Using polar coordinates in the orbital plane, the position of an object orbiting the sun is ( ( ), ( )). From Newton’s laws, to find the position it is necessary to solve...

Using polar coordinates in the orbital plane, the position of an object orbiting the sun is ((),
()). From Newton’s laws, to find the position it is necessary to solve

where
 =
 (0)(0)²
is constant, and
 is the gravitational parameter of the sun. Note that
 = 1.327 × 10¹¹
km³/sec².

(a) Write the above differential equation as a first-order system using the radial position
 and radial velocity
 =  .

(b) The approximate values for Mars are:
(0) = 2.244 × 10⁸
km,
 (0) = 0, and
 (0) = 9.513 × 10⁻⁸
radians/sec. If one instead uses astronomical units (au), then
(0) = 1.5 au and
 (0) = 0. Also, measuring time in terms of a terrestrial year (where 1 ty = 365 days), then θ (0) = 3 radians/ty. Explain why it is better to use the au, ty values rather than the km, sec values when computing the solution.

(c) Using the initial conditions from part (b), find
(), to at least three significant digits, after one terrestrial year. Make sure to state what method you used to solve the problem, why you picked that method, and how you know that the solution is correct to three significant digits.

(d) The angular coordinate of the object is determined using the following formula:

Assuming that
(0) = 0, use your results from part (c) to find the angular coordinate for the object after one terrestrial year.