E=mc² and the Sun
June 3, 2007
Roscoe, N.Y.
In response to last Thursday's blog entry I received an email from someone who contended that the Sun couldn't have been producing energy for more than 10,000 years.
Deriving the relationship between the sun's mass and its potential energy production is simple and fun because you get to use the most famous equation in physics.
According to the Wikipedia entry on the Sun, the Sun generates 1,370 watts of power per square meter of surface area at the earth's distance from the Sun.
The distance between the Sun and the earth is approximately 1.5 × 10^{11} meters. If you consider this distance as the radius (r) of a sphere, the surface of the sphere is 4πr^{2} or about 2.8 × 10^{23} square meters. That means the total radiated power of the Sun is 1,370 times that or about 3.9 × 10^{26} watts.
The watt is a unit of power equivalent to one joule of energy per second. (A joule is the energy expended by a force of one newton over one meter. A newton is the force required to accelerate a mass of one kilogram by one meter per second squared. The units of the newton are kilograms meters per second squared, and the units of the joule are kilograms meters squared per second squared.) Every second the sun produces energy of 3.9 × 10^{26} joules.
The Sun produces energy through a nuclear reaction. Two atoms of hydrogen fuse to form a helium atom. The slight loss in mass is converted to energy. This relationship can be quantified by the famous formula:

E=mc^{2}
where E is energy, m is mass, and c is the speed of light. The speed of light is about 300,000,000 meters per second, and the speed of light squared is 9 × 10^{16} meters squared per second squared. Divide the Sun's energy production by this value and you get about 4.3 × 10^{9} kilograms per second. (This figure roughly agrees with the Wikipedia entry that says "4 million tonnes" or 4 trillion kilograms of the mass of the Sun are converted to energy every second.)
So, the Sun's mass is reduced by 4.3 × 10^{9} kilograms every second. That's 2.6 × 10^{11} kilograms per minute, 1.5 × 10^{13} kilograms per hour, 3.7 × 10^{14} kilograms per day, and 1.4 × 10^{17} kilograms every year.
The mass of the Sun is about 2 × 10^{30} kilograms, and a simple division by the rate of 1.4 × 10^{17} kilograms per year indicates that the Sun has enough mass to generate energy for 14,000,000,000,000 years. However, that figure assumes that all the mass of the Sun will be converted to energy, and that won't happen. The lifetime of the Sun in its main sequence is probably more like 10,000,000,000 years.
Over the past 10,000 years, the mass of the Sun has been reduced by about 0.00000007%.