BOOK: "Computational Physics", 2nd edition, by Nicholas Giordano and Hisao Nakanishi

The problem is similar to Ex. 4.10 in the textbook.

PLEASE COMPLETE C AND D AS MARKED IN THE QUESTION.

PLEASE COMPLETE A and B only if they are required to do C and D if not then not necessary

Thanks

. Mercury '3 Orbit and General Relativity Consider Mercury in its motion around the Sun using Newton's universal law of gravitation, PC; = 4n2MMe/r2, and Newton's 2. law of motion, in astronomical base units (lAU, 1y, MG). Construct a PYTHON code using the Euler-Cromer algo- rithm to describe the planetary motion assuming the Sun to be xed in a focal point of the orbit (hint: use time steps no larger than 0. 001; as usual, attach a printout of your code and appropriate plots to document your results). (a) For an initial position at F0 = (039,0), choose the appropriate (analytically calculated) initial velocity 00 = (0,120) so that the resulting orbit is circular. Conrm this by plotting your numerical results for the orbit over ~10 periods. (b) Introduce the ellipticity e=0.206 into the orbit motion by initializing F0 2 (0.39(1+ e), 0) and 270 = (0, 110) with the appropriate (analytically determined) intial speed. Overlay the curve for the analytic orbit expression. Oc) Implement the leading-order correction term derived from general relativity (GR) into Newton's gravitational force, 17r2MMe oz am = (1+ ,3). (1) T2 Using a = 0.001, record the precession angle 6,, of Mercury's aphelion over sev- eral orbital periods, plot the results vs. time and extract the rate of precession, 9,, (linear eyeball t is ne). 0d) Repeat part (c) for two additional values of a (e.g., 0.002 and 0.0001), plot 9,,(a), and determine the slope de/da (eyeball t is ne). Use this to evaluate the precession rate (in arcsec/ 100g) for the GR prediction of oz 2 1.1 x 10'8AU2. How does your result compare to experimental observations