Why can we say that the rate at which the sun gives off energy at the surface (the luminosity) must be equal to the rate at which it produces energy deep down in the core?
So, measuring the luminosity of the sun is equivalent to measuring the rate at which the nuclear reactions produce energy in the core of the sun. But we think we know exactly how those reactions work:
4 H atoms -> 1 He atom + energy
where the energy is released because some of the mass of the Hydrogen atoms is converted to energy.
Mass of 1 Hydrogen atom: 1.673 x 10-24 grams
Mass of 1 Helium atom: 6.644 x 10-24 grams
This nuclear reaction has an input (4 H atoms) and two outputs : 1 He atom and energy. Since the He atom has less mass than 4 H atoms, that difference in mass must have been converted to energy. If the energy is produced via E=mc2, how much energy is produced by producing one He atom from 4 H atoms? (see also the units notes at the bottom). (Hint: How much mass is converted into energy?)
So, for every 4 hydrogen atoms fused, we get the amount of energy you calculated above. But we know how fast the sun has produced energy (the luminosity, according to #1), so we know how fast the hydrogen fuel in the core of the sun is being used up.
If the sun gives off 3.89 x 1033 ergs every second, how many hydrogen atoms are being destroyed every second?
The sun will remain a main sequence star until it runs out of hydrogen fuel in the core of the star. The core of the sun contains about 10% of the total mass of the star.
Why are these reactions confined to the core?
The total mass of the sun is 2x1033 gm. How long will it take until the sun has used up all the hydrogen atoms in the core (the central 10%)? That is, what is the main-sequence lifetime of the sun?
Compare the lifetime of the sun to the current age. How soon will the sun running out of fuel be a problem?