Distance to a Cepheid Variable

In the early 1900s, Henrietta Leavitt made several observations of variable stars from the Harvard College Observatory that led her to propose that the period of variation was related to the intrinsic brightness of the star. In particular, for a class of variable stars known as Cepheids, there was a strong power law relating luminosity to period, and hence a straight line on a log-log graph. Since magnitudes are ∝ log L, this can be described by the empirical relation

⟨M⟩ = -2.43(log P - 1) - 4.05

where P is the period measured in days and ⟨M⟩ is the mean absolute magnitude of the star, defined as the apparent magnitude measured from a distance of 10 pc. The apparent and absolute magnitudes are related as

m - M = 5 log d - 5

where d is the distance measured in pc.

Figure 1: Left: Colour composite view of the circumstellar nebula around RS Puppis assembled from HST images. Right: The light curve of RS Puppis. Phase 0 corresponds to when the star is at its brightest, and the data has been folded over a known period of 41.5 days. Credit: AAVSO / P. Kervella et al. (2017).

In the past, distances to stars could only be determined using something called the parallax method, where the angular shift of a star relative to the background stars was measured at two points in the Earth's orbit six months apart. The parallax angle is the same as the angle subtended by 1 au at the distance of the star. In Henrietta's time, parallax measurements only reached about 100 pc, however her method with Cepheid variables extended this range to more than 10⁶ pc, enabling much better determination of distances across the Milky Way and into nearby galaxies.

The variable star RS Puppis (see Figure 1) is one of the biggest and brightest known Cepheids in the Milky Way galaxy and has one of the longest periods for this class of star at 41.5 days.

(a) Measure the mean apparent magnitude from Figure 1, and find the distance to the star, given that absorption of the light by interstellar dust means the star appears 1.42 magnitudes fainter.

(b) In 2018 the Gaia satellite measured a parallax angle of 0.5844 milliarcseconds (where there are 3600 arcseconds in a degree). What does this suggest is the distance to the star?