The M101 Satellite Luminosity Function and the Halo–Halo Scatter among Local Volume Hosts

Here we discuss the physical and star formation properties of the two resolved dwarfs in our HST sample, DwA and Dw9; these properties are tabulated in Table 1.

To determine distances to the resolved objects, we make use of the TRGB technique (e.g., Da Costa & Armandroff 1990; Lee et al. 1993). The peak luminosity of the RGB is a standard candle in the red bands, because it is driven by core helium ignition and so it provides a useful distance estimate for galaxies with an old stellar component that are close enough that the RGB can be resolved. To determine TRGB magnitudes, we adopt the methodology described in Crnojević et al. (2019). Briefly, the photometry is first corrected to account for the color dependence of the TRGB (Jang & Lee 2017). Then the field (background + foreground) contamination as derived from a dwarf-free region of the ACS field of view is statistically subtracted from the dwarf's CMD. The LF for RGB stars with colors in the range 0.8 < (F606W − F814W)₀ < 1.3 is computed, and a model LF (convolved with the appropriate photometric uncertainty, bias, and incompleteness function derived for the observations) is fit to it with a nonlinear least squares method. The uncertainties are derived by recomputing the TRGB for 100 realizations of the statistical decontamination process. Using this method for the two objects that are resolved into stars in our HST data set, DwA and Dw9, we obtain TRGB distances of ${6.83}_{-0.26}^{+0.27}$ and ${7.34}_{-0.37}^{+0.39}$ Mpc, respectively, confirming their association with the M101 group (D ∼ 7 Mpc). We show the CMDs for DwA and Dw9 in Figure 4, along with the TRGB placement for each.

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The resolved stellar populations of DwA and Dw9 both appear to be consistent with a stellar population of old, metal-poor RGB stars (≳10 Gyr). In the bottom panels of Figure 4 we plot isochrones (Marigo et al. 2017) with an age of 12.7 Gyr. From the CMDs, both dwarfs seem to host stars with mean metallicities of [Fe/H] ≈ −1.5; the color spread seen in the RGB stars (as derived from a simple Gaussian fit to the RGB sequence) is slightly larger than the photometric uncertainties derived from the Artificial Star Tests. This could be explained by some intrinsic metallicity spread; however, the origin and implications of the spread are beyond the scope of this work. DwA has a small population of stars above the TRGB, indicative of asymptotic giant branch stars with intermediate ages (∼2–5 Gyr based on their magnitudes); such populations are often seen in similarly faint dwarf galaxies in other systems (e.g., Dw2 in Centaurus A [Cen A]; Crnojević et al. 2019).

Neither DwA nor Dw9 was detected in deep near-UV (NUV) GALEX imaging (Martin & GALEX Team 2005); this lack of NUV emission indicates that these galaxies have an upper limit of ≲1.7 ± 0.5 × 10⁻³ M_⊙ yr⁻¹ for recent star formation, obtained using the relation from Iglesias-Páramo et al. (2006). We have performed follow-up H i observations of DwA with the Robert C. Byrd Green Bank Telescope (A. Karunakaran et al. 2019, in preparation) and find no significant H i signal along the line of sight to DwA (Dw9 was not observed). We place stringent 5σ upper limits on its H i mass, log(M_{H i}/M_⊙) < 5.75, and its gas richness, M_{H i}/L_V = 0.68 M_⊙/L_⊙. The lack of both H i gas and NUV flux is consistent with the old, metal-poor stellar population of DwA.

The structural properties of the resolved candidates were determined with the maximum-likelihood technique of Martin et al. (2008) using the implementation of Sand et al. (2012). The stars selected for the structural analysis are those consistent with the RGB as seen in Figure 4. We fit a standard exponential profile plus constant background to the data, with the following free parameters: the central position (R.A.₀, decl.₀), position angle, ellipticity, half-light radius (r_h), and background surface density. Uncertainties on structural parameters were determined by bootstrap resampling the data 1000 times, from which 68% confidence limits were calculated. Key derived parameters are shown in Table 1. Our HST-derived half-light radii are slightly larger than those found in the ground-based CFHTLS data, likely due to the superior detection of outlying stars at HST depths and resolution.

The absolute magnitude of the dwarfs is derived via the procedure laid out in Crnojević et al. (2019). We simulated a well-populated CMD for each dwarf using Padova isochrones (Marigo et al. 2017) with an age of 12.7 Gyr and metallicity of [Fe/H] = −1.5 for each target, assuming a Kroupa initial mass function (Kroupa 2001). This simulated population of stars is then convolved with the photometric errors derived from the artificial star tests. We then drew stars randomly from this fake stellar population, scaling the number of stars such that it matched the number seen in the RGB region for our observed dwarfs. The flux from the drawn stars was summed along the entire LF, including stars too faint to be detectable in our HST data, in order to account for the faint unresolved component of each galaxy. This process was repeated 100 times to assess our uncertainties. Measurements were converted to the V band using the prescriptions in Sahu et al. (2014). In Table 1 we show our derived absolute magnitudes and stellar masses, and we also show our brightness measurements derived from the CFHTLS ground-based data set, as presented in Bennet et al. (2017); the data agree to within the errors.

From the properties derived above we can see that these M101 dwarf galaxies, including the dwarfs from Danieli et al. (2017), fit into the Local Group size–luminosity relation; see Figure 5.

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The 17 remaining dwarf galaxy candidates imaged with HST all had unresolved, diffuse emission and are thus in the background and not associated with M101. We remeasured the observational properties of these diffuse dwarfs with the HST data using GALFIT (Peng et al. 2002) and the procedure outlined in Bennet et al. (2017), including inserting simulated diffuse dwarf galaxies to assess our uncertainties. We also followed this procedure on the unresolved objects from Merritt et al. (2016) and found successful fits for all objects aside from DF6, where aperture photometry was used instead. When using GALFIT all parameters were left free. The HST images were often spatially binned to increase the signal in each pixel and facilitate the GALFIT measurements; the results are reported in Table 2. Comparisons between HST and the CFHTLS measurements presented in Bennet et al. (2017) are shown in Figure 6, highlighting a good agreement between the values derived from the two data sets.

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As can be seen in Figure 1, many of the 17 diffuse HST dwarfs are projected near the background NGC 5485 group. This is a bimodal group focused around the massive elliptical galaxies NGC 5485 and NGC 5473 at D ∼ 27 Mpc (Tully et al. 2015). This background group shows a large number of spectroscopically confirmed satellites among the population brighter than our diffuse dwarf sample (Makarov & Karachentsev 2011), and we further constrain its LF in Section 5.

Finally, we note that target Dw15, which was regarded as a possible M101 group member after the SBF measurements of Carlsten et al. (2019b), was found to be unresolved in our HST data (see bottom row of Figure 3). The SBF measurements successfully show that 14 of the 16 remaining unresolved objects are not M101 group members, with the final two (DwB and DwC) being regarded as unlikely M101 group members but not completely excluded. This shows that while the SBF technique reported in Carlsten et al. (2019a) is a powerful tool, it is still vital to obtain HST quality follow-up of fainter diffuse dwarf systems to verify their identity and thus refine this promising technique.

Twenty-three dwarf candidates from the Bennet et al. (2017) sample were not observed by HST and therefore have unknown distances, aside from Dw26, which was confirmed to be a background galaxy via H i observations (Bennet et al. 2017). The untargeted objects have mean properties (e.g., apparent magnitude and half-light radius) that are consistent with the mean properties of the diffuse dwarfs targeted with HST to within ∼1σ. The observational properties of the untargeted objects as derived from the CFHTLS data are reported in Table 1 of Bennet et al. (2017).

We statistically assess the membership status of the untargeted diffuse dwarfs as a population, based on the HST observations already made. For this we exclude Dw26 from the sample, as we already know its distance (D ∼ 150 Mpc) due to its H i detection.

If we assume that the 19 objects chosen for HST follow-up are a representative sample of all the LSB objects detected in Bennet et al. (2017), then we can determine the number of untargeted objects that are associated with M101 via a series of hypergeometric distributions (Zhang 2009). Hypergeometric distributions describe a population of size N with a fixed number of successes K (in this case K would be the number of LSB objects that are associated with M101), where k successes are found for every n drawings from the population. This is done without replacing previously observed objects back into the population (which would represent a binomial distribution). We use these distributions to determine the likelihood of having two successes in 19 observations for various underlying populations with N fixed at 41 (the total population of the Bennet et al. 2017 objects once Dw26 is excluded) and K allowed to have any value between 2 and 24 (with a minimum of two M101 detections from DwA and Dw9 and a maximum of 24 if all untargeted objects were M101 dwarfs). Once the likelihood of each population is determined, we construct a normalized probability function for the values of K. This function is then used to calculate the mean and standard deviation for K. These results show that the total LSB population from Bennet et al. (2017) has 5.1 ± 2.3 successes. As we have already detected two objects associated with M101, this implies that of the remaining 22 untargeted LSB objects, 3.1 ± 2.3 could be associated with the M101 group and 18.9 ± 2.3 could be unresolved and associated with either the NGC 5845 group or distant background galaxies.

Another method to estimate the distances of the untargeted LSB objects in Bennet et al. (2017) is through the SBF method reported in Carlsten et al. (2019b). These distances show a total of four possible M101 dwarfs (Dw21, Dw22, Dw23, and Dw35) among the nontargeted LSB objects from Bennet et al. (2017), as their reported distances are consistent with the M101 group to within 1σ. However, there are significant uncertainties on these distances, and no firm conclusions about the group membership of these objects were drawn by Carlsten et al. (2019b). There are a further four objects (Dw19, Dw24, Dw31, and Dw32) that have reported distances behind the M101 group but are consistent with M101 group membership within 2σ. From this we can conclude that there are plausibly approximately four M101 dwarfs among the untargeted candidates, with an extreme upper limit of eight new group members. These numbers are broadly consistent with the results from the statistical method explained above.

All of these untargeted objects fall on the Local Group size–luminosity relation if they are assumed to be at either M101 (D ∼ 7 Mpc) or NGC 5485 (D ∼ 27 Mpc) distance.

The effect that the two different methods used to statistically determine group membership have on the M101 LF is discussed further in Section 5.1 and can be seen in Figure 7. On the other hand, the LF of the NGC 5485 group shows no significant variation between the two methods, and therefore this will not be discussed further.

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We have examined the dwarf galaxy completeness around M101 on both the bright and faint ends. For a bright limit on the completeness of the Bennet et al. (2017) sample, we tested the detection algorithm used in that work via artificial dwarf galaxy simulations, to an apparent magnitude of m_g = 17 (M_g = −12.2 or M_V = −12.5 at 7 Mpc) on the bright end.

At first blush, the bright limit of Bennet et al. (2017) suggests it is possible that bright satellites of M101 (M_V <−12.5) may still be undiscovered, but this is unlikely: previous studies around M101 (e.g., Karachentsev et al. 2013) were considered complete down to M_B = −11.2 (M_V = −12.1), which overlaps with the Bennet et al. bright-end brightness limit and is explicitly complete to M_V ≈ −20 and possibly brighter (see, e.g., Figure 5 of Karachentsev et al. 2013). We are exploring additional modifications to our algorithm to probe an expanded parameter space (P. Bennet et al. 2019, in preparation): to date, preliminary results around M101 indicate no additional dwarf candidates up to a brightness of m_g = 16 (M_g = −13.2 or M_V = −13.5).

For our faint-end completeness limit to M101 dwarfs, we note that the 50% completeness limits of the Bennet et al. (2017) diffuse dwarf search were M_g = −7.2 or M_V = −7.5. Several of the faintest dwarf candidates in that work were not targeted with HST but have SBF distances that are at least consistent with M101 (Dw21, Dw22, Dw23, and Dw35); these dwarfs should be followed up with future space-based imaging. HST imaging is available for all diffuse dwarfs with SBF distance estimates consistent with M101 (Carlsten et al. 2019b) and with magnitudes M_V ≲ −8.2 mag (DwA, Dw9 and Dw15): we thus adopt this value as a conservative faint limit for our completeness.

The satellite LF for M101 was constructed by using all galaxies that are reported to be within the projected virial radius of M101 (∼250 kpc or ∼2.05° at 7 Mpc) and with confirmed distances consistent with that of M101. For this reason we do not include the proposed M101 members DDO194 (D_proj =656 kpc, Tikhonov et al. 2015) or KKH87 (D_proj = 392 kpc, Karachentsev et al. 2013) as these objects are projected too far from M101. We also do not include UGC 08882, as the reported distance (D = 8.3 ± 0.8 Mpc, Rekola et al. 2005) places the object behind M101. We do include the three faint galaxies (DF1, DF2, and DF3) found by the Dragonfly survey that were confirmed via TRGB distance measurements (Merritt et al. 2014; Danieli et al. 2017). See Table 3 for a full list of M101 members that we consider, including those confirmed by the current work.

Table 3.  Confirmed M101 Group Members within 250 kpc

Name	R.A.	Decl.	MV	Projected M101	Distance	Ongoing Star
				Distance (kpc)	(Mpc)	Formation
(1)	(2)	(3)	(4)	(5)	(6)	(7)
M101	14:03:12.5	+54:20:56	−20.8	0	6.79 ± 0.41a	Y
NGC 5474	14:05:01.6	+53:39:44	−17.6	89	6.82 ± 0.41a	Y
NGC 5477	14:05:33.3	+54:27:40	−15.3	44	6.77 ± 0.40a	Y
Holm IV	13:54:45.7	+53:54:03	−15.0	160	6.93 ± 0.48a	Y
M101 DF1	14:03:45.0	+53:56:40	−9.6	50	6.37 ± 0.35b	N
M101 DF2	14:08:37.5	+54:19:31	−9.4	97	${6.87}_{-0.30}^{+0.21}$ b	N
M101 DF3	14:03:05.7	+53:36:56	−8.8	89	${6.52}_{-0.27}^{+0.25}$ b	N
M101 DwA	14:06:49.9	+53:44:30	−9.5	100	${6.83}_{-0.26}^{+0.27}$ c	N
M101 Dw9	13:55:44.8	+55:08:46	−8.2	160	${7.34}_{-0.37}^{+0.39}$ c	N

Notes.

^aFrom Tikhonov et al. (2015). ^bFrom Danieli et al. (2017). ^cFrom this work.

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In order to place the M101 LF into context, we also compile the cumulative LFs of other Local Volume galaxies, both inside the Local Group (M31 and the MW itself) and outside of it (M94, M81, and Cen A), using data from recent resolved stellar population studies. All of the galaxy groups have been examined for satellites down to at least M_V = −9.1, which allows for the widest possible comparison down to the limiting magnitude reached in the M101 survey. These hosts span a range of masses from 2.5 × 10¹¹M_⊙ for the MW (Eadie & Harris 2016) to 9.0 × 10¹¹M_⊙ for Cen A (Woodley & Gómez 2010) using dynamic mass estimates for the innermost 40 kpc (where these estimates are available). From this sample there is no evident correlation between host mass and satellite LF, which is likely due to the relatively small mass range examined. These galaxies also span a range of environments, from very isolated groups like M94 to groups that are actively accreting smaller galaxy groups, as suggested for the MW and the LMC and possibly for M81 accreting M82 and NGC 3077 (Chiboucas et al. 2013). The galactic environment seems to be more important in predicting the galaxy LF, as will be discussed in Section 5.2 in more detail.

For the compilation of these LFs, we have only used objects where the association between the dwarf galaxy and the host has been confirmed via distance measurements. We do not include dwarfs fainter than M_g = −7.2 (M_V = −7.5), which corresponds to the 50% completeness limit for the survey of M101 reported in Bennet et al. (2017). Note that we consider our M101 sample complete down to M_V = −8.2, as discussed in Section 4.4, but for the statistical corrections for those dwarfs not imaged with HST we use the full Bennet et al. (2017) sample down to M_V = −7.5. We also limit our analysis to satellite galaxies within a projected distance of <250 kpc from the host, or with 3D distance of D < 250 kpc in the case of the MW.

We consider several sources for the satellite galaxies used to construct our cumulative LFs, following Crnojević et al. (2019). Briefly, for the MW we adopt the updated online 2015 version of McConnachie (2012)—while this excludes some recent ultrafaint discoveries, they are below the limiting magnitude of the M101 survey and are thus not considered further. For M31 we combine the catalogs from Martin et al. (2016) and McConnachie et al. (2018). For hosts beyond the Local Group, we adopt the Updated Nearby Galaxy Catalogue (Karachentsev et al. 2013) and the Extragalactic Distance Database⁸ (Jacobs et al. 2009), complementing these resources with more recent work where available. More specifically, for M81 we refer to Chiboucas et al. (2013), who performed a wide-field survey of M81 and its satellites with CFHT/Megacam along with HST follow-up imaging (we excluded objects that are considered tidal dwarfs). We also include the M81 dwarf D1005+68 (M_V = −7.94; Smercina et al. 2017). For M94 we include the new discoveries from Smercina et al. (2018), which adds two faint dwarf galaxies to the two already known M94 satellites; there are also 12 additional objects with velocities consistent with M94 (Karachentsev et al. 2013); however, these objects are outside of our 250 kpc limit. For Cen A we use the results from Crnojević et al. (2019), who compiles a Cen A LF based on the discovery of 11 new satellites within the PISCeS survey as well as on 13 previously known dwarfs. We note that Cen A is an elliptical galaxy, and it likely has a very different accretion history to that of the MW and the other spiral galaxies that we examine here; nevertheless, its mass is considered to be comparable to that of the MW within a factor of a few (e.g., Karachentsev et al. 2007). Although we reiterate that the dynamic mass estimates for these galaxies (where available) show variation of ∼3.5 between the smallest and largest galaxies, this does not appear to correlate with satellite number in the examined galaxy sample.

The completeness of the different surveys we consider is not easy to quantify. Despite efforts to ensure a uniform sample, a few caveats are inevitable: (1) MW surveys suffer from incompleteness effects, mainly due to incomplete spatial coverage, especially in the direction of the Galactic plane. Such effects could lead to an underestimate of the number of faint satellites by a factor of ∼3 (e.g., Hargis et al. 2014), meaning our constructed LF for the MW is a lower limit; (2) Crnojević et al. (2019) estimate the incompleteness of the PISCeS survey and suggest that there might be ∼5 to 10 yet undetected galaxies in the range −10 < M_V < −8 within ∼150 kpc from Cen A. This should, however, not significantly alter the slope of the derived LF. At larger galactocentric distances (between 150 and 300 kpc), 13 candidates with −12 < M_V < −8 have additionally been presented by Müller et al. (2017a), but they have not been confirmed as Cen A members yet; (3) a deep search for faint satellites has not been performed beyond the innermost 150 kpc for M94, thus the LF constructed for this host is also likely a lower limit. A search within 150 kpc of the projected distance includes over ∼80% of the virial volume for a galaxy the size of M94 (considering the 150 kpc projected radius cone that was observed, which will have sensitivity to some satellites at larger 3D radii), and the satellites themselves may be centrally concentrated and distributed like a Navarro–Frenk–White profile. Nonetheless, we do conduct a complementary analysis by comparing our Local Volume galaxies using only the innermost 150 kpc of projection in addition to our searches across the entire virial radius.

We perform a fit adopting a Schechter function for each cumulative LF using a maximum-likelihood estimator:

$$\begin{eqnarray}&&N(\lt M)={\phi }_{\ast }\gamma [\alpha +1,{10}^{0.4\left({M}_{\ast }-M\right)}],\end{eqnarray} \tag{ 2 }$$

where ϕ_* is the normalization density and γ represents the incomplete gamma function and depends on the slope (α) of the LF and on M_*, where M_* is the characteristic magnitude. We find the best-fit values for α to be −1.25 ± 0.05 for Cen A, −1.19 ± 0.06 for the MW, −1.28 ± 0.06 for M31, −1.21 ±0.05 for M81, ∼−1.18 for M94 (which was difficult to constrain, given how few satellites are in this system), and finally −1.14 ± 0.10 for M101 itself. These slopes are consistent with previous literature results and with each other. We also create the LF of the "median" Local Volume host (see Figure 7): we derive the median number of satellite members at each 0.25 mag increment in the range −20.25 ≤ M_V ≤ −7.0; these are then assembled into a "median" LF. We then fit this LF using the same method that was used for the individual galaxies, obtaining a slope of α = −1.29 ± 0.05. While the values of ϕ_* and M_* are not well constrained, variations of these parameters within reasonable limits do not significantly affect the best-fit values for α (see Chiboucas et al. 2009 and Park et al. 2017).

The above values for α were constructed using only confirmed M101 members, and we explore the effects that the unconfirmed diffuse dwarfs would have on this slope. If we include the statistically weighted objects that do not have follow-up imaging (see Section 4.3 for details of the statistical weighting), this adds a total of approximately three dwarfs over the magnitude range −7.0 ≤ M_V ≤ −10.2. The best fit obtained for this LF is α = −1.16 ± 0.12. This potential M101 LF can be seen in Figure 7. However, if instead of the statistically weighted objects we add those objects that have SBF distances consistent with M101 membership to within 1σ (i.e., Dw21, Dw22, Dw23, and Dw35) to the M101 LF, we obtain a slope of α = −1.15 ± 0.10. Thus it can be seen that the inclusion of the unconfirmed M101 members only gives rise to minor LF changes.

Figure 7 presents a direct comparison of the LFs for the considered Local Volume galaxies. At brighter magnitudes (M_V < −12), M101 appears similar to the other Local Volume galaxies; specifically, M101 has four members with M_V <−12, compared with a median value of 7.5 ± 4.7 for the other groups. However, at fainter magnitudes these LFs show substantial variation, with each of M81, Cen A, and M31 having ≳20 satellites down to M_V = −8.0, as compared with the nine satellite members of M101 down to the same magnitude. This uniqueness is borne out when comparing M101 to the median Local Volume galaxy: at M_V = −8.0, M101 has nine members compared with 24.5 ± 7.7 for the median. While there are large uncertainties due to small number statistics, it is still apparent that the M101 system is unusual and sparse. This sparseness within the M101 and M94 groups is mirrored by examinations of the innermost 150 kpc only; this shows M101 with seven members compared with a median of 17.3 ± 4.1.

The SAGA survey (Geha et al. 2017) reported on satellite galaxies around eight MW analogs with distances in the range of 20–40 Mpc, with a limiting magnitude M_r < −12.3. These results produced LFs that were comparable to the MW and M31. When we extend this comparison to the galaxies in our sample we find that that the majority of the SAGA galaxies have poorer satellite systems than the median Local Volume galaxy LF. At the SAGA survey's limiting magnitude (M_V < −12.5), the Local Volume galaxies have a median of 6.8 ± 3.8 group members; this is more group members than all but one SAGA galaxy (NGC 6181), and over half of the SAGA sample is below the 1σ limit. Therefore we conclude that the LFs found by the SAGA survey more closely resemble those of the M101 and M94 groups than the MW or M31.

Our data additionally show a lack of any M101 group members with an absolute magnitude between M_V = −15 and M_V = −10. This area of parameter space has been fully probed by previous work in the M101 system out to the projected virial radius (e.g., Karachentsev et al. 2013, 2015; Merritt et al. 2014; Bennet et al. 2017), and therefore this gap appears to be genuine—see Section 4.4 for a discussion of our bright and faint satellite magnitude limits. The addition of the 22 objects not targeted by HST does not close this gap (if any of them actually belong to the M101 system) but could potentially decrease the size, as the brightest of these objects (Dw31 and Dw32) would be brighter than DF-1 (M_V = −9.6) when placed at the distance of the M101 group. However, this still results in a gap of 4.9 mag, far larger than that observed in other comparable nearby galaxies and the SAGA survey. The mean largest magnitude gap between any two confirmed group members in this latter sample is 2.6 ± 1.1 mag, which is far below the 5.4 mag gap recorded for the M101 system (see Figure 10). A full examination of the theoretical implications of such a large gap is beyond the scope of this work (although see Ostriker et al. 2019). However, it should be noted that this behavior is not observed in galaxy simulations, which tend to produce a satellite LF close to those observed in the MW and M31, with far smaller magnitude gaps (Geha et al. 2017).

Previous work on the environmental dependence of galaxy LFs (for a summary see Ferrarese et al. 2016, and the references therein) has examined a variety of different cluster and group environments. There have been multiple contrasting results, with some studies finding consistent slopes across different mass ranges, morphologies, and environments and others finding a density dependence on the faint-end slope of the LFs. We find that the derived LF slopes for our galaxies are consistent with each other within the uncertainties and therefore in this respect do not depend upon galactic environment.

However, while the derived LF slopes are similar across all tested galaxies, M101 and M94 clearly have fewer satellite galaxies, particularly at the faint end, suggesting that the LF slope may not be the best way to compare. This sparseness might be the result of the galactic environment as M101 and M94 are in lower density regions, as we will now examine.

The relationship between environment and dwarf galaxy number can be quantified via the tidal index parameter (i.e., density contrast, Karachentsev et al. 2013):

$$\begin{eqnarray}&&{{\rm{\Theta }}}_{5}={\mathrm{log}}_{10}\left(\displaystyle \sum _{n=1}^{5}{M}_{n}/{D}_{{in}}^{3}\right)+C.\end{eqnarray} \tag{ 3 }$$

This tidal index (Θ₅) is calculated using the tidal force magnitude on a galaxy i by the neighboring galaxies n. This tidal force magnitude depends on the mass of galaxy n, M_n, and the 3D separation between the galaxies, D_in. C is a constant equal to −10.96, which has been chosen such that if Θ₅ ≤ 0 then the galaxy is isolated. The tidal index is the summation of the tidal force magnitude from the five neighbors of a galaxy where this magnitude is the highest. We find that M101 and M94 have lower tidal indices (0.5 and −0.1, respectively) than the other Local Volume galaxies examined, which are all ≥1.0 (see Figure 8). For this work we draw tidal indices from Karachentsev et al. (2013) where possible. A possible problem with the tidal index is that a satellite galaxy is more likely to be found at the apocenter of its orbit rather than at the pericenter, causing the tidal index to be underestimated. However, the use of the five most influential galaxies should lessen the potential impact of any individual galaxy's orbital positioning.

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In Figure 8 we plot the number of satellites with M_V ≤ −8 as a function of the tidal index of the central galaxy, which appears to show a relationship between tidal index and number of satellites, where the objects with larger tidal indices also have more satellites. The exception to this proposed relation is the MW, which has the largest tidal index of the galaxies examined (2.9); however, it has fewer reported satellites than M31, M81, or Cen A. This could be explained by the spatial incompleteness in surveys of the MW caused by the Galactic plane. We have also examined a more limited sample using only satellites within a 150 kpc projected distance; this showed the same trend, however, with a larger scatter. This potential relationship between satellite number and density deserves further attention in future work.

Careful examination of the CMDs for DwA and Dw9 yield no evidence for star formation and show objects that can be fit by a single old population of stars (see Figure 4 and discussion in Section 4.1). This is supported by a lack of NUV emission or H i gas associated with either DwA or Dw9 and is consistent with the Local Group, where galaxies less massive than the Magellanic Clouds within the virial radius of the MW and M31 are not star forming and have no H i gas (Spekkens et al. 2014).

The SAGA survey (Geha et al. 2017) reported star formation in most (26 of 27) of the satellite galaxies found down to magnitude M_r < −12.3 around eight MW analogs, in contrast to that observed around M31 and the MW. These dwarfs were identified as star forming using the presence of Hα. Here we measure the star-forming fraction of the satellite galaxies of Local Volume galaxies (M81, CenA, M94, and M101). We have determined which dwarfs are star forming via the presence of bright, blue main-sequence stars in resolved imaging. At magnitudes brighter than the SAGA magnitude limit (M_V ≲ −12 mag), we find that the Cen A and M81 groups have star-forming dwarf fractions that are 30% (3 of 10, Crnojević et al. 2019) and 38% (5 of 13, Chiboucas et al. 2013), respectively. This is comparable to those of the MW (40%, 2 of 5) and M31 (20%, 2 of 10). On the other hand, we find that M94 (1 of 1, Smercina et al. 2018) and M101 (3 of 3) both show star formation in all satellites above the SAGA magnitude limit. At fainter satellite magnitudes than that probed by the SAGA survey (e.g., M_V < −12), we observe no star-forming dwarfs around our target host galaxies. The one exception is M94, where we find that all recorded satellites (4 of 4) are star forming (Smercina et al. 2018) with the faintest at M_V = −9.7.

This result points to a relationship between environment and star formation fraction among the brightest satellites of the examined Local Volume galaxies. The SAGA galaxies were selected to be isolated, with no galaxies within one magnitude and 1° in projection and no massive galaxies (5 × 10¹² M_⊙) within 2 virial radii (Geha et al. 2017). These isolation requirements may have pushed the sample toward galaxies in low-density environments more akin to M101 or M94 than to the MW.

This relationship between environment and star formation can be examined directly by again using the tidal index parameter (Karachentsev et al. 2013) as described in Section 5.2. In Figure 9 we plot the star-forming fractions for the satellite galaxies against the tidal index of the host galaxy; we do this for satellites with M_V < −12 and M_V < −8. This figure shows that the objects with low tidal indices have high star-forming fractions among their satellites. Again we have also examined a sample only using satellites within a 150 kpc projected distance; this yields similar trends where galaxies with larger tidal indices have smaller star-forming fractions. However, the scatter is larger due to smaller sample sizes.

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The galaxies examined as part of the SAGA survey do not have published tidal indices, and these are hard to calculate without full distance information. However, we can construct strict upper limits, using the projected distance in place of the 3D separation in Equation (3). This gives a tidal index for the galaxy if all galaxies were at exactly the same distance. This is a poor assumption, and as a result the tidal indecies of these galaxies are likely much lower than this limit. These upper limits show that the SAGA galaxies are likely isolated as only two of the eight have upper limits greater than the tidal index of M31 (1.8). This provides additional evidence for the conclusion that a lower tidal index in a galaxy leads to a higher star-forming fraction in its satellites.

An asymmetry among LSB candidates projected around the M101 group was reported in Bennet et al. (2017). Our follow-up imaging shows that this asymmetry is caused by the presence of the NGC 5485 group to the northeast of M101, rather than being an innate property of the M101 group. However, there is still a curious asymmetry within the M101 group, with the majority of confirmed classical and LSB satellites being found to the southeast. Five of the eight satellites of the M101 group are to the southeast, compared with 2.0 ± 1.4 that would be expected in a purely random distribution. This asymmetry can be seen in Figure 1. Despite the highly asymmetric H i and optical disk of M101 (Mihos et al. 2012, 2013), which show extensive features to the northeast and east, there are no extensions to the southeast.

Examination of the spatial distribution of the NGC 5485 group, on the other hand, shows no distinct asymmetries aside from a slight overdensity to the south, which is also visible in Figure 1. This is likely the result of a selection effect, as the area to the south is closer to M101 and has therefore been more widely studied.