# Does the Sun have enough gravitation to bind its hydrogen/helium plasma?

"Does the Sun have enough gravitation to bind its hydrogen/helium plasma?"

So my question arises from the fact that high energy (or high temperature in other words) hydrogen or helium gas just explodes, instead of being bounded in one place. Then I guess the Sun has enough gravitation to bound the ionized particles. I want to calculate the gravitational potential energy and compare it with kinetic energy of the plasma, but I just don't have enough information on the density of the Sun at each radius (layer).

• Does the Sun have enough gravitation to bind its hydrogen/helium plasma? How could the answer be “No”? Sep 7 '20 at 3:55

Thermodynamics gives us the virial theorem: for any spherical, stable, self-gravitating distribution of particles, we know that its average total kinetic energy $$\langle T\rangle$$ and its average total potential energy $$\langle U\rangle$$ are related as follows:

$$2\langle T\rangle=-\langle U\rangle$$

In other words, the total kinetic energy of any such body is roughly half of what it would take to unbind the system. The Sun is a stable, self-gravitating spherical distribution of particles, so the virial theorem applies.

• I hope you don’t mind me adding the Wikipedia link. If you do, I will revert. Sep 7 '20 at 4:00
• @G.Smith Thanks for the improvement! No need to revert. Sep 7 '20 at 4:03

I just don't have enough information on the density of the Sun at each radius (layer).

According to NASA, the density profile is well-approximated by

$$D(x)=519x^4 −1630x^3 +1844x^2 −889x+155,$$

where $$D$$ is in grams per cubic centimeter and $$x$$ varies from $$0$$ at the center to $$1$$ at the surface.