Negative Dynamical Friction on compact objects moving through dense gas
Almog Yalinewich
27.6.19
Norm Group meeting
Dynamical Friction
Gravity Assist
M
b
\Delta v_{\perp}
v
v^2 \gg \frac{G M}{b}
\Delta v_{\perp} \approx \frac{G M}{b v}
\Delta v_{\parallel}
\Delta v_{\parallel} \approx \frac{\Delta v_{\perp}^2}{v} \approx \frac{G^2 M^2}{b^2 v^3}
v^2 \ll \frac{G M}{b}
\Delta v_{\parallel} \approx 2 v
Drag Force
R_b
R_b \approx \frac{G M}{v^2}
F \approx \rho v^3 R_b^2 \approx \rho G^2 M^2 /v
Why do we care?
violent relaxation in galaxies
Migration and eccentricity damping in proto - planetary discs
Negative Dynamical Friction
Standoff Distance
v_{\star}
v_w
2 R_0
R_0 \approx \sqrt{\frac{\dot{M v_w}}{4 \pi \rho_0 v_{\star}^2}}
\dot{M}
\rho_0
Equating dynamical pressure
\frac{R \left(\theta\right)}{R_0} = \frac{\sqrt{3 \left(1-\theta \cot \theta\right)}}{\sin \theta}
u = v_{\star}/v_w
\varpi = 3 \left(1- \theta \cos \theta\right)
\frac{\sigma}{R_0 \rho_0} = \frac{\left[2u\left(1-\cos \theta\right) + \varpi^2\right]^2}{2 \varpi \sqrt{\left(\theta - \sin \theta \cos \theta\right)^2 + \left(\varpi-\sin^2 \theta\right)^2}}
Wake Gravity
\rho = \sigma \delta \left(r-R\right) + \left\{ \begin{matrix} \rho_0 & r>R \\ \frac{\dot{M}}{4 \pi r^2 v_w} & {R < r} \end{matrix} \right.
a = G \int_0^{\infty} r^2 dr \int_0^{\pi} 2 \pi \sin \theta d \theta \cos \theta \frac{\rho}{r^2}
a \propto \int_0^{\pi} d \theta \cos \theta \frac{R}{R_0} \times
\left[\frac{3}{2} \left(1+\frac{2u \left(1-\cos \theta\right)}{R/R_0}\right)^2 - 2 \left(1+\frac{u^2 \sin^2 \theta}{R^2/R_0^2}\right)\right]
Wake Gravity, cont'd
a \approx 8.18 \cdot \frac{4 \pi}{3} G \rho_0 R_0
u \ll 1
a \approx - 0.975 \cdot \frac{4 \pi}{3} G \rho_0 R_0 u^2
u \gg 1
a = 0
u \approx 1.71
Numerical Verification
Accretion Powered Wind
Trajectory
\dot{M} \propto \dot{M}_B \approx \rho v_{\star} R_b^2 \approx \frac{\rho G^2 M^2}{v_{\star}^3}
a \approx G \rho R_0 \approx G \rho \sqrt{\frac{\dot{M} v_w}{\rho v_{\star}^2}} \propto v_{\star}^{-5/2}
v_{\star} \propto t^{2/7}
Super Eddington Accretion
v_k \approx v_w \approx \chi \sqrt{\frac{\dot{M}_{\rm Edd}}{\dot{M}}} c
Calibration from SS433
v_w \approx 1500 \sqrt{10^3 \frac{\dot{M}_{\rm Edd}}{\dot{M}}} \, \rm \frac{km}{s}
\dot{M} \approx \frac{L}{v_k^2} \approx \frac{L_{\rm Edd}}{c^2} \approx \dot{M}_{\rm Edd}
SS433
Condition for Acceleration
v_w > v_{\star}
\rho < 3 \cdot 10^{-10} \left(10 \frac{M_{\odot}}{M}\right) \left(\frac{v_{\star}}{100 \, \rm km/s}\right) \, \rm \frac{g}{cm^3}
Self Gravity Limit on AGN discs
\rho < 7 \cdot 10^{-12} \left(\frac{\mathcal{M}}{10^8 M_{\odot}}\right) \left(\frac{0.1 \, \rm pc}{r}\right)
Application to Grazing Accretion
Black holes in AGN Discs
Reverse migration?
Growth of Supermassive Black Holes
Negative dynamical friction can impede black hole sinking, which is necessary for SMBH growth
negative dynamical friction
By almog yalinewich
