Desperate for astrophysics help

In summary, the equation for the energy generation rate for p-p fusion in a constant density, ideal gas star can be written as E = e (p/pc) [(1-(r/R)^2)/R^3k]^4, where r is the distance from the center of the star and R is the total radius of the star. I hope this helps you with your assignment. Best of luck!
  • #1
jaidon
42
0
The energy generation rate for p-p fusion has an approximate dependence on density and temperature given by:
E= e (p/pc)(T/Tc)^4

i don't know how to write all the symbols so in the above:

E is epsilon, e is epsilon at the centre of the star, p is the density, pc is density at the centre, Tc is temp at centre.

a) find the equation for the temperature within a constant density, ideal gas star to show how E depends on the radial distance from the centre of the star. Express your answer in terms of the fraction r/R where R is the total radius of the star and r is the distance from the centre.

I am lost. We have an equation for the pressure of a constant density star P(r)= (3GM^2/(8piR^4))(1-r^2/R^2). Ideal gas law tells us P= nkt where n= density/m. The average density of a star is 3M/4piR^3.

I can't figure this out. Any help, this assignment is due tomorrow.
 
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  • #2


Thank you for your question. I am happy to assist you in understanding the equation for the energy generation rate for p-p fusion and how it relates to the temperature and density within a constant density, ideal gas star.

Let's start by breaking down the equation:

E = e (p/pc) (T/Tc)^4

where:
E = energy generation rate
e = epsilon at the center of the star
p = density
pc = density at the center
T = temperature
Tc = temperature at the center

Now, for a constant density, ideal gas star, we can use the ideal gas law to relate the pressure (P) to the temperature (T) and density (p):

P = nkT

where:
n = number density
k = Boltzmann constant

We can rearrange this equation to solve for temperature:

T = P/(nk)

Since the average density of a star is 3M/4piR^3, we can substitute this into the ideal gas law to get:

T = P/(3M/4piR^3k)

Now, let's consider the equation for pressure in terms of radial distance (r) from the center of the star:

P(r) = (3GM^2/(8piR^4))(1-r^2/R^2)

where:
G = gravitational constant
M = mass of the star
R = total radius of the star

We can substitute this equation into our previous equation for temperature to get:

T = (3GM^2/(8piR^4))(1-r^2/R^2)/(3M/4piR^3k)

Simplifying this, we get:

T = (GM/2R^3k)(1-r^2/R^2)

Now, we can substitute this expression for temperature into our original equation for energy generation rate to get:

E = e (p/pc) [(GM/2R^3k)(1-r^2/R^2)/(GM/2R^3k)]^4

Simplifying this, we get:

E = e (p/pc) [(1-r^2/R^2)/R^3k]^4

Finally, we can express this in terms of the fraction r/R as follows:

E = e (p/pc) [(1-(r/R)^2)/R^3k
 

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