Lane–Emden equation

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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.[1] The equation reads

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

where is a dimensionless radius and is related to the density, and thus the pressure, by for central density . The index is the polytropic index that appears in the polytropic equation of state, where and are the pressure and density, respectively, and is a constant of proportionality. The standard boundary conditions are and . Solutions thus describe the run of pressure and density with radius and are known as polytropes of index . If an isothermal fluid (polytropic index tends to infinity) is used instead of a polytropic fluid, one obtains the Emden–Chandrasekhar equation.

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.

Derivation

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From hydrostatic equilibrium

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Consider a self-gravitating, spherically symmetric fluid in hydrostatic equilibrium. Mass is conserved and thus described by the continuity equation   where   is a function of  . The equation of hydrostatic equilibrium is   where   is also a function of  . Differentiating again gives   where the continuity equation has been used to replace the mass gradient. Multiplying both sides by   and collecting the derivatives of   on the left, one can write  

Dividing both sides by   yields, in some sense, a dimensional form of the desired equation. If, in addition, we substitute for the polytropic equation of state with   and  , we have  

Gathering the constants and substituting  , where   we have the Lane–Emden equation,  

From Poisson's equation

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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

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For a given value of the polytropic index  , denote the solution to the Lane–Emden equation as  . In general, the Lane–Emden equation must be solved numerically to find  . There are exact, analytic solutions for certain values of  , in particular:  . For   between 0 and 5, the solutions are continuous and finite in extent, with the radius of the star given by  , where  .

For a given solution  , the density profile is given by  

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

The pressure can be found using the polytropic equation of state,  , i.e.  

Finally, if the gas is ideal, the equation of state is  , where   is the Boltzmann constant and   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  .

For n = 0

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If  , the equation becomes  

Re-arranging and integrating once gives  

Dividing both sides by   and integrating again gives  

The boundary conditions   and   imply that the constants of integration are   and  . Therefore,  

For n = 1

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When  , 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   as  . This requires that  , thus leading to the solution:  

For n = 2

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This exact solution was found by accident when searching for zero values of the related TOV Equation.[2]

We consider a series expansion around     with initial values   and  . Plugging this into the Lane-Emden equation, we can show that all odd coefficients of the series vanish  . Furthermore, we obtain a recursive relationship between the even coefficients   of the series.   It was proven that this series converges at least for   but numerical results showed good agreement for much larger values.

For n = 5

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We start from with the Lane–Emden equation:  

Rewriting for   produces:  

Differentiating with respect to ξ leads to:  

Reduced, we come by:  

Therefore, the Lane–Emden equation has the solution   when  . 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   "is complicated and involves elliptic integrals".

Srivastava's solution

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In 1962, Sambhunath Srivastava found an explicit solution when  .[3] His solution is given by   and from this solution, a family of solutions   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

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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  , which is also a singular point (fixed singularity) of the equation, and there is provided some initial data   at the centre of the star. One can prove [4][5] that the equation has the convergent power series/analytic solution around the origin of the form  

 
Numerical solution for analytical solution of the Lane-Emden equation in the complex plane for  ,  . Two movable singularities on the imaginary axis are visible. They limit the radius of convergence of the analytical solution around the origin. For different values of initial data and   the location of singularities is different, yet they are located symmetrically on the imaginary axis.[6]

The radius of convergence of this series is limited due to existence [5][7] 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  , 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.[7]

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

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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,[8]

 

Here,   is interpreted as the dimensionless mass, defined by  . The relevant initial conditions are   and  . The first equation represents hydrostatic equilibrium and the second represents mass conservation.

Homologous variables

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Homology-invariant equation

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It is known that if   is a solution of the Lane–Emden equation, then so is  .[9] 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  , which gives   and  


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

This is now a single first-order equation.

Topology of the homology-invariant equation

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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  ) and the eigenvalues and eigenvectors of the Jacobian matrix are tabulated below.[10]

Critical point Eigenvalues Eigenvectors
     
     
     
     

See also

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References

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  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 as known to terrestrial experiment". American Journal of Science. 2. 50 (148): 57–74. Bibcode:1870AmJS...50...57L. doi:10.2475/ajs.s2-50.148.57. ISSN 0002-9599. S2CID 131102972.
  2. ^ Pleyer, Jonas. "Zero Values of the TOV Equation". GitHub. Retrieved 4 January 2024.
  3. ^ Srivastava, Shambhunath (1962). "A New Solution of the Lane-Emden Equation of Index n=5". The Astrophysical Journal. 136: 680. Bibcode:1962ApJ...136..680S. doi:10.1086/147421. ISSN 0004-637X.
  4. ^ Kycia, Radosław Antoni (2020). "Perturbed Lane–Emden Equations as a Boundary Value Problem with Singular Endpoints". Journal of Dynamical and Control Systems. 26 (2): 333–347. arXiv:1810.01410. doi:10.1007/s10883-019-09445-6. ISSN 1079-2724.
  5. ^ a b Hunter, C. (2001-12-11). "Series solutions for polytropes and the isothermal sphere". Monthly Notices of the Royal Astronomical Society. 328 (3): 839–847. Bibcode:2001MNRAS.328..839H. doi:10.1046/j.1365-8711.2001.04914.x. ISSN 0035-8711.
  6. ^ Kycia, Radosław Antoni; Filipuk, Galina (2015), Mityushev, Vladimir V.; Ruzhansky, Michael V. (eds.), "On the Singularities of the Emden–Fowler Type Equations", Current Trends in Analysis and Its Applications, Cham: Springer International Publishing, pp. 93–99, doi:10.1007/978-3-319-12577-0_13, ISBN 978-3-319-12576-3, retrieved 2020-07-19
  7. ^ a b Kycia, Radosław Antoni; Filipuk, Galina (2015). "On the generalized Emden–Fowler and isothermal spheres equations". Applied Mathematics and Computation. 265: 1003–1010. doi:10.1016/j.amc.2015.05.140.
  8. ^ Hansen, Carl J.; Kawaler, Steven D.; Trimble, Virginia (2004). Stellar Interiors: Physical Principles, Structure, and Evolution. New York, NY: Springer. p. 338. ISBN 9780387200897.
  9. ^ Chandrasekhar, Subrahmanyan (1957) [1939]. An Introduction to the Study of Stellar Structure. Dover. Bibcode:1939isss.book.....C. ISBN 978-0-486-60413-8.
  10. ^ Horedt, Georg P. (1987). "Topology of the Lane-Emden equation". Astronomy and Astrophysics. 117 (1–2): 117–130. Bibcode:1987A&A...177..117H. ISSN 0004-6361.

Further reading

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