“Isochrone” is a word constructed from Greekiso, “equal”, andchronos,“time”. From this etymology one gets the essential idea ofisochronesas stellar population models : an isochrone is a model of a population of stars of uniform age (and for the sake of simplicity, of uniform composition as well.) While even globular clusters are not remotely such simple populations (though we formerly thought them to be - see the review of Gratton et al. 2012), isochrones have proven to be of immense value in understanding a range of stellar systems. Figure 1.3 shows fits of Dartmouth isochrones to the color-magnitude diagram (CMD) of globular cluster M92 from Dotter et al. 2007. These fits show us that M92’s population is consistent with a stellar population that is 13.5 Gyr old with [Fe/H] of−2.3 (modulo distance and reddening uncertainties, also modulo model uncertainties - see §6.1 through §6.2.)
Figures 1.5 through 1.7 illustrate how isochrones tell us these things. The CMD of M92 with stellar evolutionary sequences labeled is plotted in Fig. 1.4 for illustration.
All stars begin life on the main sequence (MS) burning hydrogen in their cores. As that core hydrogen is exhausted, hydrogen fusion commences in a shell around the core and the star moves off of the MS and onto the subgiant branch (SGB). Figure 1.5 shows that the clearest indication of the age of a stellar population is the position of its main sequence turnoff (MSTO), the point where this transition between the MS and SGB is occurring.
The main sequence is a mass sequence - more massive stars are further up (brighter) and to the left (bluer). A more massive star has a greater central pressure than a less massive star of the same composition because the layers above the core of the more massive star possess greater mass and therefore weight than the layers above the core of a less massive star. This
Figure 1.3:Fits of Dartmouth isochrones to M92 from Dotter et al. 2007. The two panels are for the two different synthetic color transformations available for the isochrones (§6.2.4). Reproduced by permission of the AAS.
greater pressure leads to greater temperature and density, and therefore to higher nuclear reaction rates. The greater luminosity in turn, produces a higher effective temperature Tef f and therefore a bluer color.
Because of their higher nuclear reaction rates, more massive stars exhaust their core hydrogen more quickly than less massive stars. They thus make the transition from the main sequence to the subgiant phase first. Therefore the location of the MSTO travels down a main sequence towards less massive stars with the passage of time, and this can be used to date a stellar population.
Isochrones of the same age but differing [Fe/H] and [α/Fe] are plotted in Figures 1.6 and 1.7. The most obvious differences between isochrones of different compositions are that
M92 & Isochrones
Horizontal Branch
Asymptotic Giant Branch
Main Sequence
Red Giant Branch
Subgiant Branch
Figure 1.4:CMD of globular cluster M92 with Dartmouth isochrone (red) and synthetic horizontal branch (indigo) overplotted in order to highlight major evolutionary sequences. [Fe/H] =−2.31 and [α/Fe] = 0.2 for the models. The isochrone is that of a 14.5 Gyr old population. The M92 data is from Paust et al. 2007 (Nathaniel Paust, private communica- tion).
Figure 1.5:Plots of Dartmouth isochrones (Dotter et al. 2007, Dotter et al. 2008) of varying age.
the isochrones with greater heavy element abundances, both in terms of [Fe/H] and [α/Fe], are located lower (dimmer) and to the right (redder) in a CMD, and have red giant branches (RGBs) that are less nearly vertical and curve more.
If one compares the main sequences of two isochrones of differing composition in these plots, it is tempting to conclude that the more heavy element-rich main sequence contains stars which are brighter and/or redder than their more heavy element-poor counterparts,
Figure 1.6:Plots of Dartmouth isochrones (Dotter et al. 2007, Dotter et al. 2008) of varying [Fe/H]. The squares and triangles on the [Fe/H] =−0.50 and 0.00 isochrones represent stars of the same mass on the two isochrones. The triangles represent stars of mass≈0.751 M,
Figure 1.7:Plots of Dartmouth isochrones (Dotter et al. 2007, Dotter et al. 2008) of varying [α/Fe].
based on the fact that the heavy element-rich main sequence is“above” and/or to the“right” of its heavy element-poor counterpart. But this appearance is deceptive. In fact, if one has two main sequence stars of the same mass but different metallicities, the more metal-poor one will be both brighter and bluer. Fig. 1.6 shows this for main sequence stars of two different masses.
P is proportional to a power of densityρ:
P =Kρn+1n (1.1)
These models are called polytropes. The constant K depends upon the nature of the poly- trope.
If one further assumes that n = 3 and that the opacity of the stellar material obeys a Kramer’s opacity law
k =k0ρT−3.5 (1.2)
where k0 is a function of composition (Clayton 1983), then one has what is termed the “standard model" because of its good match to the properties of the Sun. For such a model, the luminosity obeys the following relation (Clayton 1983, Carney & Harris 1998) :
LM S ∝
µ7.5M5.5 k0
(1.3)
whereµis mean molecular weight, M is the total stellar mass, and“MS” stands for“main sequence”.
A star with a greater abundance of heavy elements has both a higher mean molecular weightµand a higher opacity. The former works to increase the luminosity while the latter works to decrease it. It turns out that a given change in composition produces a much larger change in k0than inµ. Thus it is the denominator that wins, and a star with a greater abundance of heavy elements will be dimmer than a star of the same mass with a lower heavy element abundance. The lower luminosity of such a star will also lead to a lower
Another difference between isochrones of differing compositions which is immediately apparent from a casual look at Figs. 1.6 and 1.7 is that the red giant branches of populations with greater abundances of heavy elements are redder and have shallower slopes than those of populations with lower abundances of heavy elements.
The reason that more heavy element-rich red giant branches are redder than heavy element-poor ones is that more heavy element-rich stars have greater opacity, owing to the greater numbers of electrons contributed by the larger number of heavy element atoms. The layers of these stars therefore absorb more of the outgoing photon flux, experiencing greater outward photon pressure, leading to a larger radius and a cooler, redder envelope. For cool stars such as red giants, much of the opacity comes from H−, and the metals furnish the extra electrons for neutral hydrogen to capture in order to form H−.
There are several reasons for the differences in slope and curvature between RGBs of differing compositions. As explained earlier, a higher heavy element abundance leads to a larger radius, lower effective temperature, and redder color. This effect becomes more pro- nounced as one moves up the red giant branch. Additionally, line blanketing and bolometric correction increase with increasing heavy element abundance.
The bolometric correction, the difference between the V magnitude of a star and its total magnitude across all wavelengths, is larger for redder, cooler stars (such as metal-rich stars, relative to metal-poor ones) because redder stars emit less of their flux in bluer bandpasses (less flux in V, for instance, compared with say, I.) Line blanketing, the decrease in intensity at bluer wavelengths due to the presence of a large number of metallic absorption lines and the subsequent re-emission of the energy at redder wavelengths, is of course greater in stars with more metals. Molecular absorption is responsible for much of the increase in line blanketing experienced by RGB stars with higher heavy element abundances. Not only do such stars contain a greater abundance of the constituent atoms of such molecules as CH,
of more molecular species to remain bound.
When employing isochrones to study stellar systems, the general preference is for [Fe/H] and [α/Fe] to be determined by spectroscopy while the age is determined by isochrone fitting. At the MSTO and SGB, unless one or the other is determined, it would not be possible to disentangle the effects of age and composition, as both greater age and greater heavy element presence produce a dimmer and redder MSTO and SGB. (For more on degeneracy issues, see §6.3.)
In the middle of the RGB, the differences between isochrones of rather widely varying composition is small enough to be overwhelmed by issues of photometric uncertainty, membership, and general sparseness in that region of the CMD.
The differences between isochrones of varying composition at the tip of the RGB are rather great. Unfortunately, the tips of the RGBs of Leo IV and Boötes II are sparsely populated (see Chapter 5.) Given that these systems are thinly populated to begin with, and that stars spend a relatively short time on the RGB (compared to the main sequence), this is not surprising. In the case of Boötes II, in fact, the upper portions of the RGB presented in this study and previous ones are unpopulated. This study and previous ones find Leo IV’s upper RGB to be sparsely populated, with a single radial velocity member. Drawing conclusions concerning composition from such a basis is rather risky.
As such, this study will draw upon spectroscopic studies for [Fe/H] and [α/Fe] while employing isochrones to find the ages of Leo IV and Boötes II.