Turnoff Age of a Metal-Poor Globular Cluster
A globular cluster has [Fe/H] = -2.0, an apparent turnoff magnitude of V = 19.2, and a horizontal-branch magnitude of V = 15.8. The reddening is E(B-V) = 0.04 and the distance modulus from the HB is (m-M)_V = 15.6. Independent isochrone fitting gives an age of 12.5 Gyr with a turnoff absolute magnitude M_V = 4.1 and a turnoff color (B-V)_0 = 0.42. A second group uses the same photometry but assumes E(B-V) = 0.10 and derives an age of 14.5 Gyr. The cluster's RR Lyrae stars have a mean V = 15.9 and a period-metallicity relation consistent with M_V = 0.6 for the HB.
1. Check the internal consistency of the turnoff and horizontal-branch distance moduli, and identify whether the two age estimates can be reconciled by reddening alone.
Read the worked answer
From the turnoff: (m-M)_V = V_TO - M_V = 19.2 - 4.1 = 15.1, which is 0.5 mag smaller than the HB-based 15.6. From the RR Lyrae: M_V = 0.6 gives (m-M)_V = 15.9 - 0.6 = 15.3, again smaller than 15.6 but closer to the turnoff value. The discrepancy of 0.3-0.5 mag suggests either the turnoff M_V or the HB M_V calibration is off, or differential reddening is present. Reddening affects age mainly through the turnoff color: increasing E(B-V) from 0.04 to 0.10 reddens the turnoff by 0.06 mag in B-V, making the turnoff appear cooler and hence older. A 0.06 mag color shift at the turnoff corresponds to roughly 1-2 Gyr in age for a metal-poor isochrone, so the 2 Gyr difference between the two groups is broadly consistent with the reddening change. However, the distance-modulus inconsistency is independent of reddening in V (reddening in V is only ~0.12 mag for E(B-V)=0.04), so reddening alone does not fix the distance mismatch. The two ages can be reconciled by reddening, but the distance scale remains a separate problem.
2. Explain how the metallicity affects the age derived from the turnoff, and state whether the 12.5 Gyr value should be trusted over the 14.5 Gyr value.
Read the worked answer
At fixed turnoff color and magnitude, lower metallicity shifts the turnoff to bluer colors and brighter magnitudes at a given age, so an isochrone of the correct [Fe/H] is essential. For [Fe/H] = -2.0, the turnoff is bluer and the main-sequence lifetime at a given mass is slightly different; using a solar-metallicity isochrone would systematically misestimate the age by several Gyr. The 12.5 Gyr value is derived from an isochrone fit with the stated metallicity and a specific M_V, so it is internally consistent if the reddening is 0.04. The 14.5 Gyr value assumes a larger reddening, which is not supported by the stated E(B-V) = 0.04 unless independent evidence (e.g., Schlegel maps or spectroscopy) supports it. Given the stated inputs, the 12.5 Gyr value should be preferred, but the distance-modulus mismatch means the absolute age carries at least ~1 Gyr systematic uncertainty. The honest answer is that the age is 12-13 Gyr with a systematic floor set by the distance and reddening calibration.