Weighing a binary from its photometry

Adapted from IOAA 2022 (Kutaisi), theory problem 6. We have restated it in our own words and kept the original data. The official solution prints the temperature of star A as 5978 K, a slip for 5796 K, which its later steps use. It also rounds the difference of magnitudes to 0.48 and the ratio of fluxes to 0.65, which gives masses of 11.4 and 29.3 solar masses, and names the larger mass M_B where it means M_A. The numbers do not describe a real binary, since stars of these masses and temperatures would be far larger and brighter. The solution was last checked on 23 October 2026.

A binary 89 pc away has a circular orbit seen edge on, with a period of 100 days. A space telescope of diameter $D=10$ m, working at a wavelength of 364 nm, cannot resolve the two stars for a total of 38 days in each period. The spectrum of star A peaks at 500 nm and that of star B at 600 nm.

(a) Find the temperatures $T_{\text{A}}$ and $T_{\text{B}}$ of the stars. (2 points)

(b) Find the distance $l$ between the stars. (6 points)

(c) Find the total mass of the two stars. (2 points)

(d) The table gives the combined light of both stars in two configurations, with the stars side by side and with B passing in front of A. The extinction in U towards the system is 1.4 magnitudes per kiloparsec, the bolometric correction is $\text{BC}=m_{\text{bol}}-V$, and the ratio of the mean densities of the stars is ${\rho }_{\text{A}}/{\rho }_{\text{B}}=0.7$. Find the masses of the two stars. (10 points)

Answer(a) $T_{\text{A}}=5796$ K and $T_{\text{B}}=4830$ K. (b) The stars are 1.45 au apart. (c) The total mass is 40.7 solar masses. (d) Star A has 29.1 solar masses and star B 11.6 solar masses.