Lane–Emden equation

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Solutions of Lane–Emden equation for n = 0, 1, 2, 3, 4, 5.

In astrophysics, the Lane–Emden equation is a dimensionless form of Poisson's equation for the gravitational potential of a Newtonian self-gravitating, spherically symmetric, polytropic fluid. It is named after astrophysicists Jonathan Homer Lane and Robert Emden.[1] The equation reads

 \frac{1}{\xi^2} \frac{d}{d\xi} \left({\xi^2 \frac{d\theta}{d\xi}}\right) + \theta^n = 0

where \xi is a dimensionless radius and \theta is related to the density (and thus the pressure) by \rho=\rho_c\theta^n for central density \rho_c. The index n is the polytropic index that appears in the polytropic equation of state,

 P = K \rho^{1 + \frac{1}{n}}\,

where P and \rho are the pressure and density, respectively, and K is a constant of proportionality. The standard boundary conditions are \theta(0)=1 and \theta'(0)=0. Solutions thus describe the run of pressure and density with radius and are known as polytropes of index n.

Applications[edit]

Physically, hydrostatic equilibrium connects the gradient of the potential, the density, and the gradient of the pressure, whereas Poisson's equation connects the potential with the density. Thus, if we have a further equation that dictates how the pressure and density vary with respect to one another, we can reach a solution. The particular choice of a polytropic gas as given above makes the mathematical statement of the problem particularly succinct and leads to the Lane–Emden equation. The equation is a useful approximation for self-gravitating spheres of plasma such as stars, but typically it is a rather limiting assumption.

Derivation[edit]

From hydrostatic equilibrium[edit]

Consider a self-gravitating, spherically symmetric fluid in hydrostatic equilibrium. Mass is conserved and thus described by the continuity equation

 \frac{dm}{dr} = 4\pi r^2 \rho

where \rho is a function of r. The equation of hydrostatic equilibrium is

 \frac{1}{\rho}\frac{dP}{dr} = -\frac{Gm}{r^2}

where m is also a function of r. Differentiating again gives

\begin{align}
\frac{d}{dr}\left(\frac{1}{\rho}\frac{dP}{dr}\right) &= \frac{2Gm}{r^3}-\frac{G}{r^2}\frac{dm}{dr} \\
&=-\frac{2}{\rho r}\frac{dP}{dr}-4\pi G\rho
\end{align}

where we have used the continuity equation to replace the mass gradient. Multiplying both sides by r^2 and collecting the derivatives of P on the left, we can write

 r^2\frac{d}{dr}\left(\frac{1}{\rho}\frac{dP}{dr}\right)+\frac{2r}{\rho}\frac{dP}{dr}
==\frac{d}{dr}\left(\frac{r^2}{\rho}\frac{dP}{dr}\right)==-4\pi Gr^2\rho

Dividing both sides by r^2 yields, in some sense, a dimensional form of the desired equation. If, in addition, we substitute for the polytropic equation of state with P=K\rho_c^{1+\frac{1}{n}}\theta^{n+1} and \rho=\rho_c\theta^n, we have

 \frac{1}{r^2}\frac{d}{dr}\left(r^2K\rho_c^\frac{1}{n}(n+1)\frac{d\theta}{dr}\right)=-4\pi G\rho_c\theta^n

Gathering the constants and substituting r=\alpha\xi, where

\alpha^2=(n+1)K\rho_c^{\frac{1}{n}-1}/4\pi G,

we have the Lane–Emden equation,

 \frac{1}{\xi^2} \frac{d}{d\xi} \left({\xi^2 \frac{d\theta}{d\xi}}\right) + \theta^n = 0

From Poisson's equation[edit]

Equivalently, one can start with Poisson's equation,

 \nabla^2\Phi=\frac{1}{r^2}\frac{d}{dr}\left({r^2\frac{d\Phi}{dr}}\right) = 4\pi G\rho

We can replace the gradient of the potential using the hydrostatic equilibrium, via

 \frac{d\Phi}{dr}= - \frac{1}{\rho}\frac{dP}{dr}

which again yields the dimensional form of the Lane–Emden equation.

Solutions[edit]

For a given value of the polytropic index n, denote the solution to the Lane–Emden equation as \theta_n(\xi). In general, the Lane–Emden equation must be solved numerically to find \theta_n. There are exact, analytic solutions for certain values of n, in particular: n = 0,1,5. Additionally, there is an analytic expression for \theta_5 which is infinite in extent, and thus not physically realizable. For values of n between 0 and 5, the solutions are continuous and finite in extent---where the radius of the star is given by, R = \alpha \xi_1 , such that, \theta_n(\xi_1) = 0.

For a given solution \theta_n, the density profile is given by,

 \rho = \rho_c \theta_n^n

The total mass M of the model star can be found by integrating the density over radius, from 0 to \xi_1.

The pressure can be found using the polytropic equation of state, P = K \rho^{1+\frac{1}{n}} , i.e.

 P = K \rho_c^{1+\frac{1}{n}} \theta_n^{n+1}

Finally, the temperature profile can be found using the ideal gas law, P = \frac{k_B}{m} \rho T, where k_B is the Boltzmann constant and m is the mean particle mass.

 T = \frac{K \cdot m}{k_B} \rho_c^{1/n} \theta_n

Exact Solutions[edit]

There are only three values of the polytropic index n that lead to exact solutions.

n = 0[edit]

If n=0, the equation becomes

 \frac{1}{\xi^2} \frac{d}{d\xi} \left({\xi^2 \frac{d\theta}{d\xi}}\right) + 1 = 0

Re-arranging and integrating once gives

 \xi^2\frac{d\theta}{d\xi} = C_1-\frac{1}{3}\xi^3

Dividing both sides by \xi^2 and integrating again gives

 \theta(\xi)=C_0-\frac{C_1}{\xi}-\frac{1}{6}\xi^2

The boundary conditions \theta(0)=1 and \theta'(0)=0 imply that the constants of integration are C_0=1 and C_1=0.

n = 1[edit]

When n=1, the equation can be expanded in the form

 \frac{d^2\theta}{d\xi^2}+\frac{2}{\xi}\frac{d\theta}{d\xi} + \theta = 0

We assume a power series solution:

 \theta(\xi)=\sum\limits_{n=0}^\infty a_n \xi^n

This leads to a recursive relationship for the expansion coefficients:

 a_{n+2} = -\frac{a_n}{(n+3)(n+2)}

This relation can be solved leading to the general solution:

 \theta(\xi)=a_0 \frac{\sin\xi}{\xi} + a_1 \frac{\cos\xi}{\xi}

The boundary condition for a physical polytrope demands that  \theta(\xi) \rightarrow 1 as  \xi \rightarrow 0 . This requires that  a_0 = 1, a_1 = 0 , thus leading to the solution:

 \theta(\xi)=\frac{\sin\xi}{\xi}

n = 5[edit]

After a sequence of substitutions, it can be shown that the Lane–Emden equation has a further solution

 \theta(\xi)=\frac{1}{\sqrt{1+\xi^2/3}}

when n=5. This solution is infinite in radial extent.

Numerical Solutions[edit]

In general, solutions are found by numerical integration. Many standard methods require that the problem is formulated as a system of first-order ordinary differential equations. For example,

 \frac{d\theta}{d\xi}=-\frac{\phi}{\xi^2}
 \frac{d\phi}{d\xi}=\theta^n\xi^2

Here, \phi(\xi) is interpreted as the dimensionless mass, defined by m(r)=4\pi\alpha^3\rho_c\phi(\xi). The relevant initial conditions are \phi(0)=0 and \theta(0)=1. The first equation represents the hydrostatic equilibrium and the second the mass conservation.

Homologous Variables[edit]

Homology-invariant equation[edit]

It is known that if \theta(\xi) is a solution of the Lane–Emden equation, then so is C^{2/n+1}\theta(C\xi).[2] Solutions that are related in this way are called homologous; the process that transforms them is homology. If we choose variables that are invariant to homology, then we can reduce the order of the Lane–Emden equation by one.

A variety of such variables exist. A suitable choice is

U=\frac{d\log m}{d\log r}=\frac{\xi^3\theta^n}{\phi}

and

V=\frac{d\log P}{d\log r}=(n+1)\frac{\phi}{\xi\theta}

We can differentiate the logarithms of these variables with respect to \xi, which gives

\frac{1}{U}\frac{dU}{d\xi}=\frac{1}{\xi}(3-n(n+1)^{-1}V-U)

and

\frac{1}{V}\frac{dV}{d\xi}=\frac{1}{\xi}(-1+U+(n+1)^{-1}V).

Finally, we can divide these two equations to eliminate the dependence on \xi, which leaves

\frac{dV}{dU}=-\frac{V}{U}\left(\frac{U+(n+1)^{-1}V-1}{U+n(n+1)^{-1}V-3}\right)

This is now a single first-order equation.

Topology of the homology-invariant equation[edit]

The homology-invariant equation can be regarded as the autonomous pair of equations

\frac{dU}{d\log\xi}=-U(U+n(n+1)^{-1}V-3)

and

\frac{dV}{d\log\xi}=V(U+(n+1)^{-1}V-1).

The behaviour of solutions to these equations can be determined by linear stability analysis. The critical points of the equation (where dV/d\log\xi=dU/d\log\xi=0) and the eigenvalues and eigenvectors of the Jacobian matrix are tabulated below.[3]

Critical point Eigenvalues Eigenvectors
(0,0) 3 -1 (1,0) (0,1)
(3,0) -3 2 (1,0) (-3n,5+5n)
(0,n+1) 1 3-n (0,1) (2-n,1+n)
\left(\frac{n-3}{n-1},2\frac{n+1}{n-1}\right) \frac{n-5\pm\Delta_n}{2-2n} \left(1-n\mp\Delta_n,4+4n\right)

Further reading[edit]

Horedt, Georg P. (2004). Polytropes - Applications in Astrophysics and Related Fields. Dordrecht: Kluwer Academic Publishers. ISBN 978-1-4020-2350-7. 

References[edit]

  1. ^ Lane, Jonathan Homer (1870). "On the Theoretical Temperature of the Sun under the Hypothesis of a Gaseous Mass Maintaining its Volume by its Internal Heat and Depending on the Laws of Gases Known to Terrestrial Experiment". The American Journal of Science and Arts. 2 50: 57–74. 
  2. ^ Chandrasekhar, Subrahmanyan (1939). An introduction to the study of stellar structure. Chicago, Ill.: University of Chicago Press. 
  3. ^ Horedt, Georg P. (1987). "Topology of the Lane-Emden equation". Astronomy and Astrophysics 117 (1-2): 117–130. Bibcode:1987A&A...177..117H. 

External links[edit]