Widdekind Posted October 1, 2011 Posted October 1, 2011 [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).
Enthalpy Posted October 2, 2011 Posted October 2, 2011 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?
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