Hydrostatic Equilibrium

The internal structure of stars cannot be directly observed (an exception is neutrino emission from the stellar core, since neutrinos have a hard time interacting with… anything else, they are not blocked by the upper layers of the stellar body), thus it must be described by computational models in agreement with physical laws and with observable features of the stars.

The stellar evolution is controlled by the equilibrium between gravitational force, compressing the star, and hydrostatic pressure, expanding it.

For spherically symmetric objects we have the following condition for hydrostatic equilibrium:

hydrost-eq

If the acceleration can be neglected (which is the case if changes in the star happen over very long times -wouldn’t be the case for pulsing stars with periodic changes of only a few hours), then:

hydrost-eq-noacc

Note that a pressure gradient must exist to support the star (i.e. not just a pressure). The larger r is (r = distance from the center), the smaller is P.

Mass conservation equation:

masscons

Look at these differentials of mass (dM), pressure (dP), and distance (dr). To get a better idea of what we’re talking about, this is the model that we’re using:

star-model

Such pressure can be expressed as:

P-gas

This is simply the ideal gas equation, with a Maxwell-Boltzmann distribution of velocities for the particles. Since it is the pressure of an ideal gas, we may call it Pg.

Now, there are two kinds of particles in the universe: fermions (what we usually call matter, e.g. protons, electrons, neutrons, atoms, molecules, etc.) and bosons (force-carriers, e.g. photons).

If we had to take in account the Heisenberg Uncertainty Principle and the Dirac Exclusion Principle (roughly speaking, fermions don’t want to be similar to each other, or a bit more technically, two fermions cannot be simultaneously in a identically equal state), we shouldn’t take the Maxwell-Boltzmann (MB) distribution, but the Fermi-Dirac distribution. For photons, likewise, we cannot use the MB distribution, we need the Bose-Einstein distribution instead.

However, this only makes a difference in cases of extremely dense matter, e.g. white dwarfs, or neutron stars. For most of the cases (sufficiently low densities and velocities), we do just fine with the MB distribution.

Still, the radiation pressure Prad, the pressure exercised by photons, must be taken into account. Assuming blackbody radiation, this can be calculated as a function of T as:

p-rad

Remember that σ is the Stefan-Boltzmann constant.

Thus, the total pressure Pt, is:

p-tot

Here µ is something called mean molecular weight. It is simply a mass weighting factor for expressing the average mass of a gas particle (averagemass) in units of the mass of a hydrogen atom (mH):

meanmolecweigth

The radiation pressure can be larger than the pressure of the ideal gas, and even larger than the pressure caused by gravity (in which case, the star would be expanding).

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One Response to Hydrostatic Equilibrium

  1. Pingback: Energy Sources and Luminosity | The Quantum Red Pill Blog

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