Using what we have derived, let us now try and understand the birth, life and death of a star.
The birth of a star requires it to be formed from cloud of molecular gas. The reason for stars to form there is that only those clouds are large enough and cold enough to undergo gravitational collapse.
We go back to fuids to find the jeans length. \[1/k_j = \sqrt{\frac{v_s^2}{4\pi G \rho_0}} \] We can then find the mass for gravitational collpase. \[M_j = \frac{4\pi}{3}k_j^3 \rho_0\] \[M_j = \frac{4\pi}{3}(\sqrt{\frac{{v_s^2}}{ 4\pi G \rho_0}})^3 \rho_0\] \[M_j = \frac{4\pi}{3}(\frac{v_s^2}{4\pi G})^3 \rho_0^{-1/2}\] We know that the speed of sound is related to the temperature by \[v_s^2 = \frac{ kT}{\mu m_p}\] We can then find the mass for gravitational collapse. \[M_j = \frac{4\pi}{3}(\sqrt{\frac{ kT}{\mu m_p}})^3 \rho_0^{-1/2}\] \[\boxed{M_j = \frac{4\pi}{3}(\sqrt{\frac{ k}{\mu m_p}})^3 T^{3/2}\rho_0^{-1/2}}\] \[\boxed{M_j \propto T^{3/2}\rho_0^{-1/2}}\]
The life of a star stays on the main sequence and burns hydrogen in its core. The Main Sequence (MS) is defined by a state of Complete Equilibrium. A star on the MS satisfies two conditions simultaneously: Hydrostatic Equilibrium and Thermal Equilibrium (This means there is no net change in the star's entropy with time).
To understand the life of a MS star requires us to solve the stellar structure equations. But we can also get out useful scalings using Homology.
Here are the stellar structure equations: \[\frac{dM}{dr} = 4\pi r^2 \rho \tag{mass conservation}\] \[\frac{dP}{dr} = -\frac{GM}{r^2} \rho \tag{hydrostatic equilibrium}\] We can have radiative diffusion or convection as the dominant energy transport. \[\frac{dT}{dr} = -4 \pi r^2 \rho [\frac{-Gm}{4\pi r^2} \frac{T}{P} \nabla_{rad}],\quad \nabla_{rad}= \frac{3\kappa }{16 \pi a c G} \frac{lm}{mT^4} \tag{radiative diffusion}\] \[\frac{dT}{dr} = -4 \pi r^2 \rho [\frac{-Gm}{4\pi r^2} \frac{T}{P} \nabla_{ad}],\quad \nabla_{ad} \Rightarrow \frac{r}{T} \frac{dT}{dr}_{ad} = \frac{1}{\gamma}\frac{\rho}{P} \frac{dP}{dr} \tag{convection diffusion}\] \[\frac{dL}{dr} = 4\pi r^2 \rho \epsilon \tag{energy production}\] We can then use homology to get the following scalings:
Homology essentially, homology treats different stars as scaled versions of each other. They assume in HE, same opacity, same EOS and such. Example mass conservation: \[\frac{dM_1}{dr_1} = 4\pi r_1^2 \rho_1 \] \[dM_1 = dM_2 \frac{M_1}{M_2} \] \[dr_1 = dr_2 \frac{r_1}{r_2} \] We can then write the mass conservation equation as: \[\frac{dM_1}{dr_1} = \frac{dM_2}{dr_2} \frac{M_1}{M_2} \frac{r_2}{r_1} \] add in the mass continuity equation \[4\pi r_1^2 \rho_1 = 4\pi r_2^2 \rho_2 \frac{M_1}{M_2} \frac{r_2}{r_1} \] \[r_1^3 \rho_1 = \rho_2 \frac{M_1}{M_2} r_2^3 \] \[\boxed{ \frac{\rho_1}{ \rho_2} = \frac{M_1}{M_2}(\frac{r_1}{r_2})^{-3}} \]
Hydrostatic equilibrium: \[\frac{dP_1}{dr_1} = -\frac{G M_1}{r_1^2} \rho_1 \] \[dP_1 = dP_2 \frac{P_1}{P_2} \] \[dr_1 = dr_2 \frac{r_1}{r_2} \] We can then write the hydrostatic equilibrium equation as: \[\frac{dP_1}{dr_1} = \frac{dP_2}{dr_2} \frac{P_1}{P_2} \frac{r_2}{r_1} \] add in the hydrostatic equilibrium equation \[-\frac{G M_1}{r_1^2} \rho_1 = -\frac{G M_2}{r_2^2} \rho_2 \frac{P_1}{P_2} \frac{r_2}{r_1} \] Simplify \[-\frac{M_1}{r_1} \rho_1 = -\frac{ M_2}{r_2} \rho_2 \frac{P_1}{P_2} \] \[\frac{ M_1}{M_2} \frac{r_2}{r_1} \frac{\rho_1}{\rho_2} = \frac{P_1}{P_2} \] We can replace density with our previous homology result \[\frac{ M_1}{M_2} \frac{r_2}{r_1} \frac{M_1}{M_2}\frac{r_2^3}{r_1^3} = \frac{P_1}{P_2} \] \[\boxed{ \frac{P_1}{P_2} = \frac{ M_1^2}{M_2^2} \frac{r_2^4}{r_1^4} }\] If we used the equation of state \(P= nkT\) we can get \[\frac{P_1}{P_2} = \frac{\rho_1}{\rho_2} \frac{T_1}{T_2}\] \[\frac{P_1}{P_2} = \frac{M_1 /R_1^3}{M_2 /R_2^3} \frac{T_1}{T_2}\] We can then replace to get \[\boxed{ \frac{T_1}{T_2} = \frac{ M_1}{M_2} \frac{r_2}{r_1} }\]
Radiative: let us look at how to get out luminosity: \[\frac{dT_1}{dr_1} = \frac{-3 \kappa \rho_1}{4 ac} \frac{1}{T_1^3} \frac{l}{4\pi r_1^2}\] Let us slowly replace everything with our homology results: \[ \frac{T_1}{T_2} = \frac{ M_1}{M_2} \frac{r_2}{r_1} \] \[\frac{dT_1}{dr_1} = \frac{-3 \kappa \rho_1}{4 ac} \frac{1}{(T_2 \frac{ M_1}{M_2} \frac{r_2}{r_1})^3 } \frac{l}{4\pi r_1^2}\] more homology results \[\frac{\rho_1}{ \rho_2} = \frac{M_1}{M_2}(\frac{r_1}{r_2})^{-3}\] place it in \[\frac{dT_1}{dr_1} = \frac{-3 \kappa \rho_2 \frac{M_1}{M_2}(\frac{r_1}{r_2})^{-3}}{4 ac} \frac{1}{(T_2 \frac{ M_1}{M_2} \frac{r_2}{r_1})^3 } \frac{l}{4\pi (r_2 \frac{R_1}{R_2})^2}\] \[\frac{dT_1}{dr_1} = \frac{-3 \kappa \rho_2 }{4 ac} \frac{1}{ ( \frac{ M_1}{M_2})^{2}T_2 } \frac{\frac{l_1}{l_2}}{4\pi (r_2 \frac{R_1}{R_2})^2}\] \[\frac{dT_1}{dr_1} = (\frac{ M_1}{M_2})^{-2} (\frac{ r_1}{r_2})^{-2}\] We replace the derivative: \[\frac{dT_1}{dr_1} = \frac{dT_2}{dr_1} \frac{T_2 (\mu1/\mu2) (M_1/M_2)(R_2/R_1)}{r_2 R_1/R_2} = \mu (M_1/M_2) (R_2/R_1)^2\] Now we figure out the right side: \[\boxed{\frac{l_1}{l_2} \propto (\frac{m_1}{m_2})^{3} \mu_1^{4} \mu_2^{4}}\] If you add back krammers opacity you get \[\]
The key event in a star's late life is whether its core crosses the boundary of electron degeneracy in the temperature-density plane .
These stars end their lives as White Dwarfs (WDs) because their cores become degenerate before they can fuse Carbon .
These stars are massive enough that their cores remain non-degenerate during contraction.
For the vast majority of stars (the low/intermediate ones), the final state is a White Dwarf, which is described in Lecture 22 as follows:
White dwarfs are the final stage of stellar evolution for stars with masses less than 8 \(M_\odot\). And they are unique in that they are supported by electron degeneracy pressure rather than thermal pressure.
Let us find how the radius and mass scale with each other. We start with the hydrostatic equilibrium equation: \[\frac{dP}{dr} = -\frac{Gm}{r} \rho\] \[\frac{P}{r} \sim -\frac{Gm}{r} \rho\] \[P \sim -\frac{Gm^2}{r^4}\] \[P \sim K \rho^{5/3} \tag{EOS degenerate NR}\] plug in for density \[P \sim K M^{5/3} \frac{1}{R^5} \] \[ -\frac{GM^2}{r^4} \propto K M^{5/3} \frac{1}{R^5}\] \[ -\frac{G}{r^4} \propto K M^{-1/3} \frac{1}{R^5}\] \[\boxed{R \propto M^{-1/3}}\] Chandrasekhar Limit: given by when equation of state goes relativistic and is dynamically unstable. \[P \sim K M^{4/3} \frac{1}{R^4} \] \[ -\frac{GM^2}{r^4} \propto K M^{4/3} \frac{1}{R^4}\] We see that radius cancels out and we get just a mass relation: \[\boxed{M \propto (K/G)^{3/2} \sim 1.44 M_\odot}\]
Temperature and Luminosity relation The White Dwarf has a highly conductive, isothermal core (at temperature \(T_c\)) wrapped in a thin, non-degenerate, radiative envelope. The luminosity is determined by how easily heat can leak through this opaque envelope. \[\frac{dP}{dr} = -\frac{Gm}{r^2} \rho\tag{hydrostatic equilibrium}\] \[\frac{dT}{dr} = \frac{-3\kappa \rho}{16 \pi ac} \frac{L}{4\pi r^2T^3} \tag{radiative diffusion}\] We divde the two equations to get: \[\frac{dT}{dP} = \frac{\kappa L}{MT^3}\] bring in the opacity from kramers opacity law: \[\kappa \propto \rho T^{-7/2}\] \[\frac{dT}{dP} = \frac{L}{MT^{17/2}}P\] We integrate \[\int T^{17/2}dT = \int \frac{L}{M}PdP\] \[\frac{2}{19}T^{19/2} = \frac{L}{M}P^2\] \[\frac{2}{19}T^{19/4} (\frac{L}{M})^{-2}= P\] We can drop in ideal gas law \[\rho \propto \frac{P}{kT}\] \[\rho \propto \frac{\frac{2}{19}T^{19/4} (\frac{L}{M})^{-2}}{T}\] At the core the equation of state needs to match \[P = kT = \rho^{5/3}\] together we get that: \[\boxed{L \propto M T^{7/2}}\]
Cooling time: since there is no energy production, the white dwarf will cool down via radiative diffusion.
Total energy stored: \[U = MT\] We also know that the temperature luminosity relation is: \[L\propto T^{7/2}\] \[L^{2/7}\propto T\] And we know that \[L = \frac{dU}{dt}\] We rewrite \[\] \[L = M\frac{dT}{dt}\] \[L = M\frac{dL^{2/7}}{dt}\] \[L = ML^{-5/7}\frac{dL}{dt}\] \[dt= ML^{-5/7}\frac{dL}{L}\] \[\int dt= \int ML^{-12/7}dL\] \[\boxed{t\propto ML^{-5/7}}\]
Massive stars continue to fuse Helium into larger more massive elements. At the end of its nuclear burning sequence, a massive star develops an "onion skin" structure with layers of progressively heavier elements, ending with an Iron (Fe) core in the center. Iron cannot fuse to release energy; instead, fusion would consume energy.
Photodisintegration: Iron is broken up into smaller nuclei. This process is endothermic (absorbs heat), drastically reducing the core's thermal pressure. Changes the adiabatic index to 4/3 and becomes unstable.
Inverse Beta DecayElectron Capture: Due to high densities, electrons are forced into protons to form neutrons and neutrinos. And looses degeneracy pressure.
Free fall: With pressure support gone, the core collapses under gravity on a dynamical timescale free fall.
The Bounce: The collapse continues until the core density exceeds nuclear density. At this point, the strong nuclear force becomes repulsive (the Equation of State goes back to 5/3), causing the inner core to rebound or "bounce".
Shock:This bounce generates a big shock wave that moves outward.
Stalling: As the shock moves through the outer iron core, it loses energy by dissociating heavy nuclei, causing it to stall.
Neutrino Revival: The collapsing core releases a massive burst of neutrinos. These neutrinos deposit some of their energy into the stalled shock layer, reheating it and "reviving" the shock.
Key Calculations: Energetics of the core: \[\Delta E = GM_c^2/R_c \sim 10^{53} ergs\] Energy of the shock: \[\Delta E = \frac{1}{2} M_c v_s^2 \sim 10^{51} ergs\] Energy of the neutrinos should be carying away the result which is \(99\%\) of the energy. The neutrinos are fromed via inverse beta decay and electron capture.
Neutrino Trapping: We find the cross section for neutrino scattering and then find the mean free path to see that the core would be too dense and thus traps the neutrinos in it.