Orbital Mechanics Problems -

The purpose of this unit is to introduce some more sophisticated techniques for solving ODEs, to do this we will look at some orbital mechanics problems and a simple problem of particle drifts in electromagnetic fields.

PART 1: Exercises using low order techniques

1. Prove that for the Kepler problem the Euler-Cromer method conserves angular momentum. [Pencil]

2. Modify the orbit program so that instead of running a fixed number of time steps, the program stops when the satellite completes one orbit.

(a) Have the program compute the period, eccentricity, semi-major axis, and perihelion distance of the orbit. Use the Euler-Cromer method and test the program with circular and slightly elliptical orbits. Compare the measured eccentricity with

ε = √(1 + (2EL²/G²M²m³))

(b) Show your program confirms Kepler's third Law. (i.e., T² = (4π²/GN) a³)

(c) Confirm that (K) = -½(V), where (K) and (V) are the time-average kinetic and potential energy (virial theorem).

3. When a charged particle of mass m and charge q moves through a magnetic and electric field it feels the Lorentz force given by

F^→ = 1q(E^→ + v^→ × B^→)

Where v ? is the velocity of the particle. If the magnetic field is constant, then it moves in a circle of radius

R_gyro = mv/qB

Where R_gyro is the gyroradius and has a period of

T = 2πm/qB

Which is known as the gyroperiod. If the particle is in a static electric field E, then it can be shown that the center of gyration moves with a velocity

v^→_ExB = E^→ × B^→/B²

Which is commonly referred to as E x B drift. If the magnetic field is not constant, but has a small gradient, then it can be shown that the center of gyration also moves with a velocity given by

v^→_∇B = (mv²/2qB)(B^→ x ∇B/B²)

Which is known as ∇B drift. (A derivation of this equation can be found in any Plasma Physics Textbook.) By small, I mean that the gradient scale for the variation of the magnetic field is small compared to the particle's gyroradius.

Write a program using the Euler-Cromer method that solves for the trajectory of a particle in the following electric and magnetic fields. Assume that we have normalized quantities so that m=1, q=1. The initial condition for the particles is r^→ = 0, v^→ = y^{^}.

(a) E^→ = 0, V^→ = z^{^}. (Zero electric field and unit magnetic field) Verify that for a reasonably timestep, that you get the correct gyroradius and period. Use this timestep for problems (b) and (c).

(b) E^→ = x^{^} and B^→ = z^{^}. (Unit electric and magnetic field, orthogonal to each other). Verify that you get the correct E × B drift by plotting both the trajectory and the trajectory in the frame moving with the E × B velocity. In the moving frame the particle should just move in a circle.

(c) E^→ = 0, B^→ = (1 + ax)z^{^}, where a = 0.1. (Zero electric field and magnetic field in the z- direction that varies with x). Verify that you get approximately the correct gradient B drift, by plotting both the trajectory and the trajectory in the frame moving with the ∇B velocity. As in (b) in the moving frame the particle should just move approximately in a circle.

Part 2: Exercise using Runge-Kutta Methods

Suppose that our comet is subjected to a constant force in one direction (e.g., gravitational attraction of a large but distant object). Modify the orbit program to simulate this system. Set the strength of the perturbing force to be 1% of the initial gravitational force. Using the fourth-order Runge-Kutta method, show that an initially circular orbit is transformed into an elliptical orbit with the semi-major axis perpendicular to the perturbing force. Produce a graph of the angular momentum as a function of time.

Part3: Solar System Simulation using adaptive Runge-Kutta

A. Using the orbit program as a starting point, create a model of the Sun-Earth-Jupiter system. Use this model to investigate the effect of Jupiter on the orbit of the Earth. How much more massive must Jupiter be before the orbit of the Earth becomes unstable? Make a few example plots for Jupiter with 1, 10 100 and 1000 M_J.

B. Now allow the Sun to move and add Saturn to the simulation (so you now have a 4-body) simulation and see what effect Jupiter has on its orbit.

Attachment:- Assignment Files.rar