Multiwavelength Analysis of the Merging Galaxy Cluster A115

A115 was observed using the Subaru/Suprime-Cam on 2003 September 25 and 2005 October 3. We retrieved the V- and i'-band archival data from SMOKA.⁵ The total integrations are 1,530 s and 2,100 s for the V- and i'-filters, respectively. The seeings of the V- and i'-filters are FWHM = 058 and 065, respectively. Note that the V-band data set used in Okabe et al. (2010) is a subset (the total integration was 540 s) of the one used in the current study, whereas their i'-band data set is identical to ours.

The basic CCD processing (overscan subtraction, bias correction, flat-fielding, initial geometric distortion correction, etc.) was carried out with the SDFRED1⁶ (Yagi et al. 2002; Ouchi et al. 2004) pipeline. We performed the rest of the imaging data reduction using our WL pipeline, which incorporates the SCAMP,⁷ SExtractor,⁸ and SWARP⁹ packages.

We utilized the Sloan Digital Sky Survey (SDSS)-DR9 (Ahn et al. 2012) catalog to refine astrometric accuracy with SCAMP. A deep mosaic stack was produced in two steps. A median mosaic image was generated with SWARP using the alignment information output by SCAMP. This median-stacking algorithm enables us to remove cosmic rays, some bleeding trails, and some CCD glitch features. However, in terms of signal-to-noise ratio (S/N), this median-stacking result is not optimal. The final science image was created by weight-averaging individual frames, where we flagged the aforementioned, unwanted features by performing 3σ clipping on the basis of the median image generated in the first step.

We ran SExtractor in dual-image mode, which takes two images as input and uses one for detection and the other for measurement. Our detection image was created by weight-averaging the V- and i'-band mosaic images. This dual-image mode allows us to obtain identical isophotal apertures between the two filters based on the common detection image, which is deeper than either of the two images alone. These identical isophotal apertures are needed to obtain accurate object colors. Photometric zero-points were determined by using the SDSS Data Release 13 catalog that overlaps the cluster field. Because the SDSS-DR13 does not include the Johnson V-band, we performed a photometric transformation using the following relation (Jester et al. 2005):

$\begin{eqnarray}&&{V}_{\mathrm{Johnson}}={g}_{\mathrm{SDSS}}-0.59({g}_{\mathrm{SDSS}}-{r}_{\mathrm{SDSS}})-0.01.\end{eqnarray} \tag{ 1 }$

We employed isophotal magnitudes (MAG_ISO) to estimate object colors, whereas total magnitude (MAG_AUTO) was used to compute object luminosities.

We retrieved the Chandra data (ObsID: 3233, 13458, 13459, 15578, and 15581) for A115 from the Chandra archive.¹⁰ The ObsID 3233 data set was taken in 2002, while the other four were taken in 2012 November. All observations were carried out with the ACIS-I detector in VFAINT mode with total exposure time ∼360 ks. We reduced the Chandra data using the CIAO 4.9 pipeline and the CALDB 4.7.3 calibration database. The observations were reprojected to the same tangent plane and combined using the merge_obs script.

We created a broadband image by selecting the events within the energy range 0.5–7 keV with a 2 pixel × 2 pixel binning scheme. This broadband image was divided by our exposure map¹¹ to produce an exposure-corrected image. In Figure 1 this exposure-corrected image is overlaid with the VLA radio emission on our Subaru color-composite image.

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In preparation for X-ray temperature measurement, we performed our initial data reduction using the chandra_repro script. The chandra_repro script automates the instrument-dependent sensitivity corrections, Charge Transfer Inefficiency corrections, and removal of bad pixels and cosmic rays. The reduced data were reprojected to a common tangent plane using the reproject_obs script. We masked out the point sources that are detected by the wavdetect script. We then constructed a light curve and identified background flares as detections that are 3σ outliers. The flares were removed using the deflare script.

Our WL pipeline has been applied to a number of ground- and space-based imaging data (e.g., Jee et al. 2013; Finner et al. 2017), and its variant has been validated in the most recent public shear testing program (Mandelbaum et al. 2015). Readers are referred to Finner et al. (2017) for details. Here we present a brief summary of our point-spread function (PSF) model and ellipticity measurement.

3.1.1. PSF Modeling

PSF modeling is a crucial step in a WL study. Unless corrected for, the PSF not only dilutes the lensing signal, but also induces a distortion mimicking WL. In this study, we use the principal component analysis (PCA) approach (Jee et al. 2007a; Jee & Tyson 2011).

The observed PSF at a specific location on the mosaic is a combination of the PSFs from all contributing frames. Thus, to properly consider each component, we modeled the PSF for each contributing frame and then stacked them to a final PSF model.

One way to examine the fidelity of the PSF model is to compare the ellipticity pattern of the mosaic fields between observation and model as shown in Figure 2. The left panel shows the ellipticity pattern of the observed stars, and the right panel shows the pattern reconstructed by our PSF model. For the V-filter (top), both magnitude and direction of the PSFs across the mosaic field are closely reproduced. The mean residual rms is ${\left\langle \delta {e}^{2}\right\rangle }^{1/2}\sim 0.014$ per ellipticity component. The good agreement demonstrates that the PCA-based PSF model is robust. Also, it demonstrates that the image coadding alignment is performed with high fidelity; even a subpixel-level misalignment would be manifest as a noticeable PSF ellipticity pattern in the coadd image (left panel), which, however, could not be reproduced by the model (right panel) that assumes a perfect alignment. For the i'-filter (bottom), we could not make the model PSF ellipticity pattern match the observed pattern as accurately as in the case of the V-filter. The mean residual rms in this case is ${\left\langle \delta {e}^{2}\right\rangle }^{1/2}\sim 0.027$, which is nearly a factor of 2 larger. Currently, the exact source of this poor match between model and observation is unknown.

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We decide to measure WL signals from our V-band image, for which our PSF model is more accurate. An additional merit of using the V-band data rather than the i'-filter is its smaller PSF (∼11% smaller on average). Given the same PSF model accuracy, smaller PSFs provide more reliable shapes for fainter and smaller galaxies, which have higher chances of being background and thus dominate WL signals.

3.1.2. Ellipticity Measurement

We fit a PSF-convolved elliptical Gaussian to a galaxy image to determine its two ellipticity components e₁ and e₂, which we define as

$\begin{eqnarray}\begin{array}{rcl}{e}_{1} & = & e\cos 2\theta ,\\ {e}_{2} & = & e\sin 2\theta ,\\ e & = & \displaystyle \frac{a-b}{a+b}\end{array}\end{eqnarray} \tag{ 2 }$

where a and b are the semimajor and semiminor axes of the best-fit elliptical Gaussian, respectively, and θ is the position angle of the semimajor axis. Because the elliptical Gaussian is convolved with a model PSF when fitted to the galaxy image, the resulting ellipticity is corrected for PSF systematics.

The elliptical Gaussian profile contains seven free parameters: normalization, two parameters for centroid, semimajor axis, semiminor axis, position angle, and background level. We fixed the centroid and background level using the SExtractor outputs X_IMAGE, Y_IMAGE, and BACKGROUND, respectively. This reduces the number of free parameters to four, which improves convergence for faint sources. We used the χ² minimization code MPFIT¹² to fit the model to the galaxy image and estimate the ellipticity uncertainty.

In general, this raw ellipticity is a biased measure of the true shear for a number of reasons (e.g., Mandelbaum et al. 2015). The bias is often expressed as γ = (1 + m_γ)e + m_β, where m_γ and m_β are often referred to as "multiplicative" and "additive" biases, respectively. We find that although the additive bias is negligible for our WL pipeline, the multiplicative bias is not (Jee et al. 2013). From our image simulation, we determine m_γ = 0.15 for our source population. This multiplicative factor is applied to our ellipticity catalog.

Only light from galaxies located at a greater distance than the cluster is lensed by the gravitational potential of the cluster. Ideally, one can use a photometric redshift technique to enable efficient selection of background galaxies. However, this is not feasible in our case, where only two broadband filters are available. Therefore, in the current study we used a color–magnitude relation to select source galaxies.

Figure 3 shows the color–magnitude diagram of the A115 field. It is clear that a majority of the early-type galaxies of A115 show a tight color–magnitude relation. We selected galaxies that are bluer and fainter than this red-sequence to minimize the contamination of our source catalog by cluster galaxies. This selection scheme is based on the general trend that more distant galaxies are bluer and fainter than the cluster red sequence at z ∼ 0.2. Obviously, this trend is only roughly true and thus some fraction of the sources defined in this way are not behind the cluster. We estimated this fraction in our source redshift estimation (Section 3.3).

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We further refined our source catalog by imposing size and ellipticity error conditions. Objects whose semiminor axis b is smaller than 0.3 pixels were discarded because they are usually indistinguishable from stars. We require that the ellipticity error is below 0.25. This removes not only low S/N objects, but also point sources, which tend to have large ellipticity errors (in principle, stars should have no shape after PSF deconvolution). Many spurious sources are removed by the above ellipticity error and size conditions. As a further measure, we discarded sources whose ellipticities are greater than 0.9 because they are, in general, too elongated to be a galaxy. The last selection criteria that we applied is an MPFIT STATUS = 1 (a good fit).

After all selection criteria were applied, some spurious objects still survived. These objects mostly appear on diffraction spikes and reflection rings from bright stars. We removed the spurious objects by visual inspection. These spurious features are particularly important near A115N, where a bright star with diffraction spikes is located ∼4' west. Our final source catalog has ∼17,000 galaxies over the ∼600 arcmin² area. The resulting source density ∼24 arcmin⁻² is a factor of two larger than the one used in Okabe et al. (2010). We summarize our source selection criteria in Table 1.

Quantitative interpretation of a lensing signal requires information on the redshift distribution of the source population. The observed shears that are extracted from the source galaxies are expressed in units of the critical surface density Σ_c defined as

$\begin{eqnarray}&&{{\rm{\Sigma }}}_{c}=\displaystyle \frac{{c}^{2}}{4\pi {{GD}}_{l}\beta },\end{eqnarray} \tag{ 3 }$

where c is the speed of light, G is the gravitational constant, D_l is the angular diameter distance of the lens, and β is the lensing efficiency. The lensing efficiency is given by

$\begin{eqnarray}&&\beta =\left\langle \max \left(0,\displaystyle \frac{{D}_{{ls}}}{{D}_{s}}\right)\right\rangle ,\end{eqnarray} \tag{ 4 }$

where D_s and D_ls are the angular diameter distances to the cluster and from cluster to source galaxy, respectively. Note that objects with negative β values are assigned a zero value because foreground sources do not contribute to the lensing signal regardless of their redshifts. Because we do not have photometric redshifts for individual galaxies, we evaluated β for the source population statistically using a control field. This requires the assumption that the statistical properties of the control field are similar to those of the A115 field. One may be concerned that this assumption might be invalid when we compare two small fields because of the sample variance. Jee et al. (2014) investigated the issue in their mass estimation of the galaxy cluster ACT-CL J01024915. They found that, even for their 6' × 6' field, the effect of the sample variance is small, responsible for only ∼4% shift in mass. This is mainly because the image is deep and thus produces a large redshift baseline for the source galaxy distribution. In the current study, where the field is much larger with a comparable depth, we expect that the sample variance is also subdominant.

We chose the Great Observations Origins Deep Survey South (GOODS-S; Giavalisco et al. 2004) data as our control field and utilized the photometric redshift catalog of Dahlen et al. (2010). After applying the same color and magnitude selection criteria (Table 1) on the GOODS-S catalog, we compared its magnitude distribution (red bins) with that in the source population (blue bins), as shown in the top panel of Figure 4. Because the GOODS-S images are deeper, and its galaxies are better deblended, the number density of objects per magnitude bin is much higher in GOODS-S at i' ≳ 25. To account for this difference, we weighted the redshift distribution of the GOODS-S catalog for each magnitude bin by the number density ratio of our source catalog to the GOODS-S catalog (see the bottom panel of Figure 4). With the GOODS-S conformed to our source catalog, we measured the average lensing efficiency by Equation (4). The lensing efficiency obtained in this way is β = 0.72, which corresponds to the effective redshift ${z}_{\mathrm{eff}}=0.81$. This β value is similar to the estimate $\beta =0.701$ reported in Okabe et al. (2010), whose source shape measurement is based on a 1500 s $i^{\prime}$-band image. The assumption that all sources are located at this single redshift causes bias in cluster mass estimation (as discussed in Seitz & Schneider 1997; Hoekstra et al. 2000). To correct for this bias, we applied the following correction to the observed shear:

$\begin{eqnarray}\begin{array}{rcl}g^{\prime} & = & \left[1+\left(\displaystyle \frac{\left\langle {\beta }^{2}\right\rangle }{{\left\langle \beta \right\rangle }^{2}}-1\right)\kappa \right]g\\ & = & \ \left(1+0.10\kappa \right)g,\end{array}\end{eqnarray} \tag{ 5 }$

where $\langle {\beta }^{2}\rangle \sim 0.57$.

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Another concern in this procedure might be blue cluster member contamination. Given the current limited filter coverage, it is difficult to efficiently select and remove the blue cluster members. If the contamination is significant, this will lead to underestimation of the lensing signal (and, thus, underestimation of the cluster mass). However, our previous studies found that the contamination is insignificant when sources are selected on the basis of the color–magnitude relation as done in the current study. For example, in their Hubble Space Telescope WL analysis, Jee et al. (2014) compared the magnitude distribution of the sources in A520 at $z\simeq 0.2$ with those in their control fields. If the blue member contamination is significant, the source density should show an excess with respect to those in the control fields. However, no such excess was found in their study. Because the redshift of A520 is comparable to that of A115, we argue that the conclusion of Jee et al. (2014) is applicable to the current study. Note that we could not perform a similar analysis with the current Subaru imaging data because of the large difference in instrument resolution between the cluster and control fields. Instead, we examined the source density as a function of cluster-centric distance. If the blue member contamination is significant, the source density profile may show a peak near the cluster center. Figure 5 shows that for all three choices of centers, the source densities at small radii have no significant excess.

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The shapes of lensed galaxy images are sheared by a small amount, which is typically a tiny fraction of the intrinsic shape noise. Thus, measurement of these shears requires averaging over a large sample of background galaxies. The "whisker plot" in Figure 6 shows the shear in the A115 field obtained by averaging over the background galaxy ellipticities. Each whisker in the 20 × 20 grid represents the magnitude and direction of the local average ellipticity within a radius of $r=80^{\prime\prime}$.

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The shear γ can be converted to the surface mass density κ (convergence) map using the following relation (Kaiser & Squires 1993, hereafter KS93):

$\begin{eqnarray}&&\kappa ({\boldsymbol{x}})=\displaystyle \frac{1}{\pi }\int {D}^{* }({\boldsymbol{x}}-{{\boldsymbol{x}}}^{{\prime} })\gamma ({{\boldsymbol{x}}}^{{\prime} }){d}^{2}{\boldsymbol{x}},\end{eqnarray} \tag{ 6 }$

where $D{({\boldsymbol{x}})=-1/({x}_{1}-{{ix}}_{2})}^{2}$ is the transformation kernel. A number of algorithms exist for this γ-to-κ conversion in the literature.

In this study, we used the maximum entropy maximum likelihood method (MAXENT) described in Jee et al. (2007b) for our mass reconstruction. The MAXENT method utilizes the "entropy" of the pixels to regularize the mass map. This enables us to reveal high-resolution features where the S/N is high while it reduces the noise by applying large smoothing kernels in the low S/N region such as field boundaries. Color-coded in Figure 6 is the resulting κ map, which presents two prominent mass peaks. When we used the traditional KS93 inversion method, we recovered similar features near the mass peaks with a FWHM $\sim 50^{\prime\prime}$ Gaussian kernel. The κ contours are overlaid on the Subaru color-composite image in Figure 7, where we see an excellent spatial agreement between both the two BCGs and the mass peaks ($\lesssim 10^{\prime\prime}$, 32 kpc); the two mass peaks also coincide with the two X-ray peaks (Figure 8). Our bootstrapping analysis based on the KS93 reconstruction (see Section 5.2) shows that the northern and southern mass clumps are detected at a significance of 3.8σ and 3.6σ, respectively, and the two mass centroids are highly consistent with the BCGs.

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Many WL studies estimate galaxy cluster masses on the basis of the assumption that they are composed of a single halo. However, this assumption can lead to nonnegligible mass bias if substructures' masses are comparable, as in the case of A115. In this study, our main results were obtained by simultaneously fitting two Navarro–Frenk–White (NFW; Navarro et al. 1997) halo profiles to A115N and A115S. The NFW shear model derived in Wright & Brainerd (2000) was adopted. However, we also present the results from one-dimensional (azimuthally averaged) profile fitting centered on each substructure for comparison. This enables us to assess the amount of bias that would have been introduced if only the one-halo fitting method had been used. Also, the comparison provides a sanity check for the two halo fitting method, which is numerically less stable and requires more complicated procedure.

4.2.1. One-dimensional Profile Fitting

The first step in one-dimensional profile fitting is the construction of the azimuthally averaged tangential shear profile as a function of radius. Tangential shear is defined as

$\begin{eqnarray}&&{g}_{{\rm{T}}}=-{g}_{1}\cos 2\phi -{g}_{2}\sin 2\phi ,\end{eqnarray} \tag{ 7 }$

where ϕ is the position angle of the source with respect to the subcluster center, and g₁ and g₂ are the two components of the calibrated ellipticity. Figure 9 shows the three tangential shear profiles when the center is placed at the global, A115N, and A115S centers. For the two subclusters, we chose the BCG locations as their centers because the centroids of the three cluster constituents (BCG, X-ray emission, and WL mass) agree nicely. We adopted the mean of the two subcluster peaks as the global center. If our WL resolution had been poorer (e.g., if the number density of sources had been less than 10 ${\mathrm{arcmin}}^{-2}$), we would have detected only a single mass clump centered near this middle point.

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In Figure 9, WL signals are clearly detected in all three cases nearly out to the field boundary ($r\sim 900^{\prime\prime}$). The consistency of the cross shears (obtained by rotating the position angle by $45^\circ$) with zero indicates that no significant B-modes are present in our analysis.

It is a common practice to discard signals at small radii in model fitting because of a number of issues. First, the WL assumption is violated near the cluster center. Because galaxy images are sheared nonlinearly, the measurement performed without any correction can lead to cluster mass bias. Second, cluster member contamination is highest near the center, which can suppress the lensing signal. Third, the shape of the profile at small radii is sensitive to the choice of the center and the true center is unknown. Fourth, we expect baryonic effects to be nonnegligible in the central region, which can make the actual profile differ from the NFW one. Currently, no consensus exists for the choice of a cutoff radius except that it should increase with halo mass. We chose ${r}_{\mathrm{cut}}=50^{\prime\prime}$ when the center was placed on each subcluster while this threshold was increased to ${r}_{\mathrm{cut}}=200^{\prime\prime}$ for the global mass estimation. This increase is needed to reduce the impact of the cluster substructures on the tangential shear profile; the projected distance from the global center to a subcluster is $\sim 150^{\prime\prime}$. We also exclude the tangential shears at large radii if the measurements come from incomplete annuli.

We used the mass–concentration relation from Dutton & Macciò (2014) to characterize our NFW halo. From our one-dimensional NFW fitting, we determine the masses of A115N and A115S to be ${M}_{200c}={1.75}_{-0.52}^{+0.76}\times {10}^{14}\,{M}_{\odot }$ and ${3.45}_{-0.70}^{+0.90}\,\times {10}^{14}\,{M}_{\odot }$, respectively. The global mass is estimated to be ${6.17}_{-1.48}^{+2.00}\times {10}^{14}\,{M}_{\odot }$ (Table 2). Consistent masses are obtained when we assume a singular isothermal sphere (SIS) instead (Table 3). We used these SIS fitting results to evaluate inferred velocity dispersions. The reduced ${\chi }^{2}$ values show that both models describe the observed profiles reasonably well and there is no significant indication that one model is preferred over the other.

Table 2.  One-dimensional NFW Profile Fitting Result

	R200c (Mpc)	M200c (×1014 M⊙)	${\chi }_{\mathrm{red}}^{2}$
Global	${1.65}_{-0.14}^{+0.16}$	${6.17}_{-1.48}^{+2.00}$	0.84
North	${1.08}_{-0.12}^{+0.14}$	${1.75}_{-0.52}^{+0.76}$	0.44
South	${1.36}_{-0.10}^{+0.11}$	${3.45}_{-0.70}^{+0.90}$	0.93

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Table 3.  One-dimensional SIS Profile Fitting Result

	σv (km s−1)	R200c (Mpc)	M200c (×1014 M⊙)	${\chi }_{\mathrm{red}}^{2}$
Global	${922}_{-67}^{+72}$	${1.38}_{-0.10}^{+0.11}$	${5.47}_{-1.20}^{+1.29}$	0.88
North	${597}_{-49}^{+54}$	${0.90}_{-0.07}^{+0.08}$	${1.49}_{-0.37}^{+0.41}$	0.42
South	${725}_{-40}^{+42}$	${1.09}_{-0.06}^{+0.06}$	${2.67}_{-0.44}^{+0.47}$	0.96

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4.2.2. Two-dimensional Simultaneous Profile Fitting with Two Halos

The results presented in Section 4.2.1 are subject to bias if the tangential shear profile around one halo is significantly influenced by the presence of the other. Thus, for more accurate mass measurement, we must fit two halos simultaneously.

As in Section 4.2.1, we assume that each cluster's center coincides with the location of the BCG and fix the centroids of both halos in our two-dimensional fitting. Because each subcluster's mass is not large enough to produce strong signals that can constrain six free parameters (two centroid coordinates and two concentrations), fixing the centroid is required to stabilize the fitting. Because the WL mass peaks, the Chandra X-ray peaks, and the BCG positions agree excellently, it is unlikely that this centroid choice leads to any significant mass estimate bias. We model both halos with NFW profiles using the mass–concentration of Dutton & Macciò (2014) and determine the expected shear at every source galaxy position on the basis of the combined contribution from the halos. Our log-likelihood is given as

$\begin{eqnarray}&&L=\displaystyle \sum _{i}\displaystyle \sum _{s=1,2}\displaystyle \frac{{\left[{g}_{s}^{m}({M}_{A115N},{M}_{A115S},{x}_{i},{y}_{i})-{g}_{s}^{o}({x}_{i},{y}_{i})\right]}^{2}}{{\sigma }_{{SN}}^{2}+{\sigma }_{e}^{2}},\end{eqnarray} \tag{ 8 }$

where ${g}_{s}^{m}$ (${g}_{s}^{o}$) is the $s\mathrm{th}$ component of the predicted (observed) reduced shear at the $i\mathrm{th}$ galaxy position $({x}_{i},{y}_{i})$ as a function of the two clusters' masses ${M}_{A115N}$ and ${M}_{A115S}$. The ellipticity dispersion (shape noise) is ${\sigma }_{{SN}}=0.25$, whereas ${\sigma }_{e}$ is the ellipticity measurement noise of each object. Note that the evaluation of this likelihood function does not require source galaxy binning.

We used the Markov Chain Monte Carlo method to sample this likelihood. We display the resulting parameter contours in Figure 10 and list the best-fit parameters in Table 4. One may expect a degeneracy between the two parameters to exist to some extent because the two masses can trade with each other without significantly affecting the global goodness-of-the-fit. However, we find that the degeneracy is weak, which is attributed to the large distance ($\sim 900$ kpc) between the two subclusters. The masses for A115N and A115S are ${M}_{200c}={1.58}_{-0.49}^{+0.56}\times {10}^{14}\,{M}_{\odot }$ and ${3.15}_{-0.71}^{+0.79}\times {10}^{14}\,{M}_{\odot }$, respectively. These masses are consistent with our one-dimensional fitting results, although the decrease in the central value is in line with our expectation.

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Table 4.  Two-dimensional Two Halo NFW Fitting Result

	R200c (Mpc)	M200c (×1014 M⊙)
Global	${1.67}_{-0.09}^{+0.10}$	${6.41}_{-1.04}^{+1.08}$
North	${1.03}_{-0.11}^{+0.13}$	${1.58}_{-0.49}^{+0.56}$
South	${1.31}_{-0.10}^{+0.11}$	${3.15}_{-0.71}^{+0.79}$

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Table 5.  X-Ray Mass

X-ray	R500c (Mpc)	R200c (Mpc)	M500c (×1014 M⊙)	M200c (×1014 M⊙)
Global	${1.60}_{-0.07}^{+0.08}$	${2.46}_{-0.12}^{+0.13}$	${14.06}_{-1.82}^{+2.34}$	${20.48}_{-2.71}^{+3.49}$
North	${1.22}_{-0.07}^{+0.08}$	${1.87}_{-0.11}^{+0.13}$	${6.29}_{-1.01}^{+1.39}$	${9.00}_{-1.48}^{+2.03}$
South	${1.20}_{-0.07}^{+0.08}$	${1.84}_{-0.10}^{+0.13}$	${5.96}_{-0.95}^{+1.30}$	${8.52}_{-1.38}^{+1.90}$

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Table 6.  Mass Comparison

M200c (×1014 M⊙)	Global	North	South
Weak Lensing	6.41 + 1.08−1.04	${1.58}_{-0.49}^{+0.56}$	${3.15}_{-0.71}^{+0.79}$
X-ray	${20.48}_{-2.71}^{+3.49}$	${9.00}_{-1.48}^{+2.03}$	${8.52}_{-1.38}^{+1.90}$
Velocity Dispersion	${37.4}_{-5.2}^{+5.7}$	${16.3}_{-2.5}^{+2.8}$	${20.4}_{-3.3}^{+3.7}$

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The total mass of A115 is not a simple sum of the two masses of A115N and A115S if we maintain a consistent scheme in defining halo masses (i.e., a mass contained within a spherical volume as defined in Section 1). In order to derive the total ${M}_{200c}$ mass, we need to determine ${R}_{200c}$ for the total system, which requires the following two assumptions. First, we assume that the two subclusters are merging on the plane of the sky. This allows us to adopt the projected distance as the physical separation between A115N and A115S. Because our analysis (Section 5) favors the scenario wherein the merger is happening nearly in the plane of the sky, we believe that this assumption does not greatly depart from the truth. Second, we assume that the system's global center is located at the geometric mean of the two subclusters. One may argue that a better choice would be the barycenter. However, our analysis shows that this change causes a less than 10% shift in the total mass. We populate a three-dimensional grid with the sum of two densities according to the NFW parameters of both clusters. The ${R}_{200c}$ value is determined by locating the radius of the spherical volume, inside which the mean density becomes 200 times the critical density of the universe at the cluster redshift. The total mass obtained in this way is ${M}_{200c}={6.41}_{-1.04}^{+1.08}\times {10}^{14}\,{M}_{\odot }$ at ${R}_{200c}={1.67}_{-0.09}^{+0.10}$ Mpc. Comparison of these WL masses with our X-ray and spectroscopic results and the values in the literature are discussed in Section 5.1.

Our first step toward X-ray-based mass estimation is the measurement of the ICM temperature. We used the X-ray spectra within a 1–5 keV energy band and the MEwe, KAastra & Liedahl (MEKAL) plasma model (Kaastra & Mewe 1993; Liedahl et al. 1995). Exclusion of the energy band less than 1 keV is our conservative measure to minimize the impact from the well-known low-energy calibration issue of the ACIS detector (e.g., Chartas & Getman 2002). The Galactic hydrogen density, metallicity abundance, and redshift of the cluster were fixed to ${N}_{{\rm{H}}}=5.2\times {10}^{20}\,{\mathrm{cm}}^{-2}$ (Stark et al. 1992), ${Z}_{\odot }=0.3$, and $z=0.192$, respectively. Because each X-ray peak possesses a disturbed morphology and a position-dependent temperature variation, it is difficult to determine a single temperature representative of the cluster mass. We took care to avoid using too small an aperture because each peak has a cool core and to avoid using too large an aperture because the intermediate region between the two X-ray peaks has a very high temperature, which is attributed to the ongoing merger activity (Hallman et al. 2018). We used the temperature map of Hallman et al. (2018) as a guide. The resulting inner and outer radii of our annuli's are 94 kpc and 283 kpc (94 kpc and 267 kpc), respectively, for A115N (A115S).

Using the above setup, we measured ${T}_{{\rm{X}}}=7.06\pm 0.21\,\mathrm{keV}$ (${\chi }_{\mathrm{red}}^{2}=0.88$) and 6.83 ± 0.21 keV (${\chi }_{\mathrm{red}}^{2}=0.68$) for A115N and A115S, respectively. We display the X-ray spectra and fitting results in Figure 11. If we do not excise the cores, the temperatures become ${T}_{{\rm{X}}}=5.08\pm 0.08\,\mathrm{keV}$ and ${T}_{{\rm{X}}}=6.86\,\pm 0.19\,\mathrm{keV}$ for A115N and A115S, respectively. The decrease in A115N is significant and shows that the core temperature of A115N is indeed low. As shown by previous studies, we also confirm that using a smaller circular aperture leads to lower temperatures for both X-ray peaks. For example, choosing an $r=47\,\mathrm{kpc}$ aperture gives ${T}_{{\rm{X}}}=3.19\pm 0.06\,\mathrm{keV}$ for A115N and 5.12 ± 0.51 keV for A115S. These measurements are consistent with the measurements in Gutierrez & Krawczynski (2005).

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One popular method for X-ray-based mass estimation is to determine the mass using both X-ray and surface brightness measurements with the assumption that the halo follows a certain analytic profile such as NFW. We do not employ this method here, however, because the disturbed morphology prevents us from obtaining a reliable surface brightness profile. Instead, we estimate the cluster mass from a mass–temperature (M–T) relation based on the temperature measurements extracted from the aforementioned annuli.

Using the scaling relations of Mantz et al. (2016) gives ${M}_{500c}={6.29}_{-1.01}^{+1.39}\times {10}^{14}\,{M}_{\odot }$ and ${5.96}_{-0.95}^{+1.30}\times {10}^{14}\,{M}_{\odot }$ for A115N and A115S, respectively. For comparison with WL masses, we converted these ${M}_{500c}$ masses to ${M}_{200c}$ masses by extrapolation. Using the mass–concentration relation of Dutton & Macciò (2014), we obtained ${M}_{200c}={9.00}_{-1.48}^{+2.03}\times {10}^{14}\,{M}_{\odot }$ (${R}_{200c}={1.87}_{-0.11}^{+0.13}$ Mpc) for A115N and ${8.52}_{-1.38}^{+1.90}\times {10}^{14}\,{M}_{\odot }$ (${R}_{200c}={1.84}_{-0.10}^{+0.13}$ Mpc) for A115S. As mentioned in Section 4.2, the total mass of A115 is not a simple sum of the two masses. Using the method described in Section 4.2, we estimated the total X-ray mass to be ${M}_{200c}={20.48}_{-2.71}^{+3.49}\,\times {10}^{14}\,{M}_{\odot }$ or ${M}_{500c}\,={14.06}_{-1.82}^{+2.34}\times {10}^{14}\,{M}_{\odot }$ (Table 5).

We compiled our spectroscopic redshift galaxy catalog of the A115 field by combining the Golovich et al. (2017) and Rines et al. (2018) data. The Golovich et al. (2017) catalog contains 198 spectroscopic members from our own DEIMOS survey and NASA/IPAC Extragalactic Database (NED).¹³ The NED catalog has contributions from Beers et al. (1990), Zabludoff et al. (1990), Skrutskie et al. (2006), Barrena et al. (2007), and Alam et al. (2015). From their HeCS-red survey, Rines et al. (2018) provided 512 objects in the A115 field, of which 95 are A115 members. Of these 95 objects, 27 are redundant with those in the Golovich et al. (2017) catalog. We verified that the spectroscopic redshifts of these 27 common objects agree excellently to the fourth decimal point. The total number of A115 cluster members in our combined catalog is 266.

We applied the biweight estimator (Beers et al. 1990) and determined the redshift and LOS velocity dispersion of A115 to be $z=0.19216\pm 0.00032$ and ${\sigma }_{v}=1356\pm 67\,\mathrm{km}\,{{\rm{s}}}^{-1}$, respectively; we use bootstrapping to evaluate the uncertainties. Both values are consistent with the Barrena et al. (2007) measurements ($z=0.1929\pm 0.0005$ and ${\sigma }_{v}={1362}_{-108}^{+126}\,\mathrm{km}\,{{\rm{s}}}^{-1}$) and also with the Golovich et al. (2018) results ($z=0.19285\pm 0.00040$ and ${\sigma }_{v}=1439\pm 79\,\mathrm{km}\,{{\rm{s}}}^{-1}$). The top panel of Figure 12 shows the redshift distribution of the 266 members of A115. We agree with Golovich et al. (2017) that the overall redshift distribution of the A115 galaxies is well described with a single Gaussian profile.

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Assigning a galaxy to one of the two subclusters is nontrivial because their virial radii overlap. We used a GMM analysis¹⁴ to determine the membership between A115N and A115S. The analysis assigned 134 and 132 galaxies to A115N and A115S, respectively. After σ-clipping, the number of members reduced to 115 (120) for A115N (A115S). The second and third panels (light shade) of Figure 12 display the redshift distributions of A115N and A115S, respectively. The LOS difference in velocity between the two subsystems is $244\pm 144\,\mathrm{km}\,{{\rm{s}}}^{-1}$ (see the bottom panel of Figure 12). The individual velocity dispersions of A115N and A115S are ${\sigma }_{v}=1019\pm 57\,\mathrm{km}\,{{\rm{s}}}^{-1}$ and ${\sigma }_{v}=1101\pm 64\,\mathrm{km}\,{{\rm{s}}}^{-1}$, respectively.

We converted the above velocity dispersions to dynamical masses using the $M\mbox{--}{\sigma }_{v}$ scaling relation of Saro et al. (2013). The dynamical mass of the entire system was estimated to be ${M}_{200c}\,={37.4}_{-5.2}^{+5.7}\times {10}^{14}\,{M}_{\odot }$, while we obtained ${M}_{200c}\,={16.3}_{-2.5}^{+2.8}\,\times {10}^{14}\,{M}_{\odot }$ and ${M}_{200c}={20.4}_{-3.3}^{+3.7}\times {10}^{14}\,{M}_{\odot }$ for A115N and A115S, respectively.

Barrena et al. (2007) quoted a very large ($\sim 1600\,\mathrm{km}\,{{\rm{s}}}^{-1}$) velocity difference between A115N and A115S from their analysis of 88 cluster members. This claim is based on measurement of only the members within $\sim 0.25$ Mpc of the BCG. However, this measurement lacks statistical significance because only six and seven members for A115N and A115S, respectively, were found within this radius. When we repeated the analysis using our catalog, we found 15 members for each subcluster within the same radius. The redshift distributions of these galaxies are shown as the dark-shaded histograms in the second and third panels of Figure 12. The LOS velocity difference measured in this way is $838\pm 549\,\mathrm{km}\,{{\rm{s}}}^{-1}$. The central value is higher than the case where we use the GMM method to determine the subcluster membership ($244\,\pm 144\,\mathrm{km}\,{{\rm{s}}}^{-1}$). However, the two measurements are different by only $\sim 1\sigma$ because of the large uncertainty attached to the measurement from the members in the subcluster core. Nevertheless, it is interesting to note that the LOS velocity difference between the two BCGs is $\sim 853\,\mathrm{km}\,{{\rm{s}}}^{-1}$, which is close to the central value of the measurement $838\pm 549\,\mathrm{km}\,{{\rm{s}}}^{-1}$ based on the members in the core ($r\lt 0.25$ Mpc). The change in the LOS velocity happens mostly because the galaxies located in the A115N center on average have higher redshifts than the rest (see the solid versus dashed lines in the second panel of Figure 12). We do not observe this trend for A115S (the third panel of Figure 12). This radial dependence is also mentioned by Barrena et al. (2007) in Figures 13 and 14 of their paper. We defer our interpretation of the above results to Section 5.3.

M/L ratios of galaxy clusters have been used to estimate the matter density of the universe under the assumption that clusters are representative of our universe (e.g., Carlberg et al. 1997). Also, the evolution of cluster M/L values with redshift provide useful constraints on the stellar mass assembly history. Here, we present our estimation of the M/L value of A115. One of the motivations of this investigation is to examine the consistency of the resulting M/L values with results for other clusters. Although the M/L dispersion among clusters is quite large in the literature (e.g., the M/L value spans the range $50\sim 1000$ for ${10}^{14}\mbox{--}{10}^{15}\,{M}_{\odot }$ clusters according to Girardi et al. 2002), the order-of-magnitude difference between our WL and other mass estimates (Table 6) makes this comparison still statistically interesting. To measure the M/L value, we evaluated the A115 mass and luminosity within a cylindrical volume rather than a spherical volume. We used the best-fit NFW parameters presented in our two-halo simultaneous fitting (Section 4.2.2) to estimate the projected mass density as a function of radius. The projected mean surface mass density for each subcluster was computed using Equation (13) from Wright & Brainerd (2000). A two-dimensional mass density map was obtained by adding the contributions from A115N to A115S.

We constructed our A115 member catalog by combining our spectroscopic members and photometrically selected member candidates according to the color–magnitude relation. We characterized the red-sequence locus by performing a linear fit to the spec-z members and selected the candidate galaxies whose $V-i^{\prime}$ colors are within 0.05 mag from the fitted line and V-band magnitudes are brighter than $V=22$ (see photometric candidate in Figure 3). Our final member catalog contains 377 objects. We estimated B-band luminosity L_B⊙ from our V and $i^{\prime}$ magnitudes using the photometric transformation obtained by performing synthetic photometry (Sirianni et al. 2005) with a spectral energy distribution template of elliptical galaxies.

Figure 13 shows the cumulative M/L profile for our three chosen centers (two BCGs and one global). When the centers are placed at the BCGs, the M/L value is low at small radii because of the BCG's contribution to the luminosity. The M/L value is estimated high near the global center because no bright galaxies are present in this region. We find that the M/L ratio of A115N and A115S are $\sim 400$ and $\sim 650$, respectively, within their virial radii ($\sim 1$ Mpc for A115N and $\sim 1.3$ Mpc for A115S). These M/L values are higher than the mean value of the ΛCDM prediction, but can be accommodated within the distribution of the sample of 89 clusters studied in Girardi et al. (2002). This comparison shows that our WL masses, although substantially lower than the X-ray or dynamical estimates, give the most physical M/L values for A115. If dynamical masses are used instead, the implied M/L value would increase by an order of magnitude, which is difficult to accommodate within the current ΛCDM paradigm. In general, dynamics of galaxies are known to be biased in a merger (Pinkney et al. 1996; Takizawa et al. 2010).

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A115 is one of the most studied galaxy clusters. Here, we compare our WL mass estimates with those from the literature.

Global Mass. Figure 14 shows the global ${M}_{500c}$ estimates from various studies. Note that most past studies did not report separate masses for A115N and A115S. For studies that only quote ${M}_{200c}$ values, we converted them to ${M}_{500c}$ values using an NFW profile and the mass–concentration relation of Dutton & Macciò (2014). This conversion was also applied to our WL results.

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The most significant outlier in Figure 14 is the dynamical mass estimate from Barrena et al. (2007). Because our velocity dispersions from improved statistics yield a similarly high mass, we attribute the large difference not to any errors in measurement, but to a significant departure of A115 from dynamical equilibrium due to the merger. In general, velocity dispersion is believed to be boosted in epochs close to pericentric passages (e.g., Pinkney et al. 1996; Monteiro-Oliveira et al. 2017). However, it remains to be investigated by future numerical simulations whether the merger alone can inflate the velocity dispersion measurement to this extent. The dynamical mass estimate from Sifón et al. (2015) is substantially lower than the Barrena et al. (2007) result. This is because Sifón et al. (2015) treated A115 as a single halo, whereas Barrena et al. (2007) took into account the multiplicity.

The X-ray (Govoni et al. 2001; Lidman et al. 2012; Landry et al. 2013; Niikura et al. 2015) and WL mass (Pedersen et al. 2007; Oguri et al. 2010; Okabe et al. 2010; Sereno et al. 2015) estimates presented in Figure 14 seem to be consistent with our WL result. However, the caveat is that these values are obtained under the single-halo assumption.

Substructure Mass. Hoekstra et al. (2012) presented WL masses for A115N and A115S separately using Canada–France–Hawaii Telescope (CFHT) imaging data. They quoted ${M}_{500c}={3.9}_{-1.5}^{+1.4}\times {10}^{14}\,{M}_{\odot }$ and ${5.4}_{-1.2}^{+1.3}\times {10}^{14}\,{M}_{\odot }$ for A115N and A115S, respectively. These masses were derived by deprojecting their aperture masses. When they directly fit an NFW profile, they obtain ${M}_{500c}={3.2}_{-1.0}^{+1.0}\times {10}^{14}\,{M}_{\odot }$ (${3.8}_{-1.1}^{+1.2}\,\times {10}^{14}\,{M}_{\odot }$) for A115N (A115S). The deprojected values are higher than our results by a factor of 2–3; when converted to M_500c, our Subaru-base WL masses become ${M}_{500c}={1.14}_{-0.35}^{+0.40}\times {10}^{14}\,{M}_{\odot }$ and ${2.25}_{-0.50}^{+0.55}\times {10}^{14}\,{M}_{\odot }$ for A115N and A115S, respectively. When the two masses (A115N and A115S) from Hoekstra et al. (2012) are combined, the resulting global mass of A115 would be also 2–3 times higher than our WL result. In order to investigate the source of the discrepancy with the Hoekstra et al. (2012) results, we analyzed their CFHT data with our WL pipeline. The difference in depth and seeing results in a slight (∼30%) reduction in source density compared to the Subaru analysis (∼19 arcmin⁻² versus ∼24 arcmin⁻²). Nevertheless, we find that our masses derived from the CFHT data are in agreement with our Subaru-based values within ∼2%. This excellent agreement supports the repeatability of our WL mass measurement regardless of the instrument choice. Hence, we suspect that the discrepancy between Hoekstra et al. (2012) and ours may be attributed to the difference in the WL pipeline and mass estimation method.

Mass distribution. Among the few WL studies in the literature, only Okabe et al. (2010) presented a two-dimensional mass distribution for A115, which shows two mass peaks similar to ours. However, both of their mass clumps are offset toward the northeast with respect to their nearest BCGs. As mentioned in Section 3, our mass peaks coincide with the corresponding BCGs. Okabe et al. (2010) performed their WL analysis using the i'-band image, which was significantly deeper than the V-band image at the time of the analysis. Because our WL shape is derived from the V-band data, we think that the difference may be due to different systematics. To address the issue, we repeated the measurement with the i' imaging data. We find that the position-dependent PSF ellipticity pattern of the i' image is much more complex than the pattern in the V image, and our PCA-based PSF model could not reproduce the observed PSF pattern with the same fidelity (Figure 2), as mentioned in Section 3.1.1. Interestingly, the resulting mass reconstruction from this i'-band analysis resembles the one in Okabe et al. (2010), possessing similar offsets. Therefore, it is possible that the mass-galaxy offsets in Okabe et al. (2010) may be due to large residual PSF systematics in the i'-band imaging data. However, we can only be speculative regarding this issue because we do not have access to their WL catalog.

As shown in Figure 7, our mass centroids agree nicely with the BCG positions. If the BCG represents the true center of each halo, one can interpret the agreement as evidence for dark matter with negligible self-interacting cross-section. However, it is still unclear in general whether a BCG can serve as the proxy for a halo center. Alternatively, one can use smoothed galaxy distributions to define halo centers. In Figure 15, we display mass contours over galaxy number and luminosity density maps. Interestingly, the centroids of the smoothed galaxy distributions possess offsets with respect to the BCGs. For A115S, both number and luminosity density centroids are displaced south by ∼30''. Similar offsets are found for A115N except that the number density peak is at a greater distance from the BCG than the luminosity peak. Here we present our investigation of the statistical significance of the mass centroid with respect to various definitions of subcluster centers.

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We used bootstrap analysis to measure the significance of the centroids for both mass and galaxy distributions. To estimate the WL mass distribution centroid uncertainty, we bootstrapped the final source catalog and generated 5000 convergence maps using the KS93 method. From each convergence map realization, we identified peaks by determining the first moment. Then, the distribution of the resulting peak locations was processed with a Kernel Density Estimation (KDE) to define the significance regions. We also generated 5000 bootstrap realizations of the galaxy distribution by sampling the photometrically and spectroscopically selected cluster members (presented in Figure 3). Again, we identified peaks using the first moment and used KDE to define the significance regions. While we believe that the resulting centroid uncertainty of the galaxy number density is a fair measure of the significance, we argue that the centroid uncertainty of the luminosity density obtained in this way corresponds to an upper limit because the peak location in each realization is dominated by several bright galaxies. Figure 16 compares the 1σ contours among the mass, luminosity, and number density results. We find that the three centroids are highly consistent with one another (well within 1σ contours).

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A115 is a merging galaxy cluster with a number of intriguing features summarized as follows.

• 1.  
  A giant (∼2.5 Mpc) radio relic is detected at the northern edge of A115N (Govoni et al. 2001).
• 2.  
  The orientation of the radio relic is approximately perpendicular to the vector connecting A115N and A115S.
• 3.  
  The center of the radio relic is offset toward the east by ∼0.7 Mpc from this connecting vector and ∼1 Mpc from the global center.
• 4.  
  Both surface brightness and temperature jumps in X-ray are detected across the radio relic, which is translated to ${ \mathcal M }=1.4-2.0$ with systematics included (Botteon et al. 2016).
• 5.  
  The surface brightness distributions of the X-ray emission for both A115N and A115S are asymmetric.
• 6.  
  Our WL analysis shows that A115S is twice more massive than A115N and the total cluster mass is ${M}_{200c}\,={6.41}_{-1.04}^{+1.08}\times {10}^{14}\,{M}_{\odot }$.
• 7.  
  The analysis with our enhanced spectroscopic catalog with 266 members shows that the LOS velocity difference between A115N and A115S is small (244 ± 144 km s⁻¹).

Point 1 is strong evidence that the system is postmerger. Although Hallman et al. (2018) suggests a possibility that the radio relic might be a premerger shock, our numerical simulation with our WL masses as input shows that this shock would be too weak to generate such a giant radio relic even if there exists a rich population of so-called fossil electrons (W. Lee et al. 2019, in preparation). The last point supports the possibility that the merger is taking place nearly in the plane of the sky. Future radio observations can provide further insights into this viewing angle issue from polarization fraction measurements. Points 2 and 3 indicate that A115N and A115S might have collided in the north–south direction with a nonnegligible impact parameter. Point 4 can be interpreted as suggesting that the shock velocity is as high as ∼1800 km s⁻¹. Finally, we can infer from the morphology (Point 5) of the X-ray peak that A115N (A115S) might be heading southwest (northeast).

On the basis of the subset of the points above, we can carry out some consistency checks for the progression of the merger. If the impact happened near the global center, the shock traveled ∼1 Mpc. Assuming a constant shock velocity of ∼1800 km s⁻¹ derived from the Mach number, we estimate that it takes about 0.5 Gyr for the shock to reach the current location. Some simulations suggest that a shock traveling speed is a good proxy for the collision speed at the time of impact (e.g., Springel & Farrar 2007). An independent estimation of the collision velocity can be made with the following equation based on the so-called timing argument (Sarazin 2002):

$\begin{eqnarray}&&\begin{array}{l}v\sim 2930{\left(\displaystyle \frac{{M}_{1}+{M}_{2}}{{10}^{15}{M}_{\odot }}\right)}^{1/2}\\ \quad \times \ {\left(\displaystyle \frac{1-d/{d}_{0}}{1-{\left(b/{d}_{0}\right)}^{2}}\right)}^{1/2}{\left(\displaystyle \frac{d}{1\mathrm{Mpc}}\right)}^{-1/2}\,\mathrm{km}\,{{\rm{s}}}^{-1},\end{array}\end{eqnarray} \tag{ 9 }$

where the initial separation is

$\begin{eqnarray}&&{d}_{0}\ \sim \ 4.5{\left(\displaystyle \frac{{M}_{1}+{M}_{2}}{{10}^{15}{M}_{\odot }}\right)}^{1/3}{\left(\displaystyle \frac{{t}_{\mathrm{impact}}}{10\mathrm{Gyr}}\right)}^{2/3}\,\mathrm{Mpc}.\end{eqnarray} \tag{ 10 }$

We assume that ${M}_{1}+{M}_{2}=4.73\times {10}^{14}\,{M}_{\odot }$ is the sum of the individual WL cluster masses, t_impact = 11 Gyr is the time from rest to impact, b = 0 is the impact parameter, and the current separation d = 1 Mpc. Setting b = 0 is justified in this approximation because Equation (9) varies quite slowly in b/d₀ when d₀ is large. The resulting relative velocity of the clusters at impact is ∼1700 km s⁻¹. This agrees with the velocity derived from the Mach number of the shock. Furthermore, because the radio relic is close to A115N, it is unlikely that the subclusters have turned around and we are witnessing a returning phase. This is in contrast to the scenario that one might derive from the X-ray morphology.

More specific merger scenarios can be inferred when we search for merging cluster analogs in cosmological numerical simulations. Using the Wittman et al. (2018) method, we sampled cluster mergers by matching the cluster redshift, projected distance, radial velocity difference, and cluster masses. As explained in Wittman et al. (2018), this method has several advantages over the Monte Carlo Merger Analysis Code (MCMAC; Dawson 2013) method. One important advantage is that finding merger analogs in cosmological simulations allows us to consider the cases where the subcluster velocity vectors are not entirely parallel to the separation vector, while the MCMAC method always assumes that the collision is head-on. This head-on collision assumption leads to on average a larger deviation between the separation vectors and the plane of the sky. As A115 is believed to be an off-axis merger, this issue cannot be neglected.

Figure 17 shows the trajectories of each analog (top) and constraints of time since pericenter (TSP; bottom left), maximum colliding velocity (bottom middle), and velocity direction (bottom right). TSP is a useful quantity because the information helps us to distinguish between in-bound and out-bound cases. The maximum colliding velocity is the impact velocity, which can be approximated to be the shock propagation velocity. Investigation of the velocity direction allows us to infer the viewing angle of the merger. The TSP is most likely to be ∼600 Myr with a maximum collision velocity of ∼2000 km s⁻¹. These values are consistent with the above estimates based on the Mach number, position of the radio relic, and timing argument. The velocity direction (the angle between the relative velocity vector and the separation vector) is centered at ∼25°. Although not shown here, we also found that about 68% of analogs have their separation vector axis less than 19° from the plane of the sky. Therefore, our LOS velocity difference constraint 244 ± 144 km s⁻¹ only marginally favors mergers near the plane of the sky. This weak constraint is not surprising because the velocity vectors of the analogs are not perfectly aligned with the separation vectors. The trajectory plot (top) shows that the majority of the analogs are in the outgoing phase at the cluster redshift. This can also be inferred by either the short TSP or the relative velocity vector being less than 90°; the relative velocity vector is (mostly) parallel to the separation vector, rather than antiparallel. Because we do not use the radio relic in our analog search, it is interesting that this analog-based result also favors the same outgoing case. However, note that the small bump near 160° in the velocity direction panel (or near ∼1.5 Gyr in the TSP panel) shows that a small fraction of the analogs are in the returning phase.

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In summary, we find that our analysis favors A115N and A115S in an outgoing phase. However, this seems to contradict the visual impression given by the cometary tails in X-ray emission. Careful hydrodynamical simulations are needed to determine whether the observed X-ray morphologies can be reproduced in an outgoing phase.