Elliptical orbit — perihelion, aphelion and variation of speed

Problem

Elliptical orbit. Look at the graph of a planet’s elliptical orbit around the Sun, placed at one of the foci. Identify perihelion and aphelion, and describe the variation of speed along the orbit.

The Sun occupies one focus of the ellipse; perihelion and aphelion are the two ends of the major axis.

Solution

Identification. The Sun is at one focus, offset from the centre. The point of the orbit closest to the Sun is the perihelion (on the right, where the Sun is close to the edge); the farthest point is the aphelion (on the left).

Variation of speed. Kepler’s second law (the law of areas) states that the Sun-planet position vector sweeps out equal areas in equal times. Near the Sun (perihelion) the radius is short, so to cover the same area the planet must move fast; far away (aphelion) the radius is long and the planet moves slowly. $\text{Perihelion: } r\ \text{minimum},\ v\ \text{maximum} \qquad \text{Aphelion: } r\ \text{maximum},\ v\ \text{minimum}$

Why. It is conservation of angular momentum $L = m\,v\,r_\perp = \text{const}$: if $r$ decreases, $v$ must increase (and vice versa). Equivalently, from energy conservation one obtains the vis-viva formula $v^2 = GM\!\left(\frac{2}{r} - \frac{1}{a}\right)$ which gives $v$ as a function of distance $r$: maximum at perihelion (small $r$), minimum at aphelion (large $r$). Energetically, at perihelion the (negative) potential energy is minimum and the kinetic energy is maximum; at aphelion it is the opposite.

$\ev{\text{perihelion}: r_{\min},\,v_{\max};\quad \text{aphelion}: r_{\max},\,v_{\min};\quad L=mvr_\perp \text{ constant}}$

Links

Topics: Gravitation Concepts: Kepler’s laws · Conservation of angular momentum · Orbits Skills: Graph reading