Lane–Emden equation
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In astrophysics, the Lane–Emden equation is a dimensionless form of
Poisson's equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...
for the gravitational potential of a Newtonian self-gravitating, spherically symmetric, polytropic fluid. It is named after astrophysicists
Jonathan Homer Lane Jonathan Homer Lane (August 9, 1819 – May 3, 1880) was an American astrophysicist and inventor. Biography Lane's parents were Mark and Henrietta (née Tenny) Lane and his education was at the Phillips Exeter Academy in Exeter, New Hampsh ...
and
Robert Emden Jacob Robert Emden (4 March 1862 – 8 October 1940) was a Swiss astrophysicist and meteorologist. He is best known for his book, ''Gaskugeln: Anwendungen der mechanischen Wärmetheorie auf kosmologische und meteorologische probleme'' (Gas sphe ...
. The equation reads : \frac \frac \left(\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^\, 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. If an isothermal fluid (polytropic index tends to infinity) is used instead of a polytropic fluid, one obtains the
Emden–Chandrasekhar equation In astrophysics, the Emden–Chandrasekhar equation is a dimensionless form of the Poisson equation for the density distribution of a spherically symmetric isothermal gas sphere subjected to its own gravitational force, named after Robert Emden a ...
.


Applications

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


From hydrostatic equilibrium

Consider a self-gravitating, spherically symmetric fluid in
hydrostatic equilibrium In fluid mechanics, hydrostatic equilibrium (hydrostatic balance, hydrostasy) is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force. In the planetary ...
. Mass is conserved and thus described by the continuity equation : \frac = 4\pi r^2 \rho where \rho is a function of r. The equation of hydrostatic equilibrium is : \frac\frac = -\frac where m is also a function of r. Differentiating again gives : \begin \frac\left(\frac\frac\right) &= \frac-\frac\frac \\ &=-\frac\frac-4\pi G\rho \end where the continuity equation has been used to replace the mass gradient. Multiplying both sides by r^2 and collecting the derivatives of P on the left, one can write : r^2\frac\left(\frac\frac\right)+\frac\frac = \frac\left(\frac\frac\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^\theta^ and \rho=\rho_c\theta^n, we have : \frac\frac\left(r^2K\rho_c^\frac(n+1)\frac\right)=-4\pi G\rho_c\theta^n Gathering the constants and substituting r=\alpha\xi, where : \alpha^2=(n+1)K\rho_c^/4\pi G, we have the Lane–Emden equation, : \frac \frac \left(\right) + \theta^n = 0


From Poisson's equation

Equivalently, one can start with
Poisson's equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...
, : \nabla^2\Phi=\frac\frac\left( r^2\frac \right) = 4\pi G\rho One can replace the gradient of the potential using the hydrostatic equilibrium, via : \frac= -\frac\frac which again yields the dimensional form of the Lane–Emden equation.


Exact solutions

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. For n between 0 and 5, the solutions are continuous and finite in extent, with the radius of the star given by R = \alpha \xi_1 , where \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^ , i.e. : P = K \rho_c^ \theta_n^ Finally, if the gas is
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
, the equation of state is P = k_B\rho T/\mu, where k_B is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
and \mu the mean molecular weight. The temperature profile is then given by : T = \frac \rho_c^ \theta_n In spherically symmetric cases, the Lane–Emden equation is integrable for only three values of the polytropic index n.


For ''n'' = 0

If n=0, the equation becomes : \frac \frac \left( \xi^2 \frac \right) + 1 = 0 Re-arranging and integrating once gives : \xi^2\frac = C_1-\frac\xi^3 Dividing both sides by \xi^2 and integrating again gives : \theta(\xi)=C_0-\frac-\frac\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. Therefore, : \theta(\xi) = 1 - \frac\xi^2


For ''n'' = 1

When n=1, the equation can be expanded in the form : \frac+\frac\frac + \theta = 0 One assumes a power series solution: : \theta(\xi)=\sum_^\infty a_n \xi^n This leads to a recursive relationship for the expansion coefficients: : a_ = -\frac This relation can be solved leading to the general solution: : \theta(\xi)=a_0 \frac + a_1 \frac 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


For ''n'' = 5

We start from with the Lane–Emden equation: : \frac1 \left(\xi^2\frac\right) + \theta^5 = 0 Rewriting for \frac produces: : \frac = \frac1 2 \left(1+\frac\right)^ \frac 3 = \frac Differentiating with respect to ''ξ'' leads to: : \theta^5 =\frac + \frac = \frac 9 Reduced, we come by: : \theta^5 = \frac 1 Therefore, the Lane–Emden equation has the solution : \theta(\xi)=\frac 1 when n=5. This solution is finite in mass but infinite in radial extent, and therefore the complete polytrope does not represent a physical solution. Chandrasekhar believed for a long time that finding other solution for n=5 "is complicated and involves elliptic integrals".


Srivastava's solution

In 1962, Sambhunath Srivastava found an explicit solution when n=5. His solution is given by :\theta = \frac, and from this solution, a family of solutions \theta(\xi)\rightarrow \sqrt A\, \theta(A\xi) can be obtained using homology transformation. Since this solution does not satisfy the conditions at the origin (in fact, it is oscillatory with amplitudes growing indefinitely as the origin is approached), this solution can be used in composite stellar models.


Analytic solutions

In applications, the main role play analytic solutions that are expressible by the convergent
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
expanded around some initial point. Typically the expansion point is \xi=0, which is also a singular point (fixed singularity) of the equation, and there is provided some initial data \theta(0) at the centre of the star. One can prove that the equation has the convergent power series/analytic solution around the origin of the form \theta(\xi) = \theta(0) - \frac \xi^ + O(\xi^),\quad \xi \approx 0. The
radius of convergence In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series ...
of this series is limited due to existence of two singularities on the imaginary axis in the complex plane. These singularities are located symmetrically with respect to the origin. Their position change when we change equation parameters and the initial condition \theta(0), and therefore, they are called movable singularities due to classification of the singularities of non-linear ordinary differential equations in the complex plane by
Paul Painlevé Paul Painlevé (; 5 December 1863 – 29 October 1933) was a French mathematician and statesman. He served twice as Prime Minister of the Third Republic: 12 September – 13 November 1917 and 17 April – 22 November 1925. His entry into politic ...
. A similar structure of singularities appears in other non-linear equations that result from the reduction of the Laplace operator in spherical symmetry, e.g., Isothermal Sphere equation. Analytic solutions can be extended along the real line by
analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a n ...
procedure resulting in the full profile of the star or
molecular cloud A molecular cloud, sometimes called a stellar nursery (if star formation is occurring within), is a type of interstellar cloud, the density and size of which permit absorption nebulae, the formation of molecules (most commonly molecular hydroge ...
cores. Two analytic solutions with the overlapping circles of convergence can also be matched on the overlap to the larger domain solution, which is a commonly used method of construction of profiles of required properties. The series solution is also used in the numerical integration of the equation. It is used to shift the initial data for analytic solution slightly away from the origin since at the origin the numerical methods fail due to the singularity of the equation.


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 In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
s. For example, : \begin & \frac=-\frac \\ pt& \frac=\theta^n\xi^2 \end Here, \varphi(\xi) is interpreted as the dimensionless mass, defined by m(r)=4\pi\alpha^3\rho_c\varphi(\xi). The relevant initial conditions are \varphi(0)=0 and \theta(0)=1. The first equation represents hydrostatic equilibrium and the second represents mass conservation.


Homologous variables


Homology-invariant equation

It is known that if \theta(\xi) is a solution of the Lane–Emden equation, then so is C^\theta(C\xi). Solutions that are related in this way are called ''homologous''; the process that transforms them is ''homology''. If one chooses 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=\frac \varphi and : V=\frac=(n+1)\frac \varphi We can differentiate the logarithms of these variables with respect to \xi, which gives : \frac\frac=\frac 1 \xi (3-n(n+1)^V-U) and : \frac\frac=\frac 1 \xi (-1-U-(n+1)^V). Finally, we can divide these two equations to eliminate the dependence on \xi, which leaves : \frac=-\frac\left(\frac\right) 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 : \frac=-U(U+n(n+1)^V-3) and : \frac=V(U+(n+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. : \begin \hline \text & \text & \text \\ \hline (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(\dfrac,2\dfrac\right) & \dfrac & (1-n\mp\Delta_n,4+4n) \\ \hline \end


See also

*
Emden–Chandrasekhar equation In astrophysics, the Emden–Chandrasekhar equation is a dimensionless form of the Poisson equation for the density distribution of a spherically symmetric isothermal gas sphere subjected to its own gravitational force, named after Robert Emden a ...
* Chandrasekhar's white dwarf equation


References


Further reading

* *


External links

* {{DEFAULTSORT:Lane-Emden equation Astrophysics Ordinary differential equations