The things you can do with Hydrostatic Equilibrium

We are bounded in a nutshell of Infinite Space: Reading #4: The things you can do with Hydrostatic Equilibrium

In the reading of chapter 3 of Maoz’s Astrophysics in a Nutshell, we notice the several references to the Thompson Cross Section, the area of an electron which is hit by energy from the star which further causes radiation to escape the star in many ways. In fact there, are some stars which are no longer supported by the pressure the matter exerts, rather the radiation they produce counteracts the gravity! These stars, of sizes between \( 10 - 300 M_\odot\) are maintained by this radiation pressure, and as such have a different equation describing the relationship of pressure and gravity, but in a way a bit more useful to us. These stars are the examples of a higher than average mass-energy conversion efficiency in a star, which can be described mathematically so we understand its inherent brightness, its Luminosity. We know that Luminosity is simply: \[ L = \frac{energy}{time}\] and Flux is: \[ F = \frac{L}{4\pi d^2}.\]

Now we star borrowing a bit from special relativity and the definition of an electron’s cross section. So the equation with the cross section would look like: \[ \sigma_T= 6.7 \times 10^{-25} cm^2\] \[ \sigma_T \times \frac{L}{4\pi d^2} = \frac{ energy }{area \times time} \times area\] Therefore, in this equation also lies a definition for energy, and from Einstein’s equations (just accept this premise, we can go into arriving it on another blog post) we know that \[ E = pc\] where p is the momentum associated to a massless particle, i.e. a photon. Therefore, we can take our original equation and divide the energy by c to find the momentum: \[ \sigma_T \times \frac{L}{4\pi d^2 c } = \frac{ momentum }{area \times time} \times area,\] which allows for the equation to be defined for a Force: \[ \frac{ momentum }{time} = \frac{dP}{dt} = F.\] And as we saw in a previous post, the forces must equal each other to achieve hydrostatic equilibrium, in this case between gravity and radiation.

Gravitational force would be defined as: \[F_g = \frac{GM}{r^2} (m_{protons} + m_{electrons})\] and since the mass of electrons is negligible, the equations simplifies to: \[ F_g = \frac{GM}{r^2} (m_{protons}) .\]

Now we can equal our definitions of radiative and gravitational forces, and find that:  \[\frac{GM}{r^2} (m_{protons}) = \frac{L \cdot \sigma_T }{4\pi d^2 c },\] and we can now see that most of these numbers are simply constants, and we can pull out a very meaningful expression from this: \[ L = \left(\frac{4\pi G \cdot m_p \cdot c}{\sigma_T}\right) M .\] This equation directly correlates the mass of the star to its maximum inherent luminosity, the greatest brightness it could achieve were it to efficiently convert its matter into energy. This equation further simplifies down to: \[ L = 1.4 \times 10^{38} \frac{erg}{s} \left(\frac{M}{M_\odot}\right),\] all of which was discovered by Arthur Eddington, and as such this equation is the Eddington Luminosity:   \[L_E = 1.4 \times 10^{38} \frac{erg}{s} \left(\frac{M}{M_\odot}\right).\]