2017

The Cepheid relation in the infrared

Round: Data analysis
Points: 36.5
Level: Bronze
Taught in: Section D2

Adapted from IOAA 2017 (Phuket), data-analysis problem 1, part 1. We have restated it in our own words, kept the original data, and kept only the K band, since Section D2 works through the V band. The solution was last checked on 19 October 2026.

Five Cepheids in our Galaxy have parallaxes measured with the Hubble Space Telescope. The table gives each star's period $P$ , its mean apparent magnitude $⟨ K ⟩$ in the infrared K band, the extinction $A_{K}$ , which is the dimming of the star by dust in the K band, and its parallax $ϖ$ with the uncertainty $σ$ of the parallax. The magnitudes have negligible uncertainty.

Star	$P$ (day)	$⟨ K ⟩$ (mag)	$A_{K}$ (mag)	$ϖ$ (mas)	$σ$ (mas)
RT Aur	3.728	3.925	0.02	2.40	0.19
FF Aql	4.471	3.465	0.08	2.81	0.18
X Sgr	7.013	2.557	0.07	3.00	0.18
ζ Gem	10.151	2.097	0.01	2.78	0.18
l Car	35.551	1.071	0.06	2.01	0.20

Find the absolute magnitude $M_{K}$ of each star and its uncertainty, plot a suitable linear graph of the period and luminosity relation in the K band with error bars, and find the slope of the relation. In the original paper this part, together with the same graph for the V band, was worth 36.5 marks.

I solved it without the solution

AnswerThe absolute magnitudes $M_{K}$ are −4.19, −4.37, −5.13, −5.69 and −7.47, each uncertain by the same amount as $M_{V}$ , from 0.13 to 0.22. The slope of the relation in the K band is about −3.4 magnitudes for each unit of $\log (P / day)$ , steeper than the −2.5 of the V band. Section D3 shows how to find its uncertainty from the steepest and shallowest lines.

This problem also appears among the olympiad problems of Section D2, Drawing a graph by hand, which teaches the ideas it uses. Equation and figure numbers refer to the course.