[math]T^4® = \frac{3 G M \dot{M}}{8 \pi \sigma r^3} \left[1 - \sqrt{\frac{R}{r}} \right][/math]

 

[math]C_S^2 \approx \frac{k_B T}{\bar{m}}[/math]

 

[math]v_K^2 \equiv \frac{G M}{r}[/math]

 

[math]\dot{M}_{Edd} \equiv \frac{4 \pi c}{\sigma_T} \bar{m} R[/math]

Defining [math]\dot{M} \equiv \mu \dot{M}_{Edd}[/math], then w.h.t.:

 

[math]\therefore \left(\frac{C_S}{v_K}\right)^8 \approx \mu \left( \frac{k_B}{G M \bar{m}} \right)^3 \left( \frac{3 k_B c R}{2 \sigma \sigma_T} \right) r \left[1 - \sqrt{\frac{R}{r}} \right][/math]

 

[math] \approx 2 \times 10^{-10} \left( \frac{\bar{m}}{m_H} \right)^{-3} \left( \frac{M}{M_{\odot}} \right)^{-3} \left( \frac{R}{R_{\odot}} \right)^2 \mu \left[ x - \sqrt{x} \right][/math]

where we have defined [math]x \equiv r / R[/math]. Taking the eighth-root, w.h.t.:

 

[math]\frac{C_S}{v_K} \approx 0.06 \left( \frac{\bar{m}}{m_H} \right)^{-3/8} \left( \frac{M}{M_{\odot}} \right)^{-3/8} \left( \frac{R}{R_{\odot}} \right)^{1/4} \mu^{1/8} \left[ x - \sqrt{x} \right]^{1/8}[/math]

If we further assume a relativistic accretor, s.t. [math]R = \rho R_S[/math], where [math]R_S \approx 3 \; km[/math] per [math]M_{\odot}[/math], then w.h.t.:

 

[math]\frac{C_S}{v_K} \approx 0.003 \left( \frac{\bar{m}}{m_H} \right)^{-3/8} \left( \frac{M}{M_{\odot}} \right)^{-3/8} \rho^{2/8} \mu^{1/8} \left[ x - \sqrt{x} \right]^{1/8}[/math]

For NS & BH, [math]\rho \approx 3[/math], s.t.:

 

[math]\frac{C_S}{v_K} \approx 0.004 \left( \frac{M}{M_{\odot}} \right)^{-3/8} \left[ x - \sqrt{x} \right]^{1/8}[/math]

even assuming Eddington-rate accretion, of pure H plasma. For a typical NS ([math]M \approx 1.4 M_{\odot}[/math]), w.h.t.:

 

[math]\frac{C_S}{v_K} \approx 0.003 \left[ x - \sqrt{x} \right]^{1/8}[/math]

The fore-going formula only approaches parity, as [math]x \rightarrow 300[/math], at [math]\approx 1000 R_S[/math], or [math]\approx R_{\oplus} \approx R_{WD}[/math]. Inside of that radius, even as [math]v_K \rightarrow c[/math], Accretion Disk temperatures peak around 1-2 KeV << mc².

 

Does this imply, that inner-disk flows are "cold", Relativistically (and relatively) speaking ? Is this why Accretion Disks are "radiatively inefficient" s.t. "advective cooling (carrying the energy with the flow) dominates" ? As [math]v_K \rightarrow c[/math], even as [math]v_K >> C_S[/math], wouldn't most thermal emission be "headlighted" forward, with the flow ? How, then, would such inner-disk regions radiate ??

 

 

References:

 

Kolb. Extreme Environment Astrophysics.

Chris Done & Marek Gierlinski. Observing the effects of the event horizon in black holes, Mon. Not. R. Astron. Soc. 000, 1–15 (2002).

Could they radiate little once their matter has reached a near-equilibrium, and light only when new matter falls down from a direction out-of-plane?