How Do Planets Orbit Stars in Elliptical Paths?

[lmstaples]

Homework Statement

A planet orbits a star of mass M = (3*10^30)kg in an elliptical orbit.
The planet is r_1 = (0.8*10^11)m from the star at its closest approach (periastron),
and r_2 = (1.6*10^11)m at its furthest (apastron).
(G = (6.67*10^-11) Nm^2kg^-2)

(a) Calculate the semi-major axis a, and hence the period T of the planet's orbit.

(b) Use the polar equation of an ellipse to calculate the orbital eccentricity, e.

(c) Use conservation of energy and the total orbital energy per unit mass of the planet (epsilon = -GM/2a),
to calculate the orbital speeds at the periastron and apastron, v_1 and v_2.

(d) Use your values of v_1 and v_2 to show that the orbital angular momentum per unit mass is the same at periastron and apastron.

Homework Equations

r = (l)/(1+ecos(θ))
a = (l)/(1-e^2)
h=sqrt(GMl)
ε=-(GM)/2a
v=rω
1/2(dr/dt)^2 + (h^2)/(2r^2) - (GM)/r = -(GM)/(2a)

The Attempt at a Solution

So far I have:
(a) a = (r_1 + r_2)/2 = (1.2*10^11)m and T = sqrt((4pi^2(a^3))/(GM)) = (18464102.11)s = (213.7)days

(b) r = (L)/(1 + e*cos(theta)) and a = (L)/(1-e^2) => L = a(1-e^2) => r = (a(1-e^2))/(1+e*cos(theta))
get a quadratic in e and solve to get e = 1/3

(c) not getting anywhere. know that 1/2(dr/dt)^2 + (h^2)/(2r^2) - (GM)/r = -(GM)/(2a)

(d) no idea

 

[Simon Bridge]

(c) hint: total energy is the sum of kinetic and potential energy.
(d) having got the speeds in (c) use the angular momentum formula.

 

[lmstaples]

Haha thanks, I worked it out in the end - was being stupid.
Ended up with v_1 around 58km/s and v_2 around 29km/s. Then for (d) showed that specific angular momentum was the same for both periastron and apastron using r_1v_1 =r_2v_2