Input Parameters

The DSED isochrone software accepts as input age, metallicity ([Fe/H]), helium mass fraction (Y), and alpha elemental abundance ([α/Fe]). Figure 4.8 shows how adjusting the various parameters affects the shape of the isochrones. The blue lines in each panel represent the best fitting isochrone values from AM 1 by D08b. These do not agree with the

Figure 4.7 The vertical method of VBLC applied to AM 1. Cyan lines represent the ages that best fit the SGB region within the red dotted region and blue lines best fit the MSTO.

final parameters I used for AM 1, but serve rather as a staring point. Before proceeding, let us consider the physics behind each parameter’s affect on the model isochrone.

Age

Of all the parameters, age is, perhaps, the simplest to describe. Main sequence stars of mass between 0.5 and 2.0 M, though the most massive stars that remain in a globular

cluster are only< M, obey a mass luminosity relationship given by (Clayton 1968, Equation

1-50)

L∝Mν (4.4)

where ν is in the range of 3.5 to 4.0. Thus the main sequence in the CMD is really a mass sequence; higher mass stars are more luminous and consume their fuel more rapidly. Therefore, as the cluster ages, the MSTO descends the main sequence moving fainter and redward.

The elbow between the subgiant branch and RGB is governed by the size at which the outer envelope of the star is fully convective and cannot cool further (Hayashi & Hoshi 1961, the “Hayashi Track”). This feature remains nearly uniform in color as the cluster ages, and the luminosity of the SGB decreases because the stars at the TO become fainter as the cluster ages. The net effect, as shown in panel (c) of Figure 4.8 is an apparent contraction and dimming of the subgiant branch as the RGB and MS remain coincident.

Helium

The effect of helium is to increase the temperature and luminosity of a main sequence star according to the following basic argument. All other properties being equal, specifically metallicity, helium mass fraction, Y, dominates the mean molecular weight,µ, of the star as shown in Equation 22 of Carney & Harris (2001)

µ≈ 4

Once can think of main sequence stars as being in equilibrium between gravitational con- traction and the outward pressure of the gas. Rewriting the equation of state of an ideal gas in terms of mean molecular weight,

P = ρ k T µ mH

(4.6) where ρ is the mean stellar density and mH is the mass of hydrogen, it is apparent that

pressure is directly proportional to temperature and inversely proportional to µ. For the star to remain in hydrostatic equilibrium, the pressure in the core must offset the fixed gravitational weight of the envelope. Therefore, in order for pressure to remain constant as helium, and hence the µ, increase, both the temperature and density must also increase.

Nuclear reaction rates in MSTO stars undergoing the proton-proton chain reaction burn- ing hydrogen into helium are proportional to ρ2_T4_{. Greater reaction rates increases the}

luminosity, which means faster fuel consumption. Thus the helium enriched stars at the MSTO have lower mass and luminosity. The junction of the SGB and the RGB depends on the thickness of the convective envelope and hence mass. The total effect of helium enrich- ment in the observational plane, as shown in panel (d) of Figure 4.8, is a shift of the MS and RGB to hotter surface temperatures and higher luminosities, a lowering of the MSTO and a steepening of the SGB.

Metallicity

To understand the effects of metallicity on the main sequence, we must first derive an appropriate model of main sequence stars. Fortunately, Eddington (1926) supplied the classic and useful standard model that bears his name. The derivation starts with a polytropic approach to the solution to the equations of stellar structure in which pressure is a function of density. This approach invokes a relatively simple model relationship between pressure and density. A comprehensive description requires the simultaneous solution of the differential

equations of stellar structure.

For the Sun and other low mass stars, which are all that remain on the main sequence in GCs, the appropriate polytrope, of index 3, is given by Equation 2-290 of Clayton (1968)

P =K ρ43 (4.7)

Couple this with Kramer’s opacity law, an approximation appropriate for low mass stars given by

κ=κ0ρ T−3.5, (4.8)

and the equations of stellar structure may be reduced to the form given by Equation 6-20 of Clayton (1968)

LMS∝µ7.5M5.5/κ0. (4.9)

The constant term from Kramer’s opacity, κ0, in Equation 4.9 above is directly propor-

tional, though weakly, to heavy element mass fraction, Z (Clayton 1968, §6-6). Therefore, an obvious effect of increasing metallicity of a star is lowering its luminosity. The lower lu- minosity means greater stellar lifetime due to slower fuel consumption rates. Thus stars on at the turnoff in metal rich clusters tend to have higher masses due to longer stellar lifetimes. However, despite the high power on the mass term in Equation 4.9, the mass difference is small enough that the opacity term dominates leading to a lower turnoff luminosity with increased Z.

But the energy of the star must still ultimately escape the core. The increased opacity forces the star has to swell up, putting the energy into gravitational potential of the outer envelope that otherwise would radiate away as light. This increased size cools the outer atmosphere leading to a lower Tef f. The combined effect in the observational plane is that

all stars appear redder and fainter with increased metallicity, as shown in panel (a) of Figure 4.8.

Metallicity affects the RGB through line blanketing. Heavy element lines predominantly occupy the blue portion of the spectra, thus increased metallicity means smaller net flux in blue bandpasses. The effect on the (B−V) index of reduced B flux is a steeper RGB. In this way, the slope of the RGB at the level of the HB is a common indirect measurement of metallicity. Furthermore, the heavy elements serve as electron donors, increasing the production of H−, the principle continuum opacity source. This causes the entire RGB to shift toward cooler effective temperatures.

Alpha Element Enrichment

As stars ascend the RGB, they become more luminous and cooler as indicated by the general shape of the RGB in the observational plane. The principle continuum opacity source is H− which requires excess electrons in the stellar atmosphere. Several α elements, specifically Mg, Ca, Ti and Cr, have lower ionization potentials than iron and, therefore, serve as electron donors at cooler temperatures. Despite its very high abundance, oxygen has little effect because its ionization potential, 13.6 eV, is very high. The effect of increased

αenhancement, shown in panel (b) of Figure 4.8, is increased opacity at cooler temperatures resulting in a flatter RGB at higher luminosities.