Evolution of the angular momentum of protogalaxies from tidal torques: Zel'dovich approximation

The growth of the angular momentum L of protogalaxies induced by tidal torques is reconsidered. We adopt White's formalism and study the evolution of L in Lagrangian coordinates; the motion of the fluid elements is described by the Zel'dovich approximation. We obtain a general expression for the ensemble expectation value of the square of L in terms of the first and second invariant of the inertia tensor of the Lagrangian volume Γ enclosing the collapsing mass of the proto-object. We then specialize the formalism to the particular case in which Γ is centred on a peak of the smoothed Gaussian density field and approximated by an isodensity ellipsoid. The result is the appropriate analytical estimate for the rms angular momentum of peaks to be compared against simulations that make use of the Hoffman-Ribak algorithm to set up a constrained density field that contains a peak with given shape. Extending the work of Heavens & Peacock, we calculate the joint probability distribution function for several spin parameters and peak mass M using the distribution of peak shapes, for different initial power spectra. The probability distribution for the rms final angular momentum 〈Lf2〉1/2 on the scales corresponding to common bright galaxies, M≈ l011 M⊙, is centred on a value of ≈ 1067 kg m2 s−1, for any cosmologically relevant power spectrum, in line with previous theoretical and observational estimates for Lf. Other astrophysical consequences are discussed. In particular, we find that typical values 〈 λ2 〉1/2 ≈ 0.1 of the dimensionless spin parameter for peaks smoothed on galactic scales and of height v ∼ 1, usually associated with late-type galaxies, may be recovered in the framework of the Gaussian peak formalism. This partially relaxes the importance attributed to dissipative processes in generating such high values of centrifugal support for spiral galaxies. In addition, the values of the specific angular momentum versus mass — as deduced from observations of rotational velocities and photometric radii of spiral galaxies — are well fitted by our theoretical isoprobability contours. In contrast, the observed lower values for the specific angular momentum for ellipticals of the same mass cannot be accounted for within our linear-regime investigation, highlighting the importance of strongly non-linear phenomena to explain the spin of such objects.