**6.3. Angular momentum**

One of the consequences of gravitational instability is that individual proto-structures would have been set in rotation by the tidal torques from their neighbours. Hoyle (1949) was the first to suggest that this mechanism might account for the rotation of galaxies and the idea was refined further by Peebles (1969). Peebles used the linear theory (Section 5.1) to make an estimate of the angular momentum acquired by a protogalaxy and we shall follow his approach below.

The angular momentum about the centre of a sphere of volume may be written

(6.13) |

where **v** = *a*
. The origin of the
coordinates is taken to be at the
centre of the sphere and the notation throughout this section follows
that of Section 5.1. In the linear theory,
the Fourier transforms of the velocity potential
and the
fractional density contrast
are given by
Eqs. (5.7). On retaining only the growing modes,

(6.14a) (6.14b) |

Rather than take the volume
to be a sphere of radius *x*_{0}, it proves
convenient to weight the volume by a factor
exp[|**x** - **x'**| / *x*_{0}^{2}]. In this
case, Eqs. (6.13) and (6.14) give,

(6.15) |

where *f* (*y*) = exp(- *y*^{2}/4).

Equation (6.15) may be corrected for the translational motion of the centre of mass of the volume. The angular momentum about the centre of mass is then,

(6.16) |

Thus, the angular momentum grows at *t*^{5/3} in an
= 1
universe. The mean square density contrast within a typical volume
is,

(6.17) |

Hence the root mean square angular momentum may be written,

(6.18) |

where we have taken,

(6.19a) |

and the integrals *K* and *I* are,

(6.19b) |

(6.19c) |

where

and

The integral *I* diverges if *n* lies outside the range
-3 < *n* < 1/2. For
intermediate values, the combination
*I*^{1/2} *K*^{-1} has been evaluated
numerically and is listed below.

n | 0.45 | 0.25 | 0.00 | -0.25 | -0.50 | -1.00 | -2.00 |

I^{1/2} K^{-1} |
3.19 | 1.35 | 0.89 | 0.68 | 0.55 | 0.39 | 0.21 |

If *n* > 1/2, the dominant contribution to the integral *I*
occurs at
short wavelengths where the power spectrum breaks away from its
primordial shape due to non-linear effects.

From Eq. (6.18),

(6.20) |

(cf. Peebles, 1969, Eq. 24).

Most of the angular momentum is acquired during the linear regime;
in the non-linear regime galaxies recede from each other, their
separations increasing as *t*^{2/3}. The torque on any
galaxy therefore
decreases as 1/*t*^{2}. A rough estimate of the angular
momentum may,
therefore, be obtained from Eq. (6.20) by setting
<^{2}>_{x0}
1. In terms of
the dimensionless spin-parameter
defined by Eq. (6.10)
we find,

(6.21) |

for a typical protogalaxy. In deriving Eq. (6.21) we have taken
*n* 0.
As may be seen from the table above, the angular momentum is fairly
insensitive to the value of *n* for the range
-1 *n*
0 which we have
argued in Section (5.3) is required if the
theory is to account for
the observations of the two-point galaxy correlation function. This
answer must be considered fairly rough as it gives the r.m.s. angular
momentum of a randomly placed sphere whereas what is really wanted is
the angular momentum of a bound protogalaxy. Also the extrapolation of
Eq. (6.20) to density contrasts
<^{2}>_{x0} ~ 1
is somewhat uncertain. A more
accurate estimate may be obtained from *N*-body calculations
(Peebles, 1971b;
Efstathiou and Jones,
1979).
In particular, the latter authors
studied the tidal torque process using 1000 particle models with
Poisson initial conditions. For bound clumps, with density contrast
10, they find the
median value of
to be,

(6.22) |

with no significant dependence on the mass of the clump.

For a spheroidal galaxy with a Hubble-like density profile and a
flat rotation curve, the spin parameter
is approximately
0.33*v*_{m} /
_{0},
where *v*_{m} and
_{0} are the
projected rotational velocity and central
velocity dispersion respectively when the galaxy is viewed along the
equatorial plane
(White (1979a)).
Correcting this by a factor of
/4
to account for the random orientations of galaxies in space,
Eq. (6.22) predicts (*v*_{m} /
_{0})_{median}
0.14. This
result is fairly typical
of bright elliptical galaxies with *M*_{B} < - 20
(Section 2.1). Fainter
ellipticals rotate more rapidly, with *v*_{m} /
_{0}
0.2. This may
not be a problem for the tidal torque theory since the spin-parameter
is only
conserved if total energy is conserved during the collapse; it may be
that the fainter elliptical galaxies increased their binding energy in
a dissipative collapse whilst the larger ones did not. This idea will
be discussed further in Section 8.2.

The theory does, however, face a serious problem in accounting for the rotation of disc galaxies if they formed from the collapse of self-gravitating gas clouds. Consider a self-gravitating exponential disc (Section 2.2). In the absence of random motions, the total angular momentum is,

(6.23) |

and the total energy is,

(6.24) |

where
^{-1} is the
scale length of the disc. A disc galaxy, therefore, has a spin-parameter,

(6.25) |

which is far in excess of the tidal torque prediction (6.22). It is
quite clear that dissipative processes have played a major role in the
formation of disc systems so we would expect angular momentum but not
energy to be conserved during the collapse. We may therefore use Eqs.
(6.22) and (6.23) to estimate the initial radius of the protocloud at
time *t*_{i} when
/
~
1. Approximating the protocloud as a uniform
sphere of radius ** R_{i}** and total energy

(6.26) |

Hence a typical disc galaxy with scale length
^{-1}
3*h*^{-1}
kpc must have collapsed from a protocloud with
*R*_{i}
600*h*^{-1} kpc. For a mass of 10^{11}
*M*_{} this
implies a free-fall time of,

(6.27) |

which is greater than the age of the Universe. Clearly, the collapse
factor of Eq. (6.26) is unreasonably large; a more acceptable value
would be
*R*_{i}
10
(Eggen, Lynden-Bell and
Sandage, 1962).
A possible resolution of this problem will be discussed in
Section 8.2.

It is also interesting to examine the nature of the tidally induced velocity field. From Eq. 5.3(b) we find,

(6.28) |

where

and

is the tidal field. Now, if the vorticity is assumed to be zero at recombination it will remain so at any later time since Kelvin's circulation theorem holds,

Thus, in the linear regime, Eq. (6.28) may be solved giving,

(6.29) |

The root mean square value of the shear within a randomly placed
sphere of radius *x*_{0} is,

(6.30) |

The effect of shear is to elongate the perturbation. The ellipticity
will be of order
*t*. Hence,
Eq. (6.30) shows that the distortion
will grow as *t*^{2/3} and is of first order in the
amplitude
(cf. Binney and Silk
(1979)).
By the time that
1 quite significant
elongations could be produced, with
~ 1/2.