Question

compute the timescale needed for the sun to have contracted from a very large size down to todays size

Answer 1

Let us assume that the Sun is shrinking by gravity. Then from the equation that scientists have for the change of the Sun's luminosity (luminosity is an energy output) versus its radius, the Sun would be shrinking in its radius 74 centimeters per year. We would have detected such a noticeable change over the past history (over 500 years this would be a 0.005 arc seconds difference in the radius of the Sun from our viewing position on the Earth), and we haven't detected such a change. So our observations don't show the Sun to be shrinking by gravitational contraction.

What about the Sun's mass becoming less by its process of producing energy (fusion)?

The Sun actually does lose mass in the process of producing energy. Let us see how much.

We can use the following numbers from Kenneth R. Lang's book: _Astrophysical Data_:

Solar Mass = 1.989 x 1033 g
Absolute luminosity = 3.86 x 1033 erg/sec
Speed of light c   = 2.99 x 1010 cm/sec

Start with Einstein's famous equation: "E = mass times c²" and rearrange the terms to solve for the mass M:

M = E/c²

And after inputting our numbers:

        = 3.86x1033/(2.99x1010)2 
        = 4.289x1012 g/sec

we find that the Sun loses mass 4.289x10¹² g every second to energy. Or, in other units, the Sun loses mass 1.353x10²⁰ g every year to energy.

The Sun is thought to have a remaining lifetime of about 5x10⁹ years. If we assume that the Sun's rate of fuel consumption (the luminosity value given above) remains constant (it won't, but it isn't a bad assumption) in the remaining time of 5x10⁹ years, then let us see how much mass the Sun will convert to energy in its remaining lifetime.

Mass = (1.353x10²⁰ g/year) * 5x10⁹ years = 6.8 x 10²⁹ g

In units of tons, every second, the Sun's fusion processes are converting about 700 million tons of hydrogen into helium "ashes". In doing so, 0.7 percent of the hydrogen matter (5 million tons) disappears as pure energy.

(My reference for this paragraph is "The Sun" chapter in _The New Solar System_ editor: Beatty and Chaikin, Sky Publishing Press.)

Since the Sun's current mass is 1.989 x 10³³ g, the percentage of its current mass that will be converted to energy is:

6.8 x 10²⁹ g / 1.989 x 10³³ g = 0.00034 of its current mass or .034 percent.

In other words, the Sun's mass at the end of its lifetime is 99.966% of its current mass.

to answer your question we can do back calculation. For that we have to take some assumption about the initial mass of the sun, As we have shown that the Sun loses mass 4.289x10¹² g every second to energy. Or, in other units, the Sun loses mass 1.353x10²⁰ g every year to energy. So we can compute time for the sun to come down in todays size.