The Structure, Stability, and Dynamics
of Self-Gravitating Systems

Joel E. Tohline
tohline@rouge.phys.lsu.edu

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As was stated earlier, for nonrotating polytropic configurations the multidimensional statement of hydrostatic equilibrium is:

Taking the divergence of both sides of this expression we deduce, quite generally, that

Replacing the right-hand-side of this equation with the expression for Ñ2F that is provided by the Poisson equation [I.D.1], we derive

Now, enforcing spherical symmetry in the same manner as was done earlier, we deduce that for spherically symmetric polytropes,

When supplemented by appropriate boundary conditions, this single equation is sufficient to define the equilibrium structure of nonrotating, spherical polytropes.

If we define a dimensionless structural variable of the form

where rcentral is the central density of the star, the preceding equation [III.A.16] can be written as:

Now, the leading coefficient in this expression has units of length-squared, so it is natural to define a dimensionless radius of the form

Replacing r by an×x (and dr by an×dx) everywhere in the above expression for hydrostatic equilibrium [III.A.18], we derive what is familiarly known as the

i.e., r = r_central at the center of the star, and

i.e., there should be no cusp in the polytropic function (or in the density distribution) at the center of the star.

What is the relationship between the Lane-Emden function Q and the enthalpy H of a polytropic structure?

As Chandrasekhar (1967) has pointed out,¹ one method of constructing the Lane-Emden function would be to start with a series expansion near the origin. We assume a series of the form,

(The leading term in the series has been set equal to 1 in order to satisfy the boundary condition [III.A.22].) By substituting the foregoing series in the Lane-Emden equation and equating the coefficients of like powers in x, we can successively determine the coefficients C1, C2, C3, etc. For example, up to order x2, the left-hand-side of the Lane-Emden equation [III.A.21] becomes,

and the right-hand-side becomes,

Clearly from these two expressions, C₁ = 0, C₂ = -1/6, C₃ = 0, and C₄ = n/120. Hence, up to order x⁴,

With a little additional thought it also becomes clear that all coefficients of odd powers of x must be zero and, among the coefficients of even powers of x, only C₂ is not a function of the polytropic index n.

Derive the correct expressions for C6 and C8 in the series expansion for Q [III.A.24], and demonstrate explicitly that C5 = C7 = 0.

Read the accompanying discussion to see how a variety of physical properties of spherical polytropes may be calculated once the function Q(x) has been determined for a specified polytropic index n.

¹Text in green is taken verbatim from Chapter IV, § 5 of Chandrasekhar 1967.