9 The inner edge of a CND

If we assume a spectrum of clouds moving around a galactic centre forming a disk-like equilibrium structure, there are four effects that determine the physical properties of the clouds: UV radiation, tidal shear, a radially-directed wind, and selfgravitation. In the following considerations we will neglect the effects of stellar winds. Nevertheless, one has to keep in mind that they can provide an additional external pressure on the clouds at the inner edge of the CND. Only those clouds with a sufficiently high central density can resist tidal disruption. Thus, the clouds' mean density must increase with decreasing distance to the Galactic Centre. If they reach the Jeans mass they become gravitationally unstable and collapse. Taking these two effects together we obtain an efficient mechanism to create an inner edge. At this distance, the clouds that can resist tidal disruption become Jeans unstable, i.e. the dense cloud structure is lost. The UV radiation plays an important rôle in determining the radius of the clouds at each distance from the galactic centre. With an increasing UV radiation field the cloud radius decreases, because the ionization front in direction of the central star cluster is located at smaller cloud radii (Dyson 1968). At these radii the enhanced pressure of the ionized gas in the ionization front is counterbalanced by the enhanced thermal pressure due to the increasing density of the neutral gas in the isothermal cloud. Thus the cloud mass decreases with increasing UV radiation field and the cloud is less susceptible to gravitational collapse.

We can estimate this effect quantitatively in the following way. In the case of a cloud with uniform density the maximum density needed to stabilize a cloud against tidal forces is

$$% \rho_{\rm crit}^{\rm tidal}=\frac{3}{2\pi}\frac{M(R)}{R^{3}},$$ (12)

where M(R) is the enclosed mass up to the radius R from the Galactic Centre. The criterion for Jeans instability is given by

$$% \rho_{\rm crit}^{\rm Jeans}=\frac{\pi{\cal R}T}{G \mu r_{\rm cl}^{2}},$$ (13)

 where T is the temperature of the cloud, ${\cal R}$ is the gas constant, G is the gravitational constant, and $\mu$ is the molecular weight.

The cloud radius due to the balance of ionization and recombination is given by Dyson (1968)

$$% r_{\rm cl}=\xi^{2} J_{0} n_{\rm i}^{-2},$$ (14)

 where J₀ is the number of incident UV photons per cm² and s, $n_{\rm i}$ is the number density of the ionized gas in the ionization front, and $\xi=4.87\times 10^{6}$ cm $^{-\frac{3}{2}}$ s $^{\frac{1}{2}}$. With the jump condition across the ionization front $\rho_{\rm cl} {\rm e}^{-u} c_{\rm s}^{2} = 2 \rho_{\rm i} c_{\rm i}^{2}$, where $\rho_{\rm i}$ and $c_{\rm i}$ are the density and the sound velocity of the external ionized gas and u(x) is a parameter of the Lane-Emden equation, one can write

$$% r_{\rm cl}=4 \xi^{2}m_{p}^{2}J_{0}\rho_{\rm cl}^{-2}{\rm e}^{2u}(\frac{c_{\rm i}}{c_{\rm s}})^{4}.$$ (15)

Furthermore, one can approximate the function ${\rm e}^{-u(x)}\sim 3x^{-2}$ for 2 < x < 8, where $x=r_{\rm cl}/c_{\rm s}\sqrt{4\pi G \rho_{\rm c}}$. This leads to

$$% r_{\rm cl}=3.645\times 10^{15} J_{0}^{-\frac{1}{3}}c_{\rm i}^{-\frac{4}{3}} c_{\rm s}^{\frac{8}{3}}.$$ (16)

Thus, the cloud radius depends only on the number of incident UV photons, the sound speed of the ionized gas, and the sound speed of the neutral gas. If we assume that the temperature is determined by the radiation field $c_{\rm s} \propto T^{\frac{1}{2}} \propto J_{0}^{\frac{1}{8}}$, the cloud radius does not change with the cloud's distance to the Galactic Centre. This means that the cloud's radius is constant and does not depend on its central density. Consequently, the critical Jeans density $\rho_{\rm crit}^{\rm Jeans}$ is proportional to the gas temperature of the neutral gas.

Figure 2 shows the critical densities with respect to tidal shear and gravitational collapse. These graphs represent an approximation of the density variations calculated from the detailed model of VD2001 (Fig. 1). The increase or decrease of the neutral gas temperature by a factor 2 results in a variation of the location of the inner edge of $\pm$1 pc. In Fig. 2 the dashed line shows the maximum central density above which gravitational collapse occurs ( $\rho_{\rm crit}^{\rm Jeans}$ using Eq. (16)), the solid line shows the minimum density in order to resist tidal shear ( $\rho_{\rm crit}^{\rm tidal}$). The dashed surface shows the range of densities where clouds are gravitationally and tidally stable.

Figure 2: Central density of the heavy clouds versus the distance to the Galactic Centre. Dashed line: maximum central density above which gravitational collapse occurs. Solid line: minimum density in order to resist tidal shear. Dashed surface: range of densities where clouds are gravitationally and tidally stable.

The inner edge of the CND arises thus naturally as a selection effect due to external conditions of the environment on the cloud spectrum.

If there are non-negligible non-thermal pressure components, we have to add these components to the sound speed of the neutral and ionized gas. For a non-thermal linewidth c > 1 kms^-1, the critical Jeans density is always lower than the critical tidal density, i.e. no stable clouds can exist. Therefore, we suggest that only rotating clouds, which have a higher central density can survive.

9.1 A possible scenario for star formation

In the previous Sections we have investigated an isolated clumpy gas disk. However, the CND appears to interact with the surrounding gas. Coil & Ho (1999, 2000) and Zylka et al. (1999) conclude on the basis of the gas distribution and kinematics in the inner 20 pc of the Galactic Centre that there are connections between the CND and the neighbouring GMCs. They claim that there are several streamers that fall into the Galactic Centre.

From the theoretical point of view Sanders (1998) pointed out the possibility that the CND can be understood in terms of tidal capture and disruption of gas clouds falling into the Galactic Centre region. The infalling gas forms a tidally stretched filament intersecting itself. After several rotation periods the gas forms a stable ring structure which can be maintained for more than 10⁶ yr. He showed that the central star cluster can be created within the first few passages of the cloud when the long filament intersects itself at a large angle.

We will now discuss what happens when a cloud complex falls from a distance greater than 10pc into the Galactic Centre.

In VD2001 we have shown that the CND has a lifetime of $\sim10^{7}$ yr. It is thus possible that an external cloud is falling into the Galactic Centre within this period. We propose a new scenario in which a whole cloud complex is falling into the Galactic Centre where a clumpy disk structure already exists. When the cloud hits the CND, frequent partially inelastic cloud-cloud collisions will create a whole transient spectrum of clouds with different masses. Those clouds which have masses above the Jeans limit will collapse and eventually form stars. The massive stars are thus formed within a very short time during the collision of the cloud complex and the disk. Partially inelastic collisions result in a decrease of the cloud velocities. Low velocity clouds ( $v < v_{\rm rot} \sim 120$ kms^-1) form stars and approach the Galactic Centre at the same time. Thus, if an external gas cloud collides with a pre-existing CND, we expect that a part of the gas, which loses angular momentum due to cloud-cloud collisions spirals inwards with a velocity of $\sim$100 kms^-1.

If we assume an initial He I star mass of $M_{0}=30\, M_{\odot}$ with an initial velocity of $V_{0}=100\,$kms^-1, falling into an already existing old stellar cluster of density $\rho_{\rm stellar}=4\times 10^{6}\, M_{\odot}\,{\rm pc}^{-3}$ with stellar masses around 1 $M_{\odot}$, collective relaxation dominates over the particle-particle relaxation (Saslaw 1985). This leads to a relaxation time of

 

$$% \tau_{R, {\rm coll}} 7\times 10^6\times \left( \frac{V_0}{100 \,{\rm km\,s}^{-1}}\right)^3$$ =

$$\times \left( \frac{M_0}{30~M_{\odot}}\right)^{-1} \left( \frac{\... ...{\rm stellar}}{4\times 10^6 \frac{M_{\odot}}{{\rm pc}^3}}\right)^{-1} {\rm yr}.$$ (17)

A single massive star, whose velocity is approximately the Keplerian velocity of gas moving on circular orbits around the Galactic Centre, thus loses the information about its initial position and velocity within $\sim7\times 10^6$ yr.

Gerhard (2001) estimated that a star cluster of $\sim$10 $^{5}~M_{\odot}$ which is formed at a Galactic radius $R_{\rm G}=10$ pc needs several Myr to spiral into the Galactic Centre. This timescale is comparable to our collective relaxation timescale.

The star cluster relaxation time is thus comparable to the lifetime of the He I stars. These massive stars are formed at the same time and at the same distance to the Galactic Centre forming a star cluster after a few million years. The observed streaming motions of the He I stars in the Galactic Centre (Genzel et al. 1996) shows that the star cluster is not completely relaxed. Therefore, it is possible that the central He I star cluster has been built during the collision of an infalling cloud complex with an already existing CND.