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[[Image:FinitePolytropes.svg|thumb|Solutions of Lane–Emden equation for ''n'' = 0, 1, 2, 3, 4, 5.]]
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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]].<ref>{{cite journal|last=Lane|first=Jonathan Homer|title=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|journal=The American Journal of Science and Arts|year=1870|volume=50|series=2|pages=57–74|author-link=Jonathan Homer Lane}}</ref>  The equation reads
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: <math> \frac{1}{\xi^2} \frac{d}{d\xi} \left({\xi^2 \frac{d\theta}{d\xi}}\right) + \theta^n = 0 </math>
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where <math>\xi</math> is a dimensionless radius and <math>\theta</math> is related to the density (and thus the pressure) by <math>\rho=\rho_c\theta^n</math> for central density <math>\rho_c</math>. The index <math>n</math> is the polytropic index that appears in the polytropic equation of state,
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: <math> P = K \rho^{1 + \frac{1}{n}}\, </math>
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where <math>P</math> and <math>\rho</math> are the pressure and density, respectively, and <math>K</math> is a constant of proportionality. The standard boundary conditions are <math>\theta(0)=1</math> and <math>\theta'(0)=0</math>. Solutions thus describe the run of pressure and density with radius and are known as '''polytropes''' of index <math>n</math>.
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== Applications ==
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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.
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== Derivation ==
</ul>
 
=== From hydrostatic equilibrium ===
 
Consider a self-gravitating, spherically symmetric fluid in [[hydrostatic equilibrium]]. Mass is conserved and thus described by the [[continuity equation]]
 
: <math> \frac{dm}{dr} = 4\pi r^2 \rho </math>
 
where <math>\rho</math> is a function of <math>r</math>. The equation of hydrostatic equilibrium is
 
: <math> \frac{1}{\rho}\frac{dP}{dr} = -\frac{Gm}{r^2} </math>
 
where <math>m</math> is also a function of <math>r</math>. Differentiating again gives
 
: <math>\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}</math>
 
where we have used the continuity equation to replace the mass gradient. Multiplying both sides by <math>r^2</math> and collecting the derivatives of <math>P</math> on the left, we can write
 
: <math> 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 </math>
 
Dividing both sides by <math>r^2</math> yields, in some sense, a dimensional form of the desired equation. If, in addition, we substitute for the polytropic equation of state with <math>P=K\rho_c^{1+\frac{1}{n}}\theta^{n+1}</math> and <math>\rho=\rho_c\theta^n</math>, we have
 
: <math> \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</math>
 
Gathering the constants and substituting <math>r=\alpha\xi</math>, where
 
: <math>\alpha^2=(n+1)K\rho_c^{\frac{1}{n}-1}/4\pi G</math>,
 
we have the Lane–Emden equation,
 
: <math> \frac{1}{\xi^2} \frac{d}{d\xi} \left({\xi^2 \frac{d\theta}{d\xi}}\right) + \theta^n = 0 </math>
 
=== From Poisson's equation ===
 
Equivalently, one can start with [[Poisson's equation]],
 
: <math> \nabla^2\Phi=\frac{1}{r^2}\frac{d}{dr}\left({r^2\frac{d\Phi}{dr}}\right) = 4\pi G\rho </math>
 
We can replace the gradient of potential using hydrostatic equilibrium, via
 
: <math> \frac{d\Phi}{dr}= - \frac{1}{\rho}\frac{dP}{dr} </math>
 
which again yields the dimensional form of the Lane–Emden equation.
 
== Solutions ==
For a given value of the polytropic index <math>n</math>, denote the solution to the Lane–Emden equation as <math>\theta_n(\xi)</math>. In general, the Lane–Emden equation must be solved numerically to find <math>\theta_n</math>.  There are exact, analytic solution for certain values of <math>n</math>, in particular: <math>n = 0,1,5</math>.  Additionally, there is an analytic expression for <math>\theta_5</math> which is infinite in extent, and thus not physically realizable.  For values of <math>n</math> between 0 and 5, the solutions are continuous and finite in extent---where the radius of the star is given by, <math>R = \alpha \xi_1 </math>, such that, <math>\theta_n(\xi_1) = 0</math>.
 
For a given solution <math>\theta_n</math>, the density profile is given by,
 
: <math> \rho = \rho_c \theta_n^n </math>
 
The total mass <math>M</math> of the model star can be found by integrating the density over radius, from 0 to <math>\xi_1</math>.
 
The pressure can be found using the polytropic equation of state, <math>P = K \rho^{1+\frac{1}{n}} </math>, i.e.
 
: <math> P = K \rho_c^{1+\frac{1}{n}} \theta_n^{n+1}</math>
 
Finally, the temperature profile can be found using the [[ideal gas law]], <math>P = \frac{k_B}{m} \rho T</math>, where <math>k_B</math> is the [[Boltzmann constant]] and <math>m</math> is the mean particle mass.
 
: <math> T = \frac{K \cdot m}{k_B} \rho_c^{1/n} \theta_n </math>
 
=== Exact Solutions ===
 
There are only three values of the polytropic index <math>n</math> that lead to exact solutions.
 
==== <math>n=0</math> ====
 
If <math>n=0</math>, the equation becomes
 
: <math> \frac{1}{\xi^2} \frac{d}{d\xi} \left({\xi^2 \frac{d\theta}{d\xi}}\right) + 1 = 0 </math>
 
Re-arranging and integrating once gives
 
: <math> \xi^2\frac{d\theta}{d\xi} = C_1-\frac{1}{3}\xi^3 </math>
 
Dividing both sides by <math>\xi^2</math> and integrating again gives
 
: <math> \theta(\xi)=C_0-\frac{C_1}{\xi}-\frac{1}{6}\xi^2 </math>
 
The boundary conditions <math>\theta(0)=1</math> and <math>\theta'(0)=0</math> imply that the constants of integration are <math>C_0=1</math> and <math>C_1=0</math>.
 
==== <math>n=1</math> ====
 
When <math>n=1</math>, the equation can be expanded in the form
 
: <math> \frac{d^2\theta}{d\xi^2}+\frac{2}{\xi}\frac{d\theta}{d\xi} + \theta = 0 </math>
 
We assume a power series solution:
 
:<math> \theta(\xi)=\sum\limits_{n=0}^\infty a_n \xi^n </math>
 
This leads to a recursive relationship for the expansion coefficients:
 
:<math> a_{n+2} = -\frac{a_n}{(n+3)(n+2)} </math>
 
This relation can be solved leading to the general solution:
 
: <math> \theta(\xi)=a_0 \frac{\sin\xi}{\xi} + a_1 \frac{\cos\xi}{\xi} </math>
 
The boundary condition for a physical polytrope demands that <math> \theta(\xi) \rightarrow 1</math> as <math> \xi \rightarrow 0 </math>.
This requires that <math> a_0 = 1, a_1 = 0 </math>, thus leading to the solution:
 
:<math> \theta(\xi)=\frac{\sin\xi}{\xi} </math>
 
==== <math>n=5</math> ====
 
After a sequence of substitutions, it can be shown that the Lane–Emden equation has a further solution
 
: <math> \theta(\xi)=\frac{1}{\sqrt{1+\xi^2/3}} </math>
 
when <math>n=5</math>. This solution is infinite in radial extent.
 
=== Numerical Solutions ===
 
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 equation]]s. For example,
 
: <math> \frac{d\theta}{d\xi}=-\frac{\phi}{\xi^2} </math>
: <math> \frac{d\phi}{d\xi}=\theta^n\xi^2 </math>
 
Here, <math>\phi(\xi)</math> is interpreted as the dimensionless mass, defined by <math>m(r)=4\pi\alpha^3\rho_c\phi(\xi)</math>. The relevant boundary conditions are <math>\phi(0)=0</math> and <math>\theta(0)=1</math>. The first equation represents hydrostatic equilibrium and the second mass conservation.
 
== Homologous Variables ==
 
=== Homology-invariant equation ===
 
It is known that if <math>\theta(\xi)</math> is a solution of the Lane–Emden equation, then so is <math>C^{2/n+1}\theta(C\xi)</math>.<ref>{{cite book|last=Chandrasekhar|first=Subrahmanyan|title=An introduction to the study of stellar structure|year=1939|publisher=University of Chicago Press|location=Chicago, Ill.|url=http://adsabs.harvard.edu/abs/1939isss.book.....C|author-link=Subrahmanyan Chandrasekhar}}</ref>  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
 
: <math>U=\frac{d\log m}{d\log r}=\frac{\xi^3\theta^n}{\phi}</math>
 
and
 
: <math>V=\frac{d\log P}{d\log r}=(n+1)\frac{\phi}{\xi\theta}</math>
 
We can differentiate the logarithms of these variables with respect to <math>\xi</math>, which gives
 
: <math>\frac{1}{U}\frac{dU}{d\xi}=\frac{1}{\xi}(3-n(n+1)^{-1}V-U)</math>
and
: <math>\frac{1}{V}\frac{dV}{d\xi}=\frac{1}{\xi}(-1+U+(n+1)^{-1}V)</math>.
 
Finally, we can divide these two equations to eliminate the dependence on <math>\xi</math>, which leaves
 
: <math>\frac{dV}{dU}=-\frac{V}{U}\left(\frac{U+(n+1)^{-1}V-1}{U+n(n+1)^{-1}V-3}\right)</math>
 
This is now a single first-order equation.
 
=== Topology of the homology-invariant equation ===
 
The homology-invariant equation can be regarded as the autonomous pair of equations
 
: <math>\frac{dU}{d\log\xi}=-U(U+n(n+1)^{-1}V-3)</math>
 
and
 
: <math>\frac{dV}{d\log\xi}=V(U+(n+1)^{-1}V-1)</math>.
 
The behaviour of solutions to these equations can be determined by linear stability analysis. The critical points of the equation (where <math>dV/d\log\xi=dU/d\log\xi=0</math>) and the eigenvalues and eigenvectors of the [[Jacobian matrix]] are tabulated below.<ref>{{cite journal|last=Horedt|first=Georg P.|title=Topology of the Lane-Emden equation|journal=[[Astronomy and Astrophysics]]|year=1987|volume=117|issue=1-2|pages=117–130|bibcode=1987A&A...177..117H}}</ref>  
 
{| cellspacing="10"
|- align="center"
!  Critical point  !! colspan="2" |  Eigenvalues  !! colspan="2" |  Eigenvectors 
|- align="center"
|  <math>(0,0)</math>  ||  <math>3</math>  ||  <math>-1</math> ||  <math>(1,0)</math>  ||  <math>(0,1)</math>
|- align="center"
|  <math>(3,0)</math>  ||  <math>-3</math>  ||  <math>2</math>  ||  <math>(1,0)</math>  ||  <math>(-3n,5+5n)</math>
|- align="center"
|  <math>(0,n+1)</math> ||  <math>1</math>  ||  <math>3-n</math>  ||  <math>(0,1)</math>  ||  <math>(2-n,1+n)</math>
|- align="center"
|  <math>\left(\frac{n-3}{n-1},2\frac{n+1}{n-1}\right)</math>  || colspan="2" |  <math>\frac{n-5\pm\Delta_n}{2-2n}</math>  ||  colspan="2" |  <math>\left(1-n\mp\Delta_n,4+4n\right)</math> 
|-
|}
 
== Further reading ==
 
{{cite book|last=Horedt|first=Georg P.|title=Polytropes - Applications in Astrophysics and Related Fields|year=2004|publisher=Kluwer Academic Publishers|location=Dordrecht|isbn=978-1-4020-2350-7|url=http://www.springer.com/astronomy/astronomy%2C+observations+and+techniques/book/978-1-4020-2350-7}}
 
== References ==
 
{{reflist}}
 
== External links ==
*{{MathWorld | urlname=Lane-EmdenDifferentialEquation | title=Lane-Emden Differential Equation}}
 
{{DEFAULTSORT:Lane-Emden equation}}
[[Category:Astrophysics]]
[[Category:Ordinary differential equations]]

Latest revision as of 02:23, 12 January 2015

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