Irodov Problem 1.156 – Angular Momentum of Planetary Motion

Problem Statement

A planet of mass m moves in an elliptical orbit with semi-major axis a and semi-minor axis b around the Sun of mass M. Find the angular momentum.

Given Information

• Semi-major axis a, semi-minor axis b
• Planet mass m, Sun mass M

Physical Concepts & Formulas

$$L=m\sqrt{GMa(1-e^2)}=m\sqrt{GM}\cdot\frac{b}{\sqrt{a}}\cdot\sqrt{a}$$

Step-by-Step Solution

Step 1: For ellipse: b² = a²(1−e²) where e is eccentricity.

Step 2: Vis-viva: v² = GM(2/r − 1/a). At periapsis: L = mv_p·r_p.

Step 3: General result: L = mb√(GM/a).

Worked Calculation

L = mb√(GM/a)

Answer

$$\boxed{L=mb\sqrt{\frac{GM}{a}}}$$

Physical Interpretation

Angular momentum depends on both orbit size (a) and shape (b). A circular orbit (b=a) gives maximum L for given a. Radial orbits (b=0) have L=0.