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Timescale for the inwards migration of the circumnuclear disk's inner rim -- Marvin

The central stellar cluster's wind extracts angular momentum from the circumnuclear disk's inner rim, therefore the inner rim migrates

inwards. In the following I derive the timescale for the inwards migration of the CND's inner rim. Please see our paper draft for more

details.

We assume that angular momentum is extracted from a ring of radius s and radial extension \Delta l, therefore it has the mass M_{\text{D}} = 2 \pi \Sigma s \Delta l. This ring has a total angular momentum of J=j M_{\text{D}} with specific angular momentum j.

After a time dt the wind adds the mass f \dot{M} dt to the ring, where \dot{M} is the wind's outflow rate and f is the

fraction of the wind that interacts with the disk. Thus the ring has the mass M_{D} + f \dot{M} dt and the specific angular momentum

j' = \frac{J}{M_{D} + f \dot{M} dt} = \frac{j}{1+\frac{f \dot{M}}{M_{\text{D}}} dt}.

Furthermore the ring moves to s-ds, and because j = \sqrt{GMs} and j' = \sqrt{GM(s-ds)} we get

dt = \frac{M_{\text{D}}}{f \dot{M}} \left( \sqrt{\frac{s}{s-ds}} - 1\right) = \frac{M_{\text{D}}}{f \dot{M}} \frac{ds}{2 s}

where I used a taylor expansion in the last step.

Using M_{\text{D}} = 2 \pi \Sigma s \Delta l, f=\frac{h}{2s} and integrating this equation gives the time the inner rim needs to

migrate inwards:

\tau = \int dt = \int \frac{M_{\text{D}}}{f \dot{M}} \frac{ds}{2 s} = \frac{2 \pi \Sigma \Delta l}{h \dot{M}} \int s ds = \frac{\pi \Sigma \Delta l}{h \dot{M}} \left( s_0^2 - s_\text{i}^2\right),

where s_0 is the disk's initial inner rim and s_\text{i} the location of the rim after inwards migration.

In our simulations s_\text{i} = 0.5 \text{pc}, for s_0 = 1 \text{pc} we get \tau = 22 \cdot 10^4 \text{yrs}, for s_0 = 2 \text{pc} we get \tau = 112 \cdot 10^4 \text{yrs}. The actual time the inner rim needs to move inwards (as observed in the simulations) is 20 \cdot 10^4 \text{yrs} and 65 \cdot 10^4 \text{yrs}, respectively. For the s_0 = 1 \text{pc} simulation this is pretty close, for the s_0 = 2 \text{pc} simulation the deviation is less than a factor of two, which I find an acceptable deviation considering that the above mentioned derivation is only an order of magnitude estimate.

astrobear

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Timescale for the inwards migration of the circumnuclear disk's inner rim -- Marvin

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