Radio Galaxy Zoo: The Distortion of Radio Galaxies by Galaxy Clusters

For each source in the matched sample, we take its projected distance from the center of the cluster (now calculated at the redshift of the cluster to be more physically meaningful, in contrast to the matching algorithm in Section 2) and normalize it by the r₅₀₀ of the cluster, as given in the WH15 catalog. The median r₅₀₀ in WH15 is 0.65 Mpc. The histogram of normalized separations is shown in Figure 4. The distribution is bimodal, with one peak around 10⁻⁴ r₅₀₀ and the other around 10 r₅₀₀. The first peak is due to the WH15 definition of the cluster center, which is defined as the position of the BCG; because WH15 used Data Releases 8 and 9 of SDSS, whereas RGZ uses Data Release 13, there is often a small positional offset (less than 1 arcsec) between the WH15 and RGZ position of the host galaxy. Given this, we treat all sources within 0.01 r₅₀₀ of the cluster center as being the BCGs of their respective cluster and group them together in our analysis. This corresponds to a physical separation of only 7 kpc for the median cluster in our sample, well within the size of a galaxy.

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When plotting the number of sources matched to a cluster per unit area around it, the surface density scales as a power law. We perform a linear fit to the data in log−log space between 0.01 r₅₀₀ and 10 r₅₀₀, and calculate that the best-fitting power law has index α = −1.10 ± 0.03, shown as the blue points and solid line in Figure 5. We compared this result to a sample of 14,214 optically selected galaxies from SDSS. We selected the SDSS sample by drawing random R.A., decl., and redshift values following the distribution of the RGZ sample and taking the nearest corresponding galaxy in SDSS. Using the same cluster-matching algorithm, we matched 13,182 (93%) of the SDSS galaxies to WH15. Figure 5 shows the surface density distribution for the optical galaxies in orange, for which the best-fitting power law (dashed line) has index α = −0.94 ± 0.02.

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Both radio and optical distributions follow an approximately 1/r trend, consistent with an isothermal distribution within the cluster. Since a background population would have a flat profile, these distributions are likely dominated by true cluster galaxies. However, the power-law slope is slightly but statistically significantly steeper for cluster radio galaxies. A total of 271/2209 = (12.3 ± 1.0)% of non-BCG radio sources within 10 r₅₀₀ of the nearest cluster are within the innermost r₅₀₀ of that cluster, compared to 584/8823 = (6.6 ± 0.3)% of optically identified galaxies within 10 r₅₀₀. This indicates that extended radio galaxies are a biased population (consistent with Overzier et al. 2003 and references therein). We believe the cutoff in the power law beyond 10 r₅₀₀ is due to a combination of the typical size of a galaxy cluster and when galaxies begin matching to another nearby cluster, although quantifying that effect would require a more detailed analysis.

In addition, we find that BCGs are at least 2.5 times as likely to have extended radio emission as the general population of cluster galaxies: 546/2755 = (19.8 ± 1.2)% of the RGZ matches within 10 r₅₀₀ are BCGs, compared to only 743/9566 = (7.8 ± 0.4)% of the SDSS matches within the same separation. This is consistent with previous findings (Best et al. 2007).

Figure 3 shows the distribution of Δz in the RGZ−WH15 cross-match. We use this distribution to estimate the contamination and completeness of our sample. Given our redshift matching criterion of $| {\rm{\Delta }}z| \lt 0.04$, we take the mean number of matches in each $| {\rm{\Delta }}z|$ bin between 0.04 and 0.1 as the number of contaminating sources per bin. Between $| {\rm{\Delta }}z| =0$ and 0.04, 21.5% of the matched sources can be attributed to this contamination. Likewise, we take the number of sources that are above this contamination level in bins with $| {\rm{\Delta }}z|$ between 0.04 and 0.1, where the entire sample is likely from contamination, as the number of sources that should be included in our sample but are removed by the redshift threshold. Of the cross-matched sources, 1.8% are removed this way, leading to a sample that is at least 98.2% complete.

We estimate the effect of contamination as a function of cluster separation. Assuming that contaminating sources are uniformly distributed on the projected sky, we calculate the number and fraction of expected contaminating sources at each cluster separation. The top plot of Figure 6 shows the distribution of sources in our sample (solid; the same distribution as Figure 4) and the number of contaminating sources in each bin (dashed; note that this is not the cumulative contamination). In our analysis, we focus on sources within 1.5 r₅₀₀ of their nearest cluster, which also corresponds to approximately the virial radius; the contamination in the bin at this separation is only 0.01%. A more detailed contamination analysis would include the effect of galaxies at high separations matching with other neighboring clusters, which we predict contributes to the drop-off in source count that occurs beyond 10 r₅₀₀. However, we can safely say that the effect of contamination in the range of separations in which we are interested is minimal.

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Our definition of bending angle is based on the angle between the host galaxy and the peaks in the radio emission. This is a well-defined and easily measurable, but still arbitrary definition. The first thing we do is assess the measurement uncertainties in bending angle that arise from this definition and the properties of our data sample.

There are four known sources of potential error in our measurement. The first is radio component classification error, which occurs when the radio source could consist of a chance superposition of unrelated radio components. We attempted to minimize this with the 0.65 consensus threshold in the sample selection stage.

The second is astrometry error of the optical host, including a very small contribution from host misidentifications. The astrometry error typically induces less than a tenth of a degree of bending, given the angular size of the sources and the subarcsec precision in the positions of the host galaxies in SDSS. Host misidentification could also contribute an error due to a spurious match between a misidentified host and a cluster unrelated to the radio source, but as discussed in the previous section, contamination by these foreground or background sources will have a negligible effect on our results.

The third is projection effects; although the radio sources exist in three-dimensional space, we can only measure the bending angle projected onto the sky. Because sources are oriented isotropically in space, the observed bending angle for a given source could be either increased or decreased by projection; for a large sample, however, the median observed bending angle remains consistent with the true bending angle for all but the most bent sources (θ_true ≳ 60°). Even then, projection will on average tend to reduce the observed bending angle, suggesting that any trends we observe with respect to bending angle are merely lower bounds.

The fourth and largest source of error is uncertainty due to structure in the radio emission. We estimate the amount of error introduced by this uncertainty by comparing the measurement derived from our definition of bending angle to an alternative measurement of the angle between the host galaxy and the most distant point on the lowest contour on each side that does not contain the host. We calculate the running root-mean-square difference between the two measures of bending angle, shown in Figure 7. This follows an approximately linear relationship with angular size, decreasing from an estimated uncertainty of 11° for sources of size 0.2 arcmin down to 6° for sources of size 1.1 arcmin. The divergence from a linear relationship for sources larger than 1.1 arcmin is due to a small number of outliers, but does not change our results.

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From geometric arguments, the bending angle error is proportional to the error in the position of the host and inversely proportional to the angular extent of the source, because the angle will be amplified if the radio peaks are closer to the host. Bending angle is a positive definite quantity, so the errors are not symmetric; given the same "true" bending angle, a larger error will statistically make the source look more bent, so small sources have their measured bending angle amplified.

The observed bending angle as a function of angular size for sources larger than ∼0.4 arcmin is very similar to the empirically determined bending angle uncertainty. In general, we argue that this similarity in magnitude and trend shows that the relationship between bending angle and angular size is predominantly due to irreducible measurement uncertainties because of source structure, and is not reflective of a physical trend. For smaller sources, the error gets compounded by the fact that structure becomes harder to resolve in the FIRST images.

Given that the bending angle is dependent on the angular size of the source, due in part to inherent uncertainty in our measurement of the bending angle, we correct for this relationship prior to analyzing the correlation of bending with other physical quantities. In particular, source size is correlated with the separation between the source and the cluster center. We observe that sources beyond 1.5 r₅₀₀ are on average 24% larger in physical extent than sources within 1.5 r₅₀₀. (We show in Section 5.1 that beyond 1.5 r₅₀₀, the bending is statistically unaffected by the cluster, hence the cutoff here.) Although part of this correlation is physical (e.g., Machalski & Jamrozy 2006), selection effects such as redshift prevent us from further analyzing this trend.

Because we are interested in the connection between bending and separation, among others, we must remove this confounding factor. To make this correction, we take the sources that are beyond $1.5\,{r}_{500}$ and fit an exponential function to the median angular size−bending relation. We define "corrected bending angle" as the ratio of a source's measured bending angle to the predicted bending angle given this relationship (fit shown in Figure 8). We do this separately for double and triple sources for reasons explained in the next paragraph. For each morphology, we normalize it such that a source of median size was unaffected by the correction. After this correction, both the median and quartile distributions are independent of source size, except for a minor deviation of the largest sources. We update the bending angle threshold of 135° to apply to the corrected bending angle rather than to the measured bending angle. This results in our final sample of 3742 double sources and 562 triple sources (4304 total), of which 3235 double sources and 488 triple sources are matched to clusters within a projected 15 Mpc (3723 total).

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We corrected double and triple sources separately as a consistency test. Triple sources have a low rate of false-positive classifications, because they have radio emission at the position of the host galaxy. However, there may exist confusion in the classification of double sources; for example, there may be an unrelated infrared source in the WISE field equidistant between two regions of potentially unrelated radio emission that gets mistaken as the host of a double source. If the host identifications in our sample are predominantly real, we expect the size−angle trend to be the same regardless of whether they are double or triple sources. A significant misidentification rate of double sources would result in a different relationship between angular size and bending angle for double and triple sources. Figure 9 compares the size−angle trend for double and triple sources, and we find that the trend for double sources is consistent with the trend for triple sources. Therefore, we conclude that our population of double sources is dominated by correct host identifications.

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Bending angle is a positive definite quantity and therefore has a contribution due to the uncertainty alone. Using the corrected bending angle, we therefore define an "excess bending angle" for each source as

$$\begin{eqnarray}&&{\rm{\Delta }}\theta =\sqrt{{\theta }_{\mathrm{corr}}^{2}-{\theta }_{\mathrm{med},\mathrm{outer}}^{2}},\end{eqnarray} \tag{ 3 }$$

where ${\theta }_{\mathrm{med},\mathrm{outer}}=7\buildrel{\circ}\over{.} 3$ is the median corrected bending angle for sources matched to clusters beyond 1.5 r₅₀₀ (the outer region defined in Section 5.1). This value is consistent with the irreducible uncertainty in measuring the bending angle, which supports our argument that sources beyond 1.5 r₅₀₀ are outside the cluster's range of influence on bending and are thus nominally straight. For the purposes of plotting, when ${\theta }_{\mathrm{corr}}\,\lt {\theta }_{\mathrm{med},\mathrm{outer}}$ and Δθ becomes imaginary, we plot $-| {\rm{\Delta }}\theta |$ instead; a negative excess by this definition is consistent with zero excess. More sophisticated estimates of the most probable value of Δθ when θ ≲ 20° would be required if one were to compare the quantitative bending to specific physical models.

Galaxy clusters have ICM density profiles that generally peak strongly toward the center (Pratt & Arnaud 2002; Newman et al. 2013), although there exist clusters with a dynamic or clumpy ICM where this assumption may not hold (e.g., merging systems like the Bullet cluster). For a given galaxy velocity with respect to the ICM, a denser environment should cause greater distortion of the radio jets. Therefore, we expect to see a correlation between the bending of the radio source and its proximity to the cluster center.

We calculate the running median of the excess bending angle, Δθ, as a function of normalized separation from the nearest WH15 cluster, as shown in Figure 10. The shaded region represents the running interquartile range. There are three distinct regions in the distribution of bending angles as a function of separation, which we separate prior to smoothing and calculating the trends in all figures that are plotted against separation. Beyond 1.5 r₅₀₀, we do not observe a significant trend between bending and separation, and measure a median Δθ ∼ 0° using the definition given in Equation (3); we call this the outer region of the cluster. Sources that are not matched to a WH15 cluster also have a median Δθ consistent with 0°. In Section 3.1, we identified the sources within 0.01 r₅₀₀ as BCGs, so in all further analysis, we combine these sources into a single data point at r = 0.01 r₅₀₀. The rest of the sources lie in what we call the inner region of the cluster, between 0.01 r₅₀₀ and 1.5 r₅₀₀. This also corresponds to approximately the virial region of the cluster. Our data set contains 544 BCGs, 366 sources in the inner region, and 2813 sources in the outer region.

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We find a median excess bending angle of about 80 for BCGs. The Anderson–Darling (AD) test is a statistical test used to test if a data sample comes from a specific distribution; we use a two-sample AD test to rule out that BCGs and sources in the outer region are drawn from the same bending population at the 3.7σ level. Sources in the inner region have a median excess bending angle of 13°. Between 0.01 r₅₀₀ and 1.5 r₅₀₀, the sources become more bent as they get closer to the center of the cluster, ranging from Δθ ∼ 0° at r = 1.5 r₅₀₀ up to a median Δθ ∼ 20° at r = 0.05 r₅₀₀. We use a Spearman's ρ test to determine that the correlation between separation and excess bending in the inner region is significant at the 4.5σ level.

Figure 11 shows a sample of FIRST images of sources from each of the three regions. (Links to the radio/IR overlays on the RGZ forum¹⁴ for all sources in our sample are available in the supplemental data files; see the Appendix for details.) Visual inspection suggests that BCGs and sources in the inner region have similar bending properties to each other, but are different from sources in the outer region. Although there are a few approximately straight BCGs and inner region sources, there are a large number of significantly bent radio galaxies in these regions. By comparison, there are a small number of visibly bent galaxies in the outer region, but most are close to straight.

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We also expect the mass of the parent cluster to affect the bending of the radio source, because the orbital velocities of galaxies will be higher in more massive clusters. The velocity dispersion of galaxies increases with the square root of the mass of the cluster, so ram pressure will be higher even at a fixed ICM density (Becker et al. 2007; Mguda et al. 2015). We perform a linear fit between M₅₀₀ and r₅₀₀ in log–log space for the clusters in our sample and find that ${\mathrm{log}}_{10}{M}_{500}\sim (2.95\pm 0.03){\mathrm{log}}_{10}{r}_{500}$, i.e., the cluster mass is directly proportional to its volume, so we argue that the ICM density is independent of cluster mass to first order.

The influence of cluster mass on the 366 sources in the inner region of the cluster is shown in the top plot of Figure 12, because that is where we find the cluster to have a statistically detectable effect on the bending. We use a Spearman's ρ test and find that the correlation between cluster mass and excess bending angle is significant at the 4.2σ level. We observe an increase in the median Δθ of 10° between cluster masses of 5 and 20 × 10¹⁴ M_☉.

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We also look at the cluster mass relationship for the 544 BCG sources separately, shown in the bottom plot of Figure 12. There is a suggestive correlation between cluster mass and excess bending angle for the BCGs as well. We use a Spearman's ρ test to determine that the correlation between cluster mass and excess bending angle is significant at the 2.5σ level for the BCG sample, and we observe the same increase in median Δθ of 10° across the same range of cluster masses.

The bending angle of radio emission depends on both the mass of the galaxy cluster and the position of the radio source within the cluster, as shown above. We assume that the observed bending is induced by the ram pressure exerted on the galaxy by its motion through the ICM, ${P}_{\mathrm{ram}}={\rho }_{\mathrm{icm}}{v}_{\mathrm{gal}}^{2}$ (Begelman et al. 1979; Jones & Owen 1979), although this argument is equivalent if it is an ICM wind flowing past the galaxy (Burns et al. 2002). In either case, ρ_icm is the ICM density and v_gal is the relative velocity between the galaxy and the ICM. Here we use P_icm, the ICM gas pressure, as a proxy for P_ram, under the assumption that the clusters are virialized to first order. In that case, the velocity of the galaxies and the velocity of the ICM gas particles both scale with the temperature of the ICM. We justify this in more detail in Section 6.2.

Arnaud et al. (2010) derived a "universal galaxy cluster pressure profile" using a sample of local clusters observed with XMM-Newton, to which they fit a generalized NFW profile. They derive

$$\begin{eqnarray}\begin{array}{rcl}{P}_{\mathrm{icm}}(x) & = & 1.65\times {10}^{-3}\,h{\left(z\right)}^{8/3}\,{\mathbb{P}}(x)\,{{\rm{h}}}_{70}^{2}\,\mathrm{keV}\,{\mathrm{cm}}^{-3}\\ & & \times \,{\left(\displaystyle \frac{{M}_{500}}{3\times {10}^{14}{M}_{\odot }}{h}_{70}\right)}^{2/3+{\alpha }_{P}+{\alpha }_{P}^{{\prime} }(x)}\end{array}\end{eqnarray} \tag{ 4 }$$

as the best universal pressure relation, where $x=r/{r}_{500}$, ${\mathbb{P}}(x)$ is the generalized NFW model from Nagai et al. (2007), and α_P and ${\alpha }_{P}^{{\prime} }(x)$ are parameters derived from the slope of the M₅₀₀−Y_X relation.

We apply Equation (4) to our sample to get a measure of the local ICM gas pressure. Figure 13 shows the distribution of excess bending angle as a function of pressure, BCGs excluded. As shown in the previous plots of separation and mass, low-pressure ($\lt 5\times {10}^{-4}\,\mathrm{keV}\,{\mathrm{cm}}^{-3}$) environments (i.e., at large separations [low density] and/or in low-mass clusters [low velocity]) do not induce a statistical increase in bending. Above $5\times {10}^{-4}\,\mathrm{keV}\,{\mathrm{cm}}^{-3}$, increasing the pressure leads to a greater excess bending angle, up to a median Δθ ∼ 14° when the pressure reaches ${10}^{-2}\,\mathrm{keV}\,{\mathrm{cm}}^{-3}$.

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We define the orientation angle of a source by taking the difference between the angle bisector of the opening angle and the radial vector from the center of the cluster. For the purpose of this analysis, we used the "folded" orientation, ignoring whether the source opened toward or away from the cluster. This means that the orientation angle ranges from 0° if the source opened parallel to the radial vector from the cluster (either toward or away), and increases to 90° if it opened perpendicular to the radial vector.

We measure the distributions of orientation angles for two subpopulations: highly bent sources (Δθ > 21°) and less bent sources (Δθ < 21°). We choose 21° as the fiducial threshold for significant bending as this captures the 16% most bent sources in our full matched sample. Both subpopulations are restricted to sources matched between 0.01 r₅₀₀ and 10 r₅₀₀. We exclude sources beyond 10 r₅₀₀ to keep the contamination below a few percent. To determine whether these samples were randomly distributed in orientation angle, we compared the distributions to the uniform distribution using a one-sample AD test. The distributions and corresponding p-values are shown in Figure 14. The less bent sources are uniformly distributed in orientation. However, highly bent sources tentatively appear to be preferentially aligned parallel to the radial vector; we reject that they are uniformly distributed in orientation at the 2.7σ level.

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To address the question of whether buoyancy may explain the excess of radially oriented highly bent sources, we also looked at the "unfolded" orientation angles. Observationally, if the excess of radially oriented sources was due to buoyancy, then the unfolded distribution would peak near 0° and drop off near 180°. Theoretically, pressure gradients such as those that generate buoyant forces should be less important than ram pressure in the bending of radio sources.

We generate models of the two extremes of the probability density function (PDF) of orientations that could arise solely due to buoyancy, which are plotted in Figure 15 alongside the unfolded distribution. Both distributions are monotonic and peak at orientations of 0° and are constructed so as to recover the observed folded distribution in the bottom plot of Figure 14. Model 1 is a quadratic PDF that peaks for sources opening away from the cluster and decreases to 0 for sources opening toward the cluster; it represents the physical situation where buoyancy strongly dominates. Model 2 is a piecewise linear PDF that also peaks for sources opening away from the cluster but plateaus for sources that point inward of tangential; it represents the physical situation where buoyancy contributes the minimum necessary to reproduce the folded distribution. We compare the observed unfolded distribution to both models using a one-sample AD test and rule out it having been drawn from Model 1 at the 4.8σ level and from Model 2 at the 4.1σ level. As these models cover the range of buoyancy-driven PDFs, we rule out buoyancy as driving the preference for radial orientation.

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Although proximity to a cluster strongly affects the bending angle of a radio source, this is only a statistical excess. Other properties of the source also affect how the jets are bent; for example, the Mach number of the jets, which is not directly measurable, should negatively correlate with the amount of bending (Jones et al. 2017). In order to assess how much of the bending can be explained using proximity alone, we show in Figure 16 the fraction of sources in our sample above a given excess bending angle that are within 1.5 r₅₀₀ of the nearest cluster. Here we also include in the total the sources not matched to a WH15 cluster at all. Although the WH15 catalog goes out to z = 0.8, the completeness of SDSS (and thus our unmatched sample) begins to fall off beyond z = 0.6 (Beck et al. 2016); to avoid biasing our results due to this, we restrict the analysis in this section to sources with z < 0.6.

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As shown in Figure 16, (27 ± 1)% of all sources are within 1.5 r₅₀₀. If proximity to a cluster center were the dominant source of bending, we might expect almost all highly bent sources to be found within 1.5 r₅₀₀. However, for sources with Δθ > 21°, we find only (40 ± 5)% within this range. Even at the largest bending angles, most sources are beyond 1.5 r₅₀₀: the distribution peaks at Δθ > 50°, for which still only (50 ± 10)% of sources are within 1.5 r₅₀₀. These values are similar to those of Blanton et al. (2001). Simply due to the large number of radio sources in our sample that are not in the interior of clusters, any particular bent source is as likely as not to be far from a cluster. Although a highly bent radio galaxy is suggestive of a nearby cluster, its membership in a cluster requires follow-up observations of the environment.

Some mechanism must still be bending sources, even away from clusters. To take an initial look at this issue, we examined the optical environments of two subsets of radio galaxies at a median separation of $9\,{r}_{500}$ from the nearest cluster. The first sample consists of 150 highly bent sources (median Δθ ∼ 46°); the second consists of 150 less bent sources (median Δθ ∼ −7°). For each of these radio galaxies, we search SDSS for nearby galaxies within a redshift range of ±0.04(1 + z) and within a projected radius of 033, and stack the resulting counts as a function of projected physical separation.

Since we are no longer near the center of a cluster, we measure the separation between a radio galaxy and its optical neighbors in Mpc instead of scaling by r₅₀₀. We find that the surface density of galaxies neighboring the radio galaxy plateaus beyond a projected 2 Mpc from the radio galaxy, so we define the background surface density as the median surface density in an annulus between 2 and 3 Mpc. This is 6.7 Mpc⁻² around our sample of highly bent sources and 7.4 Mpc⁻² around our sample of less bent sources. Figure 17 shows the surface density as a function of separation from the radio source, normalized by the background surface density.

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The density excess rises sharply within 350 kpc of the radio sources. Between 50 and 350 kpc, we estimate an enhancement in volume density by a factor of 4.7 ± 0.6 for highly bent sources and 2.9 ± 0.5 for less bent sources compared to the local background density of SDSS galaxies. Assuming an r⁻² scaling of volume density as a function of distance to the nearest cluster center as inferred from Figure 5, these values suggest that the radio galaxies are in local densities of approximately 28ρ_crit and 18ρ_crit for highly bent and less bent sources, respectively. Such concentrations of galaxies are not included in the WH15 catalog because they are not dense enough to exceed their richness threshold of ${R}_{L* ,200}\geqslant 12$.

We look at the effect of cluster environment on the asymmetry of the radio jets. Given a radial density gradient in the ICM, the inward-pointing jet of a radially aligned radio source will be in a higher density environment and should be suppressed compared to the outward-pointing jet. Although a given cluster can have a clumpy ICM, this effect should be detectable statistically.

This analysis complements that of Rodman et al. (2019), who performed a detailed examination of the morphology and galactic environment for each source in their sample, although their sample only contained 23 sources. Here, without examining the sources individually, we measure the length of each tail using the distance between the host galaxy and the most distant point on the lowest radio contour that does not contain the host. (This is the same method we used to calibrate the bending angle uncertainty in Section 4.2.)

We separate our sample into two subpopulations: (1) the 120 inner region sources that are aligned radially with the cluster, defined as those where one tail is within 45° of the direction toward the cluster center, and the other is within 45° of the direction away from the cluster center; and (2) the 125 inner region sources that are aligned tangentially to the cluster, defined as those where both tails are within 45° of the tangential to the direction of the cluster center. We define the asymmetry of the source as the ratio of the length of the inward-pointing tail to the length of the outward-pointing tail (even tangentially aligned sources have one tail that is more inward-pointing than the other).

For radially aligned sources, we observe that the inward-pointing tail is shorter than the outward-pointing tail, and that the amount of asymmetry increases for sources closer to the center of the cluster. We fit a line to the trend and calculate that it has a slope of $0.18\pm 0.09\,{r}_{500}^{-1}$, and a Spearman's ρ test determines that the correlation between separation and asymmetry is significant at the 2.1σ level. By comparison, we do not observe a statistically significant correlation for tangentially aligned sources; Spearman's ρ test results in only a 1.5σ result.

We look at the jet asymmetry using pressure as well. The asymmetry should depend on the pressure gradient across the source, which we approximate as the difference in pressure at the ends of the two tails divided by the radial component of the source size. Using the same sample of radially aligned sources, the correlation between pressure gradient and asymmetry is lower than when looking at separation only: only 1.7σ. Our simple pressure model does not account for the asymmetry as well as just using the position within the cluster.

• 1.  
  Both extended radio galaxies and optically selected galaxies follow approximately isothermal distributions in their separations from cluster centers, but the power-law index for radio galaxies (α = −1.10 ± 0.03) is slightly but significantly steeper than the one for optically selected ones (α = −0.94 ± 0.02).
• 2.  
  BCGs exhibit extended radio emission at least 2.5 ± 0.3 times more often than the general population of cluster galaxies.
• 3.  
  The median excess bending angle (Δθ), defined in Equation (3), increases closer to the center of clusters. Sources are statistically consistent with being straight beyond 1.5 r₅₀₀, but increase to a median excess bending angle of Δθ ∼ 24° at 0.01 r₅₀₀.
• 4.  
  BCGs have median Δθ ∼ 8°. This makes them more bent than non-cluster sources, but less bent than the innermost non-BCG sources.
• 5.  
  For BCGs and sources in the inner region of clusters, Δθ correlates with cluster mass. The median Δθ for BCGs increases by ∼10° between cluster masses of 5 and $20\times {10}^{14}\,{M}_{\odot };$ for inner region sources, it also increases by ∼10° over the same range of masses.
• 6.  
  For non-BCG sources, Δθ correlates with local ICM gas pressure (a proxy for ram pressure) above $\sim 5\,\times {10}^{-4}\,\mathrm{keV}\,{\mathrm{cm}}^{-3}$. Below this pressure, sources are statistically consistent with being straight.
• 7.  
  Highly bent sources (Δθ > 21°) preferentially have opening angles oriented parallel to the radial vector to the cluster center, but there is no particular preference between opening toward or away. Less bent sources (Δθ < 21°) do not exhibit a preferential orientation.
• 8.  
  Although BCGs and sources in the inner region of clusters are more likely to be highly bent than not, for any given bending angle, there are more sources outside than inside of clusters.
• 9.  
  There is an excess of optically selected neighbor galaxies within 350 kpc around radio galaxies far from clusters (median cluster separation ∼9 r₅₀₀), and the excess is larger around highly bent sources: 4.7 ± 0.6 and 2.9 ± 0.5 times the background volume density for highly bent and less bent sources, respectively. This suggests that the radio galaxies are in local densities of approximately 28ρ_crit and 18ρ_crit, respectively.
• 10.  
  Radio tail length asymmetry exhibits a low significance (2.1σ) dependence on separation from the cluster center. Among inner region sources that are aligned with their tails parallel to the radial vector, the inward-pointing tail is shorter than the outward-pointing tail, and the difference increases for sources closer to the center of the cluster. Asymmetry exhibits a similar, but even less significant (1.7σ), dependence on the pressure gradient across the source.

In generating our sample of radio galaxies, we mapped out their projected spatial distribution around clusters and found that they appear to be distributed according to a power law similar to the distribution of optically selected galaxies in clusters. This implies that we are not catching them at special times (e.g., if they were triggered when first encountering the outskirts of clusters). However, the power-law distribution is steeper for radio galaxies (although we do not account for differences caused by mass segregation and defer this until future analysis). Previous work investigating the angular two-point correlation function, ω(θ), of radio sources on the sky and comparing this to the ω(θ) of a matched sample of radio-quiet galaxies has also found that radio sources exhibit greater clustering than radio-quiet galaxies (Wake et al. 2008).

Pimbblet et al. (2013) investigated if they could detect a preferential distribution of active galaxies (AGNs) in clusters. They did not find a difference between AGNs and the general population. This suggests that while AGN activity in general is not preferentially triggered by interactions with other galaxies (at least not on a timescale before the host galaxy has moved away from the interaction site), the AGN is more likely to be radio-loud in the proximity of other galaxies. Pimbblet et al. (2013) cross-referenced FIRST as well and found no relationship between radio luminosity and position within the cluster, but they noted that their sample size of 28 was too small to draw a firm conclusion.

The connection between radio-loud AGNs and cluster environment is also supported by Chiaberge et al. (2015), who investigated the merger history of radio-loud and radio-quiet AGNs. They found that radio-loud AGNs are almost always the product of merging systems, whereas only a fraction of radio-quiet AGNs experienced a recent merger. From a clustering perspective, galaxy interactions will happen more often when the density of galaxies is higher, thus connecting our result that radio galaxies are preferentially found near other galaxies to the result of Chiaberge et al. (2015).

The primary goal of this work was to investigate the influence of the cluster on the bending of radio galaxies. The bending angle of a radio source is not a well-defined physical quantity; the definition used in this paper is a somewhat arbitrary approximation of a source's physical radius of curvature, and even that breaks down for highly disturbed sources. Although it is an insufficient parameter to compare against physical models of individual sources, it is sufficient for determining statistical trends in morphology.

We observe two physical properties of the cluster that correlate statistically with the bending of the source: the position of the galaxy within the cluster and the mass of the cluster. Using the model of ram pressure bending first invoked by Miley et al. (1972) and quantified by Begelman et al. (1979) and Jones & Owen (1979), we interpret these correlations as physically corresponding to the two components of ram pressure: the ICM density and the relative velocity between the galaxy and ICM, respectively.

Within the inner region of the clusters ($0.01\lt r/{r}_{500}\,\lt 1.5$), which is approximately equal to the virial region but excludes the BCG, we find that sources closer to the center have a larger median excess bending angle (Δθ, defined in Equation (3)) than sources farther away. Beyond 1.5 r₅₀₀, the measured bending angle is comparable to the uncertainty in our bending measurement. Close to the center of the cluster, the density of the ICM quickly rises (Pratt & Arnaud 2002; Newman et al. 2013), which leads to the increase in Δθ of sources in the denser surroundings.

Also restricted to the inner region, sources in higher mass clusters are more bent than sources in lower mass clusters. The velocity dispersion of the galaxies in a cluster is proportional to the square root of the cluster mass via the gas temperature (Becker et al. 2007; Mguda et al. 2015), so M₅₀₀ serves as a proxy for the relative velocity between the source and the ICM. We find that ${\mathrm{log}}_{10}\,{M}_{500}\sim 2.9\,{\mathrm{log}}_{10}\,{r}_{500}$ for the clusters in our sample, so we argue that the ICM density is constant to first order as a function of cluster mass and does not play a significant role in our result: among sources that are already in a sufficiently dense medium, the ones with statistically higher velocities are statistically more bent.

Together, these two trends correspond to a statistical increase in ram pressure, causing statistically greater bending. In Section 5.3, we used the local gas pressure as another proxy for this ram pressure. The effect of ram pressure can be quantified by

$$\begin{eqnarray}&&\displaystyle \frac{{\rho }_{j}{v}_{j}^{2}}{R}=\displaystyle \frac{{\rho }_{\mathrm{icm}}{v}_{\mathrm{gal}}^{2}}{h},\end{eqnarray} \tag{ 5 }$$

where ρ_j and ρ_icm are the jet and ICM densities, respectively; v_j is the jet velocity; v_gal is the relative velocity between the host galaxy and the ICM; R is the radius of curvature of the jet axis; and h is the jet radius (Begelman et al. 1979; Jones & Owen 1979).

If the cluster is approximately virialized, then both the gas thermal velocity (v_icm) and the galaxy velocity scale similarly with the ICM temperature, ${T}_{\mathrm{icm}}\propto {v}_{\mathrm{icm}}^{2}\propto {v}_{\mathrm{gal}}^{2}$ (e.g., Lubin & Bahcall 1993), justifying our use of the static pressure ${P}_{\mathrm{icm}}\propto {\rho }_{\mathrm{icm}}{T}_{\mathrm{icm}}$ as a proxy for the ram pressure ${\rho }_{\mathrm{icm}}{v}_{\mathrm{gal}}^{2}$. As long as the jet radii (h) and lengths (ℓ) are statistically independent of the cluster properties, then the bending angle (θ) will scale with P_icm according to

$$\begin{eqnarray}&&\theta \propto \displaystyle \frac{{\ell }}{R}\propto \displaystyle \frac{{\ell }}{h}\displaystyle \frac{{\rho }_{\mathrm{icm}}{T}_{\mathrm{icm}}}{{\rho }_{j}{v}_{j}^{2}}\propto \left(\displaystyle \frac{{\ell }}{h{\rho }_{j}{v}_{j}^{2}}\right){P}_{\mathrm{icm}},\end{eqnarray} \tag{ 6 }$$

as we have found.

According to the simulations of Marshall et al. (2018), AGN activity is triggered when the ram pressure is at least a factor of 2 greater than P_icm. From this we estimate that the minimum P_ram required to induce significant bending is at least on the order of magnitude ${10}^{-3}\,\mathrm{keV}\,{\mathrm{cm}}^{-3}$. We compare this to Mguda et al. (2015), who used characteristic jet values from Freeland & Wilcots (2011) to estimate that ram pressures above $6\times {10}^{-3}\,\mathrm{keV}\,{\mathrm{cm}}^{-3}$ (in our units) were required to induce bending, which is consistent with our result. Since we can statistically detect even small amounts of bending, we expect our threshold to be lower.

From simulations, bending can also be parameterized in terms of the ratio of Mach numbers of the jet and the ICM shock (Jones et al. 2017). For our purposes, the Mach numbers of ICM flows and galaxy motions are statistically independent of environment, again assuming that clusters are virialized to first order. In addition, there may be non-environmental properties that give rise to bent radio jets, such as precession of the central black hole (Falceta-Gonçalves et al. 2010).

The BCGs in our sample are only moderately bent. Although WH15 define the center of the cluster to be the location of the BCG, which would suggest that they are stationary, in general, this is not physically accurate. Rather, a large fraction of clusters contain BCGs that are offset from the center of mass of the cluster and have a small peculiar velocity with respect to the ICM (Coziol et al. 2009; Lopes et al. 2018). Small velocities and high densities are consistent with our observation of moderate bending.

The excess bending of BCGs is approximately the same as for sources between 0.6 r₅₀₀ and 1.1 r₅₀₀, but the central density is much higher. We estimate the density ratio assuming the same generalized NFW profile as in Section 5.3. This profile diverges at r = 0, so we instead calculate the ICM density around the BCGs at r = 0.01 r₅₀₀ (as in Andrade-Santos et al. 2017). By this estimation, the central density is between 45 and 250 times the density at 0.6 r₅₀₀ and 1.1 r₅₀₀, respectively. If all galaxies in the cluster had motion with the same velocity, then BCGs should be bent the most by ram pressure. Because they are as bent as galaxies where the pressure is 45 to 250 times lower, the BCG velocity offset must be $\sqrt{\rho }=7$ to 16 times lower than the velocity for those galaxies near r₅₀₀. This leads to a typical BCG velocity between 140 and $340\,\mathrm{km}\,{{\rm{s}}}^{-1};$ this is consistent with Lopes et al. (2018), who report that 42% of disturbed clusters in their sample had a BCG with a velocity offset of at least $200\,\mathrm{km}\,{{\rm{s}}}^{-1}$.

We also find that BCGs are more likely to be extended radio galaxies than the average non-BCG by at least a factor of 2.5. For our sample, the radio luminosity distribution of both BCGs and non-BCGs has a minimum of approximately $2\,\times {10}^{23}\,{\rm{W}}\,{\mathrm{Hz}}^{-1}$. This luminosity is near the upper limit of the radio luminosity function for spiral galaxies (Mauch & Sadler 2007), which is evidence that both sets of galaxies are drawn from a comparable population of ellipticals. This result is consistent with Best et al. (2007) and Croft et al. (2007), who each performed a more detailed analysis of the connection between radio-loud AGNs and BCGs, and found that the excess of radio-loud BCGs was dependent on the stellar mass of the galaxy. Best et al. (2007) reported that the excess ranges between 2 and 10 times, depending on the mass bin.

Because most wide-angle tail (WAT) radio sources are associated with BCGs (O'Donoghue et al. 1993), they can be used to detect clusters (e.g., Blanton et al. 2000). Various surveys have attempted to use more general populations of highly bent radio galaxies to detect clusters as well (e.g., O'Brien et al. 2016; Paterno-Mahler et al. 2017). However, we find that highly bent radio sources are not necessarily associated with the virial or inner region of clusters. At every bending angle, there are more sources beyond 1.5 r₅₀₀ than near the cluster center. Observing a highly bent source is more likely to indicate that it is on the long tail of the distribution of non-cluster galaxy bending angles, as opposed to being due to cluster proximity. Of the 660 highly bent sources (Δθ > 21°) in our sample, 223 (34%) of them are within the approximately virial region (1.5 r₅₀₀), 455 (69%) are found out to 10 r₅₀₀, and 67 (10%) are not matched to a cluster within 15 Mpc at all.

We do find that radio sources in general are statistically associated with local overdensities. Within 350 kpc of the radio source, highly bent sources on the outskirts of clusters are located within statistically overdense regions of 4.7 times the background volume density of galaxies; less bent sources are in statistically overdense regions of 2.9 times the background. Other studies of the local companion density around radio-loud galaxies have also detected this excess; for example, Worpel et al. (2013) found that luminous radio galaxies have almost twice as many optical companions within 160 kpc as did galaxies in their control sample. Now we find that highly bent radio sources in particular have an excess number of companions compared to less bent radio sources, which suggests that even these small overdensities (compared to the densities near the center of clusters) in the local environment are sufficient to induce noticeable bending in the radio jets, as in the model of Venkatesan et al. (1994).

In Blanton et al. (2001), of 40 visually identified bent double radio galaxies, 54% were associated with galaxy clusters and 46% were associated with galaxy groups. Their sample was more robust than the sample in this work because all the sources had visual confirmation of bent morphology, but their general conclusion is the same as ours: radio galaxies tend to be found in overdense regions, but not necessarily in clusters.

We have used the largest sample of bent radio galaxies compiled so far to probe the orbital distribution of galaxies in the cluster, by measuring the orientation of the radio galaxy tails relative to the cluster. Under the model in which the jets are bent by ram pressure due to relative motion with respect to the ICM, the angle bisector of the opening angle would point away from the direction of motion. O'Dea & Owen (1985) have previously attempted this same analysis with a study of 70 narrow-angle tail radio sources and observed a radial preference within a projected 0.5 Mpc of the cluster center but isotropic orbits beyond that.

The distribution of orbital directions in a cluster evolves over time. When a cluster forms via mergers, galaxies and other material tend to fall in with radial orbits; over time, the cluster virializes and the orbits become isotropically distributed. Measuring the degree of isotropy reveals the evolutionary status of the cluster (Iannuzzi & Dolag 2012).

The velocity anisotropy parameter β is used to parameterize the degree of isotropy by comparing the velocity dispersions in the radial and tangential directions. We use the definition given in Binney & Tremaine (1987),

$$\begin{eqnarray}&&\beta (r)=1-\displaystyle \frac{{\sigma }_{t}^{2}(r)}{{\sigma }_{r}^{2}(r)}.\end{eqnarray} \tag{ 7 }$$

The β parameter is 0 for isotropically distributed orbits, positive for preferentially radial orbits, and negative for preferentially tangential orbits. We calculated β by decomposing the direction of motion of our sample of highly bent sources between 0.01 r₅₀₀ and 10 r₅₀₀ into radial and tangential components, disregarding the position dependence in Equation (7). We calculate β = 0.20 ± 0.02, consistent with our qualitative assessment of a radial preference.

In our analysis, we stacked our sources by folding the orientation angles around 90°. By doing so, we were able to observe the radial preference, but removed any information on whether the sources were pointing toward or away from the cluster center. If a pressure gradient (e.g., buoyancy) was the primary driver for bending rather than ram pressure, sources would preferentially open away from the center of the cluster. By unfolding the orientation angles, we observe that sources are preferentially oriented non-tangentially, but do not prefer pointing out rather than in. Therefore, pressure gradients are not the primary driver of bending in our sample.

This is also expected from theoretical considerations. Even in the absence of relative motion, a pressure gradient across the jet will cause it to bend. To first order, the transverse ram pressure force per unit volume in the jet goes as

$$\begin{eqnarray}&&{F}_{r}\sim \displaystyle \frac{{\rho }_{\mathrm{icm}}{v}_{\mathrm{gal}}^{2}}{h},\end{eqnarray} \tag{ 8 }$$

as in Equation (5). Assuming for simplicity that the ambient pressure gradient is orthogonal to the jet, the force due to it goes as

$$\begin{eqnarray}&&{F}_{p}\sim \displaystyle \frac{{P}_{\mathrm{icm}}}{z},\end{eqnarray} \tag{ 9 }$$

where z is the pressure scale height. The ratio of the two is simply then

$$\begin{eqnarray}&&\displaystyle \frac{{F}_{r}}{{F}_{p}}\sim \displaystyle \frac{{\rho }_{\mathrm{icm}}{v}_{\mathrm{gal}}^{2}}{{P}_{\mathrm{icm}}}\displaystyle \frac{z}{h}\sim {M}_{A}^{2}\displaystyle \frac{z}{h},\end{eqnarray} \tag{ 10 }$$

where M_A is the Mach number of the galaxy's motion relative to the ICM. Statistically, M_A is of order unity, so the main determining factor is the scale height of the pressure gradient. In a relaxed ICM, this will be large, but in a very dynamic ICM, local pressure gradients can be fairly steep (z of order 10 kpc). Therefore, pressure gradients are generally not expected to be the dominant cause of bending, although situations can exist where they are important.

Whereas Rodman et al. (2019) used RGZ sources to investigate the role of environment on the length of radio lobes by counting the number of nearby galaxies on both sides of the radio source, we compared the positions of each end of the source (tail or lobe) and the ICM gas pressure at each end using the analytical model of Arnaud et al. (2010). We come to the same conclusion as Rodman et al. (2019) and Malarecki et al. (2015): jet expansion into denser environments is suppressed. However, our results are more significant when asymmetry is compared to position in the cluster rather than to the pressure gradient across the source, suggesting that either the density gradient is the more important contribution or that an important piece is being left out of our simplified pressure model. In addition, Rodman et al. (2019) specifically looked at FR-II sources, whereas our population contains both FR-I and FR-II sources. Tail length is not well defined for FR-I sources, and there may be physical differences between the propagation into the ambient medium of the diffuse lobes of FR-I sources compared to the more collimated tails of FR-II sources, so the inclusion of FR-I sources in our sample may have reduced the signal.