Just a couple of quick questions, if I may.

1. Is there an easy to understand formula which tells you the speed of an orbit at a particular altitude? I can visualise the idea of an orbiting body moving laterally at x,000 metres per second while accelerating at 9.8 metres per second^2 towards the Earth, but can't come up with the maths to work things out.

2. How do you go about planning the trajectories of interplanetary probes when trying to take advantage of gravity assists? (For example, Galileo's VVEGA trajectory.) Were these sorts of things worked out stage by stage ("Hang on, what if we go past Venus TWICE?") or is there a more holistic approach?

Peter B wrote:
Just a couple of quick questions, if I may.
1. Is there an easy to understand formula which tells you the speed of an orbit at a particular altitude?

Chip:
Try "Newton's Version of Kepler's Third Law". Go here for a tutorial: http://dosxx.colorado.edu/~bagenal/1...IntoOrbit.html
(You'll find it illustraited as a problem half way down the page. Answers to the problems are linked at the bottom of that page.)

Peter B:
2. How do you go about planning the trajectories of interplanetary probes when trying to take advantage of gravity assists?

Chip:
Gravity Assist Trajectories: How NASA does it: http://www.jpl.nasa.gov/basics/bsf4-1.html

Hope this helps a bit. [img]/phpBB/images/smiles/icon_wink.gif[/img]

Chip's link has some formulas (although I am having trouble parsing and understanding what is meant by their sentence "For the Earth orbiting the Earth this speed is 30 km/s.")

But the formula is not that hard to work out. In a given amount of time t, however short, the angle in radians traversed by an object with velocity v in circular orbit of radius r is given approx. by (v x t / r). But the angle of the velocity changes by the same amount, so the acceleration is the velocity times that angle divided by time, or v x (v x t / r)/t, which is v^2/r.

If we set that acceleration equal to GM/r^2, (the acceleration due to gravity according to Newton), we come up with the formula v^2/r = GM/r^2, or

<center>v = (GM/r)^(1/2)</center>

That's the same formula as at Chip's link--except they say it takes excessive amounts of geometry or calculus to work out!

Thanks Grapes of Wrath,

I don't know what they mean by "Earth orbiting Earth", and that business about "excessive" amounts of geometry or calculus needed to work out a formula either. (Maybe to work out complex orbital patterns.) A little math is needed - especially if one is calculating a stable, circular orbit with all parameters defined, and no additional outside influences. [img]/phpBB/images/smiles/icon_wink.gif[/img]

The most useful equation I know is the vis viva equation:

v^2 = ( (2/r) - (1/a) )

where a is the semi-major axis of the orbit.

Special case 1: circular orbit. In a circular orbit, r is always equal to a, and thus,

v^2 = ( 1/r )

I'm a lazy cuss, and so I tend to use units such that the GMm balances to exactly 1.

Silas

My understanding from our mission planning guys here at APL is that there really is no short cut - you just have to try something and see if it works. I'm sure there are some heuristics involved, but there's no program you can run that will give you, say, all the ways you can get to Jupiter in a particular timeframe.

__________________
Everything I need to know I learned through Googling.

You might try Rocket and Space Technology for a (to me, at least) very understandable primer into orbital looping stuff...

See Circular Orbit Calculator for what an ametuer (err - me...)might do with that bit o' info...

And see how I get into more trouble at Simple Hohmann Transfer Orbit Calculator

Doug.

Ededited the html...

<font size=-1>[ This Message was edited by: DALeffler on 2002-03-28 18:54 ]</font>