Galaxy Evolution There are four types of galaxy evolution: morphological evolution, dynamical evolution, photometric evolution, and chemical evolution. Morphological evolution is difficult to study theoretically, since it depends sensitively on the rates of collisions between galaxies. (Collisions may turn spirals into ellipticals, and consume smaller galaxies.) One can expect that since the universe was once much density than it is today, there was an epoch where collisions were important. (Morphological evolution of spirals can also be caused by rampressure sweeping of the interstellar medium as the galaxy passes through a medium of x-ray gas. We will cover this topic later on.) The dynamical evolution of the stellar orbits in galaxies is linked to the galaxies morphological evolution, and is even harder to follow. (Very little is known about the stellar orbits in present day ellipticals, let alone the orbits of galaxies at early epochs.) However, we can make progress in the understanding of photometric and chemical evolution.
Photometric Evolution The photometric evolution of galaxies is a particularly straightforward problem. First, we need to know the Initial Mass Function of stars, or, in other words, how many stars are born as a function of mass. The IMF is usually written as a power law, i.e., φ(m)dm m (1+x) dm = φ 1 ) (1+x) d(m/ ) (15.01) where m is the stellar mass, is an arbitrary scaling constant (say, the mass of the Sun) to keep φ dimensionless, and x describes the steepness of the power law. The standard IMF is the Salpeter IMF, where x = 1.35. The constant φ 1 is simply there for normalization purposes, so that for a star cluster (or galaxy) with total mass M 0, M 0 = M 0 m φ(m/ ) d(m/ ) (15.02) 0 The second piece of information needed is the mass-luminosity relation for main-sequence stars. Again, this can be approximated with a power law. Let α be the power-law index for the mass-luminosity relation on the main sequence. If l 1 is the mainsequence luminosity of a star with mass, then l d m α = l d = l 1 ) α (15.03) where the subscript d represents the luminosities of dwarfs (i.e., main sequence stars). While the mass-luminosity relation of the main sequence is probably not a single power law, the approximation isn t too bad if you take α 3.
The third item needed is the length of time stars spend on the main sequence. The lifetime of a star is simply proportional to the energy available to the star divided by the star s luminosity. Since the available energy is proportional to the available mass, and the luminosity is proportional to mass (though the massluminosity relation), τ m l d m α ) α (15.04) Alternatively, this equation can be inverted. After a time t, the turnoff mass of a single-age stellar population will be m tn t 1 tn = m tn = ) 1 (15.05) where is the main-sequence lifetime of a star with mass. The final pieces of information that are needed are the lifetime of a typical post-main sequence (giant) star (τ g ), the average luminosity of a giant star, l g, and the mass of a typical white dwarf, w. All of these can be derived (approximately) from models of stellar evolution.
Luminosity Evolution Let s first calculate the luminosity evolution of a cluster (or galaxy) of stars with total mass M 0 all born at the same time. The total luminosity of main sequence stars is easy to compute: all we have to do is sum up the luminosity of all stars on the main sequence. The lower end of the sum is the minimum mass of an energy-generating star; the upper end is defined by the main sequence turnoff. In other words, L D = = mtn m L M 0 φ(m/ )) l d (m/ ) d(m/ ) mtn m L M 0 φ 1 ( ) (1+x) ( ) α m m l 1 d(m/ ) = M 0 φ 1 l 1 mtn = M 0φ 1 l 1 = M 0φ 1 l 1 m L { (mtn { ) α 1 x d(m/ ) ( ml } ( } tl (15.06) Note that since the exponent is negative, and low mass stars live (essentially) forever, the last term in the above equation is negligible. Thus, the total luminosity of dwarf stars is L D = M 0φ 1 l 1 ) (α x)/() (15.07)
Calculating the total luminosity of giant stars is equally simple. First, note that the timescale for giant branch evolution is much faster than that for main sequence evolution. Thus, the key is to estimate the number of stars currently turning into giants; when this number is multiplied by the length of time a typical star remains a giant, and the mean luminosity of the star as a giant, the result is the total giant star luminosity. Now consider: the rate at which main sequence stars turn into giants is defined by how many stars are at the main sequence turnoff, and how much of the main sequence is eaten away per unit time interval. According to our definition of the initial mass function, the number of stars at the main-sequence turnoff is φ(m/ ) d(m/ ) = M 0 φ 1 (m/ ) (1+x) d(m/ ) (15.08) and the main-sequence mass interval which is eaten away during a typical giant star lifetime, τ g is so (m/ ) = d(m/) dt τ g (15.09) L G = φ(m tn ) d(m tn/ ) dt τ g l g = M 0 φ 1 ) (1+x) dm tn dt l g τ g (15.10) If we now substitute t for m using (15.05), take the derivative, and combine terms, the result is L G = M 0φ 1 l g τ g (α 1) 1 (15.11)
Thus, the total luminosity of the stellar system, as a function of time, is L t = L D + L G = M 0φ 1 l 1 + M 0 φ 1 l g τ g (α 1) 1 α 1 (15.12) This can be simplified a bit if we define G(t) as the ratio of giant star luminosity to dwarf star luminosity, i.e., G(t) = L G = L D α 1 l g τ g l 1 ) 1 (15.13) Since α > 1, the exponent of time in (15.13) is significantly less than one. Hence writing the expression for G(t) in this way illustrates that the variable is only a weak function of time. This expression can be further simplified by (15.03) and (15.05) ) 1 ( ) α t 1 = = ( mtn ) α t = l 1 l tn t (15.14) Thus G(t) = α 1 { } lg τ g l tn t (15.15) Note that the first part of this equation is very close to one. Furthermore, the term in brackets is simply the ratio of the total energy generated by the star on the giant branch to the total energy generated by the star on the main sequence. Since stars turn off the main sequence when 10% of their total fuel is gone, but eventually burn 70% of their fuel, G(t) 6. (From an observers point of view, it s a bit more complicated, since most of the light a giant star produces is red, while the main sequence light may be blue. Thus, the exact value of G(t) depends on
the wavelength of observation: in the blue, G(t) 1, but in the red, G(t) > 10.) Using this notation, the luminosity of a stellar population, as a function of time, is L t = L d (t) {1 + G(t)} = M 0φ 1 l 1 ( t {1 + G(t)} (15.16)