Abstract
We would like to know how the sun is being "supported". We assume that this mechanism is hydrostatic equilibrium, but to be sure we work through the derivation.
Introduction
We know that the sun is somehow being prevented from gravitational contraction. Our theory is that it is supported by hydrostatic equilibrium, which means that the internal pressure provides an opposing support force. We calculate the gravitational force on a mass shell, the pressure required to balance it, and then derive the force equation for hydrostatic equilibrium.
Methods and Results
We first assume the Sun to be a spherical gas cloud with density ρ(r). We consider a differential mass shell of this sphere with radius r. We recall that the volume of a sphere is 4/3πr3 and that the differential volume is its derivative. Then we get a differential mass dM:
Hydrostatic Equilibrium and the Sun
- Filed under: hydrostatic equilibrium, worksheet write up
We know the equation for universal gravitation:
\rho \left ( r \right )4\pi r^{2}dr}{r^{2}}=-GM\left ( r \right )\rho \left ( r \right )4\pi dr "dF_{g}=\frac{GM\left ( r \right )\rho \left ( r \right )4\pi r^{2}dr}{r^{2}}=GM\left ( r \right )\rho \left ( r \right )4\pi dr")
=\frac{dF_{g}}{A}=-\frac{GM\left ( r \right )\rho \left ( r \right )4\pi dr}{4\pi r^{2}}=-\frac{GM\left ( r \right )\rho \left ( r \right )dr}{r^{2}} "dP\left ( r \right )=\frac{dF_{g}}{A}=\frac{GM\left ( r \right )\rho \left ( r \right )4\pi dr}{4\pi r^{2}}=\frac{GM\left ( r \right )\rho \left ( r \right )dr}{r^{2}}")
}{dr}=-\frac{GM(r)\rho (r)}{r^{2}} "\frac{dP(r)}{dr}=-\frac{GM(r)\rho (r)}{r^{2}}")
Here we let M be the total mass enclosed by the mass shell and m be the differential mass element. As a result, we get the differential gravitational force to be:
We know that pressure is equal to force divided by area. So we can say:
Now dividing by dr on both sides of the equation we arrive at the equation of hydrostatic equilibrium:
Conclusions
We have derived from simple physical laws that the equation for hydrostatic equilibrium is a plausible explanation for the way the sun is supported. A quick search shows that we are indeed correct. Hooray!
Hot!
(Sorry. Couldn't resist. :) )
what is the meaning of dP? pressure pushes in all directions. so what is this pressure force? would a star with constant pressure throughout be able to hold itself up?