2004/12/01 | 莱恩-埃姆登方程[Lane-Emden Equation]
类别(Ω〖物理〗) | 评论(2) | 阅读(347) | 发表于 14:27
A second-order ordinary differential equation that applies to polytropic profiles in density, defined as those which have the form
(1)

where P is pressure, K is a constant, is the density, and
(2)

with n is an integer (Chandrasekhar 1960).

The Lane-Emden equation is also applicable to magnetohydrodynamic fluids under the action of force-free magnetic fields.This implies that mass configurations obtained for neutral fluids through the Lane-Emden equation also exist for conducting fluids in force-free magnetic fields.Such fields are believed to exist in several astrophysical situations (Krishan 1999).

Poisson's equation for gravity states
(3)

where is the gravitational potential,G is the gravitational constant, and is the mass density. For spherical symmetry, in spherical coordinates is
(4)

Now, plug (1) into the hydrostatic law
(5)

where g is the gravitational acceleration, giving

(6)

Plugging (6) into (4),
(7)

This is actually a form of the Lane-Emden equation
(8)


(9)


(10)

Subject to the boundary conditions
(11)

(12)

These boundary conditions establish the allowed values of R and M. By appropriate change of variables
(13)

(14)

equation (10) can be transformed to the Lane-Emden equation
(15)

(16)

and the boundary conditions become
(17)

(18)

The cases n = 0 and 1 can be solved analytically; the others must be obtained numerically. The mass of a spherical body of radius R is given by the integral

(19)

so the central density is
(20)

giving


(21)

The central pressure is then given by

(22)

and the moment of inertia by


(23)

so

(24)

From the ideal gas law,
(25)

where is the mean molecular mass, and k is Boltzmann's constant. Therefore, the central temperature is

(25)

Finally, the gravitational potential energy is



(27)

For n = 0( ), the Lane-Emden equation is







The boundary condition then gives and so

so is parabolic. The first zero is found by solving






For n = 1 ()in space, the differential equation becomes


which is the spherical Bessel differential equation

with k = 1 and n = 0, so the solution is
(28)

Applying the boundary condition gives

In space,


]
(29)

The solution is
(30)

The boundary condition (11) requires B = 0. Applying (12) to (30) with B = 0,
(31)

for n = 1, 2, .... It is physically impossible for to equal 0 anywhere but at the planet's boundary. Therefore, it must be true that the first place where (30) is 0 is also the boundary. This implies that (31) must equal
(32)

which implies the surprising result that R is independent of M!
(33)

Therefore, the solution (30) is constrained by boundary conditions to
(34)

A can be expressed in terms of M using the condition
(35)

Let
(36)

(37)

The integral (35) then becomes


(38)

(39)

(40)

(41)



(42)

(43)

(44)

Let
(45)

then
(46)

To find , solve (33) for K
(47)

(48)

To find I,






(49)

(50)














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