Fluid dynamics: Hydrostatic equilibrium and density stratification

[jmz34]

Firstly, the whole question is below, for now I need help on parts of the question- I've included it all for clarity and just in case I have further problems.

An equilibrium ring of isothermal fluid orbits a star at radius R. In the plane of the ring, mechanical equilibrium results from a balance of centrifugal force and gravitational force of the central object; normal to the ring (ie.vertically) equilibrium is between the vertical component of the gravitational force of the central object and the vertical pressure gradients in the ring of gas. Show that in the limit that the ring thickness H<<R the vertical density stratification in the ring is Gaussian and determine the e-folding length in terms of the gas temperature and the angular velocity of the ring, w. Hence determine an upper limit to the temperature such that the ring is thin (H<<R) and calculate this temperature if the ring's radius is that of the Earth's orbit around the sun.

My attempt:

I first derived the angular velocity as a function of distance within the ring. I did this by balancing the centripetal force with the gravitational force between a fluid element and the star and then writing R->R+H and eliminating H^2 and higher order.

Next I wrote down the equation of hydrostatic equilibrium (setting u=0 in the momentum equation, giving):

(-1/p)dP/dr=-g where p=density, P=pressure and g is the gravitational acceleration

I can see that at the innermost fluid element, g=GM/r^2 by Gauss' law but within the fluid the elements inside the circle centred at the star and lying on the plane of the ring will contribute to g as well, and I don't know how to construct a 2D analogue of Gauss' theorem.


I would be very grateful if someone could check if my thinking is correct and if it is, help me find this 2D analogue of Gauss' theorem.

Thanks alot.

 

[eljose79]

Your thinking is on the right track. To find the 2D analogue of Gauss' theorem, we can use the concept of flux. Flux is defined as the flow of a vector field through a surface. In this case, we can consider the gravitational field as a vector field and the surface as a circle centered at the star and lying on the plane of the ring.

To find the flux, we can use the equation:

Flux = ∫∫(g x dA)

where g is the gravitational field and dA is the surface element. Since we are dealing with a 2D problem, we can consider dA as a small element of area on the circle. This can be written as:

dA = Rdθ

where R is the radius of the circle and dθ is the angular element.

Substituting this in the flux equation, we get:

Flux = ∫∫(g x Rdθ)

Since the gravitational field is pointing towards the center of the circle, we can write g as:

g = -GM/r^2

where M is the mass of the star and r is the distance from the center of the circle to the element of area.

Substituting this in the flux equation and integrating over the entire circle, we get:

Flux = ∫∫(-GM/R^2 x Rdθ)

= -GM/R ∫∫(dθ)

= -2πGM

This flux represents the total gravitational field passing through the circle. Since we are interested in the flux passing through the entire ring, we can divide this by the area of the ring to get the average gravitational field at the center of the ring. This can be written as:

g = -2πGM/(πR^2) = -2GM/R^2

Using the equation of hydrostatic equilibrium that you have written, we can now solve for the vertical density stratification in terms of g and the pressure gradient. This will give us the Gaussian density profile that we are looking for.

I hope this helps. Let me know if you have any further questions.