The <Schechter function> has logarithmic slope $\alpha\simeq-1$ for $M\ll M_*$ and an exponential cutoff for $M\gg M_*$. Thus a log-log plot is approximately a straight line of slope $-1$ at low mass and bends sharply downward near $M_*$.

For a collisionless self-gravitating system, define the inertia tensor
$$
I_{ij}=\int\rho x_ix_j\,dV.
$$
Two time derivatives, the collisionless Boltzmann equation, and integration by parts give the <tensor virial theorem>
$$
\frac12\ddot I_{ij}=2K_{ij}+W_{ij},
$$
where
$$
K_{ij}=\frac12\int\rho\langle v_iv_j\rangle\,dV,
\qquad
W_{ij}=-\int\rho x_i\partial_j\Phi\,dV.
$$
For an isolated steady state, or a long-time average of a bounded system, $\ddot I_{ij}=0$ and surface terms vanish, so
$$
\boxed{2K_{ij}+W_{ij}=0}.
$$
Taking the trace gives $2K+W=0$. Therefore
$$
\boxed{E=K+W=-K=\frac W2}.
$$

Introduce the <gravitational radius> by $W=-GM^2/R$. Virial equilibrium gives
$$
\boxed{
E_I=-\frac12M_I\langle v_I^2\rangle
=-\frac{GM_I^2}{2R_I}
}.
$$
Assume dissipationless accretion, negligible escaping mass, no external work, and final re-virialization. The accreted systems bring
$$
E_A=-\frac12M_A\langle v_A^2\rangle.
$$
With $\eta=M_A/M_I$ and $\epsilon=\langle v_A^2\rangle/\langle v_I^2\rangle$,
$$
\boxed{
E_F=E_I+E_A
=-\frac12M_I\langle v_I^2\rangle(1+\epsilon\eta)
}.
$$
Since $M_F=M_I(1+\eta)$ and $E_F=-M_F\langle v_F^2\rangle/2$,
$$
\boxed{
\frac{\langle v_F^2\rangle}{\langle v_I^2\rangle}
=\frac{1+\eta\epsilon}{1+\eta}
}.
$$
Using $E=-GM^2/(2R)$ also gives
$$
\boxed{
\frac{R_F}{R_I}
=\frac{(1+\eta)^2}{1+\eta\epsilon}
}.
$$

Many <minor galaxy mergers> have dynamically cold accreted material, $\epsilon\ll1$. If they double the mass, $\eta=1$, then $R_F/R_I\simeq4$, and
$$
\frac{\bar\rho_F}{\bar\rho_I}
=\frac{M_F/M_I}{(R_F/R_I)^3}
=\frac2{4^3}=\frac1{32}.
$$
The steep low-mass side of the <Schechter function> supplies many small satellites. Repeated <dry galaxy mergers> can therefore add stars mainly at large radius, increasing size much faster than mass and lowering <mass density>[mean density]. This explains how compact, dense redshift-two galaxies can evolve into larger present-day <elliptical galaxies> without requiring comparable in-situ <star formation>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 346 / 2 / Solution

{wiki=Galaxy_merger}

A dry galaxy merger joins galaxies with little cold gas, so dissipation and merger-triggered star formation are weak. Conservation of energy followed by <virial theorem>[re-virialization] can make repeated dry minor mergers increase a galaxy's radius faster than its mass.

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 Incoming links: Dry galaxy merger