Stellar Energy Losses
==========================



Key:

1. Cooling speed of white dwarfs, and neutron stars
2. "Delay of helium ignition in low mass red giants"[Raffelt]_
3. Helium burning lifetime


Evolution of Stars
----------------------------

.. admonition:: Videos
   :class: note

   Here is website that provides nice videos of evoution on HR digram,
   http://rainman.astro.illinois.edu/ddr/stellar/beginner.html


Main Sequence
~~~~~~~~~~~~~~~~~~~~~~~~~~

1. Formation: gas clound condense due to EM radiation.

   .. admonition:: Negative Specific Heat
      :class: note

      Gravitation dominated systems usually have negative specific heat. As the volume contracts, the star would be heated up, since specific heat is defined as

      .. math::
         C = \frac{\delta Q}{dT}.

2. Pressure, angular momentum, magnetic fields
3. IMF: [0.08, 100] :math:`\mathscr M_\odot`

   .. admonition:: Salpeter's IMF
      :class: note

      .. math::
         \frac{dN}{d\mathscr M} \propto ( \mathscr M/\mathscr M_{\odot} )^{-1.35}

      .. figure:: assets/astrophysics/stellar-energy-losses/salpeter-imf.png
        :align: center

        Salpeter's IMF

      Notice that :math:`\int x^{-n} dx = C + \frac{ x^{1-n} }{1-n}`.

4. First stars:

   1. mass fraction of hydrogen :math:`X\sim 0.75`
   2. mass fraction of helium :math:`Y\sim 0.25`

   Sun:

   metallicity: :math:`Z\sim 0.02`

5. Mass loss:

   1. stellar wind
   2. supernova explosions

6. Limit of white dwarfs: 1.4 solar masses
7. Disk of spiral galaxies: active death and birth of stars

   1. Spiral galaxies have old halo stars, globular clusters.
   2. Milky Way: 150 of them, each with :math:`10^{6}` stars.
   3. Gravitational escape velocity of them: :math:`10\mathrm{km s^{-1}}`

      .. admonition:: Escape velocity
         :class: warning

         From where?

   4. Supernova explosion ejects at :math:`10^{3}\mathrm{km s^{-1}}`.
   5. Supernova sweep whole globular cluster clean of gas.

      .. admonition:: Ah?
         :class: warning

         How?

   6. No star formation
   7. Good for stellar evolution research.

8. Viral theorem, negative specific heat -> contraction -> nuclear burning

   1. determined by mass
   2. HR diagram: :math:`\log ( L/L_\odot )` ~ :math:`\log ( T_{\mathrm{eff}}/K )`, or luminosity ~ surface temperature, scaled with solar quantities
   3. Most of the time the stars stay on the diagram
   4. Sun: 1Gyr life, half gone
   5. :math:`L\sim \mathscr M^3`
   6. :math:`\mathscr M/\mathscr M_\odot \leq 0.7 - 0.8` stars are still alive
   7. Globular cluster has turnoff: compare Fig. 2.2 and Fig. 2.3

9. MS mass-radius relation: :math:`R \sim \mathscr M^\xi`, for :math:`\mathscr M < \mathscr M_\odot`, :math:`\xi \sim 0.8`, for :math:`\mathscr M > \mathscr M_\odot`, :math:`\xi \sim 0.57`; The difference comes from the convective envolope. [rbc3]_


Red Giant
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


1. Hydrogen consumed in center
2. Next step depends on the mass of the stars
3. For :math:`\mathscr M \leq 2 \mathscr M_\odot`

   1. New configuration forms from helium ashes in the center
   2. Outer region expand -> Surface temperature drops -> Redder: red giant

      .. admonition:: Why
         :class: warning

         According to Stefan-Boltzman law?

         .. math::
            j = \sigma T^4

   3. There exists a hydrogen burning shell -> Dumps He into core
   4. He core is dense -> electrons degenerate -> mass-radius relation: :math:`R\sim \mathscr M^{-1/3}` -> Mass of core increas leads to decrease in radius
   5. Core gravitation: :math:`\Phi_c \sim - G \mathscr M_c / R_c \sim \mathscr M^{4/3}`
   6. Larger core mass -> Hotter hydrogen -> faster burning of hydrogen





References and Notes
-----------------------

.. [Raffelt] Stars as Laboratories for Fundamental Physics
.. [rbc3] http://personal.psu.edu/rbc3/A534/lec18.pdf