Lane–Emden equation
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. The equation reads where formula_2 is a dimensionless radius and formula_3 is related to the density, and thus the pressure, by formula_4 for central density formula_5. The index formula_6 is the polytropic index that appears in the polytropic equation of state, where formula_8 and formula_9 are the pressure and density, respectively, and formula_10 is a constant of proportionality. The  standard boundary conditions are formula_11 and formula_12. Solutions thus describe the run of pressure and density with radius and are known as "polytropes" of index formula_6. If an isothermal fluid (polytropic index tends to infinity) is used instead of a polytropic fluid, one obtains the Emden–Chandrasekhar equation. 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. Mass is conserved and thus described by the continuity equation where formula_9 is a function of formula_16. The equation of hydrostatic equilibrium is where formula_18 is also a function of formula_16. Differentiating again gives where the  continuity equation has been used to replace the mass gradient. Multiplying both sides by formula_21 and collecting the derivatives of formula_8 on the left, one can write Dividing both sides by formula_21 yields, in some sense, a dimensional form of the desired equation. If, in addition, we substitute for the polytropic equation of state with formula_25 and formula_4, we have Gathering the constants and substituting formula_28, where we have the Lane–Emden equation, From Poisson's equation. Equivalently, one can start with Poisson's equation, One can replace the gradient of the potential using the hydrostatic equilibrium, via which again  yields the dimensional form of the Lane–Emden equation. Exact solutions. For a given value of the polytropic index formula_6, denote the solution to the Lane–Emden equation as formula_34. In general, the Lane–Emden equation must be solved numerically to find formula_35. There are exact, analytic solutions for certain values of formula_6, in particular: formula_37. For formula_6 between 0 and 5, the solutions are continuous and finite in extent, with the radius of the star given by formula_39, where formula_40. For a given solution formula_35, the density profile is given by The total mass formula_43 of the model star  can be found by integrating the density over radius, from 0 to formula_44. The pressure can be found using the polytropic equation of state, formula_45, i.e. Finally, if the gas is ideal, the equation of state is formula_47, where formula_48 is the Boltzmann constant and formula_49 the mean molecular weight. The temperature profile is then given by In spherically symmetric cases, the Lane–Emden equation is integrable for only three values of the polytropic index formula_6. For "n" = 0. If formula_52, the equation becomes Re-arranging and integrating once gives Dividing both sides by formula_55 and integrating again gives  The boundary conditions formula_11 and formula_12 imply that the constants of integration are formula_59 and formula_60. Therefore, For "n" = 1. When formula_62, the equation can be expanded in the form One assumes a power series solution: This leads to a recursive relationship for the expansion coefficients: This relation can be solved leading to the general solution: The boundary condition for a physical polytrope demands that formula_67 as formula_68. This requires that formula_69, thus leading to the solution: For "n" = 5. We start from with the Lane–Emden equation: Rewriting for formula_72 produces: Differentiating with respect to "ξ" leads  to: Reduced, we come by: Therefore, the Lane–Emden equation has the solution when formula_77. 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 formula_77 "is complicated and involves elliptic integrals". Srivastava's solution. In 1962, Sambhunath Srivastava found an explicit solution when formula_77. His solution is given by and from this solution, a family of solutions formula_81 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 expanded around some initial point. Typically the expansion point is formula_82, which is also a singular point (fixed singularity) of the equation, and there is provided some initial data formula_83 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 formula_84. The radius of convergence  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 formula_83, 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. 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 procedure resulting in the full profile of the star or molecular cloud 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 equations. For example, Here, formula_87 is interpreted as the dimensionless mass, defined by formula_88. The relevant initial conditions are formula_89 and formula_11. The first equation represents hydrostatic equilibrium and the second represents mass conservation. Homologous variables. Homology-invariant equation. It is known that if formula_91 is a solution of the Lane–Emden equation, then so is formula_92. 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 and We can differentiate the logarithms of these variables with respect to formula_2, which gives and Finally, we can divide these two equations to eliminate the dependence on formula_2, which leaves 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 and The behaviour of solutions to these equations can be determined  by linear stability analysis. The critical points of the equation (where formula_102) and the eigenvalues and eigenvectors of the Jacobian matrix are tabulated below.