\label{sec:data}
We draw our obseravational data from Data Release 7 (DR7) \cite{SDSS} of the Sloan Digital Sky Survay (SDSS), making use of derived data products from the NYU Value-Added Galactic Catalog (VAGC) \cite{Blanton05}. The \(r\) band magnitudes from the NYU VAGC that we use have been \(K\)-corrected to \(z = 0.1\) using the code Kcorrect \cite{kcorr}, and we restrict ourselves to regions of SDSS where the spectroscopic completeness parameter FGOTMAIN exceeds 0.7. We also reject all galaxies with velocity errors greater than 25 km/s.
In selecting our sample, we adhere closely to the procudure of I14. We select a sample of hosts in the magnitude range \(-23 {<}M_{r} {<}-20\) and within a redshift range of \(0.002 {<}z {<}0.05\). A host is considered isolated if there are no brighter objects within 500 kpc on the sky in projection and within 1500 km/s in velocity space. Only isolated hosts are retained, reducing the number of sources to 22780 isolated hosts. From there we identified satellites around each host. Galaxies meet the criteria for being counted as satellites if their magnitudes fall in the range \(M_{r,host} + 1 < M_{r,sat} < -16\), they are located between 20 kpc and 150 kpc from the host in projected distance, and their velocity offset from the host lies in the range \[\rm 25 \, km/s < |v_{sat} - v_{host}| < 300 \,km/s \times exp(-(d_{proj}/300 \, kpc)^{0.8})\] This velocity bound is taken from I14, and is designed to reduce the contamination from interlopers in the satellite sample. Since our interest is in pairs of satellites, we retain only hosts with two or more satellites. Our sample contains 427 such hosts, with 965 associated satellites.
In this subsection, we investigate pairs of satellite pairs located diametrically opposite their host for signals of co-rotation. To facilitate this, we will introduce the parameter \(\alpha\), defined as the angle between the line extending from one satellite through the host and the position vector of the second satellite. This definition is sketched in Figure \ref{fig:alpha}.
We first focus on the co-rotation signal at small \(\alpha\), where for the duration of this paper we will refer to a satellite pair as “co-rotating" if they have opposite-signed velocity offsets relative to the hosts and their associated opening angle (\(\alpha\), see Figure \ref{fig:alpha}) is greater than \(90^{\circ}\), or they have same-signed velocity offsets relative to the hosts and their associated \(\alpha\) is less than \(90^{\circ}\) ; otherwise, they are deemed counter-rotating. Figure \ref{fig:zoom} shows the fraction of satellite pairs that are co-rotating as a function of \(\alpha\) for \(\alpha {<}45^{\circ}\). At \(\alpha {>}45^{\circ}\), the signal is consistent with the sample being divided equally between co-rotating and counter-rotating, i.e. a co-rotation fraction of 0.5, as is what would be expected in the absense of any co-rotating structure. At \(\alpha {<}5^{\circ}\) , the co-rotation signal increases to \(\sim 2.5 \sigma\) above 0.5, potentially indicating a relative overabundance of co-rotating pairs versus counter-rotating ones. Viewed in this way, the data would seem to indicate the presense of coherently rotating scrutures that can only be detected at small \(\alpha\).
However, viewed a different way the picture becomes less clear. Figure \ref{fig:full} shows the co-rotating fraction of satellite pairs over the full domain of \(\alpha\). It no longer seems to be the case that the data is described as being consistent with a flat line at 0.5 over most of the domain and an increase at small alpha. Rather, the spike at \(\alpha \sim 10^{\circ}\) is consistent with random noise present in the data, and that the data is consistent with a flat line. The remainder of this paper will examine the argument that the spike at small alpha is significant, and indicative of ubiquitous coherent co-rotation in satellite populations (of the kind seen in M31) by comparing the data to statistical models of satellite kinematics.