# 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

If we take a value XH = 0.80 for the young Sun, which is an estimate of the initial solar nebula composition based on the H composition by mass of the large planets, Jupiter and Saturn, then j = 0.57 when the Sun began to convert H to He. If we adopt a present-day value of XH = 0.734 for the H composition by mass (Table 5.3), corresponding to j = 0.60, then we see that the luminosity of the young Sun would have been about 72% that of today's value, assuming that the mass was the same and the radius was about 0.9 of its present value (see next section). This of course has ramifications for the Earth's early climate. Simple climate models give a 1% change in the global mean surface temperature of the Earth for a 1% change in the solar luminosity. This implies that the Earth's mean temperature would have been 28 K cooler than today. If we take a present global mean surface temperature of 288 K then the early Earth would have been frozen as its mean surface temperature would have been 260 K. However, there is geological evidence (sedimentary rocks) dating back to 4 Byr ago, suggesting the presence of water, and hence we have what is known as the faint-young-Sun paradox. We note that the age of the Earth is estimated to be about 4.5 Byr, and so it took about 100 Myr for the Earth to reach its present mass. The paradox is usually resolved by a CO2-richer primitive atmosphere than today's (see §12.2). 