Question

In your own words explain the meaning of 2EK + Ω = 0 and its importance for understanding stellar evolution. The meaning of each symbol must be fully explained, and you should indicate when the equation is valid.

Answer 1

This is a non relativistic form of the virial theorem, which establishes a relation between gravitational potential energy ( Ω) and thermal kinetic energy (EK).

The virial theorem states that, for a stable, self-gravitating, spherical distribution of equal mass objects (stars, galaxies, etc), the total kinetic energy of the objects is equal to minus 1/2 times the total gravitational potential energy. i.e.

  

As stars on the main sequence convert hydrogen into helium in their cores, the mean molecular weight of the core increases and it must contract to maintain enough pressure to supports its own weight. This contraction decreases its potential energy and, the virial theorem states, increases its thermal energy. The core temperature increases even as energy is lost, effectively a negative specific heat. This continues beyond the main sequence, unless the core becomes degenerate since that causes the pressure to become independent of temperature and the virial relation with n equals −1 no longer holds.

Consider a planet of mass m, such as the Earth, in a circular orbit around a star of mass M at a distance R. The planet’s orbital velocity is the Keplerian velocity represented by .

Therefore Kinetic Energy K = 1/2 * mv² = GMm/2R and potential energy is Ω = -GMm/R

therefore it is evident that

therefore

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