🚀 go-pugleaf

In article <BwztnB.5z3@brunel.ac.uk> mt90dac@brunel.ac.uk (Del Cotter) writes:

   In article <1992Oct30.163302.15547@rcvie.co.at> Ian Taylor writes:
   >While I'm here, anyone know if a gravity assist trajectory can be used to 
   >*reduce* speed?  

   NASA had a proposal to launch a solar probe (Starprobe) at Jupiter which
   would have put it into an orbit of (if memory serves) 0.25AU perihelion and
   a period of 4.5 years.  Whether that counts as a *slowdown* though...

	   ...but officer, I was only doing 80km a second...
					    ^^
   Then there's Ulysses, whose speed was unchanged, merely(!) redirected by
   90 degrees.

   You can get to Mercury cheaper by slowing via Venus, I think, then there
   was a scare a few (seems a million) years ago about a dastardly Russian
   plan to send a satkiller into *retrograde* Clarke orbit via the Moon.

   There's no end to the fun you can have playing interplanetary snooker; I
   think the fundamental limit for any single encounter (confirmation, please?)
   is the vector sum of spacecraft and accelerating body beforehand.

As I understand it, a gravity assist trajectory is simply a hyperbolic
path around the relevant body. Such a path is characterised as
follows:

- the departure speed is the same as the arrival speed
- the change in direction is larger for a closer approach
- the higher the speed, the smaller the change in direction.

Work out the path in the frame of the planet, then add the planet's
velocity to all the results.

From memory and the back of an envelope, here are some relevant
equations:

e is the eccentricity of the hyperbola (> 1).
d is a parameter of the hyperbola (distance to the directrix, in fact)
l is the `miss distance': by how much would you miss the planet if it
	didn't affect your path
v_0 is the speed at closest approach
v_oo is the speed `at infinity'
G is the gravitational constant: 6.67e-11 m^3/s^2 kg in SI units
M is the mass of the planet (5.976 kg for Earth)

1. closest approach:

	r_0 = ed/(1+e) (this is just from the shape of the hyperbola)

2. angular momentum conservation:

	r_0 v_0 = l v_oo

3. energy conservation:

	v_o ^2 = v_oo ^2 + 2GM/r_0

4. central force equation:

	ed = GM/((l v_oo)^2)

5. change in direction = \pi - 2 arccos(1/e) (in radians)

If you want to work out the path of some dumb body like a comet, you
observe l and v_oo, calculate ed from (4), then substitute into (1),
(2), (3), solve for v_0 and (1+e), and you're done. If you have a
probe that you want to send in some particular direction at some
particular speed, subtract the velocity of the planet (your approach
and depart speeds should now be equal, otherwise the assist is
impossible), get e from (5), substitute (1) and (2) into (3) and (4),
plug in values for e and v_oo, and solve for l (which tells you where
to aim the probe).

Nick Haines nickh@cmu.edu