The Turn Off Luminosity and the Age of the Globular Clusters
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The Turn Off Luminosity and the Age of the Globular Clusters
Jianjun Jia
1. Introduction
The determination of the age of the oldest stars gives a lower limit to the age of the
universe, and constrains the current cosmological model. Thus, it is necessary to make an
accurate measurement of the age of very metal-poor stars in the globular clusters(GCs),
which are assumed to be composed of the oldest stars and perhaps the first stars in the
universe. A typical globular cluster contains about 106 stars, and is located in the galactic
halo. The globular clusters are lack of gas and dust, and the GC stars have very low
metallicity, thus it is possible for us to study the age of them on the Hertzsprung-Russell
diagram(HRD),which will be discussed later.
We have several methods to determine the age of a globular cluster. One is the radioac-
tive decay of the thorium and uranium (Cowan et al, 2002). However, they have very weak
lines, and it gives a large error of the estimated age. Another method is to determine the
age of the old white dwarfs (Hansen, 1999). Since the white dwarfs have no nuclear energy
generation and will cool slowly in the rest of their lives, the faintest white dwarfs can be used
to estimate the age of the cluster. But the difficulty is that the age measurement depends
on the cooling mechanism and the equation of state (Shapiro & Teukolsky, 1983), and we
have no convincing theory among the various of cooling models.
The most common method used in age estimation is the turn-off Mass(M T O )-Age rela-
tion on the HRD. Assuming that he stars in a globular cluster were born contemporarily and
have the same initial metallicity, we are able to determine the age of the GC according to
the turn-off luminosity (LT O ) or the turn-off absolute magnitude (Mbol
TO
) in the HR diagram.
As we know, the massive a star, the shorter its MS lifetime. After it exhausts the hydrogen
fuel at its core, it will leave the MS branch and turn to the sub-giant branch(SGB) in the
HR diagram. If we take a snapshot of all the stars on the HRD, the stars reaching the MS
turn-off will tell us the age of the cluster according to the model of stellar evolution.
2. MS Turn-off Luminosity
For a lower mass star, the fusion of hydrogen into helium in the core, which is the MS
stage, accounts for nearly 90% of its nuclear burning time through its life. When it exhausts
the hydrogen fuel in its core, it reaches the vertex of the MS branch, and remains a helium–2–
core surrounded by a hydrogen shell. The shell hydrogen ignites while the inner helium core
shrinks, and the star expands rapidly. For a given luminosity L = 4πR2 σTef 4
f , the radius R
increases, and the effective temperature Tef f will decreases. So the star becomes cooler and
leaves the MS branch, which is so called ”main sequence turn-off”.
The theory of nuclear fusion allows us to estimate the age of a cluster from the luminosity
of stars at the turn-off point. In theoretical models, the energy generation rate and the core
hydrogen burning time are only dependent of mass and metallicity, so the mass-luminosity
relation (MLR) can be determined theoretically. However, such relation is usually given by
observations.
Fig 1 shows a simple-fitting of MLR with an index 4, but it doesn’t fit well on the
lower-mass stars. Statistically, we have
µ ¶4.0
L M
= 1.2 0 ≤ Mbol ≤ +7.5 (1)
L¯ M¯
µ ¶2.76
L M
= 0.67 + 7.5 ≤ Mbol ≤ +11, (2)
L¯ M¯
where Mbol is the absolute bolometric magnitude of the star. The MS age of a star can be
calculated by its total energy released and the mean luminosity when it stays in the main
sequence, which gives
M
τM S ≈ 1.0 × 1010 a. (3)
L
By applying MLR, we get
µ ¶2 µ ¶2/3
10 M¯ 10 L¯
τM S = 10 ≈ 10 (L < L¯ ) (4)
M L
µ ¶3 µ ¶3/4
10 M¯ 10 L¯
τM S = 10 ≈ 10 (L < L¯ ). (5)
M L
Some recent observations show more accurate MLR, especially for the lower mass stars
(Henry & McCarthy, 1993; Delfosse et al, 2000; Close et al, 2005).
The luminosity cannot be measured directly, and should be derived from the distance
(D) and the apparent magnitude (m∗ ). First, we get the absolute magnitude M ∗ of
M ∗ = m∗ + 5 − 5 log10 D. (6)
And the luminosity can be written as
∗ −M ¯ )
L = L¯ × 10−0.4(M , (7)–3– Fig. 1.— Mass-luminosity relation of the stars with the mass range from 0.1-10 M¯ . It fits well with an index of 4 for the stellar mass larger than 1 M¯ . For the lower-mass stars, another index should be chosen.
–4–
where M ¯ = 4.72 is the absolute magnitude of the Sun. Thus, the distance measure plays
an important role in the estimation of LT O . In observation, the measurement of GC distance
is based on the nearby subdwarfs and the RR Lyrae variable stars, which are so called two
standard candles.
3. Uncertainty in Turn-off Luminosity
As we know, the age of the globular clusters is determined by the turn-off luminosity.
Therefore, the uncertainty in turn-off luminosity in theoretical models will make the esti-
mated age with probable errors. In this section, a general review will be presented about
the uncertainties of the turn-off luminosity or the age in the theoretical models based on
VandenBerg and his collaborators’ papers (VandenBerg, Bolte & Stetson, 1996; Stetson,
VandenBerg & Bolte, 1996).
3.1. Convection theory
Chaboyer (1995) pointed out that our poor understanding on the convection in the
stellar model induced about 10% uncertainty of the the derived ages as inferred from LT O .
However, if the GC distance is measured by the standard candles, this uncertainty can be
significantly reduced by the coincidence of the predicted and observed SGB loci, as the
midway between the turn-off and the base of the red giant branch (RBG) on the SGB is
insensitive to the choice of convection theory, which can be used to determine the luminosity
better than the turn-off point. Thus, the treatment of the convection theory is not a serious
concern for the determination of GC ages. Fig. 2 shows the comparison of results given by
the mixing-length theory (MLT) and the Canuto & Mazzitelli model (CM) (1991), which
indicates that the derived LT O has very small shift in the two different treatments.
3.2. Nuclear reactions and opacities
The updated values of the nuclear reaction rates and opacities caused a 2% effect on the
calculated age-luminosity relation for a given evolutionary track (Harris et al, 1983). As the
opacities are completely dominated by the free-free transitions of hydrogen and helium at
low Z, the modification is not significant. VandenBerg et al (1996) illustrated that such
insensitivity of the relations to the particular generation of opacities assumed gave one
considerable confidence in the predictions for the metal-poor stars.–5– Fig. 2.— Comparison of isochrones for Z=0.0001, and two different treatments of MLT and CM convection. The difference of the predictions on the turn-off points only shift by 0.0025 in log Tef f (VandenBerg et al, 1996).
–6–
3.3. Equation of state
The interior stellar matter is approximated as ideal gas. However, when involving the
Coulomb interactions in the equation of state, the nonideal gas effect causes a 4% reduction
in age at a given LT O for GC stars (Proffitt, 1993). Also, a 6-7% reduction in age at a
TO
given Mbol is found compared with the ideal gas case with radiation pressure and electron
degeneracy assumed (Chaboyer & Kim, 1995). VandenBerg et al (1996) concluded that the
nuclear reaction rates, opacities and equation of states were well understood that future
developments in these areas would not make an improvement of the predicted ages at more
than few percent level.
3.4. Helium diffusion
The He diffusion can also introduce the uncertainty into the turn-off luminosity or the
age of the globular clusters. Noerdlinger & Arigo (1980) illustrated that the helium diffusion
could speed up a star’s main-sequence evolution, but they overestimated this effect. The
recent numerical simulation (Weiss & Schlattl, 2000) demonstrated that as soon as the star
gets cooler, the convective envelope deenpens and the helium is mixed back to the surface.
Their result shows that MS lifetimes get shorter due to the diffusion of helium into the center,
which is effectively equivalent to a faster aging of the star. Their TO-model with diffusion
reached an age about 1 Gyr less than the canonical calculations for a given MS luminosity.
3.5. Mass loss
Besides the initial metallicity, another intrinsic parameter in stellar evolution is the
mass. An interesting hypothesis (Willson et al, 1987) to explain the the Sun’s very low Li
abundance is that the Sun has an initial mass of 2 M¯ and lost half its initial mass during
the first 109 yr at a significant mass loss rate (∼ 10−9 M¯ yr−1 ). This mass loss rate seemed
too high than other models, which assumed the value only about one or two orders lower.
It was pointed out that a mass loss rate of 10−11 M¯ yr−1 applied within an instability strip
lying in the range 6600 ≤ Tef f ≤ 6900 would make GCs look ∼ 1 Gyr older than they
really are (Shi, 1995). If there is significant mass loss for the turn-off stars, there should
be a step-like feature manifest itself in the luminosity function plane at the point where the
CMD intersects the instability strip. However, no obvious bumps or steps have been foound
at turn-off luminosities. Therefore, we don’t need to worry about the uncertainty of the GC
age estimations caused by the mass loss cases.–7– Fig. 3.— Influence of diffusion on the evolutionary tracks (left panel) and lifetimes (right panel) for selected masses (0.7, 0.9, 1.1 ,1.3 M¯ ) with the same composition Y = 0.25, Z = 0.0003 (Weiss & Schlattl, 2000).
–8–
4. Summary
The theory of nuclear fusion allows us to estimate the age of a cluster from the luminosity
of stars at the turn-off point. And the age determination of globular clusters, which are the
oldest objects in the galaxies or maybe even older than the galaxies, provide numerous clues
on the evolution of the galaxies and restrict the cosmological model. The age-dependent
turn-off luminosities in the HRD give the estimation of the ages of the globular clusters
composed of low mass stars with the same age and initial chemical composition.
So far, our understanding of stellar evolution, especially the later stages of evolution,
cannot make the predictions at a very high accuracy. Therefore, the uncertainties in the
estimation of GC ages are inevitable. Fortunately, those uncertainties discussed above are
not significant so that our results are convincible at the current level of observation.–9–
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