2014

The mass function from a Doppler shift

Round: Theory
Points: Not given
Level: Bronze
Taught in: Section 8.3

Adapted from IOAA 2014 (Suceava), theory problem P4. We have restated it in our own words and kept the original data. The original calls the system a visual binary, though it is found from its spectrum, and the book gives no official solution, so the solution here is our own. The solution was last checked on 21 October 2026.

A star in a binary is observed for several months. Its spectral lines shift back and forth in a sine curve with a period of $T = 7$ days, and the largest fractional shift is $z = Δ λ / λ = 0.001$ . The orbits are circular, and the companion has mass $M_{2}$ .

(a) Show that the mass function, $f = M_{2}^{3} \sin^{3} i / (M_{1} + M_{2})^{2}$ , equals $T (v_{1} \sin i)^{3} / (2 π G)$ , where $v_{1} \sin i$ is the largest speed of the star along the line of sight.

(b) Find the mass function, using $c = 3 \times 10^{8}$ m/s and $G = 6.67 \times 10^{- 11}$ N m² kg⁻².

I solved it without the solution

Answer(a) As shown. (b) The mass function is $3.9 \times 10^{31}$ kg, about 20 solar masses, so the companion has at least 20 solar masses.

This problem also appears among the olympiad problems of Section 8.3, Spectroscopic binaries and the mass function, which teaches the ideas it uses. Equation and figure numbers refer to the course.