An Introduction to Modern Astrophysics · Problem 2.3

The Continuous Spectrum of Light — Carroll & Ostlie

Physics — Astrophysics · orbital-velocity-components

Problem

(a) Beginning with Eq. (2.3) and Kepler's second law, derive general expressions for

\mathbf{v}_r

and

\mathbf{v}_\theta

for a mass

m_1

in an elliptical orbit about a second mass

m_2

. Your final answers should be functions of

P

e

a

, and

\theta

only. (b) Using the expressions for

\mathbf{v}_r

and

\mathbf{v}_\theta

that you derived in part (a), verify Eq. (2.36) directly from

v^2 = v_r^2 + v_\theta^2