2. Tidal torques
The explanation of galaxies spins gain through tidal torques was pioneered by Hoyle (1949) in the context of a collapsing protogalaxy. Peebles (1969) considered the process in the context of an expanding world model showing that the angular momentum gained by the matter in a random comoving Eulerian sphere grows at the second order in proportion to (in a Einstein-de Sitter universe), since the proto-galaxy was still a small perturbation, while in the non-linear stage the growth rate of an oblate homogeneous spheroid decreases with time as .
More recent analytic computations (White 1984; Hoffman 1986 and R88a) and numerical simulations (Barnes & Efstathiou 1987) have re-investigated the role of tidal torques in originating galaxies angular momentum. In particular White (1984) considered an analysis by Doroshkevich (1970) that showing as the angular momentum of galaxies grows to first order in proportion to t and that Peebles's result is a consequence of the spherical symmetry imposed to the model. White showed that the angular momentum of a Lagrangian sphere does not grow either in the first or in the second order while the angular momentum of a non-spherical volume grows to the first order in agreement with Doroshkevich's result.
Hoffman (1986) has been much more involved in the analysis of the correlation of the growth of angular momentum with the density perturbation . He found an angular momentum-density anticorrelation: high density peaks acquire less angular momentum than low density peaks. One way to study the variation of angular momentum with radius in a galaxy is that followed by R88a. In this approach the protogalaxy is divided into a series of mass shells and the torque on each mass shell is computed separately. The density profile of each proto-structure is approximated by the superposition of a spherical profile, , and a random CDM distribution, , which provides the quadrupole moment of the protogalaxy. To the first order, the initial density can be represented by:
where is the background density and is given by:
being the power spectrum, while the density profile is (Ryden & Gunn 1987):
where is the height of a density peak, is the two-point correlation function, and are two spectral parameters (BBKS, Eq. 4.6a, 4.6d) while is a function given in BBKS (Eq. 6.14). As shown by R88a the net rms torque on a mass shell centered on the origin of internal radius r and thickness is given by:
where , the multipole moments of the shell and , the tidal moments, are given by:
where is the mass of the shell, and are the spherical Bessel function of first and second order while the power spectrum is given by:
(Ryden & Gunn 1987). The normalization constant A can be obtained, as usual, by fixing that the mass variance at , that is , be equal to unity. Filtering the spectrum on cluster scales, , we have obtained the rms torque, , on a mass shell using Eq. (6) then we obtained the total specific angular momentum, , acquired during expansion integrating the torque over time (R88a Eq. 35):
In Fig. 1 we show the variation of with the distance r for three values of the peak height . The rms specific angular momentum, , increases with distance r while peaks of greater acquire less angular momentum via tidal torques.
This is the angular momentum-density anticorrelation showed by Hoffman (1986). This effect arises because the angular momentum is proportional to the gain at turn around time, , which in turn is proportional to .
© European Southern Observatory (ESO) 1998
Online publication: August 6, 1998