On the Origin of the Fundamental Plane and Faber–Jackson Relations: Implications for the Star Formation Problem

We provide an explanation of the properties of the fundamental plane (FP) relation and its observed projections for a sample of nearby early-type galaxies (ETGs) in terms of a fine-tuning between the time-averaged star formation rate $⟨\Psi ⟩$ and their structural and dynamical characteristics. Their total V luminosity is linked with $⟨\Psi ⟩$ and the central velocity dispersion σ through the relation $log\left(L\right)=0.48(\pm 0.06)log(⟨\Psi ⟩)+1.00$ $(\pm 0.13)log\left(\sigma \right)+7.81(\pm 0.26)$, with an rms = 0.215 (R = 0.64 and $P<1.2\times {10}^{-16}$). This fine-tuning permits us to obtain the FP in terms of two distinct "virtual planes" in the $log\left(R_e\right)\text{--}log(⟨I_e⟩)\text{--}log\left(\sigma \right)$ space. The first one (the virial plane; VP) represents the total galaxy mass derived from the scalar virial theorem and the mass-to-light ratio M/L, while the second plane comes from the relation $L=L_0^{\prime }{\sigma }^{-2}$, where $L_0^{\prime }$ is a parameter connected with $⟨\Psi ⟩$. This is a mathematically convenient way for expressing the independence of the galaxy light from the virial equilibrium. Each galaxy in the $log\left(R_e\right)\text{--}log(⟨I_e⟩)\text{--}log\left(\sigma \right)$ space is identified by the intersection of these two planes. A posteriori, we show that the properties of the FP (tilt and scatter) and the zone of exclusion visible in the FP projections are consequences of this fine-tuning. The link between the FP properties and the SFR of galaxies provides a new view of the star formation phenomenon. The star formation history of an unperturbed galaxy seems to be driven by the initial conditions in the protogalaxies and is regulated across cosmic epochs by the variation of the main galaxy parameters (mass, luminosity, structural shape, and velocity dispersion).