Consider an observer at the center of a uniform, spherical cluster, of radius R, containing many stars. Assume that the density of stars is given by n(r) stars per cm^3
n(r)=n0* (r/r0)^-α
a. Write a formula for the flux arriving at the observer from one of the stars as radius r from the center.
b. Consider a thin shell of stars at radius r, with thickness dr. Estimate the number dN of stars in the shell.
c. Estimate the total radiation flux, from all directions, received by the observer from the thin shell of stars in part b.
d.Integrate over all of the shells to calculate the total flux received by the observer form all directions, form the entire cluster, which extends to infinity. (note that you should consider all cases, including a special case that leads to a logarithm when you integrate, not a power law.)
e. Which values of α lead to finite/infinite flux at the cluster center?
a. To calculate the flux arriving at the observer from one of the stars at radius r from the center, we need to consider the luminosity of the star and the distance between the observer and the star. Assuming the luminosity of each star is constant, the flux (F) is given by:
F = L / (4πr²)
where L is the luminosity of the star.
b. The number of stars in a thin shell of radius r with thickness dr can be estimated by multiplying the volume of the shell by the density of stars at that radius:
dN = n(r) * dV
Since the shell is thin, the volume of the shell (dV) can be approximated as the surface area of the shell multiplied by the thickness:
dV = 4πr² * dr
So, the number of stars in the thin shell is:
dN = 4πr² * n(r) * dr
c. To estimate the total radiation flux received by the observer from the thin shell of stars, we need to consider the flux (F) from each star in the shell and sum them up.
The flux from each star is given by the formula mentioned in part a:
dF = L / (4πr²)
To get the total flux, we can multiply the flux from each star by the number of stars in the shell and sum them up:
Total flux = ∫(dF * dN)
Substituting the values of dF and dN from parts a and b, we have:
Total flux = ∫((L / (4πr²)) * (4πr² * n(r) * dr))
Simplifying, we get:
Total flux = L * ∫(n(r) * dr)
d. To calculate the total flux received by the observer from all directions, from the entire cluster, we need to integrate over all the shells. However, since the density function follows a power law, we need to consider different cases based on the value of α.
Case 1: α ≠ -3
In this case, we can directly integrate the expression identified in part c:
Total flux = L * ∫(n(r) * dr) from 0 to R
Case 2: α = -3
In this special case, the density function becomes:
n(r) = n0 * (r/r0)^3
The integral in this case will lead to a logarithmic term:
Total flux = L * ∫(n0 * (r/r0)^3 * dr) from 0 to R
= L * (n0 * r0³) * ∫(1/r * dr) from 0 to R
= L * (n0 * r0³) * ln(R)
e. The value of α determines whether the flux at the cluster center is finite or infinite.
If α > 3, the density function approaches zero as r approaches zero. Therefore, the flux at the center will be finite.
If α ≤ 3, the density function does not approach zero as r approaches zero. In this case, the flux at the center will be infinite.
So, values of α greater than 3 lead to finite flux at the cluster center, while values of α less than or equal to 3 lead to infinite flux at the cluster center.