UNDER PRESSURE: STAR CLUSTERS AND THE NEUTRAL HYDROGEN MEDIUM OF TIDAL TAILS

We obtained the Hubble Space Telescope (HST) WFPC2 data for the tidal tail regions studied by Mullan et al. (2011, hereafter M11), Maybhate et al. (2007), and Knierman et al. (2003). These include contributions from cycles 7 (GO-7466) and 16 (GO-11134), with imaging performed in the F555W/F814W and F606W/F814W bands, respectively. The systems that produced these tails were originally selected to represent a large parameter space of observable and dynamical characteristics of galaxy interactions present in the local universe, from optical/H i properties, to progenitor mass ratios, to interaction ages (M11). In this study, we use the same naming convention for debris regions as these publications; e.g., NGC 1487E and NGC 1487W refer to the eastern and western tails of NGC 1487, respectively, while the three WFPC2 pointings of NGC 4038/9 are denoted by NGC 4038A, NGC 4038B, and NGC 4038C.

We also acquired the source catalogs of J2000 positions and photometry of compact objects detected in Maybhate et al. (2007) and M11; the latter contains sources detected by Knierman et al. (2003). Source photometry had been transformed from F555W/F606W and F814W to standard V and I bandpasses with the Holtzman et al. (1995) prescription¹² in these studies to facilitate comparison with the earlier photometry published in standard magnitudes. Photometry had also been corrected for Galactic extinction using Schlegel et al. (1998) and Girardi et al. (2008) (see M11 and Knierman et al. 2003 for details). Objects detected and measured with this methodology were defined as SCCs if they met the following color–magnitude criteria:

• 1.  
  M_V < −8.5. Many studies use a similar magnitude cutoff (M_V = −8 to −9; Knierman et al. 2003; Knierman 2007; Schweizer et al. 1996; Whitmore et al. 1999) because of detection limits and the threat of contamination from non-cluster sources at fainter magnitudes. Source detection within the tail regions of the M11 sample was previously found to be more than 50% complete at M_V < −8.5, so we implement this criterion here. M11 also reaffirmed the findings of Efremov & Ivanov (1987); i.e., that this magnitude limit sufficiently protects against contamination from single main-sequence and post-main-sequence stars.
• 2.  
  V − I < 1.0. This allows for a wide range of possible metallicities, extinctions, and ages in our cluster population (e.g., Peterson 1993; Ajhar et al. 1994; Whitmore et al. 1995; Kundu & Whitmore 2001). This V − I limit was also selected to reflect uncertainties or additional reddening from dynamical (Portegies Zwart et al. 2010), stochastic (Maíz Apellániz 2009), and disruptive (Anders et al. 2009) effects.

For similar reasons, M11 and Maybhate et al. (2007) used V − I < 2.0 as a color limit in their work. In practice, comparing the completeness curves of in-tail versus out-of-tail areas reveal relatively limited tail extinctions (A_V ∼ 0.5; M11), and stochastic effects should be restricted to V − I changes at the ≲5% level (Anders et al. 2009) for the M_V < −8.5, ∼10⁵–10⁶ M_☉ clusters we study. Furthermore, an examination of the M11 sources and their color–magnitude diagrams reveal that a more stringent cutoff helps reduce the number of out-of-tail and presumably background sources detected, which tend to be redder (V − I ≳ 1). These results are substantiated by the multiwavelength observations and selection criteria of Trancho et al. (2012).

Table 1 lists the tails from M11, Knierman et al. (2003), and Maybhate et al. (2007) for which archival H i data exist (22 tails total). We include the dynamical ages and mass ratios of these interacting systems for later reference. Our mass ratio values and tail ages are taken directly from M11 and sources therein. The mass ratio is defined as M₁/M₂, where M₁ and M₂ are the respective masses of the perturbing and main galaxies. Upon further review, we inverted the reported mass ratios of NGC 2782E and NGC 4747 (to 4 and 20) to reflect the understanding in the literature that the tail regions examined here are likely pulled from the smaller-massed galaxies of their interactions.

Table 1. Tidal Tail Observations

Tail	Dist.a	Ageb	Mass	b1d	b2d	Beam	Apixf	Apixg	dvh	$\epsilon _{S_i} dv$i	log $\epsilon _{N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}}$j	Instruments	Original	Project
			Ratioc			P.A.e						and Arraysk	Publications	Codesl
The Mullan et al. (2011) Sample
NGC 1487E	10.8	500	0.25	27.0	24.1	16.1	9	0.02	10	0.93	19.11	ATCA 375+750A+1.5A +1.5B+1.5D+6A	Getts (2001); English & Freeman (2001)	C531
NGC 1487W	10.8	500	0.25	27.0	24.1	16.1	9	0.02	10	0.93	19.11	ATCA 375+750A+1.5A +1.5B+1.5D+6A	Getts (2001); English & Freeman (2001)	C531
NGC 4747	20.2	320	20	61.9	48.5	0.00	400	3.84	16.4	39.47	19.09	WSRT	van der Hulst et al. (2001)	⋅⋅⋅
NGC 520	27.2	300	0.05	24.5	23.8	38.3	25	0.43	10.5	1.13	18.75	VLA C+D	Hibbard & van Gorkom (1996)	AH412/AH417
NGC 2992	36.6	100	1	23.9	17.9	−14.9	16	0.50	20.5	2.78	19.34	VLA C	Duc et al. (2000)	AD402
NGC 2993	36.6	100	1	23.9	17.9	−14.9	16	0.50	20.5	2.78	19.34	VLA C	Duc et al. (2000)	AD402
NGC 2782E	38.1	200	4	19.8	17.9	86.4	4	0.14	10.3	0.53	19.22	VLA B+C+D	Smith (1997); Smith (1994); Smith (1991)	AS389/AS453
NGC 2782W	38.1	200	0.25	19.8	17.9	86.4	4	0.14	10.3	0.53	19.22	VLA B+C+D	Smith (1997); Smith (1994); Smith (1991)	AS389/AS453
NGC 2444	58.2	100	0.50	27.9	27.1	−34.2	16	1.28	20.5	1.70	19.12	VLA C+D	Appleton et al. (1987)	AA63
NGC 2535	59.8	100	0.30	20.4	18.4	16.6	9	0.76	5.1	0.87	19.08	VLA C+D	Kaufman et al. (1997)	AK327
NGC 6872E	62.6	150	0.20	53.5	51.6	25.1	16	1.46	19.8	1.64	19.11	ATCA 750D+EW352+ 1.5G+750F	Horellou & Koribalski (2007)	C979
NGC 6872W	62.6	150	0.20	53.5	51.6	25.1	16	1.46	19.8	1.64	19.11	ATCA 750D+EW352+ 1.5G+750F	Horellou & Koribalski (2007)	C979
NGC 1614N	65.6	750	1	15.0	10.0	−83.0	9	0.91	10.6	3.09	19.63	VLA C+D	Hibbard & Yun (1996)	AH0527
NGC 1614S	65.6	750	1	15.0	10.0	−83.0	9	0.91	10.6	3.09	19.63	VLA C+D	Hibbard & Yun (1996)	AH0527
The Knierman et al. (2003)/Maybhate et al. (2007) sample
NGC 4038A	13.8	420	1	20.7	15.4	24.3	25	0.11	5.2	3.17	19.20	VLA C+D	Hibbard et al. (2001)	AG0516
NGC 4038B	13.8	420	1	20.7	15.4	24.3	25	0.11	5.2	3.17	19.20	VLA C+D	Hibbard et al. (2001)	AG0516
NGC 4038C	13.8	420	1	20.7	15.4	24.3	25	0.11	5.2	3.17	19.20	VLA C+D	Hibbard et al. (2001)	AG0516
NGC 3256E	42.8	400	1	25.7	19.3	6.94	16	0.69	33.5	13.06	20.01	ATCA 1.5B+1.5C+1.5D	English et al. (2003)	C061
NGC 3256W	42.8	400	1	25.7	19.3	6.94	16	0.69	33.5	13.06	20.01	ATCA 1.5B+1.5C+1.5D	English et al. (2003)	C061
NGC 7252E	62.2	730	1	26.9	16.1	18.6	25	2.28	42.5	4.53	19.35	VLA C+D	Hibbard et al. (1994)	AH0372/AH0412
NGC 7252W	62.2	730	1	26.9	16.1	18.6	25	2.28	42.5	4.53	19.35	VLA C+D	Hibbard et al. (1994)	AH0372/AH0412
NGC 3921S	84.5	460	1	19.3	18.1	−85.1	25	4.20	10.7	1.61	18.91	VLA C+D	Hibbard & van Gorkom (1996)	AH0417

Notes. ^aLuminosity distance (Mpc). ^bDynamical age from M11 and sources therein (Myr). ^cDefined as M₁/M₂, with M₁ and M₂ the mass of the progenitor and perturbing galaxy, respectively. ^dBeam dimensions (arcseconds). ^eBeam position angle (degrees). ^fSingle pixel area (arcsec²). ^gSingle pixel area (kpc²). ^hChannel width (km s⁻¹). ⁱ1σ detection limit in the single channel-integrated flux density (10⁻⁴ Jy km s⁻¹ channel⁻¹). ^j1σ detection limit in column density (cm⁻² channel⁻¹). ^kInstruments (ATCA, WSRT, or VLA) and array configurations used in original data acquisition (see original publications for details). ^lATCA and VLA archive project and proposal codes of acquired data.

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Neutral hydrogen data for NGC 1614, NGC 2782, NGC 2992/3, NGC 2444, and NGC 2535 were taken from the VLA archives.¹³ The data were reduced in AIPS using the standard procedures detailed in the AIPS Cookbook.¹⁴ Here, we provide a brief outline of this methodology. In addition to the program source, a bright primary (gain) calibrator and a secondary (phase) calibrator were observed. The primary calibrator was usually observed at least once in the observing program and the secondary calibrator observed every approximately 40 minutes to an hour of time on source. Using the channel-0 data, the flux density of the primary calibrator was found using setjy. Then, the amplitudes and phases of both the primary calibrator and the secondary calibrator were determined using calib. The flux density of the secondary calibrator was determined from the primary calibrator using the task getjy. The calibration was applied to the source using clcal. The channel-zero calibration solutions were copied to the spectral line data. Finally, the spectral line data were bandpass calibrated in bpass using the primary calibrator.

The calibrated data were test imaged in imagr and the line-free channels determined. Using this information, the data were continuum subtracted using uvlsf. The final, continuum subtracted images were produced in imagr; the data were imaged with varying values of the "robust" weighting parameter (Briggs et al. 1999), with values from 0 to 5. A larger robust parameter enhances detection of diffuse H i emission at the expense of angular resolution. In most cases, the diffuse emission was resolved out of the high-resolution image. In the case of two overlapping intermediate frequencies, the images in the overlapping channels were averaged and a combined cube formed.

We acquired a reduced data cube for NGC 4747 from the WSRT Web archive.¹⁵ These data were originally taken, calibrated, and reduced as a part of the WHISP program (van der Hulst et al. 2001). Here, we use the cube generated to ≈60'' resolution, which we found was an appropriate balance between spatial resolution and detecting low-signal H i in the tidal regions of NGC 4747.

The neutral hydrogen data for NGC 6872 and NGC 1487 were obtained from the ATCA archives. The NGC 1487 data were processed using Miriad (Sault et al. 1995) for all steps, including the calibration (with the tasks uvflag, blflag, and mfcal) and removal of continuum (with the task uvlin). The primary calibrators were 1934-638 and 1151-348, while the secondary was 0332-403. We produced images by combining the array configurations listed in Table 1—using standard tasks invert, clean, and restor. The data for NGC 6872 were calibrated with a combination of AIPS and Miriad. AIPS was used to flag (spflag) and image the data (imagr), while data were calibrated (using mfcal because the nearby primary calibrator 1934-638 was used as the secondary calibrator) in Miriad. The ATCA does not Doppler track, so the correct velocity definition was determined in Miriad and applied to the data in AIPS using the task cvel.

We also obtained the reduced cubes used by Maybhate et al. (2007) in their H i analysis. Their reduction strategies are similar, optimizing diffuse H i emission in the VLA-observed tails of NGC 4038, NGC 7252, and NGC 3921, and the ATCA-observed tails of NGC 3256. Data for NGC 520 were obtained from Hibbard & van Gorkom (1996). A brief description of all radio data is presented in Table 1. Table 1 includes source distances, beam dimensions (b₁ and b₂), and beam position angles. We also list angular and physical pixel areas (A_pix), channel widths (dv), 1σ detection limits in single channel-integrated flux densities $\epsilon _{S_i} dv$ (10⁻⁴ Jy km s⁻¹ channel⁻¹) and column densities $\epsilon _{N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}}$ array setups, and original publications and/or project codes for the observations. We use luminosity distances for all systems (assuming H₀ = 73 km s⁻¹ kpc⁻¹), except for NGC 4038 where we adopt the distance measurement of Saviane et al. (2004). This was done for consistency with M11. All cubes were precessed to J2000 for direct comparison with SCC positions.

A cursory glance at Table 1 shows that the pixel sizes of the cubes are not a fixed fraction of the beam sizes. The size and dimensions of the synthesized beam depend on the sampling (and weighting/tapering) of the uv plane in observing these sources, which is a function of the array configurations used, observation program design, and source declination. Works cited in the original publications column contain additional information on these issues for each observation. The pixel size, meanwhile, is chosen to avoid undersampling the beam (fewer than ≈2–3 pixels across the beam) and avoid large computational times (≳10 pixels across the beam, depending on the original map size). Different pixel sizes are selected between these extremes to maintain sufficient signal within pixels and individual channels for the varying source intensities of the tidal tail sample.

We first apply channel selection criteria to all pixels, to identify the channels in which H i emission is confidently identified. There are a variety of methods to identify signal in individual channels of an H i pixel and to mask channels without. For instance, The H i Nearby Galaxy Survey (THINGS; Walter et al. 2008) required that a signal be registered by at least three contiguous channels (each of a width ≲5 km s⁻¹). Their instrument setup was designed to Nyquist sample the warm neutral medium, and they report that this method sufficiently highlights real 21 cm emission and eliminates noise.

Our data exhibit a much wider range of channel widths, from ≈5.13 to 42.5 km s⁻¹, so we must avoid such a constraint for cubes where multiple channels would span an unrealistically broad velocity space. Moreover, our data are inhomogeneous in sensitivity, so the appearance of an H i line profile may vary depending on these instrumental constraints and orientation/projection effects. Thus, we customized the channel selection criteria for each tail after examining the spectra of many pixels across the tidal debris. Table 2 lists the channel selection requirement as a signal-to-noise ratio for single channel signal and equivalent integrated flux density for each tail, along with the minimum number of adjacent channels N_chan needed at that flux density for a pixel to count as signal.

The values in Table 2 were selected after an extensive trial-and-error process for each tail that produced reasonable spectra and maps of the properties described in Section 3.3 below. We varied both N_chan and the necessary signal-to-noise ratio until no single, isolated pixels in maps of the line-of-sight velocity dispersion (σ_los) registered physically unrealistic values within the tail regions (i.e., ≳150 km s⁻¹, depending on the tail). Pixels within a beam size are statistically correlated, so finding occasional pixels with discrepant values in velocity dispersion maps indicated that noisy channels outside the velocity space of actual emission were unwittingly selected and thus tended to increase σ_los relative to the surrounding pixels. The spectra of these locations were also visually studied to optimize our masking parameters.

We then masked out, i.e., ignored, the noise channels in the cube, with the exception of noise channels alongside channels containing signal. These were left unmasked, as were all adjacent channels until a flux density level consistent with 0 Jy (±1σ) was found along either side of those original signal channels. This helped to prevent clipping of broad spectral features present across multiple channels. If any of these additional channels registered flux densities below 0 Jy, they were then masked to discourage biasing any measurements toward lower values. When integrated, the unmasked pixels provide a total signal typically >3σ over the channel-integrated rms noise.

We display several examples of results from our signal selection procedure in Figures 1 and 2. A spectrum from one randomly selected pixel from each tail is shown in these figures, plotted in gray; the overplotted error bars indicate the rms noise. Signal channels are indicated as black points. The heterogenous channel widths and noisy quality of some of the randomly chosen H i profiles are clearly evident in these figures, highlighting the need for a varied signal selection criteria across the sample.

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We also limit our analysis to regions in our H i data encompassed by previous optical observations, i.e., selecting H i pixels that are "in-tail" and within the WFPC2 field of view (FOV) of the previous studies. Within the WFPC2 images of their data sets, M11 and Knierman et al. (2003) defined areas that were at least one count above the sky level in F606W or F555W as "in-tail." These groups used the WF2–4 chips in their analysis, choosing to ignore the planetary camera with its prohibitively high read noise. Maybhate et al. (2007) applied this definition and the optical boundaries it demarcates to the H i maps of their tidal tail sample to study the local column densities within. In M11, the expanded tidal tail sample afforded a glimpse at tails with different H i and optical morphologies, whose optically bright regions did not always coincide with accumulations of H i (cf. Hibbard et al. 2000). Consequently, we redefine "in-tail" to refer to an area of the H i data that is confined to the WFPC2 footprint of its corresponding optical observations, and is either within the optical boundaries set by M11 or met the H i criteria of Section 3.1.

For each tail, we determine the physical extent of the H i emission by convolving a map of the integrated flux density with an artificial beam twice the size of the synthesized beam and masking out pixels that do not satisfy the minimum criterion for H i signal. We found that this technique successfully suppresses artifacts in the cube that are beyond the true large-scale emission in the tidal debris, as opposed to using no smoothing kernel or kernels of sizes ≳3 times that of the beam. Regions determined to reside at the WFPC2-imaged edges of the host galaxies by M11 were rejected.

3.3.1. Primary Properties

The three-dimensional nature of the H i cubes provides an opportunity to examine not only the distribution of local H i surface densities, but also the velocities and kinematics of the gas. Thus, for each in-tail pixel, we compute the total integrated intensity I_tot (Jy beam⁻¹ km s⁻¹), the mean line-of-sight velocity v_los (km s⁻¹), and the line-of-sight velocity dispersion σ_los (km s⁻¹). In each pixel, we calculate I_tot as the zeroth moment of the velocity–intensity distribution:

$$\begin{equation} I_{{\rm tot}} = \sum _{i=1}^N I_i\, dv, \end{equation} \tag{ 1 }$$

where I_i is the intensity in each signal channel and dv is the channel width in km s⁻¹. Again, The corresponding integrated flux density S_tot can be obtained by multiplying I_tot by the ratio of pixel and beam angular sizes $A_{{\rm pix}} A_{{\rm beam}}^{-1}$ (both in arcsec²). The line-of-sight velocity and (squared) velocity dispersion is given by the first and second moments:

$$\begin{equation} \hbox{$v_{{\rm los}}$}= \frac{1}{ \sum _{i=1}^N I_i} \sum _{i=1}^N v_i I_i, \end{equation} \tag{ 2 }$$

and

$$\begin{equation} \hbox{$\sigma _{{\rm los}}$}^2 = \frac{1}{ \sum _{i=1}^N I_i} \sum _{i=1}^N (v_i - \hbox{$v_{{\rm los}}$})^2 I_i, \end{equation} \tag{ 3 }$$

respectively. In Equations (2) and (3), v_i is the velocity value for the ith channel. Summations were performed only on channels left unmasked as described in Sections 3.1 and 3.2 for all equations, and errors in I_i ($\epsilon _{I_i}$) were propagated through to all quantities. We adopt a minimum value of 50% of the given channel width for σ_los (an instrumental limit), as well as for errors in both v_los and σ_los. For pixels defined as in-tail by virtue of their optical properties alone, these moments were not computed and the H i properties were classified as "unfitted." In occasional cases of marginally detectable H i emission (a moment-zero measurement), higher moments were unable to be reliably determined, and such pixels were therefore unfitted in v_los and σ_los.

For visual reference, we display moment-zero integrated flux density maps of the full extent of each data cube in Figure 3, with WFPC2 fields of view overlaid. In all images, north is up, east is to the left of the frame, and every WFPC2 footprint is ≈247 on a side. The sizes and orientations of the synthesized radio beams are represented in the bottom left of each frame in this figure.

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We also experimented with finding the equivalent quantities by fitting the H i spectra offered by each pixel with a series of Gaussian functions. In practice, we found that our values of S_tot for both methods agreed very well. However, spectral fitting often yielded multiple Gaussian components, so in many cases it could not be easily determined which component was "best" associated with the tidal debris and which v_los and σ_los should be compared to the moment equivalents (cf. the complex line structures evinced in Figures 1 and 2). The intensity-weighted mean of the individual components facilitated a comparison of the v_los measurements, but there was no clear prescription for comparing the number of σ_los contributions from the Gaussian fits to the moment-based measurement. Many pixels throughout the tail sample can be well fit by multiple Gaussian components that overlap in varying amounts, so summing the Gaussian-fit velocity dispersions would be an inaccurate universal tracer of kinematic complexity.

In all, because the results for integrated flux density were very similar, and for simplicity of analysis we wish to use single quantities to describe the velocity structure at positions along a tidal tail, we choose to employ the moment measurements for these primary properties. Similar work for galaxy disks can be found in Tamburro et al. (2009). While this method loses sight of interesting kinematics in some locations along tidal tails, e.g., multiple overlapping Gaussian fits and extended or asymmetric profiles, complex line structures are still reflected in large second moments of the intensity distributions. Our measurement of σ_los is thus still a tracer of such tail H i kinematics.

3.3.2. Secondary Properties

Further H i characteristics can be calculated from the moment-based primary properties, e.g., the H i column density, mass surface density, and large-scale velocity gradient. The total integrated intensity I_tot is related to the H i column density $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ (cm⁻²) by the simplified relation (e.g., Verschuur & Kellermann 1988):

$$\begin{equation} \hbox{$N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$}\ ({\rm cm^{-2}}) = 1.247 \times 10^{24} \frac{I_{\rm tot}}{A_{{\rm beam}}}. \end{equation} \tag{ 4 }$$

Here, A_beam is the area of the beam in square arcseconds. The error in $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ is given by

$$\begin{equation} \hbox{$\epsilon _{N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}}$}({\rm cm^{-2}}) = 1.247 \times 10^{24} \frac{\sqrt{N_{\rm u}}}{A_{{\rm beam}}} \epsilon _{I_i} dv. \end{equation} \tag{ 5 }$$

N_u denotes the number of unmasked channels for the specified pixel, dv is the channel width (km s⁻¹), and $\epsilon _{I_i}$ is the single channel rms error in I_i. The corresponding error in S_i ($\epsilon _{S_i}$) is then simply ($A_{{\rm pix}} A_{{\rm beam}}^{-1}$) $\epsilon _{I_i}$.

We moreover measure the mass surface density, kinetic energy density, and velocity gradient of each pixel. The mass surface density $\Sigma _{M_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}}$ and its uncertainty $\epsilon _{M_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}}$ are calculated with the following equations (from Verschuur & Kellermann 1988, with some alterations to change masses to mass surface densities and include unit conversions):

$$\begin{equation} \hbox{$\Sigma _{M_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}}$}\ (\hbox{$M_{\odot }$}\,{\rm pc^{-2}}) = 1.00 \times 10^4 \frac{I_{\rm tot}}{A_{{\rm beam}}}, \end{equation} \tag{ 6 }$$

and

$$\begin{equation} \hbox{$\epsilon _{\Sigma _{M_{\scriptsize {\rm H}\,\scriptsize{I}}}}$}\ (\hbox{$M_{\odot }$}\,{\rm pc^{-2}}) = 1.00 \times 10^4 \frac{\sqrt{N_{\rm u}}}{A_{{\rm beam}}} \epsilon _{I_i} dv, \end{equation} \tag{ 7 }$$

where $\epsilon _{I_i}$ is the single channel rms error in intensity as above.

To characterize the dynamical state of the H i gas, we further define the kinetic energy density of the gas in a single pixel as $\hbox{$\Sigma _{{\rm KE}}$}= \alpha \hbox{$\Sigma _{M_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}}$}\hbox{$\sigma _{{\rm los}}$}^2$, where the factor α = 3/2 assumes the velocity distribution is isotropic. Tamburro et al. (2009) similarly calculate a pixel-by-pixel kinetic energy for their sample of non-interacting galaxies. For streaming H i features at a variety of inclinations and projection effects, this constant may differ in reality by a factor of a few. Thus, we consider this only an order-of-magnitude tracer of the actual, local turbulent kinetic energy per unit area.

Last, we produce a diagnostic of the locally centered line-of-sight velocity gradient across the plane of the sky dv_los/dr_⊥ (km s⁻¹ kpc⁻¹) from the rms of the velocity gradients between the pixel of interest and others in its immediate neighborhood. We write this as

$$\begin{equation} \hbox{$dv_{{\rm los}}/dr_{\perp }$}= \frac{1}{\sqrt{N_{{\rm pix}}}} \left(\sum _{i=1}^{N_{{\rm pix}}} \frac{(v_i - v)^2}{dr_{\perp ,i}^2}\right)^{\frac{1}{2}}, \end{equation} \tag{ 8 }$$

where v is the H i velocity for the central pixel of interest and v_i is the equivalent velocity for the ith nearby pixel with which we construct a velocity gradient over dr_{⊥, i}, the distance in kiloparsecs between the pixels. N_pix is the number of pixels that surround the central pixel of interest and are used in the calculation. Here, we use all pixels with measurable H i emission that surround the central pixel in an annulus whose inner and outer radii are set by 0.5 and 1.5 times the maximum dimension of the beam. This allows a comparison between the central pixel of interest and the closest statistically uncorrelated pixels 1 beam away. Thus, while previous quantities pertain to pixel scales (∼kpc), this velocity gradient is measured over several beams, and probes gas motions and kinematics on larger (∼10 kpc) scales. If v_los and σ_los were unable to be determined in a given pixel, then that pixel was also deemed "unfitted" by dv_los/dr_⊥.

As an example of these calculations, we present a montage of $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$, $\Sigma _{M_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}}$, σ_los, and v_los maps in Figure 4 for the tail NGC 2535. Corresponding maps of the remaining tail sample can be found in full resolution in the online version of Figure 4. As before, we include the WFPC2 FOV, but also indicate positions of SCCs with circles. We furthermore introduce maps of Σ_KE and dv_los/dr_⊥ in this figure. Contours of log $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ from 20.0 to 21.4 (cm⁻²) in steps of 0.2 dex are overlaid in all maps.

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View figure set (22 panels)

Tarball of figure set images

In this section, we concentrate on the pixels of a given tail that contain at least one SCC. The sources detected in M11 are not verified clusters, but cluster candidates, so each of these pixels has a probability that it contains at least one actual, luminous cluster. We therefore first examine the source statistics in a manner consistent with M11 to determine how likely a source is a cluster rather than a background contaminant. From this, we compute the probability that the pixel containing that SCC has at least one cluster.

Similar to M11, we determine the relative areas within and outside the tail regions surveyed within the WFPC2 footprint, A_in and A_out, respectively. The main difference in this study, aside from pixel sizes, is the optical/H i tail definition used here. Within these areas, we find the number of SCCs N_in and N_out. We divide N_in by A_in to find the overall in-tail source density Σ_in; these results are listed in Table 3 for each tail, along with the in-tail completeness limits found in M11.

Table 3. Source Statistics

	Source Detections						Star Counts			M11 Results
Tail	$M_{V,50\%}$a	Ninb	Noutc	Aind	Aoute	Σinf	Σoutg	Σout × Ainh	fSCi	Tail-wide
										SCCs?j
NGC 1487E	−4.6	1	0	40.60	5.89	0.025	0.039	1.56	0
NGC 1487W	−4.5	0	0	41.07	5.52	0	0.039	1.58	1
NGC 4747	−6.0	6	1	84.52	84.52	0.071	0.014	1.22	0.80
NGC 520	−6.4	2	0	266.21	28.66	0.008	0.007	1.87	0.06
NGC 2992	−7.2	10	3	305.49	229.87	0.033	0.009	2.77	0.72	$\checkmark$
NGC 2993	−7.3	11	1	367.49	169.38	0.030	0.009	3.34	0.70	$\checkmark$
NGC 2782E	−7.5	56	5	413.93	165.16	0.135	0.006	2.50	0.96	$\checkmark$
NGC 2782W	−7.3	6	2	472.48	105.25	0.013	0.006	2.85	0.52
NGC 2444	−8.2	5	7	504.38	850.42	0.010	0.004	1.83	0.63
NGC 2535	−8.4	36	8	871.95	554.81	0.041	0.003	2.36	0.93	$\checkmark$
NGC 6872E	−8.3	111	28	730.59	824.29	0.152	0.009	6.68	0.94	$\checkmark$
NGC 6872W	−8.5	101	14	729.12	827.22	0.139	0.009	6.66	0.93	$\checkmark$
NGC 1614N	−8.6	19	8	304.89	1408.86	0.062	0.003	1.01	0.95	$\checkmark$
NGC 1614S	−8.6	30	10	325.82	1387.93	0.092	0.003	1.07	0.96	$\checkmark$
NGC 4038A	−5.8	3	0	45.79	30.30	0.066	0.050	2.30	0.24
NGC 4038B	−5.1	1	0	46.91	29.96	0.021	0.050	2.35	0
NGC 4038C	−5.8	1	0	27.50	48.71	0.036	0.050	1.38	0
NGC 3256E	−7.8	32	9	460.19	271.43	0.070	0.035	16.12	0.50
NGC 3256W	−7.8	51	4	580.74	146.74	0.088	0.035	20.34	0.60	$\checkmark$
NGC 7252E	−8.1	11	16	627.03	928.00	0.018	0.003	2.18	0.80
NGC 7252W	−8.1	17	11	533.54	1016.92	0.032	0.003	1.85	0.89
NGC 3921S	−8.8	14	5	1718.82	1147.28	0.008	0.001	1.79	0.87

Notes. ^a50% V-band completeness limit (M11). ^bNumber of SCCs detected within the tail region. ^cNumber of SCC-like objects detected outside the tail region. ^dArea of in-tail region (kpc²). ^eArea of out-of-tail region (kpc²). ^fIn-tail SCC surface density (kpc⁻²). ^gOut-of-tail surface density of SCC-like objects (kpc⁻²). ^hProbable number of in-tail SCCs that are contaminants. ⁱOverall fraction of SCCs that are likely star clusters. ^jThis column is checked if M11 found evidence for large-scale populations of star clusters within the tail regions.

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M11 also calculated the out-of-tail density Σ_out = N_out/A_out and subtracted it from Σ_in to find the background-subtracted, average in-tail density of SCCs in these tidal tail regions. However, Table 3 shows that many, particularly close systems have extensive H i and optical coverage and small out-of-tail areas A_out. The number of detected out-of-tail objects within these regions is statistically uncertain, or not at all known (N_out = 0) in some cases.

We therefore determine Σ_out with results from star counts generated by the Galactic population synthesis model of Robin et al. (2003).¹⁶ In a 100 arcsec² solid angle centered at coordinates in each tail, we generate the number of foreground stars visible within the photometric criteria of Section 2. This includes a V − I < 1 cutoff and an apparent V-band magnitude range matching the sources detected in, and completeness limits of, M11 (19 ≲ V ≲ 26). We then determine the resulting number of contaminants per out-of-tail area Σ_out using the distance to each tail. We find that variations of solid angle (1–100 arcsec²) introduce variations of only a few percent.

In this calculation, we also roughly model the photometric errors of the returned catalogs on the errors in photometry in M11. Here, errors in magnitude m, σ_mag are calculated by a simple linear model σ_mag = A + Bm, where A and B are 0.01 and 0.0035, respectively. Moreover, we employ a mean V-band diffuse absorption of 0.7 mag kpc⁻¹, which is recommended for intermediate-to-high Galactic latitudes (Robin et al. 2003). It should be noted that this may introduce some uncertainty for NGC 3256E/W, which is a particularly low-latitude (b ≈ 11°) object. But ultimately, it is a more reliable method that is more free of statistical uncertainties with low counts of detected objects. In addition, these sources were screened for extended sources in M11, so contamination by background galaxies—already a minor issue from the stringent V − I cutoff—is minimal. In short, the primary contamination should be foreground stars, and this is better assessed with the current method.

We want to concentrate on individual in-tail SCCs and their local environments, so we then calculate the probability p_real that an SCC-containing pixel contains at least one cluster based on the probable degree of in-tail contamination by out-of-tail sources. To do so, we first find the area-adjusted number of probable contaminants in the in-tail source list, Σ_out × A_in. This quantity divided by N_in is simply the fraction of sources within the tail that are likely contaminants, or Σ_out/Σ_in. The corresponding fraction of sources that are likely SCs is then f_SC = 1 − (Σ_out/Σ_in).

We record various steps of this process in Table 3, along with the calculated V-band 50% completeness limits for each tail (M11; Knierman et al. 2003; Maybhate et al. 2007) for reference. For pixels containing one SCC, p_real is just the value of f_SC for that tail. More generally, and especially for pixels with multiple SCCs, p_real is the result of applying the binomial distribution, using n SCCs in a pixel, each with a probability p_real = f_SC. Thus, the odds that a pixel with multiple SCCs has at least one actual cluster are higher than those listed for single SCC-harboring pixels listed in the penultimate column of Table 3. We indicate in the last column whether M11 found strong statistical evidence of tail-wide cluster population in each of the tail regions. With the exception of the low-latitude (and highly foreground source-contaminated) NGC 3256W, f_SC values are all ≳0.7. In contrast, tails without global cluster populations exhibit a wider range of values.

As a final check, we compared the distribution and properties of SCs detected in NGC 2782W by Torres-Flores et al. (2012) and M11. Because of the instrumental differences between the studies (the former use Galaxy Evolution Explorer imaging and optical spectra, and the latter use HST imaging), there is ≈50% overlap between the source catalogs for that tail. That is, M11 detected about twice as many cluster candidates as Torres-Flores et al. (2012) spectroscopically characterized clusters. This agrees very well with results from the star counts model that predicted 52% of the M11 detections are contaminants.

We now compare the distributions of $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$, dv_los/dr_⊥, σ_los, and Σ_KE for between H i pixels not containing SCCs and those that do. In doing so, we must ignore the fraction of pixels that were unfitted by the measuring techniques, f_unfit; these are typically pixels that fall within the optical tail, but have ill-defined H i content. To outline potential contrasts between the pixels that are home to potential SCs and those that are not, we performed K-S tests between these distributions for each measured property. Here, p_KS values reveal the probability that the SCC-sporting and SCC-empty pixels come from the same distribution of each measurement. If p_KS ≲ 0.01, it is likely that these two categories of pixels have two separate distributions of an H i characteristic. The results of these tests are also provided in Table 4; columns indicate the results of this test performed on the indicated property. We indicate where values are below 10⁻⁴, as specific results below this cutoff are not enlightening.

Table 4. K-S Probabilities

Tail	$N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$	σlos	dvlos/dr⊥	ΣKE
NGC 1487E	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅
NGC 1487W	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅
NGC 4747	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅
NGC 520	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅
NGC 2992	1.3e-3	0.10	0.52	0.23
NGC 2993	0.17	0.039	0.45	0.086
NGC 2782E	<1e-4	<1e-4	3.7e-4	<1e-4
NGC 2782W	0.80	0.43	0.67	0.33
NGC 2444	0.53	0.37	0.090	0.61
NGC 2535	0.012	0.15	9.6e-4	0.039
NGC 6872E	<1e-4	<1e-4	0.076	<1e-4
NGC 6872W	0.24	0.26	9.8e-3	0.33
NGC 1614N	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅
NGC 1614S	0.93	0.39	0.017	0.89
NGC 4038A	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅
NGC 4038B	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅
NGC 4038C	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅
NGC 3256E	0.71	<1e-4	0.15	0.61
NGC 3256W	0.16	0.28	0.011	0.77
NGC 7252E	0.99	0.50	0.63	0.99
NGC 7252W	0.42	7.7e-3	0.044	0.39
NGC 3921S	0.41	0.27	0.87	0.25

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Overall, it appears that the distributions of our calculated H i column densities, kinematic tracers, and mechanical energy densities for both SCC-carrying and SCC-devoid pixels are indistinct (p_KS ≳ 0.01). NGC 2782E and NGC 6872E are remarkable counterexamples, with low computed K-S probabilities and two of the highest tail-wide SCC population densities reported in M11 (≈0.2 SCCs kpc⁻²). A few other tails show evidence of independent distributions of individual properties between these two pixel types (e.g., NGC 2992 with $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$, NGC 2535 and NGC 6872W with dv_los/dr_⊥, and NGC 3256E and NGC 7252W with σ_los). The implications of these results are discussed for individual tails in the Appendix. It is probable that the varying degree of contamination from foreground stars across the tail sample—displayed in Table 3—contributes to a contamination of the H i distributions for the pixels hosting at least one SCC and confuses the K-S results.

It is additionally important to consider that there are minor, inherent shortcomings to K-S tests; namely, that they are particularly sensitive to differences in the centers of cumulative distribution functions relative to their edges. Also note that many tails have small samples of SCC-carrying pixels, to which such statistical tests are particularly sensitive. as mentioned at the outset of this section, pixel size variations also introduce differences in the number of pixels considered, and the coarseness of the unbinned H i measurement distributions. This will have some, albeit minor, affect on the results of the K-S tests.

With all uncertainties considered, two distributions with K-S probabilities ≳0.01 may still be independent but are difficult to differentiate here. Regardless, it appears that the dependence of SC positions (and possibly clues to their formation) on individual H i densities and kinematics cannot be easily ascertained on a tail-by-tail, property-by-property basis like this. We therefore also explore the holistic H i parameter space of the tail pixels, to see if and how all these H i properties collectively relate to SCCs. Figure 5 shows the combined distributions of $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ and σ_los (left), and Σ_KE and dv_los/dr_⊥ (right) of NGC 2535. Similar plots for the remaining tail sample are available in the Online Figure Set 5. Bin sizes of 0.1 dex in log $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ (cm⁻²), 5.0 km s¹ in v_los, 2.0 km s⁻¹ kpc⁻¹ in dv_los/dr_⊥, and 0.25 dex in log Σ_KE (erg pc⁻²) were employed in contouring. Contour levels are scaled to show the fraction of the total number of tail pixels examined, including pixels that were unfitted by $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$, σ_los, dv_los/dr_⊥, or Σ_KE measurements. We record the fraction of unfitted SCC-carrying and SCC-devoid pixels for reference as $f_{{\rm unfit}}^{{\rm SCC}}$ and f_unfit, respectively. Median uncertainties for all unbinned properties are shown as red error bars.

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Dashed lines in the first plot show potentially important values of the plotted properties—the Maybhate et al. (2007) $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ threshold (horizontal line), and the turbulent–thermal transition σ_los ≈ 10 km s⁻¹ (vertical line; this shows where gas velocity dispersions are driven by turbulent instead of thermal motions). The vertical dashed line on the second plot indicates a constant dv_los/dr_⊥ = 15 km s⁻¹ kpc⁻¹, approximately the value of the Oort constant A, the value of the shear in the solar neighborhood. This line is purely fiducial and meant to guide distinctions between relatively shallow and steep velocity gradients. The horizontal line on that plot shows a Σ_KE cutoff of 10⁴⁶ erg kpc⁻², which considers an energy threshold set by setting a hypothetical critical H i mass surface density for star formation of 3 M_☉ pc⁻² (Schaye 2004) and threshold turbulent velocity dispersion 10 km s⁻¹. These demarcations are meant to guide the eye and highlight what fraction of a tail has relatively more dense, turbulent, etc., gas.

We indicate the fraction of the total number of tail pixels that exhibit super"critical" H i characteristics for the combined distributions with $f_{{\rm gas}}^{{\rm Q1}}$ for the left plot, and $f_{{\rm gas}}^{{\rm Q2}}$ for the right. "Q1" and "Q2" refer to particular quadrants delineated by the dashed lines, i.e., gas with log $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ > 20.6 cm⁻² and σ_los > 10 km s⁻¹ (left plot), and gas with log Σ_KE > 46 erg pc⁻² and dv_los/dr_⊥ < 15 km s⁻¹ kpc⁻¹ (right plot). Last, the SCC-laden pixels are overplotted in both plots as gray points if their calculated p_real values are ⩽0.94, and as black points if p_real > 0.94. This value of 0.94 was selected with some foresight and will be addressed further in Section 5. Recall that p_real indicates the likelihood that an H i pixel contains at least one SC. The plots in the online figure sets of Figures 4 and 5 are discussed individually for each tail region in the Appendix.

A clear correlation is evident between H i column density and line-of-sight velocity dispersion for much of our sample as seen in Online Figure Set 5. The example spectra displayed in Figures 1 and 2 show that many of these randomly selected H i profiles are kinematically complex, and many would have wide velocity dispersions as measured by the second moment of the intensity distribution. The addition of multiple H i components therefore both broadens σ_los and increases $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$. The degree of overlap between components in a spectrum may depend on the tail kinematics and projection effects, so we would perceive a different relationship—i.e., slope of contoured distributions—between these two properties in each tail of Figure Set 5. Most importantly, detecting multiple H i components indicates that the actual physical density of H i at the location of a pixel (or SCC) may not be consistently mapped by column density in all tails, so some confusion in interpreting the importance of $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ is expected.

In this section, we address the issue of heterogeneous physical resolution in our H i data and facilitate a better comparison between the tails in our sample. To make our H i cubes uniform in pixel scale, we repeated the work of Sections 3 and 4 for the tails, but with their cubes re-gridded to the largest physical scale of the tail sample, i.e., the 2.1 kpc pixel scale of NGC 3921S. We exclude the NGC 6872E/W tails in this analysis, as their physical beam sizes (≈16 kpc) are much larger than the rest of the sample (several kpc) and probe H i gas on correspondingly different spatial scales.

For this task, we used the IDL function frebin, which conserves flux for every spatial slice of the cubes. In all calculations requiring the beam size, the beam dimensions of each original cube were used, with the exception of the calculation of dv_los/dr_⊥ (Equation (8)). Here, we used an annulus whose inner and outer radii are 4 and 12 kpc in the determination of N_pix. This simulates using an 8 kpc beam, which is physically the size of the largest beams in the sample (excluding NGC 6872E/W) and allowed a more direct comparison across the tail sample.

We then sorted all the analyzed tail pixels into two categories based on pertinent characteristics of the interactions—pixels from the list of tails with widespread cluster populations as determined by M11 versus those from the tails that did not, pixels from tails that resulted from early-stage and late-stage interactions, and pixels from tails produced in interactions from different mass ratios of their progenitor galaxies. This approach allows a more significant assessment of H i and merger properties with respect to cluster populations than additional individual studies would allow, especially for the closest half of the tail sample where the large physical pixel scales prohibit meaningful individual study. In the subsections below, we therefore discuss these distinctions and the distributions of H i properties in greater detail, accompanying Figures 6–11.

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In Figures 6, 8, and 10, we outline the normalized distributions of $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$, dv_los/dr_⊥, σ_los, and Σ_KE for pixels from these types of tails. Subcategories of SCC richness, age, and mass ratio are represented as blue, filled histograms, and orange, outlined histograms. Cumulative distributions of these pixels are plotted over the histograms as blue and orange curves, respectively. In all cases, bin sizes were set by the number of pixels in each tail, and the standard deviation of their various distributions. We also indicate the fraction of pixels that were unfitted by the measuring techniques as f_unfit; as before these are typically pixels that fall within the optical tail, but have ill-defined H i content.

We also perform K-S tests between the (unbinned) blue and orange distributions in each panel of these plots. In all cases, we calculate extremely low probabilities (p_KS < 10⁻⁴) that these pairs of distributions are statistically indistinguishable. Thus, the differences in H i properties between tail types we note below are likely real and significant.

Figures 7, 9, and 11 are meanwhile analogous to the contour plots of the online figure set of Figure 5, excluding the presentation of f_SC, which changes tail to tail. Individual pixel p_real probabilities were computed as before for each pixel containing at least one SCC from each tail and are reflected here. The fraction of unfitted SCC-carrying pixels $f_{{\rm unfit}}^{{\rm SCC}}$ is also shown. Contour bins are also the same as in Figure 5, as are all definitions of H i thresholds. In each of these figures, we juxtapose the $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$–σ_los and Σ_KE–dv_los/dr_⊥ parameter space of the two types of tails considered for visual comparison.

In the following subsections, it is important to note that projection effects are still an issue. However, a reasonably random set of tail orientations would artificially change values of properties by different amounts and blur distinctions between distributions of an H i variable between two types of tails. We therefore contend that, while statistically similar distributions may be inconclusive about physical differences between types of debris, statistically different distributions are still significant even against irrevocable geometric effects. Any differences between tail types are likely telling of real physical dissimilarities in the H i content between them.

5.1.1. H i and Tail-wide SCCs

5.1.2. H i and Tail Dynamical Age

5.1.3. H i and Progenitor Mass Ratio

Supernovae (SNe), particularly Type II, are generally regarded as the dominant source of feedback in the ISM in galactic contexts (e.g., van den Bosch 2000; Tasker & Bryan 2006, 2008; Spitzer 1990). It then becomes a question for this project of whether the SNe produced in prior star formation events are the primary source of the mechanical energies observed in the neutral H i in these tidal tails. That is, do the kinetic energy densities (and presumably pressures) that we measure result from star and cluster formation, or are they the cause of star and cluster formation?

Using UV/IR-derived SFRs of a sample of relatively low inclination disk galaxies in the THINGS survey, Tamburro et al. (2009) calculated the pixel-by-pixel kinetic energy densities that would be produced from feedback from core-collapse SNe. In comparing these to measurements of the actual H i energy densities, they find that the efficiency of energy injection from SNe would have to be unrealistically high in order for their measured Σ_KE values to be supplied by star formation. Not having a uniform survey of UV, IR, or Hα diagnostics of SFRs for our sample, we instead aim to determine the potential SFRs of our tidal tails from their Σ_KE and suggest whether the results are reasonable for this kind of environment.

As in Tamburro et al. (2009), we begin by equating the kinetic energy density to that theoretically provided by SNe; i.e., Σ_KE = $\epsilon _{{\rm SN}} \dot{E}_{{\rm SN}} \tau _{{\rm SN}}$. $\dot{E}_{{\rm SN}}$ is the rate at which energy is released by SNe, which is a function of the SFR; _SN is the efficiency with which that energy is converted to the observed turbulence, and τ_SN is the turbulence dissipation rate. Combining their Equations (5)–(7), characterizing the dissipation timescale and fraction of the stellar populations that produce SNe, we derive the following expression for the pixel-by-pixel SFR (ϒ):

$$\begin{eqnarray} \frac{\Upsilon }{\hbox{$M_{\odot }$}\,{\rm yr}^{-1}} &=& 2.03 \times 10^{-3} \frac{\alpha }{\epsilon _{{\rm SN}}(1+f_{{\rm 1a}})} \left(\frac{f_{{\rm SN}}}{\langle M/\hbox{$M_{\odot }$}\rangle }\right) \nonumber\\ && \times \left(\frac{\lambda }{100\,{\rm pc}}\right)^{-1} \left(\frac{M_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}}{\hbox{$M_{\odot }$}}\right) \left(\frac{\hbox{$\sigma _{{\rm los}}$}}{10\,{\rm km\,s}^{-1}}\right)^{3}. \end{eqnarray} \tag{ 9 }$$

As before, α corresponds to the degree of isotropy (here assumed to be 3/2), and the factor 1/(1+f_1a) is a term included in our derivation to account for the fractional contribution of Type 1a SNe. Following Mannucci et al. (2005), we consider f_1a values in the range 0–0.33. The term f_SN indicates the fraction of (recently formed) stars that end as core-collapse SNe, while 〈M/M_☉〉 denotes the average mass of the stellar population within the examined region. Dividing these two terms in the equation, we consider values in the range 0.9 × 10⁻² to 1.3 × 10⁻², as shown by Tamburro et al. (2009) for an upper initial mass function (IMF) mass limit of 20–50 M_☉, with a mass function of the form $\phi (m) = m^{-\alpha _0}$, where α₀ = 1.3 for 0.1–0.5 M_☉ and 2.3 for 0.5–120 M_☉ (e.g., Calzetti et al. 2007, and references therein). We use values in the range 0.1–0.5 for the efficiency _SN; the lower limit is suggested by numerical simulations (Thornton et al. 1998), and the upper limit provides a conservative lower limit to ϒ. The turbulence driving scale is given by λ, which simulations predict to be 100 ± 30 pc (Joung & Mac Low 2006; de Avillez & Breitschwerdt 2007). The mass of H i is $M_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$, and σ_los as defined here. Note the additional factor of σ_los; this results from the implicit expression of τ_SN in terms of λ and σ_los. The constant 2.03 × 10⁻³ incorporates all unit conversions and the 10⁵¹ erg that are assumed to be released from each Type II SN event (Heiles 1987).

We then mapped out the pixel-by-pixel SFR of all the tails using our maps of $\Sigma _{M_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}}$ and σ_los, for the combinations of aforementioned constants that yield the smallest and greatest result. We performed this for both original maps and those produced at the constant pixel scale. In practice, changing the efficiency of injection _SN has the dominant effect in this equation; we added the results of all pixels belonging to individual tails and record the results in Table 5. We posit that if the tail-wide ongoing SFR (of similar timescale as the turbulence dissipation timescale) could fall between the ranges quoted in this table, it is possible that star formation and the SNe explosions that result from it generally caused the turbulent kinetic energy densities we observe rather than was an effect of it. Ideally, uniformly deep, wide-field imaging in Hα of our tail sample would provide the SFR on the corresponding timescale (∼10 Myr), allowing a direct and straightforward comparison. Unfortunately, no such survey exists.

There are a few potentially elucidating case studies, however. Knierman et al. (2012) find a Hα-derived SFR for the entire western tail of NGC 2782 of 9 × 10⁻⁶ M_☉ yr⁻¹ kpc⁻². Adjusting for the in-tail area used here, we calculate a rate ϒ ≈ 2 × 10⁻³ M_☉ yr⁻¹. This is several orders of magnitude lower than the values reported in Table 5; thus the H i turbulence is not likely caused by star formation in this case. Moreover, the Hα data of Mihos et al. (1993) indicate that most current star formation in NGC 6872 is confined to its tidal tails, and is occurring at a rate ∼3 M_☉ yr⁻¹. The individual rates for the eastern and western tails must be smaller; but in either case the rate is far below the minimum ϒ needed (∼15 M_☉ yr⁻¹ for either tail and pixel size) for the observed turbulent energy to be supplied by feedback from star formation overall.

While the turbulence we observe is purely on ∼kpc scales, it is likely crucial to the kinematic nature of the ISM on smaller, star-forming (∼pc) scales. Turbulence is injected on hundreds of parsecs to ∼kpc scales (e.g., Mac Low 1999; Elmegreen 1993; Block et al. 2010), and cascades down to smaller scales until it is dissipated into thermal energy by viscosity (Elmegreen et al. 2003; Kolmogorov 1941). There are a number of observed radiative pathways available to eliminate this energy effectively and faster than the turbulence decay timescale by several orders of magnitude; e.g., collisionally excited lines of C iv (Wolfire et al. 2003), rovibrational transitions of H₂ (e.g., Cluver et al. 2010), and radio/X-ray continuum emission (Kaufman et al. 2012).

In effect, we are left with a top–down evolutionary process, where gas kinematics at observably large scales—likely influenced by a combination of tail age and progenitor mass ratio as seen in Section 5.1—influences star formation on the otherwise unresolved, smaller scales. Turbulent gas generates a log-normal density probability density function in the local ISM (Wada et al. 2002), whose width—dispersion in density—increases mainly with turbulent Mach number (Krumholz & Thompson 2007). Thus, turbulence on large scales affects the density distribution of gas on small scales. This is exacerbated in mergers, where additional, expeditious kinematic perturbations create deviations in the log-normal distribution, favoring more high-density gas (Bournaud 2011).

In all, our results present a compelling case for the importance of interaction-established, kiloparsec-scale H i kinematics and column densities in determining the capacity of pc-scale SC formation in tidal debris. Other, currently unsurveyed interaction properties—e.g., dark matter halo potentials, interaction speeds and impact parameters, H₂ content, and dust-to-gas ratios—may help set the stage for luminous SC formation in tidal tails and may help account for the variety and distribution of SCC populations witnessed here. However, the character of the neutral hydrogen medium, described by its densities, pressures, and kinematics, also appears to be an invaluable factor in this process. We have shown that high-confidence SCC detection often occurs in tails with large fractions of high column density, turbulent gas (Section 5.1.1). Both the putative SC populations and the underlying H i medium clearly vary depending on the age (Section 5.1.2) and/or the progenitor galaxy mass ratio (Section 5.1.3) of the interaction, indicating that these dynamical properties are important in tidal tail SC formation as well.

This eastern tail of a ≈500 Myr old merger remnant between similarly massed galaxies (Lee & Lee 2005), NGC 1487E contains a single SCC that is not likely real (f_SC = 0). This system is the closest of the tail sample, so optical images from M11 reveal that star formation may have occurred in detectable, fainter structures. Some of these may be ∼10⁴ M_☉ SCs formed in situ. This tail clearly favors the formation of dispersed, lower-mass structures over more massive SCs.

Figures 4.1 and 5.1 display a high column density H i tail on ∼kpc scales, with line-of-sight velocity dispersions typically ∼20 km s⁻¹. The velocity field of this tail varies smoothly and produces line-of-sight velocity gradients within this tail all ≳5 km s⁻¹ kpc⁻¹. Figure 5.1 indicates that NGC 1487E has a high fraction of turbulent and dense gas, but M11 suspected that inclination effects artificially increase the apparent H i densities measured for this system. This can easily elevate $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ by a factor of three (0.5 dex) alone, so the actual fraction of gas above the Maybhate et al. (2007) threshold (and at high mechanical energies) may be small. Consequently, the idea of a critical H i column density for massive cluster formation may still be supported in this case.

This tail has no SCCs according to the present optical criteria. Like the eastern tail, optical detection of point sources in M11 demonstrates that this tail may preferentially exhibit a star-forming morphology devoid of massive clusters. Instead, this tail hosts a large number of fainter blue (V − I ∼ 0.0) sources across the WFPC2 FOV (M11). The distributions of $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$, σ_los, and Σ_KE are very similar to those of this tail's eastern counterpart, though again it is likely that projection effects may artificially enhance the apparent column densities and affect other measured quantities. Overall, dv_los/dr_⊥ in this tail appears constrained to values generally ≲20 km s⁻¹ kpc⁻¹, unlike in NGC 1487E, which shows several regions with larger velocity gradients. Otherwise, ∼kpc-scale mechanical energies across the H i debris are similar between these two tails.

The debris seen in NGC 4747 results from a ≈320 Myr old merger between its progenitor galaxy and the more massive NGC 4725 (Haynes 1979; Wevers et al. 1984). As in NGC 1487E/W, sources were observed at fainter magnitudes in M11, but few sources (six in tail) were detected at M_V < −8.5. These also have a good probability of being real clusters based on star counts; f_SC = 0.80 overall, while 1 pixel has a p_real value >0.94 evident in Figure 5.3.

The WHISP data used in this analysis are not as deep as the VLA/ATCA cubes for other tails, but accounting for the beam dimensions, it is still sensitive with our channel masking to log $\hbox{$N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$}$ ≳ 19.6 (cm⁻²). A small amount of low column density gas is detected toward the center of the WFPC2 FOV, and has relatively low velocity dispersions (≲30 km⁻¹) on its ≈6 kpc beam scale. We measure high velocity gradients across this H i gas (dv_los/dr_⊥ > 30 km s⁻¹ kpc⁻¹), but the few surrounding pixels for each measurement makes this quantity prone to uncertainty. A large fraction of pixels (≈50%) are marked as unfitted for at least one of the H i properties plotted, appearing in optical images but showing no clear H i emission here. This tail appears optically prominent but relatively H i-poor.

The fact that this intermediate-stage merger has hardly any H i to the best of its cube's offered resolution is peculiar. It may be possible that it has been stripped out and/or dispersed by its recent encounter with NGC 4725. What gas remains is characterized by low kinetic energy densities (∼10⁴⁶ erg pc⁻²) and low column densities. However, the compliment of blue (V − I < 0.5) point sources seen in M11 may indicate that less massive SCs (∼10⁴ M_☉) have formed in the past few hundreds of Myr. Thus, it appears that the formation of these structures would require a lower column density threshold than the fiducial log $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ ≈ 20.6 (cm⁻¹) proposed for more massive clusters (Maybhate et al. 2007), and/or the star formation that has occurred may have helped efficiently consume/disperse an existing H i reservoir over the past few hundred Myr.

The tip of NGC 520 tail, the H i and optical condensation seen here and in M11 could be a TDG from a ≈300 Myr old interaction or a dwarf galaxy entrained in the H i tidal stream (Stanford & Balcells 1991). Within, fainter sources than the SCC magnitude cutoff were originally detected. Two sufficiently bright objects of the M11 source lists can be considered SCCs and are scattered in the H i debris, but they have low probabilities of being real clusters (f_SC = 0.06).

The very center of the potential TDG contains high column density gas ($N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ ≳ 10^20.6 cm⁻¹) on ≈3 kpc scales, and has a low line-of-sight velocity gradient (dv_los/dr_⊥ < 5 km s⁻¹ kpc⁻¹). However, the Σ_KE distribution peaks barely above 10⁴⁶ erg pc⁻² from the moderately turbulent nature of the gas, and overall the fraction of the tail H i that is both of high column density and turbulent on ≈3 kpc scales is ≈17%. Once more, this environment may be somehow favorable for a smoother, diffuse star formation morphology that does not permit the formation of massive SCs.

NGC 2992 is ≈100 Myr into its interaction with NGC 2993 (Duc et al. 2000). It is heavily inclined along our line of sight, enhancing the tail column density by a factor of ≈3 (0.5 dex). The putative dwarf galaxy candidate at the tail tip was previously confirmed to be kinematically distinct and therefore is likely a real TDG (Brinks et al. 2004; Bournaud et al. 2004). Most of the 10 SCCs detected in this system can be found there, at least in projection. Though M11 found that across the debris, NGC 2992 is deficient in clustered star formation (to M_V < −8.5), individually we find that about 72% of the detected sources are probably massive clusters, especially given their nonuniform distribution.

The $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ measurements for in-tail pixels are all high in projection, with almost 80% of the gas having apparent (≈4 kpc scale) column densities log $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ ≳ 20.6 (cm⁻²). This is also evident in the $\Sigma _{M_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}}$ map, where regions with mass densities ≳10 M_☉ pc⁻² are apparent, despite the saturation of H i known to occur around that (deprojected) density. We detect a reasonably uniform σ_los distribution, with values ranging from ≈20 to 80 km s⁻¹, while our map of dv_los/dr_⊥ highlights a region with a locally shallow (<5 km s⁻¹ kpc⁻¹) velocity gradient within the TDG and its SCCs.

In terms of the SCCs, K-S tests reveal that the distributions of tail pixels with and without at least one SCC are similar for all properties except $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$, owing to the cluster candidates situated in the high column density TDG (though note the small sample size for the former category). Again, though it is not clear from Figure 5.5, these SCCs appear superimposed over a region of especially shallow ∼large-scale velocity gradient in Figure 4.5. And even if the $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ values were decreased by 0.5 dex to roughly account for tail geometry and orientation, Figure 5.5 would show that these SCCs would still be embedded in H i gas with higher column densities than the Maybhate et al. (2007) threshold.

The interacting, similarly massed counterpart to NGC 2992, NGC 2993 presents a relatively face-on orientation. High-$N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ areas on ≈4 kpc scales also register high moment-measured velocity dispersions (σ_los ≳ 40 km s⁻¹) and kinetic energy densities (log Σ_KE ≳ 47 erg pc⁻²), and there appears to be a strong dependence of σ_los on $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ in Figure 5.6. About 70% of these sources should be clusters, but only half lie in areas of high $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$. Most are still found in regions of high kinetic energy density (log Σ_KE ≳ 46 erg pc⁻²). However, K-S tests show that the distributions of SCC-empty and SCC-laden pixels are not independent for any measured quantity.

An ≈200 Myr old minor merger, NGC 2782 exhibits an eastern tail that may be the distorted remnant of the smaller galaxy (Smith 1994). Interestingly, the optical tail extends beyond the H i tail. Approximately 57% of the ≈4 kpc scale tail pixels exhibit H i of supercritical column densities and turbulent velocity dispersions. Many pixels are unfitted (f_unfit ≈ 0.24), so the fraction of high column density, turbulent gas relative to the total H i reservoir is higher.

The velocity field of the H i debris varies smoothly across the eastern side of the debris (dv_los/dr_⊥ ≲ 5 km s⁻¹ kpc⁻¹), and changes more sharply on the western side (dv_los/dr_⊥ ≳ 10 km s⁻¹ kpc⁻¹). Velocity dispersions can also range from ∼10 km s⁻¹ on the eastern side to ≳60 km s⁻¹ in the west. The Σ_KE map follows this trend accordingly. We compute a high f_SC value (0.96) for the 56 SCCs in this tail; thus most should be clusters. These nearly all populate the turbulent–high-$N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ region of Figure 5.7. They also lie in pixels with log Σ_KE > 46.5 erg pc⁻², and span a range of dv_los/dr_⊥ measurements (between 0 and 30 km s⁻¹ kpc⁻¹).

The pixels with SCCs also comprise different distributions of $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$, σ_los, Σ_KE, dv_los/dr_⊥, and Σ_KE, with K-S tests producing results <5 × 10⁻⁴ in all cases. Cluster formation may occur in regions of preferentially higher column density, higher line-of-sight velocity dispersions, high kinetic energy densities, and a broad range of ∼10 kpc scale velocity gradients in this tail. These gradients may be too broadly constructed (measuring differences over areas larger than the radio beam) to accurately trace the kinematic structure of local SCC-conducive environments, however, particularly where the velocity field varies rapidly toward the central galaxy.

If NGC 2782E is the consumed minor galaxy of the NGC 2782 interacting pair, this western tail may be the H i-rich tidal debris of the primary galaxy. While Smith et al. (1999) detect CO emission in the eastern tail, none is found in NGC 2782W by Braine et al. (2001). Perhaps correspondingly, far fewer SCCs (six) are detected in this tail. However, as mentioned in Section 4.1, the work of Torres-Flores et al. (2012) and our star counts imply that ≈52% of these sources should be SCs. In particular, Torres-Flores et al. (2012) find these sources are ≈10⁴ M_☉, ≈ a few Myr old clusters. This is consistent with Hα observations of Knierman et al. (2012) and their conclusion that ongoing star formation does occur in this tail, but at low efficiency compared to other extragalactic environments. For no properties do the distributions of SCC-carrying and SCC-devoid pixels appear statistically distinct.

The central debris shows large regions with very shallow velocity gradients (dv_los/dr_⊥ ≲ 5 km s⁻¹ kpc⁻¹), and generally a less extensive range of line-of-sight velocity dispersions (σ_los ≈ 10–30 km s⁻¹) on ≈4 kpc scales. Compared to NGC 2782E, this tail has qualitatively similar gas fractions in the turbulent–high-$N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ quadrant of Figure 5.8. The western tail has a lower fraction of unfitted pixels, however (f_unfit ≈ 0.1) and a higher percentage of gas with relatively low values of dv_los/dr_⊥. The distributions of $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$, σ_los, and Σ_KE also peak at broadly lower values for NGC 2782W than for the eastern debris.

A young (≈100 Myr old) tail, the debris of NGC 2444 formed quickly from a possible high-speed encounter (Appleton et al. 1987; cf. Koopmann et al. 2008). Gas profiles indicate the H i is kinematically disturbed (σ_los ≈ 20–80 km s⁻¹ for most of the pixels) on ≈8 kpc scales. There are no pixels with $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ above the fiducial threshold, and the dv_los/dr_⊥ map of Figure 4.9 shows the velocity gradients across the tail are mostly below ≈10 km s⁻¹ kpc⁻¹.

Five of the SCCs detected lie within the tidal debris; ≈3 of them may be SCs based on star counts (f_SC = 0.63). Moreover, the pixels upon which they appear superimposed do not seem to have different distributions of $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$, σ_los, dv_los/dr_⊥, or Σ_KE than do all tail pixels (to the extent to small sample of SCC-containing pixels allows). Most (≈95%) of the ∼8 kpc scale H i has kinetic energy densities above 10⁴⁶ erg pc⁻² and has a relatively shallow velocity gradient, but does not have a widespread cluster population. If there are indeed a few clusters in the SCC source list for this system, either they did not need kiloparsec-scale high column density H i gas to form, our resolution is too poor to see more local concentrations of H i, or previous star formation dispersed the gas efficiently in the past 100 Myr. Given the optical faintness of this tail (it does not register on WFPC2 images), it may be unlikely star formation feedback has played much of a role in this case.

The approximately face-on NGC 2535 debris results from a young (≈100 Myr) interaction with NGC 2536 (Hancock et al. 2007). Measurements of ≈6 kpc scale velocity dispersions for these profiles range from 10 to 50 km s⁻¹, typically, with the regions of highest σ_los and lowest dv_los/dr_⊥ (≲4 km s⁻¹ kpc⁻¹) closely mirroring the most optically prominent regions seen in M11. Most of the 36 in-tail SCCs also fall in these regions, ≈93% of which should be SCs. From K-S tests, the pixels that contain them appear to have generally higher values of $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$, and lower values of dv_los/dr_⊥, but do not arise from different distributions of σ_los and Σ_KE compared to all tail pixels. Several pixels have multiple SCCs "within," so their p_real values are higher (>0.94); these occupy the regions of turbulent, high $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$, low dv_los/dr_⊥, and high Σ_KE in Figure 5.10.

NGC 6872 is ≈150 Myr into an interaction with the smaller companion galaxy IC 4970 (Horellou & Koribalski 2003, 2007). NGC 6872E is the easternmost FOV of the enormous eastern tail, featuring a moderate percentage of high column density H i (about 50% of the gas has log $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ > 20.6 cm⁻² on ≈16 kpc scales). The line-of-sight velocity dispersion ranges from ≈20 km s⁻¹ from the tail edges to ≈100 km s⁻¹ in the interior.

The central region of the debris is also host to an extensive population of SCCs (over a hundred are detected here), which is also coincident with areas of shallow large-scale velocity gradients (dv_los/dr_⊥ ≲ 5 km s⁻¹ kpc⁻¹). For all considered properties except dv_los/dr_⊥, the pixels hosting SCCs comprise different distributions than those of tail pixels that do not; their distributions tend to peak at higher values of $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$, σ_los, and therefore also Σ_KE. According to Table 3, about 94% of these sources should be clusters, and pixels with multiple SCCs have even higher probabilities of containing at least one. As seen in Figure 5.11, of these pixels of highest cluster-containing probability, almost all have kinetic energy densities >10⁴⁷ erg pc⁻², and column densities >10^20.6 cm⁻². The positions of the SCCs are also aligned with regions of Hα emission (Bastian et al. 2005; Mihos et al. 1993).

There are still cluster candidates in pixels with "subcritical" $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$, however—including roughly 6%–10% of them that lie in regions unfitted by several H i properties—and many of these may be real. It is then again a question of whether this $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ threshold is important, these clusters had formed in previously high column density H i in the past 150 Myr that has since dispersed or been consumed, or instrumental effects prevent an assessment of H i properties at crucially lower (parsec–kiloparsec) scales. A non-negligible fraction of the tail pixels are unfitted (f_unfit ≈ 0.15), so these effects are certainly possible. For NGC 6872, the physical scale may be especially important, as the cube for this system has a particularly large beam, both in angular and physical extent.

This western debris highlights the base of the eastern tail of the NGC 6872 system, displaying a central clump of moderate H i emission around the Maybhate et al. (2007) limit, immersed in a tail of lower column density gas (≈80% of the in-tail pixels) that follows the optical structure seen in M11. Flanking the H i tail is bright optical debris that both contains SCCs and is devoid of detectable neutral hydrogen. This helps account for the large fractions of unfitted SCC-hosting and SCC-empty pixels (≈0.4 and 0.3, respectively). The densest regions exhibit velocity dispersions up to ≈100 km s⁻¹. Here, the velocity structure is not of a smoothly rotating, extended disk, but has a number of kinematically distinct components, as elaborated in the modeling of Horellou & Koribalski (2007). Thus, the dv_los/dr_⊥ map, showing measurements of line-of-sight velocity gradients across pixels separated by more than one beam, displays areas with seemingly steep gradients. Internally, however, the local gradients may be shallower.

In either case, the 101 SCCs detected appear as widely distributed as the kinematically varied H i debris. K-S tests demonstrate that the pixels that contain them are generally not different in terms of $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$, σ_los, or Σ_KE, and while their distribution of dv_los/dr_⊥ values appears statistically distinct, the aforementioned spatial resolution of these NGC 6872 data makes this difficult to interpret. Eight of these sources exist in exclusively optically defined debris, and the remaining sources mostly in low-$N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ areas are also likely to be actual clusters (f_SC = 0.93). Even the pixels with the highest probabilities of having at least one cluster generally have $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ values below the fiducial cutoff, while their high σ_los measurements ensure that their mechanical energy densities remain all above log Σ_KE = 46 erg pc⁻².

The northern region of the NGC 1614 tidal debris is H i-poor (f_unfit ≈ 0.89), with concrete detections only at the very northern and southern edges of the central host galaxy. It is conversely prominent in the optical, with galactic material arranged in warped rings around the merger remnant of an interaction thought to have begun ≈750 Myr ago (Neff et al. 1990 and references therein). The few pixels with detectable H i emission show high σ_los (≈75 km s⁻¹). These isolated clumps of neutral hydrogen have high column densities on ≈5 kpc scales, and appear to have larger-scale velocity gradients dv_los/dr_⊥ ∼ 10 km s⁻¹ kpc⁻¹, though the paucity of H i pixels makes this comparative measurement admittedly uncertain.

Nineteen SCCs are found interspersed in the optical debris, with only three lodged in the H i debris (several occupy the same pixels). Thus, K-S tests cannot be performed on the different pixel distributions. But the majority of the SCCs should be clusters (f_SC = 0.95), so clusters are detected in this tail regardless of H i content. NGC 1614N results from a very late stage interaction, so these sources may also be several hundred Myr old and had initially formed while surrounded by an H i medium that has since dispersed, been ionized, or consumed. While the photometry of M11 cannot age-date the SCCs, their red V − I colors (V − I ∼ 1) may imply these clusters are older (or reddened by extinction).

NGC 1614 is luminous at infrared wavelengths and contains a powerful nuclear starburst (Alonso-Herrero et al. 2001). It is possible that the H i in the northern tail has been eliminated by central winds, photoionized by the central starburst, or photoionized by past star formation. Hibbard et al. (2000) contend that a mean R-band surface brightness of ≈23 mag arcsec⁻²—depending on the H i column density and dimensions of the tidal debris—is required for the stellar population to effectively rid itself of the warm neutral medium on these scales. NGC 1614N is one of the few debris fields in the M11 sample to exceed this surface brightness (its F606W-band surface brightness is ≈22 mag arcsec⁻², which in terms of photometric bandpasses is close to R), so it is possible mass in situ photoionization of the tidal H i has occurred in the past.

The H i of the southern tail of NGC 1614 is confined to a "stub" extending from the southernmost tip of the merger remnant; most tail pixels are optically defined (f_unfit ≈ 0.89). Its H i characteristics are similar to those of the northern debris; the measurable H i here is of comparably high column density (log $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ ≳ 20.6 cm⁻²) and with line-of-sight velocity dispersions ranging from the instrumental limit to ≈75 km s⁻¹. The ranges of velocity gradients are also similar, with dv_los/dr_⊥ ≈ 5–15 km s⁻¹ kpc⁻¹.

The main difference is the number and distribution of SCCs. Thirty sources are located in the combined optical and H i tail, with most of them falling in the H i stub. Once more, a high fraction of them are probable SCs (f_SC = 0.96, with p_SC exceeding this number for many pixels that contain more than one SCC). The sources here are very blue (V − I ≲ 0; M11), indicating star formation might be relatively recent and/or ongoing. These sources trace out the high-$N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$, high-Σ_KE, and turbulent regions of the H i parameter space of Figure 5.14, and the pixels that have them do not appear different with respect to those properties than H i pixels overall.

It is however curious that the positions of these SCCs is offset by a few arcseconds from the regions of highest column density as seen in the $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ map of Figure 4.14. The H i may be in the process of being locally consumed or swept out by current star formation. Approximately 42% of the equally high-probability SCCs are found in optically bright yet H i-unfitted regions, so these sources may be further examples of past feedback and H i dispersal or consumption sometime over the course of this tail's ≈750 Myr lifetime. Unfortunately, VI observations of M11 do not permit accurate age-dating of cluster candidates to provide evidence for this age-dependent H i removal. If generous K-S probability cutoff is employed, the distributions of dv_los/dr_⊥ between pixels and without SCCs appear marginally independent. But again the low spatial resolution for this especially distant object may confuse this quantification of the large-scale velocity field.

This well-studied merger has two large, symmetric tails of ages ≈450 Myr (Knierman et al. 2003, and references therein). WFPC2 images of the ABC positions along these tails reveal them to be optically faint, diffuse, and all but devoid of SCCs (Knierman et al. 2003; M11). As a whole they are poor in H i, with very little gas at or above the Maybhate et al. (2007) threshold on ≈kpc scales. Across the WFPC2 pointings, it appears ≈30%–80% of the tail pixels are classified as unfitted by at least one H i characteristic. This gas is relatively kinematically unperturbed, with σ_los ≲ 25 km s⁻¹, and much of the gas showing dispersions less than the turbulent–thermal cutoff. Consequently, less than ≈5%–20% of the gas has kinetic energy densities >10⁴⁶ erg pc⁻².

Three SCCs are found in NGC 4038A, and one is found in the NGC 4038B and NGC 4038C FOVs each. They all have low values for f_SC and their host pixels have correspondingly low values of p_real (0–0.24); thus they are likely point source contaminants. They populate too few pixels to conduct K-S tests between distributions of H i characteristics. Unlike the active, luminous interior and bridge regions of the merger, the tails of this system are extraordinarily quiescent in terms of ongoing star formation.

Dynamically, the tails of NGC 3256 are about 400 Myr old (Knierman et al. 2003), resulting from an ongoing major merger of two similarly massed galaxies. The channel width of the NGC 3256 data cube is wide (≈33.5 km s⁻¹), so each pixel has few channels of signal. Combined with the beam size and sensitivity of the data, uncertainties in H i properties are high for pixels with $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ ≲ 10²⁰ cm⁻². The vertical "tail" of constant σ_los in Figures 5.18 and 5.19 represents the instrumental limit in determining velocity dispersions. Much of the H i gas in these tails is of high column density and line-of-sight velocity dispersions on scales of ≈5 kpc; we find that ≈72% and 88% of tail pixels have those properties in the eastern and western tails, respectively. The distribution of dv_los/dr_⊥ values in both tails is very broad, ranging from 0 to 50 km s⁻¹ kpc⁻¹, while the distribution of that property is skewed toward lower values for NGC 3256W.

The western tail has fewer pixels with at least one unfitted H i characteristic (by a factor of ≈4–10). Thirty two SCCs are located in the eastern tail, and 51 are in the west. Given the calculated values for f_SC (0.50–0.60), a little over a dozen sources in the east and about 30 sources in the west may be genuine SCs. This strong level of contamination results from this system's low galactic latitude (b ≈ 11°), and led (Knierman et al. 2003) to conclude that the eastern tail did not have a statistically significant widespread cluster population, and the western tail did. This is still true; the eastern tail may have only a handful of real clusters, but they are spread throughout the ≈1720 kpc² tail area and do not constitute a significant population.

Most of the SCCs in either tail populate the high $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ and high σ_los parameter space of the neutral medium. According to K-S tests, pixels containing SSCs in the eastern tail may have typically higher line-of-sight velocity dispersions, but do not comprise unique distributions compared to all pixels in terms of other measured properties. Conversely, the pixels hosting cluster candidates in NGC 3256W might have generally shallower velocity gradients with a generous interpretation of that K-S result. The central "spine" of the western debris is a region of low dv_los/dr_⊥ (≲5 km s⁻¹ kpc⁻¹), an attribute absent in the eastern tail. Perhaps a combination of this and the higher fraction of high column density gas contributes to the factor of ∼3 increase of possible clusters in the western tail over the east.

NGC 7252 is a late-stage (≈730 Myr old; Knierman et al. 2003) merger between two comparably massed galaxies. The channel width of this system's data cube is the widest of the debris sample (≈42.5 km s⁻¹). The H i in both debris fields is relegated to the tips of the straight and narrow eastern and western tails, and is characterized by low column and mass densities on ≈1.5 kpc scales and steep velocity gradients (dv_los/dr_⊥ ≳ 15 km s⁻¹ kpc⁻¹) on larger scales. The latter may be attributed in part to the velocities ascribed to the low-signal edge pixels, which can be uncertain. The moderately turbulent kinematics of the gas helps to offset the low mass densities and maintains the kinetic energy densities of much of the H i above 10⁴⁶ erg pc⁻².

Comparable numbers of SCCs are in either tail (11 and 17 for east and west, respectively), with fairly comparable values of f_SC (0.80 and 0.89). In 7252E, these sources are widely distributed, many of which are situated in the H i-poor optical debris ($f_{{\rm unfit}}^{{\rm SCC}} \approx 0.40$). Two neighboring sources are located in a clump of diffuse optical emission, which is slightly offset from the central high-$N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ region seen in Figure 4.20. Pixels that contain the SCCs in this tail do not seem to arise from dissimilar distributions of $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$, σ_los, Σ_KE, or dv_los/dr_⊥ than all H i tail pixels.

The same is true for the western tail, except for the velocity dispersion, for which SCC-laden pixels seem to have a more narrow range of values (σ_los ≈ 10–30 km s⁻¹). Eleven sources are dispersed throughout the primarily optically defined tidal tail, with ≈40% in especially H i-poor regions (in comparison, f_unfit ≈ 0.30). A few H i pixels in the only high mass density region at the end of the tail contain six of them. This region of the debris has the lowest dv_los/dr_⊥, and is the only part of Figure 5.21 that shows $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ > 10^20.6 cm⁻². These sources belong to the prominent optical/H i-defined TDG in the western tail (Hibbard et al. 1994). The fact that this one region of obvious clustered star formation is the only region of high column density H i and contains a large number of SCCs (the superimposed black points in Figure 5.21) indicates that, at least in this case, the Maybhate et al. (2007) $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ threshold may be important in forming SCs. These few H i pixels do not stand out in the distribution of Σ_KE; a combination of high mechanical energy densities and column densities may be observationally important in finding SCCs in situations like this.

NGC 3921 is the most distant interaction of the sample (84.5 Mpc), and accordingly its H i cube has the worst spatial resolution of the debris surveyed here (≈2 kpc for each pixel; roughly four times that for the size of the beam). The optical completeness limit for this tail as analyzed by Knierman et al. (2003) is also the worst of the sample, with a V-band 50% completeness limit at M_V = −8.8. Therefore, it is probable that some SCCs are missing from the source list.

Its H i debris has velocity dispersions on this scale of ≈5–30 km s⁻¹, and has a smoothly varying velocity structure with shallow larger-scale velocity gradients (dv_los/dr_⊥ ≲ 5 km s⁻¹ kpc⁻¹). Approximately 10% of the tail pixels have a high column density, and are confined to the long tail-shaped structure that extends from the north to the south of the $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ map. Of the 14 SCCs in the H i or optical debris, only two are embedded in this relatively higher-$N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ material, and most are dispersed throughout the debris. About 87% of these sources should be clusters, and one pixel that contains two sources—part of a short chain of sources that extends from the north of the optical image—correspondingly has a high chance of containing at least one massive cluster.

These SCC-supporting pixels do not appear to have statistically unique distributions of measured H i quantities compared to all H i pixels, and all but one of them lie in regions of low column density (log $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ < 20.6 cm⁻²). Their moderate velocity dispersions mean that their kinetic energies are still greater than the fiducial 10⁴⁶ erg pc⁻². As is the case with other tails, the low spatial resolution, feedback from star formation (certainly possible over the ≈460 Myr lifetime of the tail; Knierman et al. 2003), and/or a revised $N_{\scriptsize {{{\rm H}\,\scriptsize{I}}}}$ threshold might at least partially explain the detection of these handful of reasonably probable clusters. The particularly poor resolution makes it especially possible that sub-kiloparsec clumps of denser H i gas are being effectively diluted by the size of the beam for this system, especially when they are found in closer, better resolved tails.