A star is a glowing sphere of primarily hydrogen gas that can burn hydrogen into helium via nuclear fusion at the core. The diameter of a star can be from a few million meters to more than a trillion meters. There are stars 100 times less massive than the sun ($10^{30}$ kg) and 200 times more massive than the sun. The surface temperature of a star varies from two thousand to almost forty thousand K. And the central temperature can range from 1 million to almost 100 million K. The radius and temperature are directly related to the mass.

A star remains in equilibrium because of a perfect balance of its inward gravity and outward pressure. The equilibrium is expressed through the equation of hydrostatic equilibrium:

$$ \frac{dP}{dr} = -\rho g $$

where $P$ is the pressure, $r$ distance from the center, $\rho$ density and $g=GM_r/r^2$ the gravitational acceleration at radius $r$. Note that $G$ is the Newtonian constant and $M_r$ the mass enclosed by the radius $r$.

For the equilibrium, there must be a pressure gradient $dP/dr$ meaning the pressure must decrease with distance from the center. The pressure gradient $dP/dr$ works against $g(r)$ to keep the star in equilibrium.

The mass inside a star is distributed according to the mass conservation equation:

$$ \frac{dM_r}{dr} = 4\pi r^2 \rho. $$

The pressure $P$ given in the first equation comes from the gas pressure $P_g$ and the radiation pressure $P_r$. The gas pressure can follows a special form of the ideal gas law:

$$ P_g = \frac{\rho kT}{\mu m_H} $$

where $k$ is Boltzmann constant, $T$ temperature, $m_H$ the mass of a hydrogen atom and the mean molecular weight $\mu=\overline{m}/m_H$ where $\overline{m}$ is the average mass of a gas particle in the star.

The radiation pressure depends only on the temperature as

$$ P_r = \frac{1}{3} aT^4 $$

where $a=4\sigma/c$ is the radiation constant and $\sigma$ is the Stefan-Boltzmann constant. So the total pressure

$$ P = P_g + P_r = \frac{\rho kT}{\mu m_H} + \frac{1}{3} aT^4. $$

We get energy from the sun. Energy out of a star is measured using luminosity and the source of the luminosity are mainly gravity and the nucleus of atoms. The total mechanical energy of a star