# Intro - Stars and their Gravities At some point, gravity cannot push electrons down further thanks to the pauli exclusion principle. This means that the higher the mass you're trying to add, the higher the energy. When a degenerate gas gets heavier, it actually gets smaller --> this means that the gas gets more and more and more compressed, meaning that electrons have to fill all of the energy levels. at some point, helium can fuse, with a helium-fusing core surrounded by a hydrogen-burning shell. This shell arises from the residual heat from the degenerate helium, and is also very hot, forcing the star to puff up and turn red (as the total flux remains the same) as a star moves up the red giant phase, its temperature drops but its luminosity increases. this is because of a larger surface area + the hydrogen-burning shell is much smaller. red giants have opaque atmospheres - which will reduce their masses. # Degenerate States Strictly speaking, a degenerate state is when the electrons can only occupy 1 'energy state'. So, if you think about it, this also means that the kinetic energy of confinement HAS to be the same as the uncertainty in $p$ is 0 - so the uncertainty of $x$ can be anything! %%draw 3 panels - electron shrinking inwards... last one - ELECTRON CAN ONLY VIBRATE%% At high pressures, degeneracy can occur, therefore leading to the electron vibrating, much like the particles in a solid. The energy of vibration of each electron depends on the # Astronomy Degeneracy Case Study - The Chandrasekhar Mass Limit White dwarfs are the penultimate example of degenerate pressure around - nothing man-made even comes close to the pressures required for degeneracy to occur, and many white dwarves are both close to us and relatively bright in X-Rays. ![[degeneracy-diagram.png]] We can therefore create expressions to generalise the maximum possible mass of a white dwarf. This is given by taking the **Chandrasekhar mass** as follows: $m_{limit} = \frac{\omega\sqrt{ 3\pi }}{2}\left( \frac{\hbar c}{G} \right)^{\frac{3}{2}} \frac{1}{(\mu_{e}m_{H})^2}$ **Where:** - $\omega \approx$ 2.018, this comes from **the gradient** of the spherical harmonic (search this up!) - $\hbar$ is the **reduced planck constant**, the planck constant divided by $2\pi$. - $c$ is the speed of light - $G$ is the gravitational constant, $6.67 \times 10^{-11}$ - $\mu_{e}$ is the ratio of the **average molecular weight** of the white dwarf to each electron - the mass to charge ratio. - $m_H$ is the mass of the hydrogen atom