Angular momentum transfer and the size - mass relation in early - type galaxies

Early - type galaxies (ETGs) define a narrow strip in the size - mass plane because of the observed correlation between the effective radius $R_{eff}$ and the total stellar mass $M_{\star }$. When expressed in logarithmic units, a linear relation, $\log R_{eff}\propto \gamma \log M_{\star }$, is indeed observationally found, but the slope $\gamma$ deviates from the canonical $\gamma =1\mathrm{/}2$ value which can be naively predicted for a spherically symmetric isotropic system. We propose here that a transfer of angular momentum to the stellar component induces an anisotropy in the velocity space thus leading to a modified distribution function (DF). Assuming an Osipkov - Merritt like anisotropic DF, we derive an analytic relation between the slope $\gamma$ of the size - mass relation and the slope $\alpha$ of the angular momentum term in the DF. With this simple model, we are then able to recover the observed $\gamma$ value provided $\alpha$ is suitably set. It turns out that an anisotropy profile which is tangential inside $\sim 0.6r_a$ and radial outside, with $r_a$ the anisotropy radius, is able to reproduce the observed size - mass relation observed for massive ($M_{\star} \ge 2 \times 10^{10}
h^{-1} {\rm M}_{\odot}$) elliptical galaxies.