How long will the Sun shine?

• Assume that the rate at which the Sun radiates energy is always the same as the value today:3.8410²⁶ W (watts are the same as joules s^-1, which is the same as joules/s or joules per second).
• All of the energy comes from the conversion of hydrogen to helium.
• The mass of the Sun is2.010³⁰ kg of which 1.510³⁰ kg is hydrogen. The mass of the Sun'score is about 10% of the total mass, of which there is 1.510²⁹ kg of hydrogen, so Sun's life will basically end when thecore hydrogen is consumed.
• If all of the hydrogen in the core is ultimately converted to helium, calculate the amount of mass which is converted into energy. Einstein is famous for his equationE = mc², wherem is mass andc is the velocity of light. If four hydrogen atoms are converted to one helium atom through the reaction 4¹H⁴He + e⁺ + 2, the mass loss is as follows (neglecting the neutrinos):

Mass of 4 hydrogen atoms 6.694110^-27 kg

Mass of 1 helium atom -6.646510^-27 kg

Mass of 2 positrons -0.001810^-27 kg

Mass loss 0.045810^-27 kg

a. 60 pts Convert this mass loss into energy, using Einstein's relationshipE = mc², wherem is the mass loss in kg,c is the speed of light (3.0010⁸m s^-1), andE is in joules (J; which in MKS units is kg m² s^-2).

30 pts Calculate the length of time needed to radiate E joules at the rate of 3.8410²⁶ watts. Watts are joules per second.

c. 10 pts Convert this to years, using 1 year = 3.15410⁷s.