Weighing α Centauri A and B

Adapted from IOAA 2008 (Bandung), data-analysis problem 2. We have restated it in our own words, kept the original data, and given the table in this course's sign of the bolometric correction. The first part, plotting the table by hand, is left out, so the points add up to 80 of the original 100. The solution was last checked on 21 October 2026.

α Centauri A and B form a visual binary on the main sequence. The semi-major axis of their relative orbit is 17.59″ and the period 79.24 years. Their visual magnitudes are $-0.01$ and 1.34 and their colour indices $B-V$ are 0.65 and 0.85. For main-sequence stars, the bolometric correction $BC=M_{\text{bol}}-M_V$ is

and the mass is related to the absolute bolometric magnitude by $M_{\text{bol}}=-10.2\log (M/M_{\odot })+4.9$.

(a) Find the apparent bolometric magnitudes of the two stars. [40 points]

(b) Find the mass of each star, without using a measured parallax. [40 points]

Answer(a) $m_{\text{bol}}=-0.105$ for A and $1.105$ for B. (b) The parallax settles at 0.754″, and the masses at 1.15 solar masses for A and 0.88 for B. The official solution reads the bolometric corrections from a hand-drawn curve as 0.10 and 0.25 and reaches 1.15 and 0.88 too. Modern measurements give a parallax of about 0.75″ and masses of 1.08 and 0.91, so the simple mass relation does well.