
$$\label{magcoll}
B\geq\frac{M_{cl}}{73\; M_{\odot}} \left(\frac{R_{cl}}{0.1 \;\mathrm{pc}}\right)^{-2} \mathrm{mG}$$

We also mentioned before that the Roche mass for the cloud should be of the order of the mass of the IMBH. Hence, if we use $M_{cl}=M_{BH}$ in Eq. \[magcoll\], we can also get an estimate of the magnitude of the magnetic field needed to have an effect comparable to that of the tidal force. If we approximate again $R_{cl}$ in Eq. \[magcoll\] with S/(2$\sin\theta$) and consider the case with $(D,S)$ = (2,14) arcsec and $\Delta v=30\; \mathrm{km \; s^{-1}}$, we get $B\gtrsim 40$ mG, which is much higher than the typical magnetic field in molecular clouds, even in the Galactic Centre [where it is estimated to be maximum few mG; e.g., @Ferriere09]. Furthermore, given the estimated cloud mass, $M_{cl}=40 M_{\odot}$, we rather expect a a magnetic field of the order of 30 $\mu$G, if it were in a critical state.

Simulations
===========

Initial conditions and methods {#ic}
------------------------------

In order to test our analytical calculation, we run an hydrodynamical simulation with the Eulerian Adaptive Mesh Refinement (AMR) code RAMSES [@Teyssier02]. We adopted Cartesian coordinates and chose a cubic box with $x,y,z=[-5.5:5.5]$ pc.

The IMBH is initially put at $(x,y,z)=0$ as a sink particle, with Bondi-Hoyle accretion [@Bleuler14]. Even though no significant motion of the IMBH is expected (given the small cloud mass), we allow it to move and integrate its motion by direct force summation.

For the cloud, we chose the simplest case of a radial trajectory with $E_{orb}=0$. In this case, the time needed to reach the IMBH from a certain distance is

$$\label{tfall}
t_{fall}=\sqrt{\frac{2d^3_{BH}}{9GM_{BH}}},$$

whenever the mass of the IMBH is significantly larger than the mass of the cloud (which is the case of our model, see section. \[anmod\]). Eq. \[tfall\] then allowed us to set the initial conditions of our simulation. Specifically, we can get the initial distance of head and tail of the clou