## Solar luminosity evolution

5.4.1 The role of mean molecular weight

For the Sun we can assign an effective blackbody temperature, Tq, so that its luminosity is given by

where Rq is its radius. For a dwarf star like the Sun its luminosity during its MS evolution varies approximately as

where j is the solar mean relative molecular weight with respect to the proton mass that, as we shall see, controls the luminosity on timescales of billions of years. The mean molecular weight needs to be calculated with some care as it is not simply a weighted mean of the molecular weight of hydrogen and helium, the main solar constituents. In the interior of the Sun these constituents are fully ionized, thus electrons are a component of the particles that determine the mean molecular weight.

The density, p, of the solar plasma can be written as p = nm = njmp, (5.16)

where n is the number density of the particles and m is the mean mass of the particles, so that the relative mass with regard to the proton mass can be written as

mp where mp is the mass of the proton. The hydrogen density can be written as

and similarly for pHe, with XH + XHe = 1, (ignoring other minor elements) and X represents the fraction of each constituent by mass. The electron number density is then ne = nH +2nHe (5.19)

since atomic hydrogen has one electron and helium has two. Thus the total particle number density is n = 2nH + 3nHe (5.20)

which then can be expressed as n = 2(XHp/mp) + 3(XHeP/4mp) (5.21) on replacing p using eqn (5.16), we get