Flow of Gas around the Galactic Centre

Almog Yalinewich

ITC 9.8.18

Introduction

Journey to the Galactic Centre

Nuclear Star Cluster

Genzel et al 2010

Wind Emitting Stars

Rockefeller et al 2004

Chandra view of the Galactic Centre

Baganoff et al 2001

Other Quiescent Galactic Centres

Krolik et al 2016

Theoretical Model

Conservation Laws

\frac{1}{r^2} \frac{d}{dr} \left( \rho v r^2\right) = D r^{-\eta}
1r2ddr(ρvr2)=Dr−η\frac{1}{r^2} \frac{d}{dr} \left( \rho v r^2\right) = D r^{-\eta}
\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) =
1r2ddr(ρvr2(12v2+γγ−1pρ−GMr))=\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)
=Dr−η(12vw2−GMr)= D r^{-\eta} \left(\frac{1}{2} v_w^2 - \frac{G M}{r} \right)
\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
ρvdvdr+1ρdpdr=−GMr2ρ−Dr−ηv\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

Mass

Energy

Momentum

Integrated Forms

\rho v r^2 = D \frac{r^{3-\eta} - r_{st}^{-\eta}}{3-\eta}
ρvr2=Dr3−η−rst−η3−η\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)}
12v2+cs2γ−1=GMr3−η+(2−η)rst3−η−(3−η)rrst2−η(2−η)r(r3−η−rst3−η)\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}
rstr_{st}

Stagnation radius

Mass conservation

Energy conservation

Dimensionless Form

r_b = \frac{G M}{v_w^2}
rb=GMvw2r_b = \frac{G M}{v_w^2}
\rho_b = D \frac{r_b^{1-\eta}}{v_w}
ρb=Drb1−ηvw\rho_b = D \frac{r_b^{1-\eta}}{v_w}
\tilde{r} = \frac{r}{r_b}
r~=rrb\tilde{r} = \frac{r}{r_b}
\tilde{v} = \frac{v}{v_w}
v~=vvw\tilde{v} = \frac{v}{v_w}
\tilde{c}_s = \frac{c_s}{v_w}
c~s=csvw\tilde{c}_s = \frac{c_s}{v_w}
\tilde{\rho} = \frac{\rho}{\rho_b}
ρ~=ρρb\tilde{\rho} = \frac{\rho}{\rho_b}

Analytic Solution

\eta = 5/2, \gamma=5/3
η=5/2,γ=5/3\eta = 5/2, \gamma=5/3
\tilde{\rho} = \frac{4\sqrt{6}}{3 \tilde{r}^{3/2}}
ρ~=463r~3/2\tilde{\rho} = \frac{4\sqrt{6}}{3 \tilde{r}^{3/2}}
\tilde{v} = \frac{\sqrt{6}}{4} - \frac{1}{\sqrt{\tilde{r}}}
v~=64−1r~\tilde{v} = \frac{\sqrt{6}}{4} - \frac{1}{\sqrt{\tilde{r}}}
\tilde{c}_s^2 = \frac{5}{24} + \frac{1}{3 \tilde{r}}
c~s2=524+13r~\tilde{c}_s^2 = \frac{5}{24} + \frac{1}{3 \tilde{r}}
\tilde{r}_{st} = \frac{8}{3}
r~st=83\tilde{r}_{st} = \frac{8}{3}

Generic Case

\frac{d m}{d \tilde{r}} = f \left(\tilde{r}, m\right)
dmdr~=f(r~,m)\frac{d m}{d \tilde{r}} = f \left(\tilde{r}, m\right)

Mach number ODE

\tilde{r}
r~\tilde{r}
m
mm
\tilde{r}
r~\tilde{r}
m
mm

Wrong

\tilde{r}_{st}
r~st\tilde{r}_{st}

Right

\tilde{r}_{st}
r~st\tilde{r}_{st}

Inner Asymptote

\lim_{\tilde{r}\to0} m =- \sqrt{3}
lim⁡r~→0m=−3\lim_{\tilde{r}\to0} m =- \sqrt{3}
\gamma = 5/3
γ=5/3\gamma = 5/3

Always supersonic

\gamma < 5/3
γ&lt;5/3\gamma &lt; 5/3
\lim_{\tilde{r}\to0} m =- \infty
lim⁡r~→0m=−∞\lim_{\tilde{r}\to0} m =- \infty

Boundary condition: finite          when

\frac{dm}{d \tilde{r}}
dmdr~\frac{dm}{d \tilde{r}}
m=-1
m=−1m=-1

Inner Asymptote

v \propto -\frac{1}{\sqrt{r}}
v∝−1rv \propto -\frac{1}{\sqrt{r}}
\rho \propto r^{-3/2}
ρ∝r−3/2\rho \propto r^{-3/2}

Independent of

\eta
η\eta

Mass dominated by stagnation radius

Outer Asymptote

\lim_{\tilde{r}\to\infty } m = \sqrt{\frac{\eta-1}{\gamma \left(3-\eta\right)}}
lim⁡r~→∞m=η−1γ(3−η)\lim_{\tilde{r}\to\infty } m = \sqrt{\frac{\eta-1}{\gamma \left(3-\eta\right)}}

No steady state solution when

\eta < 1
η&lt;1\eta &lt; 1

Easiest to understand for the case

\eta = 0
η=0\eta = 0

Supersonic when

\eta > \frac{3 \gamma + 1}{\gamma+1}
η&gt;3γ+1γ+1\eta &gt; \frac{3 \gamma + 1}{\gamma+1}

Outer Asymptote

v \propto const
v∝constv \propto const
\rho \propto r^{1-\eta}
ρ∝r1−η\rho \propto r^{1-\eta}

Hydrodynamic Profiles

\gamma = \frac{5}{3}, \eta=1.9
γ=53,η=1.9\gamma = \frac{5}{3}, \eta=1.9
\gamma = \frac{4}{3}, \eta=1.9
γ=43,η=1.9\gamma = \frac{4}{3}, \eta=1.9

Hydrodynamic Profiles

\gamma = \frac{5}{3}, \eta=1.9
γ=53,η=1.9\gamma = \frac{5}{3}, \eta=1.9
\gamma = \frac{4}{3}, \eta=1.9
γ=43,η=1.9\gamma = \frac{4}{3}, \eta=1.9

Hydrodynamic Profiles

\gamma = \frac{5}{3}, \eta=1.9
γ=53,η=1.9\gamma = \frac{5}{3}, \eta=1.9
\gamma = \frac{4}{3}, \eta=1.9
γ=43,η=1.9\gamma = \frac{4}{3}, \eta=1.9

Truncation

\dot{\rho}
ρ˙\dot{\rho}
r
rr
r_t
rtr_t

Steady state always exists if mass injection is truncated

Truncation 2

\dot{\rho}
ρ˙\dot{\rho}
r
rr
r_t
rtr_t
\rho
ρ\rho
r
rr
r_t
rtr_t
r^{-2}
r−2r^{-2}

Truncation 3

\dot{\rho}
ρ˙\dot{\rho}
r
rr
r_o
ror_o
\rho
ρ\rho
r
rr
r^{-2}
r−2r^{-2}
r_i
rir_i
r_o
ror_o
r_i
rir_i

Truncation 4

\dot{\rho}
ρ˙\dot{\rho}
r
rr
r_o
ror_o
\rho
ρ\rho
r
rr
r^{-2}
r−2r^{-2}
r_i
rir_i
r_o
ror_o
r_i
rir_i
GM
GMGM
r^{-3/2}
r−3/2r^{-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)
εν≈mec2re3mec2kT(nre3)2exp⁡(−hνkT)\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
Lν≈∫ενr2drL_{\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}
ν~=hνmpvw2\tilde{\nu} = \frac{h \nu}{m_p v_w^2}
\tilde{\nu} \tilde{L} = \frac{\nu L}{\dot{M} c^2}
ν~L~=νLM˙c2\tilde{\nu} \tilde{L} = \frac{\nu L}{\dot{M} c^2}
\gamma = \frac{5}{3}, \eta = \frac{5}{2}
γ=53,η=52\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
M˙≈7.5⋅10−4M⊙&ThinSpace;yr−1×\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}
(ν0Lν01038&ThinSpace;erg&ThinSpace;s−1)1/2(M4⋅106M⊙)1/2(vw500&ThinSpace;km/s)−1\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
η=0\eta = 0

From Quataert 2004

Surface Brightness

data favours

\eta = 3
η=3\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

Berger et al 2011

Tidal Disruption Event Jet

Achromatic Jet Breaks

Break when reverse shock

reaches jet base?

Break when swept up mass is comparable to jet energy?

Chromatic Density Breaks

Truncation radius vs Bondi radius

relativistic vs Newtonian

Optically thick vs optically thin

Gas Density => Density of wind emitting stars => star density

Conclusion

Steady state, smooth, spherically symmetric flow model

prediction for spectrum from Bremsstrahlung

Resolve conflicting results for SGR A*

Infer stellar density from TDE radio signals

Plans for the Future

Transition to accretion disc at centrifugal radius

Application to larger scale flows: gravity from DM or stars

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