Stellar model fitting

Stellar model fitting (also known as isochrone fitting) as a means of stellar age estimation is fairly ubiquitous. This method’s widespread use probably originates from its adaptability, and from the relative simplicity of its implementation. At its core the method simply compares measured stellar parameters to theoretical models appropriate for the metallicity,[M/H], of the star being examined. The underlying physics, however, can be complex, as these models require detailed knowledge (or assumptions) of stellar structure. For some other methods, such as asteroseismology, this is not the case.

When used for this purpose, the model data are usually presented as isochrones, lines of constant age with varying spectral type, rather than as lines of constant mass with varying age (mass tracks). [Fe/H]is widely used, including for the work presented herein, as an easily available proxy for overall metallicity, assuming that all other elemental abundances follow Solar ratios; this is not strictly correct, and there is evidence (e.g Edvardsson et al., 1993) that the Solar [M g/Fe], [O/Fe], and [Al/Fe] differ from mean field star values. However the creation of stellar models requires that assumptions be made regarding Solar abundances, for which the commonly assumed values have changed over time, and are in fact still debated (Asplund et al., 2006; Lodders, 2010; Basu & Antia, 2013). I use a value of Z = 0.0189. There is also a well-established trend between iron abundance and overall metallicity in both field stars and exoplanet hosts (e.g Bodaghee et al., 2003; Gilli et al., 2006), although recent studies show that the latter are overabundant in metals (Neves et al., 2009; Adibekyan et al., 2012).

Traditionally, the effective temperature, Teff was measured from high resolution spectra

and, in conjunction with the absolute stellar magnitude Mv, interpolated through the theo-

retical data to obtain estimates of the stellar radius, stellar mass, and stellar age for the star (e.g. Edvardsson et al., 1993; Lachaume et al., 1999). For cases in which the distance to the object is poorly known, some studies replaced Mvwith the stellar surface gravity, log(gs)

(e.g. Konacki et al., 2005; Bouchy et al., 2005). This can often be difficult to determine pre- cisely however, even with very high quality spectra, but transiting planets offer an alternative choice of parameter space. The geometry of a transit means that the stellar density can be constrained to high precision using high signal-to-noise transit photometry (Seager & Mallén- Ornelas, 2003). Modern attempts at isochronal analysis in exoplanetary studies therefore tend to use the parameter space of[T_eff,(ρ_s/ρ)−1/3](Sozzetti et al., 2007).

Limitations

As previously mentioned, one of advantages of the stellar model fitting procedure is its sim- plicity. Very little data is required, and it is easily extendable through variation in the choice of stellar models used. The method is also applicable to a broad range of spectral types, in principal. But in reality, there are regions of parameter space in which it is difficult to obtain useful results.

In particular, it can be hard to find the ages of stars with spectral type later than mid-late G. These stars have nuclear burning timescales that are longer than the age of the Galactic disc. Such stars evolve very slowly, and have effective temperatures such that they fall into regions of parameter space in which theoretical isochrones are closely spaced (see Figure 2.1). This makes determining accurate and precise ages very difficult, even if the physical parameters of the star are well constrained, as even small error bars in Teff andρ−1/3 can cover a wide

range of ages. As far as exoplanet hosts are concerned, Triaud (2011) suggested that stars with M_s <1.2M were particularly problematic. The complex shapes of isochrones, particularly near the main-sequence (MS) turn-off, can also present issues, and interpolating through them is not always a valid approach owing to their non-uniform spacing (Soderblom, 2010). In addition, the less pronounced radius increase (and therefore density decrease) during the MS lifetime,τ_MS, of low mass stars compared to their more massive relations has an impact on age estimates.

stellar models cover, and of the format of those models. The minimum and maximum stellar mass are rarely problematic given the propensity for the host stars of transiting exoplanets to be of F or G spectral type and therefore broadly similar to the Sun (Bentley e.g. 2010, for WASP targets; Batalha et al. e.g. 2010, for Kepler targets). However the lower and upper limits on the stellarage are much more likely to come into play, and in some sets of stellar models the maximum isochrone age is greater than the currently accepted age of the Universe! In addition, most stellar model formulations struggle with very young systems, as close to the zero-age MS the isochrones compact, leading to similar problems as experienced with K- and M-dwarfs.

The choice of stellar model being used can have a large impact on the derived properties of planetary systems, particularly through the introduction of systematic errors (Southworth, 2009). This suggests that multiple sets of stellar models should be used if at all possible, in preference to relying on a single formulation.