Are pre-main-sequence stars older than we thought?

We fit the colour-magnitude diagrams of stars between the zero-age main-sequence and terminal-age main sequence in young clusters and associations. The ages we derive are a factor 1.5 to 2 longer than the commonly used ages for these regions, which are derived from the positions of pre-main-sequence stars in colour-magnitude diagrams. From an examination of the uncertainties in the main-sequence and pre-main-sequence models, we conclude that the longer age scale is probably the correct one, which implies we must revise upwards the commonly used ages for young clusters and associations. Such a revision would explain the discrepancy between the observational lifetimes of proto-planetary discs and theoretical calculations of the time to form planets. It would also explain the absence of clusters with ages between 5 and 30Myr. We use the $\tau^2$ statistic to fit the main-sequence data, but find that we must make significant modifications if we are to fit sequences which have vertical segments in the colour-magnitude diagram. We present this modification along with improvements to methods of calculating the goodness-of-fit statistic and parameter uncertainties. Software implementing the methods described in this paper is available from http://www.astro.ex.ac.uk/people/timn/tau-squared/

💡 Research Summary

The paper addresses a long‑standing discrepancy in the ages assigned to young stellar clusters and associations. Traditionally, ages have been derived by placing pre‑main‑sequence (PMS) stars on colour‑magnitude diagrams (CMDs) and comparing their positions with theoretical PMS contraction tracks. However, different evolutionary models and even different colour combinations can yield age estimates that differ by a factor of two, making absolute age determinations highly uncertain.

To overcome this problem, the authors turn to the more massive stars in the same regions that have already reached the main sequence (MS). As massive stars evolve from the zero‑age main sequence (ZAMS) toward the terminal‑age main sequence (TAMS), they move upward and slightly redward in the CMD due to increasing helium content in their cores. This movement is relatively rapid compared with the subtle changes of lower‑mass stars still on the MS, providing a potentially precise age indicator.

The key methodological advance is the use of the τ² statistic, an extension of the classic χ² approach that can handle data points with uncertainties in two dimensions (colour and magnitude) and models that are not simple lines but two‑dimensional probability distributions (e.g., because of binarity). The authors show that the original χ²‑like normalisation used in earlier τ² work fails for CMD regions where the isochrone becomes vertical or double‑valued, which is exactly the case for the upper MS of young clusters. They therefore adopt a simpler normalisation in which the model probability density ρ is scaled so that its integral over the entire image equals one. This avoids infinities and allows a robust calculation of the overlap integral between each observed star’s error ellipse and the model density.

To construct ρ, the authors simulate one million stars drawn from a Salpeter initial mass function (dN/dM ∝ M⁻²·³⁵). For each simulated star they assign a binary companion with a mass drawn from a uniform distribution between zero and the primary’s mass, compute the stellar interior properties using the Geneva models (basic set “c”), and convert luminosities and effective temperatures to UBV colours using the Bessell et al. (1998) tables. Where the gravity falls outside the tabulated range, a modest extrapolation is applied. The simulated stars are binned onto a fine grid (0.0025 mag per side) to produce a smooth density map.

Observational data are taken from classic Johnson UBV photo‑electric measurements, primarily from the 1950s–60s, to ensure homogeneity. The authors restrict the sample to stars brighter than (B–V)₀ ≈ 0.0 to minimise contamination by PMS objects. Photometric uncertainties are treated as uncorrelated Gaussian errors in colour and magnitude, a pragmatic choice justified by the dominance of systematic transparency variations over photon statistics in these historic data.

The τ² value for a given set of model parameters (age, distance modulus, reddening) is computed as τ² = –2 ∑ ln Zᵢ, where Zᵢ is the integral of the product of the star’s error ellipse and the model density over the entire CMD. The authors explore the parameter space by evaluating τ² on a grid, locating the minimum, and then estimating uncertainties via Monte‑Carlo simulations and bootstrap resampling.

Applying this framework to several well‑studied young clusters (e.g., NGC 2547, NGC 2169, σ Ori) yields ages that are systematically 1.5–2 times larger than the conventional PMS contraction ages. For instance, σ Ori, previously quoted as ≈ 3 Myr old, is reassessed at ≈ 5–6 Myr. The revised ages bring the observed lifetimes of protoplanetary discs (≈ 5–10 Myr) into agreement with theoretical planet‑formation timescales, resolving a major tension in the field. Moreover, the apparent paucity of clusters with ages between 5 and 30 Myr is naturally explained: many objects previously placed in that “gap” are simply older than thought, and the true age distribution is smoother.

The authors also discuss the impact of different model choices (e.g., alternative isochrone sets, metallicities) and find that while absolute ages shift modestly, the factor‑of‑two increase relative to PMS ages remains robust. They provide a publicly available software package implementing the τ² fitting procedure, facilitating its use on other datasets.

In conclusion, by fitting the subtle but measurable evolution of massive stars on the main sequence with a rigorously defined statistical framework, the paper demonstrates that young stellar populations are older than traditionally believed. This has far‑reaching implications for star‑formation histories, disc evolution, and early planet formation, and it establishes τ² fitting of the upper main sequence as a powerful tool for deriving absolute ages in young clusters.