Virial Equilibrium of Adiabatic Spheres (Summary)

The summary presented here has been drawn from our accompanying detailed analysis of the structure of pressure-truncated polytropes.

Figure 4: Equilibrium Adiabatic P-V Diagram

The curves trace out the function,

<math> \Pi_a = (\chi^{4-3\gamma_g} - 1)/\chi^4 \, , </math>

for six different values of <math>\gamma_g</math> (<math>2, ~5/3, ~7/5, ~6/5, ~1, ~2/3</math>, as labeled) and show the dimensionless external pressure, <math>\Pi_a</math>, that is required to construct a nonrotating, self-gravitating, uniform density, adiabatic sphere with an equilibrium radius <math>\chi</math>. The mathematical solution becomes unphysical wherever the pressure becomes negative.

The solid red curve, drawn for the case <math>\gamma_g = 1</math>, is identical to the solid black (isothermal) curve displayed above in Figure 1.

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Virial Equilibrium of Adiabatic Spheres (Summary)

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