# Case A or Case B? The effective recombination coefficient in
gas clouds of arbitrary optical thickness

Olof Nebrin  Department of Astronomy & Oskar Klein Centre,
AlbaNova, Stockholm University,
SE-106 91 Stockholm, Sweden

###### Abstract

In calculations of the ionization state, one is often forced to choose between
the Case A recombination coefficient $\alpha_{\rm A}$ (sum over recombinations
to all hydrogen states) or the Case B recombination coefficient $\alpha_{\rm
B}$ (sum over all hydrogen states except the ground state). If the cloud is
optically thick to ionizing photons, $\alpha_{\rm B}$ is usually adopted on
the basis of the ”on-the-spot” approximation, wherein recombinations to the
ground state are ignored because they produce ionizing photons absorbed
nearby. In the opposite case of an optically thin cloud, one would expect the
Case A recombination coefficient to better describe the effective
recombination rate in the cloud. In this paper, I derive an analytical
expression for the effective recombination coefficient in a gas cloud of
arbitrary optical thickness which transitions from $\alpha_{\rm A}$ to
$\alpha_{\rm B}$ as the optical thickness increases. The results can be
readily implemented in numerical simulations and semi-analytical calculations.

##  1 Introduction

If the diffuse ionizing radiation is not solved for in detail, one is forced
to choose between the Case A and Case B recombination coefficients to model
the recombination rate of ionized gas. The wrong choice can lead to
significant errors. Raičević et al. (2014) have shown that adopting either the
Case A or Case B recombination coefficient, without modelling the diffuse
radiation, can lead to significant inaccuracies in the modelling of H II
regions around stars, as well as the ionization state of the ’shadows’ behind
illuminated optically thick absorbers. Significant errors can also appear in
cosmological settings. For example, wrongly adopting the Case A instead of the
Case B recombination coefficient – as done by some authors – in the study of
chemistry in pristine atomic-cooling halos at Cosmic Dawn can lead to an error
of $80-90\%$ in the critical Lyman-Werner intensity needed to form direct-
collapse black holes (Glover, 2015), which in turn could lead to large errors
in their predicted abundance. Similarly, choosing either $\alpha_{\rm A}$ or
$\alpha_{\rm B}$ can lead to shifts of $\Delta z\sim 0.5$ in the predicted end
of reionization in models, all else being equal (Kaurov & Gnedin, 2014). It
would be useful to have a general expression for the effective recombination
coefficient $\alpha_{\rm eff}$ that interpolates between Case A and Case B as
the cloud optical thickness increase. Davidson (1977) proposed a crude
”modified on-the-spot” method for $\alpha_{\rm eff}$ which does this (see
also, e.g., Netzer & Ferland, 1984; Netzer, 1990), but it was for slab
geometry and no derivation was given for the result. Below I derive the
effective recombination coefficient for a spherical gas cloud of uniform
density and ionization state and arbitrary optical thickness. This is done by
solving for the diffuse ionizing radiation using the Eddington approximation.
The final result (Eqs. 11 and 13) can be applied in e.g. semi-analytical
models of gas chemistry in collapsing regions, or on a cell-to-cell basis in
numerical simulations of reionization.

##  2 Solving the radiative transfer equation

Consider a stationary spherical gas cloud of radius $R$, uniform hydrogen
density $n_{\rm H}=n_{\rm H^{0}}+n_{\rm H^{+}}$ and ionization fraction
$x\equiv n_{\rm H^{+}}/n_{\rm H}$. To get the effective recombination
coefficient we need to know the ionization rate as a result of recombination
coefficients to the ground state. This in turn requires us to solve for the
diffuse ionizing radiation within the cloud. Thus, we start with the radiative
transfer equation in spherical coordinates (e.g., Chandrasekhar, 1960):

$\mu\frac{\partial I_{\nu}}{\partial r}+\frac{1-\mu^{2}}{r}\frac{\partial
I_{\nu}}{\partial\mu}=j_{\rm fb,\nu}-n_{\rm H^{0}}\sigma_{\nu}I_{\nu}\hskip
1.0pt.$ (1)

Here $\sigma_{\nu}$ is the hydrogen photoionization cross-section, and $j_{\rm
fb,\nu}$ the emissivity due to recombinations to the ground state: 111The
ground state recombination line has a spectrum $\propto e^{(E_{\rm
LyC}-h\nu)/kT}$, so approximating the recombination line as a Dirac delta
function is a good approximation for temperatures $T\lesssim 0.1E_{\rm
LyC}/k\simeq 1.6\times 10^{4}\hskip 1.0pt\rm K$.

$j_{\rm fb,\nu}\simeq\frac{\alpha_{1}n_{\rm H^{+}}n_{\rm e}}{4\pi}\hskip
1.0ptE_{\rm LyC}\hskip 1.0pt\delta_{\rm D}(\nu_{\rm LyC})\hskip 1.0pt,$ (2)

where $E_{\rm LyC}=h\nu_{\rm LyC}=13.6~{}\rm eV$ is the ionization threshold,
and $\alpha_{1}=\alpha_{\rm A}-\alpha_{\rm B}$ is the recombination rate to
the ground state. We can solve Eq. (1) using the Eddington approximation,
wherein the intensity is assumed to be nearly isotropic. This approximation is
expected to be good in this case because the emissivity in Eq. (2) is
isotropic. The zeroth, first, and second moments of the intensity are:

$J_{\nu}=\frac{1}{2}\int_{-1}^{1}\textrm{d}\mu\hskip 1.0ptI_{\nu}\hskip
1.0pt,~{}~{}H_{\nu}=\frac{1}{2}\int_{-1}^{1}\textrm{d}\mu\hskip 1.0pt\mu
I_{\nu}\hskip 1.0pt,~{}~{}K_{\nu}=\frac{1}{2}\int_{-1}^{1}\textrm{d}\mu\hskip
1.0pt\mu^{2}I_{\nu}\hskip 1.0pt.$ (3)

In the Eddington approximation one make the linear approximation
$I_{\nu}(r,\mu)\simeq a_{\nu}(r)+b_{\nu}(r)\mu$, which leads to the closure
relation $K_{\nu}=J_{\nu}/3$ (e.g., Rybicki & Lightman, 1986). Taking the
zeroth and first moments of Eq. (1) and employing the closure relation yields:

$\frac{\textrm{d}H_{\nu}}{\textrm{d}r}+\frac{2H_{\nu}}{r}=j_{\rm
fb,\nu}-n_{\rm H^{0}}\sigma_{\nu}J_{\nu}\hskip
1.0pt,~{}~{}\frac{1}{3}\frac{\textrm{d}J_{\nu}}{\textrm{d}r}=-n_{\rm
H^{0}}\sigma_{\nu}H_{\nu}\hskip 1.0pt.$ (4)

These equations can be combined into a single one for $J_{\nu}$:

$\frac{\textrm{d}^{2}J_{\nu}}{\textrm{d}\tau_{\nu}^{2}}+\frac{2}{\tau_{\nu}}\frac{\textrm{d}J_{\nu}}{\textrm{d}\tau_{\nu}}-3J_{\nu}=-\frac{3j_{\rm
fb,\nu}}{n_{\rm H^{0}}\sigma_{\nu}}\hskip 1.0pt,$ (5)

where I have introduced the optical depth from the center,
$\tau_{\nu}(r)=\int_{0}^{r}\textrm{d}r^{\prime}\hskip 1.0ptn_{\rm
H^{0}}\sigma_{\nu}$ (so that it is zero at the center). The above differential
equation can be solved using the simplifying assumptions of uniform density
and ionization fraction. The general solution is:

$J_{\nu}(\tau_{\nu})=\frac{j_{\rm fb,\nu}}{n_{\rm
H^{0}}\sigma_{\nu}}+\frac{1}{\tau_{\nu}}\left(Ae^{-\sqrt{3}\tau_{\nu}}+Be^{\sqrt{3}\tau_{\nu}}\right)\hskip
1.0pt.$ (6)

The constants $A$ and $B$ can be determined from the boundary conditions:

  1. 1.

Zero net flux $F_{\nu}=4\pi H_{\nu}$ at the center ($\tau_{\nu}=0$).

  2. 2.

No incoming diffuse intensity $I_{\nu}^{-}$ at the cloud edge (at optical
depth $\tau_{\rm cl,\nu}\equiv n_{\rm H^{0}}\sigma_{\nu}R$).

The first boundary condition implies that
$\textrm{d}J_{\nu}/\textrm{d}\tau_{\nu}=0$ at $\tau_{\nu}=0$ (see Eq. 4). The
second boundary condition can be treated with the two-stream approximation
(e.g., Rybicki & Lightman, 1986), where the intensity is assumed to travel at
two angles $\mu=+1/\sqrt{3}$ (outgoing) and $\mu=-1/\sqrt{3}$ (incoming). With
our convention for the optical depth increasing from the center, this yields
$I_{\nu}^{-}=J_{\nu}+(1/\sqrt{3})(\textrm{d}J_{\nu}/\textrm{d}\tau_{\nu})$.
This should be zero at $\tau_{\rm cl,\nu}$ to satisfy the second boundary
condition. After some algebra and using Eq. (2) one ends up with the solution:

$J_{\nu}(\tau_{\nu})=\frac{\alpha_{1}n_{\rm H^{+}}n_{\rm e}E_{\rm LyC}}{4\pi
n_{\rm H^{0}}\sigma_{\nu}}\delta_{\rm D}(\nu_{\rm
LyC})\Biggl{\\{}1+\frac{1}{\tau_{\nu}}\left(e^{-\sqrt{3}\tau_{\nu}}-e^{\sqrt{3}\tau_{\nu}}\right)\left[\frac{2}{\tau_{\rm
cl,\nu}}e^{\sqrt{3}\tau_{\rm cl,\nu}}+\frac{1}{\sqrt{3}\tau_{\rm
cl,\nu}^{2}}\left(e^{-\sqrt{3}\tau_{\rm cl,\nu}}-e^{\sqrt{3}\tau_{\rm
cl,\nu}}\right)\right]^{-1}\Biggl{\\}}\hskip 1.0pt.$ (7)

This solution will determine the photoionization rate from the diffuse
ionizing radiation in the cloud.

##  3 The effective recombination coefficient

With the solution for the diffuse mean ionizing intensity $J_{\nu}$ we can now
determine the effective recombination coefficient. We can define it from the
time evolution of the ionized hydrogen number density:

$\frac{\partial n_{\rm H^{+}}}{\partial t}=\Gamma_{\star}n_{\rm
H^{0}}+\Gamma_{\rm diff}n_{\rm H^{0}}-\alpha_{\rm A}n_{\rm H^{+}}n_{\rm
e}\equiv\Gamma_{\star}n_{\rm H^{0}}-\alpha_{\rm eff}n_{\rm H^{+}}n_{\rm
e}\hskip 1.0pt,$ (8)

where $\Gamma_{\rm diff}=\int_{\nu_{\rm LyC}}^{\infty}\textrm{d}\nu\hskip
1.0pt4\pi J_{\nu}\sigma_{\nu}/h\nu$ is the photoionization rate due to the
diffuse ionizing radiation, and $\Gamma_{\star}$ the photoionization rate from
any luminous sources (e.g. nearby stars). The above definition yields an
effective recombination coefficient of:

$\alpha_{\rm eff}=\alpha_{\rm A}-\frac{\Gamma_{\rm diff}n_{\rm H^{0}}}{n_{\rm
H^{+}}n_{\rm e}}\hskip 1.0pt.$ (9)

Using Eq. (7) we find a diffuse photoionization rate of

$\Gamma_{\rm diff}(\tau)=\frac{\alpha_{1}n_{\rm H^{+}}n_{\rm e}}{n_{\rm
H^{0}}}\Biggl{\\{}1+\frac{1}{\tau}\left(e^{-\sqrt{3}\tau}-e^{\sqrt{3}\tau}\right)\left[\frac{2}{\tau_{\rm
cl}}e^{\sqrt{3}\tau_{\rm cl}}+\frac{1}{\sqrt{3}\tau_{\rm
cl}^{2}}\left(e^{-\sqrt{3}\tau_{\rm cl}}-e^{\sqrt{3}\tau_{\rm
cl}}\right)\right]^{-1}\Biggl{\\}}\hskip 1.0pt,$ (10)

where $\tau$ and $\tau_{\rm cl}$ are $\tau_{\nu}$ and $\tau_{\rm cl,\nu}$
evaluated at the Lyman limit ($\nu_{\rm LyC}$), respectively. Thus, the
effective recombination coefficient at position $\tau$ becomes (using
$\alpha_{1}=\alpha_{\rm A}-\alpha_{\rm B}$):

$\alpha_{\rm eff}(\tau)=\alpha_{\rm A}-(\alpha_{\rm A}-\alpha_{\rm
B})\Biggl{\\{}1+\frac{1}{\tau}\left(e^{-\sqrt{3}\tau}-e^{\sqrt{3}\tau}\right)\left[\frac{2}{\tau_{\rm
cl}}e^{\sqrt{3}\tau_{\rm cl}}+\frac{1}{\sqrt{3}\tau_{\rm
cl}^{2}}\left(e^{-\sqrt{3}\tau_{\rm cl}}-e^{\sqrt{3}\tau_{\rm
cl}}\right)\right]^{-1}\Biggl{\\}}\hskip 1.0pt.$ (11)

This is dependent on the position within the cloud. The cloud-averaged
recombination coefficient that determines the recombination rate of the cloud
as a whole is given by:

$\langle\alpha_{\rm eff}\rangle=\frac{\int_{0}^{R}\textrm{d}r\hskip
1.0pt\alpha_{\rm eff}(r)n_{\rm H^{+}}n_{\rm
e}r^{2}}{\int_{0}^{R}\textrm{d}r\hskip 1.0ptn_{\rm H^{+}}n_{\rm
e}r^{2}}=\frac{3}{\tau_{\rm cl}}\int_{0}^{\tau_{\rm cl}}\textrm{d}\tau\hskip
1.0pt\alpha_{\rm eff}(\tau)\tau^{2}\hskip 1.0pt,$ (12)

where I have used the assumption of homogeneity to get the final integral over
optical depth. Using Eq. (11) the average effective recombination coefficient
becomes:

$\langle\alpha_{\rm eff}\rangle=\alpha_{\rm A}-(\alpha_{\rm A}-\alpha_{\rm
B})f(\tau_{\rm cl})\hskip 1.0pt,$ (13)

where the function $f(\tau_{\rm cl})$ is given by

$f(\tau_{\rm cl})=1-\frac{1+\sqrt{3}\tau_{\rm cl}-e^{2\sqrt{3}\tau_{\rm
cl}}\left(1-\sqrt{3}\tau_{\rm cl}\right)}{2\tau_{\rm
cl}^{2}e^{2\sqrt{3}\tau_{\rm cl}}+\frac{\tau_{\rm
cl}}{\sqrt{3}}\left(1-e^{2\sqrt{3}\tau_{\rm cl}}\right)}\hskip 1.0pt.$ (14)

The effective recombination rates $\langle\alpha_{\rm eff}\rangle$,
$\alpha_{\rm eff}(\tau=0)$, and $\alpha_{\rm eff}(\tau=\tau_{\rm cl})$ are
plotted in Fig. 1. For gas clouds that are optically thin to ionizing photons
($\tau_{\rm cl}\ll 1$) we have $f(\tau_{\rm cl})\ll 1$ and hence
$\langle\alpha_{\rm eff}\rangle\simeq\alpha_{\rm A}$. For optically thick
clouds ($\tau_{\rm cl}\gg 1$) we instead have $f(\tau_{\rm cl})\simeq 1$ and
so $\langle\alpha_{\rm eff}\rangle\simeq\alpha_{\rm B}$. The cloud-averaged
effective recombination coefficient, therefore, interpolate between Case A and
Case B as expected. The same is true for the effective recombination
coefficient at the center of the cloud, $\alpha_{\rm eff}(\tau=0)$. The
optically thick limit for $\alpha_{\rm eff}(\tau=\tau_{\rm cl})$ falls right
in the middle between Case A and Case B because half of the diffuse ionizing
photons produced at the cloud edge can freely escape the cloud (as noted by
Davidson, 1977; Netzer & Ferland, 1984).

Figure 1: The predicted effective recombination coefficient at the cloud
center ($\alpha_{\rm eff}(\tau=0)$), the cloud edge ($\alpha_{\rm
eff}(\tau=\tau_{\rm cl})$), and the cloud-averaged value ($\langle\alpha_{\rm
eff}\rangle$). The Case A and Case B recombination coefficients were computed
for $T=10^{4}\hskip 1.0pt\rm K$ (Draine, 2011).

I want to thank Garrelt Mellema, Sambit K. Giri, and the Stockholm
Reionization Group at large for interesting and helpful discussions.

## References

  * Chandrasekhar (1960) Chandrasekhar, S. 1960, Radiative transfer
  * Davidson (1977) Davidson, K. 1977, ApJ, 218, 20, doi: 10.1086/155653
  * Draine (2011) Draine, B. T. 2011, Physics of the Interstellar and Intergalactic Medium
  * Glover (2015) Glover, S. C. O. 2015, MNRAS, 453, 2901, doi: 10.1093/mnras/stv1781
  * Kaurov & Gnedin (2014) Kaurov, A. A., & Gnedin, N. Y. 2014, ApJ, 787, 146, doi: 10.1088/0004-637X/787/2/146
  * Netzer (1990) Netzer, H. 1990, in Active Galactic Nuclei, ed. R. D. Blandford, H. Netzer, L. Woltjer, T. J. L. Courvoisier, & M. Mayor, 57–160
  * Netzer & Ferland (1984) Netzer, H., & Ferland, G. J. 1984, PASP, 96, 593, doi: 10.1086/131387
  * Raičević et al. (2014) Raičević, M., Pawlik, A. H., Schaye, J., & Rahmati, A. 2014, MNRAS, 437, 2816, doi: 10.1093/mnras/stt2099
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2013B-0418 | Rest | 2015A-0608 | Forster | 2017B-0907 | Munoz | 2021A-0117 | Kotulla
2013B-0421 | Rest | 2015A-0609 | Carballo-Bello | 2017B-0951 | Vivas | 2021A-0149 | Zenteno
2013B-0440 | Nidever | 2015A-0616 | Jerjen | 2018A-0215 | Carlin | 2021A-0275 | Rest
2013B-0502 | Dell | 2015A-0617 | Nataf | 2018A-0242 | Bechtol | 2021A-0922 | Nilo-Castellon
2013B-0531 | Rest | 2015A-0618 | Lidman | 2018A-0276 | Dell’Antonio | 2022A-388025 | Palmese & Wang
2013B-0612 | Chaname | 2015A-0619 | Goncalves | 2018A-0369 | Rest | 2022A-777564 | Zhao
2013B-0617 | Mackey | 2015A-0620 | Bonaca | 2018A-0380 | Rest | 2022A-975778 | Kelkar
2013B-0627 | Lima Neto | 2015B-0187 | Berger | 2018A-0386 | Zenteno | 2022B-297190 | Palmese & Wang
2014A-0157 | Favia | 2015B-0191 | Rice | 2018A-0907 | Munoz | – | –
2014A-0256 | Eckert | 2016A-0004 | Bonaca | 2018A-0909 | Puzia | – | –
Table 3: DECam public datasets used in our co-addition (sorted by Proposal
ID).

##  Appendix C Mass Map Example

We present an example of using the aperture mass map to detect the orientation
(and center) of an elliptical NFW halo. We apply the Schirmer aperture to a
mock lensing catalog derived from the study of Fu et al. (2023) (see also
Oguri et al. 2010 and Oguri 2010). The catalog spans 24k$\times$24k DECam
pixels ($6\times 6$ patches). The halo mass, source and lens redshifts, shape
noise, and source density are set to be the median values in our LoVoCCS
observations (see also Section 5.3). In Figure 12, we show the convergence of
the elliptical halo ($\kappa({\theta_{1},\theta_{2}})=\kappa_{\rm
NFW}(\sqrt{q\theta_{1}^{2}+\theta_{2}^{2}/q});~{}q=2/3$) overlaid with the
lensing mass S/N map. The orientations of the halo and the map are generally
consistent.

Figure 12: Convergence of an elliptical halo (centered in the diagram and
elongated horizontally; gray scale in the background) overlaid with the
corresponding aperture mass map (blue contours showing S/N $=2,4,6$). The
orange point and the dotted line give the detected lensing peak and the
derived orientation angle (within 1 Mpc), respectively. We vary the aperture
radius and use the aperture when the peak S/N reaches the maximum (10k pixels
$\sim 10r_{\rm s}$). The diagram spans 16k$\times$16k DECam pixels ($4\times
4$ patches; $0^{\prime\prime}.263$ per pixel), which is consistent with the
figures in Section 4.1.

##  Appendix D Red Sequence Examples

We present the examples of detecting and selecting the red sequence galaxies
in the color–magnitude diagrams in Figure 13 (A3558) and Figure 14 (A3667).
Those two clusters have enough spec-z data to show RS in CMD, and thus we use
the RS linear fit parameters detected from the spec-z selected galaxies
instead of all galaxies (though their fitting results are close). The
algorithm was presented in Section 3.4.

Figure 13: Red sequence examples in A3558 ($z=0.047$). Top left: The gray
points are all member galaxies selected by spec-z. The blue points are the
histogram peaks of those gray points in individual $i$ magnitude bins (Section
3.4), while the green triangles are the median values in those bins
(consistent with blue points generally). The orange dash line is the linear
fit of those blue points. The magenta points are the galaxies selected by the
scatter cut around the fitted line. Bottom left: The gray points are member
galaxies selected by both spec-z and $g-r$ (i.e., the galaxies denoted by
magenta points in the top left diagram) – the RS is more clear. Other markers
have similar meanings to the ones in the top left diagram. Top middle: Now the
gray points are all galaxies in the catalog with valid photometry. Other
markers have similar meanings to the top left diagram. Note the green median
points have large biases at faint magnitudes, but the blue peak points are
consistent with the trend at the bright end. Bottom middle: The gray points
correspond to the magenta ones in the top middle diagram. Other markers have
similar meanings. Again, the RS is more clear than its top counterpart because
of the first selection in $g-r$. Top right: We start from all galaxies. The
red circles show the galaxies selected by the $g-r$ fit (the spectroscopic one
in this case), while the black points are the galaxies selected by the $r-i$
fit. Here an extra $i$-magnitude cut is also included to fix the luminosity
limit. The overlap between red and black points is the RS galaxy sample.
Bottom right: Similar to the top right diagram – the red circles are selected
by the $r-i$ fit, while the black circles are selected by the $g-r$ fit. Those
diagrams indicate that a single-color selection can include
foreground/background galaxies, but a two-color selection can give a much
cleaner sample of RS galaxies.

Figure 14: Counterparts of the red sequence examples in A3667 ($z=0.055$).
They are similar to the ones for A3558 (Figure 13).

##  Appendix E Orientations and Centers

Table 4, 5 show the measured orientation angles and centers of the BCG, the RS
and mass distributions (with different radial cuts) of individual clusters
(and their types/CG counts).

We also examine the distributions of those angles (Figure 15). The
distributions are generally uniform. Large radial cuts may lead to large noise
in the angle measurement, because at large radii the number density of member
galaxies and the mass density are low.

Name | $z$ | $\alpha_{\rm B}$ | $\delta_{\rm B}$ | $\alpha_{\rm M}$ | $\delta_{\rm B}$ | $\alpha_{\rm R}$ | $\delta_{\rm R}$ | $\theta_{\rm B}$ | $\theta_{\rm M5}$ | $\theta_{\rm M10}$ | $\theta_{\rm M20}$ | $\theta_{\rm R5}$ | $\theta_{\rm R10}$ | $\theta_{\rm R20}$ | $n_{\rm CG}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
A2029 | 0.08 | 227.73 | 5.74 | 227.73 | 5.78 | 227.74 | 5.77 | -63 | 85 | 84 | -70 | -74 | -74 | -84 | a
A401 | 0.07 | 44.74 | 13.58 | 44.74 | 13.58 | 44.74 | 13.58 | -63 | -7 | 1 | -25 | -61 | -48 | -58 | a
A85 | 0.06 | 10.46 | -9.30 | 10.45 | -9.33 | 10.41 | -9.31 | 59 | -70 | -84 | -41 | 85 | 72 | 75 | b
A3667 | 0.06 | 303.11 | -56.83 | 303.14 | -56.85 | 303.15 | -56.84 | 44 | -69 | 80 | -84 | 38 | 44 | 48 | a
A3827 | 0.10 | 330.47 | -59.95 | 330.46 | -59.94 | 330.48 | -59.94 | 72 | 78 | 69 | 57 | 73 | 64 | 83 | a
A3266 | 0.06 | 67.81 | -61.45 | 67.80 | -61.43 | 67.82 | -61.45 | -10 | -74 | 80 | 86 | -25 | -24 | -44 | a
A1651 | 0.08 | 194.84 | -4.20 | 194.83 | -4.20 | 194.85 | -4.20 | -13 | 21 | 23 | -66 | -6 | -2 | -10 | a
A754 | 0.05 | 137.13 | -9.63 | 137.14 | -9.65 | 137.12 | -9.63 | 22 | 29 | 35 | 51 | 19 | 21 | 8 | a
A3571 | 0.04 | 206.87 | -32.86 | 206.89 | -32.91 | 206.85 | -32.88 | -85 | 65 | 62 | 18 | 84 | 88 | 86 | a
A3112 | 0.08 | 49.49 | -44.24 | 49.49 | -44.27 | 49.50 | -44.22 | -84 | -75 | -87 | 82 | -53 | -68 | 84 | a
A399 | 0.07 | 44.47 | 13.03 | 44.46 | 13.03 | 44.44 | 13.02 | -43 | -33 | -10 | -10 | -64 | -34 | -40 | a
A2597 | 0.08 | 351.33 | -12.12 | 351.32 | -12.13 | 351.31 | -12.11 | 50 | 68 | -67 | -66 | 52 | 81 | 76 | a
A1650 | 0.08 | 194.67 | -1.76 | 194.67 | -1.78 | 194.68 | -1.77 | 71 | 43 | 54 | 11 | 70 | 65 | 67 | a
A3558 | 0.05 | 201.99 | -31.50 | 201.98 | -31.48 | 201.97 | -31.52 | 74 | 70 | 67 | 54 | 47 | 45 | 11 | a
A3695 | 0.09 | 308.69 | -35.82 | 308.69 | -35.83 | 308.69 | -35.78 | 63 | 42 | 3 | -12 | 54 | 80 | 82 | a
A3921 | 0.09 | 342.49 | -64.43 | 342.48 | -64.35 | 342.44 | -64.43 | -5 | -76 | -39 | -23 | 0 | 5 | -11 | a
A2426 | 0.10 | 333.63 | -10.37 | 333.63 | -10.35 | 333.64 | -10.37 | -5 | 77 | 82 | 68 | -14 | -11 | 76 | a
A3158 | 0.06 | 55.72 | -53.63 | 55.75 | -53.65 | 55.75 | -53.64 | 3 | -77 | -22 | 44 | 13 | 33 | 56 | a
RXCJ1217.6+0339 | 0.08 | 184.42 | 3.66 | 184.40 | 3.64 | 184.43 | 3.67 | -82 | -29 | -32 | 21 | -22 | -45 | -56 | a
A2811 | 0.11 | 10.54 | -28.54 | 10.54 | -28.54 | 10.54 | -28.54 | 41 | -7 | 50 | 53 | 6 | 35 | 31 | a
A780 | 0.05 | 139.52 | -12.10 | 139.52 | -12.12 | 139.53 | -12.12 | 59 | 3 | 33 | 49 | 68 | 68 | 59 | a
A2420 | 0.08 | 332.58 | -12.17 | 332.57 | -12.16 | 332.56 | -12.19 | -42 | 16 | 13 | -63 | -64 | -54 | -44 | a
A1285 | 0.11 | 172.60 | -14.58 | 172.58 | -14.59 | 172.59 | -14.58 | 32 | 47 | 44 | 52 | 31 | 62 | 61 | b
A3911 | 0.10 | 341.56 | -52.72 | 341.54 | -52.72 | 341.52 | -52.69 | 46 | -25 | 3 | -4 | 40 | 55 | 55 | a
A2055 | 0.10 | 229.69 | 6.23 | 229.73 | 6.26 | 229.69 | 6.25 | 65 | 57 | -76 | -62 | 58 | 66 | 68 | c
A1750 | 0.09 | 202.71 | -1.86 | 202.78 | -1.73 | 202.71 | -1.87 | -11 | -58 | -59 | 20 | -46 | -56 | -50 | c
A3822 | 0.08 | 328.52 | -57.87 | 328.55 | -57.87 | 328.55 | -57.85 | -64 | -23 | 63 | 39 | -26 | -46 | -51 | b
A2941 | 0.12 | 26.24 | -53.02 | 26.31 | -53.06 | 26.28 | -53.02 | 3 | 85 | -78 | 33 | 14 | 23 | 30 | b
A2440 | 0.09 | 335.99 | -1.58 | 336.00 | -1.60 | 335.97 | -1.62 | -34 | 3 | 25 | 19 | -44 | -41 | -47 | c
Table 4: Cluster centers and orientation angles (Part I). We present the measurements of the BCG (B), the lensing mass distribution (M), and the RS galaxy distribution (R) of each LoVoCCS cluster studied in this work (sorted by X-ray luminosity/cluster rank). The redshift ($z$) is from SIMBAD. The coordinates ($\alpha/\delta$ for R.A./Decl. in ICRS) and angles ($\theta$) are in degrees. The angles are counterclockwise from the X-axis, with the X-axis pointing to the west and the Y-axis pointing to the north. We present the angles within 0.5 Mpc, 1.0 Mpc, 2.0 Mpc from the central peak with tags 5, 10, 20, respectively. The last column ($n_{\rm CG}$) gives the cluster type based on the number of bright central galaxies; 1, 2, $\geq$3 correspond to type a, b, c (43, 8, 7 clusters, respectively). We skip the noisy mass/RS distribution results of some clusters.  Name | $z$ | $\alpha_{\rm B}$ | $\delta_{\rm B}$ | $\alpha_{\rm M}$ | $\delta_{\rm M}$ | $\alpha_{\rm R}$ | $\delta_{\rm R}$ | $\theta_{\rm B}$ | $\theta_{\rm M5}$ | $\theta_{\rm M10}$ | $\theta_{\rm M20}$ | $\theta_{\rm R5}$ | $\theta_{\rm R10}$ | $\theta_{\rm R20}$ | $n_{\rm CG}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
A1644 | 0.05 | 194.30 | -17.41 | 194.30 | -17.41 | 194.38 | -17.25 | -44 | -80 | 83 | 52 | 75 | 89 | 76 | a
A1348 | 0.12 | 175.35 | -12.28 | 175.35 | -12.27 | 175.38 | -12.26 | -10 | -12 | -20 | 82 | -19 | -14 | -36 | a
RBS1847 | 0.10 | 334.50 | -65.18 | 334.46 | -65.14 | 334.50 | -65.17 | -7 | 62 | -84 | -17 | -46 | -46 | -69 | a
RXCJ1215.4-3900 | 0.12 | 183.86 | -39.03 | 183.87 | -39.01 | 183.83 | -39.05 | -33 | 88 | -70 | -21 | -41 | -50 | -73 | a
A2351 | 0.09 | 323.57 | -13.43 | 323.58 | -13.42 | 323.57 | -13.42 | 62 | -69 | -87 | -82 | 56 | 71 | 74 | a
RXCJ2218.2-0350 | 0.09 | 334.67 | -3.78 | 334.56 | -3.80 | 334.67 | -3.77 | 2 | -9 | -42 | 46 | -51 | -32 | -37 | c
A2443 | 0.11 | 336.53 | 17.36 | 336.54 | 17.36 | 336.52 | 17.39 | -55 | 87 | 85 | 80 | 74 | 63 | 61 | c
A2050 | 0.12 | 229.07 | 0.09 | 229.08 | 0.08 | 229.07 | 0.08 | -49 | -80 | 67 | 48 | -38 | -34 | -45 | a
A1736 | 0.04 | 201.70 | -27.14 | 201.71 | -27.11 | 201.71 | -27.13 | 77 | -59 | -38 | -29 | -77 | 85 | -85 | c
A2384 | 0.09 | 328.09 | -19.55 | 328.09 | -19.56 | 328.08 | -19.55 | -61 | -56 | -56 | -26 | 85 | -78 | -82 | b
A4059 | 0.05 | 359.25 | -34.76 | 359.21 | -34.79 | 359.25 | -34.74 | 62 | 83 | -72 | 25 | 53 | 54 | 74 | a
A3836 | 0.11 | 332.34 | -51.81 | 332.34 | -51.80 | 332.35 | -51.81 | 20 | 29 | 23 | 26 | 81 | 74 | 55 | a
A2533 | 0.11 | 346.81 | -15.22 | 346.86 | -15.23 | 346.81 | -15.21 | -48 | 44 | 52 | 76 | -49 | -67 | -72 | a
A2556 | 0.09 | 348.26 | -21.63 | – | – | 348.26 | -21.65 | 42 | – | – | – | 10 | 8 | 24 | a
A3126 | 0.08 | 52.15 | -55.71 | 52.15 | -55.71 | 52.15 | -55.71 | -66 | 85 | 87 | -75 | 76 | 45 | 76 | a
RXCJ0049.4-2931 | 0.11 | 12.35 | -29.52 | 12.36 | -29.52 | 12.34 | -29.52 | 35 | 29 | 28 | 24 | 75 | -83 | 67 | a
A2554 | 0.11 | 348.08 | -21.50 | 348.08 | -21.49 | 348.07 | -21.49 | -85 | -12 | -5 | 2 | -73 | -72 | 47 | a
A2033 | 0.08 | 227.86 | 6.35 | – | – | – | – | -36 | – | – | – | – | – | – | a
A3395 | 0.05 | 96.90 | -54.45 | 96.67 | -54.59 | 96.85 | -54.45 | 34 | 45 | 82 | -74 | 35 | 28 | 45 | b
A1606 | 0.10 | 191.15 | -11.99 | 191.15 | -12.01 | 191.15 | -12.01 | -80 | 64 | 48 | 75 | -68 | 56 | 26 | a
A2670 | 0.08 | 358.56 | -10.42 | 358.54 | -10.41 | 358.56 | -10.41 | 30 | -62 | 19 | 54 | 85 | -65 | -63 | a
A3532 | 0.06 | 194.34 | -30.36 | 194.33 | -30.36 | 194.32 | -30.37 | -8 | -27 | 3 | 58 | -40 | -26 | -13 | a
RXCJ1139.4-3327 | 0.11 | 174.85 | -33.45 | 174.83 | -33.46 | 174.85 | -33.44 | 77 | -70 | 85 | -38 | -85 | -84 | 78 | a
RXJ0820.9+0751 | 0.11 | 125.26 | 7.86 | 125.26 | 7.87 | 125.27 | 7.86 | 15 | -56 | 87 | -74 | 8 | 45 | -39 | a
A3128 | 0.06 | 52.66 | -52.62 | 52.64 | -52.53 | 52.62 | -52.56 | -81 | -80 | 78 | 50 | -86 | -87 | -64 | c
A1023 | 0.12 | 156.99 | -6.80 | 157.00 | -6.79 | 156.99 | -6.79 | 49 | 27 | -20 | -32 | 44 | 53 | 75 | a
A3528 | 0.06 | 193.59 | -29.01 | 193.61 | -29.00 | 193.59 | -29.00 | 89 | -86 | -76 | -87 | -89 | 77 | 75 | b
A761 | 0.09 | 137.65 | -10.58 | 137.65 | -10.59 | 137.68 | -10.59 | 1 | -56 | 32 | -48 | 35 | 18 | 1 | a
A3825 | 0.07 | 329.61 | -60.39 | – | – | 329.62 | -60.36 | 80 | – | – | – | -78 | -77 | -7 | b
Table 5: Cluster centers and orientation angles (Part II).  Figure 15: Angle
distributions. The labels have the same meanings as the ones in Table 4, 5.
The vertical lines give the respective median values.

##  Appendix F Significance of Orientation Alignment

Consider a set of angles $\\{\theta_{i}\\},~{}i=1,2,...,N$. Each angle
independently obeys a uniform distribution $\theta_{i}\sim U(0,\Phi)$. The
probability of an angle $\theta_{i}$ being smaller than $\phi$ is $\phi/\Phi$.
The probability of $n$ angles being smaller than $\phi$ (and the others being
larger than $\phi$) is ${N\choose n}(\phi/\Phi)^{n}(1-\phi/\Phi)^{N-n}$. Then
the probability that more than $m$ angles have their values smaller than
$\phi$ is $\sum_{i=m+1}^{N}{N\choose
i}(\phi/\Phi)^{i}(1-\phi/\Phi)^{N-i}=1-F(m;N,\phi/\Phi)$, where $F$ is the
cumulative distribution function of a binomial distribution with $m$ times of
successes in $N$ independent tests and success probability $\phi/\Phi$.

This result can be extended to the case where $\theta_{i}$ is between $\alpha$
and $\beta$. If $\alpha<\beta<\Phi$ and $\psi=\beta-\alpha$, the probability
that more than $m$ angles have values between $\alpha$ and $\beta$ is
similarly $1-F(m;N,\psi/\Phi)$.

Let $\theta_{i}$ be the angle between two orientations (e.g., BCG and RS
distribution) and $\Phi$ be the angle upper limit (90 deg). Let $N$ be the
number of clusters (e.g., 57 in our analysis after removing the noisy
cluster). If we assume the angle is randomly distributed between 0 and 90 deg,
then the probability that more than a fraction of $f$ of those clusters have
the angles smaller than $\phi$ is $1-F(57f;57,\phi/90)$. Figure 16 gives this
probability as a function of the fraction $f$ and the angle cut $\phi$. The
probability drops quickly as the angle cut drops or the fraction increases.
For example, the probability that more than 1/3 of the clusters have their
angles smaller than 30 deg is $1-F(57/3;57,30/90)\sim 0.4$, the probability
that more than half of the cluster sample have the angles below 30 deg is
$\sim 5\times 10^{-3}$, and the probability that more than half of the sample
have the angles below 15 deg is $\sim 3\times 10^{-9}$.

Figure 16: Probability of more than a fraction of the clusters have angles
smaller than an angle cut. The red solid line, the cyan dashed line, and the
white dash-dotted line correspond to probabilities of 0.5, 0.1, and 0.01,
respectively. The pink dotted line gives the angle cut divided by 90 deg for
reference.

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