A Cepheid Distance With Patchy Dust
A Cepheid in a nearby spiral galaxy has a period of 30.0 days and an apparent V magnitude of 22.0 ± 0.1. Its color excess is E(B−V) = 0.25 ± 0.05, and you adopt R_V = 3.1. The period–luminosity relation is M_V = −2.43(log10 P − 1) − 4.05. The galaxy's heliocentric redshift is 180 km/s; for comparison, H0 = 70 km/s/Mpc gives a redshift distance of 2.6 Mpc. A team reports a distance of 2.8 Mpc and says it agrees with the redshift distance, so no extinction correction is needed. You are reviewing the distance and its uncertainty. The Cepheid's light curve is well sampled, but the galaxy is inclined and has patchy dust lanes. The PL relation has an intrinsic scatter of about 0.2 mag, which is not included in the photometric error.
1. Compute the extinction-corrected distance to the Cepheid. Show the PL relation evaluation and the distance modulus.
Read the worked answer
First evaluate the PL relation: log10(30.0) = 1.477, so M_V = −2.43(0.477) − 4.05 = −1.16 − 4.05 = −5.21. Extinction: A_V = R_V E(B−V) = 3.1 × 0.25 = 0.775 mag. Corrected apparent magnitude m0 = 22.0 − 0.775 = 21.225. Distance modulus μ = m0 − M_V = 21.225 − (−5.21) = 26.435. Then log10 d = (μ + 5)/5 = 31.435/5 = 6.287, so d = 1.94 Mpc. Without extinction, μ = 27.21 and d = 2.76 Mpc. The extinction correction therefore matters at the 40% level, far larger than the 0.1 mag photometric error.
2. Evaluate the team's 2.8 Mpc claim. What error did they make, and why is agreement with the redshift distance not a validation?
Read the worked answer
The team's 2.8 Mpc is essentially the uncorrected distance; it ignored A_V = 0.775 mag. Because extinction makes the star appear fainter, omitting it makes the star seem more distant. The correction factor is 10^(A_V/5) = 10^0.155 = 1.43, so the true distance is about 1.94 Mpc, not 2.8 Mpc. The apparent agreement with the redshift distance (2.6 Mpc) is therefore misleading: the corrected Cepheid distance is lower, and the difference corresponds to a peculiar velocity of roughly 40–50 km/s, which is plausible for a nearby galaxy. Agreement with redshift is not a validation when the redshift distance itself has peculiar-velocity error and H0 uncertainty.
3. Assess the uncertainty budget. Is the corrected distance consistent with the redshift distance?
Read the worked answer
Uncertainty budget: E(B−V) ±0.05 gives A_V ±0.155 mag, hence distance modulus ±0.155 and fractional distance error (ln10/5)×0.155 ≈ 7%. Photometry ±0.1 mag gives about 4.6%. The PL intrinsic scatter of 0.2 mag gives about 9.2% and is not included in the team's error bar. Combining in quadrature: sqrt(7² + 4.6² + 9.2²) ≈ 12.5%. So d ≈ 1.94 ± 0.24 Mpc. The redshift distance 2.6 Mpc is about 2.7σ higher if only this error is used, but peculiar velocities of a few hundred km/s are common, so the two are not necessarily inconsistent. A larger Cepheid sample and a measured reddening law would be needed to make the comparison decisive.