Why Stars are Stable

By Michael on September 18, 2006 at 3:35 pm | In Astrophysics, Blog Posts | No Comments

Hey slackers!

So I am taking a class about the structure and evolution of stars. I have been taking classes for 6 years (albeit one class at at a time) just so I would have the math and physics background to take this class. As someone that got into the quantitative side of astronomy via variable stars, I want to understand stars. Lucky you, I’m going to take you along on some of the concepts as they are introduced in this class. Today I am going to talk about hydrostatic equilibrium. It’s a fancy term that explains why stars are stable most of their lives. It goes like this…

We all know that gravity is an attractive force. So if I have a little blob of material in a star, gravity is going to try to pull that blob towards the center of the star. But we just said that stars are generally stable, meaning that the material can’t be free-falling into the star — something is resisting the force of gravity. In physics if things aren’t moving we call them “static”. So if this blob of material in a star or the sun is static, there must be some force opposing gravity. Mathematically we can write this as an equality:

Here the “g” stands for gravity and the “b” stands for buoyancy. We don’t know the buoyancy force is yet, but we know something is pushing that material outward.

We know how to write the force of gravity:

Newton gave us this equation. The G is the gravitational constant, the big M is the mass of the sun or star and the small m is the mass of the little blob of material.

Pressure is force per unit area. So if I push my hand on your back, the pressure you feel is the amount of force I am exerting divided by the area of my hand. If I used the same amount of force but with a much smaller area, say the point of a knife, the pressure is much greater and I stab you in the back. Much more force can be applied on your back, say by your seat on an airplane as it takes off, provided the area is greater.

We know that gas has pressure, just like the atmospheric pressure here on the surface of the earth. There is a pretty simple formula for gas pressure:

Ignore the n and and the R for now — the P stands for pressure and the T stands for temperature. So the pressure is proportional to the temperature. The force from pressure is a little different in this case, though, because in order for gas pressure to exert a force it has to be unbalanced. In a star (and on Earth) the pressure is greater the closer you get to the center. So even for a small blob of material, the pressure at the bottom is greater than the pressure at the top. Physicists call this a gradient. So the force from gas pressure in this case is related to the difference in pressure between the top and the bottom of the blob. We are calling the “buoyancy” force above by its true name now, the gas pressure.

If you are not familiar with calculus, the small “d” in front of the P (for pressure) and A (for area) means a vanishing small change. So the equation above is stating that the gas pressure is equal to a small change in pressure times the tiny little area of our blob of gas.

With me so far? Now the point: we set the two forces to equal each other, since the material is static and not moving.

One last little bit of algebra, replacing the small “m” on the left with the density and the volume (because density is mass divided by volume so mass is density times volume) lets us cancel some terms and rearranging others we get:

So the gravitational force is equal to the ratio of the change in pressure (dP) to the change in radius (dr). The forces balance and the star is stable for millions of years. Were these two forces to become unbalanced it would cause sizable changes to the star in a matter of hours!

This is all a little deep, perhaps, but congrats for wading through it. This is an extremely important concept in stellar structure. Next time I’ll talk about some cool calculations you can do with this to learn things about the Sun and stars.

More info and illustrations of hydrostatic equilibrium are here: http://jersey.uoregon.edu/~imamura/208/jan27/mech.html