## 2. A220A: Stellar Structure and Evolution – J. Fortney

One can derive a characteristic timescale for a process by dividing a given quantity by the time rate of change of that quantity. One important timescale for stars is the “dynamical” or “free-fall” timescale. Here our quantity of interest is the radius of a star. With that as a starting point,

a) Derive a simple relation for the dynamical timescale. Then, for parts b) and c) just below, order the objects from the shortest to the longest dynamical time.

b) Main Sequence Sun, Red Giant Sun, white dwarf Sun.

c) Main sequence stars: $0.2M\odot, 1M\odot, 5M\odot$

d) Another timescale is the nuclear timescale, which is basically the main sequence lifetime. Where does the approximate main sequence lifetime relation, $t_{MS} \propto M^{-2.5}$, come from?

Solution(incomplete):

a) the most common way to derive the dynamical timescale is from the Kepler’s laws. But there is another shorter version based on the fact that the dynamical timescale is basically the ratio of a characteristic length and the characteristic speed: $\tau_{dyn}=\frac{characteristic (R)}{characteristic (v)}=R/v_{esc}$ where the escape velocity is: $v_{esc}=[2GM/R]^{1/2}$ so that $\tau_{dyn}=[\frac{R^{3}}{2GM}]^{1/2}$

b) We can assume that the Sun’s mass stays the same from the main sequence till the white dwarf phase(in reality it probably will eject significant amount of mass due to stellar winds during the red giant branch).  So the only varying parameter is the radius.

At the main sequence the radius is just $1R\odot$. When the sun enters the red giant phase the sun will not be burning Hydrogen in the core anymore. The hydrostatic equilibrium will be lost so the core will contract heating up the shell outside it. And because of this shell burning the radius of the sun will swell up to around $\sim 200R\odot$ in order to adjust to the increase in luminosity. When the sun is done with the giant phase most of the shell material will be blown into space leaving the hot Carbon-Helium core to cool down. The size of this core or the white dwarf Sun will be on the order of $R_{whitedwarf Sun} \sim R_{earth} = 0.009R\odot$. So the timescales are:

$\tau_{dyn.WD} < \tau_{dyn.MS} < \tau_{dyn.RGB}$

If you take the mass loss into account, the WD sun will retain roughly the half of the mass after RGB. But even with this mass, the result above will still hold.

c) There is an empirical relation ship between the radius and mass at the main sequence:

$\frac{R}{R\odot}=(\frac{M}{M\odot})^{0.8}$

So the radii for main sequence $0.2M\odot, 1M\odot, 5M\odot$ stars are $0.3R\odot,R\odot,3.6R\odot$ respectively, thus:

$\tau_{dyn. 0.2M\odot} < \tau_{dyn. 1M\odot} < \tau_{dyn. 5M\odot}$

d) The heat released by fusing a mass $\Delta M$ of Hydrogen into Helium is approximately $0.007\Delta Mc^{2}$. So for the Sun the time required to burn all its Hydrogen for the respective current solar luminosity is:

$\tau_{nuc}=\frac{0.007M\odot c^{2}}{L\odot} \sim 10^{10} years$

Another empirical relation between the luminosity and the mass at the main sequence is:

$\frac{L}{L\odot}=(\frac{M}{M\odot})^{3.5}$

Combining these 2 you get the main sequence time scale:

$\tau_{MS} \sim (\frac{M}{M\odot})^{-2.5}$

## 1. 220A: Stellar Structure and Evolution – J. Fortney

$\frac{dT}{dm}=\frac{-3}{64\pi^{2} ac}\frac{\kappa L_{r}}{r^{4}T^{3}}$

a) In a post-main sequence Sun-like star, ascending the red giant branch, why does the helium core have a smaller temperature gradient than the hydrogen shell above it? Ignore any issue of energy loss due to neutrino emission.

b) What particular term is important for causing outer convective zones to grow in low-mass stars? What specifically is going on?

c) What particular term is important to causing inner convective zones to grow in high-mass stars? What specifically is going on?

Solution(incomplete):

a) another way to write equation above is:

$F=\frac{\lambda c}{3}\nabla a T^{4}=\frac{ac}{3\rho\kappa}\nabla T^{4}$

here you can see that in order to generate the same amount of Flux, for lower opacity $\kappa$ you need lower temperature gradient. Since $\kappa (He) < \kappa (H)$ the temperature gradient of the Helium core has lower temperature gradient corresponding to its opacity.

b) the growth of the outer convective zones of low-mass stars is due to the Hydrogen recombination. The photon sees a lot of ‘possibilities’ due to the availabilities of higher states, thus increasing the opacity. So the opacity is the important term in this situation.

c) For high-mass stars the inner convective zones grow in order to compensate the increasing luminosity. So the important term in this situation is the Luminosity.