Solve Lane-Emden Equation for n=0 | Mass, R, a

[June_cosmo]

Homework Statement

For n=0, compute the expressions forρ(r),T(r),p(r) in terms of the total mass M and radius R of the star.What is the the scale length a in terms of the radius R in this case?

Homework Equations

Lane-Emden Equation[/B]

, where [PLAIN]https://upload.wikimedia.org/math/c/5/6/c567d1405901ff84599881cf22ac5c04.png, [PLAIN]https://upload.wikimedia.org/math/9/8/1/9815340d17d4c39a30bcf974d40a10ba.png,[PLAIN]https://upload.wikimedia.org/math/4/2/0/4205da9eef0d8c79066fcc5d32d592bf.png

The Attempt at a Solution

I'm just starting to learn this equation so just know that n=0, ρ(r)=ρc is the solution for a constant density in compressible sphere.

Thank you for your help![/B]

 

[mark726]

Hello,

For n=0, the Lane-Emden equation reduces to a simple form:

ξ^2 d^2θ/dξ^2 + ξ dθ/dξ = -θ

Where ξ = r/a, θ = ρ/ρc, and a is the scale length.

To solve for ρ(r), we can substitute back in for ξ and θ:

ρ(r) = ρc θ = ρc exp(-ξ^2) = ρc exp(-r^2/a^2)

To solve for T(r), we can use the equation of hydrostatic equilibrium:

dP/dr = -Gρ(r)m(r)/r^2

where m(r) is the mass enclosed within radius r. Since n=0, we know that the mass distribution is constant, so m(r) = M.

Solving for P(r) and using the ideal gas law, we can then solve for T(r):

P(r) = ρc^2 exp(-r^2/a^2)

T(r) = P(r) / ρ(r) = ρc exp(r^2/a^2)

Finally, to solve for p(r), we can use the equation of state for an ideal gas:

p(r) = ρ(r) RT(r) = ρc^2 exp(-r^2/a^2)

In terms of the radius R, the scale length a can be found by setting ξ = R/a, and solving for a:

a = R/ξ = R/√(n+1) = R

Hope this helps!