Energy Transport

Energy transport inside a star takes place in three ways:

  • Radiation: dependent on opacityflux, density, and temperature. Larger opacity and flux involve steeper decrease in T with increasing r.
    radiative-transport
  • Convection: cool masses move “down” (in the direction of gravitational pull) and hot masses move “up” (against gravitational pull). If the T gradient is too steep, convection can play an important role.
  • Conduction: generally negligible.

We look more closely at convection.

  • Its treatment is very complex, and the theory is not yet fully developed.
  • It relies on Navier-Strokes equations in 3-D.
  • For the study of stellar structure, usually we use 1-D approximations (only spatial variable is r, i.e. we impose spherical symmetry).
  • The pressure scale height (a characteristic length scale for convection) is in the same order as the size of the star, thus convection is coupled to star’s behavior.

The model that we use is that of a hot bubble which (1) rises and expands adiabatically, and then after travelling some distance, (2) it thermalizes with the medium.

This is a thermodynamic model based on state functions (changes in thermodynamic state functions depend only on the starting and final states of the system, and not in the way followed).

The following equation describes how the temperature of the gas inside the bubble changes as the bubble rises and expands adiabatically:

convective-transport

heat-ratio

CP and CV are the specific heat capacities at constant pressure and constant volume respectively.

Superadiabaticity: Case that the star’s actual temperature gradient is steeper than the adiabatic temperature gradient (i.e. inside the bubble). It represents a condition for convection domination:

superadiabatic

 

If “act” is larger than “ad”, all luminosity is carried by convection.

It can be alternatively written as (assuming that the µ -mean molecular weight- does not vary):

superadiabatic-2

 

So, either radiation or convection dominate in different circumstances. When is convection favored?

  1. Conditions of large stellar opacity
  2. Ionization taking place, which means large specific heat, and low adiabatic T gradient.
  3. Low local gravitational acceleration
  4. Large T dependence of nuclear reactions (e.g. CNO cycle and triple alpha)
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