MATH 4981L Spacecraft Trajectory Design
26 Feb, 2013
Homework 1 – due next lecture
1. Assume a system of 3 particles (or bodies that can be represented as particles for
gravitational purposes)
(a) Write out the expression for F1, i.e., the gravity forces on m1.
(b) Write out the equations of motion for the system, that is, write the differential
equations that govern the position vectors r1, r2, r3.
(c) Let r1 = x1 i + x2 j + x3 k, where i j k are unit vectors in an inertial frame. Write out
the results of part (b) in first-order state equation form, e.g. xɺ1 = x2 ; xɺ2 = x1 − x2 .
(d) [Optional] Determine an expression for the gradational potential function U for
this system. Show that F1 = ∇1U and the results is the same as that in (a).
2. For each system below, assume they all lie on the same plane and on a straight line.
Calculate the distance between the center of the first body and the center of mass of the
system. Express the answer in radius of the first body (e.g. solar radius, Earth’s radius,
etc.)
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Sun-Earth
Sun-Earth-Jupiter
Sun-Jupiter
Sun-All 8 planets
Earth-Moon
Saturn-Titan
Jupiter-Io-Europa-Ganymede-Callisto
Pluto-Charon-Nix-Hydra