L18 Interiors (the energy transport)

L18 Interiors (the energy transport)#

Week 10, Thursday

Material covered and references#

  • Now that we found the temperature structure in the atmosphere, let’s turn back to finding the temperature structure in the interior.

We started by recasting the radiative energy transport in terms of radius instead of optical depth, in the large optical depth regime.

In the textbooks:

  • Leblanc 5.2.3

  • Kip 5.1

  • Hansen 4.2

  • We first use the high optical depth case, and recast our equation in terms of the stellar radius.

  • We discuss the use of the Rosseland mean opacity, as a way to simplify the wavelength dependent calculations.

  • We then go back to spherical geometry, as stellar interior cannot be approximated as flat. This means that the flux is no more constant. We introduce a quantity \(L_r(r)\), which describe the enclosed luminosity (or power) at a radius coordinate \(r\). This quantity should be constant as long as there is no production of energy. The relation between \(L_r(r)\) and the temperature gradient is called the “energy transport equation”.

  • But we know that energy is created by nuclear fusion in the center of the sun. We therefore construct a new differential equation to relate the gradient of \(L_r(r)\) with the energy production epsilon. This relation is called the “energy conservation equation”.

Now we have a system of equation we can solve. In the next few lectures, we will discuss some caveats:

  • Is the energy always transported by radiation?

  • How can we find the energy production rates for nuclear fusion?

  • Is the ideal gas law always valid in the interior of stars?