- #1

iamalexalright

- 164

- 0

## Homework Statement

In brief, I am working on a stellar model.

We started with these four equations:

[tex]\delta q/\delta x = px^{2}/t[/tex]

[tex]\delta p /\delta x = -pq/tx^{2}[/tex]

[tex]\delta t /\deltax = -Cp^{1.75} / x^{2}t^{8325}[/tex]

[tex]\delta t /\delta x = -2q/5x^{2}[/tex]

In the model you switch to the 4th equation instead of the 3rd when the ratio of the 3rd to the 4th is less than 1.

Because the variables p and t vary over orders of magnitude we switch to logarithmetic variables.

[tex]g_{1}=log(p)[/tex]

[tex]g_{2}=log(q)[/tex]

[tex]g_{3}=log(t)[/tex]

[tex]y=log(x)[/tex]

The 4 equations transform to:

[tex]log[-\delta g_{1}/\delta y] = g_{2} - g_{3} - y[/tex]

[tex]log[\delta g_{2}/\delta y] = g_{1} + 3y - g_{2} - g_{3}[/tex]

[tex]log[-\delta g_{3}/\delta y] = log(C) + 1.75g_{1} - y - 9.25g_{3}[/tex]

[tex]log[-\delta g_{3}/\delta y] = log(2/5) + g_{2} - g_{3} - y[/tex]

log(C) = -5.51674

The initial conditions are:

x = 1, q = 1, t = 0, p = 0

Since we encounter problems we use:

[tex]p^{1.75} = 1.75t^{8.25}/8.25C[/tex]

[tex]t = 1.75/8.25(1/x - 1)[/tex]

to find these initial values.

We then work on way inward (from x = 1 to x = 0)

**2. The attempt at a solution**

I worked out

g1 = -8.37512874

g2 = 1

g3 = -2.36361205

y = -0.00877392431

to be initial values.

Now I am having troubles doing the actual numerical integration...for instance if I try to find g1 at a point x-h(where h I have .02) i do this:

[tex]g_{1 at x-h} = g_{1 at x} + (1/6)(k_{1} + 2k_{2} + 2k_{3} + k_{4})}[/tex]

Now [tex]k_{1}[/tex] is simply the derivative of [tex]g_{1}[/tex] at x.

For [tex]k_{2}[/tex] I use [tex]x + .5hk_{1}[/tex] instead of x in the derivative equation(and the same method for k3 and k4).

Is this correct? Also I don't know what to do about that log outside(meaning, do I raise 10 by the right-hand side to get the actual derivative and not the log of it?).

Thanks in advance