be able to cut it.)
How long that takes can vary widely. The distance between Mars and Earth varies from about 40 million to 250 million miles depending on where they are in their orbits. And as you're thrusting towards Mars, it's still zipping along at almost 15 mps in it's orbit, so you'll have to make an Artillery roll (not Gunnery ;-) )
Note that this all glosses over plane changes (Mars orbits in a different plane than Earth. If you don't start at a node where they cross, you need to spend delta-v to haul yourself into the proper plane), and things like the Oberth effect (you get more delta-V from your reaction mass the closer to Earth you are), interactions with other bodies--either in passing or for assists-- and some other things I'm probably forgetting.