University of Illinois at Urbana-Champaign
Department of Aerospace Engineering
AE 402
Homework No. 3
Prof. PrussingDue 21 sep 2007
11. (Based on a problem by W.E.Wiesel) A spacecraft of constant mass mis in orbit
about the sun. Aperfectly-reflecting flat solar sail is deployed and controlled so that
its normal is always aligned with the radius vector from the sun. The acceleration
due to solar radiation pressure is then
a=2SA
cm
r
r3
where Sis the solar intensity at 1 au, Ais the sail area, and cis the speed of light in
avacuum.
a) Showthat the equation of motion of the spacecraft is
¨
r+(
µ
−2SA
cm )r
r3=00
b) What types of (conic) orbits are possible if
i) 2SA
cm <
µ
ii) 2SA
cm =
µ
iii) 2SA
cm >
µ
c) Showthat in one of the cases (i)−(iii) the spacecraft always escapes, and that in
the other twocases escape is possible, depending on initial conditions.
d) Showthat for all escape orbits the hyperbolic excess speed v∞is givenby
v
2
∞=v
2
o−2
r
o(
µ
−2SA
cm )
where roand voare the radius and speed when the sail is deployed.
e) Determine the magnitude of the solar gravitational acceleration on a spacecraft
located a distance of 1 au from the sun. Express your answer in mm/s2.
f) If the acceleration due to solar radiation at 1 au is 2 mm/s2,which of the above
cases (i)−(iii) applies?
12. Problem 1.12
13. Problem 1.14
14. Problem 1.17.
a) Solvethe problem in the book by assuming msat<<m..
b) Showthat if values for asatandTsatare known, the value ofmcan be deter-
mined.
c) If asat and Tsat are measured to be 6878 km and 5677 s, respectively,compute the
mass of the earth in kg.Compare with the value in Appendix 1. (The value of Gis
givenonp.6ofthe text.)
15. Problem 1.15*
September 2007