Mass of a Sphere with varying Density

[johnaaronrose]

What is the mass of a sphere of radius a*R where the density at any point distance x*R from the sphere's centre is f(x) where f(x) is an algebraic function of x? R is the radius of a sphere with a common centre to this one but of larger or equal radius and a thus takes a value between 0 & 1.

My solution:
Consider sphere made up of infinite number of spherical shells. Assume that a spherical shell is at distance x*R from centre with infinitesimal thickness dx.
Volume of spherical shell = its surface area * its thickness = 4Pi((x*R)^2) * d(x*R)
Mass of spherical shell = mass of spherical shell * density of spherical shell
= 4Pi((x*R)^2) * d(x*R) * f(x) = 4Pi(R^3)(x^2)f(x)dx
Mass of sphere = Integral from a to 0 of 4Pi(R^3)(x^2)f(x)dx = 4Pi(R^3) * Integral from a to 0 of (x^2)f(x)dx

The reason I ask this is that Brian Cox & Jeff Forshaw in the footnote on p235 of their book "The Quantum Universe: everything that can happen does happen" state:
g(a) = 4Pi(R^3)p * Integral from a to 0 of (x^2)f(x)dx
where g(a) is the fraction of a star's mass lying in a sphere of radius aR (e.g. a=0.5 means that this sphere has half the radius of the star), R is the radius of the spherical star, p is the average density of the star, f(x) is the density of the star at a point distance x*R from the star's centre.

My view is that:
g(a) = 4Pi(R^3) * Integral from a to 0 of (x^2)f(x)dx / ((4Pi(R^3)/3) * p)
= Integral from a to 0 of (x^2)f(x)dx / (1/3) * p)

Am I correct in this result?

PS apologies for not using the mathematical symbols for Integral, Pi etc: I haven't figured out how to get them into this webpage.

 

[hapefish]

I approached the problem as a triple integral in spherical coordinates and got $$\int_0^{aR} \int_0^{2\pi} \int_0^{\pi} f(\frac{r}{R})r^2 \sin (\phi) d\phi d\theta dr$$
Doing a simple substitution of $x= \frac{r}{R}$ and doing the two inner integrals gives the same result as you: $$4\pi R^3 \int_0^a f(x)x^2 dx$$

Your second assertion about the fraction of the star's mass seems quite reasonable, but I would probably need to look at the book (which I do not own) to be 100% certain that you interpreted it correctly.

Good Luck!

 

[pasmith]

What is the mass of a sphere of radius a*R where the density at any point distance x*R from the sphere's centre is f(x) where f(x) is an algebraic function of x? R is the radius of a sphere with a common centre to this one but of larger or equal radius and a thus takes a value between 0 & 1.

My solution:
Consider sphere made up of infinite number of spherical shells. Assume that a spherical shell is at distance x*R from centre with infinitesimal thickness dx.
Volume of spherical shell = its surface area * its thickness = 4Pi((x*R)^2) * d(x*R)
Mass of spherical shell = mass of spherical shell * density of spherical shell
= 4Pi((x*R)^2) * d(x*R) * f(x) = 4Pi(R^3)(x^2)f(x)dx
Mass of sphere = Integral from a to 0 of 4Pi(R^3)(x^2)f(x)dx = 4Pi(R^3) * Integral from a to 0 of (x^2)f(x)dx

This is the right idea. For a spherically symmetric star of radius R with density \rho(r) varying with distance from the centre, the star's mass will be
M = 4\pi \int_0^R r^2\rho(r)\,\mathrm{d}r.

If we set x = r/R and \rho(r) = f(x) then we obtain
M = 4\pi R^3 \int_0^1 x^2 f(x)\,\mathrm{d}x.

I suppose, having non-dimensionalised distance it would be strange not to non-dimensionalise density as well, so we should have set \rho(r) = \rho_0 f(x) for some reference density \rho_0. Doing that we obtain
M = 4\pi \rho_0 R^3 \int_0^1 x^2 f(x)\,\mathrm{d}x.

The reason I ask this is that Brian Cox & Jeff Forshaw in the footnote on p235 of their book "The Quantum Universe: everything that can happen does happen" state:
g(a) = 4\pi R^3 p \int_0^a x^2f(x)\,\mathrm{d}x
where g(a) is the fraction of a star's mass lying in a sphere of radius aR (e.g. a=0.5 means that this sphere has half the radius of the star), R is the radius of the spherical star, p is the average density of the star, f(x) is the density of the star at a point distance x*R from the star's centre.

"The fraction of a star's mass lying in a sphere of radius aR" is (taking \rho_0 = p)
<br /> \frac{1}{M} 4\pi \int_0^{aR} \rho(r)r^2\,\mathrm{d}r<br /> = \frac{4\pi p R^3}{M} \int_0^a x^2f(x)\,\mathrm{d}x<br /> = 3 \int_0^a x^2f(x)\,\mathrm{d}x<br />
using the fact that M = (4\pi p R^3)/3, which, allowing for the fact that I rescaled f, is what you obtained.

Why, then, do Cox and Forshaw obtain a different result? I think there are some inaccuracies in the quoted passage: firstly they seem to be taking \rho(r) = pf(x), which as explained above is reasonable, and secondly g(a) appears to be the total mass within the distance aR of the origin, rather than the fraction of the star's mass within that distance. Making those changes, we find
4\pi \int_0^{aR} \rho(r)r^2\,\mathrm{d}r<br /> = 4\pi pR^3 \int_0^a x^2f(x)\,\mathrm{d}x<br />
which is the given g(a).