Tilak of SDM College points out that there is a continuous flow of energy from the Sun, and that needs an expenditure of some mass as a result, and thus the gravitational pull of the Sun should decrease and all the planets should move away from the Sun; is this correct?

There is an extremely useful technique in science (and especially so in astronomy) called the order-of-magnitude estimate, or more informally, the back-of-the-envelope calculation. Let’s try that here. The luminosity of the Sun is 4 10^{33} ergs/s. This energy is radiated away, and clearly originates from the nuclear fusion reactions going on in the core, so it is fair to compute how much this is in mass. Dividing by the square of the speed of light, *c*=3 10^{10} cm/s (because *m=E/c*^{2}), the rate of mass loss is approximately 4 10^{12} gm/s, or a million metric tons a second. That sounds like a lot, so let us compare it to the mass of the sun, which is 2 10^{33} gm. So the Sun is losing roughly 2 10^{-21} of its mass every second, or about 6 10^{-14} of its mass every year. Over its entire lifetime of 10 billion years == 3 10^{17} s, this amounts to a fraction of 6 10^{-4} of the total mass. Less than one tenth of one percent. Quite negligible.

As a matter of fact, the Sun also loses mass directly, via the solar wind. It loses 10^{-14} of its mass every year this way. As already demonstrated above, this is a tiny fraction of the available mass, and won’t amount to a hill of beans in the big picture.

That said, I must point out that there are some types of massive stars that have significantly higher mass loss rates due to the wind, sometimes as high as 10^{-5} solar masses a year. Such stars can lose a significant fraction of their mass over their lifetimes, and this must be properly accounted for in stellar evolution models.