SCALING LAWS FOR DARK MATTER HALOS IN LATE-TYPE AND DWARF SPHEROIDAL GALAXIES

The H i rotation curves of giant spirals typically extend out to a few tens of kiloparsecs in radius. Over this region, we model the DM halo as a nonsingular isothermal sphere (ITS) with velocity dispersion σ, central density ρ_◦, and core radius r_c. These three parameters are not independent; they are related by

$$\begin{eqnarray}&&{\sigma }^{2}=(4\pi G{\rho }_{\circ }{r}_{{\rm{c}}}^{2})/9\end{eqnarray} \tag{ 1 }$$

(e.g., King 1966), where G is the gravitational constant. In the core of the ITS ($r\ll {r}_{{\rm{c}}}$), the density is roughly constant and the rotation curve of a massless disk is

$$\begin{eqnarray}&&V(r)\simeq {(4\pi G{\rho }_{\circ }/3)}^{1/2}\;r.\end{eqnarray} \tag{ 2 }$$

For $r\gg {r}_{{\rm{c}}},$ the density ρ(r) of the ITS is ∝ r⁻² and the rotation curve becomes flat,

$$\begin{eqnarray*}&&V(r)\simeq \sqrt{2}\;\sigma .\end{eqnarray*}$$

If we have rotation data only in the inner V ∝ r part of the ITS, then we can measure ρ_◦ but not r_c or σ. Therefore ρ_◦ is often the only halo parameter that we can measure for low-luminosity galaxies in which the rotation data do not extend to large radii. On the other hand, if the measurements extend out into the flat part of the rotation curve, then all three halo parameters can be estimated.

In low-luminosity Sc–Im galaxies, the visible matter contributes only a small fraction of the total mass. In brighter systems, the visible matter appears to dominate the radial potential gradient in the inner parts of the disk, and a multicomponent mass model (stars + gas + DM) is needed. The shape of the rotation curve due to the visible stellar matter is calculated from the observed surface brightness distribution assuming that the mass-to-light ratio is constant with radius. Values of M/L are adjusted to fit as much of the inner rotation curve as desired. The contribution from the H i gas, including a correction for He, is calculated separately. Molecular gas is assumed to follow the light distribution (e.g., Regan et al. 2001), so its contribution is included in the adopted M/L. Given the total rotation curve V_vis due to the potential of the visible matter, the DM halo rotation curve is ${V}_{\mathrm{DM}}(r)={({V}^{2}-{V}_{{\rm{vis}}}^{2})}^{1/2}.$ A mass model such as the nonsingular ITS is then fitted to V_DM to derive the halo asymptotic velocity ${V}_{\infty }=\sqrt{2}\sigma ,$ r_c, and ρ_◦.

Rotationally supported late-type dwarfs are especially suitable for rotation curve decomposition. They are easy to measure: they are rich in H i, and the gas extends to large radii. They are easy to interpret: they contain a disk and a halo but not a bulge. They are interesting: they give leverage to the derivation of any correlations between DM parameters and luminosity. And they are relevant: the visible matter density is low, so it has not greatly modified the halo parameters.

A note on total DM masses: The referee emphasizes that the total mass of a DM halo with a ρ ∝ r⁻² density profile does not converge. However, analysis of cold dark matter (CDM) n-body simulations shows that the volume density is best fitted by an "Einasto profile" (Einasto 1965; Einasto & Haud 1989) d ln (ρ(r))/d ln r ∝ $-{r}^{1/n}$ with n ≃ 5.8 (Merritt et al. 2006). Its total mass converges as $r\to \infty .$ Point 3 in Section 2.2 notes that Navarro et al. (1996) functional forms with outer ρ ∝ r⁻³ are also popular in fitting n-body halos, even though they—unlike Einasto halos—logarithmically diverge in mass as $r\to \infty .$ We find that the above Einasto and Navarro–Frenk–White (NFW) profiles agree remarkably well over ∼7 orders of magnitude in ρ before the Einasto profile decreases more rapidly at the largest r. Comparison of decompositions made with isothermal, NFW, and Einasto DM halos is postponed to a future paper. Here we note that, in using isothermal halos over modest ρ ranges at small r, we do not mean to imply that they should be extrapolated to large radii where total masses diverge. In fact, we mention total DM masses only briefly in this paper.

Lake & Feinswog (1989) point out that few observations of rotation curves reach large enough radii to determine halo parameters if measurement errors are interpreted strictly. Further assumptions are required.

1. We assume that observed rotation curves that become flat stay flat at larger radii.

2. The mass-to-light ratio of the disk is unknown and cannot be determined from the rotation curve alone. As van Albada et al. (1985) showed, halo parameters and disk mass-to-light ratios are degenerate. A wide range of DM and disk parameters can give a good fit to the observed rotation curve. As the amount of visible mass is reduced, the required central DM density increases and its core radius decreases. Their extreme models (van Albada et al. 1985, Figures 4 and 8) provided almost equally acceptable fits to the rotation curve. At one extreme is a maximum disk, with the largest M/L that does not require the halo to have a hollow core. This assumption gives the minimum ρ_◦ for the halo and the maximum value of r_c consistent with the rotation curve. The other extreme has a disk with M/L = 0.

Fortunately, we have additional constraints. Disk masses cannot be arbitrarily small. The presence of bars and spiral density waves requires that disks be self-gravitating. ABP used Toomre's (1981) swing amplifier theory and the properties of the observed spiral structure (i.e., two arms but not one) to constrain the density of the disk. For 18 of 21 Sc–Im galaxies, the outcome favored maximum disk decompositions. We adopt maximum disk decompositions, recognizing that this is still a controversial subject.

Many authors try to settle the question of maximal versus submaximal disks using a range of dynamical observations and arguments. Some evidence favors maximal disks (e.g., Visser 1980; Taga & Iye 1994; Sackett 1997; Debattista & Sellwood 1998, 2000; Bosma 1999; Salucci & Persic 1999; Sellwood & Moore 1999; Weiner et al. 2001; Athanassoula 2004; Weiner 2004). Other evidence suggests that some disks are submaximal (e.g., Bottema 1993, 1997; Courteau & Rix 1999; Herrmann & Ciardullo 2009; Bershady et al. 2011; van der Kruit & Freeman 2011). Our choice of maximum disks affects only giant galaxies; dwarfs are so dominated by DM that M/L uncertainties have little effect. With submaximal disks, DM parameter correlations with galaxy luminosity would be shallower. The referee notes that, with M/L = 0 minimum disks (see Figure 10 of de Blok & McGaugh 1997), the sense of the ρ_o(M_B) and r_c(M_B) correlations in Section 5 would be reversed. This would be inconsistent with constraints summarized in Section 9.11. In any case, minimum disks cannot make self-gravitating structures such as bars and density waves.

3. For the DM halo model, we adopt the nonsingular (cored) ITS (Section 2.1). Ryder et al. (2004) and de Blok (2010) review the well-known collision (Moore 1994) between the cuspy central density profiles ρ(r) seen in CDM simulations and the observational evidence that dwarf galaxies (at least) have flat cores. The Navarro et al. (1996, 1997; NFW) profile, with its ρ ∝ r⁻¹ cusp at small radii, has its roots in DM-only simulations and is still widely used. See Diemand & Moore (2011) for a review of cusp structure. CDM simulation papers widely agree that CDM halos are expected to have cuspy centers rather than isothermal-like cores (e.g., Moore et al. 1998, 2001; Klypin et al. 2001, 2011; Colín et al. 2004; Diemand et al. 2004, 2005, 2008; Hayashi et al. 2004; Navarro et al. 2004, 2010; Klypin et al. 2011).

De Blok (2010) reviews the controversial observational situation. The observed rotation curves, especially for fainter galaxies, usually indicate that their DM halos have cores. Some authors have tried to "save cusps" by emphasizing systematic uncertainties like beam smearing in the H i rotation data and noncircular motions in the higher-resolution optical rotation data; these could create the appearance of a core. Over the past decade, rotation curve data and mass modeling have improved, and cores in the DM halos of dwarf and low-surface-brightness galaxies are consistently favored (e.g., de Blok & Bosma 2002; Marchesini et al. 2002; Weldrake et al. 2003; Gentile et al. 2004, 2009; de Blok et al. 2008; Oh et al. 2008, 2011a, 2011b; Spano et al. 2008; Donato et al. 2009; Plana et al. 2010; Chen & McGaugh 2010; Walker et al. 2010). De Blok (2010) concludes that almost all dwarf disk galaxies that have been studied have cored halos. The situation is less clear for giant disks like M101. A careful review by Sellwood (2009) concludes that the core-cusp problem is unsolved for giant galaxies. We note, however, that hydrodynamic simulations of galaxy formation with feedback (e.g., Governato et al. 2010) show that violent feedback from rapid star formation in the inner regions of disk galaxies can transform a cusped DM halo into a cored halo. We adopt cored isothermal halos in this paper.

The dSph galaxies are in pressure equilibrium, and the dynamical analysis is based on stellar radial velocities rather than rotation. Many authors use Jeans equation analyses. Another promising approach is Schwarzschild (1979, 1982) orbit superposition modeling of the observed spatial and velocity distributions of individual stars in dSph galaxies. The Nuker team orbit superposition code (e.g., Gebhardt et al. 2003; Richstone et al. 2004; Thomas et al. 2004, 2005; Siopis et al. 2009) has been applied to six dSph companions of our Galaxy by Jardel and Gebhardt. They conclude that these galaxies have a heterogeneous range of DM profiles: dSph galaxies with cores include Fornax (Jardel & Gebhardt 2012) and probably Carina (Jardel & Gebhardt 2013); dSph galaxies with cuspy, NFW-like profiles include Draco (Jardel et al. 2013) and Sextans (Jardel & Gebhardt 2013); and Sculptor is uncertain but most consistent with an NFW profile (Jardel & Gebhardt 2013); see Jardel (2014) for a summary. However, (1) the features in the line-of-sight velocity distributions that differentiate between best-fit NFW profiles and best-fit cored halos are far from obvious in these papers; (2) in the formally NFW galaxies, cored halos are excluded by not much more than 1σ, and (3) the cored halo machinery used in our paper provides an adequate fit to all the data. Other authors using different analysis techniques favor cored halos for dSph galaxies (e.g., Amorisco & Evans 2012).

In Section 6.1, Figure 5, and the Appendix, we show that the baryons in both dSph and dIm galaxies mostly have Gaussian density profiles, as expected for an isothermal, isotropic distribution of test particles lying within the constant-density core of a cored DM distribution. The consistency of these assumptions encourages us to adopt cored halo models for all galaxies.

In the present paper, we do not further discuss the issue of halo cores and cusps, other than to note again that baryonic physics can affect CDM density distributions and produce cores when none were present initially. The most promising mechanism is rapid ejection of most baryons via star formation feedback (Navarro et al. 1996; Gnedin & Zhao 2002; Governato et al. 2010; de Souza et al. 2011; Oh et al. 2011a; Di Cintio et al. 2014; Madau et al. 2014). The idea is especially compelling for tiny dwarfs. Surface brightnesses decrease rapidly with decreasing galaxy luminosity. Galaxies like Draco and UMi are so DM dominated that if the present stars were spread out into primordial gas with a density profile like that of the stars, then—given metal-poor stellar populations like those of M/L ∼ 2 globular clusters but measured dynamical M/L ∼ 10²—that gas would not self-gravitate. It would be difficult to engineer instabilities that make stars. The suggestion has long been that these galaxies contained more baryons when the stars formed and then blew most of the baryons away via the first supernovae (see Kormendy 1985, 1987b; Kormendy et al. 2009; Kormendy & Bender 2012 for the observations, and Dekel & Silk 1986 for the theoretical concepts). Observational evidence that is consistent with this picture is presented in Section 8.

For the purposes of this paper, the difference between cored and cusped DM profiles is relatively benign. The history of work on the luminous density profiles of elliptical galaxies is relevant here. Ground-based observations of the central parts of ellipticals were made during the 1980s and revealed results that include the fundamental plane correlations (Kormendy 1984, 1987b, 1987c; Lauer 1985). Then the Hubble Space Telescope (HST) showed that the brightest ellipticals have cores with surface brightnesses ${\rm{\Sigma }}(r)\propto {r}^{-m}$ at small radii, with m ≃ 0–0.25 (e.g., Lauer et al. 1995). Earlier ground-based observations of core radii and central densities proved to probe the same physics as HST observations of break radii and densities (Kormendy et al. 1994). Thus, most of the ground-based results remained valid. In the future, when the details of halo density profiles are better known, we expect that our DM parameters will continue to measure the relevant physics. Our core radius may turn out to be a measure of a break radius in the halo density profile, and the central density will probably be a measure of some average density inside the break radius, independent of whether the halos have flat cores or not.

For rotation-dominated galaxies, DM halo parameters are taken from rotation curve decompositions in the literature. While some authors use the nonsingular ITS model for their dark halos, most others use a simple analytic pseudo-isothermal sphere. These models are significantly different, and we need to derive the transformations between DM halo parameters from the two models (Section 3), using the nonsingular ITS as our reference model. Other than this transformation, we have made as few changes as possible, only adjusting the data to a uniform distance scale and adopting the following selection criteria:

1. As mentioned above, we include only late-type, Sc–Im and dSph systems. It now seems likely that dSph galaxies are related to late-type galaxies. The structural parameters of their visible matter are similar to those of the lowest-luminosity Im galaxies (Kormendy 1985, 1987c; Binggeli & Cameron 1991; Ferguson & Binggeli 1994; Kormendy & Bender 2012; see also Figure 9 of this paper). Most of the classical nearby dSph galaxies show evidence of star formation in the past 1–8 Gyr (e.g., Aaronson & Mould 1985; Mighell 1990; Mighell & Butcher 1992; Lee et al. 1993; Beauchamp et al. 1995; Weisz et al. 2011; see Da Costa 1994; Mateo 1998; Tolstoy et al. 2009 for reviews). At the time of their most recent star formation, they would have been classified as irregular galaxies (Kormendy & Bender 1994, 2012). See also Section 9.6 in this paper. We therefore include Sc–Im and dSph galaxies in the same parameter correlation diagrams.

2. Most galaxies with inclinations i < 40° are excluded. As Broeils (1992) remarks, H i rotation curves are less accurate for galaxies that are nearly face-on. Similar criteria were used by, e.g., Begeman (1987) and de Blok et al. (2008). Oval distortions (Bosma 1978; Kormendy 1982; Kormendy & Kennicutt 2004) can lead to incorrect estimates of inclinations for the more face-on galaxies. We retained four i < 40° galaxies from ABP (M101, NGC 5236, NGC 6946, and IC 342) because they provide leverage at high luminosities. Although we did apply this inclination cut to our sample, it turns out that most nearly face-on galaxies satisfy the DM parameter correlations.

3. The rotation curves must extend to large enough radii to constrain the DM parameters. The adopted criterion is that the observed rotation curve extends to at least 4.5 exponential scale lengths of the disk. For a self-gravitating exponential disk, the rotation curve peaks at 2.2 scale lengths (Freeman 1970), so the rotation curve contribution from the disk drops significantly over 4.5 scale lengths. A flat observed rotation curve then gives useful constraints on the DM parameters. The particular choice of 4.5 scale lengths allows us again to keep a few galaxies that provide leverage at high luminosities. This radius criterion was not rigidly applied: we retained a few galaxies whose rotation data extend out to 3.5–4.4 scale lengths, because the DM halo was dominant enough so that halo parameters were well determined. These galaxies are DDO 127, DDO 154, DDO 168, NGC 247, NGC 925, and IC 2574. We found that the radius cut is important. Without it, the sample is larger and mostly consistent with the correlations, but with larger scatter than seen in Figures 2–4 and 6–7. This selection criterion is an important difference between the present work and previous investigations of DM scaling laws.

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4. In the late stages of the selection process, various "sanity checks" on physically plausible decompositions resulted in the omission of additional objects:

(i) Some recent papers do not use maximum disk V(r) decompositions but rather use disk M/L ratios based on optical or infrared colors (e.g., using Bell & de Jong 2001). These decompositions are usually plausible, but we need to be consistent in using maximum disk decompositions if we hope to measure the scatter in the DM parameter correlations.

(ii) A few galaxies were omitted because the adopted M/L was much too large for a blue, star-forming disk (e.g., UGC 5716 with M/L_B ≃ 8.9 on our distance scale; van Zee et al. 1997).

(iii) Several authors model the DM halo with the pseudo-isothermal sphere (PITS; see Section 3). The PITS has a well-defined central density but differs significantly in its structure and rotation curve from the nonsingular ITS. Galaxies were omitted when the published PITS ρ_◦ and r_c values implied asymptotic outer rotation velocities that are much larger than the outermost velocities that are actually observed.

(iv) NGC 1705 (Meurer et al. 1998) and NGC 2915 (Meurer et al. 1996; Elson et al. 2010) were omitted because (1) intense starbursts drive strong winds that may affect the inner rotation curves; (2) H i beam smearing in NGC 1705 affects the central V(r) point that, in essence, determines M/L_B; and (3) both galaxies are sometimes classified as amorphous (I0) rather than as Magellanic irregulars (Im). See, however, Elson et al. (2010, 2013).

(v) The edge-on galaxy NGC 801 is classified as Sc in de Vaucouleurs et al. (1991, hereafter RC3), but photometry in Kormendy & Bender (2011, see Figures S6 and S7) shows that it has a pseudobulge that makes up 23% of the total light. This is clearly evident in the photometry of Kent (1986), Andredakis & Sanders (1994), and Courteau (1996). Including the pseudobulge in any decomposition makes the results very uncertain. NGC 801 was the lowest-ρ_◦, largest-r_c point in Figure 4 of Kormendy & Freeman (2004). Here we conclude by comparison with (P)B/T ratios in Simien & de Vaucouleurs (1986) that NGC 801 is a misclassified Sbc. On this basis only—and not because of its DM parameters—we discard it everywhere except in Figure 8.

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(vi) Finally, the ultrathin, edge-on Sd galaxy UGC 7321 was omitted because its rotation curve is better fitted with a NFW DM profile than by a cored halo (O'Brien et al. 2010). Given the form of V(r) for an NFW halo, UGC 7321 deviates from our DM correlations in the direction of small r_c and large ρ_o.

In addition to the selection cuts, we adopt the following procedures to make parameters from different sources be as consistent with each other as possible.

More accurate distances are now available than the Kraan-Korteweg (1986) Virgocentric flow model distances used in Kormendy & Freeman (2004). We adopt distance measurements and sources as listed in Table 1 (Section 4). Accurate distances based on primary standard candles (e.g., Cepheid variables) are available for 62% of our sample. Another 12% of our sample galaxies are far enough away (D = 37–84 Mpc) that distances based on recession velocities and a Hubble constant of H_◦ = 70 km s⁻¹ Mpc⁻¹ (Komatsu et al. 2011) are accurate. For the remainder of our sample, a more complicated strategy was required (see Notes to Table 1).

Table 1.  Dark Matter Halo Parameters for Rotationally Supported Galaxies

Galaxy	D	Source	MB	$\mathrm{log}\sigma$	$\mathrm{log}{\rho }_{\circ }$	$\mathrm{log}{r}_{{\rm{c}}}$	DM	Source
	(Mpc)			(km s−1)	(M⊙ pc−3)	(kpc)	Model
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
NGC 24	8.1	1	−18.10	1.8129	−1.8020	0.8242	ITS	Chemin et al. (2006)
NGC 45	7.1	1	−18.54	1.7404	−2.0444	0.8728	ITS	Chemin et al. (2006)
NGC 55	2.17	1	−19.25	1.7559	−2.4073	1.0719	ITS	Puche et al. (1991)
NGC 55	2.17	1	−19.25	1.8732	−2.5263	1.2498	PITS	Broeils (1992)
NGC 247	3.65	1	−19.17	1.8785	−2.5158	1.2471	ITS	ABP
NGC 247	3.65	1	−19.17	1.8424	−2.5836	1.2476	PITS	Broeils (1992)
NGC 253	3.94	1, 2	−21.03	2.0734	−2.3110	1.3396	ITS	ABP
NGC 925	9.2	1, 2	−20.17	1.8813	−1.9606	0.9719	PITS	de Blok et al. (2008)
NGC 300	2.00	1	−18.43	1.7348	−2.1715	0.9348	ITS	ABP
NGC 300	2.00	1	−18.43	1.8294	−2.2403	1.0650	PITS	Broeils (1992)
NGC 598	0.85	1, 2	−19.00	1.8075	−2.1518	0.9937	ITS	ABP
NGC 598	0.85	1, 2	−19.00	1.9417	−2.0961	1.0971	PITS	Corbelli (2003)
NGC 1003	9.2	1	−19.15	1.8327	−2.3112	1.0986	PITS	Broeils (1992)
NGC 1560	3.45	1, 2	−17.36	1.8327	−2.2712	1.1131	PITS	Broeils (1992)
NGC 2366	3.19	1, 2	−16.53	1.4521	−1.6732	0.3990	PITS	Oh et al. (2011b)
NGC 2403	3.22	1	−19.39	1.9025	−1.8747	0.9502	ITS	ABP
NGC 2403	3.22	1	−19.39	1.8964	−2.0379	1.0288	PITS	Broeils (1992)
NGC 2998	71.1	3, 4	−21.76	2.0926	−2.8692	1.6375	PITS	Broeils (1992)
NGC 3109	1.34	1, 2	−16.44	1.6884	−1.9226	0.7600	ITS	Jobin & Carignan (1990)
NGC 3109	1.34	1, 2	−16.44	1.8581	−2.1709	1.0561	PITS	Broeils (1992)
NGC 3198	13.6	1, 5	−20.53	1.8976	−2.4601	1.2379	ITS	Blais-Ouellette et al. (1999)
NGC 3198	13.6	1, 5	−20.53	1.9020	−2.4966	1.2611	PITS	Broeils (1992)
NGC 3274	6.5	1, 2	−16.46	1.6360	−0.9209	0.2067	PITS	de Blok & Bosma (2002)
NGC 3359	19.5	3, 4	−20.90	1.9128	−2.6141	1.3300	ITS	ABP
NGC 3621	6.64	1	−19.99	1.9020	−1.9591	0.9919	PITS	de Blok et al. (2008)
NGC 3726	18.6	1, 6	−21.19	1.9367	−2.2067	1.1503	PITS	Verheijen (1997)
NGC 3741	3.19	1	−13.86	1.4928	−1.7555	0.3958	Burkert	Gentile et al. (2007)
NGC 3917	18.6	1, 6	−19.82	1.8023	−1.8857	0.8554	PITS	Verheijen (1997)
NGC 4183	18.6	1, 6	−19.56	1.7300	−2.3661	1.0233	PITS	Verheijen (1997)
NGC 4244	4.49	1, 2	−18.77	1.7528	−2.4294	1.0771	ITS	ABP
NGC 4395	4.61	1, 2	−18.19	1.7889	−2.4741	1.1362	ITS	ABP
NGC 4455	7.8	1	−17.49	1.7245	−2.0936	0.8815	PITS	de Blok & Bosma (2002)
NGC 5033	17.2	3, 4, 7	−21.06	2.0885	−2.5221	1.4598	ITS	ABP
NGC 5033	17.2	3, 4, 7	−21.06	1.9393	−2.2059	1.1507	PITS	Broeils (1992)
NGC 5204	4.65	1, 2	−16.90	1.6435	−1.3596	0.4176	ITS	Sicotte & Carignan (1997)
NGC 5204	4.65	1, 2	−16.90	1.6190	−1.0419	0.2502	PITS	Swaters et al. (2003)
NGC 5236	4.47	1, 2	−20.63	1.9890	−2.0209	1.1125	ITS	ABP
NGC 5457	7.2	2, 9	−21.41	1.9450	−2.4479	1.2775	ITS	ABP
NGC 5585	7.2	1, 2, 10	−18.47	1.7202	−1.4656	0.5633	ITS	Blais-Ouellette et al. (1999)
NGC 5585	7.2	1, 2, 10	−18.47	1.7045	−1.4426	0.5360	PITS	Broeils (1992)
NGC 5907	15.15	11	−20.82	2.0693	−2.4750	1.4170	ITS	ABP
NGC 5907	15.15	11	−20.82	2.0473	−2.1380	1.2246	PITS	Miller & Rubin (1995)
NGC 6015	17.7	3, 4	−20.12	1.9619	−2.3587	1.2511	ITS	Verdes-Montenegro et al. (1997)
NGC 6503	5.27	1, 2	−18.57	1.8407	−1.7056	0.8057	ITS	ABP
NGC 6503	5.27	1, 2	−18.57	1.7695	−1.3832	0.5714	PITS	Broeils (1992)
NGC 6822	0.50	1, 2	−15.67	1.5596	−1.5577	0.4488	PITS	Weldrake et al. (2003)
NGC 6946	5.9	1, 2	−21.06	2.0885	−2.0322	1.2166	ITS	ABP
NGC 7793	3.91	1, 2	−18.86	1.6107	−1.5467	0.4946	ITS	Carignan & Puche (1990)
IC 342	3.28	2, 9	−20.90	2.0488	−2.2782	1.2988	ITS	ABP
IC 2574	4.02	1, 2	−18.24	1.6990	−2.4480	1.0302	ITS	Martimbeau et al. (1994)
UGC 191	17.8	3, 4	−18.23	1.6773	−2.4867	1.0310	PITS	van Zee et al. (1997)
UGC 2259	9.2	1	−16.63	1.7404	−1.9768	0.8390	ITS	Carignan et al. (1988)
UGC 2259	9.2	1	−16.63	1.8456	−2.0352	0.9778	PITS	Broeils (1992)
UGC 2885	84.0	3, 4	−22.50	2.2909	−3.0104	1.9004	PITS	Broeils (1992)
UGC 3174	8.1	1	−15.29	1.5539	−1.1367	0.2326	PITS	van Zee et al. (1997)
UGC 4499	17.2	12, 13	−18.08	1.6365	−2.0805	0.7871	PITS	Swaters et al. (2003)
UGC 5005	56.0	3, 4	−18.00	1.6717	−1.9561	0.7572	PITS	de Blok & McGaugh (1997)
UGC 5764	7.1	1	−14.47	1.4797	−1.3038	0.2418	PITS	van Zee et al. (1997)
UGC 6446	18.6	1, 6	−18.49	1.5781	−1.4200	0.4130	PITS	Verheijen (1997)
UGC 6983	18.6	1, 6	−18.76	1.7844	−2.6574	1.2233	PITS	Verheijen (1997)
UGC 11820	18.8	3, 4	−17.44	1.6383	−2.3430	0.9200	PITS	van Zee et al. (1997)
UGC A442	4.27	1, 2	−15.16	1.5441	−1.7091	0.5112	ITS	Côté et al. (2000)
ESO 287-G13	37.5	3, 4	−20.90	2.0962	−2.4740	1.4429	PITS	Gentile et al. (2004)
ESO 381-G20	5.44	1	−15.11	1.5315	−2.0820	0.6829	ITS	Côté et al. (2000)
ESO 444-G84	4.61	1, 2	−13.82	1.6021	−1.2755	0.3501	ITS	Côté et al. (2000)
F568-V1	84.1	3, 4	−18.40	1.7844	−2.3460	1.0677	PITS	Swaters et al. (2000)
F583-1,D584-4	36.9	3, 4	−17.35	1.6861	−2.0362	0.8145	PITS	de Blok et al. (2001)
F583-1,D584-4	36.9	3, 4	−17.35	1.7049	−2.0454	0.8117	PITS	Kuzio de Naray et al. (2008)
F583-1,D584-4	36.9	3, 4	−17.35	1.6955	−2.0408	0.8131	PITS	mean of the above
DDO 127	6.9	1, 2	−14.98	1.4230	−1.9212	0.4941	ITS	Bosma: Kormendy & Freeman (2004)
DDO 154	4.3	1, 2	−14.75	1.4624	−1.9142	0.5263	ITS	Carignan & Purton (1998)
DDO 154	4.3	1, 2	−14.75	1.4797	−1.9460	0.5523	PITS	Broeils (1992)
DDO 161	7.2	3, 4	−16.84	1.6128	−2.2648	0.8573	ITS	Côté et al. (2000)
DDO 168	4.33	1, 2	−16.27	1.7001	−1.8688	0.7437	PITS	Broeils (1992)
DDO 170	16.1	3, 4	−16.15	1.5839	−2.0398	0.7093	PITS	Broeils (1992)

Notes. Notes on distances—Adopted distances (Column (2)) are from sources listed in Column (3). To save space, most of our sources are compilations for many galaxies rather than the papers in which distance measurements are reported. Our primary source is the Kennicutt et al. (2008) list of galaxies in the local 11 Mpc volume. For D ≤ 11 Mpc, we use his distances whenever possible. Our second main source is the Karachentsev et al. (2004) Catalog of Neighboring Galaxies. These two sources almost always agree exactly; for our sample, they never disagree significantly. For 37 of our 60 galaxies (i.e., 62%), we use direct distance measurements in order of preference, as derived from Cepheid variables, from TRG, i.e., the tip of the red giant branch in the color–magnitude diagram, from brightest stars, or from membership in groups whose distances were measured either using the above direct methods or (in the case of UGC 4499) using surface brightness fluctuations in NGC 5866. Beyond this volume, five galaxies are in the Ursa Major cluster; we adopt D = 18.6 Mpc from Kennicutt et al. (2008) and Tully & Pierce (2000). When no direct measurement is available, we base D on a derivation of the large-scale flow field of galaxies as given in the listed sources. For four nearby galaxies, we adopt Kennicutt's flow-field distances based on a Hubble constant of H₀ = 75 km s⁻¹ Mpc⁻¹. We then have a significant gap in distance between these objects at D = 7–8 Mpc and galaxies at D ≥ 15 Mpc for which we use flow-field distances from references (3) and (4). For the latter galaxies, we adopt H₀ ≃ 70 km s⁻¹ Mpc⁻¹, consistent with Wilkinson Microwave Anisotropy Probe (WMAP) determinations (Komatsu et al. 2009, 2011). It may seem unrealistic to use different Hubble constants for nearby and distant galaxies, but Karachentsev & Makarov (1996) show that the apparent value of H₀ reaches a maximum at D ∼ 2 Mpc and then decreases out to larger distances. Kennicutt et al. (2008) adopt the Karachentsev & Makarov flow-field solution, having engineered the currently best possible consistency between direct distance measurements and ones determined from flow fields. On the other hand, at large distances (seven galaxies have D = 37–84 Mpc), H₀ is robustly smaller; we adopt the WMAP value. In between are seven galaxies at D = 15–19 Mpc for which flow-field results are particularly uncertain.

Distance references—(1) Kennicutt et al. (2008); (2) Karachentsev et al. (2004); (3) Mould et al. (2000) Virgocentric flow field solution as implemented in Reference (4); (4) NED "D (Virgo Infall only)" with H₀ = 70 km s⁻¹ Mpc⁻¹; (5) Freedman et al. (2001); (6) Tully & Pierce (2000); (7) flow-field solution (3, 4) applied to the Can Ven Spur (group 43−1 in Reference 8); (8) Tully (1988); (9) Kormendy et al. (2010); (10) Drozdovsky & Karachentsev (2000); (11) flow-field D = 14.9 Mpc for group 44−1 (Reference 8) averaged with SBF D = 15.4 Mpc for group member NGC 5866 (reference 12); (12) Tonry et al. (2001); (13) in group 15−10 (reference 8) with NGC 2681, which has SBF D = 17.2 Mpc (reference 12).

Notes on DM models—The DM model used in rotation curve decomposition is given in Column (8). ITS means the nonsingular isothermal sphere; PITS means the pseudo-isothermal sphere (Equation (3)). NGC 3741 was analyzed (Gentile et al. 2007) using a Burkert (1995) analytic DM profile. The Burkert profile has a constant-density core and a volume density ρ(r) which closely resembles that of the PITS over a factor of >20 in density. We scaled its parameters to PITS parameters and then used our standard scaling to convert PITS parameters to ITS equivalents. All decompositions used in this paper are listed in the table, one per line. Their sources are given in Column (9). For the 15 galaxies with two independent decompositions—almost always ones with different DM models—comparison of the results gives estimates of the 1 s.d. measurement errors in the parameters; their averages are $\epsilon (\mathrm{log}\sigma )=0.033,$ $\epsilon (\mathrm{log}{\rho }_{{\rm{o}}})=0.073,$ and $\epsilon (\mathrm{log}{r}_{{\rm{c}}})=0.064.$ These include errors in calibrating PITS to ITS results, but they do not include systematic errors that result if some disks are submaximal. These mean parameter errors for the 1/4 of our galaxies that have two decompositions are representative of the errors for galaxies that have only one decomposition.

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DM parameters were corrected to our adopted distances by assuming that ${r}_{{\rm{c}}}\propto D$ and that ρ_◦ ∝ D⁻². This is not strictly correct, because gas and dynamical masses scale differently with distance. But the errors are small on the scales of Figures 2–4 and 6–7. Estimates of σ are independent of galaxy distance.

Galactic absorption corrections are from Schlegel et al. (1998) as listed in NED. Total magnitudes B_T are generally means of RC3 values listed in NED, values listed in the main catalog of Hyperleda, and values given in the "Integrated photometry" link at Hyperleda. For galaxies with rotation curve decompositions from van Zee et al. (1997), that paper is also the source of B_T. Absolute magnitudes listed in Table 1 are corrected for internal absorption as in Tully & Fouqué (1985).

For dSph galaxies, the data that we can use to estimate the central DM densities are the projected stellar velocity dispersion profiles σ(R) and the surface density distributions I(R) of the stars. Even if we assume that the galaxies are spherically symmetric, these observables alone do not provide strong constraints on the central DM density. For example, we do not know the form of the underlying DM distribution or how the anisotropies of the stellar velocity distributions change with radius (see Walker et al. 2009 for more details). Some assumptions are needed. These should be consistent with the assumptions that we used in Section 2 for the spiral galaxies.

We use the spherical Jeans equation to estimate central DM densities from σ(R) and I(R). The measurable I(R) distributions for these low-surface-brightness dSph galaxies do not usually extend over a large range of surface density, and the star counts are often represented by simple approximations such as Plummer and King models. Because the Jeans equation involves derivatives of the density distribution, the outcome can be sensitive to details of how I(R) is modeled (e.g., Evans et al. 2009). We estimate the form of the baryon distribution I(R) expected in faint dwarf galaxies under a set of simple assumptions:

We assume that DM halos have constant-density cores. The issue is not settled (Section 2.2), though some authors support this assumption with their dynamical analyses (e.g., Amorisco & Evans 2012). We adopt this assumption to be consistent with our analysis of the Sc–Im galaxies.

How do the radial extents of the stars in dSph galaxies compare with the likely sizes of their DM cores? Typical half-light radii r_h for dSph galaxies are a few hundred parsecs. From Table 1, the core radii for the DM halos in the faintest of our Sc–Im galaxies are about 1.6–2 kpc. Although the DM core radii of dSph galaxies may be smaller than those of faint, rotation-dominated Sc–Im systems, it seems reasonable to consider an approximation in which the stars are immersed entirely within the uniform-density core of the DM halo. Section 7 shows that this produces results that are internally consistent.

The projected velocity dispersion profile σ(R) also appears in the Jeans equation. The observed σ(R) profiles for the classical dSph galaxies are almost isothermal well beyond their half-light radii r_h, out to R ∼ 1000 pc. Even for the ultrafaint system Seg 1, with r_h ≃ 30 pc, the σ(R) profile is approximately isothermal out to about 60 pc. We assume (i) that each system is spherically symmetric, (ii) that the stars in a dSph galaxy lie entirely in the constant-density DM core, (iii) that the σ(R) distribution is isothermal, (iv) that the velocity distribution is isotropic, and (v) that the contribution of the stars to the total density is negligible. Then the spherical Jeans equation takes the simple form

$$\begin{eqnarray}&&\displaystyle \frac{{\sigma }_{*}^{2}}{{\rho }_{*}}\displaystyle \frac{d{\rho }_{*}}{{dr}}=-\displaystyle \frac{{GM}(r)}{{r}^{2}}=-\displaystyle \frac{4\pi G{\rho }_{\circ }}{3}\;r,\end{eqnarray} \tag{ 34 }$$

where ρ_* and σ_* are the volume density and velocity dispersion of the stars, M(r) is the total mass enclosed within radius r, and ρ_◦ is the central density of the DM. Then the stellar volume density is Gaussian,

$$\begin{eqnarray}&&{\rho }_{*}(r)\propto \mathrm{exp}(-{r}^{2}/{a}_{*}^{2}),\end{eqnarray} \tag{ 35 }$$

and the volume density of the DM core is

$$\begin{eqnarray}&&{\rho }_{\circ }=3{\sigma }_{*}^{2}/2\pi {{Ga}}_{*}^{2}.\end{eqnarray} \tag{ 36 }$$

The projected surface density Σ_*(R) of a Gaussian volume density distribution ρ_*(r) is also Gaussian with the same scale length a_*. In fact, the star-count distributions of the classical dSph galaxies and some of the ultrafaint systems are well represented by Gaussians over most of their radial extent. As an example, Figure 5 shows the Gaussian radial profiles of the projected star counts for the Carina dSph system. Similar plots for all dSph galaxies used in this paper are shown in Figure 11 in the Appendix. The radii in Figures 5, 11, and 12 are along the major axes of the galaxies.

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To compare the DM parameters of dSph galaxies with those for the rotationally supported disk galaxies, we use the halo density ρ_◦ derived from the Gaussian star-count distribution and σ_* via Equation (36) uncorrected for any contribution from the stars. While the near-isothermality of the velocity dispersion makes the analysis simpler, it does mean that there is little structure in the radial distribution of the kinematics from which one could estimate the core radius of the DM. At the present time, the central density is the only halo parameter that we can measure directly for dSph galaxies.

For the halo size parameter in rotationally supported disk galaxies, we used the core radius r_c of the DM distribution derived from the shape of the rotation curve, following the King model scaling ${r}_{{\rm{c}}}^{2}=9{\sigma }^{2}/4\pi G{\rho }_{\circ }.$ In order to combine data for dark halos of galaxies over a wide range of luminosities, we would like to be able to measure King core radii for the halos of the dwarf galaxies. Because we cannot derive a size parameter for the DM in dSph galaxies from the stellar kinematics, we begin by tabulating the corresponding ${{r}_{{\rm{c}}}}_{*}$ value for their stellar distributions. King model fits are not available for all of our dSph galaxies, so we need to derive empirically the relation between the King ${{r}_{{\rm{c}}}}_{*}$ and the Gaussian a_* using eight dSph galaxies for which both lengths have been measured. The two lengths are tightly correlated:

$$\begin{eqnarray}&&\mathrm{log}{{r}_{{\rm{c}}}}_{*}=(0.752\pm 0.073)\mathrm{log}{a}_{*}-(0.273\pm 0.038).\end{eqnarray} \tag{ 37 }$$

With this calibration, Table 2 lists the distance, absolute magnitude, stellar velocity dispersion, DM core density, and stellar r_c for 11 dSph companions of the Milky Way for which the star-count profiles are close to Gaussian. For CVn I and Coma 1, only a mean value of σ is available, and we have to assume that these systems are isothermal. We exclude two faint systems (Her and Wil 1) for which the star-count profiles are not Gaussian.

Table 2.  Dark Matter Central Densities and Baryonic Velocity Dispersions and Core Radii for Spheroidal and Irregular Galaxies

Galaxy	D	Source	MB	Source	$\mathrm{log}\sigma$	$\mathrm{log}{\rho }_{\circ }$	$\mathrm{log}{r}_{{\rm{c}}}$	Source
	(Mpc)				(km s−1)	(M⊙ pc−3)	(kpc)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
Carina	0.094	NED	−8.0	W09	0.820 ± 0.079	−1.245 ± 0.158	−0.676 ± 0.005	IH95
Draco	0.076	NED	−8.1	W09	0.959 ± 0.057	−0.687 ± 0.292	−0.780 ± 0.101	IH95
Fornax	0.138	NED	−12.4	W09	1.068 ± 0.033	−1.557 ± 0.069	−0.371 ± 0.006	C05
Leo I	0.270	NED	−10.9	W09	0.964 ± 0.066	−1.142 ± 0.133	−0.606 ± 0.007	IH95
Leo II	0.205	NED	−9.0	W09	0.820 ± 0.046	−0.952 ± 0.094	−0.786 ± 0.007	C07
Scl	0.088	NED	−9.9	W09	0.964 ± 0.052	−1.098 ± 0.105	−0.622 ± 0.004	IH95
Sextans	0.086	NED	−8.6	W09	0.898 ± 0.071	−1.647 ± 0.145	−0.466 ± 0.010	IH95
UMi	0.069	NED	−7.8	W09	0.978 ± 0.055	−1.101 ± 0.111	−0.611 ± 0.007	IH95
CVn I	0.218	M08	−8.0	W09	0.881 ± 0.023	−1.729 ± 0.061	−0.448 ± 0.015	M08
Seg I	0.023	M08	−0.9	W09	0.633 ± 0.121	0.332 ± 0.247	−1.408 ± 0.018	M08
Coma	0.044	NED	−3.5	W09	0.663 ± 0.076	−0.784 ± 0.153	−0.967 ± 0.009	M08
GR8	2.13	TRG	−12.0	B06	0.954 ± 0.039	−1.270 ± 0.080	−0.565 ± 0.007	B03A
SDIG	3.21	TRG	−11.7	C00	0.954 ± 0.082	−1.576 ± 0.165	−0.450 ± 0.006	C00
Cam B	3.34	TRG	−11.8	B03B	0.845 ± 0.062	−2.103 ± 0.090	−0.281 ± 0.002	B03B
Leo T	0.42	HB	−6.5	I07	0.839 ± 0.063	−0.900 ± 0.132	−0.791 ± 0.015	R08
LGS3	0.76	TRG	−9.4	M98	0.929 ± 0.051	−1.114 ± 0.114	−0.642 ± 0.018	Y97
Leo A	0.75	TRG	−11.4	B06	0.978 ± 0.059	−1.853 ± 0.125	−0.328 ± 0.014	Y96
DDO 210	0.99	TRG	−11.0	B06	0.813 ± 0.067	−1.163 ± 0.107	−0.675 ± 0.003	Y03
DDO 216	1.10	TRG	−13.0	Z00	1.021 ± 0.041	−1.544 ± 0.076	−0.396 ± 0.004	Y03
UGCA 292	3.60	TRG	−11.7	Z00	0.978 ± 0.046	−1.742 ± 0.079	−0.342 ± 0.008	Y03
DDO 53	3.60	TRG	−13.4	B06	0.978 ± 0.073	−1.723 ± 0.133	−0.358 ± 0.010	B06
UGC 7298	4.20	TRG	−12.3	B06	0.929 ± 0.066	−1.670 ± 0.135	−0.431 ± 0.009	B06
KK230	1.90	TRG	−9.5	B06	0.875 ± 0.029	−1.426 ± 0.064	−0.566 ± 0.010	B06

Notes. Notes for dSph galaxies: Column (2) lists distances D based on stellar indicators from NED or M08 (Column (3)). Column (4) absolute magnitudes M_B are from V-band luminosities in W09 (Column (5)) converted to B band adopting $B-V=0.60$ for all systems. Column (6) velocity dispersions are from W09. Column (7) $\mathrm{log}{\rho }_{\circ }$ is from Equation (36). Errors do not include distance errors. Column (8) lists $\mathrm{log}{r}_{{\rm{c}}}$ for the stars, not the DM, derived from Gaussian lengths a_* via Equation (37). Errors are from fits to Gaussian surface density distributions and do not include distance errors. Column (9) gives the sources of the star-count data. dSph galaxy references: C05—Coleman et al. (2005); C07—Coleman et al. (2007); IH95—Irwin & Hatzidimitriou (1995); M08—Martin et al. (2008); W09—Walker et al. (2009).

Notes for dIm galaxies: Column (2) lists distances D based on stellar indicators from NED given in Column (3) (mostly TRG stars; in one case, horizontal branch stars). Column (4) absolute magnitudes M_B are adopted from sources in Column (5) with some adjustment for distance. Column (6) H i velocity dispersions are adopted from sources in Column (9) with some adjustment for rotation. Column (7) $\mathrm{log}{\rho }_{\circ }$ is from Equation (40) including the small contribution from rotation (not separately tabulated). Errors do not include distance errors. Column (8) lists $\mathrm{log}{r}_{{\rm{c}}}$ for the H i gas, not the DM, derived from Gaussian scale lengths a_* via Equation (37). Errors are from fits to Gaussian surface density distributions and do not include distance errors. Column (9) gives the sources of the H i data. dIm galaxy references: B03A—Begum et al. (2003); B03B—Begum & Chengalur (2003); B06—Begum et al. (2006); C00—Côté et al. (2000); I07—Irwin et al. (2007); M98—Mateo (1998); R08—Ryan-Weber et al. (2008); Y96—Young & Lo (1996); Y97—Young & Lo (1997); Y03—Young et al. (2003); Z00—van Zee (2000).

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In the faintest dIm galaxies, the H i is supported by its velocity dispersion. We use the H i as our mass tracer. The H i velocity dispersion is usually observed to be close to isothermal and is likely to be isotropic. Assuming that the H i lies within the uniform-density core of the DM, arguments like the ones that we used for dSph galaxies lead us to expect that the distribution of projected H i surface density is again Gaussian. This is observed. The observed Gaussian length scale a_* for the H i distribution and the observed H i velocity dispersion σ then give the central volume density of the DM halo core, ${\rho }_{\circ }=3{\sigma }^{2}/2\pi {{Ga}}_{*}^{2},$ as above. As an example, Figure 5 shows the Gaussian profile of the H i surface density for the dIm galaxy DDO 210. Because Plummer models are a convenient and often-used represention of the surface densities in dwarf systems, the figure also shows the best Plummer model fitted to the H i surface density profile. The Gaussian provides a better fit when the surface density data extend over more than a few e-foldings. The near-Gaussian surface density distributions for the other dIm galaxies used in this paper are shown in Figure 12 (Appendix). These H i surface density distributions are derived from the published H i surface density contour maps: the contours are digitized, and the major-axis radius for each contour is calculated from the area of the contour and its mean ellipticity.

As we did for dSph galaxies, we consider the equilibrium of a spherical, isothermal, isotropic system of H i test particles in a uniform-density DM core. For the test particles, ρ_*(r) is the volume density, Σ_*(R) is the surface density, and c_* is the isothermal sound speed or velocity dispersion. Some dIm galaxies show slow, near-solid-body H i rotation that we can readily include in our derivation. We take Ω_* to be the angular velocity, assumed constant. For the DM, M(r) is the enclosed mass and ρ_◦ is the core density.

The equation of hydrostatic equilibrium is then

$$\begin{eqnarray}&&\displaystyle \frac{{c}_{*}^{2}}{{\rho }_{*}}\displaystyle \frac{d{\rho }_{*}}{{dr}}-{{\rm{\Omega }}}_{*}^{2}r=-\displaystyle \frac{{GM}(r)}{{r}^{2}},\end{eqnarray} \tag{ 38 }$$

and the H i volume density again is Gaussian,

$$\begin{eqnarray}&&{\rho }_{*}(r)\propto \mathrm{exp}(-{r}^{2}/{a}_{*}^{2}).\end{eqnarray} \tag{ 39 }$$

The central volume density of the halo is

$$\begin{eqnarray}&&{\rho }_{\circ }=\displaystyle \frac{3{c}_{*}^{2}}{2\pi {{Ga}}_{*}^{2}}+\displaystyle \frac{3{{\rm{\Omega }}}_{*}^{2}}{4\pi G}.\end{eqnarray} \tag{ 40 }$$

The correction to the dispersion c_* from the channel spacing of the receiver (∼1.5 km s⁻¹) is negligible. We estimate the error in the DM density ρ_◦ derived from Equation (40) as follows. The contribution to ρ_◦ from the ${{\rm{\Omega }}}_{*}^{2}$ term is small. Then the error in ρ_◦ is

$$\begin{eqnarray}&&{(\displaystyle \frac{{\sigma }_{{\rho }_{\circ }}}{{\rho }_{\circ }})}^{2}=4{(\displaystyle \frac{{\sigma }_{{c}_{*}}}{{c}_{*}})}^{2}+4{(\displaystyle \frac{{\sigma }_{{a}_{*}}}{{a}_{*}})}^{2},\end{eqnarray} \tag{ 41 }$$

and the error in $\mathrm{log}{\rho }_{\circ }$ is

$$\begin{eqnarray}&&{\sigma }_{\mathrm{log}{\rho }_{\circ }}=\displaystyle \frac{1}{2.303}\displaystyle \frac{{\sigma }_{{\rho }_{\circ }}}{{\rho }_{\circ }}.\end{eqnarray} \tag{ 42 }$$

The projected H i surface density is Gaussian with the same scale length a_* as that of ρ_* (Figures 5 and 12),

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{*}(R)={{\rm{\Sigma }}}_{*}(0)\mathrm{exp}(-{R}^{2}/{a}_{*}^{2}).\end{eqnarray} \tag{ 43 }$$

We need to correct a_* for the telescope beam width. For a Gaussian telescope beam of width W = FWHM/2.355, the corrected value of a_* for the galaxy is

$$\begin{eqnarray}&&{a}_{{*}_{{\rm{c}}}}^{2}={a}_{*}^{2}-2{W}^{2}.\end{eqnarray} \tag{ 44 }$$

If the error in a_* from the least-squares fit is σ_a*, then the error in a_*c is given by

$$\begin{eqnarray}&&\displaystyle \frac{{\sigma }_{{a}_{{\ast }_{{\rm{c}}}}}}{{a}_{{\ast }_{{\rm{c}}}}}=\displaystyle \frac{{a}_{*}^{2}}{{a}_{*}^{2}-2{W}^{2}}(\displaystyle \frac{{\sigma }_{{a}_{*}}}{{a}_{*}}).\end{eqnarray} \tag{ 45 }$$

As we did for dSph galaxies, we first use the core radius of the visible matter. We will see that this is much smaller than the core radius of the DM halo. We transform the beam-corrected H i Gaussian scale length a_* (Equation (44), dropping the subscript c from here on) using the relation (Section 6.1) $\mathrm{log}\;{r}_{{\rm{c}}}=0.752\mathrm{log}{a}_{*}-0.273.$ The error in $\mathrm{log}\;{r}_{{\rm{c}}}$ is $0.327({\sigma }_{{a}_{*}}/{a}_{*}).$

Table 2 lists the distance, absolute magnitude, H i velocity dispersion, DM core density, and r_c of the H i distribution for 12 nearby dIm galaxies for which the dynamical contribution of the H i rotation is negligible and the H i surface density distribution is close to Gaussian. The effect of the beam correction is small: the correction increases $\mathrm{log}{\rho }_{\circ }$ by between 0.01 and 0.11 and decreases $\mathrm{log}{r}_{c}$ by between 0.003 and 0.04.

Figure 9 compares the baryonic and DM surface densities of galaxies as functions of luminosity. It shows quantitatively a conclusion that is well known in the literature: the ratio of baryonic mass to DM mass within the radial extent of the baryons decreases strongly with decreasing galaxy luminosity. Like Kormendy et al. (2009, hereafter KFCB) and Kormendy & Bender (2012), we conclude that at M_V ≳ −18, the dwarf spiral, Im, and Sph galaxies in Figures 2–9 form a sequence of decreasing baryon content in smaller galaxies.

In contrast, lower-luminosity bulges and elliptical galaxies have much higher surface densities at or inside the effective radius. If mergers of disks make bulges and if we ignore baryonic pulling corrections, then the ratio of visible to DM surface density increases toward lower bulge luminosity. Like Kormendy (1989), like KFCB—which reviews much additional evidence—and like many other authors, we conclude that bulges and ellipticals form a sequence of increasing dissipation during the formation of smaller galaxies.

For the brighter M_V < −18 galaxies of all kinds, the effective density in stars is similar to the DM density at or inside the same radius. For Sc–Im systems, this is at least partly a consequence of using maximum-disk decompositions and of the rotation curve conspiracy (Bahcall & Casertano 1985; van Albada & Sancisi 1986; Sancisi & van Albada 1987), i.e., the observation that rotation curves of giant galaxies are roughly flat and featureless, so the parts of galaxies that are controlled by DM are not easily distinguished from the parts that are controlled by visible matter. The total range of $\mathrm{log}{{\rm{\Sigma }}}_{{\rm{e}}}$ for the brightest disks is large, ranging from less than 2ρ_or_c to substantially larger than 2ρ_or_c for the DM. Most are sufficiently visible matter dominated to support self-gravitating structure such as bars and spiral density waves. A few are DM dominated; then the arguments of Ostriker & Peebles (1973) apply.

There is a caveat. For bulges and ellipticals, the high baryon densities at $r\ll {r}_{{\rm{e}}}$ may pull gravitationally on DM halos by enough to increase their central densities over the values for Sc–Im galaxies that are shown in Figure 9. But bulges and ellipticals have central projected densities that are more than 3 dex higher than the effective densities shown in Figure 9. The central parts of early-type galaxies are very baryon dominated (see also Cappellari et al. 2013b, 2015; Cappellari 2015, who reach similar conclusions). The central densities of disks are 0.7 dex (for an exponential) higher than the effective densities shown in Figure 9. So even pure disks are moderately dominated by visible matter near their centers. Both results are qualitatively as expected: Visible matter needs to dissipate, sink inside the DM, and become self-gravitating enough to form stars and visible galaxies (e.g., Gunn 1987; Ryden & Gunn 1987). Disks also need to be self-gravitating in order to be unstable to the formation of bars and spiral density waves (ABP). And a great deal of dissipation happens in the wet mergers that make coreless-disky-rotating ellipticals (KFCB): their densities rise above the DM densities by larger amounts at fainter M_V.

At M_V > −18, tinier dwarfs are more DM dominated. By M_V ≳ −10, they are almost-dark galaxies with just enough of a frosting of stars so that they can be detected. We conclude the following: (1) The differences between dIm and dSph galaxies in all parameter correlations shown in this paper are small. Whether or not a galaxy retains cold gas and can still form stars in today's universe is a second-order effect. This argues—as Dekel & Silk (1986) emphasized—that the baryon retention sequence seen for the fainter galaxies in Figure 9 is generated primarily by supernova-driven baryon loss or another process (such as a failure to capture baryons after reionization) that has the same effect. (2) We suggest that there exists a large population of tiny halos that are essentially dark—that the discoverable galaxies at M_V ≳ −13 represent a smaller and smaller fraction of tinier and tinier halos. This has been suggested as the solution to the problem that the fluctuation spectrum of CDM predicts more dwarfs than are observed in environments like the Local Group (e.g., Klypin et al. 1999; Moore et al. 1999).

Figure 10 shows one of the most important new results in this paper: the baryon content of galaxies goes to nearly zero robustly at halo V_circ = 42 ± 4 km s⁻¹. This is another representation of the galaxy baryon retention sequence. The inference is that still-lower-mass halos are mostly dark; they contain a discoverable frosting of baryons only if they manage at least slightly to avoid the physical processes that created the baryon retention sequence.

Figure 10 shows that rotation curve decompositions reveal a linear correlation between the maximum rotation velocities V_circ,disk of galaxy disks and the outer, circular-orbit rotation velocities V_circ of massless test particles in their halos. Baryons become unimportant where this extrapolates to V_circ,disk = 0, i.e., at V_circ ≃ 42 ± 4 km s⁻¹.

The black and red points in Figure 10 show the maximum rotation velocities given by the rotation curve decompositions for the disk and bulge, respectively. The decompositions results are from van Albada et al. (1985), Athanassoula et al. (1987), Kent (1987, 1989), Carignan & Freeman (1988), Jobin & Carignan (1990), Begeman et al. (1991), Puche & Carignan (1991), Puche et al. (1991), Broeils (1992), Martimbeau et al. (1994), Miller & Rubin (1995), Rhee (1996), Sofue (1996), de Blok & McGaugh (1997), Sicotte & Carignan (1997), van Zee et al. (1997), Verdes-Montenegro et al. (1997), Verheijen (1997), Carignan & Purton (1998), Meurer et al. (1998), Blais-Ouellette et al. (1999), Côté et al. (2000), Swaters et al. (2000), de Blok & Bosma (2002), Corbelli (2003), Weldrake et al. (2003), Chemin et al. (2006), Gentile et al. (2007), Noordermeer et al. (2007), de Blok et al. (2008), Dicaire et al. (2008), Noordermeer (2008), Yoshino & Ichikawa (2008), Sofue et al. (2009), Elson et al. (2010), Puglielli et al. (2010), and Oh et al. (2011a, 2011b).

The disk and bulge fits in Figure 10 are, respectively,

$$\begin{eqnarray}&&{V}_{{\rm{circ,disk}}}=(1.16\pm 0.03)({V}_{{\rm{circ}}}-200)+(183\pm 3);\end{eqnarray} \tag{ 46 }$$

$$\begin{eqnarray}&&{V}_{{\rm{circ,bulge}}}=(1.73\pm 0.29)({V}_{{\rm{circ}}}-200)+(166\pm 9),\end{eqnarray} \tag{ 47 }$$

where all quantities are in km s⁻¹. The important result is that both fits robustly have intercepts at V_circ > 0. Little extrapolation is needed: baryonic disks disappear at V_circ ≃ 42 ± 4 km s⁻¹. This result is insensitive to the choice of maximum-disk decompositions: V_circ is much less sensitive to the assumed M/L than are ρ_o and r_c, and in any case, only the high-V_circ points would be affected if disks were not maximal, because the V_circ ≃ 50 km s⁻¹ dwarfs are so DM dominated that the choice of M/L scarcely matters. The extrapolation of the black line in Figure 10 to the V_circ,disk = 0 intercept is so small that this intercept would remain almost unchanged if the slope of the line changed slightly.

It is well known that smaller galaxies are more completely dominated by DM (Figure 9), but Figure 10 provides a measure of the halo mass at which baryons become dynamically unimportant. Dwarf spheroidal and irregular galaxies with M/L values of ∼10–10² (the objects in Table 2) are missing from Figure 10; they do not rotate rapidly enough for rotation curve decomposition. But they are examples of objects with trace amounts of baryons that lie below the black points in Figure 10. The message is that dark galaxies are likely to have V_circ ≲ 42 ± 4 km s⁻¹ and σ ≲ 30 ± 3 km s⁻¹. These values are remarkably similar to the values that we derive for DM halos of dSph and dIm galaxies from their baryonic velocity dispersions (Section 7). Thus, we conclude that, at the V_circ where galaxies get dim, their properties are just like the (dim) dSph galaxies in our sample. We regard this V_circ agreement as a sign that our conclusions are robust.

In fact, Figures 9 and 10 together make a strong case that the sequence of almost-empty halos may extend to smaller V_circ and σ than the above values. So we again suggest that, at lower luminosities, the galaxies that are plotted are increasingly the "tip of an iceberg" of undiscovered, still-darker dwarfs, as predicted by the CDM density fluctuation spectrum.

Important: these extreme dwarfs apparently remain luminous in the Virgo cluster. There is no "missing dwarfs problem" in the Virgo cluster (Moore et al. 1999). Perhaps the much richer proto-Virgo environment was so dense with jostling, small protogalactic halos and turbulent gas that small halos could capture gas more easily than (say) in the proto Local Group, where there likely was more empty space and where gas ejected from galaxies by supernovae could be thrown farther away from tiny DM halos.

The main observational result of this paper is that DM halos of Sc–Im galaxies satisfy well-defined scaling laws. Halos in less luminous galaxies have smaller core radii, higher central densities, and smaller velocity dispersions. This confirms the results of our previous analyses of smaller samples (Kormendy 1988, 1990; Kormendy & Freeman 1996, 2004). Scaling laws provide new constraints on the nature of DM (e.g., Tremaine & Gunn 1979) and on galaxy formation and evolution. Most of these remain to be explored. Simple implications and other correlation results are summarized in this section.

The surprisingly high DM densities in dwarf spheroidals are normal for galaxies of such low luminosity (Kormendy 1987a). This implies (Kormendy 1988, 1990) that dSph galaxies are real galaxies that collapsed out of the DM fluctuation spectrum. They are not tidal fragments. Tides plausibly can pull bound fragments out of more luminous galaxies, but they cannot retain even the relatively low DM densities in those progenitors (Barnes & Hernquist 1992), much less increase ρ_◦ to the high values characteristic of dwarf spheroidals.

Figure 6 shows that the DM halo density of the faintest dwarfs can exceed 1 M_⊙ pc⁻³. This halo density is much higher than the density of the Galactic disk near the Sun (∼0.1 M_⊙ pc⁻³). Although the low surface brightnesses of the faintest dwarfs may make them appear fragile, their high DM densities make them in fact among the most robust to tidal disruption by their parent galaxy. We could expect that accreted faint dwarfs would survive orbital decay and tidal disruption and would end up, possibly intact, in the inner regions of the parent system.

In this context, we can compare the location of the DM-dominated dSph galaxies in the plane of half-light radius r_h versus luminosity L with the location of the baryon-dominated globular clusters, non-dwarf spheroidals (sometimes called dwarf ellipticals), and normal, small (i.e., compact) ellipticals (e.g., Kormendy & Bender 2012; Brodie & Romanowsky 2015). The dSph galaxies have r_h values that are one to two orders of magnitude larger than the baryon-dominated objects at similar L. Again, this is an indication that the baryons in the dSph systems are a minor tracer component, almost irrelevant to the equilibrium of these galaxies.

Dwarf spheroidal galaxies are not included in the least-squares fits in Figure 7 because only ρ_◦ can be derived for their halos. However, we show that, with a simple shift in baryonic M_B, $\mathrm{log}{r}_{{\rm{c}}},$ and $\mathrm{log}\sigma ,$ the dwarfs lie on the extrapolation of the DM scaling laws for brighter galaxies in all panels of Figure 7. Section 9.4 interprets the shifts. If it is correct, then DM halos of dSph and Sc–Im galaxies form a single physical sequence as a function of DM core density, velocity dispersion, or—by inference—DM mass.

The central DM densities ρ_◦ of dwarf galaxies are, within the approximations of our analysis that ρ(r) ≃ constant, derived correctly, independent of the unknown DM core radius r_c and velocity dispersion σ. This provides us with a way to estimate both the baryon loss for these galaxies and the true values of r_c and σ for their DM halos:

To shift the dwarf galaxies onto the extrapolation of the ${\rho }_{\circ }-{M}_{B}$ correlation for brighter galaxies, we need to assume that they have or once had enough additional baryons such that, if these were converted into stars, the galaxies would be substantially more luminous. These shifts are ΔM_B ≃ −4.6 mag for dSph galaxies and ΔM_B ≃ −3.5 mag for dIm galaxies, both relative to brighter galaxies that themselves probably lost a significant fraction of their baryons. To shift the dwarfs onto the other DM correlations then implies that the core radius of the DM is larger than the core radius of the visible matter by ${\rm{\Delta }}\mathrm{log}{r}_{{\rm{c}}}\simeq 0.70$ for Sph galaxies and by ${\rm{\Delta }}\mathrm{log}{r}_{{\rm{c}}}\simeq 0.85$ for dIm galaxies. And the DM particle velocity dispersion is larger than the velocity dispersion of the stars in dSph galaxies by ${\rm{\Delta }}\mathrm{log}\sigma \simeq 0.40$ and larger than the velocity dispersion in the gas in dIm galaxies by ${\rm{\Delta }}\mathrm{log}\sigma \simeq 0.50.$ With these shifts, dwarfs of both kinds lie on the extrapolation of the DM correlations that we derived for galaxies with rotation curve decomposition in all panels of Figure 7.

The above shifts are consistent with the hypothesis that fainter dwarfs have lower visible matter densities (Figure 9) and higher M/L_V ∼ 10²–10³ because they lost more of their baryons early. One possible reason is galactic winds (e.g., Dekel & Silk 1986). In the absence of DM, the loss of most baryons would unbind the few stars that had already formed. But since faint galaxies have dense DM halos, the systems remain bound despite any baryon loss. An alternative to baryon blowout is the difficulty of capturing or holding on to baryons in low-mass DM potential wells when the universe was reionized (Klypin et al. 1999; Bullock et al. 2000; Cattaneo et al. 2011). In addition, dIm galaxies are relatively dark in part because some of their baryons are still in H i gas.

The inferred shifts of ${\rm{\Delta }}\sigma \simeq 0.40$ in dSph galaxies and 0.50 in dIm galaxies imply that these almost-dark dwarfs have higher DM masses than we thought. They imply that the typical velocity dispersions of their DM halos are σ ∼ 30 km s⁻¹. Thus, these galaxies lie near the V_disk = 0 km s⁻¹ intercept at V_circ ≃ 42 ± 4 km s⁻¹ of the black line in Figure 10. It is not clear that we have discovered any galaxies with DM ${V}_{{\rm{circ}}}\ll 40$ km s⁻¹.

Figure 9 robustly illustrates the well-known result that the ratio of baryon surface density to DM surface density decreases dramatically at M_V ≳ −18. By M_V ∼ −10, galaxies typically have M/L_V ∼ 10². The most extreme dwarfs barely contain enough baryons to be discoverable. Specifically, Figure 10 shows that the baryon content of spiral and irregular galaxies goes to zero at DM halo V_circ > 0. Instead, the V_circ,disk = 0 km s⁻¹ intercept of the remarkably linear and very well determined correlation of the black points in Figure 10 happens at halo V_circ = 42 ± 4 km s⁻¹. Smaller galaxies are robustly faint. In Section 9.4, we suggest that fainter dSph and dIm galaxies form a sequence of decreasing baryon retention or capture in tinier galaxies. Note again that some "apparently missing" baryons in dIm galaxies are present in H i gas.

As the DM ${V}_{{\rm{circ}}}$ decreases in Figure 10, bulges get less massive more rapidly than disks. Bulges disappear entirely at V_circ ∼ 104 ± 16 km s⁻¹. This is consistent with the well-known observation that galaxies such as M33 usually do not have bulges. Much fainter galaxies never have bulges. At high V_circ, we note that there exist bulgeless galaxies at all V_circ up to at least 300 km s⁻¹ (see Kormendy et al. 2010 and Fisher & Drory 2011 for statistics).

The differences between S+Im galaxies (blue points in Figure 9) and Sph galaxies (green points in Figure 9) are small. In particular, both galaxy kinds have similarly decreasing baryon-to-DM density and mass ratios at lower galaxy luminosities. And among extreme dwarfs, some dSphs contain no gas and no young stars; other dSphs contain no gas but formed stars relatively recently, and dIm galaxies such as M81 dwarf A contain modest amounts of gas and still form stars gently. But all of these galaxies have similar structure (Figure 5 and Appendix) and structural parameter correlations (Kormendy 1985, 1987c; KFCB; Kormendy & Bender 2012; Figure 9, here).

This circumstance extends to higher galaxy masses, too: Kormendy & Bender (2012) show that Sph galaxies are essentially bulgeless S0 galaxies, that S0 disks extend the Sph correlations to higher luminosities, and that S0 disks and spiral galaxy disks also have similar structural parameter correlations. Like Kormendy & Bender (2012) and more explicitly Kormendy (2015), we conclude that the differences between z ∼ 0 galaxies that still contain gas (and that still can form stars) and those that do not (and that can not) are relatively minor.

This is consistent with the general picture advocated in papers like Dekel & Silk (1986) and Kormendy & Bender (2012) that the most important physical processes (probably more than one) that control the formation of dwarf galaxies are the ones that determine the baryon depletion. Whether all or just most of the cold gas gets ejected, gets used up, or never gets accreted is secondary.

It is interesting to note that almost all galaxies detected in the HIPASS blind H i survey for galaxies with velocities ≲12,000 km s⁻¹ were also detected in starlight. Galaxies with H i gas and DM but no stars appear to be very rare in the nearby universe (Doyle et al. 2005).

We suggested in our earlier papers (Freeman 1987; Kormendy 1988, 1990; Kormendy & Freeman 2004) that there may exist a large population of galaxies that are completely dark. This was based on the observation that, as luminosity decreases, dwarf galaxies become much more numerous and much more nearly dominated by DM. Similar suggestions were made for different reasons by Tully et al. (2002).

This hypothesis is greatly strengthened by Figure 10. Rotation curve decompositions reveal a linear correlation between the maximum rotation velocities V_circ,disk of galaxy disks and the outer, circular-orbit rotation velocities V_circ in their halos. It extrapolates to V_circ,disk = 0 (i.e., baryons become unimportant) at V_circ ∼ 42 km s⁻¹. In Section 7, we conclude that dSph galaxies also live in halos with V_circ ≃ 42 km s⁻¹. This means that the range of baryon content of such DM halos is very large, extending from galaxies that allow rotation curve decomposition to extreme dwarfs that can only be found by detecting tiny enhancements in star counts.

Our results suggest that empty halos are likely to be darker versions of Draco, UMi, and the ultrafaint dSph galaxies. DM halos that contain H i gas but no stars are rarely detected. Tiny dSph galaxies have only a frosting of baryons because they lost or never captured more. But there is nothing magic about owning a few percent of the normal baryon content. Draco and UMi do not know or care that they contain just enough baryons to be discovered by us 12 billion years after they formed. If many DM halos captured or kept even fewer baryons, they could escape detection and form a population of dark dwarfs.

Undiscovered dark dwarfs could solve the problem that the spectrum of initial density fluctuations predicted by CDM predicts too many dwarf satellites of giant galaxies (Klypin et al. 1999; Moore et al. 1999; Bullock et al. 2000; Simon & Geha 2007; Diemand et al. 2008; Diemand & Moore 2011; Kirby et al. 2008; Springel et al. 2008; Tollerud et al. 2008; Bovill & Ricotti 2009, 2011a, 2011b; Walsh et al. 2009). The favored explanation for why these dwarfs are not seen is that they virialized early, before or during the reionization of the universe, and so lost or never captured the canonical fraction of baryons because the baryons were too hot to be confined in shallow DM potential wells.

Our suggestion is borne out by the discovery of ultrafaint dwarfs with M_B as faint as −1 (see Simon & Geha 2007; Tolstoy et al. 2009 for reviews and Table 2 for references on galaxies used here). Large surveys such as the Sloan Digital Sky Survey (York et al. 2000) have made it possible to detect ever-darker dwarfs via ever-smaller star-count excesses. The DM densities of these galaxies are in the range seen for brighter dSph galaxies and for the smallest galaxies with rotation curve decompositions. They are real galaxies in the sense of Section 9.2. They are darker than Draco and UMi. And they hint that still-darker galaxies probably exist.

We emphasize that there is no sign of any discontinuity in baryon fractions as a function of galaxy mass. Only the names that we give to galaxy types change as baryon fractions decrease. The largest baryon fractions belong to spiral galaxies whose baryons are self-gravitating enough to dissipate into disks and to make internal structures such as bars and spiral arms. Medium-large baryon depletions can apply to either dSph or dIm galaxies, depending on their environments. These include galaxies that still have gas but that have low enough baryon content so that the baryons have not organized themselves into disks. In these dIm galaxies, some of the factor-of-25 apparent "baryon depletion" (up to a factor of 10) is not due to low baryon content but rather is due to the fact that the baryons are still in gas. That is, in various dIm galaxies with various H i/stellar mass fractions, the true baryon depletion factor ranges from ∼25 (in galaxies with little H i) to ∼2.5 (in galaxies with H i/stellar mass ratios of ∼10). More extreme galaxy environments are so hostile to dwarfs that they cannot retain gas; then we call them dSph galaxies and necessarily measure even larger baryon depletions. And then all of the apparent baryon depletion is associated with baryon absence, not baryon "hiding" in H i gas. At all small DM masses, there is a large range in baryon depletions. There is substantial overlap in M_B between spiral galaxies, Im or dIm galaxies, and Sph or dSph galaxies. It is likely that there is overlap in DM mass between each of the above groups and dark galaxies. We see no reason to postulate a dichotomy in baryon properties at any DM mass.

S. Faber (2015, private communication) points out that our results suggest at least a partial solution to another problem with ΛCDM galaxy formation, TBTF. It is closely related to the problem that the spectrum of initial density fluctuations predicts more halo substructure than we observe in dwarf galaxy companions of (e.g.) our Galaxy or M31. For the smallest and most numerous such halos, the solution may be that they never formed visible galaxies (Section 9.7). But many authors (e.g., Boylan-Kolchin et al. 2011, 2012; Garrison-Kimmel et al. 2013, 2014) have pointed out that intermediate-mass halos with V_circ ∼ 30–70 km s⁻¹ (i.e., galaxies that are a little less massive than the Magellanic Clouds) are TBTF in that they are too massive to fail to accrete or to hold on to discoverable masses of baryons. Whether this problem is or is not new and separate from other dwarf galaxy formation problems depends in part on whether or not these dwarfs have cored or cuspy (i.e., NFW) density profiles (see Garrison-Kimmel et al. 2013, 2014).

We conclude (Section 9.4) that dSph and dIm galaxies whose stellar and gas velocity dispersions are σ ∼ 10 km s⁻¹ really live in bigger halos with σ ∼ 10^1.5 km s⁻¹ and V_circ ∼ 42 km s⁻¹. This is probably an underestimate. Our ${\rm{\Delta }}\mathrm{log}\sigma$ estimate was based on an extrapolation of correlations that, at the low-mass end, include galaxies that are dominated by DM and that have M/L values of ∼10¹. These objects (e.g., DDO 154; Carignan & Freeman 1988) almost certainly also lost baryons. If we could correct for this loss, then these galaxies would be plotted at brighter M_B and the derived DM parameter correlations would get slightly steeper. We would derive a slightly larger value of ${\rm{\Delta }}\mathrm{log}\sigma .$ This correction cannot be made too large or the total masses of these dwarfs become uncomfortable large. But if ${\rm{\Delta }}\mathrm{log}\sigma \sim 0.5$–0.6, then the larger dSph and dIm galaxies in our sample are in the mass range of the TBTF "missing galaxies." Therefore, we suggest that the solution to the TBTF problem is at least in part that intermediate-mass dwarfs are present in the Local Group but are masquerading as smaller galaxies. Brook & Di Cintio (2015) reach a similar conclusion via a different route.

Virialized DM density depends on collapse redshift z_coll, ${\rho }_{\circ }\propto {(1+{z}_{{\rm{coll}}})}^{3}.$ The DM densities for classical dSph galaxies like Draco and UMi are about 100 times higher than those for the brightest spirals. The collapse redshifts of the halos of Draco and UMi and those of the brightest spiral galaxies are then related by $(1+{z}_{{\rm{dwarf}}})/(1+{z}_{{\rm{spiral}}})\simeq 5$. For the faintest dSph galaxies, this ratio is about 8. If there is a correction for baryonic DM compression in the giant galaxies, then it would make their DM ρ_◦ smaller, and this would increase the above ratios still further.

Djorgovski (1992) compared an earlier version of the DM parameter correlations to the scaling laws predicted by hierarchical clustering (Peebles 1974; Gott & Rees 1975). For a power spectrum of initial density fluctuations that is a power law in wavenumber k, $| {\delta }_{k}{| }^{2}\propto {k}^{n},$ the size R, density ρ, and velocity dispersion σ of a bound object are related approximately by

$$\begin{eqnarray}&&\rho \propto {R}^{-3(3+n)/(5+n)};\end{eqnarray} \tag{ 48 }$$

$$\begin{eqnarray}&&\rho \propto {\sigma }^{-6(3+n)/(1-n)};\end{eqnarray} \tag{ 49 }$$

$$\begin{eqnarray}&&\sigma \propto {R}^{(1-n)/(10+2n)}.\end{eqnarray} \tag{ 50 }$$

Here we have used the relation $\rho \propto {\sigma }^{2}{R}^{-2}$ for an ITS. Djorgovski pointed out that the DM parameter correlations in Kormendy (1990) imply that n ≃ −2.45, close to the value n ≃ −2 expected for giant galaxies in CDM theory. The more accurate fits in Equations (31)–(33),

$$\begin{eqnarray}&&{\rho }_{\circ }\propto {r}_{{\rm{c}}}^{-1.109\pm 0.066};\end{eqnarray} \tag{ 51 }$$

$$\begin{eqnarray}&&{\rho }_{\circ }\propto {\sigma }^{-1.821\pm 0.274};\end{eqnarray} \tag{ 52 }$$

$$\begin{eqnarray}&&\sigma \propto {r}_{{\rm{c}}}^{0.528\pm 0.035},\end{eqnarray} \tag{ 53 }$$

give n = −1.83 ± 0.17, n = −2.07 ± 0.07, n = −2.08 ± 0.17, respectively. These values are not independent. Their average, n = −2.0 ± 0.1, is significantly improved over earlier determinations. It is remarkably close to the value n ≃ −2.1 expected in ΛCDM theory at a halo mass of 10¹² M_⊙ (Shapiro & Iliev 2002). Note that the slopes have not been corrected for baryonic DM compression. The above comparison provides a measure of the slope of the fluctuation power spectrum on mass scales that are smaller than those accessible to most other methods.

Shapiro & Iliev (2002) have made a more detailed comparison of the DM parameter correlations published by Kormendy & Freeman (1996) with their predictions based on COBE-normalized CDM fluctuation spectra. They found that the agreement between predictions and observations is best for ΛCDM.

It is interesting to note a consequence of the theoretical prediction that the slope n gets steeper at smaller mass scales. If n ≃ −2.6 for the smallest dwarfs (Shapiro & Iliev 2002; Ricotti 2003), then the straight lines in the left panels of Figure 6 should curve downward toward the visible matter parameters of dSph galaxies. It will be important to look for curvature in the correlations as more data become available for dwarf galaxies.

Some authors (e.g., van der Kruit 2010) have suggested that late-type galaxy disks are very submaximal, with M/L of ∼0.5–0.7 of maximum-disk values. We addressed this issue in Section 2.2. Here, having derived DM correlations based on maximum-disk decompositions, we can use these results to help justify our assumptions "after the fact," as follows:

If the disks of giant spirals (M_B < −19) are typically 2/3 maximal, then DM core radii must be decreased and DM central densities must be increased to maintain good fits to the rotation curves. The inevitable consequence is that ρ_◦ for the biggest galaxies becomes almost as big as the values for dwarf galaxies that we illustrate in Figure 6. That is, the ${\rho }_{\circ }-{M}_{B}$ correlation would get very shallow, with only a small range in ρ_◦. This has three uncomfortable or even untenable consequences:

1. The derived differences in the formation redshifts of giant and dwarf galaxies would get much smaller than the current values in Section 9.9. This would be inconsistent with current thinking that the smallest known dwarfs formed (i.e., their halos collapsed and virialized) at z ≳ 10.

2. Our derived ∣ΔM_B∣ of baryon loss would get much smaller: $| {\rm{\Delta }}{M}_{B}| \ll 3.5.$ The inferred baryon deficiency would then be too small to be consistent with the large M/L ratios of ∼10² that are almost universally measured for these dwarfs.

3. Shallower DM parameter correlations would force us to derive a slope of the power spectrum of initial density fluctuations (Section 9.10) that is very different from −2. That derivation is an important consistency check that nothing has gone very wrong with our derivation of the parameter correlations. If DM halos are so cuspy that our analysis is invalid, or if disks are so submaximal that our derivations of the correlation slopes are very wrong, then the Section 9.10 consistency with the slope of the ΛCDM fluctuation spectrum vanishes. The success of the comparison in Section 9.10 is an argument in favor of our assumptions. This includes our omission of any corrections to the correlation slopes that result from taking account of the compression of DM halos in giant galaxies that is caused by the gravity of their baryons.

We emphasize that the scatter in Figures 2–4 and 6–8 has surely been increased by problems with the data.

1. Distance errors are smaller than in previous derivations of the correlations, but they are not negligible.

2. The Table 1 parameters are very heterogeneous. Inconsistencies in rotation curve decomposition procedures used by different authors inflate the scatter. For example, the Broeils (1992) galaxies define the same correlations with fewer points but smaller scatter than the sample as a whole.

3. If some disks are submaximal, then this affects the scatter in the correlations. If the degree to which they are submaximal depends on M_B (Kranz et al. 2003; Bosma 2004; van der Kruit & Freeman 2011), then this also affects the correlation slopes.

4. The assumption that DM halos have isothermal cores is challenged by CDM theory, although it is suppported by many observations. It will be important to see how the correlations are affected if NFW or Einasto halos are used.

5. DM compression by the visible matter (Sellwood & McGaugh 2005) may affect some galaxies more than others. ABP found early versions of the present DM correlations; even in their small sample, they already observed that many early-type galaxies deviate toward small r_c and large ρ_◦ compared to the DM correlations for late-type galaxies. They interpreted this as a result of the compression of DM halos by the gravity of the visible components. Similar deviations were similarly interpreted by Kormendy (1988, 1990), Thomas et al. (2009), Thomas (2010), Jardel et al. (2011), and Bender et al. (2015). Correction for baryonic compression of DM likely would decrease the measured ρ_o values slightly for giant but not dwarf galaxies. This would increase the difference between low ρ_o values in giant galaxies and high ρ_o values in dwarf galaxies.

If disks are submaximal and if their gravity compresses DM, then the corrections for these effects change ρ_o in opposite directions. It is possible that these corrections are modestly significant but fortuitously cancel.

6. We have consistently used the absolute magnitude M_B as a measure of baryonic mass against which to scale the DM halo parameters. For some of the fainter dIm galaxies, the baryonic mass is dominated by the H i mass, and it would make sense to use a total baryonic mass as is done in deriving the baryonic Tully–Fisher relation. This affects our interpretation of the ΔM_B shift for some dIm galaxies in Figure 7 as discussed in Section 7. We are grateful to J. S. Gallagher and to the referee for this suggestion.

7. In selecting dIm galaxies for this sample, we were probably overcautious in choosing galaxies that have small solid-body rotation. The Gaussian surface density and related apparatus for estimating the core density of the DM halo core are expected to work for regions of these galaxies in which the rotation is solid body.

We also acknowledge that the shifts ΔM_B, ${\rm{\Delta }}\mathrm{log}{r}_{{\rm{c}}},$ and ${\rm{\Delta }}\mathrm{log}\sigma$ determined in Section 7 and discussed further in Section 9.4 are probably underestimated. The smallest dS + dIm galaxies that have rotation curve decomposition results in Table 1, in Figures 2–4, and in the correlation derivations in Equations (21)–(33) have mass-to-light ratios of M/L_V ∼ 10¹. Consistent with the rest of our discussion, they probably also have lost a substantial fraction of their baryons. Certainly they are faint enough to be on the baryon depletion sequence illustrated in Figure 9. If we could correct for this baryon loss, then the correlations that we derive would get steeper. Consequently, all parameter shifts for the extreme dSph and dIm galaxies would get somewhat larger. None of our conclusions would change qualitatively, but correlations and shifts both would get tweaked. We would infer that the smallest detected DM halos have particle velocity dispersions that are slightly bigger than σ ∼ 30 km s⁻¹.

We will address these issues in future papers.