2007

A binary exchanging mass

Round: Theory
Points: 10
Level: Silver
Taught in: Section 8.5

Adapted from IOAA 2007 (Chiang Mai), theory problem 3. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

Two stars of masses $M_{1}$ and $M_{2}$ , a distance $D$ apart, move on circular orbits round their centre of mass with angular speed $ω$ . Mass flows steadily from one star to the other, which slowly changes the period and the separation. Treat the stars as points and ignore their spins. For small $x$ and $y$ , $(1 + x)^{n} \approx 1 + n x$ and $(1 + x) (1 + y) \approx 1 + x + y$ .

(a) Find the total angular momentum and the kinetic energy of the system. [2 points]

(b) Find the relation between $ω$ and $D$ . [2 points]

(c) In a time $Δ t$ , star 1 gains a mass $Δ M_{1}$ . Find $Δ ω$ in terms of $ω$ , $M_{1}$ , $M_{2}$ and $Δ M_{1}$ . [3 points]

(d) In one binary $M_{1} = 2.9$ solar masses, $M_{2} = 1.4$ solar masses and the period is 2.49 days. After 100 years the period has grown by 20 s. Find $Δ M_{1} / (M_{1} Δ t)$ , per year. [1.5 points]

(e) Which way is mass flowing? [0.5 points]

(f) Find $Δ D / (D Δ t)$ , per year. [1 point]

I solved it without the solution

Answer(a) $L = M_{1} M_{2} D^{2} ω / (M_{1} + M_{2})$ and the kinetic energy is $M_{1} M_{2} D^{2} ω^{2} / (2 (M_{1} + M_{2}))$ . (b) $ω^{2} = G (M_{1} + M_{2}) / D^{3}$ . (c) $Δ ω = 3 ω Δ M_{1} (M_{2} - M_{1}) / (M_{1} M_{2})$ . (d) $2.9 \times 10^{- 7}$ per year. (e) From $M_{2}$ to $M_{1}$ , the lighter star to the heavier. (f) $6.2 \times 10^{- 7}$ per year.

This problem also appears among the olympiad problems of Section 8.5, Times of minima and changing periods, which teaches the ideas it uses. Equation and figure numbers refer to the course.