MERGERS IN GALAXY GROUPS. II. THE FUNDAMENTAL PLANE OF ELLIPTICAL GALAXIES

Throughout this paper, we will take the liberty of assuming a consistent set of units of size (kpc), velocity (${\rm km}\;{{{\rm s}}^{-1}}$), mass (${{M}_{\odot }}$), luminosity (${{L}_{\odot }}$), and surface brightness (${\rm mag}/{\rm arcse}{{{\rm c}}^{2}}$), and omit these units from logarithms, writing:

$$\begin{eqnarray}&&{\rm log} R=a{\rm log} \sigma +b\mu +c,{\rm or}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{\rm log} R=\alpha {\rm log} \sigma +\beta {\rm log} L+\gamma ,{\rm where}\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\alpha =a/(1-5b),\beta =-2.5\;b/(1-5b),\end{eqnarray} \tag{ 3 }$$

and γ and c are related by a unit-dependent constant. These relations can be tied to the scalar virial theorem (SVT) for a relaxed object, which states that $2T+W=0$, where T and W are the total kinetic and potential energy, respectively. We distinguish between the SVT and the full (or tensor) virial theorem, which contains additional terms. Now if $T\propto M{{\sigma }^{2}}$, and $W\propto {{M}^{2}}/R$, then $M{{\sigma }^{2}}\propto {{M}^{2}}/R$ and $R\propto M/{{\sigma }^{2}}$. This yields the "virial FP" or virial scaling:

$$\begin{eqnarray}&&{\rm log} R=-2{\rm log} \sigma +{\rm log} M+{{c}_{v}}.\end{eqnarray} \tag{ 4 }$$

A simple dimensional equality of the form ${{\sigma }^{2}}=kG{{M}^{2}}/R$ yields the same result, but with ${{c}_{v}}-{\rm log} (kG)$. The term k is often referred to as the "structural non-homology" parameter because it is constant for self-similar (homologous), virialized systems, but can vary if the assumptions that yielded Equation (4) do not hold. It is also referred to as the "virial parameter," because its definition is an equality between kinetic and potential energy. Equation (4) can also be written as:

$$\begin{eqnarray}\begin{array}{rcl} {\rm log} R & = & -2{\rm log} \sigma +{\rm log} L+{\rm log} (M/{{M}_{\star }}) \\ {} & {} & +{\rm log} ({{M}_{\star }}/L)+{\rm log} (kG), \\ \end{array}\end{eqnarray} \tag{ 5 }$$

where ${{M}_{\star }}/L$ is the stellar mass-to-light ratio (also written as ${{\Upsilon }_{\star }}),$ whereas ${{M}_{\star }}/M$ is the stellar mass fraction within R. If these two terms are constant, and if the assumptions of virial equilibrium and the scalings of T and W that yielded Equation (4) all hold, then equating Equations (4) and (5) implies that $\alpha =-2$ and $\beta =1$. These values are the so-called "virial" coefficients, which define the "virial" FP. If the assumption of homology is broken, then these terms could vary with L, R or σ, and α and β will likely differ from −2 and 1, indicating a tilt in the FP.

Up to this point, we have not been specific about the exact definitions of M, L, R or σ. To some extent the definitions are arbitrary, especially for the size R. We choose to adopt the parametric Sérsic (1968) profile fit for consistency with many observational catalogs, and because it provides a good fit to the surface brightness profiles of elliptical galaxies (Caon et al. 1993; Graham & Colless 1997). The Sérsic profile defines a total luminosity L for the galaxy, as well as an effective radius R_e containing half of the light, and has been shown to be a suitable parameterization for the surface brightness profiles of elliptical galaxies (for a reference to Sérsic-related quantities, see Graham & Driver 2005). In keeping with convention, the dispersion σ is the central, luminosity-weighted, line of sight dispersion within ${{R}_{e}}/8$, whereas the luminosity-weighted dispersion within R_e is written as ${{\sigma }_{e}}$.

Observers often prefer to use surface brightnesses rather than luminosities, as surface brightnesses are (nearly) distance-independent and allow for the calibration of the FP as a distance indicator. We use the conventional mean surface brightness within R_e:

$$\begin{eqnarray}&&{{\mu }_{e}}=-2.5{\rm log} [(L/2)/(\pi R_{e}^{2})]+21.572+15+{{M}_{P,\odot }},\end{eqnarray} \tag{ 6 }$$

where ${{M}_{\odot ,P}}$ is the absolute magnitude of the Sun in a given photometric band P. The factor of 15 is for units of ${{L}_{\odot }}$ and kpc. Substituting into Equation (5) gives:

$$\begin{eqnarray}\begin{array}{rcl} {\rm log} {{R}_{e}} & = & 2{\rm log} \sigma +0.4{{\mu }_{e}}-{\rm log} ({{M}_{\star }}/L)-{\rm log} k \\ {} & {} & +{\rm log} ({{M}_{\star }}/M)-{\rm log} (G) \\ {} & {} & -0.4\;{{M}_{\odot ,P}}-15.427. \\ \end{array}\end{eqnarray} \tag{ 7 }$$

This is exactly the FP relation, plus four tilt terms and a fixed constant offset to return it to the virial scalings. The tilt terms are not unique and can be further subdivided, or combined. For example, Hyde & Bernardi (2009a) write the non-homology term k as a ratio between dynamical and total mass, ${{M}_{{\rm dyn}}}/{{M}_{{\rm tot}}}$ (see also D'Onofrio et al. 2006).

Finally, one can use a rotation-corrected velocity V_rms (e.g., Cappellari et al. 2006) in place of σ. We define a similar term S as ${{S}^{2}}=\sigma _{e}^{2}(1+{{\langle v/\sigma \rangle }^{2}})$, where ${{\sigma }_{e}}$ is the luminosity-weighted dispersion within R_e rather than the central dispersion (S is much like the S₁ of Weiner et al. 2006). The mean $v/\sigma$ can be measured in a number of different ways depending on the source of the data; we opt for a similar luminosity-weighted average within R_e for consistency with Cappellari et al. (2011). Incorporating S adds a tilt term to Equation (5):

$$\begin{eqnarray}\begin{array}{rcl} {\rm log} {{R}_{e}} & = & -2{\rm log} \sigma +{\rm log} L+{\rm log} ({{M}_{\star }}/L)+{\rm log} {{k}_{S}} \\ {} & {} & -{\rm log} ({{M}_{\star }}/M)+2{\rm log} (\sigma /S)+{\rm log} (G), \\ \end{array}\end{eqnarray} \tag{ 8 }$$

where ${{k}_{S}}={{S}^{2}}R/(GM)=k{{(S/\sigma )}^{2}}$.

Since Djorgovski & Davis (1987), numerous works have expanded on the interpretation of the FP and its tilt, both from theoretical considerations and observationally attempting to decompose the various contributors to the tilt. Bender et al. (1992) proposed an alternative coordinate system dubbed k-space, emphasizing the need to consider both face-on and edge-on projections of the FP, while Guzman et al. (1993) quantified the finite extent of the FP. Bender et al. (1993) and Guzman et al. (1993) were among the first to detail the importance of stellar populations (age and metallicity) in determining the ${{M}_{\star }}/L$ term's contribution to the tilt. It is now commonly accepted that the stellar mass-to-light ratio term is a significant contributor to the tilt, while some form of structural non-homology (Prugniel & Simien 1996) can explain the remainder—possibly rotational support, non-universal stellar mass profiles (Prugniel & Simien 1997; Busarello et al. 1997; Trujillo et al. 2004) and/or variable dark matter fractions (Hyde & Bernardi 2009b; Graves & Faber 2010).

More recently, total masses have been inferred by studies using spatially resolved kinematics and dynamical models (Rix et al. 1997; Gerhard et al. 2001; Cappellari et al. 2006, 2013a,e.g.,) or from strong gravitational lensing (Bolton et al. 2008; Auger et al. 2010). Such analyses have been complicated by the fact that stellar masses are sensitive to the assumed form of the initial mass function (IMF), which recent evidence suggests may be non-universal (e.g., van Dokkum 2008; Conroy & van Dokkum 2012), impacting the tilt of the FP (Dutton et al. 2011). Stellar masses can also be sensitive to the star formation history of each galaxy (Allanson et al. 2009; Graves & Faber 2010), which is difficult to constrain. Even if the stellar mass is well-constrained, dynamical and lensing models necessarily make non-trivial assumptions to determine the total mass.

Nonetheless, Auger et al. (2010) modeled strong lensing galaxies to find a relation between projected mass density Λ within ${{R}_{e}}/2$, and the mean σ within ${{R}_{e}}/2$, ${{\sigma }_{e/2}}$, of ${\rm log} {{R}_{e}}\propto (1.83\pm 0.13){\rm log} {{\sigma }_{e/2}}-(1.30\pm 0.06){\rm log} {\rm \ \Lambda\ }$. This is equivalent to ${\rm log} R\propto \alpha {\rm log} {{\sigma }_{e}}+\beta {\rm log} M$ with $\alpha =-1.14$ and $\beta =0.81$, neither pair being exactly the virial FP coefficients.

In an independent sample of local early type galaxies, Cappellari et al. (2013a) inferred total masses from dynamical modeling of spatially resolved stellar kinematics. This yielded a best-fit plane of the form ${\rm log} {{M}_{{\rm model}}}\propto \alpha {\rm log} {{\sigma }_{e}}+\beta {\rm log} {{R}_{e}}$, with $\alpha =1.928\pm 0.026$ and $\beta =0.991\pm 0.024$, which is very nearly the virial FP. While this would seem to imply that the FP is simply a reflection of the fact that galaxies are virialized, recall that Equation (4) required the assumptions that $M{{\sigma }^{2}}\propto T$, and that ${{M}^{2}}/R\propto T$, neither of which have been validated. Indeed, Cappellari et al. (2013a) point out that the fact that k appears to be constant for their particular mass and radius definitions is not necessarily an obvious extension of the virial theorem, since real galaxies can be complex, multi-component systems. We will return to this point later.

Galaxy mergers have long been thought to form elliptical galaxies, and numerical simulations—especially N-body simulations—have been a particularly powerful tool for testing this hypothesis (see Struck 1999 and Struck & Mason 2006, p. 115 for excellent reviews). However, a much smaller fraction of the literature has attempted to predict or explain the nature of the FP and its tilt. Weil & Hernquist (1996) presented detailed simulations of mergers in groups, but their predictions for the FP were left unpublished (see Weil 1995). Bekki (1998) presented one of the first analyses of the FP tilt in dissipational binary spiral merger simulations, suggesting that dissipation could establish non-homology. A number of studies alternatively focused on mergers of spheroidal galaxies, including Capelato et al. (1995), Nipoti et al. (2003a), Boylan-Kolchin et al. (2005), Borriello et al. (2006), and more recently Hilz et al. (2012). Most of these studies have focused on determining whether the FP and its tilt can be preserved in subsequent mergers of spheroids—so far, the answer is "maybe," depending on the number of mergers, mass ratios, orbital parameters, and properties of the merging systems. However, such simulations typically do not address how spheroidal galaxies are formed in the first place.

Binary, major mergers of spiral galaxies are still viewed as a plausible formation channel for elliptical galaxies, although this mechanism is not universally accepted (Naab & Ostriker 2009). Aceves & Velázquez (2005) presented simulations of binary mergers of spirals scaled to follow the observed Tully–Fisher relation (Tully & Fisher 1977). These merger remnants were similar to ellipticals and followed a tilted FP even in the absence of gas dissipation, owing to the precise scaling of the progenitor spirals. By contrast, Robertson et al. (2006) reported that dissipational binary mergers were sufficient to create a tilted FP (a = 1.55 and b = 0.33) with 0.06 dex scatter. Hopkins et al. (2008) went a step further, claiming that dissipation is both necessary and sufficient to create the FP. However, such binary merger simulations did not incorporate truly hierarchical cosmological merger histories and so did not consider the impact that repeated, mainly minor mergers may have on the size and mass growth of ellipticals. Although a few consecutive re-mergers of merger remnants were included, binary mergers alone do not capture the full range of possible merger histories of ellipticals, and so are at risk of overestimating the role of dissipation over structural non-homology from dissipationless merging, as reported by Aceves & Velázquez (2005).

A relatively new class of cosmological "zoom" simulations can in principle follow realistic merger histories and form elliptical galaxies self-consistently. Zoom simulations have been used to study 2D scaling relations (Naab et al. 2009; Oser et al. 2012) and detailed kinematics (Naab et al. 2014), but not the FP, as far as we are aware—perhaps due to the necessarily limited resolution of full cosmological simulations. Oñorbe et al. (2006) presented scaling relations from hydrodynamical cosmological simulations in small volumes (10 Mpc side boxes), but with low resolution by today's standards (64³ gas and dark matter particles each). Feldmann et al. (2011) and Dubois et al. (2013) compared their central group galaxies to the projected FP, but with six to seven galaxies each, neither study had the statistics to fit scaling relations. Hybrid simulations are an alternative to full ab initio conditions, whereby halos in a dark matter only simulation are seeded with realistic galaxies at intermediate redshifts and evolved to the present (e.g., Dubinski 1998). Nipoti et al. (2003b) simulated the evolution of five massive spheroidal galaxies in a z = 0.59 cluster, while Ruszkowski & Springel (2009) analyzed several dozen merger dry merger remnants in a rich cluster seeded with 50 galaxies; in both cases, the initial galaxies contained a stellar bulge and dark halo but not disks. Both papers presented plots of the projected FP, finding evidence for a tilt—in Ruszkowski & Springel (2009), driven by variable dark matter fractions—but neither quoted best-fit FP parameters.

Taranu et al. (2013, hereafter Paper I) presented controlled, collisionless N-body simulations of mergers of spirals in groups, incorporating cosmologically motivated merger histories and measuring accurate sizes, dispersions and luminosities for the central merger remnants. We now aim to extend the analysis of the morphologies and kinematics of these galaxies to assess the impact of dissipationless merging of spiral galaxies on the FP.

The simulations of Paper I were run with the PARTREE N-body tree code (Dubinski 1996). Briefly, they consist of over a hundred galaxy groups, initially comprising three to thirty spiral galaxies sampled from a realistic LF. Each group was designed to collapse like a high redshift (z = 2) group at the turnaround radius, inducing mergers and forming a central elliptical. Unlike previous merger simulations, the groups were meant to sample a variety of plausible cosmological accretion histories, albeit not an unbiased sample. The initial galaxies were also chosen to follow the observed Tully–Fisher relation (Tully & Fisher 1977) with no intrinsic scatter.

The initial conditions in the simulations were tightly controlled, allowing only a few parameters to vary. All of the galaxies were self-similar, re-scaled models of M31 (Widrow et al. 2008). Two versions of each simulation were run: one where the spirals began with an exponential bulge (the "bulge Sérsic index = 1" or $B.{{n}_{s}}=1$ sample), and another otherwise identical run with de Vaucouleurs profile bulges ($B.{{n}_{s}}=4$). This roughly brackets the range of realistic bulge profile in spiral galaxies, allowing us to test the impact of input galaxy structure on the properties of the final remnant. Additionally, groups of a given mass were seeded with varying numbers of galaxies, in order to test the impacts of multiple, mostly minor mergers versus a few mostly major mergers. We distinguish between these subsamples as "Many" (M) and "Few" (F), where the former groups began with a larger-than-average number of galaxies for their mass. Similarly, the main group sample had galaxy luminosities drawn from a realistic "LF," whereas a smaller control sample featured equal-mass mergers ("Eq").

Although collisionless, the sample size and resolutions (typically 1–2 million stellar particles with a 100 pc softening radius) of the Paper I simulations are both higher than in typical zoom simulations (Naab et al. 2009, e.g.,). Each central remnant was imaged along 10 randomly oriented, evenly spaced projections, as well as the three principal axis vectors, creating surface brightness and kinematic maps. A full account of the morphology, kinematics, and structural properties of the simulated galaxies in this catalog is available in Paper I, along with a more detailed description of the methodology.

We use 2D Sérsic fits from (Simard et al. 2011, hereafter S+11) and visual morphological classifications from (Nair & Abraham 2010, hereafter N+10), both based on SDSS data. We also use spatially resolved kinematics for 65 nearby ellipticals from ATLAS3D (Cappellari et al. 2011), including kinematic measures from Cappellari et al. (2013a), and stellar mass-to-light ratios and dark matter fractions from Cappellari et al. (2013b). In Paper I, we used 2D GALFIT fits described in the appendix of Krajnović et al. (2012). In this paper, we will use the values of L and R_e tabulated in Cappellari et al. (2013a), as recommended by the authors of that paper. Although these are not derived from 2D Sérsic fits, they are generally similar for most galaxies and allow for more direct comparison to Cappellari et al. (2013a). Since spatially resolved kinematics are unavailable for most SDSS galaxies, we use central velocity dispersions σ unless otherwise specified.

Wherever possible, we have opted to use similar methodologies in all cases. The simulation sizes and luminosities are based on single Sérsic fits using GALFIT (Peng et al. 2002, 2010). SDSS data use single Sérsic fits from GIM2D (Simard et al. 2002). SDSS stellar masses are based on fits to photometry (Mendel et al. 2014), using a Chabrier (2003) IMF. These stellar masses are on average marginally but not significantly different from those used in Paper I, and we refer to Mendel et al. (2014) for full details. A3D stellar masses are derived by multiplying the model L_r with the $M/{{L}_{r}}$ derived from fits to spatially resolved spectra assuming a Salpeter (1955) IMF (Cappellari et al. 2012, 2013b). To maintain consistency with the normalization of $M/L$ values with different IMFs, we divide the A3D $M/{{L}_{r}}$ by a factor of 1.7, which compensates for the systematically larger $M/{{L}_{r}}$ in a Salpeter IMF for an old, solar-metallicity population. This does not entirely account for systematic differences between stellar masses from the two samples, and Mendel et al. (2014) caution that stellar masses derived from photometric fits are subject to up to 60% systematic errors. We similarly caution against over-interpretation of trends based on stellar masses with a fixed IMF.

The FP relation is typically written as ${\rm log} {{R}_{e}}=a{\rm log} \sigma +b\mu +c$, as in Equation (1). We derived this relation from the SVT in the traditional way in Section 2.1. The FP can also be expressed in terms of L instead of μ. This formulation is equivalent to using μ, and Equation (3) gives simple transformations between the luminosity and μ FPs. While luminosity is arguably a more fundamental variable, being independent of R_e, we will use μ for the moment for consistency with previous studies.

Figure 1 shows the projected FP, with two variables (σ and ${{\mu }_{e}}$) collapsed onto the x-axis, as in Hyde & Bernardi (2009a). This visually demonstrates the exceptionally small orthogonal scatter of the FP, even for the exponential bulge ($B.{{n}_{s}}=1$) sample, which is not shown exactly edge-on. Similarly, the tilt does depend on the structure of the progenitors, since it is larger in magnitude for the $B.{{n}_{s}}=4$ sample, as is the extent of the FP.

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Table 1 tabulates best-fit FP parameters for simulations and observations alike. Several trends are clear from the simulations. First, the tilt is smaller than observed but definitely in the correct sense: $a\lt 2$ and $b\lt 0.4$. The scatter in the simulations is exceptionally small (0.02 dex), even when combining different progenitor subsamples, and is considerably smaller than for the observations (0.06–0.08 dex). For the most part, the different observational samples are consistent with one another, other than unusually small a parameters for the N+10 weighted subsample and b for the unweighted (both possibly due to the undersampling of faint and small ellipticals in N+10). Nonetheless, S+11 is largely consistent with the completely independent measurements from A3D, although the A3D intercepts are slightly lower and scatter slightly larger. It is also clear that the FP of S0 galaxies is quite different from that of ellipticals; since the simulated galaxies show no signs of being genuine bulge plus disk systems, we will not consider further comparisons with S0s. It should be noted, however, that the S+11 sample is likely contaminated with S0s, which is unavoidable since no automated classification can separate them (see Paper I for details).

Table 1.  Sérsic Model Fundamental Plane Fits

Simulations: 10 Equally Spaced Projections, Randomly Oriented
$B.{{n}_{s}}$		Subsample	a	b	Intercept	rms
1		All	1.74	0.32	−9.50 ± 0.06	0.02
1		Many	1.80	0.31	−9.49 ± 0.10	0.02
1		Few	1.70	0.31	−9.26 ± 0.11	0.02
4		All	1.64	0.29	−8.65 ± 0.04	0.02
4		Many	1.67	0.28	−8.60 ± 0.06	0.02
4		Few	1.64	0.29	−8.62 ± 0.08	0.02
All		All	1.69	0.29	−8.89 ± 0.04	0.02
All		Many	1.75	0.28	−8.71 ± 0.05	0.02
All		Few	1.66	0.29	−8.85 ± 0.07	0.02
Principal Axis Projections, Unweighted
$B.{{n}_{s}}$		Projection	a	b	Intercept	rms
1		Major axis	1.76 ± 0.03	0.30	−9.22 ± 0.25	0.03
1		Minor axis	1.71 ± 0.02	0.31	−9.30 ± 0.29	0.02
4		Major axis	1.71 ± 0.02	0.28	−8.74 ± 0.14	0.02
4		Minor axis	1.58 ± 0.01	0.28	−8.36 ± 0.12	0.01
Observations
Cat.	T.	W.	a	b	Intercept	rms
S+11	E	N	1.35 ± 0.00	0.28	−7.89 ± 0.01	0.06
S+11	E	Y	1.28 ± 0.00	0.29	−7.95 ± 0.01	0.06
N+10	E	N	1.36 ± 0.04	0.22	−7.88 ± 0.12	0.06
N+10	S0	N	1.08 ± 0.02	0.27	−7.11 ± 0.08	0.07
N+10	E	Y	1.08 ± 0.10	0.28	−7.40 ± 0.28	0.06
A3D	E	N	1.29 ± 0.09	0.33	−9.12 ± 0.55	0.07
A3D	S0	N	0.96 ± 0.07	0.30	−7.63 ± 0.38	0.07
A3D	E	Y	1.31 ± 0.09	0.33	−9.09 ± 0.52	0.07

Notes. Slopes are given in log space, i.e., for ${\rm log} {{R}_{e}}$ as a function of ${\rm log} L$. Simulation data are from analyses after 10.3 Gyr, including various subsamples of randomly oriented (but equally spaced) projections, as detailed in the text, as well as principal axis projections. Simulated data include subsamples for relatively many or few mergers. Observational data for each catalog (Cat.) and Hubble type (T.) are 1/V$_{{\rm max} }$ corrected, with fits optionally weighted (W.) or not by the difference between the simulated and observed luminosity functions. Rms lists the Rms orthogonal scatter of all points from the best-fit relation. Errors on a for simulations and on b for simulations, N+10 and S+11 are uniformly 0.01 or smaller, and are omitted for brevity; for A3D, errors on b are 0.02.

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In Paper I, we presented optional weighting schemes, which can have a significant impact on 2D scaling relations—i.e., projections of the FP. For the simulations, we weighted the "Many" merger sample more heavily at the more luminous end, and the "Few" merger sample at the faint end. This weighting does not impact the simulated FP fits, because all of these points really do lie on a single plane with minimal scatter. No weighting scheme will make a significant difference unless it changes the weights where there is curvature in a scaling relation, and there is no curvature in the simulated FP. We omit these weighted fits from Table 1, as all of the parameters are within the errors of the unweighted fits.

A second weighting scheme in Paper I weighted the observed data to match the much flatter LF of the simulations. This scheme can make a significant difference (especially for N+10), which is possible if there is some curvature in the observed plane at the high mass end. In the N+10 case specifically, it may also be due to the sample selection, which selected roughly a log-normal distribution around L* rather than a magnitude- or volume-limited sample.

The FP parameters also depend on the projection angle. Choosing only minor axis projections yields the steepest FP, whereas major axis projections minimize the tilt. They can also be sensitive to the orbits of the merging galaxies—we leave this analysis to the Appendix. Regardless, the tilt in the simulations is weaker than observed. This difference can be quantified in various ways. One method is to project the vector between the unit normal of the simulated plane (−1, $a\approx 1.69$, $b\approx 0.29$) and the unit normal of the virial plane (−1, a = 2, b = 0.4) onto a similar vector between the observed (−1, $a\approx 1.3$, $b\approx 0.29$) and virial plane's unit normals. This gives a "tilt fraction" of 37%. Any other quantification of the tilt fraction would yield a similar result, so this difference requires an explanation.

The discrepancy between the predicted and observed tilt is mainly in the a parameter and is lessened if one considers the stellar mass FP (Table 2). This is justifiable because the simulations effectively do not include any stellar population variations and have a constant ${{M}_{\star }}/L$, whereas luminous ellipticals tend to have larger ${{M}_{\star }}/L$ in general (and ${{M}_{\star }}/{{L}_{r}}$ in particular), assuming a universal IMF. This implies that a substantial portion of the tilt is due to variations in stellar populations along the FP. Once this is accounted for, the gap between the observed tilt and the $B.{{n}_{s}}=4$ sample's tilt is considerably smaller. In fact, for A3D, the observed tilt in the stellar mass FP is smaller than in the $B.{{n}_{s}}=4$ sample, as A3D has a significantly larger value of $b\approx 0.36$ than the other samples. This b value is consistent with the near-infrared J-band FP fits from the 6dF survey (Magoulas et al. 2012) of $a=1.52\pm 0.03$ and (effectively) $b=0.36\pm 0.02$. This is close to a stellar mass FP, since $M/{{L}_{\star ,J}}$ is less sensitive to age and metallicity than $M/{{L}_{\star ,r}}$.

Table 2.  Sérsic Model Stellar Mass Fundamental Plane Fits

Simulations: 10 Equally Spaced Projections, Randomly Oriented
$B.{{n}_{s}}$		Subsample	a	b	Intercept	rms
All		All	1.72	0.31	−9.07 ± 0.04	0.02
All		Many	1.78	0.30	−8.91 ± 0.05	0.02
All		Few	1.69	0.32	−9.05 ± 0.07	0.02
All		All−S	1.78	0.29	−8.69 ± 0.05	0.02
Observations
Cat.	T.	W.	a	b	Intercept	rms
S+11	E	N	1.48 ± 0.00	0.28	−7.87 ± 0.01	0.06
S+11	E	Y	1.42 ± 0.01	0.28	−7.96 ± 0.01	0.06
N+10	E	N	1.56 ± 0.04	0.27	−8.09 ± 0.11	0.06
N+10	S0	N	1.26 ± 0.02	0.28	−7.76 ± 0.06	0.06
N+10	E	Y	1.25 ± 0.06	0.28	−7.67 ± 0.20	0.06
A3D	E	N	1.71 ± 0.08	0.36	−10.13 ± 0.49	0.05
A3D	S0	N	1.56 ± 0.09	0.28	−8.29 ± 0.45	0.06
A3D	E	Y	1.73 ± 0.08	0.36	−10.10 ± 0.48	0.06
A3D-S	E	N	1.78 ± 0.11	0.37	−10.04 ± 0.67	0.06
A3D-S	S0	N	1.87 ± 0.14	0.30	−9.42 ± 0.61	0.07
A3D-S	E	Y	1.83 ± 0.11	0.37	−10.50 ± 0.61	0.06

Note. Column headings and notes are as in Table 1, but surface brightnesses/magnitudes are now based on stellar masses instead of luminosities. See text for details on sources of stellar masses. A3D-S and All-S use the kinematic measure S in place of σ, correcting for ordered rotation—see Equation (8).

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It is clear that multiple dissipationless mergers can produce a tilt in the FP, and that this tilt could be a significant fraction of the observed tilt. However, we caution again that the stellar mass FP is sensitive to the assumption of a universal IMF and/or star formation history. Furthermore, the stellar masses for A3D are based on fits to spatially resolved spectra assuming a Salpeter IMF, whereas SDSS stellar masses are based on fits to photometry assuming a Chabrier IMF instead; both methods have their caveats and we make no judgment as to which are more likely to be accurate.

Finally, Table 2 includes a fit to the FP using S in place of σ, for A3D and the simulations. Using S lowers the tilt slightly in A3D, although only significantly for the S0s—not surprisingly, since they show more rotational support and are more affected by the $v/\sigma$ correction. Using S in the simulations makes little difference in a and actually lowers b to 0.29. This is likely because the simulated ellipticals generally have less rotational support than A3D ellipticals, as shown in Paper I.

Having established that the FP of the simulations is tilted from the virial relation, we will now decompose the FP and fit the various tilt terms from Equation (5). We use ${{M}_{r,\odot }}=4.68$ throughout, which sets the constant in Equation (5) to −11.93.

The full tilt of the FP can be characterized from the equality ${{\sigma }^{2}}={{\Upsilon }_{t}}GL/{{R}_{e}}$, where the tilt factor ${{\Upsilon }_{t}}=k(M/L)$ is a mass-to-light ratio that encompasses the entirety of the tilt, including non-homology (k), stellar population variations (${{\Upsilon }_{\star }}$) and variable dark matter fractions (${{M}_{\star }}/{{M}_{{{R}_{e,3{\rm D}}}}}$). Figure 2 shows ${{\Upsilon }_{t}}$ as a function of luminosity for the entire sample. The term clearly varies with luminosity in both observations and simulations, although the variation is shallower for simulations (unsurprisingly, since the tilt is also smaller). Nonetheless, the intercept and median values of ${{\Upsilon }_{t}}$ are consistent with S+11, though somewhat offset from A3D. This is mainly because the simulated galaxies have lower σ than observed, and A3D galaxies have slightly larger σ than the other samples. Sample selection is evidently important, since the N+10 is a subset of the S+11 samples and uses the same σ, R_e and L measures, but shows an anomalous break in ${{\Upsilon }_{t}}$ at $2\times {{10}^{10}}\;{{L}_{\star }}$. This break is likely to be the cause of the difference between weighted and unweighted N+10 fits. We do not speculate further on the causes of these systematic differences and simply re-assert that the simulated tilt is at least broadly consistent with the observed intercept and shallower in the slope.

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We now turn to identifying which of the tilt terms contribute to the variable ${{\Upsilon }_{t}}$. Figure 3 shows projected dark matter fractions, i.e., $1-{{M}_{\star ,{{R}_{e}},2{\rm D}}}/{{M}_{{{R}_{e}},2{\rm D}}}$. Similar fractions of order 2%–5% smaller are found for $1-{{M}_{\star ,{{R}_{e}},3{\rm D}}}/{{M}_{{{R}_{e}},3{\rm D}}}$. Dark matter fractions increase for larger galaxies because the effective radii extend further out to regions dominated by the halo. This is evidenced by the fact that independent projections of the same galaxy with larger R_e have larger dark matter fractions—unsurprisingly, since the dark matter fraction will increase with the radius as long as the stellar density profile is steeper than the halo profile.

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The non-trivial discovery here is that ${{M}_{\star }}/{{M}_{{{R}_{e}},2{\rm D}}}$ is significantly smaller in larger/more luminous galaxies, driving what would appear to be a large fraction of the tilt. This is partly caused by the fact that, as shown in Paper I, galaxies undergoing many mergers (M) are larger at fixed luminosity than those formed from few mergers (F); similarly, at a fixed size, the M subsample is less luminous and so contains a smaller stellar mass fraction. For the whole sample, we find that ${{M}_{{\rm DM},{{R}_{e}},3{\rm D}}}\propto R_{e}^{2}$ nearly exactly, whereas ${{M}_{\star ,{{R}_{e}},3{\rm D}}}\propto R_{e}^{1.72}$, with a slightly shallower slope for the most luminous galaxies. The enclosed dark matter mass in different galaxies scales more steeply with R_e than does the stellar mass.

The full explanation for the origin of this trend in dark matter fractions is complicated by the fact that neither the stellar density profiles nor the dark halo profiles are self-similar in the merger remnants. Nonetheless, we have demonstrated that merging multiple self-similar galaxies produces a particular scaling relation for ${{M}_{\star }}/M$ within R_e, rather than retaining the same constant fraction that they began with (roughly 30%–35%, depending on how one defines R_e for a spiral galaxy). We will discuss the absolute values of these dark matter fractions and compare them to observations further in Section 7; for now, what matters is that the trend lies in the correct sense to cause some part of the tilt in the FP.

The three remaining terms that can contribute to the tilt are shown in Figure 4, beginning with the virial parameter k_S. For a dispersion-supported, uniform unit sphere, ${{k}_{S}}\simeq k=0.2$, so most of the simulated galaxies have ${{k}_{S}}\gt 0.2$ because they are centrally concentrated. For the most part, k_S does not vary strongly with any of the FP parameters, although it tends to be larger for more luminous galaxies. At the most luminous end, a number of outliers appear with unusually low values of k_S. These tend to be systems which were shown to have overestimated R_e in Paper I, and their k_S values are small only because the $M_{{{R}_{e}},3{\rm D}}^{2}$ term in the denominator increases more rapidly than R_e. The outliers with large k_S tend to be triaxial systems with strongly projection-dependent S.

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The remaining two tilt terms are also shown in Figure 4 and contribute little to the tilt in simulations. The $\sigma /S$ term (left panel) is nearly constant at unity in both simulations and in A3D. Again, the main outliers are groups for which R_e has been overestimated. Since projected dispersions tend to drop with the radius, ${{\sigma }_{e}}/\sigma$ becomes significantly smaller than unity if R_e is overestimated. While ${{\sigma }_{e}}$ is typically quite small in A3D, it is often offset by a substantial rotation correction in $v/\sigma$. This correction is much smaller in the simulated ellipticals, since most are slow rotators (see Paper I).

In the simulations, ${{M}_{\star }}/L$ is nearly constant by construction; the only outliers are groups for which a massive satellite appears within R_e, since we have not masked or fit satellites in the stellar mass maps. By contrast, both observed data sets show a trend of increasing ${{M}_{\star }}/L$ with L. The A3D ellipticals appear to have a steeper slope, which may be partly due to systematics. A3D ${{M}_{\star }}/L$ are derived from fits to spatially resolved spectra, whereas the SDSS ${{M}_{\star }}/L$ is based on fits to broadband colors. The A3D fits also assume a Salpeter IMF rather than the Kroupa IMF used in SDSS; we have adjusted the normalization of the A3D ${{M}_{\star }}/L$ by dividing by a factor of 1.7 to correspond to an old, Kroupa IMF, but this normalization may not be self-consistent.

Having established that the dark matter fraction is likely a major contributor to the tilt, we will now fit various permutations of the FP, including each tilt parameter to determine which are closest to the virial FP. To do so, one can fit an FP with R_e, a velocity (σ, ${{\sigma }_{e}}$ or S), and a third parameter other than the usual L or ${{\mu }_{e}}$. The values of the parameters are then compared to the expected virial coefficients a = 2, b = 0.4 or $\alpha =-2$ and $\beta =1$, depending on whether the third parameter is a mass/luminosity or surface density. In principle, the fits should be convertible using Equation (3), but in practice the choice can be significant. As an example, one fit close to the virial plane in the simulations is:

$$\begin{eqnarray}&&{\rm log} {{R}_{e}}=\alpha {\rm log} S+\beta {\rm log} {{M}_{{{R}_{e}},3{\rm D}}}+\gamma ,\end{eqnarray} \tag{ 9 }$$

with $\alpha =-1.97\pm 0.03$, $\beta =1.02\pm 0.01$, and just 0.023 dex scatter. This fit is tabulated as fit #3 in Table 3, along with the dynamical mass coefficient c, which will be discussed further in Section 4.4. The fit is almost exactly the virial FP in Equation (4), and should yield parameters of a = 1.89, b = 0.39. Fitting the same FP with $-2.5{\rm log} ({{M}_{{{R}_{e}},3{\rm D}}}/R_{e}^{2})$ as the third parameter yields $a=1.90\pm 0.01$, $b=0.379\pm 0.003$. These are nearly precisely the expected values, but the errors on a and b are smaller, so that the FP tilt is small but significant. Using ${{\sigma }_{e}}$ in place of S gives very similar results, so both are acceptable choices for relatively slowly rotating galaxies. However, it is not always the case that a and b transform to α and β exactly. For example:

$$\begin{eqnarray}&&{\rm log} {{R}_{e}}=a{\rm log} \sigma -2.5b{\rm log} ({{M}_{{{R}_{e}},3{\rm D}}})/R_{e}^{2}+c,\end{eqnarray} \tag{ 10 }$$

yields $a=1.94\pm 0.01$, $b=0.43\pm 0.01$, and 0.028 dex scatter. However, fitting a similar FP as in Equation (9) returns $\alpha =-1.55\pm 0.02$, $\beta =0.90\pm 0.01$, with 0.025 dex scatter (fit #1 in Table 3). This is quite far from the virial plane and even more discrepant from the simple conversion using Equation (3). It is not generally the case that fits using mass versus mass surface density are identical, and it appears that using ${{\sigma }_{e}}$ or S is necessary to obtain a virial FP in both cases.

Table 3.  Mass Fundamental Plane and Dynamical Mass Estimators

Simulations: All $B.{{n}_{s}}$, 10 Equally Spaced, Randomly Oriented Projections
#	M	R	K	c	${\Delta}c$	α	β	γ	rms
1	${{M}_{{{R}_{e}},3{\rm D}}}$	Re	σ	2.550 ± 0.010	0.065	−1.553 ± 0.022	0.899 ± 0.007	−5.640 ± 0.030	0.024
2	${{M}_{{{R}_{e}},3{\rm D}}}$	Re	${{\sigma }_{e}}$	2.626 ± 0.006	0.053	−1.979 ± 0.032	1.012 ± 0.009	−5.957 ± 0.036	0.021
3	${{M}_{{{R}_{e}},3{\rm D}}}$	Re	S	2.575 ± 0.006	0.058	−1.969 ± 0.031	1.017 ± 0.009	−6.016 ± 0.038	0.023
4	${{M}_{1/2}}$	${{r}_{1/2}}$	σ	2.669 ± 0.013	0.080	−1.220 ± 0.019	0.803 ± 0.005	−5.269 ± 0.024	0.027
5	${{M}_{1/2}}$	${{r}_{1/2}}$	${{\sigma }_{e}}$	2.741 ± 0.008	0.061	−1.548 ± 0.022	0.886 ± 0.006	−5.498 ± 0.024	0.023
6	${{M}_{1/2}}$	${{r}_{1/2}}$	S	2.676 ± 0.012	0.065	−1.535 ± 0.023	0.888 ± 0.006	−5.539 ± 0.026	0.025
7	${{M}_{1/2}}$	Re	σ	5.045 ± 0.027	0.141	−3.764 ± 0.189	1.390 ± 0.045	−6.665 ± 0.119	0.052
8	Mmodel	${{R}_{e,{\rm maj}}}$	σ	5.295 ± 0.016	0.061	−1.918 ± 0.022	1.007 ± 0.007	−6.342 ± 0.032	0.024
9	Mmodel	${{R}_{e,{\rm maj}}}$	${{\sigma }_{e}}$	5.447 ± 0.013	0.060	−2.509 ± 0.037	1.167 ± 0.011	−6.889 ± 0.046	0.021
10	Mmodel	${{R}_{e,{\rm maj}}}$	S	5.367 ± 0.014	0.067	−2.509 ± 0.038	1.176 ± 0.011	−6.974 ± 0.051	0.024
11	Mmodel	Re	σ	6.296 ± 0.018	0.059	−1.873 ± 0.020	1.013 ± 0.006	−6.573 ± 0.029	0.021
12	${{M}_{{{R}_{e}},2{\rm D}}}$	Re	σ	4.272 ± 0.019	0.068	−1.319 ± 0.013	0.829 ± 0.004	−5.568 ± 0.020	0.021
13	${{M}_{{{R}_{e}},2{\rm D}}}$	Re	${{\sigma }_{e}}$	4.438 ± 0.011	0.045	−1.650 ± 0.017	0.916 ± 0.005	−5.826 ± 0.020	0.017
14	M200	${{r}_{1/2}}$	σ	53.46 ± 0.32	0.107	−2.938 ± 0.053	1.382 ± 0.016	−9.819 ± 0.087	0.029
15	Mgroup	${{r}_{1/2}}$	σ	68.37 ± 0.34	0.096	−2.783 ± 0.047	1.315 ± 0.014	−9.462 ± 0.078	0.028

Note. Each dynamical mass estimator is based on a mass (M), radius (R), and kinematic (K) tracer. For each estimator, the median c is tabulated, from $c=(R{{K}^{2}}/G)/M$, along with the error on the median and the scatter ${\Delta}c$ about the median. Also shown is the best-fit ${\rm log} R=\alpha {\rm log} K+\beta {\rm log} M+\gamma$, with the rms scatter about this relation.

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Even using the projected mass ${{M}_{{{R}_{e}},2{\rm D}}}$ and ${{\sigma }_{e}}$ in Equation (10) returns $a=1.98\pm 0.01$ and $b=0.43\pm 0.01$, with 0.020 dex scatter; however, the values (α, β) = (−1.65, 0.92) are even further from the virial FP (fit #13 in Table 3). Notably, they also differ from the values quoted by Auger et al. (2010) using ${{\sigma }_{e/2}}$ and ${{M}_{{{R}_{e}}/2}}$. These are effectively (a,b) = (1.83, 0.52), or (α, β) = (−1.13, 0.81), if transformed with Equation (3). It is not clear if systematics or sample selection contribute to the differences, as we have not attempted to match methods with this particular sample or model gravitational lensing.

The one common conclusion from these fits is that most FPs using a total mass of some sort are close to the virial plane; however, reproducing ($\alpha ,\beta$) = (−2, 1) is more difficult than ($a,b$) = (2, 0.4). If the goal is to get as close to the virial FP as possible using both mass and mass surface density, then R_e, ${{M}_{{{R}_{e}},3{\rm D}}}$ and S or ${{\sigma }_{e}}$ are appropriate choices for FP variables.

However, this conclusion is specific to the conditions of our simulated galaxies, and may not apply to galaxies with different formation histories. There is no guarantee that any given mass measure will follow a virial FP. Similarly, even a mass estimator that follows a virial FP cannot give an accurate mass without knowing the dynamical mass coefficient (see Section 4.4). This is because the virial FP is not, in fact, a trivial consequence of the virial theorem. Although every merger remnant becomes very nearly virialized with a virial ratio of 1 ± 1%, biased subsets of a virialized galaxy need not have unity virial ratios. The local virial ratio of the total mass within ${{R}_{e,3{\rm D}}}$ in the merger remnants ranges from 0.4 to 0.5. This is expected because the interior mass of a galaxy (stellar or total) is a very biased subset, residing in the deepest part of the potential well. Likewise, stars are biased tracers of the potential of any galaxy with an extended dark halo. In principle, any variation of the local virial ratio within a given size measure R can translate into variation of the virial parameter k and hence a tilt in the FP. Only a homologous population of galaxies can be expected to have constant local virial ratios within R, while the value of the local virial ratio (which is reflected in the dynamical mass coefficient) should be less than unity.

The existence of a tight mass FP with the virial coefficients implies that one can extract a dynamical mass of the form ${{M}_{{\rm dyn}}}=c{{\sigma }^{2}}R/G$ as long as c is known. In Section 4.3, we showed that Equation (9) is very nearly a virial FP, and so one should be able to derive an accurate value for ${{M}_{{{R}_{e}},3{\rm D}}}$ from S and R_e alone. However, the precise value of c depends on the structure of the galaxy, while the constancy of c relies on the assumption of homology. We already showed in Figure 4 (left panel) that the virial parameter is not exactly constant and has significant projected-dependent scatter. We also demonstrated in the previous section that stellar virial ratios and virial ratios within R_e are neither exactly constant nor unity, so these assumptions must be tested.

Values of c for a variety of dynamical mass estimators are shown in Figure 5. For the projected mass ${{M}_{{{R}_{e}},2{\rm D}}}$, values of c = 3.5–5 are appropriate when using σ and R_e; for ${{M}_{{{R}_{e}},3{\rm D}}}$, c = 2.5–3 is more suitable. The tightest correlations are found by replacing σ. Using S gives a median c of 2.58 and 0.06 dex scatter, while ${{\sigma }_{e}}$ produces c = 2.63 with 0.05 dex scatter. As Figure 5 demonstrates, using S lowers the projection-dependent scatter but cannot remove it entirely—more extreme projections require values of c as small as 1.5 or as large as 3.5. While the correlation with R_e and S is fairly tight, there are several notable outliers (again, in galaxies where R_e is overestimated) and a hint of a shallow, luminosity-dependent slope in c. Nonetheless, for a typical random projection, R_e and S or ${{\sigma }_{e}}$ can be used to estimate ${{M}_{{{R}_{e}},3{\rm D}}}$ to within 10%–15%. at least partly due to the fact that these three variables define a tight virial FP with 0.02 dex scatter.

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Other dynamical mass definitions have been proposed in the literature. On theoretical grounds, Wolf et al. (2010) advocate the use of ${{M}_{1/2}}$, where ${{M}_{1/2}}$ is the total mass within ${{r}_{1/2}}$, and ${{r}_{1/2}}$ is the 3D radius containing half of the total luminosity of the galaxy. Using Jeans models, they derive a relation ${{M}_{1/2}}=3{{r}_{1/2}}{{\sigma }^{2}}$, arguing that ${{r}_{1/2}}$ is a unique radius at which the effects of anisotropy are minimized. For a Sérsic profile with $0.5\lt {{n}_{s}}\lt 10$, ${{r}_{1/2}}=1.34{{R}_{e}}$ is an excellent approximation (Ciotti 1991), and so one can also write ${{M}_{1/2}}=4{{R}_{e}}{{\sigma }^{2}}$ without having to infer ${{r}_{1/2}}$.

Another alternative dynamical mass was defined by Cappellari et al. (2013a). They applied a variant of Jeans modeling ("JAM" models) to A3D galaxies to obtain a mass M_JAM, where ${{M}_{{\rm JAM}}}=L[{{(M/L)}_{{\rm JAM}}}(r\lt {{R}_{e}})]$. Their Figure 14 suggests that ${{M}_{{\rm JAM}}}/{{M}_{{\rm vir}}}$ shows minimal scatter (0.08 dex), provided that ${{M}_{{\rm vir}}}=3.9{{R}_{e,{\rm maj}}}\sigma _{e}^{2}$, where ${{R}_{e,{\rm maj}}}$ is the major axis (not circularized) R_e. Similarly, Cappellari et al. (2013a) argue that M_JAM is an appropriate dynamical mass, because it defines a tight virial FP with ${{R}_{e,{\rm maj}}}$ and ${{\sigma }_{e}}$ (see their Figure 12).

We have tested both of these dynamical mass estimators, listing the values of c and the mass FP parameters α and β for each estimator in Table 3. ${{r}_{1/2}}$ is measured as the radius containing half of the stellar mass within r₂₀₀. This is equivalent to the half-luminosity radius, since ${{M}_{\star }}/L$ is nearly constant, and for most galaxies, almost all of the bound star particles lie within r₂₀₀. The median c for ${{M}_{1/2}}$, ${{r}_{1/2}}$, and σ (fit #4 in Table 3) is 2.67, with 0.08 dex scatter. This is within 10% of the value of 3 quoted by Wolf et al. (2010), but the scatter with ${{M}_{1/2}}$ is larger than for ${{M}_{{{R}_{e}},3{\rm D}}}$. The scatter is lower if using ${{\sigma }_{e}}$, at 0.06 (fit #5), while the median of 2.74 remains lower than 3.

Using R_e in place of ${{r}_{1/2}}$ gives a median c = 5.05 with 0.14 dex scatter (fit #7 in Table 3). This value of c is a full 25% larger than the nominal value of 4. Furthermore, the best-fit FP is wildly different from the virial FP, and has much larger scatter than any other mass FP in Table 3 (using ${{\sigma }_{e}}$ or S does not improve either fit). This is mainly because the approximation ${{r}_{1/2}}=1.34\;{{R}_{e}}$ does not hold for our galaxies—instead, ${{r}_{1/2}}\approx 1.88{{R}_{e}}$, with 0.13 dex scatter. This limits the usefulness of ${{M}_{1/2}}$, since ${{r}_{1/2}}$ is not directly measurable. The tight relation between R_e and ${{r}_{1/2}}$ only applies for pure, spherical Sérsic profiles. Realistic merger remnants and ellipticals alike are not perfectly spherical. Systematic effects in SDSS-quality imaging limit the ability to recover R_e and n_s, even if ellipticals have perfect Sérsic profiles, while our merger remnants evidently only approximately follow Sérsic profiles.

We define an analogous mass to M_JAM as ${{M}_{{\rm model}}}=L[M/L(r\lt {{R}_{e,3{\rm D}}})]$, but using the actual $M/L$ within ${{R}_{e,3{\rm D}}}$. M_JAM equals M_model if the JAM model derives the correct $M/L$. Note that M_model does not really have a physical meaning, as it is a projected luminosity multiplied by a mass-to-light ratio within a sphere. If R_e is correct and truly the half-light radius, and if $M/L(r\lt {{R}_{e,3{\rm D}}})=(M/L)(r\lt {{r}_{1/2}})$, then ${{M}_{{\rm model}}}=2{{M}_{1/2}}$. In practice, $M/L$ should increase with radius, and so ${{M}_{{\rm model}}}/2$ is really a lower limit on ${{M}_{1/2}}$.

Fit #9 in Table 3, using M_model, ${{R}_{e,{\rm maj}}}$, and ${{\sigma }_{e}}$, results in a median c = 5.45 with 0.06 dex scatter (note that c = 2.72 for M_model/2). M_model can be recovered nearly as accurately as ${{M}_{{{R}_{e}},3{\rm D}}}$; however, the value of c is considerably larger than the value of 3.9 quoted by Cappellari et al. (2013a). This is mainly due to systematics—the values of R_e used by Cappellari et al. (2013a) are not derived from Sérsic profile fits and tend to be larger than those from 2D, single Sérsic GALFIT fits. Perhaps for similar reasons, we are also unable to recover a tight virial FP using M_model. Our best-fit relation for M_model is ${{R}_{e,{\rm maj}}}\propto \sigma _{e}^{-2.51\pm 0.04}M_{{\rm model}}^{1.17\pm 0.01}$ (fit #9 again).

We conclude that ${{M}_{{{R}_{e}},3{\rm D}}}$ is the only dynamical mass that satisfies two broad conditions. First, it can be accurately recovered within 10%–15% using R_e and S or ${{\sigma }_{e}}$. second, it is linked to the FP, in that ${{M}_{{{R}_{e}},3{\rm D}}}$, S and R_e define a nearly exact virial plane (Equation (9)). This latter criterion was advocated by Cappellari et al. (2013a), but using M_model; we do not confirm this result using M_model and ${{R}_{e,{\rm maj}}}$ based on 2D Sérsic fits. M_model still defines a tight mass FP, just not with exactly the virial scalings—unless σ is used in place of ${{\sigma }_{e}}$, in which case fit #8 comes close.

Still, a nagging question remains—does the fact that ${{M}_{{{R}_{e}},3{\rm D}}}$, R_e and S (or ${{\sigma }_{e}}$) define a virial FP hold any fundamental significance? Similarly, can the value $c\;\approx$ 2.6 be derived from the SVT? To answer this, we begin with the virial ratio ${{q}_{50}}=-2{{T}_{{{R}_{e}},3{\rm D}}}/{{W}_{{{R}_{e}},3{\rm D}}}$, and note that ${{q}_{50}}\approx 0.45$ in typical merger remnants. We emphasize that q₅₀ depends on the dynamics of the stars and halo, generally being lower than unity, and must not be assumed to equal unity. Since q₅₀ is not constant, it is not surprising that ${{M}_{{\rm dyn}}}/{{M}_{{{R}_{e,3{\rm D}}}}}$ varies slightly too.

Next, we need relations between kinetic and potential energies within R_e and observables. Let ${{q}_{W,50}}\;=-{{W}_{{{R}_{e}},3{\rm D}}}/(G{{M}^{2}}/{{R}_{e}})$; ${{q}_{W,50}}\approx 3.0$ in the simulations. Then if S traces the total kinetic energy, $2{{T}_{{{R}_{e}},3{\rm D}}}=3{{\sigma }^{2}}\;M{{(S/\sigma )}^{2}}$, where $S/\sigma \approx 0.95$ (Figure 4). Inserting these values into the virial ratio yields:

$$\begin{eqnarray}&&M=[3{{(S/\sigma )}^{2}}/({{q}_{50}}{{q}_{W,50}})]{{\sigma }^{2}}{{R}_{e}}/G.\end{eqnarray} \tag{ 11 }$$

Thus, the dynamical mass coefficient should be $c=3{{(S/\sigma )}^{2}}/({{q}_{50}}{{q}_{W,50}})$. For typical simulations, this is $3{{(0.95)}^{2}}/(0.45\times 3.0)=2.0$. This is somewhat lower than the value of 2.6 shown in Figure 5. The main reason for this is that $\langle {{\sigma }_{{\rm DM}}}/\sigma \rangle =1.34$ in the simulations, and so both σ and S underestimate T, which includes the dark matter kinetic energy. This is irrelevant if one simply uses an empirical value of c, but important if one wishes to derive a true virial mass using the SVT. If one is not really deriving a virial mass using the SVT, then it is not clear why one should favor a mass estimator that is best fit by the virial FP in the first place. In principle, any correlation will do, even if it is not strictly a dimensional mass defined by a virial FP.

Finally, having gone through this exercise, it is worth pointing out that none of these dynamical masses are even remotely close to the total mass of the galaxy. Using ${{r}_{1/2}}$ and σ to estimate M₂₀₀ (including substructure), we obtain c = 53.5 with 0.11 dex scatter (fit #14 in Table 3); using R_e yields still larger c and scatter. The dynamical mass estimators we have discussed here are mainly useful for setting limits on the dark matter content within elliptical galaxies, given rather strict assumptions.

As we have shown, most of the tilt of the FP in the simulated galaxies is caused by varying dark matter fractions, and while the virial parameter k (or k_S) is not exactly constant, it does not contribute much to the tilt. We check the consistency of this result by creating a mock FP from simple, spherical bulge plus halo systems, using the same GalactICS code (Widrow & Dubinski 2005; Widrow et al. 2008) as was used to generate the model spirals in Paper I.

Each galaxy consists of a Sérsic profile bulge and a dark halo. The stellar component follows a scaling relation of ${{R}_{e}}\propto M_{\star }^{0.7}$, with σ scaled to create a virial FP and R_e and σ normalized to followed the observed relations for ellipticals in Paper I. This results in a scaling ${{\sigma }^{2}}\propto M/R\propto {{M}^{0.3}}$. The halo scale radius ${{r}_{s}}=2{{R}_{e}}$, and the scale velocity $v=1.2\sigma$, roughly consistent with the simulated ellipticals in this paper. The dark halo has an inner density profile of $\rho \propto {{r}^{-1}}$, an outer profile of $\rho \propto {{r}^{-2.5}}$, and truncates smoothly to $\rho =0$ from 30r_s to 35r_s. This is much like a truncated Navarro–Frenk–White profile (Navarro et al. 1997). The total halo mass is $37\;{{M}_{\star }}$, equivalent to assuming that ${{{\Omega}}_{\star }}\lt 0.01$.

We create 49 such galaxies with bulge ${{n}_{s}}=4$, ${{M}_{\star }}/L=2$, and R_e ranging from 1.4 to 27 kpc. Modest intrinsic scatter is added by making some galaxies slightly over- or under-massive for their size. We then generate mock images using the same methodology as Paper I; the FP for the galaxies has $a=2.06\pm 0.02$, b = 0.42 and negligible scatter. The small deviation from the virial plane is entirely due to systematics. Direct, one-dimensional fits to the surface brightness profile without a variable sky background give $(a,b)\;=(2.00\pm 0.01,0.39)$.

We then repeat this process, introducing non-homology by scaling n_s by a slope of 1 per dex in ${{M}_{\star }}$, roughly consistent with the observed and simulated relations in Paper I. Each galaxy keeps the same R_e, ${{M}_{\star }}$, and M. Since ${{n}_{s}}\gt 4$ profiles are more centrally concentrated, σ is larger than in the ${{n}_{s}}=4$ case (or smaller if ${{n}_{s}}\lt 4$), and so σ scales more steeply with ${{M}_{\star }}$. For this sample, we find $(a,b)=(1.94\pm 0.04,0.41\pm 0.01)$—as expected, the a parameter is lower, since σ scales more steeply with R_e as well. However, a is not significantly smaller than 2, and while the difference between a in this variable n_s sample to the ${{n}_{s}}=4$ sample is marginally significant, it is mainly systematic. One-dimensional fits give $(a,b)=\;(1.98\pm 0.01,0.40\pm 0.01)$, at best a barely significant change from the ${{n}_{s}}=4$ case.

In conclusion, the effect of structural non-homology in galaxies with Sérsic stellar mass profiles and fixed halo profiles is negligible. However, this may not necessarily be the case if the stellar profile deviates from a Sérsic law, or if the dark halo profiles scale differently from the stellar profiles. For example, one could induce non-homology with otherwise self-similar dark halos if $d{\rm log} {{R}_{s}}/d{\rm log} {{R}_{e}}\gt 0$, which would result in larger dark matter fractions in larger galaxies (see Borriello et al. 2003, who conducted a similar exercise). Unfortunately, this interpretation is overly simplistic—the halos in the simulated galaxies are not completely self-similar, and for the common profiles we have fit, $d{\rm log} {{R}_{s}}/d{\rm log} {{R}_{e}}\lt 0$ and the central dark matter density ${{\rho }_{0,{\rm DM}}}$ is not constant either. We leave further analysis of dark halo profiles to simulations with fully cosmological merger histories and halo profiles.