text:
\section{Introduction}
	\label{sec:intro}

	Einstein's theory of General Relativity (GR) is perceived to be the best description of gravity on all scales. GR has been very well tested in the Solar System, but requires exotic dark matter (DM) to explain larger structures such as galaxies, clusters, and the cosmic microwave background (CMB) within the GR-based $\Lambda$CDM cosmological model \citep{Planck_2015}. At present time, there is no decisive direct detection of DM (e.g. \citet{LUX_2016}); though there are many lines of indirect evidence, primarily via flat rotation curves (RCs) of galaxies, gravitational lensing, galaxy clustering and the CMB. Moreover, the $\Lambda$CDM picture leads to several small-scale discrepancies between numerical simulations based on DM and observations on sub-galactic scales \citep{Kroupa_2015, Bullock_2017}, perhaps most seriously with the Local Group satellite planes \citep{Pawlowski_2014}.

	This leaves room for modified gravity (MG) theories such as the notable MOdified Newtonian Dynamics (MOND) \citep{Milgrom_1983} as an alternative to GR.  MOND \citep[reviewed in][]{Famaey_McGaugh_2012} attempts to explain
	the observations usually attributed to dark matter by a suitable modification of standard GR.
	\begin{wscomm}
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	The original MOND formulation was non-relativistic and really a
	fitting function rather than a realistic theory;
	it has later been incorporated into relativistic theories following from
	the well-known Tensor-Vector-Scalar (TeVeS) theory proposed by
	\citet{Bekenstein_2004}. There are various versions of MG, but figure 10 from \citet{Famaey_McGaugh_2012} indicates that the most promising should involve a departure from Newtonian gravity at accelerations below some threshold $a_{_0}$, as occurs in MOND.

	\begin{wscomm}
		Removed chunk of "pro-MOND" above.
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	The current situation of $\Lambda$CDM vs MOND remains contentious, with many successes for $\Lambda$CDM on large scales but some anomalies on galactic and sub-galactic scales; while MOND appears mostly successful  on galactic scales but has
	no clear-cut predictions for cosmology.
	\begin{cpcomm}
		Test comment from CP.
	\end{cpcomm}

	As noted in \citet[][hereafter \citetalias{Pittordis_2019}]{Pittordis_2019}, a clear convincing direct detection of dark matter
	in underground experiments would be the most decisive scenario, but the converse is not
	true: null results from dark matter detectors can never
	rule out the paradigm, because the DM interaction cross-section might simply be too small for any practical experiment (or, the cross-section could be weak-like but the DM particle masses could be very large $\sim 10^9\,$GeV, implying a local number density far below the value for conventional TeV-scale WIMPs). Therefore, in the absence of a DM direct detection,
	new tests which can discriminate between DM and modified gravity from direct tests of gravity at the relevant very low
	accelerations are highly desirable.

	A promising test of MG is via wide binary (WB) stellar systems in the Solar neighbourhood \citep[within $\approx 250$~pc,][]{Hernandez_2012}. WBs are isolated stellar pairs with separations $\ga 7 \kau$ in order to have orbital accelerations $\la a_{_0} = 1.2 \times 10^{-10}$~m/s\textsuperscript{2}.
	The small size of WBs means no significant amount of DM should be distributed within them, so Keplerian dynamics should apply in GR+DM models.
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	In a MOND context, local WBs must feel the Galactic external field which is slightly larger than $a_0$, and the external
	field effect results in
	WB orbital velocities exceeding the Newtonian prediction by $\sim 20\%$ \citep{Banik_2018_Centauri} for an interpolating function consistent with galaxy RCs \citep{McGaugh_Lelli_2016}. This excess $\approx 20\%$ remains approximately constant at larger WB separations, although WB tests are more prone to observational
	caveats at larger separations. Studies of WBs in general have been explored by e.g.
	\citet{Yoo_2003},
	\citet{Lepine_2007},
	\citet{Kouwenhoven_2010},
	\citet{Jiang_2010},
	\citet{Dhital_2013},
	\citet{Coronado_2018}, and others.

	Previous work concerning
	tests of MOND-like gravity has been done by
	\citet{Hernandez_2012},
	\citet{Hernandez_2012_2}
	\citet{Hernandez_2014},
	\citet{Matvienko_2015},
	\citet{Scarpa_2017}
	and \citet{Hernandez_2019}; these typically give hints of deviations in
	the direction expected from MOND-like gravity, though due
	to the limited precision of the pre-GAIA data used, these hints
	were not decisive. If modelled appropriately, WBs can provide a very `clean' model-independent test of gravity in the extremely weak-field regime relevant to galactic outskirts and large scale structure.

	Note that the orbital periods of WBs are $\sim$ Myr, so accelerations are undetectable and full orbit modelling is not possible, thus each single binary is strongly under-constrained; however the statistical distribution of relative velocities is readily predicted from simulations assuming random phases and inclinations, so a large sample of thousands of WBs  can provide a sensitive test.

	In our first paper in this series \citep[][hereafter \citetalias{Pittordis_2018}]{Pittordis_2018}, we
	used simulations to explore the prospects for
	this wide-binary test, in anticipation of the much improved data
	from the GAIA spacecraft \citep{Gaia_2016}; \citetalias{Pittordis_2018} used
	simulated wide-binary orbits for a variety of acceleration
	laws, including Newtonian and various MOND models both
	with and without an external field effect (hereafter ExFE);
	the general conclusion was that GAIA data provides promising prospects for such a test, since MOND-like models without an ExFE should produce large and obvious deviations
	towards larger velocity differences.
	In MOND-like models with the ExFE included, as theoretically preferred, the local Galactic acceleration field substantially suppresses MOND-like effects, but does not eliminate
	them. These models with ExFE give predicted relative velocities much
	closer to Newtonian, but do still show subtle deviations,
	most notably a significantly larger fraction of binaries with
	pairwise velocities in the range $(1.1 - 1.5) \times \vc(\rp)$, where
	$\vc(\rp)$ is the Newtonian circular velocity at projected separation $\rp$.
	Here we recall two of the main conclusions from \citetalias{Pittordis_2018}:
	MOND-like theories can allow bound binaries with relative
	velocities above the Newtonian ceiling, $v_{3D}/\vc(\rp) >\sqrt{2}$,
	(where $v_{3D}$ is the 3-D pairwise relative velocity); but with
	the ExFE included the fraction of such systems is predicted
	to be very small, typically 1 percent or less; so simply counting such systems is unlikely to be a practical test due to possible contamination, observational errors, and small-number
	statistics. However, the upper percentiles of this velocity ratio, or similarly the fraction of binaries with velocity ratio
	between $\sim 1.1 - 1.5$ are more promising statistics.

	The second main conclusion from \citetalias{Pittordis_2018} was that, in
	MOND theories including the ExFE, there is an optimal window
	of binary projected separations, $5 \la \rp \la 20 \kau$, for practical application of the test.
	Even wider separations are not favoured
	in practice because the inclusion of the ExFE causes the
	MOND-like effects to almost saturate at a fixed percentage at $\rp \ga 10 \kau$, while
	several observational issues become proportionally worse at even wider separations $\ga 20 \kau$.

	\begin{wscomm}
		Snipped a sentence above: new paras on PS19  below.
	\end{wscomm}

	After the release of GAIA Data Release 2,
	we explored an observational application of the test with that dataset  \citep[][hereafter PS19]{Pittordis_2019} .
	Since the 3D separation $r$ is not a practical observable (since the line-of-sight separation is typically well below
	the precision of distance measurements, even for GAIA) we have to replace
	it with projected separation $r_p$ as a proxy, which shifts the
	ratios to lower values depending on viewing angles; but this
	effect can be readily included in simulations.   The main conclusions of PS19 were that
	there is a sufficient population of bound binaries to carry out the test in future, but also that
	there exists a "tail" of pairs with large velocity differences, too numerous to explain by random alignments.
	This may be accounted for by common-origin unbound pairs (as modelled in PS19), or perhaps a more likely
	explanation by \cite{Clarke_2020} is hierarchical triples with an unresolved or faint third star in a closer orbit,
	which boosts the observed velocity difference between the two visible stars.

	In this paper we provide an updated version of PS19, using the recent GAIA Early Data Release 3 \citep{GaiaEDR3_2020a, GaiaEDR3_2020b} to provide a $\sim 3 \times $ larger sample of candidate WBs, with more precise proper motions.
	We also update our modelling here to fit the observed
	distributions with a mixture of binary, triple and flyby systems, using both Newtonian and MOND gravity laws and
	various eccentricity distributions.

	The plan of this paper is as follows: in Section 2 we
	describe the methods for selecting candidate wide-binaries from the
	GAIA Early Data Release 3
	(hereafter, EDR3) data and cleaning the sample; we then compare to the earlier sample from \citet{Pittordis_2019},
	and the EDR3 sample from  \citet{Badry_2021}.  In Section 3 we discuss various simulations of the velocity-ratio distributions
	for Newtonian and the realistic MOND-like binary orbits (with ExFE) from \citep[][hereafter \citetalias{Banik_2018_Centauri}]{Banik_2018_Centauri}; and we
	also simulate triple star systems, which appear to provide a reasonable fit to the high-velocity tail of the distribution, $\vratio \geq \sqrt2$, and unbound flyby systems.   In Section 4 we fit the observed distributions with a mixture of simulated binary,  triple and flyby model populations, for both GR and MOND cases; and
	we summarise our conclusions in Section 5.

	\section{GAIA EDR3 and sample selection}
	\label{sec:edr3}

	\subsection{Preliminary selection}
	\label{sec:preliminary selection}

	Our starting point is the public GAIA Early Data Release 3 dataset (EDR3),
	\citep{GaiaEDR3_2020a, GaiaEDR3_2020b} released on 2020 December 3.
	We initially select all stars with measured parallax $\omega > \frac{10}{3} \mas$
	(i.e. estimated distance $< 300 \pc$) with a GAIA broadband magnitude $G < 17$, and cutting out the Galactic plane with absolute latitude $\vert b \vert \le 15 \deg$ yielding a preliminary EDR3 sample of 2,101,920 stars (hereafter PEDR3).
	(Data quality cuts are applied at a later stage, in order that these
	may be adjusted post-selection).  These criteria are expanded from $200 \pc$ and $G < 16$ used by
	\citetalias{Pittordis_2019} from Gaia Data Release 2 (hereafter, DR2), since the improved data quality in EDR3 allows
	an expanded volume while giving slightly smaller transverse velocity errors.

	The parallax and magnitude cuts above are chosen to provide a
	large enough volume to contain a
	usefully large statistical sample of wide binaries; while the moderate
	distance limit and relatively bright magnitude limit ensures that
	GAIA provides high precision on distances and transverse velocities.

	We then used the same search method as described in section 2 from \citetalias{Pittordis_2019}

	\begin{wscomm}
		Removed ref to GDR2 - do later
	\end{wscomm}
	to search this nearby-star sample for pairs of stars
	with projected separation $\le 50 \kau$ (calculated
	at the mean distance of each candidate pair),
	difference in parallax distances of both stars $\vert d_1 - d_2 \vert \leq \, 8.0 \pc$, and distances also
	consistent with each other within $4\times$
	the combined uncertainty e.g., $ \vert d_1 - d_2 \vert \leq 4\sigma_{d}$,  and projected velocity difference $\Delta v_p \le 3 \kms$
	as inferred from the difference in proper motions; here, the projected velocity difference is computed
	assuming {\em both} stars in each candidate pair are
	actually at the mean of the two estimated distances.

	We note here that this common-distance assumption is important:
	if the relative velocities are calculated using
	individual parallax distances, then an example random 1 percent
	difference in parallax for a system
	with transverse velocity $40 \kms$ scales to a $0.4 \kms$ transverse
	velocity difference, which is similar to or larger than the orbital
	velocities of interest below.
	However, since we are almost entirely
	interested in the velocity {\em difference}
	within a binary, the common-distance assumption leads to an error
	in estimated relative velocity proportional to
	the unknown {\em true} fractional distance difference, $(d_1 - d_2)/ d$
	(see also \citet{Shaya_2011}, and Section 2.4 of \citetalias{Pittordis_2018}
	for related perspective effects).
	For true binaries with random orientation we
	expect $ \vert d_1 - d_2 \vert \le \rp$ for 71 percent of systems,
	and $\le 2 \rp$ for 90 percent.
	Then for a typical binary (see below)
	with $\rp \sim 10 \kau$ and $d \sim 180 \pc$
	we have $\rp/d \sim 2.7 \times 10^{-4}$; this is
	much smaller than the fractional uncertainty of the parallaxes,
	so choosing the common-distance assumption yields a much more
	precise estimate of the relative velocity for {\em genuine} binaries.

	This search results in a first-cut
	sample of 92,677 candidate EDR3 wide binaries (hereafter WB-EDR3); this sample is then pruned with
	additional cuts as described in the following subsections.
	The sky distribution of these candidates is shown in Figure~\ref{fig:aitoff} (left panel).

	\subsection{Triple and higher systems}
	\label{sec:triples}
	To reject the majority of ``moving groups'' or similar,
	we searched our WB-EDR3 sample for any star in common between two
	or more candidate binaries; if so, both or all those binaries
	were rejected,  reducing the WB-EDR3 sample to 83,884 candidate binaries with no
	star common to more than one candidate binary.

	\subsection{Sky cuts}
	\label{sec:skycut}
	Inspection of the WB-EDR3 sample showed a roughly uniform
	distribution across the sky,

	with some enhancement near around some well-known open clusters,
	i.e. the Pleiades, Praesepe, Upper Sco, and the Blanco 1 open cluster.
	Similar to \citetalias{Pittordis_2019}, we applied sky cuts to eliminate regions around the above clusters;
 the details of the exclusion regions are listed in Table~\ref{tab:openclus}.
	These sky cuts reduce the sample to 83,100 candidate binaries.

\begin{table}
\begin{tabular}{lcc}
\hline
Cluster  & RA limits (deg) & Dec limits (deg) \\
\hline
Blanco 1 & (0, 2) & (-31,  -29) \\
Pleiades & (54,  59) & (+22,  +27) \\
Praesepe & (129, 131) & (+18, +22) \\
Upper Sco & (238, 248) & (-29, -19) \\
\hline
\end{tabular}
\caption{ List of coordinates for excluded regions around open clusters.}
\label{tab:openclus}
\end{table}

	\subsection{Removing faint companions}
	\label{sec:faintstars}
	Similar to the methods of \citetalias{Pittordis_2019}, we also searched for additional co-moving companion stars to a fainter limit: we selected a ``faint star" sample of GAIA stars with $G < 20$ and a measured parallax $\omega > \frac{10}{3} \mas$
	(i.e. estimated distance $< 300 \pc$); for each star in each candidate binary, we then searched for faint-star companions with the following criteria:
	\begin{enumerate}
		\item Parallax consistent with the main star at $4\sigma$.
		\item Angular separation less than $2/3$ of the main-binary separation
		(since hierarchical triples are expected to be unstable for inner-orbit
		separation above $\sim 0.4 \times$ the outer separation);
		and angular separation above $0.5$ arcsec to avoid barely-resolved
		companions.
		\item Measured projected velocity difference from the main star
		$\le 5 \kms$.
	\end{enumerate}
	If any such ``third star'' was found, (in 698 cases), we rejected the
	candidate binary since a hierarchical triple will generally
	boost the projected velocity difference of the wide pair;
	this left a de-tripled sample of 82,402 candidate binaries from the WB-EDR3 sample.

	Clearly, the third-star search above will not reject third stars
	which are either very faint or unresolved from one of our
	binary members;  this will need to be considered for possible
	follow-up observations later. We consider the effects of contamination from undetected triple systems later in section \ref{sec:triple-sim}, showing this may be the dominant source of contamination.

	\subsection{Data quality cuts on wide binary candidates}
	\label{sec: Data quality cuts}
	We applied data-quality cuts to the WB-EDR3 sample based on the GAIA
	parameters, as \citet{Arenou_2018}  Equation 1 as follows:
	\begin{eqnarray}
		\chi^2 & \equiv  & {\tt astrometric\_chi2\_al}  \nonumber \\
		\nu & \equiv & {\tt astrometric\_n\_good\_obs\_al } - 5 \nonumber \\
		u & \equiv & \sqrt{\chi^2 / \nu} \nonumber \\
		u & \le & 1.2 \times {\rm max}(1, \exp\left[ -0.2(G-19.5) \right] )
		\label{eq:arenou}
	\end{eqnarray}

	We rejected candidate pairs from the WB-EDR3 sample, where at least one star did not satisfy Eq.~\ref{eq:arenou};
	this rejected 9,243 candidate pairs, leaving a cleaned sample of 73,159 candidate wide binaries,
	hereafter labelled CWB-EDR3: this is the main sample used for analysis below.

	\begin{wscomm}
		Slight re-order above
	\end{wscomm}

	\begin{figure*}

		\begin{center}
			\includegraphics[width=\linewidth]{aitoff.png}
			\caption{
				The sky distribution in Galactic coordinates for candidate
				binaries in EDR3: the left panel shows the initial set of candidate binaries WB-EDR3, the right panel is
				the cleaned sample CWB-EDR3 after applying all cuts in Section~\ref{sec:edr3}.  Note that the holes from
				limited sky coverage in PS19 no longer appear with the improved coverage of EDR3.  }
			\label{fig:aitoff}
		\end{center}
	\end{figure*}

	\subsection{Results and scaled velocities}

	For the surviving 73,159 candidate binaries, we show
	a plot of projected velocity difference vs projected separation
	in Figure~\ref{fig:rpvp}; this shows a clear excess approximately
	as expected for bound binaries, with an overdense cloud following
	a locus $\vp \sim 1 \kms (\rp / 1 \kau)^{-0.5}$.
	We note that our sample starts to miss true binaries at
	projected separations below  $\rp \la 0.6 \kau$, due to the
	$3 \kms$ velocity threshold,
	but this $\rp$ is much  smaller than the separations of interest
	below. At $\rp > 5 \kau$ the threshold includes pairs with velocity
	difference far above the bound limit,
	which are interesting for assessing sample contamination as seen below.

	It is more informative to rescale to the typical Newtonian
	orbit velocity, so we next estimate masses for each binary
	using an estimated mass/luminosity relation:  here,
	we adopt the main-sequence $M_I(mass)$ relation of \textit{Version 2021.03.02} from \citet{Pecaut_2013},
	and the $V - I, M_I$ colour relation from the same,
	where $M_I$ denotes absolute magnitude.  From those we
	apply the colour relation given in Table~C2 of \citet{GaiaEDR3_2021} to predict
	$G$ magnitude from $V$ and $I$ magnitudes as
	\begin{eqnarray}
		G & \simeq & V \; - \; 0.01597 \; + \; 0.02809 (V-I) \;
		- \; 0.2483 (V-I)^2 \nonumber \\
		& & + 0.03656 (V-I)^3 \; - \; 0.002939 (V-I)^4
		\label{eq:gvi}
	\end{eqnarray}
	to obtain
	a predicted relationship between absolute GAIA magnitude
	$M_G$ vs mass; we then fit to this to obtain an approximate
	mass/$M_G$ relation
	\begin{equation}
		\frac{M}{\msun} = 10^{0.074 (4.69 - M_G )}
		\label{eq:mass-mg}
	\end{equation}
	Then, for each star we have $M_G$ directly from $G$ and
	parallax distance,
	hence an estimated mass follows.   Since the luminosity(mass)
	relation is rather steep, small errors in $G$ or distance
	have relatively little effect on mass estimates below.

	\begin{figure*}
		\begin{center}
			\includegraphics[width=16cm]{Gaia_Dvp_logrp.png}
			\caption{Scatter plot of projected relative velocity $\vp$ (y-axis)
				vs projected separation (log scale, x-axis) for the CWB-EDR3 binary candidates.
				The main selection cuts are visible at top and right. }
			\label{fig:rpvp}

			\includegraphics[width=16cm]{DvpVcN_logrp_Gaia.png}
			\caption{Scatter plot of projected velocity relative to Newtonian,
				$\vp/v_c(\rp)$, vs projected separation for CWB-EDR3 sample.  The dashed line at
				$\sqrt{2}$ indicates the Newtonian limit. The upper cutoff is
				now slightly fuzzy due to the additional dependence of velocity ratio on mass.
				Note that in later analysis we only use the sample at $\rp > 5 \kau$ so $\log_{10} r_p > 3.7$,
				so incompleteness in the upper-left region is not an issue.   }
			\label{fig:rp_vratio}
		\end{center}
	\end{figure*}

	For each candidate binary we then define
	\begin{equation}
		v_c(\rp) \equiv \sqrt{G M_{tot} / r_p}
		\label{eq:vc}
	\end{equation}
	as the estimated circular-orbit velocity at the current
	{\em projected} separation;
	for each candidate binary, we then divide the measured projected velocity
	difference by the above to obtain a dimensionless ratio
	\begin{equation}
		\vtilde \equiv v_p / v_c(\rp) \ ;
	\end{equation}
	a scatter plot of this ratio is shown in
	Figure~\ref{fig:rp_vratio}, and
	various histograms of this ratio  are compared with models below.

	\subsection{Transverse velocity errors}

	We have estimated relative-velocity errors assuming
	uncorrelated errors between the two components of the binary,
	simply from the root-sum-square of the quoted rms errors in $\mu_\alpha$
	and $\mu_\delta$ for each of the two stars in each binary,
	and multiplying by distance to obtain the transverse-velocity
	error. (This should be reasonable as long-range correlated errors should
	mostly cancel between the two stars).

	Table \ref{tab:GDR2ToEDR3TransVelErrors} shows the comparison of the transverse velocity random errors between the CWB-DR2 and CWB-EDR3. We can see with the better quality of data from DR2 to EDR3, and a larger binary sample, the median has decreased to an impressively small value of $\sigma(\vp) \approx 0.06 \kms$.
	We also see a decrease in values when converting to the ratio to circular-orbit velocity, $\sigma(\vp)/v_c(\rp)$, where the median for the full candidate sample has now reduced to 0.06 and the 80th percentile is 0.1;
	for the ``wide'' subsample with $5 < \rp < 20 \kau$, the
	median is 0.14 and the 80th percentile is 0.26.
	A scatter plot of $\sigma(\vp)$
	versus distance is shown in Figure~\ref{fig:dsigv}; the trend
	with distance is clear, but
	most systems have $\sigma(\vp) \la 0.15 \kms$ even near our
	$300 \pc$ limit.

	\begin{table}
		\caption{Table comparing percentiles of transverse velocity errors
			and relative to circular velocity,  between CWB-DR2 (PS19) and CWB-EDR3 data sets.
			Upper table for the full sets, lower table for subsample with $5 < \rp < 20 \kau$. }
		\label{tab:GDR2ToEDR3TransVelErrors}
		\centering
		\begin{tabular}{l|l|c|}
			\hline
			All $\rp$   &   CWB-DR2   & CWB-EDR3    \\

			\hline

			\multicolumn{1}{|p{3cm}|}{\centering $\sigma(\vp) \  [\kms ]$ \\ (50\%, 80\%, 90\%) \\} & $\approx\,[0.09, 0.14, 0.2]$ &  $\approx\,[0.06, 0.1, 0.13]$   \\

			\multicolumn{1}{|p{3cm}|}{\centering $\sigma(\vp)/v_c(\rp)$ \\ (50\%, 80\%, 90\%) \\} & $\approx\,[0.08, 0.17, 0.26]$ &  $\approx\,[0.06, 0.12, 0.19]$   \\
			\hline
			& & \\
			\hline
			\multicolumn{1}{|p{3cm}|}{\centering ($5 < \rp < 20 \kau$)\\}
			&  CWB-DR2  & CWB-EDR3     \\
			\hline

			\multicolumn{1}{|p{3cm}|}{\centering $\sigma(\vp)\ [\kms ]$ \\ (50\%, 80\%, 90\%) \\} & $\approx\,[0.09, 0.14, 0.19]$ &  $\approx\,[0.05, 0.09, 0.11]$   \\

			\multicolumn{1}{|p{3cm}|}{\centering $\sigma(\vp)/v_c(\rp)$ \\ (50\%, 80\%, 90\%) \\} & $\approx\,[0.23, 0.39, 0.54]$ &  $\approx\,[0.14, 0.26, 0.34]$   \\
			\hline
		\end{tabular}
		\begin{wscomm}
			Have moved km/s into rows, got rid of TVE acronym.
		\end{wscomm}

	\end{table}

	\begin{figure*}
		\begin{center}
			\includegraphics[width=\linewidth]{vError_vs_distance.png}
			\caption{Scatter plot of rms velocity uncertainty $\sigma(\vp)$ versus
				mean distance for candidate binaries in CWB-EDR3.}
			\label{fig:dsigv}
		\end{center}
	\end{figure*}

	The latter values are significantly smaller than 1, but not very small,
	so the effect of random proper motion errors will affect the detailed
	shape of the distributions below.  However, in future GAIA
	data releases these values are expected to reduce by factors
	of at least 2--4 as  proper motion precision scales
	as $\propto t^{-3/2}$,  so the random errors in proper motions
	are likely to become relatively unimportant in the medium-term
	future.

	We note that for a ``typical'' binary below
	at $\rp \sim 10 \kau$ and $d \sim 180 \pc$,
	the angular separation is $0.27$ mrad
	or 55 arcsec, so these are very well resolved and the uncertainty
	on $\rp$ is essentially the same as the error on the mean distance,
	typically well below 1 percent and almost negligible.
	The error on $\vp$ is dominated
	by random errors on the proper motions, assuming that
	correlated systematic errors mostly cancel between the two components
	of the binary.   Since we are mostly interested in statistical
	distributions, the effect of random errors is modest as long
	as these are not larger than $\sim 0.25$ in $v_p / v_c(\rp)$.
	Note that for systems with small observed ratios
	$\vp/v_c(\rp) \sim 0.5$, the {\em fractional} uncertainty in this
	ratio is rather large;
	however such systems still have a high probability
	of the true ratio being $\la 0.8$, so this scatter is relatively
	unimportant.   For systems with $\vp/v_c(r_p) \ga 1$, the fractional
	uncertainty is relatively modest; though possible non-Gaussian
	errors in the GAIA data remain a concern, this should improve in
	future GAIA releases as more observing epochs become available
	to reject outliers.

	\begin{wscomm}
		New para:
	\end{wscomm}
	In summary, we note that the velocity precision for our current sample has improved over the PS19 sample, since
	the improved precision of GAIA EDR3 over DR2 outweighs our expanded distance and magnitude limits.
	The precision will continue to increase with future DR4 and beyond in the extended mission,
	so GAIA random errors will become negligible
	compared to other sources of uncertainty (especially contamination from triple systems, see below).

	\subsection{Comparison with El-Badry, Rix and Heintz Wide Binary EDR3 sample}

	Here we note that a catalogue of candidate binaries
	in GAIA EDR3 has been published by \citep[][hereafter \citetalias{Badry_2021}]{Badry_2021}.
	Much of our sample selection was completed independently
	prior to the appearance of \citetalias{Badry_2021}, but here we give a brief comparison.

	\begin{table*}
		\caption{Comparison of Wide Binary selection between our criteria and ERH21  }
		\label{tab:WBEDR3compERH21}
		\centering
		\begin{tabular}{l|r|r|r|}

			Criteria  &   WB-EDR3  & ERH21     \\
			\hline

			Magnitude $G$  & $<17$ & $\neq {\tt NULL}$    \\
			Parallax $\bar{\omega}\,[mas]$ ($d\,[\pc]$)  & $>\frac{10}{3}\,mas$ ($<300\,\pc$) & $>1\,mas$ ($<1\,\kpc$) \\
			Projected Separation $r_p$  & $\leq50\,kAU$ ($\sim 0.242\,\pc$) & $\leq206.265\,kAU$ ($1\,\pc$)   \\
			Parallax of both members $|\bar{\omega}_1 - \bar{\omega}_2|$ &$\leq4\sqrt{\sigma^2_{\bar{\omega}_1} + \sigma^2_{\bar{\omega}_2} }$ & $ \displaystyle <b\sqrt{\sigma^2_{\bar{\omega}_1} + \sigma^2_{\bar{\omega}_2} } \
			\begin{cases}  b=3\, , \, \theta_p\geq\,4\arcsec \\ b=6\, , \,\theta_p<\,4\arcsec  \end{cases}  $ \\
			Projected velocity threshold $v_p$  & $\le 3 \kms$ & $\le 2.1 \kms (\rp / 1\kau)^{-0.5}$   \\
			\hline
		\end{tabular}

	\end{table*}

	As seen in Table \ref{tab:WBEDR3compERH21}, \citetalias{Badry_2021} chose  $1 \kpc$ distance limit with Gaia magnitude cuts at $G\neq {\tt NULL}$, while we selected our primary sample at $300 \pc$ with $G<17$.  Note that
	\citetalias{Badry_2021} do use other cuts on relative errors,
	which do introduce some implicit magnitude-dependence, as fainter stars are more likely to fail those cuts.
	Our selection adopts a fixed projected separation of $r_p\leq50\kau $ with parallaxes consistent within 4 times the combined uncertainty $\leq4\sqrt{\sigma^2_{\bar{\omega}_1} + \sigma^2_{\bar{\omega}_2} }$ and a projected velocity
	threshold $v_p \le 3 \kms$ (the same as in \citetalias{Pittordis_2019}). Whereas \citetalias{Badry_2021} use $r_p\leq1\pc$, with parallaxes to both members satisfying \citetalias{Badry_2021} Equation 2 as follows:
	\begin{eqnarray}
		|\bar{\omega}_1 - \bar{\omega}_2| & \leq  & b\sqrt{\sigma^2_{\bar{\omega}_1} + \sigma^2_{\bar{\omega}_2} }  \nonumber \\
		b = 3 & \theta_p \geq 4\arcsec \nonumber \\
		b = 6 & \theta_p < 4\arcsec
		\label{eq:badry21}
	\end{eqnarray}
	In addition, they apply the same separation-dependent
	threshold for projected velocities $v_p$ from \citet{Badry_2018}, which translates to $v_p \le 2.1 \kms (\rp / 1\kau)^{-0.5}$
	(equivalent to the Newtonian bound limit $\sqrt{2} \,\vc$ for
	a $2.5 \msun$ system, or just above at
	$1.83 \,\vc$ for a more typical system with mass
	$1.5 \msun$).
	This means that our sample extends to substantially higher
	(unbound) velocity ratios at $\rp \ga 3 \kau$, which turns out to be
	useful below for modelling the tail of high-velocity systems.

	There are additional differences in how we cut for clusters,
	triples, etc, but these turn out to be relatively less important. We cut out known clusters after removing triples and hierarchical systems as described in sections \ref{sec:skycut}, \ref{sec:triples} and \ref{sec:faintstars}. In \citetalias{Badry_2021}, they begin cleaning by counting the number of phase-space neighbours per source from their primary selection that are brighter than $G=18$, consistent with the size and velocity dispersion of a typical cluster. The neighbours per source are defined satisfying projected separations $r_p\leq 5 \pc$ and proper motions $\Delta \mu \leq 5\kms$, with parallaxes to both sources within
	$\Delta \bar{\omega}\leq2\sqrt{\sigma^2_{\bar{\omega}_1} + \sigma^2_{\bar{\omega}_2} }$
	Any source (or either component from the binary candidate) that contains more than 30 neighbours is removed. This leads to removing all candidates where either component of the binary is a member in another binary candidate; also removing candidates that might be members of small clusters or moving groups that were no caught in the initial cleaning, performed by applying the same method in counting the number of phase-space neighbours but without the magnitude cut.

	We have done a cross-match of our sample to \citetalias{Badry_2021} as follows: we first select the subset of \citetalias{Badry_2021} binaries which could in principle have passed our sky, distance and magnitude criteria, i.e., where both stars have $G < 17$, galactic latitude $\vert b \vert > 15 \deg$, difference in parallax distances $\vert d_1 - d_2 \vert \leq \, 8.0 \pc$, projected separation $\rp < 50 \kau$, distance to both stars $d\leq 300 \pc$, parallaxes of both stars consistent with each other within $4\times$ the combined uncertainty e.g., $ \vert d_1 - d_2 \vert \leq 4\sigma_{d}$, projected velocity difference $\Delta v_p \le 3 \kms$, and applying cuts from \citet{Arenou_2018}. The combination of these cuts gives
	a subsample of 65,234 candidate binaries from \citetalias{Badry_2021} which could
	in principle pass our other cuts, here called the ERH-cut sample.  Cross-matching those against
	our CWB-EDR3 sample of 73,159 candidates, we find that 64,498
	are in common with ERH-cut, while there are 8,661 candidates in our CWB-EDR3 but not found in ERH-cut, and 752 candidates in ERH-cut are not found in CWB-EDR3.
	\begin{wscomm}
		Clarify when matching CWB-EDR3 vs ERH-cut,  or whole ERH ..
	\end{wscomm}

	Figures \ref{fig:ERHaitoff} and \ref{fig:ERH_VpRp_scatter} illustrate the comparisons of the 8,661 candidates in our CWB-EDR3 but not found in ERH-cut, and 752 in ERH-cut not found in CWB-EDR3.
	As seen in Figure \ref{fig:ERH_VpRp_scatter}, most of the additional binaries in our sample are either at larger velocity differences (above the \citetalias{Badry_2021} limit and below our $3 \kms$ limit), or at smaller separations $\rp < 0.5 \kau$ which we do not consider below.

	As seen in Figure~\ref{fig:ERHaitoff}, the binaries in ERH-cut but not in CWB-EDR3 are largely removed
	due to our additional sky cuts around open clusters.

	\begin{wscomm}
		New sentence:
	\end{wscomm}
	In summary we find good agreement between our sample and the ERH-cut sample,
	and differences in selection criteria have relatively little effect; the additional "binaries" in our sample are mostly
	potential triples at large velocity differences, which are useful for modelling in later sections.

	\begin{figure*}
		\begin{center}
			\includegraphics[width=\linewidth]{ERHaitoff.png}
			\caption{Aitoff projection of CWB-EDR3 binaries not found in ERH-cut sample (red points), and binaries from ERH-cut  not found in CWB-EDR3 (blue points).}
			\label{fig:ERHaitoff}
			\includegraphics[width=\linewidth]{ERH_VpRp_scatter.png}
			\caption{Scatter plot of projected velocity difference $\Delta\vp$
				vs projected separation for CWB-EDR3 binaries not found in ERH-cut sample (red points),
				and binaries from ERH-cut not found in CWB-EDR3 (blue points). }
			\label{fig:ERH_VpRp_scatter}
		\end{center}
	\end{figure*}

	\section{Orbit simulations}
	\label{sec:orbits}

	In this Section we describe the simulations for predicting distributions of projected velocities in Newtonian and MOND models,
	for various eccentricity distributions, and for binaries, triple systems
	As noted in Section 3.1 of \citetalias{Pittordis_2019}, we described how velocity ratios are handled within our orbit simulations of observables for the 2D sky-projected velocities as well as 2D projected separations; this is simply because
	radial velocities are not yet available for the large majority
	of our candidate WBs;  though radial velocities may become available in future
	with large spectroscopic surveys
	such as 4MOST, PFS, WEAVE and MSE, and targeted follow-up of selected systems.

	\subsection{Binary orbits}
	\label{sec:orbsmg}
	Similar to Section 3 from \citetalias{Pittordis_2019},
	here we simulate a large sample of $\sim 5 \times 10^6$ orbits
	with random values of $a, e$ in both Newtonian gravity and
	one specific modified-gravity model from \citet{Banik_2018_Centauri} (?).
	\begin{wscomm}
		Changed to one MG model here
	\end{wscomm}
	We then study the joint distribution of observables,
	in particular projected separation
	$r_p$ and relative velocity ratio $\vp / v_c(r_p)$, using several different eccentricity distribution functions.

	In the case of modified gravity models, the orbits are generally
	not closed ellipses, so they are not strictly defined by
	the standard Keplerian parameters $a, \, e$. As in \citetalias{Pittordis_2018} and \citetalias{Pittordis_2019}, modified-gravity orbits are parametrised using the ``effective'' orbit size $\hat{a}$ and quasi-eccentricity $\hat{e}$ as follows:
	we define $\hat{a}$ to be the separation at which the simulated relative velocity
	is equal to the circular-orbit velocity (in the current
	modified-gravity model), then we define $\theta_{\rm circ}$ to be the angle between the relative velocity vector and the tangential direction when the orbital separation crosses
	$\hat{a}$, and then $\hat{e} \equiv \sin \theta_{\rm circ}$;  these definitions
	coincide with the usual Keplerian $a, e$ in the case of standard gravity.

	After integrating these orbits using either of the above
	gravity laws (Newton/GR, Realistic MoND Model with ExFE as described in \citetalias{Banik_2018_Centauri}) and a chosen
	value for external field $g_e \sim 1.2a_0$,  we ``observe'' the
	resulting binaries at many random times and random inclinations to the
	line-of-sight.

	For each simulated orbit/epoch snapshot, we produce simulated
	observables including the projected separation $r_p$, projected relative
	velocity $\vp$, and also $\vp/v_C(r_p)$ corresponding
	to our observables from Gaia.

	The radial acceleration law is chosen according to the
	selected gravity theory under consideration with the
	External Field Effect (ExFE) on (see below).
	For the Newtonian/GR case, we have the standard
	\begin{equation}
		g_{N} = \frac{G(M_1+M_2)}{r^2}
		\label{eq:g_n}
	\end{equation}

	For the MOND-like case with ExFE,
	we use the fitting function of \citet{McGaugh_2016} (hereafter MLS),
	sometimes known as the ``radial acceleration relation'',
	given by
	\begin{eqnarray}
		g_{MLS} & = & g_N \, \nu(g_N/a_0) \ ; \nonumber \\
		\nu(y) & = & \frac{1}{1 - \exp(-\sqrt{y})} \ ;
		\label{eq:g_mls}
	\end{eqnarray}
	we refer to this $\nu$ function as the MLS interpolating function below.
	This function is shown by MLS to produce a good fit to rotation curves
	for a large sample of disc galaxies spanning a range of masses;
	it also has the desirable feature that the function $\nu(y)$ converges
	very rapidly to 1 when $y \ga 20$, so deviations on Solar System
	scales are predicted to be vanishingly small, consistent
	with observational limits.

	The only modified-gravity model we consider to compare against our CWB-EDR3 sample is the MOND model with ExFE, using the approximation
	of \citetalias{Banik_2018_Centauri}; this is given by
	\begin{eqnarray}
		g_{N,int} & = & G(M_1 + M_2)/r^2  \\
		g_{N,gal} & = & 1.2 \, a_0 \\
		g_{N,tot} & = & \left( g_{N,int}^2 + g_{N,gal}^2 \right)^{1/2} \\
		g_{i, EFE} & = & g_{N,int} \nu(g_{N,tot}/a_0) \left(1 +
		\frac{\kappa(g_{N,tot})}{3} \right) \\
		\kappa & \equiv & \frac{\partial \ln \nu }{\partial \ln g_{N} }
		\label{eq:exfe}
	\end{eqnarray}
	where $g_{N,int}$ is the internal Newtonian acceleration of the binary;
	$g_{N,gal}$ is the external (Galactic) Newtonian acceleration,
	$g_{N,tot}$ is the quadrature sum of these, $\nu$ is the MLS function
	from Equation~(\ref{eq:g_mls})
	and $g_{i, EFE}$ is our model MOND-ian
	internal acceleration, approximating the
	application of the external field effect.
	(This is not quite an exact solution of the MOND-like equations,
	but is shown by \citetalias{Banik_2018_Centauri} to be a good approximation to the full
	numerical solution).

	Above,  the observed  Galactic rotation values
	$v_{LSR} \simeq 232 \kms$ and $R_0 \simeq 8.1 \kpc$ imply
	a total Galactic acceleration close to $1.75 \, a_0$, hence we
	require $g_{N,gal} \nu(g_{N,gal}/a_0) \approx 1.75 \, a_0$. Solving this
	leads to $g_{N,gal} \approx 1.16 \, a_0$ as above and
	$\nu \approx 1.51$, in reasonably
	good agreement with the estimated baryonic contribution to the Galactic
	rotation (as expected, since the MLS fitting function was derived by
	fitting to a sample of external spiral galaxies with well-observed rotation
	curves, so this is consistent with our Galaxy being typical).

	As in \citetalias{Pittordis_2019}, the modified gravity model produces
	typical binary orbital velocities boosted by $\sim 16 \%$ at separations
	beyond $7 \kau$.
	\begin{wscomm}
		Replaced para with short sentence
	\end{wscomm}

	\begin{figure*}
		\begin{center}
			\includegraphics[width=\linewidth]{Hist_Dvpvcrp_EDR3AndGDR2.png}
			\caption{
				Histograms of velocity ratio, $\vratio$, for observed
				binaries: the four panels show four bins of projected separation as
				labelled above each panel.  The histograms shows our observed
				CWB-EDR3 (red) and the CWB-DR2 (blue) sample; numbers in the latter are
				renormalised to match the total number in the CWB-EDR3 sample.}
			\label{fig:Gdr2hist}
		\end{center}
	\end{figure*}

	\begin{figure*}
		\begin{center}

			\includegraphics[width=\linewidth]{Hist_Dvpvcrp_GaiaAndGravModels_CutXaxisTo2.png}
			\caption{ Histograms of velocity ratio, $\vratio$, for observed and simulated
				binaries: the four panels show four bins of projected separation as
				labelled in the legend. The shaded pink histogram shows our observed
				CWB-EDR3 (red); solid lines show Newtonian orbits, and dashed lines for the MOND model.
				Line colours show eccentricity distribution: flat $f(e) = 1$ (red), Tokovinin (green) and $f(e) = 2e$ (blue).
				Simulated histograms are normalised to the number of observed systems at ratio $\le 1.6$.    }
			\label{fig:mghistCut2}
		\end{center}
	\end{figure*}

	In Figure~\ref{fig:mghistCut2}, we show histograms of
	the ratio $\vratio$ for four bins of projected separation,
	and two gravity acceleration models: standard Newtonian,
	and the Banik$\sim1.2a_0\,$Realistic ExFE from \citetalias{Banik_2018_Centauri}; and a 3-Body Newtonian model compared with the GAIA CWB-EDR3 and CWB-DR2 samples..
	Model histograms are scaled to match the total of the data at
	velocity ratio $\le \sqrt{2}$.

	On inspection of Figure~\ref{fig:mghistCut2}, several features are notable:
	\begin{enumerate}
		\item
		The observed histograms all show a clear peak at a ratio $\sim 0.6$, in rough agreement with
		either Newtonian gravity or our MOND model.  The shape of the peak is  sensitive to both the eccentricity distribution
		than the gravity model, with flat $f(e)$ giving a slightly pointed peak, while $f(e) = 2e$ makes the peak
		flatter and shifted to a slightly lower value.
		\item Comparing to results from PS19 (not reproduced here), we found that
		simulated histograms for MOND {\em without} ExFE showed a large and obvious
		shift of the peak to larger velocity ratios $\ga 1$, especially in the
		wider separation bins where the Newtonian acceleration
		is well below $a_0$;
		such a large shift in the peak appears clearly inconsistent with the data,
		so we confirm the conclusion of PS19 that MOND {\em without} ExFE is firmly excluded (though this
		was already disfavoured on theoretical grounds).
		\item
		The simulated histograms for MOND {\em with} ExFE show a distinctly larger fraction of binaries
		at velocity ratio $1.1 - 1.5$, compared to the Newtonian cases; that excess is rather insensitive
		to the $f(e)$ distribution, increasing	 only moderately with projected separation.
		So as in PS18, this region is the key discriminant between Newtonian vs MOND models.
		\item The actual data show a clear peak at $\sim 0.6$ as above,
		then a fall towards $\sqrt{2}$; however this is followed by
		a prominent tail which slowly declines to much larger ratios $\ga 5$.
		Clearly, the presence of this tail  makes it hard to decide a preference
		between Newtonian or MOND-with-ExFE: a smooth downward extrapolation of
		the tail below $\sqrt{2}$  can  account for a significant fraction of observed systems at
		velocity ratio $1.1 < \vratio < 1.4$, so
		the tail is not well understood but
		has a major impact on the statistics.
		This tail is discussed and modelled in the next Section.
	\end{enumerate}

	\subsection{Triple system simulations}
	\label{sec:triple-sim}
	As we can see from the histograms in Figures \ref{fig:Gdr2hist}
	and \ref{fig:mghistCut2}, the observed CWB-EDR3 sample has a velocity distribution described qualitatively by a ``hump+tail'' shape very similar to \citetalias{Pittordis_2019}.  In \citetalias{Pittordis_2019}, we
	selected random "binaries" from copies of the DR2 catalogue with sky positions randomised by a few degrees;
	this showed that the "tail" is much too populous to
	explain as random chance projections of unassociated stars; we verify this for our new sample in \ref{sec:Randoms} below. The next alternative was the possibility of co-natal stars born in the same open cluster, which is currently dissolving therefore, having similar velocities and undergoing chance flybys. Although this provided acceptable $\chi^2$ values, there was a potential inconsistency in that the fitted number of flyby events was decreasing with $r_p$, while simulations predict a rising distribution, due to the increasing phase-space volume at larger projected separation.  It was then suggested by \citet{Clarke_2020} that triple systems with an unseen or unresolved third star could account for the tail, and the simulated distribution was found to produce a good fit to the tail, though the fraction of triples required is quite large (fixed at 50 percent in \citet{Clarke_2020}).
	Below, we will model the observed distribution with a mixture of binary, triple and flyby systems.

	To simulate the triple systems, we generate two binary orbits made up of three stars. We label star 1 as the single star, and stars 2 \& 3 as the inner binary, so the outer orbit is star 1 orbiting the barycentre of stars 2 and 3.
	We choose random masses for the three stars with a flat distribution up to $0.7 \msun$, then declining
	as $m^{-2.35}$; we pick three random masses, relabel so that $M_2 > M_3$, and apply the constraint that stars 1 and 2 are $\ge 0.5 \msun$,  while star 3 is $\ge 0.01 \msun$ and may be unobservable.

	For orbit sizes, we choose the outer orbit size $\hat{a}_{out}$ from a flat distribution in $\log_{10} (\hat{a}_{out} / 1 \kau) \in (0,2)$, and the inner orbit size $\hat{a}_{inn}$ is chosen from a flat distribution in $\log_{10} \hat{a}_{inn}$, with a minimum at 0.1 AU and a maximum at $0.3 \times$ the outer orbit; this is an approximate  criterion for long-term stability.

	We generate eccentricities from one of three distributions: either flat $f(e) = 1$, an intermediate case
	from \citet{Tokovinin_2016} (hereafter Tokovinin), and $f(e) = 2e$.
	Both inner and outer eccentricities use the same distribution function, but drawn independently of each other.

	For simplicity, we treat the inner and outer orbits as independent.
	We solve for the two orbits independently in their own planes, and then apply a random 3D rotation matrix,
	$\mathbf{R}$,  to the inner orbit relative velocity to generate a random relative alignment between the two orbits.
	Next the system is "observed" at 10 random phases and 10 random viewing directions for each phase. At each simulated
	"observation" we evaluate the projected separation, and the 3D velocity difference between star 1 and
	the "observable center" of stars 2+3, as follows.
	\begin{equation}
		\mathbf{v}_{3D,obs} = \mathbf{v}_{out} - f_{pb} \, \mathbf{R}\mathbf{v}_{inn}
	\end{equation}
	where $\mathbf{v}_{out}$ is the outer orbit velocity (star 1 relative to the barycentre of 2+3), and $\mathbf{v}_{inn}$ is
	the relative velocity between stars 2+3.   The latter velocity is rotated and scaled down by a dimensionless factor
	$f_{pb} \le 0.5$, defined as the fractional offset between the "observable centre" and the barycentre of stars 2+3,
	relative to their actual separation.
	The "observable centre" is defined according to the angular separation: if less than 1 arcsec, we assume stars 2+3 are
	detected by GAIA as a single unresolved object, and use the luminosity-weighted centroid (photocentre) of the two.
	Otherwise for separation $> 1\,$arcsec, we assume stars 2 and 3 are detected as separate objects, or star 3 is unobservably faint, and we take the position of star 2 alone as the observable centre.  Therefore, this results in a fractional offset
	\begin{equation}
		f_{pb} = \begin{cases}
			\frac{M_3}{M_2+M_3} - \frac{L_3}{L_2 + L_3}  \ (\theta < 1 \text{ arcsec}  ) \\
			\frac{M_3}{M_2+M_3}   \ (\theta \ge 1 \text{ arcsec} )
		\end{cases}
	\end{equation}
	where the $L_{2,3}$ are the model luminosities.

	The 3D velocity above is then converted to 2D projected velocity according to the random viewing direction, and observables are saved to create simulated histograms for triple systems; this procedure is repeated for both of the gravity models (Newtonian / MOND) and each of the three eccentricity distributions. These simulated triple results are then used in the fitting procedure below.

	\subsection{Random Samples}
	\label{sec:Randoms}
	Using the same method as described in Section 3 of \citetalias{Pittordis_2019}, we estimated the level of contamination of our CWB-EDR3 sample by random chance projections of unrelated stars which just happen to contain small velocity differences. We constructed several randomised samples by starting with our single-star PEDR3 sample, removing one star from each binary, then randomising the RA/Dec values individually by a few degrees (e.g., randomising within $ \in \pm 3 \deg$);  these position shifts preserve global distributions with respect to galactic coordinates, but large enough to eliminate most truly associated stars; we then re-ran the binary search on the position-randomised samples. The number of random "binaries" was small in each single run, so we repeated 9 times with different random seeds and averaged.

	We show an example histogram for a set of randomised samples compared to our observed CWB-EDR3 data in Figure \ref{fig:histRands}; here the mean of random samples is multipled by $10\times$ for visibility, since the number of randoms is much smaller than the observed data and difficult to see in the histogram.

	The main result from our randomised samples is that they contain far fewer candidate ``binaries'' than the observed CWB-EDR3, hence observed data greatly outnumber the randoms. As we can see, the randomised sample show no peak at small velocity ratios, but a fairly smooth distribution with a gradual rise towards larger velocity ratios. We get the same result as
	\citetalias{Pittordis_2019} that the ``tail'' distribution of the observed data is much more populous than our randomised samples, hence the ``tail'' cannot be due to chance projections of unrelated stars.
	\begin{figure*}
		\begin{center}
			\includegraphics[width=\linewidth]{Hist_Dvpvcrp_EDR3AndRands.png}
			\caption{
				Histograms of velocity ratio $\vratio$ for observed and randomised sample. The four panels show four bins of projected separation as titled. The histograms show our observed CWB-EDR3 (red), and the mean of 9 randomised samples (blue), with the mean random sample artificially scaled-up by $10 \times$ to enhance visibility.  }
			\label{fig:histRands}
		\end{center}
	\end{figure*}

	\subsection{Flyby simulations}
	\label{sec:flyby}
	We find below that fitting purely with binary+triple systems gives reasonable fits, but requires a somewhat improbable
	result that the ratio of triples to binaries is steeply increasing towards larger projected separations. As in \citetalias{Pittordis_2019},  we therefore
	model an additional population of "flyby" systems which have small velocity differences (e.g. co-natal stars born in the same birth cluster) on unbound hyperbolic flyby trajectories.

	Similar to \citetalias{Pittordis_2019},
	our randomised-position samples show that
	the tail is much too populous to be explained by
	random chance projections of unrelated stars;
	one plausible explanation appears to be pairs of co-natal stars
	born in the same open cluster, which therefore have similar
	velocities and are currently undergoing a chance close flyby.
	Evidence for a population of such ``cold streams'' has been
	given by \citet{Oh_2017}.

	In this case, we would expect two of the three velocity components
	(perpendicular to the escape direction)
	must be similar in order to get a close flyby, while the velocity
	difference in the escape direction should approximately reflect
	the distribution in ejection velocities from the cluster
	and the time difference between the two ejections.

	During a flyby, the relative velocity will speed up
	according to a hyperbolic flyby orbit.  To simulate this,
	we generated random flyby encounters as follows:
	\begin{enumerate}
		\item We chose a distribution of impact parameter $b$
		with $dn/db \propto b$ up to a maximum value of $300 \kau$ or 1.45 pc.
		\item We chose a distribution of asymptotic velocity difference
		$\vinf$  uniform up to $2 \kms$.
		\item Given random values of $b, \vinf$ generated
		as above, each such pair produces one flyby encounter.
		We sampled each hyperbolic flyby
		at random times while the 3D separation was $\le 300 \kau$,
		and computed the 3D relative separation and relative velocity vectors. We
		``observed'' these from random viewing angles to produce $\vp, \rp$.
		\item We truncated the sample as for the DR2 data,
		at $\rp < 40 \kau$ and  $\vp \le 3 \kms$,
		and computed the resulting velocity ratio $\vp / v_c(\rp)$.
	\end{enumerate}
	For hyperbolic flyby encounters the 3D velocity ratio $v_{3D}/v_c(r)$
	is always  $\ge \sqrt{2}$, with the simulations showing a modest
	pile-up in the distribution just above this value; this pile-up
	arises from flybys with eccentricity
	not much larger than 1, which speed-up substantially but have
	velocity ratios decreasing towards $\sqrt{2}$ as
	they approach pericenter.
	The projection to 2D separation and velocity smears this distribution towards lower ratios,
	and therefore fills-in the
	gap below $\sqrt{2}$; the result is that simulated fly-bys with a flat
	distribution of $\vinf$ produce a smooth
	maximum in the distribution at a ratio below 1.0,
	with a gently declining  tail at larger ratios, as shown in the next section.

	\section{Data vs model comparisons}
	\label{sec:comp}

	In this section we fit the observed distributions of $\vp/\vc(r_p)$ in our EDR3 wide-binary sample
	as a mixture of binary, triple and flyby systems, where the velocity distribution for each is given by the results of
	the simulations in Section~\ref{sec:orbits}.  (Thus, this is an generalised case between \citet{Clarke_2020} who fitted binaries+triples, and \citetalias{Pittordis_19} which fitted binaries + flybys. )
	We take the four bins as above in projected separation, 	and fit independently to the observed distribution in each separation bin.

	As noted earlier, we found that fitting with only binary+triple systems can produce reasonable fits to the data
	(in agreement with \citet{Clarke_2020}),  but has
	a problem in that the fitted ratio of triple to binary systems is steeply increasing with $r_p$; the
	fitted triple/binary ratio is 1.25  in the
	smallest $r_p$ bin, but increases to around 2.75 in the widest $r_p$ bin; this is because the "tail" becomes
	relatively more populous relative to the "hump" with increasing $r_p$, and the tail is completely fitted by triples
	since there are no binaries above ratio $1.41$ in Newtonian gravity, or 1.6 in MOND.
	We therefore modified the fitting to
	include an additional population of "flyby" systems as in Section~\ref{sec:flyby},
	which naturally tends to increase with $r_p$.

	This leads to an issue in that the triple and flyby populations are somewhat degenerate with each other since they
	both produce similar shapes in the "tail", so we then fixed the triple/binary population ratio at a constant value;
	we explored fits with triple/binary ratio set to 0.5, 0.8 or 1.0, and found that 1.0 is marginally preferred, so
	results below are for this value.
	We then repeat the fitting for each separation bin with two gravity models and three
	eccentricity distributions, for a total of $2 \times 3 \times 4$ independent fits.
	Each fit has only two adjustable parameters,
	the total number of pure binary systems and the total number of flyby systems;
	we hold the ratio of triple to binary systems fixed at a selected constant value, here 1.0.  The shape of
	each of those is constrained at the output of the simulated orbits, for the corresponding gravity model and
	eccentricity distribution. We then simply fit the number in each population
	to minimise the $\chi^2$ residuals between binary+triple+flyby
	model and the data.

	The fitting range is $0 \le \vtilde \le (5,6,7,7)$ respectively in the four $r_p$ bins; these upper limits
	are chosen because our $3 \kms$ velocity
	cutoff  leads to some incompleteness for higher-mass binaries at smaller $r_p \sim 5 \kau$ and high $\vtilde \ge 5$. At our
	90th percentile system mass of $2.02 \msun$, $3 \kms $ maps to velocity ratio $\vtilde = 5.0 \sqrt(r_p/5 \kau)$
	so our $3 \kms$ threshold includes nearly all systems up to the above $\vtilde$ limits.

	The results of the fitting procedure are shown in Figures~\ref{fig:3BodyNewtonFlat} - \ref{fig:3BodyBanik2Ecc}  below.
	Here Figures~\ref{fig:3BodyNewtonFlat} - \ref{fig:3BodyNewton2Ecc} show the fits with Newtonian
	gravity and the three simulated eccentricity distributions (flat, Tokovinin, $2e$) respectively,
	and Figures~\ref{fig:3BodyBanikFlat} - \ref{fig:3BodyBanik2Ecc} show the
	corresponding results for the  specific modified-gravity model described in Sec.~\ref{sec:orbsmg},
	and the same three eccentricity distributions.

	Some selected values from the fitting results are listed in Tables~\ref{tab:3BodyNewtonFlat} - \ref{tab:3BodyBanik2Ecc};
	for each separation bin and model, this lists the observed numbers of
	EDR3 systems and fitted number of binary, triple and flyby systems, in three selected ranges of $\tilde{v}:$
	firstly $ 0 \le \tilde{v} < 7$, secondly $0 \le \tilde{v} \le 1.4$, and also $1.1 \le \tilde{v} \le 1.4$,
	with the latter range being particularly sensitive to modified-gravity effects.
	\begin{wscomm}
		New chunk above introducing fits
	\end{wscomm}

	\begin{figure*}
		\begin{center}
			\includegraphics[width=0.8\linewidth]{Fit_NewtonFlat_3BodyNewton_InOutFlat.png}
			\caption{ Fits of velocity ratios using the Binaries+Triples+FlyBys model for Newtonian gravity and
				all orbits drawn from a flat eccentricity distribution.  The four panels show four bins of projected separation as labelled.
				The red histogram shows the CWB-EDR3 data. The blue-dotted line is the fitted triple population; black half-dashed line are the FlyBys; green dashed line is the total, so the difference is fitted true binaries.
				\label{fig:3BodyNewtonFlat}   }
			\includegraphics[width=0.8\linewidth]{Fit_NewtonTok_3BodyNewton_InOutTok.png}
			\caption{ Same as Figure~\ref{fig:3BodyNewtonFlat}, except that model orbits are drawn from a Tokovinin eccentricity distribution. \label{fig:3BodyNewtonTok} }
		\end{center}
		\begin{center}
			\includegraphics[width=0.8\linewidth]{Fit_Newton2Ecc_3BodyNewton_InOut2Ecc.png}
			\caption{Same as Figure~\ref{fig:3BodyNewtonFlat} with Newtonian gravity, but model orbits are drawn from a $f(e) = 2e$ eccentricity distribution.  \label{fig:3BodyNewton2Ecc}  }
		\end{center}
	\end{figure*}

	\begin{figure*}
		\begin{center}
			\includegraphics[width=0.8\linewidth]{Fit_BanikFlat_3BodyBanik_InOutFlat.png}
			\caption{ Same as Figure~\ref{fig:3BodyNewtonFlat}, but using the realistic MOND gravity model of Sec.~\ref{sec:orbsmg} and all orbits drawn from a flat eccentricity distribution.
			}
			\label{fig:3BodyBanikFlat}
		\end{center}

		\begin{center}
			\includegraphics[width=0.8\linewidth]{Fit_BanikTok_3BodyBanik_InOutTok.png}
			\caption{ Same as Figure~\ref{fig:3BodyBanikFlat}, with realistic MOND gravity model and model orbits drawn from a Tokovinin eccentricity distribution.
			}
			\label{fig:3BodyBanikTok}
			\includegraphics[width=0.8\linewidth]{Fit_Banik2Ecc_3BodyBanik_InOut2Ecc.png}
			\caption{ Same as Figure~\ref{fig:3BodyBanikFlat}, with realistic MOND gravity model and model orbits drawn from
				an $f(e) = 2e$ distribution.
			}
			\label{fig:3BodyBanik2Ecc}
		\end{center}
	\end{figure*}

	The overall goodness of fit $\chi^2$ values are listed in the Tables and plotted versus projected separation bin
	in Figure~\ref{fig:ChiSqrdFit}; since the bins are width 0.1 and $\vtilde$ upper limits are respectively 5,6,7,7, there
	are respectively 50, 60, 70, 70 datapoints in the four separation bins.
	The result seen in Figure~\ref{fig:ChiSqrdFit} is that all four separation bins show a common ordering, with
	Newtonian gravity and $2e$ eccentricity distribution giving the best fit,, and MOND with flat $f(e)$ giving the worst.
	Interestingly, all three Newtonian fits give a lower $\chi^2$ value than the best of the MOND models (also $f(e) = 2e$).
	Therefore, since our three eccentricity models span the range expected for any reasonably smooth distribution,
	the fact that we do not know the true eccentricity distribution seems not to be a serious problem for the future development
	of this test.

	On inspection of the data and model histograms in Figs~\ref{fig:3BodyNewtonFlat} - \ref{fig:3BodyBanik2Ecc},
	it is clear that a substantial part of the difference is due to the region around velocity ratios $1.0 \le \tilde{v} \le 1.5$,
	where all of the MOND fits overshoot the observed data values by a significant margin.  This region was
	already highlighted as a key discriminant in \citetalias{Pittordis_2018}, since MOND predicts a substantially
	higher fraction of true binaries in this range compared to Newtonian gravity. Therefore, the predicted "MOND excess" in this
	range appears {\bf not to be present in the data}, which will become a serious problem for MOND {\bf if} our modelling of
	the triple + flyby populations proves realistic. We are not yet confident in our modelling of the triple + flyby or other
	populations, but hope to improve this in future work.

	The formal significance of the preference for Newtonian gravity over MOND is very high, with the best
	Newtonian model ($2e$) having $\Delta\chi^2 \simeq -182, -146, -69, -47$
	respectively in the four bins compared to the best  MOND model (also $2e$);  the absolute $\chi^2$ values are acceptable for our best Newtonian model in the two wider bins,	but not in the two narrower bins, so the formal significance should not be overstated.

	The major caveat here is that our triple model is only a single realisation for each eccentricity distribution: for all other parameters we have taken a principle of "maximum randomness" with uncorrelated masses, $e$ values, inclinations and orbit ratios between inner and outer orbits.  If the real triple population deviates from this simple assumption, then the predictions for the velocity distribution in triples will vary accordingly, and this will change the fit results.  However, we note that in order to improve the MOND fits, it is hard to change the shape of the model distribution for the pure binaries;
	for pure binaries, assuming random phases and inclinations the model distribution is entirely determined given the $f(e)$ distribution function,  and we have already explored reasonable limiting cases for this.  Therefore, to improve the MOND fits we would need the triple+flyby models to produce fewer systems than our model in the range $1.0 \le \tilde{v} \le 1.5$, while maintaining the shape in the tail at $\tilde{v} > 1.5$; this may be challenging, but remains to be explored.

	One other concern from a referee was that the apparent-magnitude limit of our sample leads to a distance-dependent
	mass limit in our selected systems; to leading order this should cancel since the estimated masses are included
	in the definition of velocity ratio. To address this more directly, we constructed a volume-limited subsample
	by discarding binaries where either star has absolute magnitude $M_G > 9.61$, corresponding to apparent $G > 17$ at 300 pc;
	results for this volume-limited subsample are shown in Appendix~\ref{sec:vollim}, and are qualitatively similar to the full
	sample.

	\begin{figure*}
		\begin{center}
			\includegraphics[width=\linewidth]{ChiSqrdFittingModels.png}
			\caption{ Plot comparing the $\chi^2$ fitting results for each of the six orbit models (lines labelled in legend) against the four  bins in projected separation bin $r_p$.  \label{fig:ChiSqrdFit} }
		\end{center}
	\end{figure*}

	\begin{table*}

		\caption{Number of candidate binaries in selected ranges of
			projected separation and velocity ratio: data, and model fits for
			combined Binary (B), Triples (T) and FlyBys (F) populations,
			for Newtonian gravity and Flat eccentricity distribution.  Rows are range of projected separation, as
			in Column 1.    Columns 2-4 are for all velocity ratios,
			columns 5-7 for ratios $< 1.4$, and columns 8-10 for
			ratios between $1.1$ and $1.4$. Rightmost column 11 is the $\chi^2$ value of the fit.}
		\label{tab:3BodyNewtonFlat}
		\centering
		\begin{tabular}{l|rrrr|rrrr|rrrr|r}
			\phantom{        } &  \multicolumn{4}{|c|}{$\vratio < 7$} &
			\multicolumn{4}{|c|}{$\vratio < 1.4$} &
			\phantom{        } &
			\multicolumn{3}{|c|}{$1.1 < \vratio < 1.4$}
			\phantom{        } & \\
			\hline
			$\rp$ range  &   Data & Fit(B) & Fit(T) & Fit(F)  & Data & Fit(B) & Fit(T) & Fit(F) & Data & Fit(B) & Fit(T) & Fit(F) & $\chi^2$ \\
			\hline

			$5-7.1 \kau$   & 3055 & 1359.18 & 1359.18 & 358.91 & 2251 & 1359.18 & 809.72 & 142.15 & 276 & 113.63 & 185.15 & 33.26 & 217.24 \\
			$7.1-10 \kau$  & 2471 & 1000.88 & 1000.88 & 496.65 & 1655 & 1000.88 & 561.82 & 162.14 & 222 & 74.94  & 128.91 & 42.21 & 181.45 \\
			$10-14.1 \kau$ & 1894 & 756.34  & 756.34  & 375.27 & 1235 & 756.34  & 401.75 & 106.71 & 190 & 59.09  & 93.01  & 29.32 & 113.82 \\
			$14.1-20 \kau$ & 1643 & 596.12  & 596.12  & 411.86 & 981  & 596.12  & 310.11 & 103.16 & 162 & 50.39  & 73.34  & 32.82 & 95.13 \\
			\hline
		\end{tabular}

		\caption{Same as Table~\ref{tab:3BodyNewtonFlat}, but for model orbits drawn from Tokovinin eccentricity distribution. }
		\label{tab:3BodyNewtonTok}
		\begin{tabular}{l|rrrr|rrrr|rrrr|r}
			\phantom{        } &  \multicolumn{4}{|c|}{$\vratio < 7$} &
			\multicolumn{4}{|c|}{$\vratio < 1.4$} &
			\phantom{        } &
			\multicolumn{3}{|c|}{$1.1 < \vratio < 1.4$}
			\phantom{        } & \\
			\hline
			$\rp$ range  &   Data & Fit(B) & Fit(T) & Fit(F)  & Data & Fit(B) & Fit(T) & Fit(F) & Data & Fit(B) & Fit(T) & Fit(F) & $\chi^2$ \\
			\hline

			$5-7.1 \kau$   & 3055 & 1335.01 & 1335.01 & 363.40 & 2251 & 1335.01 & 804.63 & 143.93 & 276 & 108.42 & 175.40 & 33.68 & 128.93 \\
			$7.1-10 \kau$  & 2471 & 983.39  & 983.39  & 503.95 & 1655 & 983.39  & 557.20 & 164.52 & 222 & 79.69  & 124.89 & 42.83 & 125.67 \\
			$10-14.1 \kau$ & 1894 & 742.48  & 742.48  & 389.47 & 1235 & 742.48  & 399.55 & 110.75 & 190 & 64.25  & 90.63  & 30.43 & 86.26  \\
			$14.1-20 \kau$ & 1643 & 582.64  & 582.64  & 428.38 & 981  & 582.64  & 310.53 & 107.30 & 162 & 47.85  & 71.16  & 34.13 & 73.45 \\
			\hline
		\end{tabular}

		\caption{Same as Table~\ref{tab:3BodyNewtonFlat}, but model orbits drawn from $2e$ eccentricity distribution. }
		\label{tab:3BodyNewton2Ecc}
		\begin{tabular}{l|rrrr|rrrr|rrrr|r}
			\phantom{        } &  \multicolumn{4}{|c|}{$\vratio < 7$} &
			\multicolumn{4}{|c|}{$\vratio < 1.4$} &
			\phantom{        } &
			\multicolumn{3}{|c|}{$1.1 < \vratio < 1.4$}
			\phantom{        } & \\
			\hline
			$\rp$ range  &   Data & Fit(B) & Fit(T) & Fit(F)  & Data & Fit(B) & Fit(T) & Fit(F) & Data & Fit(B) & Fit(T) & Fit(F) & $\chi^2$ \\
			\hline

			$5-7.1 \kau$   & 3055 & 1316.35 & 1316.35 & 381.73 & 2251 & 1316.35 & 807.82 & 151.19 & 276 & 119.27 & 172.12 & 35.38 & 90.62 \\
			$7.1-10 \kau$  & 2471 & 960.18  & 960.18  & 535.86 & 1655 & 960.18  & 559.54 & 174.94 & 222 & 78.63  & 121.39 & 45.54 & 95.91 \\
			$10-14.1 \kau$ & 1894 & 727.53  & 727.53  & 409.39 & 1235 & 727.53  & 401.16 & 116.42 & 190 & 62.06  & 89.18  & 31.99 & 64.91 \\
			$14.1-20 \kau$ & 1643 & 571.99  & 571.99  & 441.07 & 981  & 571.99  & 309.06 & 110.48 & 162 & 47.53  & 68.25  & 35.14 & 54.61 \\
			\hline
		\end{tabular}
	\end{table*}

	\begin{table*}
		\caption{As Table~\ref{tab:3BodyNewtonFlat}, but for realistic MOND gravity model
			and all orbits drawn from a flat eccentricity distribution.  Table shows number of candidate binaries in selected ranges of
			projected separation and velocity ratio: data, and model fits for Binary(B), Triples (T) and FlyBys (F) populations.
			Rows are range of projected separation, as
			in Column 1.    Columns 2-4 are for all velocity ratios,
			columns 5-7 for ratios $< 1.4$, and columns 8-10 for
			ratios between $1.1$ and $1.4$. }
		\label{tab:3BodyBanikFlat}
		\centering
		\begin{tabular}{l|rrrr|rrrr|rrrr|r}
			\phantom{        } &  \multicolumn{4}{|c|}{$\vratio < 7$} &
			\multicolumn{4}{|c|}{$\vratio < 1.4$} &
			\phantom{        } &
			\multicolumn{3}{|c|}{$1.1 < \vratio < 1.4$}
			\phantom{        } & \\
			\hline
			$\rp$ range  &   Data & Fit(B) & Fit(T) & Fit(F)  & Data & Fit(B) & Fit(T) & Fit(F) & Data & Fit(B) & Fit(T) & Fit(F) & $\chi^2$ \\
			\hline

			$5-7.1 \kau$   & 3055 & 1432.17 & 1432.17 & 324.17 & 2251 & 1403.88 & 821.73 & 128.39 & 276 & 284.35 & 210.41 & 30.04 & 440.28 \\
			$7.1-10 \kau$  & 2471 & 1080.01 & 1080.01 & 441.05 & 1655 & 1058.38 & 578.80 & 143.98 & 222 & 246.91 & 156.96 & 37.48 & 388.00 \\
			$10-14.1 \kau$ & 1894 & 816.03  & 816.03  & 323.16 & 1235 & 794.93  & 412.01 & 91.90  & 190 & 193.42 & 115.20 & 25.25 & 252.31 \\
			$14.1-20 \kau$ & 1643 & 651.20  & 651.20  & 347.80 & 981  & 634.78  & 318.44 & 87.12  & 162 & 145.36 & 90.29  & 27.71 & 191.88 \\
			\hline
		\end{tabular}

		\caption{Same as Table~\ref{tab:3BodyBanikFlat}, with realistic MOND gravity model, but model orbits
			drawn from Tokovinin eccentricity distribution. }
		\label{tab:3BodyBanikTok}
		\begin{tabular}{l|rrrr|rrrr|rrrr|r}
			\phantom{        } &  \multicolumn{4}{|c|}{$\vratio < 7$} &
			\multicolumn{4}{|c|}{$\vratio < 1.4$} &
			\phantom{        } &
			\multicolumn{3}{|c|}{$1.1 < \vratio < 1.4$}
			\phantom{        } & \\
			\hline
			$\rp$ range  &   Data & Fit(B) & Fit(T) & Fit(F)  & Data & Fit(B) & Fit(T) & Fit(F) & Data & Fit(B) & Fit(T) & Fit(F) & $\chi^2$ \\
			\hline

			$5-7.1 \kau$   & 3055 & 1407.09 & 1407.09 & 309.72 & 2251 & 1367.90 & 824.92 & 122.67 & 276 & 244.63 & 203.49 & 28.70 & 310.78 \\
			$7.1-10 \kau$  & 2471 & 1053.24 & 1053.24 & 445.39 & 1655 & 1028.03 & 573.56 & 145.40 & 222 & 211.76 & 148.49 & 37.85 & 290.38 \\
			$10-14.1 \kau$ & 1894 & 793.60  & 793.60  & 326.06 & 1235 & 769.61  & 406.41 & 92.72  & 190 & 157.91 & 107.49 & 25.48 & 168.44 \\
			$14.1-20 \kau$ & 1643 & 630.18  & 630.18  & 361.62 & 981  & 611.93  & 315.44 & 90.58  & 162 & 126.55 & 84.90  & 28.81 & 134.53 \\
			\hline
		\end{tabular}

		\caption{Same as Table~\ref{tab:3BodyBanikFlat} with MOND gravity model, but model orbits drawn from $f(e) = 2e$ eccentricity distribution.  }
		\label{tab:3BodyBanik2Ecc}
		\begin{tabular}{l|rrrr|rrrr|rrrr|r}
			\phantom{        } &  \multicolumn{4}{|c|}{$\vratio < 7$} &
			\multicolumn{4}{|c|}{$\vratio < 1.4$} &
			\phantom{        } &
			\multicolumn{3}{|c|}{$1.1 < \vratio < 1.4$}
			\phantom{        } & \\
			\hline
			$\rp$ range  &   Data & Fit(B) & Fit(T) & Fit(F)  & Data & Fit(B) & Fit(T) & Fit(F) & Data & Fit(B) & Fit(T) & Fit(F) & $\chi^2$ \\
			\hline

			$5-7.1 \kau$   & 3055 & 1388.33 & 1388.33 & 328.25 & 2251 & 1343.12 & 828.50 & 130.01 & 276 & 236.31 & 202.87 & 30.42 & 273.36 \\
			$7.1-10 \kau$  & 2471 & 1030.85 & 1030.85 & 466.33 & 1655 & 997.42  & 573.73 & 152.24 & 222 & 195.67 & 142.85 & 39.63 & 242.15 \\
			$10-14.1 \kau$ & 1894 & 780.61  & 780.61  & 336.02 & 1235 & 750.77  & 407.54 & 95.55  & 190 & 139.01 & 104.81 & 26.25 & 134.96 \\
			$14.1-20 \kau$ & 1643 & 613.04  & 613.04  & 380.43 & 981  & 595.11  & 312.86 & 95.29  & 162 & 102.92 & 81.47  & 30.31 & 102.47 \\
			\hline
		\end{tabular}
	\end{table*}

	\section{Conclusions}
	\label{sec:conc}
	Following on from previous modelling work in PS18, and our observational study from GAIA DR2 in PS19, we have selected a new sample of candidate wide binaries from the recent GAIA Early Data Release 3 (EDR3); our cleaned sample contains 73,159 candidate binaries with distance $\le 300 \pc$, projected separation $\le 50 \kau$ and projected velocity difference $\le 3 \kms$.
	Comparisons with our previous PS19 sample and the independent binary sample of El-Badry et al (2021) show generally good consistency, after allowing for the different selection criteria used by El-Badry et al.
	We then estimate masses for each component using a main-sequence mass/luminosity relation, and our key observable is the projected velocity difference in each binary, from the GAIA proper motions assuming a common distance,
	divided by the velocity for a circular orbit at the observed projected separation;
	the statistics of this ratio can be reliably predicted from simulations, assuming random inclinations and orbital phases, given
	an assumed eccentricity distribution.

	We have then focused on the subset of 9063 candidate wide binaries with projected separations between $5 \kau$ to $20 \kau$, the most promising range for testing modified gravity. In agreement with PS19, the distributions show a clear "hump plus tail" shape, with a hump consistent with pure binaries and a tail to much larger velocity ratios.
	Dividing the range into four separation bins, we then fit each observed distribution as an arbitrary mixture of binary plus triple populations; we repeat the fitting for Newtonian gravity and one specific MOND model, and three assumed $f(e)$ eccentricity distributions.
	The fitting results show a clear preference for Newtonian gravity over MOND, with a high formal significance; this is
	largely because the MOND model fits overshoot the observed number of systems in the range $1.0 \le \tilde{v} \le 1.5$.
	Varying the model eccentricity distributions has little effect on this preference for standard gravity. However, this conclusion is only tentative at present due to using a somewhat over-simplified maximally-random model for the triple population. Further studies are clearly needed with a larger variety of models for the triple and flyby populations.

	The prospects for future development of this test seem to be very good: GAIA EDR3 used only the first 33 months of GAIA survey data, while there are 7.5 years existing at present, and projected GAIA operations extend to 10 years, so the precision will substantially improve and allow even larger samples in future. This will be helpful,  but the statistical uncertainties seem to be already sub-dominant now; the major uncertainty at present appears to be in the modelling and fitting of the presumed triple and flyby population contributing the tail of high-velocity systems.

	There are interesting prospects for observationally detecting or constraining the triples, including for example: searches for faint companions in deep sky surveys; adaptive-optics or speckle imaging to directly resolve inner binaries in triples; searches for astrometric accelerations or excess residuals with future GAIA data; and high-precision radial velocities from ground-based data, including possible time-dependence.  We plan to expand on these possibilities in future work.  Such programs may require substantial amounts of telescope time to build up good statistics of triples and other contaminants, but has a potentially large payoff in a decisive test of acceleration-based modified gravity models similar to MOND.

	\section*{Acknowledgements}
	We thank an anonymous referee for comments which significantly improved the paper.
	We thank Indranil Banik for many helpful discussions, and
	Zizhen Lin for contributions to the triple-system simulations.
	CP has been supported by Colin Lee, Gary Barr, and Adel Szabo Malcolm (Legal \& General Investment Management), Peter Jackson (Exasol) and Konstantin Orfenov (BNP Paribas).

	\section*{DATA AVAILABILITY}
	The original data used in this paper is publicly available.
	The Gaia data can be retrieved through the Gaia archive
	(\url{https://gea.esac.esa.int/archive}). The binary catalogue CWB-EDR3
	will be made available on \url{https://zenodo.org} on acceptance of this paper.

	\bibliographystyle{mnras}

redpajama_set_name: RedPajamaArXiv


text:
Etsu Nupe Calls For Improved Welfare Of Workers
News, Press Release
Winifred Oyo-Ita
HoS of The Federation
Mohammed Manga; Abuja
His Royal Highness, the Emir of Bida, Alhaji (Dr) Yahaya Abubakar has advised officials of the National Council on Establishment (NCE) to ensure that they take decisions that will help improve the welfare of workers across the country to boost their morale for efficient service delivery to Nigerians.
The Emir made the call yesterday at his palace in Bida when officials of the technical session of the National Council on Establishment (NCE) lead by the Permanent Secretary, Service Policy and Strategy Office in the Office of the Head of the Civil Service of the Federation, Hajiya Nuratu Batagarawa visited him.
He said that though the present economic realities facing the country are so challenging and may serve as a setback to improving the welfare of workers, the issue according to him can easily be addressed through the proper adoption of information communication technology (ICT) techniques in the Public Service to acquire the necessary data base of all Public Servants in the country so as to adequately cater for their welfare.
He expressed worry over how government's scarce resources are often diverted by few individuals rather than using it for capital projects which will enhance the well-being of citizens of Nigeria.
The Etsu Nupe reminded the officials of the fact that Public Service is a trust and must be seen as so by all Nigerians particularly those in authority.
Earlier, the Permanent Secretary, Service Policy and Strategy Office in the Office of the Head of the Civil Service of the Federation, Hajiya Nuratu Batagarawa said that the visit was part of the familiarisation tour of some tourist sites in Niger State by the officials and other delegates of the council with the view to further exploring ways of ensuring that such tourist sites are best put to public use for improved social and economic development of the country.
Posted by Abdul Jimoh
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redpajama_set_name: RedPajamaCommonCrawl


text:
package helloworld.behavioral.chain_of_responsibility;

import org.junit.Test;
import static org.hamcrest.MatcherAssert.assertThat;
import static org.hamcrest.Matchers.is;

/**
 * @author yihua.huang@dianping.com
 */
public class HelloWorldHandlerTest {

    @Test
    public void testHelloWorldHandler(){
        HelloWorldHandler helloWorldChainOfResponsibility = new HelloWorldInterjectionHandler().setNext(new HelloWorldObjectHandler());
        assertThat(helloWorldChainOfResponsibility.helloWorld(),is("Hello Chain of Responsibility!"));
    }
}

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Travel Inspiration From
Tennesse Dept. of Tourist Development
Romantic Spaces in East Tennessee Historic Places
February is the month for love and East Tennessee combines history and romance for one powerful elixir.
February is the month for love and East Tennessee combines history and romance for one powerful elixir that entices couples to return year after year. Let your favorite historic romance novel come alive at an inn or hotel located in these cities and towns that are brimming with charm. After all, romance like this can only be Made in Tennessee.
The Oliver Hotel in Knoxville
A romantic stay at Oliver Hotel in Knoxville, Tennessee
In 1876, German immigrant Peter Kern built a three-story building in the heart of downtown Knoxville to house his bakery, confectionery creations, and a venue for special events. The building is now The OIiver Hotel and is listed on the Register of Historic Places. The boutique hotel is idea for a romantic getaway.
The Oliver is an original expression of the city, and celebrates its atmosphere with chic, clean, Southern designs that pay subtle homage to the location's rich local culture. The 28 elegant rooms feature handcrafted furniture, original local artwork and amenities that makes for a luxurious experience. Downstairs is the Peter Kern Library, now a popular "watering hole" for guests and visitors to the city.
The hotel is on the corner of Market Square, a popular destination for dining and shopping. Its location is ideal for romantic strolls through Krutch Park and dining in award-winning restaurants such as Tupelo Honey Café and Knox Mason. Live entertainment occurs at the Tennessee or Bijou theaters just a block away.
During the day, bird-eye views of the city from the observation level of the Sunsphere can be enjoyed. If you are looking for inspiration and learning, the East Tennessee History Center and many galleries are also nearby. The shopping experiences range from the country feel of Mast General Store to the more trendy places such as Urban Outfitters. And do not forget to take home a Tennessee souvenir from Nothing Too Fancy which is in proximity.
Hale Springs Inn
Romantic charm and deep history entices guests to Hale Springs Inn. Photo credit: Hale Springs Inn
The Hale Springs Inn in downtown Rogersville is not only a place with romantic charm, but historic appeal as well. Built in 1824, this three-story Federal style tavern and inn played host to many distinguished guests like United States presidents Andrew Jackson, Andrew Johnson, and James K. Polk. Jackson particularly loved the balcony over the inn's front porch, where he stood to address the townspeople.
The inn underwent a multi-million dollar renovation and was fully restored in 2008 to its 1820-30s glory days. The bed and breakfast has three presidential suites all named by its previous guests and six additional larger rooms, filled with furniture from the historic period. There is a bar and upscale dining downstairs at McKinney's Restaurant & Tavern, named after the inn's original owner, John McKinney.
Rogersville is a treasure of sites with historic significance. The Hawkins County Courthouse's brick columns and paladin windows over the front door, is the oldest original courthouse still in use in the state. The first Tennessee newspaper, the Knoxville Gazette, was printed in Rogersville in 1791. The Tennessee Newspaper and Printing Museum is open in the old Southern Railway Train Depot nearby.
Rogersville is 90 minutes north of Knoxville and only 30 miles south of Kingsport on Hwy 11W and is also a designated stop along the Sunny Side Trail, one of 16 self-driving trails throughout Tennessee.
General Morgan Inn & Conference Center
The opulence pulls you in at the General Morgan Inn for a wonderful stay.
One of my personal favorite getaways is to downtown Greeneville and the historic General Morgan Inn. The hotel is named after General John H. Morgan who lost his life after fleeing from the Dickson-Williams Mansion located behind the property.
The General Morgan's close proximity to Knoxville off I-81 means you spend less time traveling and more time relaxing and sightseeing. The lobby is grand without being intimidating and the rooms provide a place of solitude. Plush bedding and comfortable furniture decorate the rooms. The view of downtown is classic America.
Plan to eat downstairs in Brumley's Restaurant and Bar, named after the Brumley family who operated the then Hotel Brumley from the 1920's to 1980's. Their seasonal menu of Southern American cuisine comes with a creative flair.
Greeneville is also the home of President Andrew Johnson, so plan to visit the Andrew Johnson National Historic Site and the National Cemetery.
The Read House Hotel
Soaring columns and lavish adornments can be found at the Read House in Chattanooga.
There are many architectural landmarks in downtown Chattanooga. One is The Read House Historic Inn and Suites, built in 1926. This 10-story brick and terracotta building designed in the Georgian style was built with lavish adornments now too costly to duplicate: terrazzo floors inlaid with marble; paneling of quarter sawed black walnut, carved and gilded woodwork, mirrors recessed in massive arches; and a lobby beautifully defined by its soaring columns. The Read House is listed on the National Register of Historic Places as a prime example of period architecture and decorative art.
The Read House is continually updated with today's modern conveniences while preserving the ambience of days gone by. I have enjoyed my stay in both the traditional rooms and the suites, both providing a luxurious seclusion with plush bedding and heritage style furniture. Club Level Rooms and Presidential Suites are also available.
Be sure and eat at their on-site restaurant, Porter's Steakhouse, which has earned the Best of the Best Award by readers of the Chattanooga Time Free Press six years running. Walk through downtown Chattanooga for a cup of coffee at Mean Mug, Cadence Coffee Company, or conveniently walk to The Read House lobby where Chattanooga's only downtown Starbucks is located.
The Read House is right downtown and within walking distance to the Tennessee Aquarium, Chattanooga Market, and scores of restaurants that make the city a top destination for a romantic retreat. The city also provides a free shuttle that runs right in front of the hotel.
Places in this article
Tupelo Honey Cafe
Knox Mason
In the history-rich 100 block of Knoxville's Gay Street, Knox Mason is a hub for foodie fervor and home to a celebrated chef.
Bijou Theatre
The Bijou Theatre is one of East Tennessee's oldest and most revered performance venues
East Tennessee History Center
Find your story at the East Tennessee History Center!
Visit one of Tennessee's most treasured historic relics!
Andrew Johnson National Historic Site
Andrew Johnson served as our Nation's 17th President, tasked with restoring the Nation after the American Civil War. Visit the places that shaped him on his path to the Presidency.
Located in the heart of downtown Chattanooga on the beautiful riverfront, the Tennessee Aquarium is the top-rated aquarium for guest satisfaction.
Desserts & Bakeries
Inspiration just for you
We have a hunch these experiences might be right up your alley.
Tennessee Food Festivals That Leave Mouths Watering
When the leaves are ablaze, forks around Tennessee are raised to endlessly eat around the state. Take a bite out of the Volunteer State.
Some of the Best Waterfalls on Tennessee's Cumberland Plateau
Tennessee offers some of the best waterfalls (and swimming holes) in the country.
7 Places to Take the Kids This Spring Break in West Tennessee
Stories of music legends, untouched nature and educational fun can all be found in West Tennessee.
Waterfalls in East Tennessee Just Wait to Be Discovered
Hike or drive to these Made in (East) Tennessee waterfalls.
Tennessee's Rural Roads: Home to the Appalachian Quilt Trail
The stories of Tennessee's rural communities can be found along The Appalachian Quilt Trail.


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He Was Caught Trying To Join ISIS, Now He's In Jihadi Rehab
By Dina Temple-Raston
Protesters gathered outside the federal courthouse in Minneapolis where three young Somali-Americans are on trial for allegedly planning to go to Syria to join the Islamic State. The demonstrators say the FBI and local law enforcement is targeting and entrapping Somali-Americans in terrorism cases. Six men have already pleaded guilty.
All eyes were on 20-year-old Abdullahi Yusuf when he stepped on the stand in federal court last week in Minneapolis to testify for the prosecution in America's largest ISIS recruitment trial.
As Yusuf began to speak, his words provided a rare, behind-the-scenes look at how more than a dozen young men convinced themselves that the way to prove they were good Muslims was to travel to Syria and fight for ISIS.
The radicalization process in this case was frightening simple. It began with a meal at a Somali restaurant and a pickup game of basketball. Then, he testified, smart phones came out of their pockets and the young men began watching ISIS videos on a YouTube channel called "Enter the Truth."
They stayed up until 2 a.m., Yusuf said, talking about the killing in Syria and watching ISIS videos.
"The channel is saying that the West is corrupt and you can follow religion perfectly over there [in Syria]," Yusuf told the court. "And you don't have to conform to anyone else's traditions."
Yusuf said he had never seen videos like that before and when he got home that night, he stayed up for another couple of hours watching the videos on his own.
In retrospect, Yusuf said that it was that Friday night in 2014 that his ISIS recruitment began.
Testifying Against His 'Former Friends'
He testified he was introduced to radical Islam by the three men now on trial, Guled Omar, 22, Abdirahman Daud, 22, and Mohamed Farah, 21.
"My former friends," he called them. Asked why he was testifying against them, Yusuf stopped for a moment and said, "Because I realize what I did was wrong ... and I am thinking for myself for once."
Yusuf is not only a key witness in this case. He's also the charter member of America's first jihadi rehabilitation program. He was arrested last year, months after the FBI stopped him from boarding a flight that would eventually have taken him to ISIS in Syria.
He pleaded guilty in February to providing material support – essentially himself, as a fighter – to a terrorist organization. While he awaits sentencing, he is getting counseling so he can better understand what lead him to embrace ISIS.
His counselor is a local high school social studies teacher named Ahmed Amin. Amin was born in Somalia and his family moved to the U.S. when he was 12. His English is unaccented. He wears designer shirts. He the kind of teacher with the creative lesson plans who holds classes outside on sunny days.
He teaches at Roosevelt High School during the day and began working with a Minneapolis group called Heartland Democracy that has created a counseling program for young men like Yusuf.
"I understand the difficulties of identity that lead people to join organizations like ISIS," Amin told NPR in an interview. "It is hard trying to live in two worlds. From 9 to 5 these kids have to live one way when they are at school, they are socialized to be American. And then they go home, learn to be religious and are trying to cope with that. It is harder than you'd think."
A Lengthy Reading List
Abdullahi Yusuf was just 18 years old when he decided he would go to Syria to fight with ISIS. Amin is trying to help him understand why he made that choice.
"Initially, I wanted him to read some texts that would get him to really question his own personal views, to question his understanding of the world," said Amin. "One of the first things I had him read was Sherman Alexie's Reservation Blues."
The 1995 novel is about the difficulties of navigating two worlds at once, in this case, the life of an American Indian off the reservation. Amin also had Yusuf writing papers about French philosopher Michel Foucault's ideas about prisons and power and society.
"I'm introducing him to many different texts that could really challenge how he thinks and sees the world," Amin said.
"We try to give him as much work that gets him to be critical in hopes of finding out what really led him to do this," Amin added. "I think that's the first step, for him to really have an answer as to why he did what he did."
In his testimony, Yusuf said the three defendants in the trial helped goad their friends into joining a complicated plan to go to Syria.
Commotion In The Courtroom
This ISIS case has rocked Minneapolis. The courtroom gallery erupted last week during Yusuf's testimony after some young Somalis on the back benches told Yusuf's mother that her son was a spy.
U.S. District Court Judge Michael J. Davis ended up throwing some people out of the courtroom to restore order. Yusuf was subjected to blistering cross-examination on Monday as defense attorneys claimed there were inconsistencies in his story.
The prosecution plans to call two other young men who say they were involved in the alleged conspiracy. One of them backed out of the plan, told the FBI about the plot, and became a confidential informant.
Defense attorneys say the FBI used him to entrap their clients and that has sparked protests outside the courthouse.
"Somali youth are under attack," one chant begins. "Stand up, fight back."
Because all the men involved in the case are all in their early 20s, there has been a push to develop a program that helps young men like Yusuf who have pleaded guilty to terrorism charges.
Amin says watching this happen in Minneapolis is heartbreaking and brings back painful memories. The Twin Cities struggled with young men from the Somali community leaving to join al-Qaeda's arm in Somalia, al-Shabab, nearly 10 years ago.
"You know, these are really young kids," Amin said, "and in my heart I really believe that they fell for something. They need a chance to correct, to undo, what they did."
Judge Davis has appointed a German radicalization expert, Daniel Koehler, to assess the men who have already pleaded guilty in the case. Koehler runs the German Institute on Radicalization and de-Radicalization Studies. He wouldn't discuss the individual assessments he's made of the young men in Minneapolis, though he did tell NPR that he thought that some of the young men who pleaded guilty, though not all of them, could be helped with counseling.
Koehler is in negotiations with six states, including Minnesota, to set up de-radicalization programs.
"There's a real desire to figure something out to fix this," he said.
Corrected: May 16, 2016 at 12:00 AM EDT
Previous audio and Web versions of this story misidentified one of the men on trial as Adnan Farah. The defendant is Mohamed Farah.
Dina Temple-Raston
Dina Temple-Raston is a correspondent on NPR's Investigations team focusing on breaking news stories and national security, technology and social justice.
See stories by Dina Temple-Raston


Today cybersecurity researchers have published a detailed way about how can you scam the scammers that are running phishing operation in which they pretend to be Virgin Media support on Twitter.
Same cybersecurity researchers are saying that this particular type of phishing cyber attack is relatively new on the cyberspace.
It starts with a scammer registering multiple fake Twitter accounts that are used to impersonate legitimate customer support. Then the scammer starts monitoring the real support accounts for irate customer messages.
When bad customer messages arrive, a fake Twitter account quickly starts to exploit the customer's frustration by sending messages back to them, which are typically loaded with malicious links.
Sadly and happily at the same time, this is what happened to a member of the cyber security team when he complained to Virgin Media via Twitter.
Immediately after receiving replies from the official account and from the fake account he decided to trick the trickster.
At first, it was tested how gullible the scammers were by providing a fake name Wade Wilson, which for those who don't know is a comic book character also known as Deadpool. Then when the scammer asks for an address it was given to him the address of a police station.
After getting the address without having any suspicion the scammers requested Wade Wilson's credit card details that were linked to the official Virgin Media account. At this point, it was offered to scammers a set of test credit card details.
When the credit card authorization failed to work for scammers, they tried to convince the victim to provide the details of another card. The scammers were told that a new credit card details will be given shortly but at the same time, the cybersecurity researcher provided them a link with the intention of trying to trick them into clicking on it in order to expose their IP address.
To finish the scammer off the cybersecurity researcher faked a screenshot of an AmEx fraud alert SMS featuring its own phishing link requesting that the user click to verify their card details.
After sending a fake SMS message the researchers received a click on his web server. At this point, the researchers reported this all back to Twitter, that suspended the fake account, and also alerted the Police in the UK to find and apprehend those responsible for the cyber attack.


UQ Taiko is Brisbane's newest and only not-for-profit performance taiko drumming team. Based at UQ, it was established in March of 2014. Their goal is to add cultural and musical diversity to the UQ community, as well as to the greater Brisbane community, through performing and giving workshops on the traditional Japanese art and music of taiko drumming.
Taiko drumming is a traditional form of Japanese art and music that is characterised by powerful beats, dynamic movements, and overflowing energy!


Links to business pages for food hygiene inspections and reports. Tips for food safety in the home.
Search for a food hygiene inspection report and find out how we carry out our inspections.
Food information regulations came into force in Scotland in 2014.


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California Police Concerned About Marijuana Deliveries from Nearby Cities
October 8, 2018 • by POL Staff
Morgan Hill—an upscale suburb of San Jose, California—has banned marijuana dispensaries. So the prospect of delivery vans—akin to those operated by online retailers—coming into town has Police Chief David Swing concerned.
Swing is opposed to a proposal that would allow marijuana businesses to send home-delivery vans into communities where retail stores are prohibited, according to the Los Angeles Times.
"This will make it easier and more lucrative to rob a delivery person than a liquor store," Swing said, noting that under the proposal, delivery drivers would be allowed to carry up to $10,000 in cash.
"Robberies are the tip of the iceberg. They can lead to other crimes, including aggravated assaults and homicides," Swing said.
Proposition 64 bars the state from issuing a license to pot shops that have not been given city approval to operate. But city council members throughout the state say that authority is undermined if deliveries are legal statewide.
Read more about Drug Enforcement Marijuana Elected Leaders California the west patrol