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student
2004-Jan-19, 05:22 AM
Hi,

I have a question on a S/C flyby problem and hoping someone could
help me on it. Here's the problem:
An interplanetary S/C is approaching Venus but is still outside the
planet's sphere of influence. Its solar orbit is coplanar to that of
Venus. Relative to Venus, the S/C speed is 5 km/s. The flight path
angle of the S/C solar orbit as it approaches Venus is -10 deg.

What is the speed of the S/C relative to the Sun before interaction
with the planet?

The S/C flies by Venus in such a way as to maximize the energy gain
due to its interaction with Venus. What is the impact parameter
corresponding to this maximum-energy-gain approach? What is the speed
of the S/C, relative to the Sun, after leaving the sphere of influence
of Venus? During the swingby, through what angle does the S/C turn,
in both Venus's and the Sun's frames of reference?

Gravitational parameters for the Sun and Venus are
muS=1.327E11 km^3/s^2
muV=324859 km^3/s^2
RV=.723327*AU=1.0821E8 km radius of Venus from Sun

Below is my solution:

V=Sqrt[muS/RV]=35.0189 speed of Venus relative to Sun
Vinf=5
vi=V+Vinf=40.0189 speed of the S/C relative to the Sun before
interaction with the planet
b=10*Pi/180 Angle between the positive directions of V and vi
a=-muV/Vinf^2=-12994.4 semi major axis of passage

d=ArcTan[vi*Sin[b]/(vi*Cos[b]-V)]=1.00717=57.7064 deg
d is the angle S/C turn in Venus's frames of reference

e=1/Sin[d/2]=2.07225 eccentricity of passage
B=-a*e*Sin[Pi/2-d/2]=23584.7 impact parameter corresponding to
this maximum-energy-gain approach

vf=Sqrt[vi^2+2*V*(V*(1-Cos[d])+vi*(Cos[b-d]-Cos[b]))]=43.2397
vf is the speed of the S/C, relative to the Sun, after leaving the
sphere of influence of Venus.

I was wondering if my solution to this problem is correct. This
is a problem on my PhD qualifier and I will not be getting a solution
back for it, so any help is greatly appreciated.

Thank you!