# The L Proportional To Sigma(8) Correlation for Elliptical Galaxies With Cores: Relation With Black Hole Mass

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We construct the Faber-Jackson correlation between velocity dispersion sigma and total galaxy luminosity L-V separately for elliptical galaxies with and without cores. The coreless ellipticals show the well-known, steep relationship d log sigma/d log L-V = 0.268 or L-V proportional to sigma(3.74). This corresponds to d log sigma/d log M = 0.203, where M is the stellar mass and we use M/L proportional to L-0.32. In contrast, the velocity dispersions of core ellipticals increase much more slowly with L-V and M: d log sigma/d log L-V = 0.120, L-V proportional to sigma(8.33), and d log sigma/d log M = 0.091. Dissipationless major galaxy mergers are expected to preserve sigma according to the simplest virial-theorem arguments. However, numerical simulations show that sigma increases slowly in dry major mergers, with d log sigma/d log M similar or equal to +0.15. In contrast, minor mergers cause sigma to decrease, with d log sigma/d log M similar or equal to -0.05. Thus, the observed relation argues for dry major mergers as the dominant growth mode of the most massive ellipticals. This is consistent with what we know about the Formation of cores. We know no viable way to explain galaxy cores except through dissipationless mergers of approximately equal-mass galaxies followed by core scouring by binary supermassive black holes. The observed, shallow sigma proportional to L-V(+ 0.12) relation for core ellipticals provides further evidence that they formed in dissipationless and predominantly major mergers. Also, it explains the observation that the correlation of supermassive black hole mass with velocity dispersion, M-circle proportional to sigma(4), "saturates" at high M-circle such that M-circle becomes almost independent of sigma.