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4.10. Evolution along the Hubble sequence and growth of the black hole

Non-axisymmetric instabilities like bars force the galaxies to evolve towards large mass concentrations, large bulge-to-disk ratios, lower gas content through consumption by star formation. This means that a galaxy born as a late-type, will progressively evolve towards early-types, with a somewhat chaotic path, from barred to unbarred, and sometimes moving backwards, when the disk accretes mass. The gross lines of this evolution are sketched in fig 14.

Figure 14

Figure 14. Cartoon of galaxy evolution along the Hubble sequence. A galaxy, with a small mass distributed mainly in a disk, without bulge (late-type), is unstable with respect to spiral and bar formation (steps 1 and 2). The bar drives the gas towards the center, and the bulge is building up (see in each frame the edge-on projection). When there is too much mass concentrated in the center, the bar is destroyed (step 4), and the gas coming from the outer parts, enrich the disk, and re-establish a larger disk to bulge ratio. Later on (steps 5 and 6), another bar will form, when the disk to bulge ratio is favorable. A secondary bar (cf step 3) may help the primary one to drive the mass towards the center. At the end, the galaxy may be classified early-type.

To summarize this bar-driven evolution, and gather the main features obtained through N-body simulations, and supported by observations, it is interesting to test a toy model, in a semi-analytical way, including:

- star formation, with a combination of a quiescent rate, proportional to the gas density, in a time scale of 3 Gyr, and a bar-driven contribution, with a threshold (Q < 1) and a rate equal to (1 - Q) / t*, with t*, the star-formation time-scale (proportional to the dynamical time-scale for gravitational instabilities).

- radial flows: when a bar is formed, gravity torques produce gas inflow, therefore with a threshold Q < 1 also, and rate (1 - Q) / tvis, with tvis, the ``gravitational viscosity'' time-scale, ~ 1 / Omega (Mtot / Md)2.

- bulge formation: the inflowing gas (and stars) are assumed to form the bulge through star-formation and vertical resonances

- death of bars: when Q > 1 (central concentrations, lack of gas and self-gravitating disk)

- gas infall: possibility of a continuous small infall or a periodically substantial one (from companions).

- black hole formation: a fixed fraction beff of the radial gas flow is taken to contribute to its formation, i.e. dMbh / dt = beff Mg(1 - Q) / tvis, with a threshold Q < 1.

Figure 15 displays some results of the toy model (Combes, 2000). The most striking feature is the self-regulation of the stability parameter Q towards 1. Although the galaxy initially starts almost completely gaseous, the gas mass fraction soon stabilises to 10% of the total. Also the mass of the central concentration (or black hole) stabilises to a constant fraction of the bulge mass, as observed (Magorrian et al. 1998).

Figure 15

Figure 15. Model of periodic gas accretion in a galaxy with star formation, birth and death of bars, radial flows and black hole formation taken into account: Top left Full line: gas mass versus time; dash line: gas mass fraction. Top middle Full line: stellar mass; dash lines: disk stellar mass at top, and bottom bulge mass. Top right Full line: total mass; dash lines: total disk mass at top, and bottom bulge mass. Bottom left Disk star formation rate versus time. Bottom middle Toomre Q parameter. Bottom right Full line: mass of the central black hole, and dash line: mass ratio between the black hole and the bulge.

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