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Gravitational Potential Energy

We can obtain some useful insights into the energy released during a supernova explosion, by considering the release of gravitational potential energy.

Gravitational potential energy is defined as P="-GMm/r" , where m is the mass of a particle, at a distance r, from a large object of mass M; G is the gravitation constant.

We can see that P is always negative and becomes more so the closer we are to the attracting body. This should not be seen as a problem as it conforms to our experience that to launch a rocket into space and escape from the Earth's 'potential well', work must be done and energy expended. It also means that far from the attracting body the gravitational potential energy becomes zero.

To experience the gravitational potential energy of the Earth tie a string to a brick and lift the brick off the ground. This requires some effort and the expenditure of a certain amount of energy. Raising the brick off the ground increases the bricks gravitational potential energy compared with its value on the ground. The potential energy can be converted back into a more obvious form of energy by dropping the brick. If, instead, we let the brick fall while the string runs through our fingers then the resulting friction burn should convince anyone of the reality of gravitational potential energy!

When an object is released above the ground, it accelerates in response to the gravitational force of the Earth and its potential energy is converted into kinetic energy. When the brick strikes the ground the kinetic energy is converted, partly into the sound of the impact, but mostly into heat, slightly warming the ground and the brick. The most important principles of physics are the conservation laws. This means that as the brick falls its increase in kinetic energy is exactly balanced by its decrease in gravitational potential energy.

It is possible to calculate the amount of energy required to completely take the Earth or the Sun apart and move all the bits very far away (to infinity). The amount of energy required to do this is identical to the energy released by assembling the Sun from a large gas cloud, which is the way the Sun formed. The equation giving the amount of energy is very simple (omitting a factor of 0.6 for a sphere of uniform density)

Some units

Solar mass 2.0x1030 kg
Gravitational Constant, G 6.67x10-11 N m2 kg-1
Solar luminosity 4.0x1026 J s-1
Age of Sun 4,560 million years
Age of universe 14,000 million years

Gravitational potential energy released by one solar mass collapsing from infinity

Final Radius Energy Released Years of Sunshine
Sun 690,000 km 3.9x1041 J 31 million
White dwarf 6,000 km 4.4x1043 J 3,500 million
Neutron star 10 km 2.6x1046 J 2 million million
Rotating black hole 1.5km 1.8x1047 J 14 million million

Other sources of energy for comparison

Exploding 1 solar mass of TNT 8.2x1036 J 651
Converting 1 solar mass of hydrogen to helium 1.2x1045 J 100,000 million
Converting 1 solar mass of helium to iron 3.1x1044 J 25,000 million
Converting 1 solar mass to energy 1.8x1047 J 14 million million

In order to assemble the Sun from its initial cloud of gas we see that the equivalent of 31 million years worth of present-day solar radiation will be released. Some of this energy will have been radiated away by the steadily growing Sun and some will have gone into raising its internal temperature. The temperature of a gas is just another measure of the kinetic energy of the gas molecules. The higher the temperature the faster the motion of the gas molecules and the greater is their kinetic energy.

The fact that gravitational potential energy would only power the Sun for 31 million years highlights one of the problems faced by Victorian astronomers, before the discovery of nuclear energy, when they realised that the Solar system was much older than 31 million years. It is interesting to see that even in forming the Sun the release of gravitational potential energy, far exceeds the release of chemical energy. One solar mass of the high explosive TNT could only power the Sun for 651 years.

As a star evolves and becomes a giant it creates a dense core that is in many respects similar to a white dwarf. The potential energy released is now comparable to the energy so far released by the Sun. The star is however far more luminous than the Sun at this stage of its life so gravitational potential energy is still not of major importance.

During a Type II supernova explosion an iron core of mass and size comparable to a white dwarf collapses to become a neutron star in about 0.1 seconds. The gravitational potential energy released is equal to the difference between that taken to form the white dwarf and that taken to form the neutron star. We now see that the amount of gravitational potential energy released far exceeds the nuclear energy which would be produced if the entire Sun was converted from hydrogen to helium. Although the energy released is much greater than seen in the light coming from the supernova it is comparable to the total energy of the explosion, which is mostly released as neutrinos.

As a matter of interest, you will see that the gravitational potential energy released forming a black hole is equal to the entire mass energy of the Sun, given by E="Mc2." More exact theory shows that the maximum energy released by matter falling into a rotating black hole is one half this value. This still makes feeding a black hole the most efficient method known of generating energy.

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