What, temperature would the Sun need to be if all gas particles had the same vel

Question

What, temperature would the Sun need to be if all gas particles had the same velocity and there were no quantum tunneling? Remember that the strong nuclear force takes over at r TildeTilde;10^13 cm What is the ratio of this temperature to the temperature we find for the Sun's central temperature (10^7 K) If the Sun were really at the temperature that you found, but quantum tunneling worked and not all particles have the same velocity, how much brighter would the Sun be than now? Express your answer as a ratio

Explanation / Answer

The sun would continue to shrink , releasing gravitational potential energy in the form of radiation and by heating the core.

The virial theorem tells us that the central temperature is roughly proportional to M/R (mass/radius). So for a fixed mass, as the star contracts, its core gets hotter.

There are then (at least) two possibilities.

The core becomes hot enough for protons to overcome the Coulomb barrier and begin nuclear fusion. For this to happen, the protons need to get within about a nuclear radius is 10^15m.

The potential energy is

e2/(40r)=1.44 MeV or 2.3×10^13 J.

The protons in the core will have a mean kinetic energy of 3kT/2, but some small fraction will have energies much higher than this according to a Maxwell-Boltzmann distribution.

The fusion will take place when protons with energies of 10kT exceed the Coulomb potential energy barrier. There will be a small numerical uncertainty on this, but because the reaction rate would be highly temperature sensitive it will not be an order of magnitude out.

This means that fusion would not begin until the core temperature reached about 1.5×10^9K.

2. Ratio = 1.5 * 10^9 / 10^7

= 150 : 1

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