2017 D1 | Calibrating the distance ladder to the LMC

An accurate trigonometric parallax calibration for Galactic Cepheids has long been sought but is hard to achieve. All known classical Cepheids are more than 250 pc away, so a 10% distance needs parallax uncertainties of 0.2 milliarcsec or better, which requires space-based observations. The Fine Guidance Sensor on HST measured parallaxes better than 10% for 10 Cepheids, with periods from 3.7 to 35.6 days and distances of about 300 to 560 pc.

Table 1 gives periods $P$, mean magnitudes in V, K and I, extinctions $A_V$ and $A_K$, and parallaxes with their uncertainties. The apparent magnitudes have negligible uncertainty.

(D1.1) The period–luminosity (PL) relation is $L\propto P^\beta$; in practice it is written in terms of absolute magnitude (the "PL relation"). Use Table 1 to plot a suitable linear graph to derive the PL relation in V and in K, on separate graphs. Determine the slope of the best-fit line. (You may find $\Delta(\log_{10}x)\approx\Delta x/(x\ln10)$ useful.) (36.5)

The reddening is a function of wavelength and varies between Cepheids, so the PL relation is really a period–luminosity–colour relation. The reddening-free "Wesenheit" magnitude removes it using the star's own colour:

(D1.2) Plot and derive the reddening-free PL relation using $W_{VI}$. Estimate its slope and the slope's uncertainty. (14.5)

(D1.3) Use the PL relations from (D1.1) and (D1.2) with the LMC Cepheids in Table 2 (with extinction-corrected $\langle V_{\rm corr}\rangle$ and $\langle W_{VI}\rangle$). Estimate the distance modulus of each star, then the distance to the LMC (in pc) and its standard deviation for each band. Are the two distances statistically different (YES/NO)? Are the standard deviations different (YES/NO)? Which band is more accurate for the LMC distance? (24)