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.
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.