The times of minima of V1107 Cassiopeiae

Adapted from IOAA 2011 (Katowice), data-analysis problem 1. We have restated it in our own words and kept the original data. The solution was last checked on 20 October 2026.

V1107 Cas is an eclipsing binary of the W Ursae Majoris type, whose two stars touch, so that its light changes all the time and it shows two nearly equal minima in each orbit. The table lists twelve observed times of minimum as heliocentric Julian dates, with their uncertainties.

(a) Find a first value of the period, assuming that it is constant and that the observations on any one night were continuous. (b) Using $M_{\text{calc}}=M_0+PE$, where $E$ is a whole or half number of cycles counted from a chosen first minimum $M_0$, draw an O − C diagram, with $E$ across and $O-C=M_{\text{obs}}-M_{\text{calc}}$ up. (c) From the diagram, improve $M_0$ and $P$ and estimate their uncertainties. (d) Predict the times of the minima between 19 h UT on 1 September 2011 and 2 h UT on 2 September 2011. The whole problem was worth 25 marks.

AnswerThe improved ephemeris is $M_{\text{calc}}=2454092.4115+0.2734127E$, with the period uncertain by 0.0000001 days and $M_0$ by 0.0002 days. The minima in the window fall at about 19 h 11 m and 22 h 28 m UT on 1 September 2011 and 1 h 45 m UT on 2 September. The official solution reads the line by eye and finds a period of 0.2734129 days and times one minute later, within the uncertainty of the prediction.