The pulsations of β Doradus

Adapted from IOAA 2016 (Bhubaneswar), theory problem 6. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

The Cepheid β Doradus pulsates with a period of 9.84 days. Assume that it is brightest when most contracted, with radius $R_1$, and faintest when most expanded, with radius $R_2$, that it stays spherical and that it radiates as a blackbody. Its bolometric magnitude varies from 3.46 to 4.08. Doppler measurements show that the surface moves in or out at an average speed of 12.8 km/s. The peak of its spectrum moves between 531.0 nm and 649.1 nm. Take the Sun's luminosity as $3.826\times {10}^{26}$ W.

(a) Find the ratio $R_1/R_2$. [7 points]

(b) Find $R_1$ and $R_2$ in metres. [3 points]

(c) Find the bolometric flux $F_2$ from the star when it is most expanded. [5 points]

(d) Find the distance of the star, in parsecs. [5 points]

Answer(a) $R_1/R_2=0.890$. (b) $R_1=4.42\times {10}^{10}$ m and $R_2=4.96\times {10}^{10}$ m. (c) $F_2=5.88\times {10}^{-10}$ W/m². (d) About 315 pc. The official solution uses $-26.72$ for the Sun, close to its visual magnitude, in place of its bolometric magnitude, which gives $6.51\times {10}^{-10}$ W/m² and 298 pc.