# Increase in Luminosity of the star

The Luminosity of a star is proportional to the 4 th power of its temperature and square of its radius. Also the mass-luminosity relationship says that the luminosity of star is proportional to approximately M^3.5.

Consider a star in main sequence. It is in hydrostatic equilibrium so there is no collapse or expansion. Thus the Radius does not change much. Also the star is thermal equilibrium so the energy it generates equals the energy it radiates away. Thus the temperature must not change significantly. Moreover the star is losing mass as energy due to fusion. Now if we consider the mass luminosity relationship here, then why is there a slight increase in luminosity in star during its stay in the main sequence. If it's losing mass then it should lose its luminosity as well, right?

Please tell me where I am going wrong in my reasoning.

## 1 Answer

The luminosity mass relation is inexact. The luminosity also depends on the composition of the star, particularly in and around the nuclear burning regions.

The composition changes during the main sequence lifetime as hydrogen gets turned into helium. The average mass per particle goes up and the number of electrons per mass unit goes down. The former means the temperature increases to maintain a similar pressure, thus increasing the fusion rate. The latter decreases the opacity of the gas leading to a shallower temperature gradient and so a bigger star. A star like the Sun gets about twice as luminous over the course of its main sequence life.

The loss of mass due to radiation from the star's surface is negligible for the Sun, but more important in higher mass stars.

For the Sun, the mass loss rate due to this process is about $$6\times 10^{-14}$$ of its mass per year, so only about 0.6% of its mass is lost in this way over its main sequence life. The mass lost via the solar wind is probably a bit more important but the total is insufficient to counteract the compositional change effect. The radiation mass loss rate goes up as $$M^{3.5}$$ for higher mass stars, but of course their lives are also shorter by $$M^{-2.5}$$. The net effect is that the fraction of mass lost is proportional to the mass, so about 6% for a ten solar mass star. On its own this would only reduce the luminosity by about 20%; the compositional change effect is much stronger.

• FWIW, Wikipedia claims that the Sun's mass loss through radiation is nearly equal to the mass loss through the solar wind & CMEs. – PM 2Ring Oct 29 '20 at 9:28
• @Rob Jeffries Okay, so you mean the increase in the mean molecular mass dominates the decrease in Mass of the star and thus as a result of this the Luminosity of star increases very slowly in MS as compared to when it goes off the MS, right? – Dhruv Deshmukh Oct 29 '20 at 10:53
• @PM2Ring The Radiation mass loss rate is $3.83\times 10^{26}/c^2$ per second = $6\times 10^{-14}M_\odot$/yr. The current solar wind mass loss rate is a few $10^{-14}M_{\odot}$/yr, but this would have been an order of magnitude or more higher in the first Gyr of the Sun's life. My statement is correct. – ProfRob Oct 29 '20 at 12:30
• I agree that your answer is correct (as usual). And Wikipedia is only talking about current mass losses. And of course in the far future, when the Sun goes through its red giant phases, it will certainly shed a lot more mass through wind etc than through radiation. :) – PM 2Ring Oct 29 '20 at 12:37