Starstruck: Hydrostatic Equilibrium and the Virial Theorem

Introduction

Here we consider how to show the equation for hydrostatic equilibrium. The methodology used is a force balance followed by some algebra.

Solutions

Question: Find the differential equation describing hydrostatic equilibrium.

a. Write an expression for the differential element dM contained in the cell.

Considering a mass shell with thickness dr and density p(r), we can write the equation for the differential element contained in this cell by finding the volume of the shell and multiplying this by the density, as follows:

$\bg_black dM = p(r)4\pi r^2dr$

b. What is the gravitational force, F_g, acting on the shell?

We know the differential element for the mass (that is, the mass of an infinitely thin shell). We can use the general equation for gravitational force:

$\bg_black F_{g}=-\frac{{GMm}}{r^2}$

When M and m describe the two masses. However! We do not have two masses here. We have a mass, M, and the differential element dM that represents the mass of the shell. So, we shold rewrite the expression for F_gas:

$\bg_black F_{g}=-\frac{{GM(r)dm}}{r^2}$

When M(r) describes the mass of the sun up to a radius r.

c. Assuming there is a pressure P(r) acting to balance F_g, write the net force equation in terms of the relevant qualities.

We know:

$\bg_black P(r)=\frac{force}{area}$

We want to show that the forces balance, so we can set this equal to the expression we found for F_gin part b. Solving for the F_p, force of pressure, and substitute this expression in:

$\bg_black P(r){4\pi r^{2}}=F_{p} = F_{g} = -G\frac{M(r)p(r)dm}{r^{2}}$

It is correct that the signs are opposite, because the forces are acting in opposite directions.

d. Show the equation of hydrostatic equilibrium.

To show that this equation fits the form of the equation of hydrostatic equilibrium, we can substitute in the value of dm that we found in part a, and then manipulate the equation algebraically:

$\bg_black P(r){4\pi r^{2}} = -G\frac{M(r)dm}{r^{2}}$

$\bg_black P(r){4\pi r^{2}} = -G\frac{M(r))p(r)4\pi r^{2}dr)}{r^{2}}$

$\bg_black P(r) = -G\frac{M(r)p(r)dr}{r^{2}}$

$\bg_black \frac{P(r)}{dr} = -G\frac{M(r)p(r)}{r^{2}}$

This is the equation of hydrostatic equilibrium!!

Conclusion

We have shown the derivation for the equation of hydrostatic equilibrium, using only simple physics and algebra. Pretty cool!!

Acknowledgements

Thanks to Iryna and Monica for working with me on this worksheet – see their blogs for the discussions of other questions!

Also, I would like to thank my classmates whose blogs I have been reading - they gave me the idea to use an online Latek editor instead of typing equations in plain text. It looks way better this way! Thanks guys!

Starstruck

Thursday, October 27, 2011

Hydrostatic Equilibrium and the Virial Theorem

No comments:

Post a Comment