Planet between two equal stars — unstable equilibrium

Problem

Challenge: planet between two equal stars. A planet sits exactly halfway between two equal stars of mass $M$. Show in what sense the equilibrium is unstable and state the type of drift that results.

At the midpoint the two attractions balance, but a small perturbation unbalances them.

Solution

Equilibrium at the midpoint. Let $2d$ be the distance between the stars. At the midpoint the planet is a distance $d$ from each; the two forces are equal and opposite: $F_1 = F_2 = \frac{GMm}{d^2} \quad\Longrightarrow\quad F_\text{net} = 0$ This is a point of equilibrium.

Why it is unstable. Shift the planet by $\delta x$ towards star 2 (to the right). It is now a distance $d - \delta x$ from star 2 and $d + \delta x$ from star 1: $F_2 = \frac{GMm}{(d-\delta x)^2}\ \text{increases}, \qquad F_1 = \frac{GMm}{(d+\delta x)^2}\ \text{decreases}$ The force towards the nearer star grows (the $1/r^2$ factor blows up as $r$ decreases), while the force towards the farther star falls. The resultant is no longer zero but points towards star 2, i.e. in the direction of the displacement. The net force pushes the planet further away from the equilibrium point: this is the signature of an unstable equilibrium.

Quantitatively, the resultant for small $\delta x$ is $F_\text{net} \approx \frac{4GMm}{d^3}\,\delta x$ proportional to, and in the same direction as, the displacement: an “anti-restoring” force that amplifies the perturbation (the opposite of a spring).

Resulting drift. The planet gets captured by the nearer star: it falls towards it with increasing acceleration. The equilibrium exists only in theory, for one perfect position. This is entirely analogous to the collinear Lagrange points $L_1, L_2, L_3$, which are unstable along the line joining the two bodies.

$\ev{\text{unstable equilibrium: } \delta x \Rightarrow F_\text{net}\propto +\delta x \Rightarrow \text{capture by the nearer star}}$

Links

Topics: Gravitation Concepts: Law of universal gravitation Skills: Analysis of limiting cases and thought-experiment physics