Solutions of the Lane-Emden Equation

Raghav Govind Jha & Soumen Roy

May 13, 2013

1 An Effective Stellar Model

Polytropes are a good way to provide simple solutions and estimates of various stellar parameters. Theyare much simpler to use and implement numerically than the full stellar equations. It assumes a powerlaw relation between pressure and density which should be valid throughout the star.

For a polytrope one assumes that gas pressure varies as P = K which inturn means P = Kn+1

n ,where is the adiabatic index, n is the polytropic index and K is a constant.

We assume that the star is hydrostatic equilibrium. The two stellar equations are as follows :

dP

dr=

GMrr2

(1)

dMrdr

= 4pir2 (2)

The above equations when combined together gives,

1

r2d

dr

(r2

dP

dr

)= 4piG (3)

with P = Kn+1

n

and, = cDn , r = .

is the length constant given by,

=

(K(n+ 1)

4piG

1n

n

c

) 12

(4)

Combining it all, we have the following

1

2d

d

(2dDnd

)= Dn

n(5)

This is the Lane-Emden equation for convective polytropic stars.

1

Short Primer : Lane-Emden Equation 2

The above equation can be written in form of unit coefficients for the second derivative term as,

d2Dnd2

+2

dDnd

= Dnn

(6)

Also, we have two boundary conditions 1 at the center i.e at = 0

dDnd

= 0 (7)

Dn = 1 (8)

For only n = 0, 1, 5, we can solve this equation analytically. In this letter, we will will resort to thepossible numerical simulations for n = 0 to 10 . We also note that for n 5, the star is physically notpossible since the radius diverges and there are issues with the binding energy of the formation of thenuclear reactions taking place in the sun. Whereas, the n = 0 actually corresponds to incompressiblefluid since desnity is always constant. This solution can also be taken as a rough estimate of the internalstructure of earth. The most intersting cases are for n = 1.5 and n = 3 and these correspond to thecases where = 5/3 (non-relativistic) and = 4/3 (relativistic) respectively.

2 Numerical Solutions of the Lane-Emden Equation

The program to solve the equation for different values of n = 0 to n = 10 was written in Matlabusing the ode45 inbuilt operation which employs the fourth & fifth order Runge-Kutta method. Thedetails are included in the Appendix at the end.

The plot of Dn and was obtained for different n and is shown in the Fig(1) below :

3 Appendix

funct ion [ Dzeta ] = po l y t r o p e ( zeta ,D)Dzeta = zeros ( 2 , 1 ) ;Dzeta (1 ) = D( 2 ) ;Dzeta (2 ) = (2/ z e t a )D(2) D(1) n ; % Change n a c c o r d i n g l y

funct ion [ Dzeta ] = f i l e w r i t e ( zeta ,D)[ zeta ,D] = ode45 ( @po ly t rope , zeta ,D)f i d = fopen ( p o l y t r o p e ( n ) . dat , w ) ; % Change n as pe r you r equ i r emen td1 = D( : , 1 ) ;fo r i = 1 : length ( z e t a )

f p r i n t f ( f i d , %f \ t , z e t a ( i ) , d1 ( i ) ) ;f p r i n t f ( f i d , \n ) ;

end

1Though, they are both at the centre, can be strictly called as initial conditions at the time of formation of the star

Short Primer : Lane-Emden Equation 3

Figure 1: Solutions to the Lane-Emden Equation

f c l o se ( f i d )

% You have to change the i nd e x a c c o r d i n g l y .

References

[1] S. Chandrasekhar, An Introduction to the Study of Stellar Structure, 1939.

[2] Rudra Pratap, Getting Started with Matlab, Oxford University Press

An Effective Stellar ModelNumerical Solutions of the Lane-Emden EquationAppendix