The Stellar Mass Fundamental Plane: The virial relation and a very thin
  plane for slow-rotators
  Early-type galaxies -- slow and fast rotating ellipticals (E-SRs and E-FRs)
and S0s/lenticulars -- define a Fundamental Plane (FP) in the space of
half-light radius $R_e$, enclosed surface brightness $I_e$ and velocity
dispersion $\sigma_e$. Since $I_e$ and $\sigma_e$ are distance-independent
measurements, the thickness of the FP is often expressed in terms of the
accuracy with which $I_e$ and $\sigma_e$ can be used to estimate sizes $R_e$.
We show that: 1) The thickness of the FP depends strongly on morphology. If the
sample only includes E-SRs, then the observed scatter in $R_e$ is $\sim 16\%$,
of which only $\sim 9\%$ is intrinsic. Removing galaxies with
$M_*<10^{11}M_\odot$ further reduces the observed scatter to $\sim 13\%$ ($\sim
4\%$ intrinsic). The observed scatter increases to the $\sim 25\%$ usually
quoted in the literature if E-FRs and S0s are added. If the FP is defined using
the eigenvectors of the covariance matrix of the observables, then the E-SRs
again define an exceptionally thin FP, with intrinsic scatter of only $5\%$
orthogonal to the plane. 2) The structure within the FP is most easily
understood as arising from the fact that $I_e$ and $\sigma_e$ are nearly
independent, whereas the $R_e-I_e$ and $R_e-\sigma_e$ correlations are nearly
equal and opposite. 3) If the coefficients of the FP differ from those
associated with the virial theorem the plane is said to be `tilted'. If we
multiply $I_e$ by the global stellar mass-to-light ratio $M_*/L$ and we account
for non-homology across the population by using S\'ersic photometry, then the
resulting stellar mass FP is less tilted. Accounting self-consistently for
$M_*/L$ gradients will change the tilt. The tilt we currently see suggests that
the efficiency of turning baryons into stars increases and/or the dark matter
fraction decreases as stellar surface brightness increases.
