Scales

  • Black holes are described by 3 parameters: Mass, spin, and charge
    • In practice, astrophysical black holes have no charge, since they are embedded in a plasma which “shorts out” the hole
  • We can define some characteristic scales of black holes
    • $R_{s} = \frac{2GM}{c^{2}}$ is the Schwartzchild radius
    • $\Delta t_{LC} = \frac{R_{s}}{c}$ is the light crossing time (roughly the time needed to cross the black hole, up to a scale factor)
  • How bright can be a black hole be?
    • We would want to convert all of the black hole’s mass into radiation in the relavent time scale
      • $L_{max} = \frac{Mc^{2}}{\frac{2GM}{c^{3}}} \approx \frac{c^{5}}{G} \approx 10^{59} \frac{erg}{s}$
        • For reference, supernovae are around $10^{52}$
      • What is the luminosity of the observable universe (excluding the blackholes)?
        • This is Fermi estimation problem: Find the mass of the sum, calculate the mass to energy conversion, and calculate the time needed for a photon to escape the center
          • The escape time is roughly a billion years, which gives a luminosity of around $10^{37}$
          • There are roughly $10^{10}$ stars in the universe. Assume all stars are like the sun
          • There are also roughly $10^{10}$ galaxies in the universe
          • Combining these gives that the luminosity of the universe is $10^{53}$
      • The point of this is that gravity is very efficient at converting mass to energy
    • If the black hole is moving close to the speed of light towards you (pileup of photons), then the observed luminosity can exceed this threshold
      • Gamma ray bursts (GRBs) are the prime example of this boosted luminosity

Eddington Limit

  • Imagine that you have a star which is gravitationally bound and emits radiation.
    • These two forces oppose each other: gravity attracts to the center, while radiation pressure pushes the star boundary outwards
    • Larger stars have more mass, but they also tend to produce more radiation pressure
    • The Eddington limit is when these two forces balance each other
      • Define $\dot{m}$ as the accretion rate (gas falling onto a compact object or star)
      • The luminosity is $L = \frac{GM\dot{m}}{r}$, where $\frac{M}{r}$ is the compactness
        • The associated Eddington luminosity is an upper bound on the brightness of these objects (there are exceptions though)
  • Let’s look at the simplest case: We have a fully ionized Hydrogen gas (plasma) accreting isotropically onto a star/ compact object
    • Electrons getting accelerated around protons produces radiation, which creates the outward pressure

Thomson Scattering

  • Imagine a EM wave propagating along the $\hat{z}$ direction. This will oscillate an electron in a direction transverse to z (fix this direction to be x)
    • This oscillating generates the radiation
  • Newton’s 2nd Law, and letting $E(z,t) = exp(i\omega t) \hat{x}$, the resulting motion is $x(t) = A exp(i\omega t)$ where $A = \frac{qE(z)}{m\omega^{2}}$
    • The cross section scales like $A^{2}$, which means that the cross section of the proton is roughly 1 million times smaller than the electron
      • $\frac{\sigma_{Tp}}{\sigma_{Te}} = \frac{m_{e}^{2}}{m_{p}^{2}}$
  • The radiation force is defined as the energy flux times the cross section divided by the speed of light
  • $F = \frac{L}{4\pi r^{2}} \rightarrow f_{rad} = \frac{\sigma_{T} L }{4\pi r^{2} c}$
  • Setting the gravitational force (ignoring the electron mass) equal to the radiative force (ignoring the proton cross section) gives the target luminosity (re: Eddington Luminoosity) as $L_{edd} = \frac{4\pi c G M m_{p}}{\sigma_{T}}$
    • Setting the mass scale to that of the sun, we get that $L_{Edd} = 1.3E38 (\frac{M}{M_{*}}) \frac{ergs}{s}$
    • For a black hole, $L_{edd} = 1.5E45 M_{7} \frac{erg}{s}$ where $M_{7} = \frac{M}{1E7 M_{*}}$
  • If we assume that the object is a perfect blackbody, we can define an effective temperature $L = 4\pi R^{2} \sigma_{B} T_{eff}^{4}$

Variability

  • The acceleration is $\frac{GM}{r^2}$. Suppose that the radiation pressure suddenly vanished. How long would it take for the star to collapse?
    • Naively, one would say that $\frac{1}{2} a t_{dyn}^{2} = R \rightarrow t_{dyn} = \sqrt{\frac{2R^{3}}{GM}}$
      • Assuming that force at star radius is the same for all smaller radius
      • This is roughly $t_{dyn} = \frac{1}{\sqrt{G\rho}}$
    • $t_{dyn}$ sets the timescale for how quickly the luminosity of the star can change