Assume a test star in circular orbit: [ \fracmv^2r = \fracG m M(r)r^2 \quad \Rightarrow \quad v(r) = \sqrt\fracG M(r)r ] For (v(r)) constant, we need (M(r) \propto r).
The maximum steady luminosity is set by radiation pressure balancing gravity (Eddington limit): [ L_\textEdd = \frac4\pi G M_\textBH m_p c\sigma_T \approx 1.3 \times 10^38 \left(\fracM_\textBHM_\odot\right) \text erg/s ] For (M_\textBH = 10^8 M_\odot), (L_\textEdd \approx 10^46) erg/s. Over (10^8) years, total energy released (\sim 10^61) ergs – consistent with observations. physics galaxy discussion questions solutions
Run through the "Big Three" before writing a single equation: Assume a test star in circular orbit: [