Śrīdhara Brāhmaṇa: Chandrasekhar Limit

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I absolutely loved this sentence from the Epilogue of Cox and Forshaw’s The Quantum UniverseEverything that can Happen does Happen (2012):

We could present a very broad overview of how Chandrasekhar mass comes about, but instead we’d like to do a bit more: we’d like to describe the actual calculation because that is what really makes the spine tingle.👍

I simply could not resist following them along, and extracting the essence of it. Here we go!

Steps 1 and 2: Lay out the notation and state the simplifications assumed.

Step 3: Calculate the Electron Degeneracy Pressure for Relativistic Speed of the electron, rather than for the “low velocity” regime (as was done by Fowler), but take this important fork where Special Relativity is invoked.  This last step fundamentally changes the asymptotic formula for pressure: the exponent becomes 4/3 instead of 5/3. (Try it out like Fowler by replacing momentum p with mv, following Newton’s Classical Mechanics rather than Einstein’s Special Relativity.)

  Savor how Pauli Exclusion Principle and Heisenberg Uncertainty Principle are used in such a straightforward manner.

Step 4: Stick with Newton and Gauss to calculate Gravitational Attraction. (No General Relativity.)

Step 5: See what happens at Tipping Point when the relentlessness of gravity overpowers the maximum pushback that electrons can provide.

The Nobel Prize in Physics 1983 was divided equally between Subramanyan Chandrasekhar “for his theoretical studies of the physical processes of importance to the structure and evolution of the stars” and William Alfred Fowler “for his theoretical and experimental studies of the nuclear reactions of importance in the formation of the chemical elements in the universe”.

Of course, this does not mean a Black Hole will be formed after the Tipping Point. As the gravitational attraction compresses the star, the protons and electrons fuse and become (new) neutrons, and join the existing neutrons, and we get a neutron star. The neutrons also obey Pauli Exclusion Principle and so pushback with neutron degeneracy pressure. But again, this also has limits, as the agitation of the neurons as they get confined – due to Heisenberg Uncertainty Principle – reaches its maximum, due to Einstein’s Special Relativity. Yes, at some point, gravity wins and….we get Not-so Black Holos. 😏 In the opposite direction, if a neutron star loses mass somehow, all that pent-up pressure explodes into a Supernova.

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