2017

A Cepheid relation free of dust

Round: Data analysis
Points: 14.5
Level: Bronze
Taught in: Section D3

Adapted from IOAA 2017 (Phuket), data-analysis problem 1, part 2. We have restated it in our own words and kept the original data. The solution was last checked on 20 October 2026.

Dust between us and a star makes it fainter and redder. Astronomers remove the dimming without measuring it by using the Wesenheit magnitude, which for the V and I bands is $W = V - R (V - I)$ , with $R = 2.45$ for the dust in our Galaxy. The table gives the period $P$ , the mean magnitudes $⟨ V ⟩$ and $⟨ I ⟩$ , and the parallax $ϖ$ with its uncertainty $σ$ of the five Cepheids of Section D2. The magnitudes have negligible uncertainty.

Star	$P$ (day)	$⟨ V ⟩$ (mag)	$⟨ I ⟩$ (mag)	$ϖ$ (mas)	$σ$ (mas)
RT Aur	3.728	5.464	4.778	2.40	0.19
FF Aql	4.471	5.372	4.510	2.81	0.18
X Sgr	7.013	4.556	3.661	3.00	0.18
ζ Gem	10.151	3.911	3.085	2.78	0.18
l Car	35.551	3.732	2.557	2.01	0.20

Work out the absolute Wesenheit magnitude $M_{W}$ of each star, plot the relation between $M_{W}$ and the period with error bars, and estimate its slope and the uncertainty of the slope. In the original paper this part was worth 14.5 marks.

I solved it without the solution

AnswerThe slope of the Wesenheit relation is −3.4 ± 0.3 magnitudes for each unit of $\log (P / day)$ . The official solution gives −3.4 ± 0.2 and accepted uncertainties from 0.1 to 0.3, so this answer falls within the range the official solution accepted.

This problem also appears among the olympiad problems of Section D3, Slope, intercept and their uncertainties, which teaches the ideas it uses. Equation and figure numbers refer to the course.