Flow of Gas around the Galactic Centre
Almog Yalinewich
Introduction
Journey to the Galactic Centre
Nuclear Star Cluster
Wind Emitting Stars
Chandra view of the Galactic Centre
Simplified Model
Other Galactic Centres
Theoretical Model
Conservation of Mass
\frac{1}{r^2} \frac{d}{dr} \left( \rho v r^2\right) = D r^{-\eta}
Conservation of Energy
\frac{1}{r^2} \frac{d}{dr} \left( \rho v r^2 \left(\frac{1}{2} v^2 + \frac{\gamma}{\gamma-1} \frac{p}{\rho} - \frac{G M}{r} \right)\right) =
= D r^{-\eta} \left(\frac{1}{2} v_w^2 - \frac{G M}{r} \right)
Conservation of Momentum
\rho v \frac{d v}{d r} + \frac{1}{\rho} \frac{d p}{d r}= -\frac{G M}{r^2} \rho - D r^{-\eta} v
Integrated Forms
\rho v r^2 = D \frac{r^{3-\eta} - r_{st}^{-\eta}}{3-\eta}
\frac{1}{2} v^2 + \frac{c_s^2}{\gamma-1} = GM \frac{r^{3-\eta} + \left(2-\eta\right) r_{st}^{3-\eta} - \left(3-\eta\right) r r_{st}^{2-\eta}}{\left(2-\eta\right) r \left(r^{3-\eta}-r_{st}^{3-\eta}\right)}
r_{st}
Stagnation radius
Mass conservation
Energy conservation
Dimensionless Form
r_b = \frac{G M}{v_w^2}
\rho_b = D \frac{r_b^{1-\eta}}{v_w}
\tilde{r} = \frac{r}{r_b}
\tilde{v} = \frac{v}{v_w}
\tilde{c}_s = \frac{c_s}{v_w}
\tilde{\rho} = \frac{\rho}{\rho_b}
Analytic Solution
\eta = 5/2, \gamma=5/3
\tilde{\rho} = \frac{4\sqrt{6}}{3 \tilde{r}^{3/2}}
\tilde{v} = \frac{\sqrt{6}}{4} - \frac{1}{\sqrt{\tilde{r}}}
\tilde{c}_s^2 = \frac{5}{24} + \frac{1}{3 \tilde{r}}
\tilde{r}_{st} = \frac{8}{3}
Generic Case
\frac{d m}{d \tilde{r}} = f \left(\tilde{r}, m\right)
Mach number ODE
\tilde{r}
m
\tilde{r}
m
Wrong
\tilde{r}_{st}
Right
\tilde{r}_{st}
Inner Asymptote
\lim_{\tilde{r}\to0} m =- \sqrt{3}
\gamma = 5/3
Always supersonic
\gamma < 5/3
\lim_{\tilde{r}\to0} m =- \infty
Boundary condition: finite when
\frac{dm}{d \tilde{r}}
m=-1
Inner Asymptote
v \propto -\frac{1}{\sqrt{r}}
\rho \propto r^{-3/2}
Independent of
\eta
Mass dominated by stagnation radius
Outer Asymptote
\lim_{\tilde{r}\to\infty } m = \sqrt{\frac{\eta-1}{\gamma \left(3-\eta\right)}}
No steady state solution when
\eta < 1
Easiest to understand for the case
\eta = 0
Supersonic when
\eta > \frac{3 \gamma + 1}{\gamma+1}
Outer Asymptote
v \propto const
\rho \propto r^{1-\eta}
Hydrodynamic Profiles
\gamma = \frac{5}{3}, \eta=1.9
\gamma = \frac{4}{3}, \eta=1.9
Hydrodynamic Profiles
\gamma = \frac{5}{3}, \eta=1.9
\gamma = \frac{4}{3}, \eta=1.9
Hydrodynamic Profiles
\gamma = \frac{5}{3}, \eta=1.9
\gamma = \frac{4}{3}, \eta=1.9
Truncation
\dot{\rho}
r
r_t
Steady state always exists if mass injection is truncated
Truncation 2
\dot{\rho}
r
r_t
\rho
r
r_t
r^{-2}
Truncation 3
\dot{\rho}
r
r_o
\rho
r
r^{-2}
r_i
r_o
r_i
Truncation 4
\dot{\rho}
r
r_o
\rho
r
r^{-2}
r_i
r_o
r_i
GM
r^{-3/2}
Radiation
Bremsstrahlung
\varepsilon_{\nu} \approx \frac{m_e c^2}{r_e^3} \sqrt{\frac{m_e c^2}{k T}} \left(n r_e^3\right)^2 \exp \left(- \frac{h \nu}{k T} \right)
L_{\nu} \approx \int \varepsilon_{\nu} r^2 dr
Truncation necessary to avoid divergence
Synthetic Spectrum
\tilde{\nu} = \frac{h \nu}{m_p v_w^2}
\tilde{\nu} \tilde{L} = \frac{\nu L}{\dot{M} c^2}
\gamma = \frac{5}{3}, \eta = \frac{5}{2}
Retrieving the Mass accretion rate
\dot{M} \approx 7.5 \cdot 10^{-4} M_{\odot} \, yr^{-1} \times
\left(\frac{\nu_0 L_{\nu_0}}{10^{38}\,{ erg\,s^{-1}}}\right)^{1/2} \left(\frac{M}{4\cdot 10^6 M_{\odot}}\right)^{1/2} \left(\frac{v_w}{500 \, km/s}\right)^{-1}
Applications
Milky Way Galactic Centre
Density Comparison
Data favours
\eta = 0
From Quataert 2004
Surface Brightness
data favours
\eta = 3
Surface Brightness 2
Different normalisation
All models agree
outside Chandra PSF
Unresolved central source?
Stellar Density
Genzel et al 2010
Other Galactic Centres
Swift 1644+57
Achromatic break
at 10 days
Tidal Disruption Event Jet
Jet Break?
Break when reverse shock
reaches jet base?
Break when swept up mass is comparable to jet energy?
Density Break?
Break radius
t_b \cdot c \approx 0.01 \, pc
fallback rate implies
M \approx 10^6 M_{\odot}
Bondi radius
r_b \approx 0.004 pc \frac{M}{10^6 M_{\odot}} \left( \frac{v_w}{1000 km/s} \right)^{-2}
3 r_b \approx t_b \cdot c
Truncation radius?
Conclusion
Steady state, smooth, spherically symmetric flow model
prediction for spectrum from Bremsstrahlung
Resolve conflicting results for SGR A*
Proposition for achromatic breaks in TDEs
Flow of Gas around the Galactic Centre TAU
By almog yalinewich
