Mass-Luminosity Relationship

The Mass-Luminosity relationship, interestingly enough, can be roughly dervied without specifying how stars generate energy! It turns out that the form of the Mass-Luminosity relation depends upon how energy is moved around inside of a star, i.e., it depends upon how easily energy can flow from the core of the star to the surface of the star. To get a feel for why this comes about, we will perform a rough and ready analysis of the energy flow between regions of differing temperatures.

• Region 1 has T1 and Region 2 has T2 where T1 > T2 and so

  energy flows from Region 1 ===> Region 2

• The hotter (more energetic) region 1 mixes with the lower temperature (lower energy) material in region 2 and heats it. This energy mixing leads to the transport of energy.

• The rate at which energy moves (is mixed) is
  • Flux ~ - c x l x dE/dR

  Now note that:

  • Flux ~ L/R**2
  • l = distance traveled before interacting ~ 1/(opacity x density)
  • Radiation energy ~ T**4 and so dE/DR ~ T(central)**4/R

    We assumed that the star only produces energy near its center from where the energy then simply works its way outward to the surface of the star.

• Substituting these things into the Flux relation leads to
  • Flux ~ L/R**2 ~ T**4/(opacity x density x R) ~ (M**4/R**4) x (R**2/(opacity x M))

    So that we have

    L ~ M**3/opacity !!!!

  • That is, depending upon the opacity, how strongly the photons interact with matter, we have a Mass-Luminosity relation which is not too bad when compared to the empirical one.

    We arrived at this relation without having to specify how the star produced energy, we only required that the star produce energy the bulk of its luminosity in its core.