The sun has a mass of 1.99E30 kg and is located 1.5E11 m from the earth. The ave

Question

The sun has a mass of 1.99E30 kg and is located 1.5E11 m from the earth. The average energy per unit time per unit area that reaches the earth’s upper atmosphere from the sun is 1.35 kW/m2. This energy is released by the sun in a series of fusion reactions that fuse four hydrogen nuclei together to form a single helium nucleus. This process releases 4.27 E- 12 J per helium nucleus created. If we assume the sun to be composed of hydrogen and that it will continue to ‘burn’ until it has consumed 10% of the available hydrogen, estimate the lifetime of the sun. Answer in units of years. Hint: The surface area of a sphere is 4r2. Work with units to see how to proceed.

Explanation / Answer

We know that,

Mass of the sun= Ms = 1.99 X 1030 kg

Mass of the hydrogen atom= Mh = 1.67X10-27kg

Total no of hydrogen atom present in the sun

= Ms/Mh= 1.19X 1057 kg

Total no of hydrogen atoms consumed by the sun in its entire life time = 10% of the total hydrogen atoms

= 10% of 1.19X 1057

= 1.19X 1056

Total energy released by 4 hydrogen atoms= 4.27X 10-12J

So for 1.19X1056 hydrogen atoms energy released will be =

4.27X 10-12 X 1.19X1056/4 = 1.27 X1p44 J = E (say)

Sun released this energy in all 4pi direction uniformly.

If radius of Earth is = re then,

earth receives ( pi re2)/( 4pi r2) of the total energy.

r= distance between earth and sun.

Now, area of the earth = 4 pi re2

So total power received by earth =

E X (pi re2)/(4 pi r2) X (1/(t X 4pi re2)

Where,

E=total energy released by the sun during its entire life time

t= life time of the sun

So, power = 1.35 X 103 watt /m2

Putting value of E we get,

t=2.64X 109 years

Which is the life time of the sun.

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