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Polytropic Stars

In physics we develop insight by distilling a problem down to its essence. For stellar structure, this essence is found by expressing the pressure within a star in terms of density alone and solving the resulting equation of hydrostatic equilibrium. This type of model is referred in the scientific literature as a polytropic stellar model. In this type of analysis, we are ignoring the details of how energy is transported out of the star. Temperature is present, but only implicitly.

The advantage of this type of analysis is that we find a simple solution for the stellar density with radius that is not too far off from the results found by solving the complete set of equations that describe the structure of a star. Because of this, polytropic stellar models are used as the initial models in computer codes that recursively solve for stellar structure. We also learn from this type of analysis that not all physically realized equations for pressure permit a static stellar solution—this points to the mechanism that drives core collapse in dying stars, which leads to supernovae and the creation of neutron stars and black holes.

The polytropic pressure law used in defining a polytropic stellar model is

P = K ργ,

where P is the pressure, ρ is the denisty, K is a constant, and γ is the adiabatic index.

There is a tremendous amount of physics buried in the adiabatic index. If we consider the case of a fully ionized plasma that is at the stability limit to convection, we find a pressure law with γ = 5/3. If we consider a star that is pressure supported in part by radiation, with the fraction of pressure provided by radiation held constant, we find that γ = 4/3. These two values mark the most general range of values seen in real stars. These two value are also bound the range of γ for degenerate gases, which are gases of low enough temperature that the effects of quantum mechanics, particularly the effects of the Pauli exclusion principle, determine the pressure-density relationship. For a non-relativistic degenerate gas, γ = 5/3, and for a relativistic degenerate gas, γ = 4/3. But while 5/3 and 4/3 mark the bounds of the most general range of values seen in real stars, there are circumstances when γ can go much lower, below the critical value of 1.2, for example, when a gas is becoming ionized.

This plot gives the solution of the polytropic equation in terms of either density or ϕ as a function of radius, which is expressed either in terms that keep the stellar mass constant or as the variable ξ. The adiabatic index in a table can be changed by the user by clicking on a cell with the mouse. The index must be a value between 1.2 and 2. More information on how to control the applet is given by the Applet Control Guide.

The figure on this page shows the density as a function of radius for four different values of γ. With the radio buttons set to “Density” and “Fixed Mass,” which are the default settings, the plot shows the density in units of central density as a function of normalized radius for stars of a single mass (the units of radius are explained at the end of this page). The primary point to notice is that there is very little difference among these models for the core density. Most of the mass is confined within about the same radius, regardless of the value of γ. Where the differences occur is at the outer edge of each star. For stars with a large value of γ, the density drops rapidly to 0. As γ becomes smaller, the core of the star becomes slightly smaller while the envelope of the star becomes much larger. At the value of γ goes to 1.2, radius of the star's surface goes to infinity, even though the mass of the star remains constant; this is best seen in the figure by plotting ϕ, which is related to density by the equation below. Stars with γ ≤ 1.2 have no static solution.

From these solutions, we see that when a star maintains a particular adiabatic index as its core shrinks, the outer surface of the star also shrinks by the same factor. But if the adiabatic index of a star becomes smaller as the star's core shrinks, the radius of its surface may increase. This happens when a star moves from the main sequence into the giant phase; the core shrinks until the nuclear fusion of helium commences, and the radius of the star expands.

The absence of a hydrostatic solution for γ ≤ 1.2 is interpreted as a condition for stellar collapse. This is the orgin of core collapse in massive stars that have burned most of their nuclear fuel. As the core of such a star shrinks and becomes hotter, atoms at the core disintegrate and combine with electrons to form free neutrons, causing the adiabatic index to drop below 1.2. The core of such a star will collapse until either a state is reached with γ > 1.2, which forms a neutron star, or the core forms a black hole.

A similar situation is encountered in protostars of un-ionized gas. Once the core temperature reaches a temperature that allows ionization, the adiabatic index drops below 1.2, and the protostar collapses until the whole star ionizes, returning the adiabatic index to a value above 1.2.

Figure Notes

Three pieces of physics enters into the derivation of the polytrope equation: the hydrostatic equation, which is the equation for pressure balancing gravitational force, the polytropic pressure equation, which is given above, and the mass of the star inside a given radius. The equation that one finds is a nonlinear second-order differential equation. Rather than being written as an equation of density versus radius, the equation is written as an equation for ϕ in terms of ξ, where ϕ is related to density by

ρ = ρ0 ϕ1/( γ - 1 ),

and ξ is related to radius r by

r = a ξ.

The term a in this equation is a function of mass, central density, and adiabatic index.

With the “Fixed Mass” radio button selected, the the radius in the diagram is normalized so that each curve represents the same mass and the curve for γ = 1.2 has a = 1.

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