Download ASTR450 Homework #9: Perturbations in Astronomy - Prof. Douglas P. Hamilton and more Assignments Astronomy in PDF only on Docsity! ASTR450 Homework # 9 – Perturbations Due Thursday, November30 1. Impulse Approximation to a Drag Force. Consider a satellite on an inclined elliptical orbit acted on by a drag force of the form F = −kvrel, where vrel is the velocity of the satellite relative to the atmosphere, and k is a positive constant. Recall that Earth’s atmosphere decays exponentially with height (scale height ≈ 10km). a) Approximate the range of eccentricities for which the drag force can be approximated by an impulse at pericenter. b) Consider a rotating Earth. Start by making a qualitative estimate of the error made in neglecting rotation. How does the Earth’s rotation affect an equatorial orbit (i = Ω = 0)? Describe how the orbital elements a, e, i,Ω,$ vary in time. c) Now imagine orbits with i 6= 0,Ω 6= 0. Using the perturbation equations and other physical arguments, describe qualitatively how these orbits will evolve (i.e. how a, e, i,Ω,$ vary in time). 2. Radial Perturbation Forces. Consider a radial perturbation force of the form F = Rr̂, where R is a function of the distance r. a) Apply the perturbation equations to this force and obtain simplified expressions for da/dt, de/dt, di/dt, dΩ/dt, and $/dt. b) A radial perturbation to gravity, which is itself a radial force, is an example of a central force. So angular momentum must be conserved. Show that your equations conserve the angular momentum vector and describe the constraints that this imposes on these orbits. c) Now let R = Arn where A is a constant. Take the time average of your expressions over a single unperturbed Keplerian orbit (this step assumes that the perturbation is small). Show that < rn sin ν >= 0 and argue, on physical ground, that < cos ν > is negative (or zero) and that < r−2 cos ν >= 0. It can be shown that < rn cos ν > is negative for n > −2 and positive for n < −2. Use this fact to determine how the sign of your time-averaged d$/dt depends on A and n. Use the Central Force Integrator to check your results numerically. d) Finally, consider the General Relativistic (GR) Perturbation R = Ar−4 where A is a small negative constant. The integral < r−4 cos ν >= a−4e(1 − e2)−5/2. Solve, analytically, for the value of A that will give 30 degrees of precession per orbit for e = 0.5 (The true effects of GR on Mercury’s orbit are almost exactly a million times weaker). Convert your prediction into the proper initial conditions for the Central Force Integrator, and test it! Start your orbit at pericenter and turn in a copy of your plot.
