Orbital Frequency ○◂｜encounter｜ENC｜20260908163317-00-•

In an astrophysical environment, a BH is completely specified in GR by two numbers, its mass $M$ and its specific angular momentum or spin $a=J/cM$, where $J$ is the BH angular momentum and $c$ is the speed of light. The spin value is conveniently expressed in terms of a dimensionless spin parameter, $a_{\ast }=a/R_g$, where the gravitational radius is $R_g\equiv GM/c^2$. The mass simply supplies a scale, whereas the spin changes the geometry. The value of $a_{\ast }$ lies between 0 for a Schwarzschild hole and 1 for a maximally-rotating Kerr hole. A defining property of a BH is its event horizon, the immaterial surface that bounds the interior region of space-time that cannot communicate with the external universe. The event horizon, the existence of an innermost stable circular orbit (ISCO), and other properties of BHs are discussed in many texts (e.g., Shapiro & Teukolsky 1983; Kato et al. 1998). The radius of the event horizon of a Schwarzschild BH ($a_{\ast }=0$) is $R_S=2R_g=30$ km$(M/10M_{\odot })$, the ISCO lies at $R_{ISCO}=6R_g$, and the corresponding maximum orbital frequency is ${\nu }_{ISCO}=220$ Hz$(M/10M_{\odot }{)}^{-1}$. For an extreme Kerr BH ($a_{\ast }=1$), the radii of both the event horizon and the ISCO (prograde orbits) are identical, $R_K=R_{ISCO}=R_g$, and the maximum orbital frequency is ${\nu }_{ISCO}=1615$ Hz$(M/10M_{\odot }{)}^{-1}$.

Source: Remillard & McClintock 2006 ┃ X-Ray Properties… ⚬𓂃

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