THE L∝σ⁸ CORRELATION FOR ELLIPTICAL GALAXIES WITH CORES: RELATION WITH BLACK HOLE MASS

This Letter is about the Faber & Jackson (1976) correlation between velocity dispersion σ and total luminosity L_V, absolute magnitude M_V, or stellar mass M for elliptical galaxies. We construct this correlation separately for galaxies with and without cores. We define a "core" to be the central region interior to the radius where the outer, Sérsic (1968) function brightness profile log I(r)∝r^1/n shows a downward break to a shallow, central profile, as we have done in previous papers (Kormendy 1999; Kormendy et al. 2009, hereafter KFCB; Kormendy & Bender 2009). We find that σ increases with galaxy L_V or M much more slowly in core galaxies than it does in coreless galaxies.

It is well known that the Faber–Jackson relation L∝σ⁴ "saturates" in this way at high luminosities (e.g., Lauer et al. 2007a; Cappellari et al. 2012a). These highest-luminosity galaxies preferentially contain cores. That is, they form the high-luminosity side of the "E–E dichotomy" of elliptical galaxies into two types (see Kormendy & Bender 1996 and KFCB for reviews). The two types overlap in luminosity (Lauer et al. 1995; Faber et al. 1997; Section 3 in this work), but the change occurs at M_V ≃ −21.6 (H₀ = 70 km s⁻¹ Mpc⁻¹) in the Virgo cluster (KFCB). Then:

Giant ellipticals contain "cores"; i. e., the central profile is a shallow power law separated by a break from the outer, steep Sérsic function profile. That is, cores are missing light with respect to an inward extrapolation of the outer Sérsic profile. Core galaxies rotate slowly, so their dynamics are dominated by anisotropic velocity dispersions. They are modestly triaxial. Their isophotes show boxy distortions from best-fit ellipses. Their stars are both old and enhanced in α element abundances with respect to the Sun. Two other differences from smaller ellipticals are especially diagnostic of formation mechanisms: they often contain powerful radio sources, and they are massive enough to hold onto hot, X-ray-emitting gas. KFCB review evidence that they are made in dissipationless ("dry") mergers. In contrast:

Normal- and low-luminosity ellipticals are coreless; in fact, they have extra central light with respect to an inward extrapolation of the outer Sérsic profile. They are rapid rotators and are more nearly isotropic, although the axial velocity dispersions can be smaller than the other two components. Their isophotes are disky-distorted with respect to best-fit ellipses. Their stars show larger ranges in ages and little or no α-element enhancement. They rarely contain strong radio sources, and they contain little or no X-ray gas. KFCB review evidence that they form in wet mergers with starbursts.

We emphasize that the division of ellipticals into two kinds is not statistical—it is not based on separating galaxies into two groups using some probabilistic recipe and then asking whether the resulting Faber–Jackson relations are statistically different enough to justify the separation. Rather, it is based on a clear-cut distinction of galaxy profiles into two (core and coreless) varieties with few intermediate cases (Gebhardt et al. 1996; Lauer et al. 2007b; KFCB). Moreover, the core–coreless distinction separates the galaxies remarkably accurately by other physical features, including those listed above. For example, Faber et al. (1997) and Lauer (2012) show that core (coreless) ellipticals rotate slowly (rapidly). Faber et al. (1997) show further that core (coreless) ellipticals tend to have boxy (disky) isophotes. The E–E dichotomy is not based on L_V, σ, or any correlation between them. Only after the division is made on the above physical grounds do we ask whether the two classes of ellipticals have the same or different Faber–Jackson relations.

When we do this, we find an additional difference between core and coreless galaxies. They have different Faber–Jackson relations.

The galaxies that cause the Faber–Jackson relation to saturate are core galaxies. Our picture that they form in dry mergers makes a prediction about the Faber–Jackson relation that we wish to check. The simplest arguments are based on the virial theorem, on the assumption that no orbital energy is added during the merger, and on the assumption that progenitors and remnants have homologous structures. Under these idealized circumstances, mergers preserve σ unchanged (e.g., Lake & Dressler 1986; Nipoti et al. 2003; Hilz et al. 2012). These circumstances motivate the test made in this Letter.

Figure 1 shows the Faber–Jackson correlations for core and coreless ellipticals. Our data source is Lauer et al. (2007b), who tabulate V-band absolute magnitudes M_V, velocity dispersions σ, and classifications of central profiles into core or "power law" (i.e., no core). This sample is especially useful here because it contains many first-ranked cluster ellipticals. We make the following additions or corrections.

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We add M 32 from KFCB.

For elliptical galaxies in the Virgo cluster, we use M_V and profile classifications from KFCB. For example, the low-luminosity ellipticals NGC 4458, NGC 4478, and NGC 4486B show "extra light" above the inward extrapolation of the outer Sérsic profile, exactly as in other coreless ellipticals (Figures 19, 17, and 22 in KFCB, respectively). Therefore, even though the profiles flatten slightly near the center, we classify these as extra light ellipticals. In contrast, the Nuker definition of cores (Lauer et al. 1995, 2007b) depends on fitting a double-power-law "Nuker function" to the profile at radii r ≲ 20''. The profile is classified as having a core if the inner cusp slope dlog I/dlog r < 0.3. This procedure finds cores in NGC 4458, NGC 4478, and NGC 4486B. It generally cannot find extra light components, because the radius range used is too small. But the agreement between the Nuker procedure and ours is almost always good. The exceptions are a few small galaxies in which there is a substantial extra light component and only a small central break in the light profile.

We add velocity dispersions for low-luminosity M 32 analogs in the Virgo cluster from Hopkins et al. (2009). These have substantial leverage on the fits for coreless galaxies.

We omit S0 galaxies unless we know that they have bulge-to-total ratios of almost 1. For example, we retain NGC 3115. Rather than deal with heterogeneous published bulge–disk decompositions in a Letter, we restrict ourselves to ellipticals.

For all galaxies, we adopt measurement errors of 0.1 in M_V and 0.03 in log σ. This leaves an intrinsic physical scatter in the Faber–Jackson relations of 0.06 in log σ for core galaxies and 0.10 in log σ for coreless galaxies at a given M_V. These were derived by adding the intrinsic scatter in quadrature to the measurement errors before fitting and then iterating the intrinsic scatter until the reduced $\chi ^2_r$ of the fits equals 1.

Figure 1 shows that the Faber–Jackson L_V∝σ^m correlation is different for core and coreless ellipticals.

Coreless galaxies approximately satisfy the Faber–Jackson relation that we are used to. Absent cores, there is no significant slope change at high luminosities as there is when core and coreless galaxies are combined (Tonry 1981; Davies et al. 1983; Matković & Guzmán 2005; Cappellari et al. 2012a, 2012b). The red line is described by

$$d \log \sigma /d \log L_V = 0.268\pm 0.015$$

or

$$\begin{equation} {L_V \over {10^{11} L_{V,\odot }}} = (0.67\pm 0.10)\left ({\sigma \over {250\, {\rm km\, s^{-1}}}}\right)^{3.74\pm 0.21}. \end{equation} \tag{ 1 }$$

If we use M/L∝L^{0.32 ± 0.06} (Cappellari et al. 2006), then

$$d \log \sigma /d \log M \simeq 0.203\pm 0.040,$$

which corresponds to M∝σ^{4.94 ± 0.97}. Note that the relatively large error in the relation between σ and mass is driven by the large uncertainty in the relation between M/L and L.

The core galaxies clearly follow a much shallower Faber–Jackson relation than the coreless galaxies. The fit result is

$$d \log \sigma /d \log L_V = 0.120\pm 0.018$$

or

$$\begin{equation} {L_V \over {10^{11} L_{V,\odot }}} = (0.79\pm 0.11)\left ({\sigma \over {250\, {\rm km\, s^{-1}}}}\right)^{8.33\pm 1.24}. \end{equation} \tag{ 2 }$$

If we again use M/L∝L^{0.32 ± 0.06}, then

$$d \log \sigma /d \log M \simeq 0.091 \pm 0.022,$$

corresponding to M∝σ^{11.0 ± 2.7} (again, the large error in the exponent is due to the uncertainty in M/L). The shallowness of these relations accounts for all of the "sigma saturation" in the Faber–Jackson relation.

Core and coreless galaxies overlap over a factor of ∼10 in luminosity, i.e., over the range −20.50 > M_V > −22.85. If we restrict the fits to just this region (Figure 2), then we obtain, for the coreless galaxies,

$$d \log \sigma /d \log L_V \simeq 0.30 \pm 0.06,$$

and for the core galaxies,

$$d \log \sigma /d \log L_V \simeq 0.16 \pm 0.04.$$

Thus, the Faber–Jackson relations of core and coreless galaxies are different even in the overlap region and are consistent there with the corresponding fits over the full luminosity range. This test was not made by Lauer et al. (2007a).

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Our results for core galaxies agree well with predictions of numerical simulations of dissipationless major mergers. These simulations show that the virial-theorem prediction is not quite accurate. Some exchange in energy between visible and dark components and some escape of stars both result in merger remnants that are slightly more compact (smaller size, higher density, and higher velocity dispersion) than the virial-theorem arguments predict.

If galaxies are treated as single-component systems, both Nipoti et al. (2003) and Hilz et al. (2012) consistently obtain dlog σ/dlog M ≈ 0.05.

If stars and dark matter are treated separately, the slope gets slightly steeper. As Hilz et al. (2012) show, equal-mass mergers lead to an increase in the velocity dispersion of the stellar component, because violent relaxation broadens the energy distribution. Bound particles become more strongly bound, and some weakly bound particles escape. Hilz et al. (2012) obtain for the evolution of the effective line-of-sight velocity dispersion of the stellar component in equal-mass mergers that result from radial or nearly radial orbits:

$$d \log \sigma /d \log M \approx +0.15 \quad {\rm (major\; mergers),}$$

a three-times-steeper slope than in the single-component case.

Essentially the same result was obtained for the same physical reasons by Boylan-Kolchin et al. (2006). They simulated a larger range of collision geometries than Hilz et al. (2012) and showed that the slope m in the M∝σ^m correlation is steepened very little when the collision impact parameter is large but is steepened by large amounts when the collision is nearly radial. Even one head-on, equal-mass merger makes the slope similar to what we observe.

Hilz et al. (2012) also present results for the growth of massive ellipticals via minor mergers. As expected, such growth happens mostly at large radii; it actually leads to a decrease of the effective projected velocity dispersion:

$$d \log \sigma /d \log M \approx -0.05 \quad {\rm (minor\; mergers).}$$

Thus, the observed dlog σ/dlog M relation for core galaxies is consistent with their formation predominantly in major mergers. Minor mergers may play a role, too, but this role has to be mostly confined to radii that are large enough to have little effect on measurements of projected σ. Otherwise, we would see a shallower or even negative dlog σ/dlog M slope.

We cannot expect observations and simulations to agree perfectly, because the n-body mergers are not part of a hierarchical clustering simulation and because the simulations depend on the mass and velocity distributions of both the visible and dark components and the latter are poorly observed.

Therefore, we can benefit from a "sanity check" of our results against observations of the fundamental plane correlations. Analytical derivation of the fundamental plane in a way that incorporates gas dissipation and starbursts is difficult. But the correlations are reasonably well reproduced by simulations of wet mergers (Boylan-Kolchin et al. 2006; Robertson et al. 2006; Hopkins et al. 2008, 2009). Therefore we use the observed correlations. We emphasize that any virial-theorem-based derivation is approximate, because virial properties are total properties, and we have accurate measurements only of the stellar galaxies. In the following, we assume that the central parts of visible galaxies are self-gravitating systems interior to and largely independent of their dark matter halos, so the virial theorem can be applied to the visible galaxies without considering surface terms. This is an approximation, and results from it should be viewed with caution. Then, the virial theorem tells us that M = (M/L_V)L_V∝σ²r_e, where r_e is the "effective radius" that encloses half of the light of the galaxy. The most accurate measurements of the r_e–M_V projection of the fundamental plane (KFCB; Kormendy & Bender 2012) give $r_e \propto L_V^{0.76}$ for all bulges and ellipticals. Substituting this, together with M/L∝L^0.32 from Cappellari et al. (2006), gives L∝σ^3.55. This is in good agreement with our Faber–Jackson relation for coreless ellipticals. In contrast, if we begin with our observation of that L_V∝σ^8.33 for core ellipticals and use $r_e \propto L_V^{0.76}$ to derive the dependence of mass-to-light ratio on L, then we get $M/L_V \propto L_V^{-0.003}$. This is remarkably consistent with our assumption that core ellipticals form in dry mergers with little further star formation.

Is this last result consistent with dynamical M/L measurements of core ellipticals? No study addresses the question in the way that we need, but the closest ones—and arguably the most reliable dynamical studies of mass-to-light ratios—are the ATLAS3D papers by Cappellari et al. (2012a, 2012b). Figure 14 in the second paper shows correlations of M/L_r against σ separately for their rapidly rotating and slowly rotating early-type galaxies. Lauer (2012) shows that the above division is closely related to the E–E dichotomy, but it is not the same. In particular, some "slow rotators" have smaller σ than any core galaxy. A detailed comparison of their results and ours is postponed to a future paper. But they find a shallower variation in M/L_r for slow rotators than for rapid rotators and interpret it as a signature of dry mergers. More importantly, for σ > 160 km s⁻¹ (slightly smaller than the smallest σ for our core galaxies), the slow rotators in their Figure 14 show no variation of M/L_r with σ and therefore L. We conclude that the ATLAS3D results are fully consistent with our results in what the measurements show and in what the authors conclude. These sanity checks are reassuring.

In summary, our division into two kinds of ellipticals is robust. Notwithstanding the above caveats, we are encouraged to interpret the sense of the difference between core and coreless galaxies in Figures 1 and 2 as further evidence that the most recent (one or several) mergers that made core galaxies were major and dry.

The emphasis on major mergers that characterized galaxy research in the 1990s (Wielen 1990) has in recent years been moderated as the feeling has grown that major mergers are rare whereas minor mergers happen in large numbers to every big galaxy (e.g., Hirschmann et al. 2012). Relaxation oscillations in our understanding are natural. However, we have the impression that the pendulum has now swung too far in the other direction. The discussion in the previous section supports this. If minor mergers were the dominant growth mode of massive ellipticals, we would expect a different Faber–Jackson relation for core galaxies than we observe. Minor mergers must happen, and they presumably affect galaxy structure at large radii, but the inner parts of core galaxies seem to have formed mainly in major mergers.

Another key to this belief is our understanding of the formation of galaxy cores:

Many n-body simulations show that, when galaxy merge, the highest densities in the merger remnant are similar to the highest densities in (one or both) progenitor galaxies. Because central densities are higher in lower-luminosity galaxies (Kormendy 1985; Lauer 1985; Faber et al. 1997), major mergers tend to destroy the correlations between near-central parameters (Faber et al. 1997). That is, cores are not a natural result of major mergers. So, why do virtually all of the most luminous ellipticals have cores? The now-favored explanation is that cores are manufactured by black hole (BH) binaries that form naturally in mergers (e. g., Ebisuzaki et al. 1991; Faber et al. 1997; Milosavljević & Merritt 2001; Milosavljević et al. 2002). When a BH binary with total mass M_• interacts with stars in the galaxy, the tendency toward energy equipartition causes the stars to gain energy and therefore to be thrown away from the center. Therefore, the stellar density is decreased and a core is excavated. Multiple mergers have cumulative effects; if one merger creates a mass deficit of fM_•, then N successive mergers should make a mass deficit of ∼NfM_•. Theoretical predictions are that f ∼ 0.5–2. Published measurements of mass deficits M_def are approximately consistent with these predictions; M_def∝M_• and Nf ∼ 1–10, consistent with core formation by several successive dry mergers (Milosavljević & Merritt 2001; Milosavljević et al. 2002; Ravindranath et al. 2002; Ferrarese et al. 2006; Merritt 2006; Kormendy & Bender 2009).

The important point is that, for core scouring to lift enough stars, mass ratios of ∼1:1 are favored. The most accurate measurement of the mean relative mass deficit is 〈M_def/M_•$\rangle = 4.1^{+0.8}_{-0.7}$, and the largest values are ∼9 (Kormendy & Ho 2013). The most accurate n-body determination of how much mass is lifted in one merger is 〈M_def/M_•〉 ≃ 0.5 (Merritt 2006). The fractional mass lifted decreases only slowly with decreasing BH mass ratio. Nevertheless, for observed mass deficits to be explained by reasonably small numbers of mergers, major mergers with mass ratios near 1:1 are favored.

Other evidence points in the same direction (e.g., van der Wel et al. 2009).

The conclusion from hierarchical clustering simulations that major mergers are rare is not a problem. Giant core ellipticals are rare. The Virgo cluster contains only eight of them among thousands of smaller galaxies. If we do not conclude that core ellipticals formed in a rare process, then we are deciding on the wrong process.

The Faber–Jackson correlation between velocity dispersion and luminosity is different for core and coreless ellipticals. Coreless ellipticals show the well-known, steep relation, approximately σ∝L^1/4. In contrast, core ellipticals have a shallower correlation, σ∝L^0.12. The implied, still shallower dependence on galaxy mass, σ∝M^0.09, is approximately similar to n-body predictions for dissipationless major mergers. Like previous authors (e.g., Desroches et al. 2007; von der Linden et al. 2007; Lauer et al. 2007a; Cappellari et al. 2012a, 2012b), we interpret this as evidence that brightest cluster galaxies formed dissipationlessly.

This has implications for the estimation of the masses M_• of supermassive BHs at galaxy centers. Galaxy luminosities and velocity dispersions are both commonly used as proxies for M_•. However, it is well known that they give different answers for the biggest galaxies (Lauer et al. 2007a). Our interpretation of the different Faber–Jackson relation for core ellipticals now tells us why: major dry mergers increase σ only a little, but the final BH mass after the merger is the sum of the input BH masses, and this increases M_• more rapidly than σ. That is, unlike the situation at low M_•, where M_• correlates tightly with σ (e.g., Tremaine et al. 2002; Kormendy & Ho 2013), luminosity is the better M_• predictor for the largest galaxies. This confirms the conclusions of Lauer et al. (2007a). In particular, it explains the observation that the correlation of BH masses with velocity dispersions "saturates" at high masses such that M_• becomes almost independent of σ (e.g., McConnell et al. 2011; McConnell & Ma 2013; S. P. Rusli et al. 2013, in preparation; see Kormendy & Ho 2013 for an update and review).

It is a pleasure to thank Thorsten Naab for helpful discussions and the referee for a very helpful report. J.K.'s work is supported by the Curtis T. Vaughan, Jr., Centennial Chair in Astronomy at the University of Texas at Austin.