Denuded Dwarfs Demystified: Gas Loss from dSph Progenitors and Implications for the Minimum Mass of Galaxies

TL;DR

This paper investigates how gas loss in dwarf spheroidal galaxies influences their placement on the Potential Plane, revealing a critical potential threshold linked to photoionization-driven gas expulsion and implications for the minimum galaxy mass.

Contribution

It identifies a critical potential related to gas loss in dwarf galaxies and links photoionization to their baryonic mass depletion, refining understanding of galaxy formation limits.

Findings

Gas loss causes deviations from the Potential Plane in dSphs.

Critical escape velocity for gas loss is about 50 km/s.

Minimum galaxy baryonic mass is estimated between 500 and 10,000 solar masses.

Abstract

The placement of early-type dwarf galaxies (dSphs and dEs) with respect to the Potential Plane defined by pressure-supported late-type dwarf galaxies (dIs and BCDs) has been determined from surface brightness profiles. dEs and the most luminous dSphs lie on the Plane, suggesting that they emerged from late-type dwarfs that converted most of their gas into stars. However, there is a critical value of the potential at which dSphs start to fall systematically below the Plane, with the deviation growing as the potential becomes shallower. The displacements are attributed to depletion of baryons through gas loss, smaller galaxies having lost proportionately more gas. The critical potential corresponds to an escape velocity of 50 $\pm$ 8 km/s, which is what is expected for gas with a temperature of 13,000 $\pm$ 4,000 K, typical of a low-metallicity HII region. This suggests that photoionization was responsible for instigating the loss of gas by galaxies with potentials shallower than the critical value, with evacuation occurring over a few tens of millions of years. Extreme ratios of dynamical to luminous masses observed for the smallest dSphs are an artifact of mass loss. Because the efficiency with which gas was converted into stars was lower for dSphs with shallower potentials, there should be a minimum baryonic mass for a galaxy below which the stellar mass is negligible. Gross extrapolation of the trend of inferred gas masses with stellar masses suggests a value between 500 and 10,000 M$_\odot$. The corresponding dynamical mass is below 10$^6$ M$_\odot$.

Figures

Tables

Table 1. Table 1: Observed Properties of Dwarf Spheroidal Galaxies

Galaxy	$E$	${\tau }_1$	$v_{\odot }$	$\mathrm{𝐷𝑀}$	Method	Foundation	[Fe/H]	Source	${\sigma }_{\text{los}}$	Source	${\mu }_0$	$r_0$	$q$	Foundation
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)	(14)	(15)
And I	0.054	0.061	$-375.8$	$24.366\pm 0.031$	TRGB	mcco04,mcco05	$-1.11\pm 0.12$	var14	$10.2\pm 1.9$	tol12	22.20	92	0.78	mcco06
And II	0.062	0.070	$-193.6$	$24.086\pm 0.021$	TRGB	mcco04,mcco05	$-1.37\pm 0.12$	var14	$9.3\pm 0.6$	amo14	22.75	189	0.80	mcco06
And III	0.057	0.065	$-345.6$	$24.424\pm 0.041$	TRGB	mcco05	$-1.81\pm 0.12$	var14	$9.3\pm 1.4$	tol12	22.42	57	0.48	mcco06
And V	0.124	0.141	$-403.0$	$24.483\pm 0.041$	TRGB	mcco05	$-1.71\pm 0.12$	var14	$10.5\pm 1.1$	tol12	23.14	51	0.82	mcco06
And VI	0.064	0.073	$-354.0$	$24.490\pm 0.041$	TRGB	mcco05	$-1.50\pm 0.10$	col13,mcco16	$12.4\pm 1.4$	col13	21.09	58	0.59	mcco06
And VII	0.196	0.223	$-309.4$	$24.442\pm 0.050$	TRGB	mcco05	$-1.24\pm 0.12$	var14	$13.0\pm 1.0$	tol12	21.89	138	0.87	mcco06
Bootes I	0.018	0.020	$99.0$	$19.193\pm 0.051$	RR	dal06	$-2.51\pm 0.03$	lai11	$4.6\pm 0.7$	kop11	25.17	384	0.78	oka12
Carina	0.063	0.072	$222.9$	$20.154\pm 0.055$	TRGB	sme94	$-1.69\pm 0.17$	koc08	$6.6\pm 1.2$	wal09a	23.01	326	0.67	irw95
Cetus	0.029	0.033	$-87.0$	$24.508\pm 0.029$	RR	ber09	$-1.90\pm 0.10$	lew07	$8.3\pm 1.0$	kir14	22.52	74	0.67	mcco06
Coma Ber	0.017	0.019	$98.1$	$18.235\pm 0.054$	RR	mus09	$-2.25\pm 0.04$	kir13	$4.6\pm 0.8$	sim07	25.70	231	0.64	mun10
CVn I	0.014	0.016	$30.9$	$21.702\pm 0.028$	RR	kue08	$-1.91\pm 0.01$	kir13	$7.6\pm 0.4$	sim07	25.55	276	0.70	oka12
CVn II	0.010	0.011	$-128.9$	$21.066\pm 0.068$	RR	gre08,oka12	$-2.12\pm 0.05$	kir13	$4.6\pm 1.0$	sim07	23.73	58	0.77	oka12
Draco	0.027	0.031	$-291.0$	$19.643\pm 0.046$	RR	kin08,bon04	$-1.98\pm 0.01$	kir13	$9.1\pm 1.2$	wal09b,wal10	22.98	284	0.71	irw95
Fornax	0.020	0.023	$55.3$	$20.764\pm 0.021$	TRGB	riz07b	$-1.04\pm 0.01$	kir13	$11.7\pm 0.9$	wal09a	20.92	570	0.70	irw95
Leo I	0.036	0.041	$282.5$	$21.985\pm 0.040$	TRGB	bel04	$-1.45\pm 0.01$	kir13	$9.2\pm 1.4$	wal09b,wal10	19.96	118	0.79	irw95
Leo II	0.017	0.019	$78.0$	$21.889\pm 0.031$	TRGB	bel05	$-1.63\pm 0.01$	kir13	$6.6\pm 0.7$	koc07	21.49	90	0.87	irw95
Leo IV	0.025	0.028	$132.3$	$21.092\pm 0.056$	RR	mor09	$-2.45\pm 0.07$	kir13	$3.3\pm 1.7$	sim07	25.28	81	0.96	oka12
NGC 147	0.173	0.197	$-193.1$	$24.396\pm 0.028$	TRGB	han97,now03,con12	$-0.49\pm 0.12$	var14	$16.0\pm 1.0$	geh10	19.25	95	0.69	jar03
NGC 185	0.182	0.207	$-203.8$	$24.072\pm 0.061$	TRGB	now03,riz07a,con12	$-0.92\pm 0.12$	var14	$24.0\pm 1.0$	geh10	18.28	66	0.89	jar03
NGC 205	0.062	0.070	$-246.0$	$24.709\pm 0.100$	TRGB	but05	$-0.87\pm 0.05$	ho15	$35.0\pm 5.0$	geh06	17.95	110	0.59	jar03
Sculptor	0.018	0.020	$111.4$	$19.647\pm 0.031$	TRGB	riz07a	$-1.68\pm 0.01$	kir13	$9.2\pm 1.1$	wal09a	21.23	331	0.68	irw95
Sextans	0.050	0.057	$224.2$	$19.979\pm 0.041$	TRGB	lee03	$-1.94\pm 0.01$	kir13	$7.9\pm 1.3$	wal09a	24.63	809	0.65	irw95
Ursa Min	0.032	0.036	$-246.9$	$19.369\pm 0.085$	RR	bel02,car02,nem88	$-2.13\pm 0.01$	kir13	$9.5\pm 1.2$	wal09b,wal10	23.40	510	0.44	irw95
\insertTableNotes

Table 2. Table 2: Observed Properties of Dwarf Elliptical Galaxies

Galaxy	Nucleation	$E$	${\tau }_1$	$v_{\odot }$	[Fe/H]	${\sigma }_{\text{los}}$	${\mu }_0$	$r_0$	$q$
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
PGC 32348	N	0.039	0.044	$541.0$	$\dots$	$15.4\pm 1.6$	19.39	8.3	0.71
VCC 0009	N	0.038	0.043	$1674.0$	$-0.5\pm 0.1$	$26.0\pm 3.9$	19.46	20.3	0.85
VCC 0033	N	0.037	0.042	$1179.3$	$-0.9\pm 0.2$	$20.8\pm 4.9$	18.21	5.2	0.89
VCC 0437	N	0.031	0.035	$1412.3$	$-1.3\pm 0.3$	$40.9\pm 4.0$	19.49	19.9	0.54
VCC 0543	nN	0.031	0.035	$977.1$	$-0.7\pm 0.1$	$35.1\pm 1.4$	19.19	14.2	0.57
VCC 0634	N	0.027	0.031	$484.5$	$-0.4\pm 0.0$	$31.3\pm 1.6$	19.18	14.5	0.78
VCC 0750	N	0.027	0.031	$1058.8$	$-0.3\pm 0.1$	$43.5\pm 2.9$	19.59	10.5	0.72
VCC 0846	N	0.028	0.032	$-730.0$	$\dots$	$19.0\pm 5.0$	20.45	9.8	0.76
VCC 0917	nN	0.032	0.036	$1244.8$	$-0.6\pm 0.1$	$28.4\pm 1.4$	18.43	6.1	0.52
VCC 1087	N	0.026	0.030	$658.6$	$-0.4\pm 0.1$	$42.0\pm 1.5$	18.59	12.2	0.68
VCC 1261	N	0.028	0.032	$1825.3$	$-0.6\pm 0.2$	$44.8\pm 1.4$	18.25	14.2	0.57
VCC 1355	N	0.034	0.039	$1245.0$	$-0.7\pm 0.1$	$20.3\pm 4.7$	20.47	17.5	0.96
VCC 1407	N	0.031	0.035	$1007.2$	$-1.1\pm 0.1$	$31.9\pm 2.1$	18.70	7.1	0.83
VCC 1431	N	0.054	0.061	$1489.4$	$-0.3\pm 0.3$	$52.4\pm 1.6$	17.66	6.0	0.96
VCC 1453	N	0.035	0.040	$1880.0$	$-0.2\pm 0.0$	$35.6\pm 1.4$	19.22	13.0	0.82
VCC 1528	nN	0.027	0.031	$1615.4$	$-0.2\pm 0.0$	$47.0\pm 1.4$	18.14	6.3	0.92
VCC 1549	N	0.030	0.034	$1389.3$	$-0.4\pm 0.3$	$36.7\pm 2.3$	18.60	8.2	0.83
VCC 1826	N	0.017	0.019	$2033.0$	$\dots$	$22.3\pm 2.2$	18.52	5.3	0.69
VCC 1861	N	0.030	0.034	$629.7$	$-0.3\pm 0.1$	$31.3\pm 1.5$	19.04	11.1	0.93
VCC 1895	nN	0.018	0.020	$970.2$	$-0.9\pm 0.1$	$23.8\pm 3.0$	18.40	8.2	0.47
VCC 2083	N	0.018	0.020	$867.9$	$-0.7\pm 0.4$	$28.4\pm 2.4$	19.61	7.9	0.85
\insertTableNotes

Table 3. Table 3: Derived Properties of Dwarf Spheroidal Galaxies

Galaxy	$M_{\text{Ks}}$	$\log r_0$	$\log P^{\text{ pre}}$	$\log P^{\text{ obs}}$	$\log {ℳ}_{\text{str}}$	$\log {ℳ}_{\text{gas}}$	$\log {ℳ}_{\text{bar}}$	$\log {ℳ}_{\text{dyn}}$	$v_{\text{esc}}$
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
And I	$-14.36$	2.52	4.96	4.50	7.02	7.30	7.48	8.06	25
And II	$-15.13$	2.78	4.81	4.54	7.32	7.26	7.59	8.24	23
And III	$-12.64$	2.32	4.77	4.00	6.33	7.01	7.10	7.79	23
And V	$-12.32$	2.29	4.80	3.91	6.20	7.03	7.09	7.86	26
And VI	$-14.32$	2.35	5.22	4.65	7.00	7.43	7.57	8.06	30
And VII	$-15.76$	2.71	5.17	4.86	7.58	7.59	7.88	8.47	32
Bootes I	$-9.33$	2.11	3.98	2.89	5.00	6.06	6.09	6.96	11
Carina	$-11.93$	2.23	4.56	3.81	6.04	6.70	6.79	7.40	16
Cetus	$-13.57$	2.46	4.77	4.24	6.70	7.07	7.23	7.82	20
Coma Ber	$-6.52$	1.70	3.84	2.18	3.88	5.52	5.53	6.55	11
CVn I	$-10.61$	2.47	4.13	3.05	5.52	6.56	6.60	7.75	19
CVn II	$-8.51$	1.66	4.27	3.02	4.68	5.90	5.93	6.51	11
Draco	$-11.22$	2.07	4.73	3.69	5.76	6.76	6.80	7.51	22
Fornax	$-15.88$	2.59	5.26	5.03	7.63	7.47	7.85	8.26	29
Leo I	$-14.78$	2.15	5.36	5.03	7.18	7.24	7.51	7.61	23
Leo II	$-12.68$	2.02	4.91	4.33	6.35	6.80	6.93	7.19	16
Leo IV	$-7.97$	1.81	3.84	2.65	4.46	5.63	5.66	6.38	8
NGC 147	$-17.28$	2.54	5.74	5.64	8.19	7.59	8.28	8.48	39
NGC 185	$-17.42$	2.32	6.19	5.92	8.24	8.17	8.51	8.61	59
NGC 205	$-19.04$	2.67	6.35	6.22	8.89	8.43	9.02	9.28	86
Sculptor	$-13.25$	2.14	5.07	4.44	6.57	7.10	7.21	7.59	23
Sextans	$-12.07$	2.59	4.32	3.51	6.10	6.83	6.91	7.91	19
Ursa Min	$-11.28$	2.27	4.57	3.51	5.78	6.80	6.84	7.75	23
\insertTableNotes

Table 4. Table 4: Derived Properties of Dwarf Elliptical Galaxies

Galaxy	$M_{\text{Ks}}$	$\log r_0$	$\log P^{\text{ pre}}$	$\log P^{\text{ obs}}$	$\log {ℳ}_{\text{str}}$	$\log {ℳ}_{\text{dyn}}$
(1)	(2)	(3)	(4)	(5)	(6)	(7)
PGC 32348	$-17.59$	2.63	5.70	5.68	8.31	8.53
VCC 0009	$-20.51$	3.19	5.99	6.29	9.48	9.54
VCC 0033	$-18.84$	2.59	6.13	6.21	8.81	8.76
VCC 0437	$-19.95$	3.18	6.10	6.07	9.25	9.93
VCC 0543	$-19.58$	3.03	6.10	6.07	9.10	9.65
VCC 0634	$-19.99$	3.04	6.11	6.22	9.27	9.56
VCC 0750	$-18.77$	2.90	6.18	5.88	8.78	9.70
VCC 0846	$-17.84$	2.87	5.61	5.54	8.41	8.96
VCC 0917	$-18.39$	2.66	6.13	5.96	8.63	9.10
VCC 1087	$-20.05$	2.97	6.35	6.32	9.29	9.74
VCC 1261	$-20.51$	3.03	6.41	6.44	9.47	9.86
VCC 1355	$-19.31$	3.12	5.69	5.87	8.99	9.26
VCC 1407	$-18.99$	2.74	6.23	6.13	8.87	9.27
VCC 1431	$-19.79$	2.66	6.71	6.53	9.19	9.62
VCC 1453	$-19.74$	3.00	6.18	6.17	9.17	9.62
VCC 1528	$-19.37$	2.68	6.55	6.34	9.02	9.55
VCC 1549	$-19.38$	2.80	6.32	6.23	9.03	9.45
VCC 1826	$-18.29$	2.61	6.05	5.98	8.59	8.83
VCC 1861	$-19.73$	2.93	6.18	6.24	9.16	9.44
VCC 1895	$-18.98$	2.80	6.02	6.07	8.86	9.08
VCC 2083	$-18.32$	2.78	6.00	5.82	8.60	9.21
\insertTableNotes

Equations

$$W_{20}=(8\ln 5{)}^{1/2}{\sigma }_{\text{los}}$$

$$I(r)=\frac{2I_0}{e^{r/r_0}+e^{-r/r_0}}$$

$$m_{\text{sech}}=-2.5\log \left[11.51036q{r}_0^2{I}_0\right]$$

$$M_{\text{I,TRGB}}$$

$$+(0.217\pm 0.020)[(V-I{)}_{\text{TRGB}}-1.6]$$

$$M_{\text{V,RR}}$$

$$+(0.214\pm 0.047)[[Fe/H]-(-1.5)]$$

$$P^{\text{obs}}\equiv |-{\mathcal{M}}_{\text{bar}}/r_0|$$

$${\mathcal{M}}_{\text{bar}}={\mathcal{M}}_{\text{gas}}+{\mathcal{M}}_{\text{str}}$$

$${\mathcal{M}}_{\text{str}}={{\rm Y}}_{\text{str}}{\mathcal{L}}_{\text{sech}}$$

$$\log P^{\text{pre}}$$

$$+(1.134\pm 0.080)(\log W_{20}-1.8)$$

$$+(-0.198\pm 0.018)({\mu }_0^{\text{i=}0}-20)$$

$${{\rm Y}}_{\text{str,Ks}}=0.88\pm 0.20$$

$$P^{\text{pre}}\propto {\left[W_{20}\right]}^{1.1}{\left[I_0^{\text{i=}0}\right]}^{0.5}$$

$${\mu }_0^{\text{i=}0}={\mu }_0-2.5\log q$$

$${\left[\delta \log P^{\text{obs}}\right]}^2$$

$$+{\left[\delta \log q\right]}^2+{\left[0.4\delta {\mu }_0\right]}^2$$

$$\log P^{\text{obs}}(dSph)$$

$$+(1.750\pm 0.133)\left[\log P^{\text{pre}}-5.5\right]$$

$$\left(\log P^{\text{pre}}\le 5.5\right)$$

$$P^{\text{obs}}/P^{\text{pre}}$$

$$=f_{\text{str}}$$

$$=1-f_{\text{gas}}$$

$${\mathcal{M}}_{\text{bar}}={\mathcal{M}}_{\text{str}}/\left[P^{\text{obs}}/P^{\text{pre}}\right]$$

$${\mathcal{M}}_{\text{gas}}$$

$$={\mathcal{M}}_{\text{str}}\left[\frac{1-P^{\text{obs}}/P^{\text{pre}}}{P^{\text{obs}}/P^{\text{pre}}}\right]$$

$${\left[\delta {\mathcal{M}}_{\text{gas}}/{\mathcal{M}}_{\text{bar}}\right]}^2$$

$$+{\left[\delta q/q\right]}^2{\left[2.5c+P^{\text{obs}}/P^{\text{pre}}\right]}^2$$

$$+{\left[\delta {\mu }_0\ln 10\right]}^2{\left[c+0.4P^{\text{obs}}/P^{\text{pre}}\right]}^2$$

$$+{\left[b\delta W_{20}/W_{20}\right]}^2+{\left[s\ln 10\right]}^2$$

$${\left[\delta {\mathcal{M}}_{\text{bar}}/{\mathcal{M}}_{\text{bar}}\right]}^2$$

$$+{\left[2.5c\delta q/q\right]}^2+{\left[(c\ln 10)\delta {\mu }_0\right]}^2$$

$$+{\left[b\delta W_{20}/W_{20}\right]}^2+{\left[s\ln 10\right]}^2$$

$$\delta ({\mathcal{M}}_{\text{gas}}/{\mathcal{M}}_{\text{bar}})=\delta (P^{\text{obs}}/P^{\text{pre}})$$

$${\left[\delta \log (P^{\text{obs}}/P^{\text{pre}})\right]}^2$$

$$+{\left[(1+2.5c)\delta \log q\right]}^2$$

$$+{\left[(0.4+c)\delta {\mu }_0\right]}^2$$

$$+{\left[b\delta \log W_{20}\right]}^2+s^2$$

Full text

black The placement of early-type dwarf galaxies (dSphs and dEs) with respect to the Potential Plane defined by pressure-supported late-type dwarf galaxies (dIs and BCDs) has been determined from surface brightness profiles. dEs and the most luminous dSphs lie on the Plane, suggesting that they emerged from late-type dwarfs that converted most of their gas into stars. However, there is a critical value of the potential at which dSphs start to fall systematically below the Plane, with the deviation growing as the potential becomes shallower. The displacements are attributed to depletion of baryons through gas loss, smaller galaxies having lost proportionately more gas. The critical potential corresponds to an escape velocity of $50\pm 8\,\rm km\,s^{-1}$, which is what is expected for gas with a temperature of $13,000\pm 4,000\,\rm K$, typical of a low-metallicity HII region. This suggests that photoionization was responsible for instigating the loss of gas by galaxies with potentials shallower than the critical value, with evacuation occurring over a few tens of millions of years. Extreme ratios of dynamical to luminous masses observed for the smallest dSphs are an artifact of mass loss. Because the efficiency with which gas was converted into stars was lower for dSphs with shallower potentials, there should be a minimum baryonic mass for a galaxy below which the stellar mass is negligible. Gross extrapolation of the trend of inferred gas masses with stellar masses suggests a value between 500 and $10,000\,\rm\mathcal{M}_{\odot}$.\colorblack The corresponding dynamical mass is below $10^{6}\,\rm\mathcal{M}_{\odot}$.

††pubyear: 2018††pagerange: Denuded Dwarfs Demystified: \colorblack Gas Loss from dSph Progenitors and Implications for the Minimum Mass of Galaxies–Denuded Dwarfs Demystified: \colorblack Gas Loss from dSph Progenitors and Implications for the Minimum Mass of Galaxies

1

Introduction

\color

black

1.1

Definition of “Dwarf”

\color

black The term "dwarf" is loosely used to refer to a galaxy with an elliptical or irregular morphology that has a luminosity that is very much smaller than that of the Milky Way. How much smaller is generally not quantified. Since this paper draws upon a sample of “dwarfs” to discern the origin of differences between different kinds, it is important first to establish a quantitative criterion to pinpoint what “small” means. \colorblack In that regard, it should be noted that throughout this paper distances and quantities derived from them are anchored to the scale of McCall (2014), on which the Virgo Cluster has a distance modulus of 30.98.

Lelli et al. (2014) felt that any criterion defining a dwarf should eliminate gas-rich disk galaxies with bulges. Therefore, they defined a dwarf galaxy to be any galaxy for which the square root of the kinetic energy per unit mass equates to an asymptotic rotational velocity at or below $100\,\rm km\,s^{-1}$. The corresponding limit to the absolute magnitude in $K_{s}$ is about $-21$. By this definition, the most luminous gas-rich dwarfs are rotationally-supported and lie on the Tully-Fisher relation. Morphologically, though, they are spiral galaxies.

\color

black There is another way to define dwarfs energetically, one that is consistent with morphological typing. Figure 1 displays the baryonic Tully-Fisher relation for a sample of dwarf irregular galaxies and blue compact dwarfs (McCall et al., 2012) and a sample of generally more luminous but especially gas-rich galaxies that are rotating (Stark et al., 2009), all converted to the distance scale adopted in this paper (see §3.4).

Line-of-sight velocity dispersions measured for systems believed to be pressure-supported have been converted to equivalent rotational velocities by multiplying by $\sqrt{3}$. The majority of galaxies in each sample have gas fractions that are 40% or greater, so uncertainties in mass-to-light ratios adopted for stars have only a minor effect on baryonic masses. There is a break in the slope of the relationship beginning at an energy equating to a rotational velocity of $\sim 50\,\rm km\,s^{-1}$ (McCall 2012), which for pressure-supported objects is a velocity dispersion of $\sim 30\,\rm km\,s^{-1}$. The corresponding absolute magnitude in $K_{s}$ is about $-18.0$ ($-15.5$ in $V$). The correlation for galaxies with less energy (lower luminosity and mass) is half as steep, and the scatter far exceeds observational errors (McCall, 2012). Notable also is the observation by Lee et al. (2007) that the distribution of H$\alpha$ equivalent widths for galaxies widens by a factor of two at absolute magnitudes fainter than about $-16$ in $V$, implying that star formation becomes regulated differently than in more luminous galaxies. Such galaxies are believed to be predominantly supported by pressure (McCall et al., 2012), and are classified as irregular galaxies, not spirals.

It seems somewhat arbitrary to regard the absence of a bulge as a criterion to define a dwarf because the Tully-Fisher relation is independent of bulge strength. It is more logical to define a dwarf on the basis of dynamical support, especially considering the mesh with morphology. Pressure-supported galaxies are the primary subject of this paper, and the term "dwarf" is used to refer to a galaxy with a velocity dispersion of around $30\,\rm km\,s^{-1}$ or less.

\color

black Although the definition adopted for a dwarf is based upon the properties of gas-rich galaxies, in this paper it is extended to include gas-poor ones. In so doing, many galaxies traditionally regarded as dwarf ellipticals on the basis of their surface brightness profiles Sandage & Binggeli (1984) can no longer be viewed as “true” dwarfs (see, e.g., Toloba et al. 2014). Much of our knowledge about them is based upon studies of the Virgo Cluster, in which galaxies labeled as dEs exhibit diverse properties, including in many cases rotation. However, Ryś et al. (2013) point out that some rotators may be stripped or harassed lenticulars or spirals that retain at least a fraction of the angular momentum of their progenitors. From an evolutionary standpoint, then, they aren’t really dwarfs. Nevertheless, there exist low-luminosity pressure-supported galaxies, such as VCC 1261, that do not qualify as dwarfs based on the criterion laid down above.

\color

black

1.2

Dwarf Nomenclature and Characteristics

\color

black Fundamentally, there are two kinds of pressure-supported dwarfs: gas-rich and gas-poor. \colorblack Gas-rich versions fall into two categories: dwarf irregular galaxies (dIs) and blue compact dwarfs (BCDs). Gas-rich galaxies display surface brightness profiles that are exponential at large radii, but towards their centres profiles of dIs flatten whereas those of \colorblack many BCDs continue to brighten \colorblack (Vaduvescu et al., 2005; Cairós et al., 2015). Gas-poor dwarfs also have been classified into two groups: dwarf spheroidal galaxies (dSphs) and dwarf elliptical galaxies (dEs). The former, which are the least luminous, display surface brightness profiles like those of dIs \colorblack (Kormendy, 1985; Vaduvescu & McCall, 2008). \colorblack The latter, originally recognized in the Virgo cluster, are defined to be objects with surface brightness profiles flatter than a de Vaucouleurs law, which distinguishes them from giant ellipticals (Sandage & Binggeli, 1984). Core profiles often rise steeply (with many being nucleated), but the behaviour at large radii can be close to exponential (Toloba et al., 2014; Janz et al., 2014). The various structural similarities amongst all of these galaxies suggest evolutionary connections, and admit the possibility that dSphs and/or dEs are dIs and/or BCDs that have lost their gas.

It is useful to draw upon terminology describing positioning in the Hubble sequence to distinguish between gas-rich and gas-poor dwarfs. In this paper, dIs and BCDs together will be called late-type dwarfs (LTDs) and dSphs and dEs as a group will be termed early-type dwarfs (ETDs).

Among LTDs, the \colorblack photometric distinction between dIs and BCDs can be ascribed to star formation activity. By regarding the enhancement of surface brightness in the core of a BCD to be merely the consequence of a burst of star formation, Vaduvescu et al. (2005) demonstrated \colorblack through surface photometry in $K_{s}$ that the two kinds of galaxies are structurally indistinguishable.

\color

black Lelli et al. (2014) showed that BCDs for which rotational motions are evident (the most luminous) generally have steeper inner velocity gradients than dIs displaying rotation, suggesting that their starburst is linked to the inner shape of the potential well. However, there are many such galaxies that are morphologically distorted, even with multiple starburst sites, possibly as a consequence of an interaction (Cairós et al., 2015). Nevertheless, Lelli et al. (2014) recognized compact dIs with inner velocity gradients similar to those seen in BCDs, and concluded that BCDs must evolve into such galaxies provided that mass is not redistributed.

The distinction between dSphs and dEs is more complex, but appears to be anchored to mass. Grebel (2001) considered dSphs to be fainter than $M_{V}\sim-14$. \colorblack In keeping with Sandage & Binggeli (1984), she recognized dEs to be fainter than $M_{V}\sim-17$, consistent with what is expected using the modern distance modulus for the Virgo Cluster adopted by McCall (2014). However, an apparent two-magnitude gap between the brightest dSphs and faintest dEs has been shown to be a selection effect. Forbes et al. (2011) identified faint dEs in the Virgo cluster with structural scaling relations extending close to the realm of dSphs. \colorblack As a function of increasing luminosity, the Sersic index for dSphs stays more or less constant around unity and then seamlessly transitions to a slowly rising trend for dEs (Misgeld et al., 2008, 2009; Lieder et al., 2012). \colorblack However, velocity dispersions show that dark matter predominates over baryons in the inner regions of dSphs (Wolf et al., 2010), but that stars make up half of the mass \colorblack within the half-light radius of dEs (Toloba et al., 2014). \colorblack Furthermore, Lieder et al. (2012) presented evidence for a break in the slope of the colour-magnitude relation for dEs in the Virgo cluster at $M_{\textit{V}}\sim-14$, with fainter galaxies uniformly blue and brighter galaxies trending redward with increasing luminosity. They proposed that the break signifies a critical mass required for a galaxy to retain gas, and explained the blue colours of dEs with lower masses to be a consequence of less efficient enrichment. \colorblack Such an explanation is attractive because it simultaneously allows structural trends for dSphs and dEs to join. Perhaps dEs fainter than the break luminosity should be regarded as the dSphs of the cluster. From an evolutionary standpoint then, the logical luminosity in the near-infrared at which to distinguish dSphs from dEs is $M_{\textit{Ks}}\sim-16.5$ (see also Forbes et al. 2011), since the mean $V-K_{s}$ colour of dEs in Virgo is 2.5 (see §3.2.1).

Often the classification “dSph” is applied to ETDs located in the Local Group, whereas “dE” is applied to similar galaxies beyond, most commonly in clusters. The environmental distinction is highlighted in this paper, in part because of the different approaches to the analysis of the two categories of galaxies. All but three of the “dSphs” in the Local Group sample to be discussed satisfy the luminosity condition, too, having absolute magnitudes fainter than $-16.5$ in $K_{s}$.

1.3

Relationship of Early-Type Dwarfs to Late-Type Dwarfs

It has long been known that there exists a Fundamental Plane connecting measurements of radii, velocity dispersions and surface brightnesses of luminous early-type galaxies (Djorgovski & Davis, 1987; Dressler et al., 1987). ETDs connect to it in a two-dimensional Fundamental Manifold (Zaritsky et al., 2006a, b). Tollerud et al. (2011) even showed that \colorblack pressure-supported bodies spanning eight orders of magnitude in luminosity scatter around a Fundamental Curve in mass-radius-luminosity space.

Because they are structurally similar to dIs, it has been proposed that ETDs are LTDs that are deficient in gas (Kormendy, 1985). Further support for this idea came from the recognition of a Fundamental Plane for \colorblack pressure-supported LTDs, which attributes scatter in the relationship between the near-infrared luminosity and internal motions to variations in surface brightness (Vaduvescu et al., 2005). It revealed that the correlation between dynamical and structural properties for dSphs is similar to that for \colorblack pressure-supported dIs and BCDs (Vaduvescu & McCall, 2008). Also, in the Virgo Cluster, structurally compact ETDs display SDSS light profiles similar to the low surface brightness component of BCDs located there, and the more diffuse ETDs have properties akin to irregulars (Meyer et al., 2014). In another near-infrared study, Young et al. (2014) confirmed that underlying structural properties of dIs in the Local Volume overlap with those of ETDs in both the Local Volume and the Virgo Cluster. \colorblack Furthermore, Lelli et al. (2014) showed that rotating ETDs in the Virgo Cluster have inner velocity gradients that correlate with central surface brightnesses similarly to LTDs, and that the gradients themselves are comparable to those of rotating BCDs and compact dIs. However, correlations with luminosity observed \colorblack by Young et al. (2014) for Local Volume galaxies and Virgo Cluster dEs showed significant differences in both slope and scatter, suggestive of environmental regulation. A principal components analysis by Woo et al. (2008) also showed that the relationships between stellar masses \colorblack (derived using stellar population models from observations in $V$), radii, and internal motions for ETDs and LTDs overlap. Since the dispersion in properties along one principal axis was eight or more times that along the other two, Woo et al. (2008) felt that dwarfs might better be regarded as populating a "Fundamental Line" in their parameter space, \colorblack a suggestion previously made by Dekel & Woo (2003) for galaxies with masses between $6\times 10^{5}$ and $3\times 10^{10}\,\rm M_{\odot}$.

Kormendy & Freeman (2016) examined the placement of dIs and dSphs with respect to scaling relations for dark matter in rotationally-supported spiral and irregular galaxies (i.e. disk galaxies with weak or non-existent bulges). Specifically, they focussed on connections among the central density, core radius, velocity dispersion, and absolute magnitude in $B$. From the standpoint of continuity, they contended that the relationships observed for rotating systems should extend into the realm of pressure-supported systems, and sought a path for unification. Particularly revealing was the relationship between the central density of the dark matter halo and the absolute magnitude. Relative to the extrapolated trend defined by the bright galaxies, absolute magnitudes of dIs and dSphs proved to be fainter than expected for their densities by $\sim 3.5\,\rm mag$ and $\sim 4.6\,\rm mag$, respectively. The implication is that the dwarfs have proportionately higher surface mass densities of dark matter relative to baryons compared to \colorblack galaxies that Kormendy & Freeman (2016) regard as rotationally-supported systems. The offsets were attributed to baryon depletion, which could have been a consequence of proportionately less efficient capture of gas in the early universe or proportionately greater loss of gas later on, say by supernovae-driven winds. This finding overlaps that of Tollerud et al. (2011), who argued that the weak correlation of mass with luminosity that they observed for the faintest dSphs was evidence that growth of small galaxies is limited by the shallowness of their potential wells.

With the presumption that trends for rotationally-supported galaxies are extensions of trends for dIs and dSphs, Kormendy & Freeman (2016) utilized the observed offsets to gain insights into relative baryon depletion factors. They concluded that the ratio of the surface density of baryonic matter to dark matter decreases with decreasing luminosity for galaxies fainter than $-18$ in $V$.

Although Kormendy & Freeman (2016) found dSphs to be a magnitude fainter than dIs with the same central density, a comparison of baryonic masses does not easily follow. In $B$, star formation enhances the brightess of dIs with respect to dSphs, possibly explaining much of the difference. Furthermore, the authors did not account for gas, which augments the baryonic total for dIs substantially. Because of the small samples, the comparison could be biased by selection effects, too. Nevertheless, within the scatter, scaling relations for the two types of galaxies largely overlap, verifying at least their structural similarities, especially from the standpoint of dark matter. In fact, Kormendy & Freeman (2016) consider the presence or absence of gas to be a secondary issue.

Using measurements of [Fe/H] in individual stars, Kirby et al. (2013) showed that LTDs and dSphs follow the same relation between metallicity and overall stellar mass. However, the \colorblack metallicity distributions of stars within dSphs display a sharp cutoff at high metallicity, leading Kirby et al. (2013) to conclude that star formation was curtailed, presumably through the loss of gas. Many dEs have distinct nuclei and/or enhanced cores, and the chemical properties of dSphs indicate that much of the star formation occurred in a burst (Richer & McCall, 1995). Thus, ETDs might more appropriately be linked to BCDs. Yet, Weisz et al. (2011) concluded that dwarfs of all types formed most of their stars before $z\sim 1$, and that the mean histories of star formation were comparable up until a few Gyr ago.

ETDs \colorblack (and even LTDs with gas fractions below 0.4 – see Geha et al. 2006b) are generally found in environments where the mean density of matter is high, either \colorblack in the vicinity of giant hosts or in clusters. In this sense, many may be considered to be the visible manifestation of subhaloes. This has led many researchers to suggest that gas deficiencies stem from tidal effects (Mayer et al., 2001) or ram-pressure stripping (Einasto et al., 1974; Lin & Faber, 1983). Indeed, star formation histories elucidated from studies of resolved stellar populations suggest that gas loss was stimulated by an external mechanism (Weisz et al., 2011). Furthermore, there are known to be rotationally-supported dE galaxies in the Virgo Cluster that fit the Tully-Fisher relation, suggesting that they are stripped versions of rotating star-forming galaxies (Toloba et al., 2011). However, measurements of proper motions show that some dSphs around the Milky Way, such as Fornax and Leo I, are on orbits with pericentres substantially greater than those of the LMC and the SMC (Lux et al., 2010; Sohn et al., 2013), both of which still host a lot of gas. Furthermore, no neutral hydrogen has been found around dSphs in the Local Group (Bailin & Ford, 2007; Spekkens et al., 2014). Although it is clear that environment plays a role in distinguishing the gaseous content of LTDs from ETDs, stripping need not be the only way. Conceivably, the conditions under which ETDs developed led to more rapid star formation, in which case gas loss could have been promoted through faster consumption and/or more vigourous winds (Dekel & Silk, 1986; Dekel & Woo, 2003; Robles-Valdez et al., 2016).

\color

black Based upon their internal motions, dSphs have dynamical masses that are extremely large compared to their luminosities (Wolf et al., 2010). In fact, mass-to-light ratios for ultra-faint dSphs are larger than observed for any other kind of galaxy, including ultra-diffuse galaxies (Wolf et al., 2010; van Dokkum et al., 2016; Toloba et al., 2018). At first glance, this distinction might argue against a connection with LTDs. However, dynamical mass-to-light ratios are misleading, because the light only accounts for baryons in stellar form. What really needs to be examined is the ratio of the dynamical mass relative to the total baryonic mass prior to gas loss. If the stellar mass in a dSph today represents a small fraction of the baryonic mass originally present, then a large mass-to-light ratio would be an artifact of inefficient conversion of gas into stars.

1.4

A Path Forward: The Potential Plane

The Fundamental Plane for LTDs suffers from significant scatter over and above what can be attributed to observational errors (Vaduvescu & McCall, 2008). McCall et al. (2012) surmised that if LTDs were virialized, there might be an observable relationship between the kinetic energy per unit mass conveyed by the velocity dispersion (as conveyed by the HI line width) and the potential energy per unit mass indicated by baryons, if the baryonic mass scaled with the mass of dark matter. This is the precept of the Fundamental Manifold envisioned by Zaritsky et al. (2011). Indeed, a strong relationship was found, although not that expected for virialized bodies. The potential as judged from the total baryonic mass correlates with a linear combination of the velocity dispersion and near-infrared surface brightness (corrected for projection), so tightly in fact that scatter can be entirely attributed to measurement uncertainties. This relationship is referred to as the “Potential Plane”.

By virtue of being founded upon baryonic masses, the Potential Plane opens up a possible avenue for recognizing how much gas ETDs once had. If it is hypothesized that ETDs were once dIs or BCDs and that dark matter dominates the mass of all dwarfs, then the primary visible effect of expelling gas would be a reduction in the baryonic mass. Thus, ETDs would be expected to display baryonic potentials systematically below those of LTDs with similar structural and dynamical properties. How much gas they lost could be gauged from the offset. The constraint would be invaluable for determining how the galaxies evolved prior to losing their gas and how much of an impact they have had upon the intergalactic medium.

In this paper, light profiles of ETDs with a wide range of luminosities are modeled in an identical way to LTDs in order to identify the placement of the galaxies with respect to the Potential Plane defined by LTDs. \colorblack Unlike Lelli et al. (2014), the primary focus is on faint dwarfs, i.e., those believed to be predominantly supported by pressure, and the intent is to gain insights into why galaxies that are so structurally similar have come to have such different gaseous properties. Unlike Kormendy & Freeman (2016), the paper emphasizes photometric comparisons in $K_{s}$, thereby overcoming any biases that might come from variations in star formation activity in LTDs. In comparing baryonic properties, gas is accommodated. Furthermore, the paper extends the analysis to dEs, i.e. more luminous gas-poor dwarfs than are typically found locally.

In §2, the sample of ETDs to be studied is constructed. In §3, surface brightness profiles for sample galaxies are fitted in the same way as for LTDs. In addition, distances and absolute magnitudes for sample galaxies are evaluated homogeneously. In §4, stellar masses are estimated, and the location of ETDs with respect to the Potential Plane is pinpointed. Implications for evolution follow in §5, where the mass of gas lost by dSphs is quantified, the effect of gas loss on dynamical mass-to-light ratios is analyzed, and the energy at which ETDs diverge from the Potential Plane is utilized to constrain the cause of gas loss. As well, an estimate for the minimum baryonic mass of a galaxy is extrapolated. \colorblack Results are discussed in the context of past research in §6, and conclusions are presented in §7.

2

Sample

In order to locate an ETD with respect to the Potential Plane, it is necessary to quantify its surface brightness, size, luminosity, and internal motions. Thus, candidates for study were restricted to those galaxies for which both the surface brightness profile and the velocity dispersion of stars \colorblack were published, and for which the distance could be reliably determined from published constraints \colorblack (such as the tip of the red giant branch, RR Lyrae stars, or membership in a galaxy group or cluster).

In all, 23 dSphs were identified for study, of which five are classed as ultra-\colorblackfaint dwarfs. \colorblack The galaxies are identified in Table LABEL:tab_obsdsph. \colorblack None have a velocity dispersion more than one sigma higher than $30\,\rm km\,s^{-1}$. Extant data on stellar constituents enables the determination of distances from either the tip of the red giant branch (TRGB) or RR Lyrae stars (see §3.4). The three most luminous galaxies, namely NGC 147, 185, and 205, might be considered to be dEs on the basis of their \colorblack absolute magnitudes and light profiles (see §1.2), but they are grouped together with fainter dSphs because of their location in the Local Group and the nature of the data available for them. \colorblack All of them exhibit rotation (Geha et al., 2010), but the associated energy in all cases is less than 40% of that in random motions.

Given the difficulty of determining accurate distances individually to unresolved dwarfs, it was advantageous to restrict the dE sample to clusters or groups. Recently, high-quality velocity dispersions and Sersic models of deep $H$-band surface photometry were presented for a large collection of dEs in the Virgo Cluster (Toloba et al., 2014; Janz et al., 2014), so they are the primary focus of attention here. The simplest objects, namely those classified as dE(N) and dE(nN) (nucleated and non-nucleated featureless dwarfs, respectively; see Lisker et al. 2007) were selected for study. The former are distributed like E/S0 galaxies in Virgo, while the latter appear to be part of the unrelaxed population of spirals and star-forming dwarfs (Lisker et al., 2007).

Of the dEs selected, the faintest are substantially brighter than the bulk of the dSphs. To extend the sample to lower luminosities, three pressure-supported dEs studied by Forbes et al. (2011) were added (PGC 032348, VCC 846, and VCC 1826). \colorblack All are nucleated (Trentham & Hodgkin, 2002; Forbes et al., 2011). Two of these are in the Virgo Cluster, while the third is a member of the Leo I Group (Paturel et al., 2003).

\color

black In all, 21 dE galaxies were selected for study, four of which are non-nucleated. They are identified in Table LABEL:tab_obsde. Nine have a velocity dispersion exceeding $30\,\rm km\ s^{-1}$ by more than one sigma. \colorblack For 16 of the 18 galaxies for which rotational motions have been constrained (Toloba et al., 2014), the energy exhibited by rotation is only 20% or less of that associated with random motions, and in that sense the dEs can be regarded as predominantly supported by pressure. For the other two, the energy in rotation is 50% of that in random motions.

3

Measurements

3.1 Kinematics

With the exception of NGC 147, 185, and 205, for which internal motions have been gauged from spectroscopy of integrated light, velocity dispersions for sample dSphs come from measurements of the radial velocities of individual stars. Adopted values \colorblack and uncertainties along with sources are provided in Table LABEL:tab_obsdsph. For dEs, velocity dispersions come exclusively from \colorblack long-slit spectroscopy of the integrated light (Toloba et al., 2014; Forbes et al., 2011). \colorblackRegardless of technique, measurements are reflective of motions within the half-light radius, which is where velocity dispersion profiles are relatively flat. Adopted values of velocity dispersions \colorblack and their uncertainties and sources are provided in Table LABEL:tab_obsde.

\color

black Because two kinds of measurements of velocity dispersions are employed, it is important to be aware of factors that might bias comparisons. First, mass segregation (if it exists) should not be a factor because both discrete and integrated measurements preferentially sample the giant population. Stellar binarity inflates velocity dispersions measured by any method, but probably by less than 30% for galaxies with velocity dispersions between 4 and $10\,\rm km\,s^{-1}$ (Minor et al., 2010). Although it is conceivable that dEs host central black holes whereas dSphs do not, measurements of velocity dispersions were made in such a way that they are unlikely to be inflated significantly by associated dynamical effects. For globular clusters, measurements of velocity dispersions derived from integrated light can be biased by “shot noise” from bright giants if a few contribute a significant fraction of the light in the sampling aperture (Dubath et al., 1997; Lanzoni et al., 2013; Lützgendorf et al., 2015). That is not expected to be a problem here because the galaxies for which there are such measurements are generally unresolved, so sampling apertures encompass many more stars (Dubath et al., 1997). Lützgendorf et al. (2015) showed that in the extremely dense core of the massive globular cluster NGC 6388 discrete measurements of velocities from observations acquired with adaptive optics are biased towards the mean cluster velocity as a result of contamination by the wings of the point spread function for other stars and by the unresolved background, thereby lowering the velocity dispersion. Velocity dispersions adopted here for dwarf spheroidal galaxies were not measured using adaptive optics, and in any case such galaxies are nowhere near as concentrated as NGC 6388. What matters most is that velocity dispersions derived by integrated and discrete measurements agree in globular cluster fields where the aforementioned problems are minimal or appropriately corrected (Lützgendorf et al., 2015).

In establishing the Potential Plane, motions in LTDs were quantified by \colorblack $W_{20}$, the width of the HI line profile at 20% of the peak. HI line profiles for most of the galaxies in the LTD sample are Gaussian (see, for example, Huchtmeier et al. 2003). Therefore, to examine the placement of ETDs with respect to the Potential Plane, a Gaussian velocity distribution was assumed \colorblack in order to estimate $W_{20}$ from the measured line-of-sight stellar velocity dispersion $\sigma_{\textit{los}}$ \colorblack (Verheijen & Sancisi, 2001):

[TABLE]

\color

black For three pressure-supported LTDs (Leo A, DDO 210, and Sag DIG), Kirby et al. (2017) compared stellar velocity dispersions determined from spectroscopy of individual stars to intensity-weighted velocity dispersions derived from HI maps. Considering uncertainties, they concluded that the two methods deliver consistent results. Thus, it is not expected that positioning of ETDs with respect to the Potential Plane of LTDs would be biased significantly by \colorblack the use of a different tracer of kinematics.

3.2 Photometry

3.2.1 Dwarf Spheroidal Galaxies

In order to reduce the impact of young stars on light profiles, the Potential Plane for LTDs was defined using surface photometry in the $K_{s}$ band (Vaduvescu et al., 2005). However, in the dSph sample, useful images in this band were available for only three galaxies (NGC 147, NGC 185 and NGC 205). For the rest, it was necessary to fit data in $V$ and transform photometric parameters to $K_{s}$. Fortunately, dSphs are not forming stars, so optical profiles should resemble those in the near-infrared.

To estimate $V-K_{s}$, Goldmine data for 40 dEs in the Virgo cluster (Gavazzi et al., 2003) were examined. Goldmine $K$ magnitudes were transformed from the ARNICA to the 2MASS system using equations provided by Cutri et al. (2008), and both $V$ and $K$ magnitudes were corrected for extinction and redshift using the York Extinction Solver (see §3.3). The 32 galaxies with absolute magnitudes fainter than $-17$ in $V$ display no clear trend of $V-K_{s}$ with luminosity. For them, the mean reddening-free $V-K_{s}$ colour is $2.49\pm 0.27$. This value was adopted to convert dSph photometry in $V$ to $K_{s}$. The quoted uncertainty is the standard deviation, which is relevant if colour variations are exclusively cosmic in origin. However, some of the dispersion must be due to observational errors, so for any single galaxy it is best to regard any error estimate stemming from it as an upper limit.

For many galaxies, surface brightness profiles have been conveyed via star counts (Irwin & Hatzidimitriou, 1995; Muñoz et al., 2010; Okamoto et al., 2012). To convert $\hbox{counts}/\hbox{arcmin}^{2}$ to $\hbox{mag}/\hbox{arcsec}^{2}$, the published central surface brightness or apparent total magnitude (with extinction added) was compared with the counts expected at the centre or overall from the published fit of a King law (King, 1962) to the stellar densities.

\color

black The Sersic function (Sérsic, 1963) has proved to be attractive for modeling light profiles of dwarfs because the extra degree of freedom provided by the Sersic index offers the means to fit a wide range of shapes. For example, Young et al. (2014) chose to model dI profiles with a Sersic function, mainly because of its success in modeling dEs, and partly because of interest in studying variations in the Sersic index itself. However, this freedom has the potential to obscure underlying simplicity. BCDs in the field are a case in point. Their light profiles are typically very different from those of dIs in their cores but similar at large radii. Rather than fitting the two kinds of galaxies with different Sersic functions, it is simpler and perhaps more logical to regard a BCD as a dI with a starburst on top of it. In $K_{s}$, the surface brightness profile of the diffuse component of a dI is approximated well by a hyperbolic secant function (Vaduvescu et al., 2005). Modeling of the light profile of a BCD can be accomplished by superimposing a Gaussian component on a sech profile (Vaduvescu et al., 2006), which accounts for the burst of star formation. Extracting underlying structural parameters accordingly, McCall et al. (2012) showed that the two kinds of galaxies are indistinguishable in the Potential Plane (although the relationship between BCDs and dIs in clusters may be more complex: see Meyer et al. 2014).

Except at their extremities, surface brightness profiles for dSphs are morphologically similar to those of dIs and BCDs. \colorblack In this paper, rather than utilizing a King law, Sersic function, or Gaussian (Kormendy & Freeman, 2016), they have been modeled in the same way, \colorblack partly because they may be related to LTDs, but especially to minimize bias in the comparison of fitted parameters. Thus, the surface brightness $I$ as a function of distance $r$ along the major axis was represented by

[TABLE]

where $I_{0}$ is the central surface brightness of the sech model and $r_{0}$ is the scale length. To fit the profiles of Andromeda II and VII, a Gaussian component was added. To fit the steeper core profiles of NGC 147, NGC 185, and NGC 205, an exponential component had to be added instead of a Gaussian. \colorblack To ensure that each fit was not skewed by anomalies in surface brightness in the core or periphery, inner and/or outer boundaries were set as necessary by eye.

Fits to surface brightness profiles are shown in Figure 2, and the parameters of the fits are given in Table LABEL:tab_obsdsph. \colorblack Because of the disparate ways in which data were presented in the literature, indicative uncertainties only are provided (in the table notes). Surface brightnesses in $K_{s}$ for galaxies observed in $V$ typically have errors of $0.4\,\rm mag$, $0.3\,\rm mag$ of which being due to background subtraction and $0.27\,\rm mag$ to the conversion from $V$ to $K_{s}$. In comparison, photometric zero-point errors ($0.02\,\rm mag$) and fitting errors ($0.02\,\rm mag$) are inconsequential. For the three galaxies observed in $K_{s}$, the uncertainty in the central surface brightness is estimated to be $0.15\,\rm mag$. Based upon experience with fitting, errors in scale lengths are typically 5%.

Generally, profiles are fitted well by the models. At radii beyond the exponential zone, dSphs often show an enhancement in surface brightness. \colorblack For some, this may be due to underestimating the background level. However, for those objects whose surface brightnesses steadily grow fainter, albeit at a slower rate, rather than leveling off (e.g., And V, Cetus, CVn II, and Sculptor), the enhancement may be a result of tides (Irwin & Hatzidimitriou, 1995). The only galaxy with a dubious fit is And II, which has been shown to be undergoing a merger (Amorisco et al., 2014). In the centre of Sculptor there is a poorly-sampled spike in surface brightness analogous to what is seen in NGC 147, 185, and 205. And II is excluded from analyses of early-type dwarfs below, although it is marked in all figures.

To establish luminosities and masses of dSphs directly comparable to those of the LTDs used to define the Potential Plane, the apparent “sech magnitude” $m_{\textit{sech}}$ was computed by integrating the sech model out to infinity:

[TABLE]

Here, $q$ is the axis ratio, i.e., the ratio of the length of the semi-minor to semi-major axis of the isophotes. As for LTDs, any core component was ignored. \colorblack Adopted axis ratios and sources are listed in Table LABEL:tab_obsdsph. Uncertainties in axis ratios are estimated to be 10%. Propagating all sources of error, uncertainties in the $K_{s}$ magnitudes are typically $0.43\,\rm mag$ if they originate from observations in $V$ and $0.21\,\rm mag$ otherwise (predominantly driven by the errors in the surface brightnesses).

3.2.2 Dwarf Elliptical Galaxies

The distinction between the surface brightness profiles of dSphs and dEs is reminiscent of the difference between the profiles of dIs and BCDs. For both kinds of galaxies, the surface brightness outside of the core declines more or less exponentially with radius (for the dEs studied here, Sersic indices range from 0.8 to 1.7, with an average of 1.1). It is hypothesized here that the framework of a dE is similar to that of a dSph, and that the foundational surface brightness profile resembles that described by a sech function. Any excess in surface brightness towards the centre is regarded as an attribute which is not fundamental, but rather more like the excess seen in a BCD. Even if this is not correct, it remains important that any comparison of dEs with dSphs be founded upon characterizations established in a like manner.

The sech parameters for each dE were derived by matching both the slope and level of the surface brightness defined by a published Sersic model at a radius of 2.5 sech scale lengths (on average, 1.7 times the half-light radii measured by Janz et al. (2014)). This radius is the minimum out to which a profile needs to be sampled to determine sech parameters robustly (McCall et al., 2012).

Janz et al. (2014) carried out photometry of Virgo dEs in 2MASS $H$, and published Sersic indices, effective radii, axis ratios, and integrated magnitudes for Sersic models. For galaxies with multi-component fits, the Sersic profile for the global/outer component was adopted to establish sech parameters. Forbes et al. (2011) employed photometry in SDSS $r$ for Virgo Cluster galaxies (Janz & Lisker, 2008) and their own observations in $B$ and $R$ for the galaxy in the Leo I Group. Sech parameters for the Virgo galaxies were derived from Sersic indices, effective semi-major axis lengths, axis ratios, and model magnitudes that were kindly conveyed by J. Janz (2015, personal communication). For the Leo galaxy, Forbes et al. (2011) fitted a Sersic model, but only the Sersic index and scale length were published. To set the surface brightness scale in $r$, the integrated Sersic magnitude in $r$ was estimated from the Petrosian magnitude published in SDSS DR12 (Graham et al., 2005).

With sech parameters established from the Sersic profiles at 2.5 sech scale lengths, resulting sech profiles deviate on average from the Janz et al. (2014) profiles by only $0.14\,\rm mag$ at 5 sech scale lengths. In the centres, the profiles differ on average by $1.0\,\rm mag$, mostly because of the depression of the sech function (by $0.7\,\rm mag$ with respect to a pure exponential). Despite the different core behaviours, integrated sech magnitudes deviate on average from the integrated Sersic magnitudes by only $0.14\,\rm mag$, all but three being fainter.

For all but one of the dEs in the sample from Janz et al. (2014), an estimate of $H-K$ in the ARNICA system was obtainable from the GoldMine database (Gavazzi et al., 2003). To estimate surface brightnesses and integrated sech magnitudes in $K_{s}$, the reddened colours were converted to the 2MASS system using the transformation equation provided by Cutri et al. (2008). For the remaining galaxy, the reddening-free $H-K_{s}$ colour in the 2MASS system was adopted to be $0.16\pm 0.07$, which is the average for the other dEs. For every galaxy studied by Forbes et al. (2011), the sech magnitude in $K_{s}$ was estimated by approximating the reddening-free $r-K_{s}$ colour to be $2.37\pm 0.21$, which is the mean for the dEs from the Janz et al. (2014) sample as evidenced by their SDSS Petrosian magnitudes in $r$.

Parameters of the sech models for dEs are summarized in Table LABEL:tab_obsde. \colorblack Estimated uncertainties are given in the table notes. By propagating errors, uncertainties in apparent sech magnitudes are judged to be $0.26\,\rm mag$ typically.

3.3

Extinction Corrections

Magnitudes and colours of galaxies and the stars within them were corrected for extinction using the York Extinction Solver (YES) after first removing any corrections included in published values. In estimating the extinction of a target, YES properly accounts for the effective wavelength shifts plaguing observations through broad-band filters by accounting for the spectral energy distribution (SED) of both the probe of extinction and the target (McCall, 2004).

For each ETD, the optical depth due to dust in the Milky Way was computed from $E(B-V)$ colour excesses mapped by Schlegel et al. (1998), which are based upon colours of elliptical galaxies corrected to a redshift of zero. Adopted colour excesses and optical depths at $1\,\rm\mu m$ are listed in Tables LABEL:tab_obsdsph and LABEL:tab_obsde. The extinctions and K-corrections in $V$, $r$, and $K_{s}$ were evaluated from the optical depth and the redshift using the spectral energy distribution of an elliptical galaxy (McCall, 2004). Surface brightnesses were also corrected for redshift dimming. The extinction in $I$ and reddening $E(V-I)$ of stars at the TRGB were evaluated using the spectral energy distribution of an M0 giant. The extinction in $V$ of RR Lyrae stars was judged using the spectral energy distribution of an F0 giant (Smith, 1995). SEDs for the stars were adopted from the library of Pickles (1998).

3.4

Distances and Absolute Magnitudes

Distances to dSphs were determined from the TRGB or RR Lyrae stars. RR Lyrae distances were adopted for galaxies lacking a sufficient number of stars to locate the TRGB reliably (Draco, Ursa Minor, Cetus, Canes Venatici II, Leo IV, Canes Venatici I, Bootes I, and Coma Berenices). TRGB distances were adopted for the rest because they don’t explicitly depend on [Fe/H], the measurement of which is very difficult.

Just as in the analysis of LTDs by McCall et al. (2012), the absolute magnitude in $I$ of the TRGB, $M_{\textit{I,TRGB}}$, was computed from

[TABLE]

where $(V-I)_{\textit{TRGB}}$ is the colour of TRGB stars corrected for extinction and redshift. The slope is from Rizzi et al. (2007b). Following McCall (2014), the zero-point is set by the maser distance to NGC 4258, which was adopted to be $7.6\,\rm Mpc$ (Humphreys et al., 2013). The apparent magnitude in $I$ for the TRGB was extracted for each galaxy from the literature. When the corresponding $V-I$ colour was not explicitly given, it was judged from a colour-magnitude diagram.

The absolute magnitude in $V$ of the RR Lyrae stars, $M_{\textit{{V,RR}}}$, was computed from

[TABLE]

where [Fe/H] is the iron abundance relative to the solar value. The slope is from Gratton et al. (2004). To ensure that distances aligned with the TRGB scale, the zero-point was determined by comparing TRGB and RR Lyrae distances to And I, II, and III, the only three galaxies in the sample for which data could be guaranteed to be homogeneous (the red giant tip was measured by McConnachie et al. 2004 and McConnachie et al. 2005, RR Lyrae mean magnitudes were measured by Pritzl et al. 2004 and Pritzl et al. 2005, and iron abundances were measured by Vargas et al. 2014). The adopted value is $0.08\,\rm mag$ brighter than the widely-used calibration of Cacciari & Clementini (2003).

Adopted distance moduli for dSphs are listed in Table LABEL:tab_obsdsph. Typically, the error in TRGB distances is $0.04\,\rm mag$ (excluding the zero-point uncertainty, which is $0.11\,\rm mag$). Distances derived from RR Lyrae stars are somewhat more uncertain, $0.06\,\rm mag$, because they require explicit knowledge of the metallicity.

For each dE, the distance was assigned to be that of the host cluster or group. The distance modulus for the Virgo Cluster was adopted to be $30.98\pm 0.12$. This is the weighted mean of the Cepheid moduli derived for NGC 4535, 4536, 4548, 4321, and 4496A computed using the methodology of McCall (2014), converted to the maser scale described above. The uncertainty here is the dispersion expected as a result of depth, given that the angular radius of the cluster is $6^{\circ}$. The distance modulus for the Leo Group was adopted to be $30.11\pm 0.07$ on the maser scale, which is the mean of the distance moduli for NGC 3351, 3368, 3377, 3379, and 3384 published by McCall (2014).

Absolute magnitudes of ETDs in $K_{s}$ were calculated from distance moduli and the apparent sech magnitudes \colorblack computed from equation 3. They are given for dSphs in Table LABEL:tab_comdsph and dEs in Table LABEL:tab_comde. \colorblack Uncertainties were estimated by propagating the errors in the apparent sech magnitudes, extinctions, and distances. Typical values for dSphs are $0.43\,\rm mag$ for those observed in $V$ and $0.22\,\rm mag$ for those observed in $K_{s}$. They are $0.34\,\rm mag$ for dEs.

4

Analysis

4.1

Establishment of the Potential Plane

The Potential Plane for LTDs is the relationship between the potential, the HI line width, and the central surface brightness of the sech model. The index of the potential, i.e., referred to here as the observed potential $P^{\textit{\,obs}}$, is the ratio of the baryonic mass $\mathcal{M}_{\textit{bar}}$ to the scale length $r_{0}$ of the sech component:

[TABLE]

The baryonic mass is given by the sum of the gaseous and stellar masses,

[TABLE]

where the mass of stars is estimated from the stellar mass-to-light ratio $\Upsilon_{\textit{str}}$ and the luminosity $\mathcal{L}_{\textit{sech}}$ of the sech component:

[TABLE]

McCall et al. (2012) found that a prediction $P^{\textit{\,pre}}$ for $P^{\textit{\,obs}}$ that is accurate to within observational errors for dIs and BCDs alike with absolute magnitudes fainter than $-20$ in $K_{s}$ is given by

[TABLE]

with

[TABLE]

given $M_{\textit{Ks},\odot}=3.315$ (Holmberg et al., 2006; Flynn et al., 2006). Here, $\mu_{0}^{\textit{i=}0}$ is the central surface brightness in $K_{s}$ of the sech model at zero tilt (judged from the axis ratio by assuming an oblate spheroidal geometry) and $W_{20}$ is the HI line width as observed, i.e., with no correction for tilt. The units of $P^{\textit{\,pre}}$, $W_{20}$, and $\mu_{0}^{\textit{i=}0}$ are $\mathcal{M}_{\odot}\,pc^{-1}$, $\rm km\,s^{-1}$, and $\rm mag\,arcsec^{-2}$, respectively. Note that the zero-point is different from that in McCall et al. (2012); it has been updated to account for an improved distance to NGC 4258 (Humphreys et al., 2013).

The solution for $\Upsilon_{\textit{str,Ks}}$ was made possible by its impact on the balance between gaseous and stellar masses. Comparable estimates have been obtained in $H$ from population syntheses (see Young et al. 2014 and references therein). There was little improvement in the fit if line widths were allowed to vary with tilt, prompting McCall et al. (2012) to conclude that motions within the mass range sampled are primarily disordered and isotropic.

The relationship between the observed and predicted potentials for LTDs is displayed in Figure 3.

The 48 galaxies that McCall et al. (2012) employed in the fitting, which are marked by open black circles, delineate the Potential Plane from the side (the black line). The dispersion ($0.10\,\rm dex$) is completely ascribable to observational errors (McCall et al., 2012). \colorblack Although five of the galaxies have velocity dispersions exceeding $30\,\rm km\,s^{-1}$ by more than one sigma, they do not deviate from the trend.

Equation 4.1 reveals that for LTDs,

[TABLE]

where $I_{0}^{\textit{i=}0}$ is the central surface brightness in linear units. For a virialized system, $P^{\textit{\,pre}}$ should vary as $\left[W_{20}\right]^{2}$. McCall et al. (2012) suggested that the observed behaviour might be an artifact of varying mass-to-light ratios, compensation for which may be coming through the surface brightness term. However, uncomfortably large fluctuations in the mass-to-light ratios are required to develop the virial expectation. Regardless of its origin, the Potential Plane offers a baseline against which other kinds of dwarfs can be compared.

4.2

Placement of ETDs Relative to the Potential Plane

The main objective of this paper is to establish the positioning of ETDs with respect to the potential plane. ETDs lack gas, so $\mathcal{M}_{\textit{bar}}$ is determined by $\mathcal{L}_{\textit{sech}}$ alone. Hence, $P^{\textit{\,obs}}$ is a proxy for $r_{0}\,I_{0}^{\textit{i=}0}$, so the relationship between $P^{\textit{\,obs}}$ and $P^{\textit{\,pre}}$ conveys, in effect, the relationship between $r_{0}$ and $\left[W_{20}\right]^{1.1}\left[I_{0}^{\textit{i=}0}\right]^{-0.5}$.

To estimate $\log P^{\textit{\,pre}}$ for ETDs, central surface brightnesses were corrected to face-on by adopting an oblate spheroidal geometry, which is suggested by the distribution of axis ratios of dEs (Lisker et al., 2007; Sánchez-Janssen et al., 2016). Just as for LTDs, then,

[TABLE]

where \colorblack $\mu_{0}$ is the central surface brightness of the sech model in magnitude units and $q$ is the ratio of the semi-minor to semi-major axis of the isophotes.

Since ETDs lack gas, it is necessary to utilize the line-of-sight velocity dispersion of stars to gauge $W_{20}$. Comparisons with gas-based values for LTDs are valid, though, because dark matter is the predominant constituent of both dSphs and dIs (Kormendy & Freeman, 2016), its gravitational field establishing the motions of baryons of all forms \colorblack (see Kirby et al. 2017). As for LTDs, motions in ETDs were assumed to be predominantly random and isotropic\colorblack. Even rotating ETDs (which tend to be the most luminous) are pressure supported in their inner regions (Lelli et al., 2014). Thus, values of $W_{20}$ determined from equation 1 were not adjusted for $q$. The resulting values of $\log P^{\textit{\,pre}}$ for dSphs and dEs are provided in Tables LABEL:tab_comdsph and LABEL:tab_comde, respectively. \colorblack Typical uncertainties arising from the errors in measurements are given in the table notes. They were estimated by propagating the errors in the velocity dispersions, surface brightnesses, and axis ratios through equation 4.1.

Equations 6, 7, and 8 were utilized to estimate $P^{\textit{\,obs}}$ for ETDs for an appropriate choice of $\Upsilon_{\textit{str,Ks}}$ and with the approximation that $M_{\textit{gas}}$ is zero. One way to evaluate $\Upsilon_{\textit{str,Ks}}$ and its range of variation is to use stellar population models. However, an initial mass function must be assumed. Martin et al. (2008) derived mass-to-light ratios this way for faint satellites of the Milky Way. For 11 dSphs with $\mathcal{L}_{V}/\mathcal{L}_{V,\odot}$ exceeding 1000 ($-2.7<\mathcal{M}_{V}<-8.8$), the average value for $\Upsilon_{\textit{str,Ks}}$ in solar units is $0.99\pm 0.17$ using the Salpeter IMF (Salpeter, 1955) and $0.50\pm 0.07$ using the Kroupa IMF (Kroupa et al., 1993) based upon the $V-K_{s}$ colour adopted in §3.2.1. Considering uncertainties, the range of variation for either choice of IMF is small. Thus, it is approximated in this paper that $\Upsilon_{\textit{str,Ks}}$ for ETDs has a single value.

There is cause to believe that the actual value \colorblack of $\Upsilon_{\textit{str,Ks}}$ should be similar to that of LTDs. The resolved population of nearby dIs (i.e., stars with $M_{\textit{Ks}}\leq-7.5$) typically contributes only 5% of the light in $K_{s}$, with the bulk of the flux coming from stars more than $4\,\rm Gyr$ old (Vaduvescu et al., 2005). Furthermore, the luminosity of an LTD inferred from the sech model must be representative of the old population, because there is no offset between BCDs and dIs in the Potential Plane (McCall et al., 2012). Besides requiring no assumption about the IMF, the value for $\Upsilon_{\textit{str,Ks}}$ derived for LTDs (equation 10) has the benefit of being tied directly to the observations establishing the Potential Plane. Also, it is consistent with the mean value derived for dSphs by Martin et al. (2008) using the Salpeter IMF. Therefore, it is adopted for all ETDs.

Computed stellar masses and quantities which follow from them are listed for dSphs in Table LABEL:tab_comdsph and dEs in Table LABEL:tab_comde. \colorblack Typical uncertainties, which are listed in the table notes, were computed by propagating random errors in the measurements defining them. Systematic errors common to both LTDs and ETDs, like those for the mass-to-light ratio and the zero-point of the distance scale, were ignored. So, the error in $\mathcal{M}_{\textit{str}}$ was derived from the error in the absolute magnitude alone. The error in $P^{\textit{\,obs}}$ was derived from the errors in the surface brightness, axis ratio, angular scale length, and distance rather than the errors in the stellar mass and physical scale length in order to correctly account for the partial cancellation of the distance dependencies of the latter two quantities:

[TABLE]

Figure 3 shows where the ETDs lie with respect to the Potential Plane \colorblack defined by LTDs. Remarkably, dEs fall right on the plane. The brightest dSphs (NGC 147, 185, and 205), all of whose absolute magnitudes are brighter than $-16.5$ in $K_{s}$, fall near the Plane, too, in the realm of the dEs. However, below $\log P^{\textit{\,pre}}=5.5$, dSphs form a sequence that diverges below the Plane and its extrapolation. Taking into account uncertainties in both the abscissae and ordinates, the relationship defined by the dSphs with absolute magnitudes fainter than $-16.5$ (and excluding And II) is given by

[TABLE]

The root mean square deviation is only $0.17\,\rm dex$, comparable to observational errors.

Because dEs and the most massive dSphs lie so close to the Potential Plane, it is natural to posit that all dSphs started there. Then, the positioning of dSphs that deviate is most easily explained if they are simply LTDs that have lost gas. The trend observed in Figure 3 implies that proportionately more gas was lost by dSphs with shallower potentials, a result that could not be extracted from the two-parameter correlations studied by Kormendy & Freeman (2016).

\color

black Observed and predicted potentials for ETDs are coupled because the central surface brightness was employed to establish apparent magnitudes. In Figure 3, the trajectory that any of those galaxies would take if their surface brightness were fainter by $2\,\rm mag$ is marked with an arrow. The slope is close to that of the locus of dSphs with $\log P^{\textit{\,pre}}\leq 5.5$. This means that errors in surface brightness are not responsible for the dispersion. It also means that any aspects of evolution that solely affect the surface brightnesses of the galaxies are indiscernible.

Unfortunately, LTDs in the sample do not extend to potentials as shallow as observed for dSphs. It is conceivable that such galaxies don’t exist if the point where dSphs begin to deviate represents an energy threshold below which LTDs begin to have difficulty retaining their gas. Thus, inferences made about the smallest dSphs are at the mercy of extrapolation. Nevertheless, the trend for LTDs is so tightly defined (as evidenced by the $3\sigma$ confidence boundaries displayed in Figure 3) that any downturn within reason would be unlikely to eliminate the discrepancy with dSphs altogether. In §5.3, the location of the break-point is analyzed to gain insights into how dSphs lost their gas.

Figure 4 displays the correlation of stellar mass with the predicted potential.

At any given value of the potential, dEs typically have more mass in stars than LTDs. Because dEs lie on the Potential Plane, the excess is comparable to the mass of gas remaining in LTDs. Thus, the stellar masses of dEs today should be close to their original baryonic totals. On the other hand, the three most massive dSphs (NGC 147, 185, and 205) appear to be somewhat deficient in stars relative to dEs. However, their proximity to the potential plane suggests that they were more efficient than most smaller dSphs in converting gas into stars, and in that sense they are like dEs.

5

Implications

The analysis above motivates the hypothesis that the Potential Plane of LTDs and its extrapolation define the starting point for ETDs, and that today’s deviations of dSphs from the Plane are a consequence of gas loss. In this section, implications are explored.

Given the prevailing evidence that dark matter controls dynamics (Wolf et al., 2010; McCall et al., 2012; Kormendy & Freeman, 2016), it is approximated here that the deviations of dSphs from the Potential Plane are close to vertical, i.e., that any loss of gas did not lead to a substantial perturbation of either the velocity dispersion or surface brightness of constituent stars. If there were any response of the dark matter to the loss of gas, it likely would have been in the sense of becoming more dispersed. The outcome would have been a lower velocity dispersion and/or a fainter surface brightness, reducing the predicted potential. Consequently, the vertical offset from the Potential Plane that is observed today probably can be regarded as the \colorblack maximum possible deviation \colorblack that could arise from gas loss.

5.1

Gas Masses for dSphs

If the only distinction between dSphs and LTDs is the gas content, then for a dSph the ratio of the observed potential to the predicted potential at a given value of the predicted potential can be used to evaluate the fraction $f_{\textit{str}}$ of baryonic matter that was in stellar form \colorblack immediately before gas loss, and in turn the gas fraction $f_{\textit{gas}}$. \colorblack Accordingly,

[TABLE]

\color

black Thus, \colorblack just prior to gas loss, the baryonic mass was

[TABLE]

and the mass of gas was

[TABLE]

The ratio of potentials (the gap between dSphs and LTDs in Figure 3) can be expressed in terms of $P^{\textit{\,pre}}$ using equation 4.2. A black horizontal dashed line in Figure 3 marks where the ratio drops to 0.1.

Figure 5 demonstrates that $\mathcal{M}_{\textit{gas}}/\mathcal{M}_{\textit{str}}$ was higher in dSphs with shallower potentials, and suggests that those galaxies with \colorblack $\log P^{\textit{\,pre}}<5.5$

lost more than half of their baryons.

A system which loses half or more of its mass becomes gravitationally unbound (Hills, 1980), so the mere existence of such objects supports the contention that dark matter is the dominant mass constituent (see also Kormendy & Freeman 2016).

\color

black Inferred gas masses computed from equation 5.1 are listed in Table LABEL:tab_comdsph. Representative uncertainties, given in the table notes, were computed from

[TABLE]

where $b$ and $c$ are the coefficients in equation 4.1 defining the dependence of $\log P^{\textit{\,pre}}$ on $\log W_{20}$ and $\mu_{0}$, respectively, and where $s$ is the standard error of the mean for the fit ($0.01\,\rm dex$). The corresponding uncertainty in the baryonic mass is given by

[TABLE]

The uncertainty in the gas fraction is given by

[TABLE]

where

[TABLE]

Figure 6 compares the inferred gas masses for dSphs with those determined from HI fluxes for LTDs.

The gas masses for dSphs increase as $P^{\textit{\,pre}}$ increases, despite the declining gas fractions. Especially significant is that the trend for dSphs meshes very well with that for LTDs, lending support for the hypothesis that the deviations of dSphs below the potential plane are a consequence of evacuation of gas from LTDs.

5.2

Dynamical Masses

Dynamical mass-to-light ratios for the smallest dSphs can approach a thousand, giving the impression that baryons were incorporated into them with very low efficiency. However, given the evidence that the galaxies lost much of their gas, it makes more sense to blame the extremes on the failure of luminosities to gauge the baryonic masses of the precursors.

Wolf et al. (2010) showed that a “dynamical” mass $\mathcal{M}_{1/2}$ within the radius $r_{1/2}$ of the volume enclosing half of the light can be determined accurately from the measured line-of-sight velocity dispersion $\sigma_{\textit{los}}$ regardless of how mass is distributed. Specifically, from the spherical Jeans equation,

[TABLE]

In this paper, a dynamical mass $\mathcal{M}_{\textit{dyn}}$ more indicative of the total is defined by

[TABLE]

How closely this dynamical mass comes to the total mass depends upon the density profile; it will underestimate the total mass if dark matter is more widely distributed than luminous matter. Nevertheless, a \colorblack reliable value of $\mathcal{M}_{\textit{dyn}}/\mathcal{M}_{\textit{str}}$ \colorblack within $r_{1/2}$ should be given by $\mathcal{M}_{\textit{dyn}}/\mathcal{L}_{\textit{sech}}$ with an appropriate choice for $\mathcal{M}_{\textit{str}}/\mathcal{L}_{\textit{Ks}}$ (equation 10). Then, $\mathcal{M}_{\textit{dyn}}/\mathcal{M}_{\textit{bar}}$ follows from a choice for $\mathcal{M}_{\textit{gas}}/\mathcal{M}_{\textit{str}}$ (for dSphs, from equation 5.1).

The semi-major axis of the isophote encompassing half of the light from the stars, normally referred to as the effective radius $r_{\textit{eff}}$, is tightly correlated with the radius $r_{1/2}$. To within 2% (Wolf et al., 2010),

[TABLE]

Also, for a sech profile with scale length $r_{0}$,

[TABLE]

Then, the dynamical mass to associate with a sech profile is given by

[TABLE]

Results for dSphs and dEs are given in Tables LABEL:tab_comdsph and LABEL:tab_comde, respectively. Values for dSphs range from $2\times 10^{6}$ to $2\times 10^{9}\,\rm\mathcal{M}_{\odot}$. \colorblack Representative uncertainties listed in the table notes were computed from the errors in the angular scale lengths, distances, and velocity dispersions. The uncertainty in the ratio of the dynamical mass to the stellar mass is given by

[TABLE]

Figure 7 shows, as a function of the predicted potential, how $\mathcal{M}_{\textit{dyn}}/\mathcal{M}_{\textit{bar}}$ for dSphs depends upon the choice for $\mathcal{M}_{\textit{gas}}/\mathcal{M}_{\textit{str}}$. Results are compared with estimates for LTDs, for which $\mathcal{M}_{\textit{gas}}/\mathcal{M}_{\textit{str}}$ has been gauged from HI fluxes.

If it is approximated that dSphs did not lose any gas (pink circles), i.e., that stellar masses today are close to the baryonic masses of the precursors, then $\mathcal{M}_{\textit{dyn}}/\mathcal{M}_{\textit{bar}}$ rises rapidly as $\log P^{\textit{\,pre}}$ drops below 5.0, approaching 500 for the least massive systems. Accounting for gas (blue squares) tempers the rise to match that observed for LTDs (open circles), because the less massive systems had proportionately more baryons in the past than the more massive ones. Thus, the extremes exhibited by the least luminous dSphs should be regarded as an artifact of an incomplete census of baryons.

Combining the data for LTDs with the gas-corrected results for the dSphs with $\log P^{\textit{\,pre}}$ below 5.5, a least squares fit gives

[TABLE]

with an rms deviation of $0.22\,\rm dex$. The fit will be used in §5.3 to estimate escape velocities.

5.3

How Gas Was Lost by dSphs

Figure 3 shows that there is a critical value of the predicted potential, $P^{\textit{\,pre}}_{\textit{crit}}$, below which ETDs break from the Potential Plane. The corresponding escape velocity offers the means to discriminate why. The escape velocity $v_{\textit{esc}}$ at $r_{1/2}$ is given by

[TABLE]

\color

black If the mass within radius $r$ grows linearly with $r$, as observed for rotating galaxies, then the escape velocity computed from equation 29 should be applicable over a wide range of radii. From the definition of the potential (equation 6), $P^{\textit{\,pre}}_{\textit{crit}}$ sets $\mathcal{M}_{\textit{dyn}}/r_{1/2}$ as follows:

[TABLE]

The value of $\mathcal{M}_{\textit{dyn}}/\mathcal{M}_{\textit{bar}}$ can be evaluated from $P^{\textit{\,pre}}_{\textit{crit}}$ using equation 5.2, and $r_{\textit{eff}}/r_{1/2}$ and $r_{0}/r_{\textit{eff}}$ are defined by equations 24 and 25, respectively.

In Figure 3, the trend for dSphs intersects the Potential Plane at $\log P^{\textit{\,pre}}_{\textit{crit}}=5.74\pm 0.19$ ($M_{\textit{Ks}}\cong-17.5$), where the units of $P^{\textit{\,pre}}_{\textit{crit}}$ are $\mathcal{M}_{\odot}\,\rm pc^{-1}$. The corresponding value of $\log\mathcal{M}_{\textit{dyn}}/\mathcal{M}_{\textit{bar}}$ is $0.41\pm 0.06$. Thus, the escape velocity signified by $P^{\textit{\,pre}}_{\textit{crit}}$ is $50\pm 8\,\rm km\,s^{-1}$.

Khare (1953) and Pack (1953) showed that a gas for which the ratio of specific heats is $\gamma$ expands into a vacuum at $2/(\gamma-1)$ times the speed of sound in the gas. For a monatomic gas, then, the terminal velocity should be 3 times the speed of sound (see also Chevalier & Clegg, 1985; Murray et al., 2005; Oppenheimer & Davé, 2006). Therefore, the escape velocity corresponds to that of a gas at a temperature of \colorblack $13,100\pm 4,400\,\rm K$.

The temperature is characteristic of a low-metallicity HII region, and it suggests that the ultraviolet radiation field stemming from a starburst is behind the evacuation of gas from dSphs with a potential shallower than $P^{\textit{\,pre}}_{\textit{crit}}$. The gas would have been lost quickly. For the galaxies studied here, the time required for \colorblack uniformly expanding gas with a temperature of $13,000\,\rm K$ to traverse $r_{1/2}$ ranges from 10 to 47 million years.

5.4

dSph Starburst Masses

To evacuate gas via photoionization, the mass of the instigating starburst would have had to have been great enough to ionize the bulk of the gas \colorblack left but not exceed the mass of stars seen today. An idea of the minimum mass of the burst, $\mathcal{M}_{\textit{burst}}$, can be judged from ionization equilibrium, because the luminosity of photons with energies sufficient to ionize hydrogen is tied to the number of stars that have spectral type O. In equilibrium, the luminosity by number of photons energetic enough to ionize neutral hydrogen, $Q(\hbox{H}^{0})$, is related to the recombination rate by

[TABLE]

where $\alpha_{B}$ is the Case B recombination coefficient, $n_{e}$ is the number density of electrons in the ionized gas, $X$ is the hydrogen mass fraction, $\mathcal{M}_{\textit{gas}}$ is the mass of gas in the galaxy (equation 5.1), and $m_{p}$ is the mass of the proton. The inequality arises from the fact that there is a finite amount of gas to ionize. Case B is applicable to a galaxy in which $Q(\hbox{H}^{0})$ is just sufficient to ionize all of the available gas.

An estimate for $n_{e}$ can be judged from the scale length of the light distribution. Define $n_{H}$ to be the local number density of hydrogen atoms and ions. If a fraction $f$ of the gas were confined to a volume with a radius equal to $r_{1/2}$, then the mean density $\langle n_{H}\rangle_{1/2}$ over that volume would be

[TABLE]

Everywhere, ionization would be nearly complete, so $n_{e}\cong n_{H}$. However, because the interstellar medium is clumpy, $n_{e}>\langle n_{H}\rangle_{1/2}$. Fortunately, as can be seen from equation 32, adoption of $\langle n_{H}\rangle_{1/2}$ as a lower limit to $n_{e}$ still guarantees a lower limit to $Q(\hbox{H}^{0})$ and, in turn, a lower limit for the mass of the starburst.

For each dSph, the minimum value of $Q(\hbox{H}^{0})$ required to ionize all pre-existing gas was computed from equations 32 and 33 using $X=0.738$ (appropriate for metallicity $Z=0.002$), $\alpha_{\textit{B}}=2.04\times 10^{-13}\,\rm cm^{3}\,s^{-1}$ (appropriate for a temperature of $13,100\ \rm K$: Pequignot et al. 1991), and $f=1/2$. Estimates for ${\langle n_{H}\rangle}_{1/2}$ range from 0.05 to $2.8\,\rm\hbox{cm}^{-3}$, and results for $\log Q(\hbox{H}^{0})$ span 49.5 to 51.8 ($\,\rm s^{-1}$). Starburst99 v7.0.1 was employed to scale $\log Q(\hbox{H}^{0})$ to $\mathcal{M}_{\textit{burst}}$ (Leitherer et al., 1999; Vázquez & Leitherer, 2005; Leitherer et al., 2010; Leitherer et al., 2014). For a zero-age starburst of mass $1\times 10^{6}\,\rm\mathcal{M}_{\odot}$ with $Z=0.002$ and with stellar masses distributed according to a Salpeter IMF, the predicted value of $\log Q(\hbox{H}^{0})$ is 52.444 (2,040 O stars).

Computed lower limits for $\mathcal{M}_{\textit{burst}}/\mathcal{M}_{\textit{str}}$ are plotted against the predicted potential in Figure 8.

Typically, the estimates for the starburst masses are only about 1% of the stellar masses of dSphs today. Thus, it is plausible that photoionization \colorblack initiated the loss of gas from dSphs. The results for Canes Venatici II and Coma Berenices are 30% of their stellar masses, suggesting that the star formation events required to evacuate their gas could have been responsible for the formation of a large fraction of their stars.

5.5

The Minimum Mass of a Galaxy

Celestial bodies that are defined to be galaxies contain stars. \colorblack They contain stars because at some time in the past star formation ignited in a body of gas that had accumulated in a dark matter halo. It is fair to ask, then, “Are there galaxies that failed?". More pointedly, “Could there \colorblack have been a threshold for the baryonic mass below which stars \colorblack could not form?”.

Figure 5 demonstrates that baryons in dSphs were not converted to stars with 100% efficiency, and Figure 9 shows that the amount of gas remaining just prior to gas loss correlated strongly with the stellar mass.

Both Figures 5 and 9 reveal that the efficiency of conversion \colorblack diminished with shallower potentials, a result that extends downward by three orders of magnitude in stellar mass the trend established from reconstructed star formation histories for galaxies (Behroozi et al., 2013). This opens up the possibility that there was \colorblackindeed a minimum \colorblack baryonic mass below which stars did not form. Even if the efficiency were invariant (but less than 100%), there would have been a threshold in the baryonic mass below which the stellar mass could be considered negligible. With knowledge of how the gas masses inferred for the progenitors of dSphs correlate with stellar masses, it is possible in principle to evaluate what the minimum baryonic mass of a galaxy might be. Gross extrapolations are required, but it is interesting to see what present data confer.

A linear least squares fit to the dSphs \colorblack in Figure 9 gives \colorblack

[TABLE]

where masses are in solar units. The rms deviation of points about the fit is $0.12\,\rm dex$. The minimum baryonic mass of a galaxy might be considered to be the baryonic mass at which the stellar mass is $1\,\rm\mathcal{M}_{\odot}$. In Figure 9, the fit and $3\sigma$ confidence boundaries are shown extrapolated to that value. Of course, underlying the extrapolation is the assumption that the relationship between $P^{\textit{\,obs}}$ and $P^{\textit{\,pre}}$ observed for LTDs can be extrapolated not only into the realm of the faintest dSphs studied in this paper, which was the premise behind evaluating their gas masses in the first place, but even beyond to the stellar mass minimum. The corresponding gas mass, which can be regarded as the minimum baryonic mass of a galaxy, lies between 500 and $10,000\,\rm\mathcal{M}_{\odot}$, \colorblack the preferred value being $2,000\,\rm\mathcal{M}_{\odot}$.

\color

black Determining the halo mass associated with the minimum baryonic mass is more problematic, because surveys and simulations used to relate stellar masses to halo masses through density matching aren’t deep enough or have insufficient resolution to include galaxies like those that are the focus of this paper (see Behroozi et al. 2013). Dynamical masses estimated from equation 26 gauge total masses within $r_{1/2}$, but it is difficult to extrapolate these results to determine halo masses because neither the mass distributions nor their edges are known.

To constrain at least the dynamical mass of the halo housing the minimum baryonic mass, dynamical masses for dwarfs are plotted against stellar masses in Figure 10.

A linear least squares fit to the dSphs gives

[TABLE]

where masses are in solar units. The rms deviation of points about the line is $0.27\,\rm dex$. Extrapolating the $3\sigma$ confidence boundaries, the dynamical mass of a galaxy with a stellar mass of only $1\,\rm\mathcal{M}_{\odot}$ lies between $10^{3}$ and $10^{6}\,\rm\mathcal{M}_{\odot}$.

Figure 10 also displays with a dotted black line the trend of $\mathcal{M}_{\textit{dyn}}$ with $\mathcal{M}_{\textit{str}}$ that would be expected if $\mathcal{M}_{\textit{dyn}}/\mathcal{M}_{\textit{str}}$ were a constant. Stellar masses fall farther and farther to the left of this line as dynamical masses decrease, showing that haloes with lower \colorblack dynamical masses produce stars less efficiently. \colorblack In fact, there is reason to believe dSphs with lower halo masses produce stars less efficiently, because the correlation of halo \colorblack masses with stellar masses gleaned for much larger galaxies (Behroozi et al., 2013), the extrapolation of which is displayed by the dashed arrow in Figure 10, has a slope similar to that for the trend of $\mathcal{M}_{\textit{dyn}}$ with $\mathcal{M}_{\textit{str}}$.

Given the extrapolations required, results \colorblack for minimum masses are at best tentative and at worst meaningless. Perhaps with more confidence, equation 5.5 provides a means to estimate the cumulative effects of gas loss from an ensemble of dSphs described by a luminosity function.

6

Discussion

\color

black Supernovae have been flagged often as being responsible for evacuating gas from ETDs, either individually or through the cumulative effects of many. However, models by Ferrara & Tolstoy (2000) showed that complete loss of gas (blowaway) only occurs for galaxies with a total mass at or below $5\times 10^{6}\,\rm\mathcal{M}_{\odot}$. As was recognized by \colorblack those authors, the hitch is that most dSphs, including all of those studied in this paper, have higher dynamical masses. Recent models of the Ursa Minor dSph confirm that supernovae do not completely remove gas (Caproni et al., 2017). Although supernovae must have played a role in driving some gas out of Ursa Minor, the mass of gas \colorblack predicted to remain after 3 Gyr exceeded the observational upper limit by two orders of magnitude. Notably, in the first 600 Myr, supernovae-driven winds pushed more than half of the original gas out to large radii, where it would have been less strongly bound. Caproni et al. (2017) concluded that an additional mechanism was necessary to completely remove the gas.

Dekel & Woo (2003) examined the role of feedback in determining scaling relations for low surface brightness galaxies and dwarfs (i.e., galaxies with \colorblack stellar masses down to $6\times 10^{5}\,\rm\mathcal{M}_{\odot}$). By assuming that the energy input from supernovae was proportional to the mass of stars and then equating it to the binding energy of the gas, they were able to reproduce well the observed scaling relations. They argued that the binding energy distinguishes ETDs from LTDs, with the former \colorblack housed in haloes with \colorblack a mass, expressed as a circular velocity, between 10 and $30\,\rm km\,s^{-1}$ ($4.3<\log P^{\textit{\,pre}}<5.6$) \colorblack and the latter residing in more massive haloes with a velocity above $30\,\rm km\,s^{-1}$. This is because the intergalactic UV radiation field established by the first stars and galaxies would have photo-evaporated gas in bodies with \colorblack a velocity less than about $20\,\rm km\,s^{-1}$ ($\log P^{\textit{\,pre}}<5.1$) and halted infall of gas into systems with \colorblack a velocity less than $30\,\rm km\,s^{-1}$. Consequently, ETDs would have had to form most of their stars prior to reionization, whereas LTDs could have retained gas to the present. The problem with this scenario is that it cannot readily explain how there can be ETDs and LTDs today with overlapping values of $P^{\textit{\,pre}}$. Photo-evaporation initiated internally by a starburst could have been the impetus for evacuation of gas from the precursors of larger ETDs.

Wyithe & Loeb (2013) developed a model for the rate density of star formation at high redshift incorporating a prescription for the suppression of star formation by supernovae feedback. They found that individual starbursts in low-mass galaxies came to an end after only $2\times 10^{7}$ years, a time short enough to suggest that subsequent star formation was quenched by massive stars. How is open to question. Supernovae might have been involved, but photoionization \colorblack probably played a role, too.

Spaans & Norman (1997) demonstrated the importance of a multi-phase interstellar medium to the evolution of dwarfs. They claim that heating by supernovae created a hot phase with a temperature of $\sim 10^{7}\,\rm K$ and a filling factor that reached close to unity by a redshift of $\sim 1$ (see also Clarke & Oey 2002). \colorblack The gas was so hot that it would have been able to escape. \colorblack However, the work presented here reveals that gas in dSphs could have begun to escape before a hot phase developed. \colorblack Nevertheless, the loss of gas could have been assisted or even accelerated by supernovae, either directly by the pressure of ejecta or indirectly through additional heating.

Tides and/or ram pressure stripping have been proposed as an alternative to outflows driven in one way or another by star formation. Simulations by Mayer et al. (2001) of the evolution of a small disk galaxy in an NFW halo (Navarro et al., 1996) orbiting a massive host revealed that tidal stirring could lead to structural and kinematic properties comparable to \colorblack those of dwarf spheroidals after just a few orbits (see also Arraki et al. 2014). \colorblack Ram pressure had to be invoked to completely evacuate gas, though (Mayer et al., 2006), which worked for galaxies with a circular velocity below $30\,\rm km\,s^{-1}$ ($\log P^{\textit{\,pre}}<5.6$). Interestingly, it is the ETDs in that velocity range that are observed to deviate from the Potential Plane. To evacuate a galaxy with a circular velocity exceeding $30\,\rm km\,s^{-1}$,

a heating source appears to have been necessary to counteract inflow stimulated by tides and to maintain the extension of the gas. Mayer et al. (2006) propose that the heating could have been provided by the cosmic UV background. \colorblack However, the issue may be moot. ETDs this massive do not deviate significantly from the Potential Plane, suggesting that most of their gas was converted into stars.

Besides naturally leading to a relationship between morphology and environment, \colorblack the combination of tidal stirring and ram pressure admits the possibility of star formation over several gigayears (one or more orbits), which indeed has been observed for dwarfs in the Local Volume (Weisz et al., 2011). In fact, Weisz et al. (2011) goes so far as to say that the transformation from gas-rich to gas-poor could not have been caused by stellar feedback alone because internal processes cannot explain the connection to environment.

Figure 5 showed that for dSphs there was a strong correlation between the star formation efficiency and the baryonic mass. It seems unlikely that the combination of tides and ram pressure could have led to such a relationship because orbital parameters vary from galaxy to galaxy. \colorblack Also, it is hard to see how a dSph could be a tidally stirred LTD given the structural similarities of the two kinds of galaxies today. For \colorblack these reasons, it is preferable to \colorblack surmise that the absence of gas in dSphs stems from a star formation history different from that of LTDs.

The three most massive dSphs, all satellites of M31, as well as dEs, have \colorblack circular velocities close to or above $30\,\rm km\,s^{-1}$. All have potentials deeper than $P^{\textit{\,pre}}_{\textit{crit}}$ and congregate near the potential plane. Their positioning is consistent with the theoretical evidence that their gas would have been difficult to strip by any means. As suggested in §4.2 (see Figure 4), such galaxies appear to have converted a much higher fraction of gas into stars than objects with a potential shallower than $P^{\textit{\,pre}}_{\textit{crit}}$.

\color

black If gas loss from dSphs stemmed from photoionization by a starburst, it remains to be explained why there is such a tight relationship between $\mathcal{M}_{\textit{str}}/\mathcal{M}_{\textit{bar}}$ and the potential at the time of gas loss (Figures 3 and 5). It is as if the progenitor of a dSph \colorblack “knew”what fraction of its baryons it had to convert into stars before losing its gas. In fact, relating gas loss to photionization offers some insights. For gas to be ejected, most of it had to be ionized, so the mass of the instigating starburst had to be connected at some level to the total mass of gas expelled. Since most baryons were gaseous, then $\mathcal{M}_{\textit{str}}/\mathcal{M}_{\textit{bar}}\propto Q(\hbox{H}^{0})/\mathcal{M}_{\textit{gas}}$. Based upon equation 32, the trend in $\mathcal{M}_{\textit{str}}/\mathcal{M}_{\textit{bar}}$ could be explained if $n_{e}$ were higher in dwarfs with deeper potentials. \colorblack However, no correlation of ${\langle n_{H}\rangle}_{1/2}$ with $P^{\textit{\,pre}}$ is apparent for dSphs. \colorblack Nevertheless, the trend still could have developed if gas in more massive dwarfs tended to be clumpier \colorblack (i.e., if the filling factor were lower), \colorblack since $n_{e}/{\langle n_{H}\rangle}_{1/2}$ would be raised.

\color

black Those LTDs with $P^{\textit{\,pre}}$ below $P^{\textit{\,pre}}_{\textit{crit}}$ risk losing their gas quickly if they form enough stars at once to photoionize most of their gas. It is because there is no long-lived intermediate stage that LTDs and ETDs with potentials shallower than $P^{\textit{\,pre}}_{\textit{crit}}$ are so cleanly separated. So long as LTDs form stars gradually, they should be able to remain LTDs until their gas supply is exhausted. Indeed, Chisholm et al. (2015) observed that the bulk of the low-ionized gas in dwarf irregular galaxies is not moving fast enough to escape. In fact, Tassis et al. (2008) was able to reproduce scaling relations for LTDs without invoking gas loss at all, but rather through a combination of increasingly inefficient star formation towards lower masses and efficient mixing of metals into haloes.

Kormendy & Freeman (2016) found that baryons are dynamically unimportant in galaxies in which the outer circular velocity in the dark matter halo is below $42\pm 4\,\rm km\,s^{-1}$. The escape velocity at $P^{\textit{\,pre}}_{\textit{crit}}$ corresponds to a circular velocity of $35\pm 6\,\rm km\,s^{-1}$. Although one may question how well the estimated escape velocity is gauging the circular velocity in the outer halo, the agreement \colorblack may arise because dSphs with smaller circular velocities have expelled most of the baryons that they once had.

\color

black The minimum mass of a galaxy has a bearing on the missing satellite problem. A cosmological simulation of the Local Group by Sawala et al. (2016) showed that reionization would have prevented stars from forming in haloes with masses below about $10^{7.5}\,\rm\mathcal{M}_{\odot}$ and would have led to the loss of gas by photo-evaporation. This is because molecular hydrogen would have been dissociated and, in the absence of metals, cooling would have been curtailed (Dekel & Woo, 2003). Galaxies that survived to the present were interpreted to be those with the most concentrated distributions of baryons.

In §5.5, the dynamical mass corresponding to the estimated minimum baryonic mass of a galaxy was estimated to be at most $\sim 10^{6}\,\rm\mathcal{M}_{\odot}$. The value rises by a factor of six if conditions were such that the mass of stars formed in the first star formation event was $10^{3}\,\rm\mathcal{M}_{\odot}$ rather than $1\,\rm\mathcal{M}_{\odot}$ (within a factor of seven of the stellar mass in the least luminous dSphs studied in this paper). Also, the limit would be higher if halo masses are larger than inferred from $\mathcal{M}_{\textit{dyn}}$, \colorblack as Behroozi et al. (2013) indicate (see Figure 10). Thus, the observations suggest that the theoretically-predicted minimum \colorblack halo mass is not unreasonably high.

7

Conclusions

The positioning of dSphs and dEs (early-type dwarfs) with respect to the Potential Plane defined by dIs and BCDs (late-type dwarfs) has been examined. With the mass-to-light ratio in $K_{s}$ fixed at the fitted value for late-type dwarfs, dEs and massive dSphs are clustered very near the Plane, \colorblack suggesting that they evolved from late-type dwarfs that managed to transform the bulk of their gas into stars. There is a critical value of the potential at which dSphs with shallower potentials fall away \colorblack from the Plane. The deviations are attributed to an incomplete census of baryons, which can be interpreted to be a consequence of gas loss. As such, the magnitude of the offset of a dSph from the plane gauges the amount of gas that the galaxy once had. The observations show that dSphs of lower mass must have \colorblack converted baryons into stars with lower efficiency and lost proportionately more gas. Ratios of dynamical to baryonic masses fall in line with observations of late-type dwarfs once the missing gas is accommodated.

The escape velocity defined by the critical potential is $50\pm 8\,\rm km\,s^{-1}$, which implies that the temperature of escaping gas would have been $13,000\pm 4,000\,\rm K$. The temperature is typical of metal-poor HII regions, suggesting that \colorblack near-complete photoionization \colorblack of neutral gas following a starburst \colorblackplayed a role in driving gas loss. This is plausible because, for every dSph sampled, the minimum mass of a starburst required to ionize the entirety of the gas is smaller than the observed mass of stars today. \colorblack If unimpeded, evacuation of gas following such a starburst would have occurred over a few tens of millions of years \colorblack (less if aided by stellar winds or supernovae). This can explain why it is that dwarfs in a state of transition are rare and consequently why it is that dwarfs with a potential shallower than the critical value form two distinct sequences. The escape velocity implies a circular velocity of $35\pm 6\,\rm km\,s^{-1}$, close to the halo value below which baryons are believed to be dynamically irrelevant.

Because smaller dSphs were less efficient at converting gas into stars, there should be a minimum baryonic mass for \colorblack galaxies, i.e., below which no stars ever formed. Grossly extrapolating the trend of the inferred gas mass with stellar mass down to a stellar mass of $1\,\rm\mathcal{M}_{\odot}$, the minimum baryonic mass comes out to be between 500 and $10,000\,\rm\mathcal{M}_{\odot}$. \colorblack The corresponding dynamical mass is less than $10^{6}\,\rm\mathcal{M}_{\odot}$. \colorblack Even though there may have been smaller bodies of baryonic matter in the early Universe, it is possible that only their dark matter haloes remain today if the gas was photo-evaporated during reionization.

Acknowledgements

MLM conducted part of this research while a Visiting (York) Scholar at Massey College, and is grateful to the College for the use of its facilities. The Natural Sciences and Engineering Research Council of Canada is thanked for its continued financing. Thanks are conveyed to B. Schuman for his advice about titling. MLM is particularly grateful to B. Nathoo, F. Shariff, H. Tanzer, and J. Vecchiarelli for their dedication to preserving his health long enough to complete this paper. This research has made use of the databases GOLDMine and HyperLEDA, as well as the software Igor Pro and Starburst99.