text:
\section{Introduction}

The Hyades, an open cluster
whose every member contributes to our knowledge of stellar evolution, open cluster formation, and open cluster evolution, remains an important rung on the extragalactic distance scale ladder \citep{An07b,deB01}.
Much effort has resulted in an ever improving Hyades distance, from convergent point methods \citep{Bue52, Han75} using only proper motions,
to directly measured parallax \citep{Per98, McA11}, through recent efforts using a combination of parallax,  proper motion, and radial velocities \citep{Lee17}, with cluster membership determination always a critical issue. Hence the first goal of this project is an unambiguous determination of cluster membership for {vA\,351}=H346 = L42 = V805 Tau = HG7-203 = LP415-65, RA $04^{\rm h} 25^{\rm m} 13.54^{\rm s}$ DEC $+17\arcdeg 16' 05\farcs48$ (2000), V=13.27, K=8.27.

This Hyad was first angularly resolved \citep{Fra94} as a result of a blind search of faint   members of the Hyades,  using the {\it Hubble Space Telescope} ({\it HST~}) Fine Guidance Sensor \#3 ({\it FGS}\,3) in the transfer function or fringe scanning (TRANS) mode. \cite{Fra98} details this {\it FGS~} observation mode. Further observations  yielded a relative orbit, and, depending on an assumed parallax based on Hyades membership, a total mass for the system \citep{Fra98b} inconsistent with the colors and spectral types suggesting M dwarf components. A white dwarf component was posited to bring the total mass into agreement with the dynamical mass. Subsequently, we acquired {\it FGS}\,3 POS mode \nocite{Ben16} observations (the fringe tracking POS mode described in Benedict et al. 2016) with which to derive a mass fraction and independent parallax. Once {vA\,351~} was identified as a binary we added it to our then ongoing Mass-Luminosity Relation radial velocity program, also described in \cite{Ben16}. Thus our second goal: to establish the physical properties of all {vA\,351~} components, determining how many stars comprise this system, and derive their individual component masses.

Observational astronomers are hard-pressed to devise experiments. Nature provides these by presenting us with binary stars. Put one star next to another and see what happens. Extreme cases provide laboratories within which to test our knowledge of stellar magnetic fields
and mass transfer.
{vA\,351~} provides such a laboratory. Hence,  a detailed examination of the complex behavior of a subset of {vA\,351~} components provides a third goal,
that of generating a qualitative model explaining the observations.

These observations include {\it FGS~} TRANS which use the entire fringe to
match a superposition of the fringe of each component to the fringe of a known single star, resulting in position angles and separations with which to derive a relative orbit. They also provide component
magnitude differences. {\it FGS~} POS mode produces positions
of the zero crossing of a fringe for each star observed. Measuring a series of {vA\,351~}  positions relative to a known reference frame provides a parallax, establishes a proper motion for the center of mass of the system, and yields a mass ratio for the supposed two components. For many of the M dwarf binary systems investigated in \cite{Ben16},
adding radial velocities (RVs) to the astrometry allowed us to
completely sample the motion of a binary system. The three
complementary data sets improve the accuracies of the final component masses.   For {vA\,351~} all assist in dissecting the system, yielding masses for the four components.

This paper summarizes our efforts to understand the {vA\,351~} system. We first present the results of our analysis of the TRANS-mode astrometric data, yielding a binary orbit and total mass (Section~\ref{AstRel}).  Next we present a radial velocity (RV)  analysis using  absorption lines, yielding  a $\cal{M}_{\rm C}$/$\cal{M}_{\rm B}$~ mass ratio (Section~\ref{spec}) and a confirmation of the component A RVs with respect to the AD-BC barycenter.  Finally we fold in the POS-mode data and derive a mass fraction, a proper motion, and a parallax for the system (Section~\ref{POSast}).  Section~\ref{allphot} presents evidence of small amplitude but precisely measured variability from photometry acquired with the POS mode observations, and recent ground-based $V-$band photometry. In Section~\ref{BS} we use to  these results to  confirm masses for the three M dwarf  components and to derive a WD mass  required to satisfy the derived total mass and the ($\cal{M}_{\rm A}$+$\cal{M}_{\rm D}$)/($\cal{M}_{\rm B}$+$\cal{M}_{\rm C}$) mass ratio. Half our spectra include emission lines of both H$\alpha$~ and He\,\text{\small{I}}~ ($\lambda 5876.6\AA$).  We derive equivalent widths and  RVs for both H$\alpha$~ and He\,\text{\small{I}}~ emission lines (Section~\ref{specem}).
We describe the structure of this quadruple system and posit some possible emission line formation mechanisms (Section~\ref{SciFi}), offer conclusions (Section~\ref{summ}), and summarize our results in Section~\ref{sum}.

Unless otherwise noted, times are modified Julian Date, mJD = JD-2400000.5. We abbreviate millisecond of arc as mas throughout.

\section{The {vA\,351}~ Relative Orbit}  \label{AstRel}

We  have  13 position angle and separation
TRANS mode measurements of component B relative to component A acquired over 4.7 yr.
Five  speckle measures, taken with the DSSI speckle camera on the 4.3m Lowell Observatory Discovery Telescope (LDT; formerly DCT), extend the relative orbit astrometric coverage to 26.0 yr. The LDT speckle program \citep{Hor12,Hor15}, in operation since 2014,  obtained data for vA 351 on four separate nights in October and November of that year. A more recent observation, under very poor weather conditions, was secured in early February 2020. Analysis of these data files proceeded using the same methodology as described most recently in \cite{Hor17}, using a weighted least-squares fit to the summed spatial frequency power spectrum of the speckle data frames.

We transformed all position angle and separation observations into x and y positions and assigned a 1.5 mas error to each x, y, this to yield a reduced $\chi^2$ near unity for the relative orbit modeling. Figure~\ref{Rel} contains the resulting low-inclination, high eccentricity relative orbit with  observations and residuals presented in Table~\ref{tbl-TR} and orbital parameters listed in Table~\ref{tbl-TSORB}. The second largest residual in Figure~\ref{Rel} (\#18) comes from the February 2020 speckle measurement, secured under very poor weather conditions. We excluded this and one early TRANS observation (\#2, components unresolved along one {\it FGS~} axis) in the final orbit determination.

The relative orbit appears quite plausible. However, see the Appendix  for an arduous and tortuous path traversed, and for a  blind alley  out of which we backed, to eventually arrive at Figure~\ref{Rel}.

\section{Absorption-line Spectroscopy} \label{spec}

\subsection{McDonald Cassegrain Echelle Spectrograph}
We obtained  RV data with the McDonald 2.1m Otto Struve telescope and
Sandiford Cassegrain Echelle spectrograph \citep{McC93}, hereafter CE.
The CE delivers a dispersion equivalent to 2.5 km s$^{-1}$/pix ($R =
\lambda/\Delta\lambda = 60,000$) with a wavelength range of 5500
$\leq$ $\lambda$ $\leq$ 6700 \AA~spread across 26 orders.
The McDonald data were collected during thirty-three observing runs
from 1995 to 2009 and reduced using the standard IRAF \citep{Tod93}
{\tt echelle} package tools. CE RVs (listed in Table~\ref{tbl-CE}) were derived using the IRAF cross-correlation tool {\tt
  fxcorr}.

  \subsection{Center for Astrophysics Digital Speedometers}
vA 351 was monitored spectroscopically at the CfA with the Digital Speedometers (Latham 1992) between January of 1982 and October of 1999, on two telescopes: the 1.5m Tillinghast reflector at the Fred L.\ Whipple Observatory on Mount Hopkins (AZ), and the 4.5m-equivalent Multiple Mirror Telescope (also on Mount Hopkins) before its conversion to a monolithic 6.5m telescope. With a resolving power, $R \approx 35,000$,  the spectra consist of a single echelle order centered on the \ion{Mg}{1}~b triplet (5187~\AA) and spanning 45~\AA. Reductions were carried out with a dedicated pipeline. The average signal-to-noise ratio for the 53 observations is about 12 per resolution element of 8.5~km~s$^{-1}$. We identify this source as CfA, hereafter, and provide the CfA RVs in Table~\ref{tbl-CfA}.

\subsection{IGRINS and other Sources of {vA\,351~} RVs}
On UT 2016 February 23 and 25 we obtained two epochs with the R$=$45,000, near-infrared spectrograph IGRINS \citep{Mac16} on the McDonald Observatory Harlan J. Smith 2.7m telescope. An additional observation was taken on UT 2019 April 11 at the 4.3m LDT.  Exposure times were 16--24 minutes, divided into 2 or 4 nodded pairs at two distinct slit positions. The resulting signal to noise was $>$100 per resolution element of $\sim$6 km s$^{-1}$. Data reduction was carried out using standard processing with the IGRINS pipeline \citep{LGK17}\footnote{https://github.com/igrins/plp/tree/v2.1-alpha.3}. The pipeline yields telluric-corrected, one-dimensional spectra with wavelength dispersion solutions determined from OH night sky emission lines and atmospheric absorption lines.  Barycentric velocity corrections were applied to the target and RV standard star data as part of the cross-correlation analysis (Section~\ref{WTF?}).

Hartmann et al. (1987) \nocite{Har87} report a single RV of (presumably) component A with the MMT.  \cite{Sta97} present two RV measurements of vA 351 components A, B, and C obtained with the NOAO 4m Mayall telescope at two epochs separated by $\sim$50 days.  We list dates, RVs, and phases for these three epochs from the literature and the three IGRINS epochs  in Table~\ref{tbl-oRV}.

\subsection{Absorption Line  Analysis} \label{WTF?}
For the CE observations we used the same high-resolution, high signal-to-noise template spectrum of GJ623\,AB
as in \cite{Ben16}  for all absorption line cross-correlations. The quality of the GJ623\,AB  orbit yields an  absolute RV uncertainty at the epoch of template observation of less than 0.025 km s$^{-1}$ \citep[][table 4]{Ben16}. GJ623\,AB has a $\Delta V=5.2$, rendering the effect of the secondary on the cross-correlation
function (CCF) negligible. We visually inspected
the resulting CCF to select the better of the 26 apertures, typically
using half to obtain an average velocity.  The CCF resulting from the absorption line template had
from one to three peaks (Table~\ref{tbl-CE}), indicating the existence of at least three, not just
two components (hereafter, components A, B, and C).  We identify component A with the strongest peak in the CCF,
and list all CE component RVs with internal $1-\sigma$ errors
in Table~\ref{tbl-CE}. For components A, B, and C we find average internal per epoch errors;
0.3, 0.6, and 0.8 km s$^{-1}$.

Radial velocities from the CfA and IGRINS spectra were measured with algorithms for three-dimensional cross-correlation such as
 TRICOR  \nocite{Zuc95} (Zucker et al. 1995). For the IGRINS data, we used the same template for the three vA 351 components (A, B, and C), a high S/N exposure of the slowly rotating M2 star GJ 725A (Prato 2007; Mann et al. 2015)\nocite{Man15, Pra07}. For the CfA data,
  rotational broadening was applied to achieve the best overall match, resulting in $v \sin i$ estimates of 11$\pm$2 km s$^{-1}$ for stars B and C, and 3$\pm$3 km s$^{-1}$ for the brighter star A.  For the IGRINS observations, the cross-correlation coefficient was optimized
   when we used the GJ 752A template for the A component, with no broadening applied, and the M2.5 dwarf standard GJ 436 for the B and C components with $v \sin i$ of 12$\pm$2 km s$^{-1}$ (Prato 2007; Mann et al. 2015). For both the optical and IR data, these
    $v \sin i$ values  agree with the independent determinations of Stauffer et al. (1997). \nocite{Sta97

From CfA measurements the light ratio of each of the fainter components relative to star A was found to be $0.53 \pm 0.02$ at the mean wavelength of these observations. This yields a flux ratio BC/AD = 1.06 $\pm$ 0.03, or $\Delta$m = 0.06 $\pm$ 0.03, with BC being marginally brighter at this wavelength.  The bandpass is somewhat close to V though on the blue side and much narrower (45 \AA). It is similar to the D51 filter used for the Kepler Input Catalog \citep{Bro11}, which was modeled after the Dunlap Observatory DDO51 filter. The CfA $\Delta$m = 0.06 $\pm$ 0.03 is consistent with the measurements listed in Table~\ref{tbl-TR}.  The IGRINS $H-$band spectra yielded light ratios of B/A$=$0.66$\pm$0.02 and C/A$=$0.64$\pm$0.08, hence a BC/AD flux ratio of 1.3$\pm$0.1 and $\Delta H=0.28$ (combined components AD fainter than combined components BC), a puzzle given the $V-$ and $K-$band lower $\Delta m$ values (Table~\ref{tbl-TR}).

The two fainter components, B and C, exhibit the largest RV
variation. When fit to an orbit these yield the elements listed in Table~\ref{tbl-RVO}.
We plot component velocities phased to a BC period, $P_{\rm BC}=0.7492425^{\rm d}$
in Figure~\ref{RVabs}. The rms RV residual to these orbits is 2.7 km s$^{-1}$. A ratio of the RV amplitudes yields a mass ratio for components C and B; $K_B / K_C =$ $\cal{M}$$_{\rm C}/$$\cal{M}$$_{\rm B}=0.945\pm0.005$. The average
component A RV is $\langle \rm{V}_{\rm A}\rangle=+40.7\pm2.0$ km s$^{-1}$. An average of the BC RV $\gamma$ values yields
$\langle \rm{V}_{\rm BC}\rangle=+40.2\pm0.2$ km s$^{-1}$.
We have consistency with the expected radial velocity of a Hyades member at the location of {vA\,351}, V=+39.80 km s$^{-1}$, computed from a convergent point solution with the location of the convergent point as reported by the \cite{Lee17} ($\alpha_{\rm J2000} = 97\fdg73$, $\delta_{\rm J2000} = +6\fdg83$), adopting a space velocity for the cluster of V$_0$ = +47.13 km s$^{-1}$, based on ongoing radial velocity measurements at the CfA.

In Figure~\ref{fig-WTF?} we plot the RV variation
attributable to the motion of A  about a common center of mass with BC, phased to the period
(Table~\ref{tbl-RVORB}) determined through a  solution incorporating only RVs. While noisy, the component A velocities show a clear signature of a highly eccentric orbit. We were unable to extract a clear signature of the BC center of mass motions around the AD-BC system barycenter.

\section{POS mode Astrometry}\label{POSast}
\cite{Ben16}  describes the acquisition and reduction of {\it FGS~} POS  mode data. Figure \ref{fxy} shows the distribution on the sky of
the four reference stars (numbers 2, 3, 4, 5) for the POS
mode measurements, with number  1 indicating
{vA\,351}.    Not all reference stars were
measured at each of the seven epochs of observation that included POS mode.
We acquired all {\it HST~} astrometry data sets with {\it FGS}\,3.  For POS mode we obtained a total of 50 reference star
observations and 19 observations of {vA\,351}, secured over 1.8 years.

\subsection{Prior Knowledge and Modeling Constraints} \label{MODCON}
As in our previous astrometry projects, e.g., \cite{Ben01,Ben07,
  Ben11, McA11, Ben16} we include as much prior information as possible in
our {\it FGS~} fringe-tracking modeling.
In contrast to much of our previous relative astrometry
(e.g., Harrison {\sl et~al.~} 1999, Benedict {\sl et~al.~} 2011)\nocite{Har99,Ben11}, for this project we
obtain  reference star absolute parallaxes and proper motion priors from {\sl Gaia~} DR2 \citep{Bro18}.
A comparison of the Figure~\ref{CCD} color-color diagram with the Table~\ref{tbl-pis} input parallaxes confirms the giant nature of
reference stars 4 and 5, labeled ref-4 and ref-5 in the figure.

We did not simply adopt the {\sl Gaia~} DR2 parallax for {vA\,351~}
because the {\sl Gaia~} astrometry does  not yet account for over 80 mas \citep{Fra98b} of orbital motion. Similarly, rather than using the highly precise  {\sl Gaia~} DR2
proper motion from measures taken over a limited multiple of the AD - BC orbit, we introduce no {vA\,351~} proper motion priors.

In a quasi-Bayesian approach we input all priors as observations with
associated errors, not as hardwired quantities known to infinite
precision.
The lateral color
calibration (see section
3.4 of Benedict {\sl et~al.~} 1999)\nocite{Ben99} and the $B-V$ color indices are also treated as
observations with error.

\subsection{{vA\,351~} POS Mode Photocenter Corrections} \label{PCC}
{\it FGS~} POS mode measures the zero-crossing of a fringe.
Thus, for a binary the observed fringe becomes a linear superposition of two fringes.
For a nearly equal brightness component binary (Table~\ref{tbl-TR}) the measured zero crossing can
be significantly perturbed from a position coincident with either component.
\cite{Ben01}, section 4.2, discusses the generation of such corrections at some length.
Figure~\ref{pcor} demonstrates the determination of corrections along the  x and y axes of {\it FGS}\,3
for a typical POS mode observation of {vA\,351}. Corrections ranged from 4 mas to 80 mas, the latter required
because for that observation the {\it FGS~} locked on the other component.

\subsection{The POS Model}

From the reference star astrometric data we determine the rotation and offset
``plate constants" relative to an arbitrarily adopted constraint epoch
(the so-called ``master plate") for each observation set. The
{vA\,351~} reference frame contains five stars, but only three were observed
at each epoch. Hence, we constrain the scales along $x$ and $y$ to
that provided by the {\it FGS~} Optical Field Angle Distortion calibration \citep{Ben99}, and the two axes to orthogonality.
The consequences of this
choice are minimal. For example, imposing these constraints on the
Barnard's Star astrometry discussed in \cite{Ben99} results in an
unchanged parallax and increases the error by 0.1 mas, compared to a
full 6 parameter model  \citep[e.g.][equations 6,7]{Ben17}.

Our reference frame model becomes, in terms of standard coordinates,
\begin{equation}
        x'  =  x + lc_x(\it B-V)
\end{equation}
\begin{equation}
        y'  =  y + lc_y(\it B-V)
\end{equation}
\begin{equation}
x''=x'+corrx
\end{equation}
\begin{equation}
y''=y'+corry
\end{equation}
\begin{equation}
\xi = {\rm cos}(A)*x'' +{\rm sin}(A)*y'' + C  \\ - \mu_\alpha \Delta t  - P_\alpha\varpi - ORB_\alpha
\end{equation}
\begin{equation}
\eta = -{\rm sin}(A)*x'' + {\rm cos}(A)*y'' + F  \\ - \mu_\delta \Delta t  - P_\delta\varpi - ORB_\delta
\end{equation}

\noindent
, where $\it x$ and $\it y$ are the measured coordinates from {\it HST}, $corrx$ and $corry$ are
the photocenter corrections derived in Section~\ref{PCC} above,
$\it lc_x$ and $\it lc_y$ are lateral color corrections , and $B-V$ represents the
color of each star, either from SIMBAD or estimated from the
spectral types suggested by Figure~\ref{CCD}.  $A$  is a field rotation in radians,
$C$ and $F$ are offsets, $\mu_\alpha$ and
$\mu_\delta$ are proper motions, $\Delta$$t$ is the epoch difference from
the mean epoch, $P_\alpha$ and $P_\delta$ are parallax factors, and
$\it \varpi$  is the parallax.
$\xi$ and $\eta$ are
relative positions that (once rotation, parallax, the proper motions
 are determined)
should not change with time. All Equation
5 and 6 subscripts are in RA and DEC because the master constraint plate was rolled
into the RA DEC coordinate system before the analysis. We obtain
the parallax factors from a JPL Earth orbit predictor (Standish
1990\nocite{Sta90}), upgraded to version DE405.  We obtain orientation to the
sky for the master plate using ground-based astrometry from
the PPMXL \citep{Roe10} with uncertainties in the field orientation of
$\pm 0\fdg1$.  This orientation also enters the modeling as an
observation with error.

We derive $A$, $C$, and $F$ for each epoch only from reference star measurements.
Applied as constants, we solve  for a  position
within our reference frame, a parallax, a proper motion, and a major axis measurement, $\alpha_{\rm A}$, for the {vA\,351A~} orbit about the system center of mass.   To {vA\,351~} we apply $ORB$  as a
correction term that is a function of the traditional astrometric and RV
orbital elements listed in Tables~\ref{tbl-TSORB} and \ref{tbl-RVORB}.
These include the
orbital period (P), the epoch of passage through periastron in
modified Julian days (T$_0$), the eccentricity ($e$), the inclination, $i$, and the position
angle of the line of nodes ($\Omega$).

 We also constrain the angle
($\omega$) in the plane of the true orbit between the line of nodes
and the major axis to differ by 180\arcdeg~for the component A and B
orbits.

\subsection{Assessing Reference Frame Residuals}
 From histograms of the astrometric residuals for 50 reference star and
19 {vA\,351~} position measurements (Figure~\ref{his}), we conclude that we
have obtained corrections at the $1.2$ mas level in the region
available at all {\it HST} roll angles (an inscribed circle centered
on the pickle-shaped {\it FGS~} field of regard).  The resulting reference
frame ``catalog'' in $\xi$ and $\eta$ standard coordinates (Equations
5 and 6) was determined with median absolute errors of $\sigma_\xi=
0.9$ and $\sigma_\eta = 1.0$ mas in $x$ and $y$, respectively.

\subsection{Results of {vA\,351~}  Modeling}

In Table~\ref{tbl-SUM} we list our final values for the absolute parallax and proper motion  (with $1\sigma$ errors), where
the {\sl Gaia~} parallaxes from both DR2 and EDR3 \citep{Lin20a} are also included for comparison.  We included neither parallax as a prior. Extending the {\sl Gaia~} observational time baseline has brought their parallax closer to our value, now only slightly more than $1\sigma$ different. \cite{Sta21} find that the {\sl Gaia~} $RUWE$ (renormalised unit weight error) robustly predicts unmodeled photocenter motion, even in the nominal "good" range of 1.0--1.4 \citep[see also][]{Bel20}. For {vA\,351~} EDR3 lists a value very much larger, $RUWE$=9.33, clearly indicating the {\it Gaia} astrometry was affected.
The median $RUWE$ value for the 9 stars (with a Gaia $G-$band magnitude $G<16$, not used as astrometric reference stars) in EDR3 within $7\hbox{$^\prime$}$ of {vA\,351~} is 1.05. Reference stars (Table~\ref{tbl-pis}) have a median $RUWE$ value  of 1.38, a partial explanation for our poorer than average \citep[see the 105 parallax results in][]{Ben17} parallax result for {vA\,351}

Table~\ref{tbl-ORB}
contains the final orbital parameters with formal ($1 \sigma$)
uncertainties.
Figure~\ref{vAorb} illustrates component A and B astrometric
orbits.

 That the FGS POS measures did not sample the entire
  orbit and  that the {\sl Gaia~} solution thus far neglects orbital motion could provide a partial explanation for the  proper motion discrepancies seen in Table~\ref{tbl-SUM}.
 The TRANS and speckle astrometric residuals (Table~\ref{tbl-TR}, RMS=2.8 mas) suggest an unmodeled perturbation to the TRANS measured separation
  between components A and B.  We discuss further  evidence for this
  perturbation in Section~\ref{BS}.

  In the past \citep[e.g.][]{Ben16} we have employed a relationship between astrometry and RV \citep[c.f.][]{Pou00}
 to narrow the range of possible combinations of parallax, $\varpi$, and orbit semi-major axis, $\alpha$:
\begin{equation}
\displaystyle{{\alpha_{\rm A}\,{\rm sin}\,i \over \varpi_{\rm abs}} = {P K_{\rm A} \sqrt{(1 -\epsilon^2)}\over2\pi\times4.7405}}
\label{PJeq}
\end{equation}
However, the significant inclination error (Table~\ref{tbl-TSORB}) renders this constraint less useful. The right hand (RHS) and left hand (LHS) quantities calculated from results listed in Tables~\ref{tbl-TSORB}, \ref{tbl-RVORB}, and \ref{tbl-SUM} do agree within their errors
(RHS$=0.267\pm0.154$, LHS$=0.220\pm0.004$).

\subsection{{vA\,351~} Component Masses} \label{MT}
Our orbit solution and derived absolute parallax provide an orbital
semimajor axis, $a$ in AU, from which we can determine the system mass
through Kepler's Third Law.
We find (in solar units) ${\cal M}_{\rm tot} = 2.06 \pm 0.24{\cal M}_{\sun}$.
At each instant in the orbits of the two components around the common
center of mass,

\begin{equation}
{\cal M}_{\rm A} / {\cal M}_{\rm B} = \alpha_{\rm B} / \alpha_{\rm A}
\end{equation}

\noindent a relationship that contains only one observable,
$\alpha_{\rm A}$, the {vA\,351~} semi-major axis measurement. Instead, we calculate
the mass fraction

\begin{equation}
f = {\cal M}_{\rm B}/({\cal M}_{\rm A}+{\cal M}_{\rm B}) = \alpha_{\rm A} / (\alpha_{\rm A} + \alpha_{\rm B}) = \alpha_{\rm A} /a, \label{frac}
\end{equation}

\noindent where $\alpha_{\rm B}$ = $a-\alpha_{\rm A}.$ This parameter,
$f$, also given in Table~\ref{tbl-ORB}, creates a ratio of the two quantities
directly obtained from the observations: the {vA\,351A~} semi-major axis
 ($\alpha_{\rm A}$ from POS mode, Figure~\ref{vAorb}) and the relative orbit size ($a$
from TRANS mode, Figure~\ref{Rel}),  listed in
Table~\ref{tbl-ORB}. From these we derive a mass fraction of 0.443
$\pm$ 0.022.  Equations 5, 6, and 7 yield ${\cal M}_{\rm A} = 1.14
\pm 0.14{\cal M}_{\sun}$ and ${\cal M}_{\rm B} = 0.91 \pm 0.11
    {\cal M}_{\sun}$.

    The mass errors of $\sim$12\% exceed any of our past  \citep{Ben16,Ben01} determinations.  We ascribe this to a combination of orbit low inclination and high eccentricity, increasing the errors on relative orbit size and perturbation size, and primarily to the parallax error on what is the smallest parallax yet measured for a binary by the FGS.
The median parallax in our previous MLR work was 113 mas, 6 times larger than
that measured for {vA\,351}. The median percent error in our MLR study was 0.4\%, significantly better
than the vA 351 3.5\% parallax error.

Nonetheless, {vA\,351B~} with $\cal{M}_{\rm B}$ $= 0.91{\cal M}_{\sun}$ is consistent with a G5V star, and {vA\,351A~}  with
$\cal{M}_{\rm A}$ $= 1.14{\cal M}_{\sun}$  an F8V star \citep{Cox00}. This is not consistent with each component being a single low mass red
    dwarf as suggested by the Figure~\ref{CCD} color-color diagram, which  cleanly and unambiguously identifies {vA\,351~} as such. To make up for the excess mass we now identify component A to have a companion, component D. Component B also must have a companion, component C. We now assert that $\cal{M}_{\rm A}$+$\cal{M}_{\rm D}$~= 1.14$\cal{M}_{\odot}~$ and $\cal{M}_{\rm B}$+$\cal{M}_{\rm C}$~= 0.91$\cal{M}_{\odot}$.

\section{Photometry} \label{allphot}

\subsection{{\it FGS~} POS Mode Photometry} \label{Fphot}
{\it FGS}\,3 has demonstrated relative photometric precision approaching the milli-magnitude level for two other M dwarf stars, Proxima Cen and Barnard's Star \citep{Ben98a}.  The {\it FGS~} acquires data at a 40Hz rate with each observation lasting from 80 to 130 seconds. With similar exposures and a brightness two magnitudes fainter than Proxima Cen, measures of {vA\,351~} have lower S/N. We flat-fielded  the {vA\,351~} observations using a sum of the average counts for astrometric reference stars ref-2 and ref-3 identified in Figure~\ref{fxy}. Figure~\ref{FP} contains flat-fielded average counts as a function of BC orbital phase.
We observed {vA\,351~} three times during each scheduled {\it HST~} observation set, hence, the {vA\,351~} groupings of three.
The observations used the {\it FGS}\,3 F583W filter. The bandpass center, 583nm, has a width 234nm, which includes
both the H$\alpha$~ and He\,\text{\small{I}}~ emission lines discussed below (Section~\ref{specem}).
Figure~\ref{FP} includes a sine wave with a period constrained to the BC orbital period. We find a F543W ($V$+H$\alpha$) amplitude of 0.021$\pm$0.005 mag. The sampling is too sparse to make definitive statements,
but the fit
to the BC orbital period indicates only one dimming event per cycle, when {vA\,351B~} is more distant than {vA\,351C}. This might
require additional complications (such as eclipses or a dark spot on the back side of component C). Alternatively, because {vA\,351~} varies (V* V805 Tau, flare star), we might attribute the  larger signal  at
$\Phi_{\rm BC} \simeq 1$ to flaring activity.

 \subsection{$V$-band Photometry} \label{BSP}
To test the eclipse hypothesis, one of us (BS) obtained absolute
$V$-band photometry in autumn 2019 using the Lowell Observatory
0.7-m robotic telescope.  {vA\,351~}  was observed with 100-second exposures,
typically several visits each on a total of 13 nights, using
 ``Canopus" software \footnote{http://bdwpublishing.com/mpocanopusv10.aspx} to reduce the data, yielding
conventional aperture photometry.  The zero-point was adjusted
to within a few percent of standard V using magnitudes of
four field comparison stars.

Rather than providing additional evidence to assist in choosing between the interpretations above (Section~\ref{Fphot}),  a periodogram analysis of the newer $V$-band photometry yields a
highly significant peak  near, $P_V=1.7^{\rm d}$ with a lower peak near the BC orbital period, $P_{\rm BC}=0.7^{\rm d}$. Fitting a sine curve to the $V$ photometry yielded $P_V=1.783\pm0.003^{\rm d}$ and an amplitude $\Delta V = 0.011\pm 0.001$ magnitude with $\langle V \rangle= 13.27$.

The left side of Figure~\ref{BSPfig} presents the $V$ measurements phased to  the 1.783$^{\rm d}$
period with the FGS F543W measurements superposed. The fit includes only the Lowell $V$ measurements,
The right side of Figure~\ref{BSPfig} phases
FGS and $V$ photometry to the BC orbital period with the single cycle fit from Figure~\ref{FP} superposed.
The variation in $V$ only does not correlate with the {vA\,351~} BC orbit. The bandpass containing H$\alpha$~ does.

\section{Dissecting {vA\,351}}\label{BS}
Our astrometry  (Section~\ref{MT}) yields masses for the component A,D and the component B,C pairs. Our RV measurements (Section~\ref{WTF?}) yield a mass ratio for the B,C components. From $\cal{M}_{\rm BC}$=$0.91 \pm 0.11$$\cal{M}_{\odot}~$ and
$\cal{M}_{\rm C}$/$\cal{M}_{\rm B}$=$0.945\pm0.005$ we find $\cal{M}_{\rm B}$=$0.47\pm0.06$$\cal{M}_{\odot}~$ and $\cal{M}_{\rm C}$=$0.44\pm0.06$$\cal{M}_{\odot}$. Assuming for the AD-BC pair a $\Delta K=0$,  that $K_B\simeq K_C$, $A_K\simeq0$, and from our parallax, a distance modulus m-M=3.68, we find
absolute magnitudes $M_K=6.09$ for each. The \cite{Ben16} $K-$band MLR predicts $\cal{M}$=0.46$\cal{M}_{\odot}$, assuming equal brightness components B and C. The agreement provides validation for the astrometry and RV results. We now lack only individual masses for components A and D.

To establish their masses
 we use
the data and sources listed at the top of Table~\ref{tbl-RIP}. This table also
provides a line by line guide to the process resulting in a mass estimate for components A  and component D, the (suspected) white dwarf required to satisfy the
total mass resulting from the astrometry (Section~\ref{MT}).

Table~\ref{tbl-RIP} begins with photometry of the total system, $V$-band from Section~\ref{BSP} and $K$-band from 2MASS \citep{Skr06} on {\bf line 1}.
We transform these into arbitrary intensities ({\bf line 2}), and assert a magnitude difference between components AD and BC on
{\bf line 3}.
In support of the {\bf line 3} data, Table~\ref{tbl-TR} contains magnitude differences in various bandpasses.
The {\it FGS~} TRANS $F583$ measures are close to $V$-band \citep{Hen99}.
The speckle magnitudes are similar to $r$ and $i$ \citep{Hor12}.
Some of the components of {vA\,351~} are quite active, but, without the flare events
on mJD 50481 and 56934, the measures
are consistent with $\Delta V = 0.0\pm0.1$. Typically $\Delta K$ is less than $\Delta V$ for M dwarfs \citep{Ben16},
supporting an estimate of $\Delta K = 0.0\pm 0.03$. Finally we introduce ({\bf line 4}) estimates of interstellar absorption, taking as priors a little less than half the  total absorption of $A_V=1.17$ mag (with $A_K/A_V=0.11$) along this line of sight \citep{Sch11}.
Including an absorption parameter significantly reduced the Table~\ref{tbl-RIP} solution residuals, and brought the $V$ and $K$ results into closer agreement.

Next we introduce ({\bf line 5})  data provided by the RV measures, the $\cal{M}_{\rm C}$/$\cal{M}_{\rm B}$~ mass ratio from the absorption
line results (Figure~\ref{RVabs}, Table~\ref{tbl-RVO}), $\cal{M}$$_{\rm C}/$$\cal{M}$$_{\rm B}=0.945\pm0.005$. {\bf Lines 6-10}  introduce the mass and parallax data from the astrometric modeling results (Table~\ref{tbl-ORB}).

We then appeal to three external relationships, allowing us
to estimate a mass from an absolute magnitude. The first two relate M dwarf intrinsic luminosity to mass.
Rather than use the
fifth-order polynomial Luminosity to Mass relations (LMR) in \cite{Ben16},
we simplify to a linear LMR for the $V$-band and a parabolic LMR for the $K-$band,
 \begin{equation}
{\cal M} = C_0 + C_1 (M_{V,K}) + C_2 (M_{K} ) ^2 \label{LMRf}
\end{equation}
with coefficients given in Table~\ref{tbl-LMR}. The total dynamical mass of {vA\,351~} precludes very low
component masses, where the considerable non-linearity in the LMR requires higher order terms.
The third relation maps  $K-$band absolute magnitude as a function of mass for
white dwarf stars. We obtain this mapping from  Bergeron cooling models
\citep{Hol06,Kow06,Tre11,Ber11}, assuming a Hyades age of 670 My \citep{Gos18}.
Figure~\ref{WDM} shows this mapping, and a relation  fit with a third order polynomial with offset,
\begin{equation}
{\cal M} = C_0 + C_1 (M_{K} - X_0) + C_2 (M_{K}  - X_0) ^2  +C_3 (M_{K}  - X_0) ^3\label{LMRWD}
\end{equation}
with the coefficient zero-point  at the bottom of Table~\ref{tbl-LMR}. Figure~\ref{WDM}
shows a multi-valued relation between mass and luminosity  (in a very narrow luminosity range) for the $V-$band. Hence, in the least-squares analysis below
we only use the $K-$band to determine the WD mass.

We next search for component intensity values, system parallax, and line of sight absorption ({\bf lines 11-16}) that minimize  the
sum of the squares of the O-C residuals, where O are the {\bf line 1-10} values, and C are the {\bf line 17-24}
final parameter values and residuals for both $V$ and $K$ bandpasses. We transform the {\bf line 11-14} intensities
into the {\bf line 26 - 30} magnitudes (corrected for the estimated absorption) and then with the parallax (yielding a distance modulus, $m-M=3.68$) into the {\bf line 31 - 35} absolute magnitudes. Using
Equations~\ref{LMRf}~and~\ref{LMRWD} we derive the {\bf line 36 - 39} masses. This sequence iterates
(separately for both the $V-$ and $K-$bands) until  minima in the
sums of the squares of the {\bf line 17 - 24} residuals are found ({\bf line 25}). The $K$-band yields a significantly better fit with smaller residuals.

We find final masses by applying the above least-squares process to both the $V-$ and $K-$band data, then averaging
the results ({\bf lines 36-40}). We find $\cal{M}$$_{\rm A} = 0.53\pm0.10$$\cal{M}_{\odot}$, $\cal{M}$$_{\rm B} = 0.43\pm0.04$$\cal{M}_{\odot}$, $\cal{M}$$_{\rm C} = 0.41\pm0.04$$\cal{M}_{\odot}$, and $\cal{M}$$_{\rm D} = 0.54\pm0.04$$\cal{M}_{\odot}$, where the errors are from the intrinsic scatter in the $V-$ and $K-$band MLR. The WD mass, $\cal{M}$$_{\rm D}$, comes only from the $K-$band photometry.
The major mass disagreement between the $V-$ and $K-$band is for component A. None of these M dwarf masses can contribute new
data points for the \cite{Ben16} MLR, having been derived by appeal to that MLR.

 The {\bf line 25} fit quality parameter is much worse for the $V$-band. We find those values are decreased to 1.31 for $V$ and 0.001 for $K$ by including a larger absorption prior, $A_V=0.59$, $A_K=0.05$. This change also improves the A,B,C component mass agreement, yielding $\cal{M}$$_{\rm A} = 0.56\pm0.07$$\cal{M}_{\odot}$, $\cal{M}$$_{\rm B} = 0.46\pm0.01$$\cal{M}_{\odot}$, $\cal{M}$$_{\rm C} = 0.44\pm0.01$$\cal{M}_{\odot}$, and $\cal{M}$$_{\rm D} = 0.53\pm0.04$$\cal{M}_{\odot}$.
However, \cite{Tay06} concludes that Hyades interstellar extinction  is negligible. Thus, if real, this extinction must be local to {vA\,351}. However, photometry with the Wide-field Infrared Survey Explorer (WISE) \citep{Wri10} shows no excess far-IR flux  indicating heated dust that might increase local absorption.
We note that the Table~\ref{tbl-RIP} WD $V-K=+1.18$ far exceeds that predicted for a 670My old 0.54$\cal{M}_{\odot}$ WD, $V-K=+0.3$. The fit including a higher absorption yields the same WD mass, but a $V-K=+0.66$ much nearer to that predicted.

We estimate the WD mass error from the scatter in the Figure~\ref{WDM} LMR, and have ignored any contributions from the WD and Hyades age uncertainties. The inputs to our mass derivation ({\bf lines 1 - 10}) provide very little leverage with which to assign an age to the WD. For example, our assumed WD age of 670My and a mass, $\cal{M}$$_{\rm D} = 0.53$$\cal{M}_{\odot}$, yields from the Bergeron cooling models  an absolute magnitude M$_K=12.01$. An age of 400My for that same mass WD yields M$_K=11.75$. However the other components have M$_K\sim 6$, over 100$\times$ brighter, rendering the WD component age ineffective at changing the mass results ({\bf lines 36 - 40}).

The IGRINS spectra yielded an AD-BC $\Delta H=0.28\pm0.10$. Relatively normal M dwarf binary stars would evidence $\Delta K \simeq \Delta H$. Redoing the modeling outlined in  Table~\ref{tbl-RIP}, substituting in {\bf line 3} $\Delta K =0.28 \pm 0.1$ for $\Delta K =0.0 \pm 0.03$ yields slightly higher masses for components B, C, and D (each up by 0.02$\cal{M}_{\odot}$), and a slightly lower mass for component A (down by 0.04$\cal{M}_{\odot}$). However the $K-$band goodness of fit parameter ({\bf line 25}) increases by a factor of 20. Thus, a lower $\Delta K$ value is a better fit to our aggregate information.

From the Table~\ref{tbl-RIP} component B and C masses
and the RV amplitudes, $K_{\rm A}$ and $K_{\rm B}$ from Table~\ref{tbl-RVO}, we can derive an inclination of $i_{\rm BC} \simeq  23\arcdeg$ and a separation, $a_{\rm BC} \simeq 0.016$ AU. Estimating component radii using the \cite{Boy12} eclipsing binary mass-radius relation, components B and C are separated by $\sim4$ stellar diameters. Furthermore, if we assume that B and C are synchronized in the 0.75-day orbit, and spin-orbit aligned, as it seems very likely, we can predict a 30 km s$^{-1}$~ equatorial rotational velocity for a typical radius of ~0.46$R_\odot$ for a 0.46$\cal{M}_{\odot}~$ star \citep{Boy12}. Our analysis of the  CfA and IGRINS spectra (Section~\ref{WTF?}) yielded $v\,sin\,i \simeq11$ km s$^{-1}$, implying $ i_{\rm BC} \simeq 20 \arcdeg$.

We have established that the {vA\,351~} system contains four components with components A and D orbiting a common center of mass and
components B and C orbiting a common center of mass.
The AD subsystem and BC subsystem orbit a common center of mass.
Radial velocity measurements easily detect and characterize the BC motion (Figure~\ref{RVabs}) and the AD-BC orbit (Figure~\ref{fig-WTF?}).
The residuals to the BC RVs cannot contain any signature of the
AD RVs. However RV measures of component A might evidence AD motion. Additionally, when measured by FGS TRANS and speckle the separation between A and the BC subsystem
should change because of the AD motion. Component A dominates the TRANS fringe for AD, because
A is $> 3$ magnitudes brighter than component D. Component A will thus be measured to be nearer to or more distant from
BC because it orbits a barycenter with a WD, component D.

In an attempt to tease out possible effects of component D on
component A we  appeal to periodograms. Figure~\ref{LSRVT} presents  Lomb-Scargle periodograms
for (top) the time series of x and y TRANS residuals to the relative orbit shown in Figure~\ref{vAorb}, and (bottom) for the time series of residuals to the
component A RV orbit  (Figure~\ref{fig-WTF?})
We find one significant peak in the component A RV power spectrum at $P=21^d$ with a false alarm probability  $<0.1$\%. As expected, the Figure~\ref{RVabs}  B and C RV residuals  have no significant power in the range 500$^d > P > 8^d$.
The TRANS X and Y residuals contain a  peak produced by a slight increase of power in both
axes at $P\sim25^d$ with a false alarm probability  $> 50$\%. That the only peak simultaneously present in both the TRANS astrometric residuals and the component A RV occurs
at near $P\sim 21^d$ suggests an AD period near  that value.

A formal fit of a sine wave to the RV A residuals yields $P_{\rm AD}=20.82\pm 0.01^{\rm d}$, and $K_{\rm A}=0.30\pm0.09$ km s$^{-1}$.
Our $\cal{M}_{\rm A}$, $\cal{M}_{\rm D}$, and a circular orbit with this period yield a {vA\,351A~} perturbation of 1.5 mas,  a total separation, $a_{\rm AD} \simeq 0.15$ AU, and would require a nearly face-on orientation, $i_{\rm AD} > 89\arcdeg$, to produce the small RV amplitude detected. Additional evidence for this  orientation includes the near zero {vA\,351A~} $Vsin\. i$ (Section~\ref{WTF?}).

\section{Emission Lines} \label{specem}

\subsection{H$\alpha$} \label{specHa}
Early in our spectroscopic data collection process it became apparent that {vA\,351~} exhibited a strong and quite
variable H$\alpha$~ emission with between one and four components. Given that we obtained the many observations
under a large range of weather conditions, any inter-comparisons of H$\alpha$~ behavior are best carried out using normalized data.
We transformed all H$\alpha$~ data to normalized counts, allowing the determination of equivalent width (hereafter, EW).

To measure EW we transformed the intensities, I($\lambda$) thusly;
\begin{equation}
AW = (1-(I(\lambda)/\langle I(\lambda)\rangle) ) \label{EW}
\end{equation}
where $\langle I(\lambda)\rangle$  averages  two ranges, I(6547\AA-6556\AA) and I(6572\AA-6584\AA). To ensure high enough S/N for inclusion in subsequent analysis,
the measured background, $\langle I(\lambda)\rangle$, had to exceed 40 counts per pixel (S/N$\sim9$).
We used a Gaussian multi-peak fitting routine in the
GUI-based commercial package {\it IGOR}\footnote{https://www.wavemetrics.com} to derive component wavelength (for some RV determinations we used a Voigt function) and EW from a 6\AA ~span centered on the H$\alpha$~ region of each spectrum. Component FWHM typically ranged from $0.3< {\rm FWHM}< 0.7$\AA.
Figure~\ref{Qew} provides an example of the fitting process.
We tabulate the resulting H$\alpha$~ EW in \AA ~ in Table~\ref{tbl-EWs} with component 1,4 ({vA\,351C~} and B, see below) velocities  in Table~\ref{tbl-HaBCRVs} and component 2,3 velocities in Table~\ref{tbl-Ha23RVs}. While the average total H$\alpha$~ EW, $\langle{\rm EW}_{\rm tot} \rangle = -6.9\pm1.0$, indicates  a highly active {\it single} star \citep{New17}, {vA\,351~} is not a single star. The individual EW for components B and C ($\langle{\rm EW}_{\rm B} \rangle =-1.1\pm0.2 , \langle{\rm EW}_{\rm C} \rangle = -0.9\pm 0.3$), when placed on the Newton {\sl et~al.~} (2017) figure 2,  argue  only slight component over active for their masses. \cite{Alo15} find an EW$_{tot}=-7.0$ on mJD=55935, consistent with our measures in the range 49962$<$mJD$<$54484

We probed H$\alpha$~ velocity behavior close to syzygy (phases $\Phi_{\rm BC}=0.25~ {\rm and} ~0.75$) on those nights we were able to obtain observations closer to quadrature ($\Phi_{\rm BC}=0 ~{\rm and} ~0.5$). We constrained  peak parameters (amplitude, width, and Voigt shape)  to near quadrature values, solving only for position in wavelength, as demonstrated in Figure~\ref{HaRVprobe}. This approach depends on an assumption of profile stability for EW2, EW3 over hours, and provides no independent EW information at syzygy for any component.

When we first observed four components in H$\alpha$, these were ascribed to components A, B, C, and now D (3, 4, 1, and 2 in Figure~\ref{Qew}). B and C were easily identifiable with peaks 4 and 1,
given their known, short-period RV behavior from absorption lines (Figure~\ref{RVabs}).  Figure~\ref{HaRV} shows component B and C  RVs from H$\alpha$~ emission as a function of BC absorption line orbital phase.
Table~\ref{tbl-RVO} includes the results of orbit fitting to all resulting component B and C H$\alpha$~ RVs. The  $K_{\rm B}$ and $K_{\rm C}$ amplitudes from H$\alpha$~ significantly exceed those derived from the absorption lines. The strikingly systematic residuals seen in Figure~\ref{HaRV} might be explained by some sort of flow pattern.  For example the  component B positive residuals  in the ranges $0.1<\Phi<0.25$ and $0.75<\Phi <0.9$  indicate outflow from component B. This  flips to negative RV residuals towards  component B (inflow) for $0.25<\Phi<0.35$ and $0.75<\Phi<0.9$. The residuals for component C invert this sequence, indicating that a similar process works on both stars

The bottom panel of Figure~\ref{BCrules} plots H$\alpha$~ EW values for components B and C, which show strong correlation with BC orbital phase. The top of Figure~\ref{BCrules} demonstrates that the sum of EW2 and EW3 show no such correlation, arguing that these components have little association with {vA\,351B~} and C. Returning to the lower panel, EW B is slightly larger than EW C at quadrature phase $\Phi_{\rm BC}=1.0$ or 0.0.  EW C is significantly
larger than  EW B at the quadrature phase $\Phi_{\rm BC}=0.5$. These observations are consistent with the leading hemispheres of both stars having increased H$\alpha$~ activity, component C more so than B.  In general component B and C H$\alpha$~ EW is highest near quadrature ($\Phi_{\rm BC} =0.0, 0.5, 1.0$), consistent with the wide-band photometry (Figure~\ref{FP}), showing a brighter total magnitude near those phases. With an estimated $20\arcdeg$ inclination,
apparently the H$\alpha$~ emission suffers  attenuation at syzygy ($\Phi_{\rm BC} =0.25, 0.75$). While we have no direct H$\alpha$~ EW measures for {vA\,351B~} and C at syzygy, the patterns in Figure~\ref{BCrules} show a reduction in EW as either component approaches syzygy,  supporting  that hypothesis.

The period derived from the H$\alpha$~ B and C components agrees with that from absorption lines (Figure~\ref{RVabs}). The far larger
$K_{\rm B}$ and $K_{\rm C}$ amplitudes do not agree. We offer a working hypothesis to explain this inconsistency.
H$\alpha$~ becomes excited at some point between components B and C (labeled 4 and 1 in Figure~\ref{Qew}).
If a ring or cloud exists around each star, the hydrogen gas gets excited between the two components, then diffuses around to the opposite side, where shadowing cools
the gas, allowing the hydrogen to emit H$\alpha$.
To produce the larger $K_{\rm B}$ and $K_{\rm C}$ amplitudes requires cloud/ring radii approximately $1.7\times$ the star radius. Location in each shadow  increases the distance between gas and barycenter. Exhibiting a similar period,
 the emitting gas travels further in the same time, meaning it has a higher velocity.

We provide an animation (Figure~\ref{Disney}, $HalphaEWbetter.mp4$) illustrating the behavior of the various components of the H$\alpha$~ emission as a function of B-C orbital phase. We construct the animation  using only results of the constrained probing exemplified by Figure~\ref{HaRVprobe}. The traces have been shifted in wavelength to minimize  component 2 position changes (Figure~\ref{Qew}) from frame to frame. Originally produced to highlight the RV behavior of components B and C (4 and 1 in Figure~\ref{Qew}), the animation also shows the positional (RV) and EW constancy of components 2 and 3. For the 129 values in Table~\ref{tbl-Ha23RVs} peaks 2 and 3 have  $\Delta$RV=$35.4\pm3.4$ km s$^{-1}$ and for the 140 values in Table~\ref{tbl-EWs}, a $\Delta$EW=$0.02\pm0.29$
 This consistency argues for their having the same source, presumably associated with either {vA\,351A~} and/or D. Figure~\ref{HaRVAD}, showing the average of component 2 and 3 RVs plotted against the AD-BC orbital period phase, $\Phi_{\rm AD-BC}$, lends support to this assertion. The RVs exhibit significant scatter, but  agree with the orbit derived from the Figure~\ref{RVabs} absorption-line orbit. The bifurcation of the central H$\alpha$~ component (labeled 2 and 3 in Figure~\ref{Qew}) could be  either self-absorption \citep[e.g.][]{Sem13, Pav19} in a rotating cloud or emission from a rotating ring, either associated with A, or D, or both.

In Section~\ref{BS} we  provided evidence for a roughly 21$^{\rm d}$ periodicity associated with {vA\,351A~} RV residuals to the AD-BC orbit, presumably because of the orbit of {vA\,351A~} around the AD barycenter. Figure~\ref{HaRVAD} identifies H$\alpha$~ components V2 and V3 with the AD pair. The periodograms of V2 and V3 in Figure~\ref{LSRVV2V3} solidify that association, showing a very strong signal, $P\sim25^{\rm d}$ in the sum of the individual periodograms.
Fitting  sine waves to the V2 and V3 RVs yield $P_{\rm V2}=25.6\pm 0.1^{\rm d}$, $K_{\rm V2}=2.3\pm0.4$ km s$^{-1}$,
$P_{\rm V3}=25.8\pm 0.1^{\rm d}$, and $K_{\rm V3}=1.4\pm0.4$ km s$^{-1}$. Again, if a consequence of the A-D orbit, the low $K$ values suggest a nearly face-on orbit.

\subsection{He\,\text{\small{I}}} \label{specHe}

In the CE spectra the $\lambda$ 5876.6\AA ~He\,\text{\small{I}}~ line appears in emission with from one to three components.
This line typically appears in stellar absorption spectra in  B and hotter stars, requiring T$>$10,000K \citep{Cox00}. Re-inspection of CE M dwarf spectra
acquired in support of our  MLR project \citep{Ben16} shows the He\,\text{\small{I}}~ line in emission for systems with primary mass less than 0.27$\cal{M}_{\odot}~$ (Gl\,234, Gl\,791.2, Gl\,473, Gl\,65) and absent for systems with primary mass $\cal{M}$$\ge0.33$$\cal{M}_{\odot}~$ (Gl\,22, Gl\,469, Gl\,623, Gl\,831, GJ\,1081, G\,250-029). Thus, it is surprising that {vA\,351B~} and C, with $\cal{M}$$ \simeq 0.50$ $\cal{M}_{\odot}$, show He\,\text{\small{I}}~ emission.

 Figure~\ref{HeQew} contains a plot of {\it AW} (Equation~\ref{EW}, with background local to the He\,\text{\small{I}}~ line) as a function of wavelength for some of the higher S/N observations (same epochs of observation as for Figure~\ref{Qew}).  We produced He\,\text{\small{I}}~ EW
 and RVs (as  for H$\alpha$), listed in
Table~\ref{tbl-HeEWs}  and Table~\ref{tbl-HeIRV}
and plotted in Figure~\ref{HeRV} (RVs) and Figure~\ref{HeIABC} (EW). We label the central He\,\text{\small{I}}~ component EW2 and   V$_{\rm cen}$. Our He\,\text{\small{I}}~ EW and RV measures contain significantly more noise than for H$\alpha$.

Again, the RV behavior (Figure~\ref{HeRV}) of the higher and lower velocity peaks identify them as coming from components B and C. The lower S/N precludes probing in closer to syzygy as we did for H$\alpha$~ (Figure~\ref{HaRVprobe}).
However, unlike H$\alpha$, the component B and C RVs from  He\,\text{\small{I}}~ more closely track the absorption line results, yielding similar $K_{\rm B}$ and $K_{\rm C}$ RV amplitudes, with results listed in Table~\ref{tbl-RVO}. This argues that excitation and re-emission of He\,\text{\small{I}}~ occurs closer to each stellar surface. Residuals indicate that, as for H$\alpha$, simple, zero eccentricity orbital motion is an inadequate model for the motions of the He\,\text{\small{I}}~ emission.  We ascribe EW2 and V$_{\rm cen}$ to the {vA\,351~} AD components (see below for rationale).
 We derive He\,\text{\small{I}}~ EW averages from the   Table~\ref{tbl-HeEWs} values: $\langle {\rm EW}_{\rm B} \rangle$=-0.29$\pm$0.12, $\langle {\rm EW}_{\rm C}\rangle$=-0.34$\pm$0.14, $\langle {\rm EW}_{\rm cen}\rangle$=-0.64$\pm$0.45, $\langle {\rm EW}_{\rm tot}\rangle$=-1.17$\pm$0.48. Similar to H$\alpha$, most of the He\,\text{\small{I}}~ emission comes from the central source presumably associated with  components A and D.

Figure~\ref{HeIRV} shows the Table~\ref{tbl-HeIRV} V$_{\rm cen}$ RVs plotted against the AD-BC orbital phase with the absorbtion line RV orbit overplotted. While quite noisy, the V$_{\rm cen}$ velocities argue for the association of the central He\,\text{\small{I}}~ emission with the AD components of {vA\,351}.

An initial supposition, that proximity heating (components B and C separated by only 0.016 AU) excites both H$\alpha$~ and He\,\text{\small{I}}~ emission can easily be  falsified.  Our Sun with $T_{\rm eff} = 5777$K has a surface emissivity $F_\odot=6.32\times10^{10}$ erg cm$^{-2}$s$^{-1}$ \citep{Cox00}. With masses near 0.5$\cal{M}_{\odot}~$ (Table~\ref{tbl-RIP}, {\bf lines 35-36}), components B and C would have $T_{\rm eff} \simeq 3\,800$K \citep{Cox00} and $F_\odot=1.18\times10^{10}$ erg cm$^{-2}$s$^{-1}$, assuming $F\propto T^4$. For a separation r=0.016 AU (Section~\ref{BS}) the energy received by either component from the other is insufficient to raise the surface temperature to the
10\,000K necessary to excite the He\,\text{\small{I}}~ emission. This excitation level requires a non-thermal source, perhaps a flare-induced mechanism
\citep{Kow18} or magnetic fields.

Relative to the latter, \cite{Don08} and \cite{Mor08b,Mor10} have mapped the magnetic fields of a set of M dwarf stars. They find that most M dwarfs have poloidal fields which increase in intensity with rotation rate. Presuming tidal locking for components B and C, their sub- one day rotation
period and 0.5$\cal{M}_{\odot}~$ masses would suggest a magnetic field for each of $\sim 500$ Gauss (e.g. Morin et al. 2011, figure 2). \nocite{Mor11} Perhaps an interaction of the  component B and C magnetic fields excites the H$\alpha$~ and He\,\text{\small{I}}~ emission. Additionally, component poloidal fields might serve to focus energy to polar locations, similar to terrestrial aurorae. We have no concrete evidence that the He\,\text{\small{I}}~ emission arises near there. Alternatively, perhaps the magnetic field is
a mix of poloidal and toroidal, with He\,\text{\small{I}}~ emission scattered near the stellar surface.
Table~\ref{tbl-HeEWs} has  time-resolved data with which to tease out more hints as to excitation and emission. Doing so is beyond the scope of this paper. We hope that our observational data will stimulate the derivation of details of some magnetohydrodynamic mechanism operating in this system.

Lastly, we explore some relations between the various emission line EW. Figure~\ref{HaHeCor}, left, shows the strong correlation between H$\alpha$~ component 2 and 3 EW from Table~\ref{tbl-EWs}. We plot the log of the absolute value of EW2, $log |{\rm EW2}|$, against  $log |{\rm EW3}|$. The maximum values in the upper right all occur within 4 days of AD-BC periastron. We have weaker evidence that the same energy source(s)  power both the    H$\alpha$~ and He\,\text{\small{I}}~ emission associated with {vA\,351~} AD, shown by their correlation (Figure~\ref{HaHeCor}, right). The one strongest He\,\text{\small{I}}~ point also occurred near AD-BC periastron

\subsection{X-ray Emission}
 \cite{DEl13} catalog transient 0.3-10keV X-ray bursts observed by Swift-XRT \citep{Geh04}. During the seven years of operation documented in the catalog,
{vA\,351~} was detected three times, listed in Table~\ref{tbl-SX}. We tag each observation with both the BC and AD-BC orbital phases. In a $5\arcdeg$-radius subset of the catalog centered on {vA\,351}, it is among the brightest
with an average 0.3-10keV flux, $\langle {\rm Flux} \rangle = 1.62 \times 10^{-12}$ mW m$^{-2}$.
Two detections occurred close to BC syzygy, phase $\Phi_{\rm BC} \sim  0.25, 0.75$.
One detection occurred very close to AD-BC periastron.
The BC syzygy correlation suggests the presence of active (coronal?) regions on the side of each star facing the other
(i.e., facing the observer at syzygy), perhaps associated with dark spots.
The BC system is fainter at  phase $\Phi_{\rm BC} = 0.75$ (Section~\ref{Fphot},
Figure~\ref{FP}), when {vA\,351B~} is closest to us. Dark spots on the hemispheres of each star facing the other
might explain both features. From admittedly small number statistics we abstract that some X-ray behavior  comes from BC, some from periastron interaction of AD with BC

\section{Discussion} \label{SciFi}
With all of these (sometimes conflicting) results we now search for a coherent description of the entire {vA\,351~} system.
To do so, we  summarize measured results, identify inferences drawn from these results, clearly label
conjectures, and end with constructing a provisional picture of  {vA\,351}.

 {\it FGS~} TRANS mode and speckle camera astrometry confirm {vA\,351~} binarity with a high eccentricity, low inclination orbit,  producing a total mass requiring more than just two M dwarfs. TRANS and speckle measures provide a near-zero magnitude difference between the the originally identified components A and B. RVs acquired from two primary sources yield a component A-B orbit agreeing with the TRANS/speckle result. These RVs also identify two M dwarfs in a very short period, circular orbit, $P_{\rm BC}=0.749^{\rm d}$.
CfA and IGRINS analyses yield $v\,sin\,i \simeq11$ km s$^{-1}$~ for each component. The $K$ values for  {vA\,351B~} and C provide
 $\cal{M}_{\rm C}$/$\cal{M}_{\rm B}$=0.945. {\it FGS~} POS mode astrometry yields an independent parallax, a proper motion, and  a mass fraction,  $f$. The mass fraction shows $\cal{M}_{\rm A}$~far too large to be only an M dwarf. To component A we add component D, presumed to be a white dwarf. $\cal{M}_{\rm B}$, too large to be a single M dwarf, now acquires component C, known from RV spectra to be another M dwarf. We  derive individual masses for all components, using the distance, total masses, masses of detected components, AD-BC magnitude difference, an estimate of interstellar extinction,
$\cal{M}_{\rm C}$/$\cal{M}_{\rm B}$~ mass ratio, $V-$ and $K-$band Mass-Luminosity relations, and a $K-$band WD Mass-Luminosity relation (assuming a 670 My age for the Hyades).

The values for $\cal{M}_{\rm B}$ ~and $\cal{M}_{\rm C}$ ~and the observed RV amplitudes yield an orbital inclination, $i_{\rm BC} \simeq 25\arcdeg$ and a physical separation,
$a_{\rm BC}\simeq 0.016$ AU. This inclination is consistent with an observed $v\,sin\,i \simeq11$ km s$^{-1}$,  plausible B and C radii, spin-orbit alignment, and tidal locking. Periodograms of {vA\,351A~} absorption line RVs and variations in AD-BC separation suggest $P_{AD}\sim22^d$.  $\cal{M}_{\rm A}$, $\cal{M}_{\rm D}$ ~and a presumed astrometric detection limit yield an orbital inclination, $i_{\rm AD} \simeq 90\arcdeg$, and for a circular orbit, a physical separation, $a_{\rm AD}\sim 0.2$ AU.

We observe H$\alpha$~ in emission with  from two to four components. Components B and C are easily identified through RV variation with the same period, $P_{\rm BC}=0.749^{\rm d}$. These variations have  larger velocity amplitudes than for the absorption lines.  The  H$\alpha$~ BC velocity variation with larger amplitude than seen for the absorption line measures likely places the H$\alpha$~ emission
further from the BC system barycenter, behind, in the shadow of each component, where it can cool sufficiently to emit. The
EW behavior, phased to the BC orbital period and residual patterns seen for the H$\alpha$~ RV BC orbits offers supporting evidence.
The two central H$\alpha$~ peaks have an RV signature agreeing with that for component AD-BC orbit. Periodograms of their RVs show a strong peak at $P\sim 25^{\rm d}$.

We observe He\,\text{\small{I}}~ in emission at $\lambda 5875$\AA ~with from one to three components. Again, identification of components B and C come from RV variation with the same period, $P_{\rm BC}=0.749^{\rm d}$, this time with  velocity amplitudes similar to those from absorption lines. This suggests closer proximity of the He\,\text{\small{I}}~ emission regions  to each component  B and C. We ascribe B, C  emission, which requires
an excitation temperature, T$\sim10,000$K, to non-thermal  heating effects, either near continuous micro-flaring or magnetic field interactions.
The RVs of the strongest He\,\text{\small{I}}~ peak identifies it with either A and/or D.

X-ray emission has been detected when B-C are near syzygy, phase $\Phi_{\rm BC} \sim  0.25, 0.75$. One outburst occurred very near AD-BC periastron.
Based on sparse sampling, FGS photometry (with an F583W bandpass containing H$\alpha$) suggests that the system is fainter at B-C orbital syzygy, B nearer to us than C. Spots on each hemisphere of components B and C facing the other could satisfy the photometry. Spots associated with stellar activity could then explain
the X-ray coincidence with BC syzygy. A remaining photometric mystery is that the $V$-band photometry yields $P_V=1.78^{\rm d}$, a variation unidentifiable with any previously determined activity.

When initially formed, all components of {vA\,351~}  could have had radii significantly larger than at present. For example the 0.4$\cal{M}_{\odot}~$ components of PAR 1802 in the Orion Nebula Cluster (age 1 My), have radii $R\simeq1.65R_{\odot}$ \citep{Che12}, four times larger than drawn in Figure~\ref{LooneyTunes2}. This raises the interesting possibility that the AD and BC separations have changed over time. \cite{Nao14} and  \cite{Nao16} demonstrate that the Kozai-Lidov Effect could force BC into a smaller separation, effected by  interactions with the more distant AD components, which might be treated as a single mass. On the other hand $\cal{M}_{\rm BC}$ ~acting as a single mass could have rearranged components A and D into a smaller separation orbit. This latter orbital evolution almost certainly had to occur, given that the WD component D, if similar to other known WD Hyads, had a $\sim 3$$\cal{M}_{\odot}~$ \citep{Sal18} early A to late B progenitor, whose red giant phase would have engulfed component A at the inferred, present A-D separation.

\section{Conclusions} \label{summ}
These results support a final  picture: a binary subsystem with M dwarf component A and white dwarf component D with $P_{\rm AD}\sim22^{\rm d}$ lies 0.5-4.4 AU from a binary subsystem with components B and C, both M dwarf stars. The AD-BC inclination, $i_{\rm AD-BC}=14\arcdeg\pm8\arcdeg$, compared to our inferred B-C inclination, $i_{\rm BC}\sim20\arcdeg$ suggests coplanarity.

The cartoon in Figure~\ref{LooneyTunes2} shows the B and C components of the total system as seen from Earth, with an inclination
inferred from the derived masses (Table~\ref{tbl-RIP}) and absorption line $K_{\rm B}$ and $K_{\rm C}$, with parallax (Table~\ref{tbl-SUM}) impressing a scale in AU. In the absence of any measure, we have constrained the position angle of the line of nodes, $\Omega$, to that of the AD-BC orbit. We take the component stellar sizes  from the \cite{Boy12} mass-radius relation for eclipsing binaries, they typically having short orbital periods, as does {vA\,351~} BC.
The H$\alpha$~   EW$_{\rm B,C}$ trend towards smaller values as $\Phi_{\rm BC}$ approaches $0.25{\rm ~or~} 0.75$ in  Figure~\ref{BCrules}, demonstrating some eclipse (or darker spot) behavior in whatever
excites the H$\alpha$~  emission. Seen from the other one, both the B and C components subtend only 20$\arcdeg$ on the sky.

It is difficult to choose among accretion, stellar winds, continuous microflaring,  and magnetic field interactions to stimulate the H$\alpha$~ and He\,\text{\small{I}}~ emission lines.
We illustrate our hypothesis in Figure~\ref{LooneyTunes2}. The H$\alpha$~ emission, once excited anywhere,
 can sufficiently cool
enough to emit H$\alpha$~ in shadowed regions near the triangles (always close to components B and C, respectively), at the distance from the BC system center of mass required to produce the observed H$\alpha$~ RV $K$ values. We estimate their placement by solving Equation~\ref{PJeq} for $\alpha$ values   constrained  by our now known parallax and period, and the higher H$\alpha$~  $K$ values  (Figure~\ref{HaRV}, Table~\ref{tbl-RVO}). The fat arrows in Figure~\ref{LooneyTunes2} indicate notional coronal loop-like flows or circulation that we infer from the
H$\alpha$~ RV residuals in Figure~\ref{HaRV}. We have made no attempt to provide detailed flow patterns.
The He\,\text{\small{I}}~ emission (from EW, Figure~\ref{HeIABC})
varies little and its velocity matches that from absorption lines, so is likely produced near the stellar surface. The He\,\text{\small{I}}~ RV residuals in Figure~\ref{HeRV}  hint at a  flow pattern similar to that of H$\alpha$.
 These data do not constrain an exact He\,\text{\small{I}}~ excitation location (other than near the stellar surfaces), but observations of active M stars would suggest the polar regions of each
component.  We offer no hypothesis explaining the genesis of H$\alpha$~ and He\,\text{\small{I}}~ emission associated with the AD pair, other than to point out that a 670 My old, 0.53$\cal{M}_{\odot}~$ WD would have a surface temperature capable of producing such emission.

\section{Summary}\label{sum}

\begin{enumerate}
\item Fringe scans (TRANS mode) spanning 4.7 years and a few more recent speckle observations yield a
low inclination, highly eccentric orbit. They also provide component $\Delta V$ measurements,
indicating nearly equal brightness components.

\item Absorption-line radial velocities from the McDonald Observatory 2.1m
  telescope and Cassegrain Echelle spectrograph and a long series of observations from the CfA Digital Speedometer
   yield an RV orbit which agrees with that derived from astrometry. They also yield a period
  and mass ratio for two, short orbital period M dwarf components (now identified as B and C) of this system, with $P_{\rm BC}=0.7492425\pm0.0000003$ days.

\item Spanning 1.8 years, fringe tracking (POS) observations of  {vA\,351A~}, including
  photocenter corrections, provide an absolute parallax with
  a 4\% error, $\varpi_{\rm abs} = 18.37\pm0.65$ mas, a proper motion,
  and a mass fraction relative to the  astrometric reference frame. This parallax establishes Hyades membership for {vA\,351},
  satisfying our first goal.

\item The astrometric solution yields  period, $P=2.705\pm0.004$ y, and a total mass for the system, ${\cal M}_{tot} = 2.06 \pm 0.24{\cal M}_{\sun}$. The mass fraction provides ${\cal M}_{\rm A} = 1.14
\pm 0.14{\cal M}_{\sun}$ and ${\cal M}_{\rm B} = 0.91 \pm 0.11
    {\cal M}_{\sun}$, masses  inconsistent with M dwarf components. We assume that component A is comprised of components A and D, and component B
    is component B plus component C. The {vA\,351~} system is now ${\cal M}_{\rm AD} = 1.14$$\cal{M}_{\odot}~$ and  ${\cal M}_{\rm BC} = 0.91$$\cal{M}_{\odot}$.

\item We have achieved our second goal, that of establishing component masses. Combining $V$ and $K$ photometry with a parallax  derived from our astrometry, $\cal{M}_{\rm AD}$, $\cal{M}_{\rm BC}$, the AD - BC magnitude difference, $\cal{M}_{\rm C}$ /$\cal{M}_{\rm B}$ ~mass ratio from absorption line RVs,  an estimate for interstellar absorption, M dwarf Luminosity-Mass relations, and a WD Luminosity-Mass relation for an assumed Hyades age of 670 My we solve for individual component masses that would best satisfy all the priors. We find $\cal{M}$$_{\rm A} = 0.53\pm0.10$$\cal{M}_{\odot}$, $\cal{M}$$_{\rm B} = 0.43\pm0.04$$\cal{M}_{\odot}$, $\cal{M}$$_{\rm C} = 0.41\pm0.04$$\cal{M}_{\odot}$, and $\cal{M}$$_{\rm D} = 0.54\pm0.04$$\cal{M}_{\odot}$. The WD mass, $\cal{M}$$_{\rm D}$, comes only from the $K-$band photometry. Including an extinction, $A_V=0.6$, lowers the Table~\ref{tbl-RIP} fit residuals, improving the consistency between the $V$ and $K$ solutions, yielding $\cal{M}$$_{\rm A} = 0.56\pm0.07$$\cal{M}_{\odot}$, $\cal{M}$$_{\rm B} = 0.46\pm0.01$$\cal{M}_{\odot}$, $\cal{M}$$_{\rm C} = 0.44\pm0.01$$\cal{M}_{\odot}$, and $\cal{M}$$_{\rm D} = 0.53\pm0.04$$\cal{M}_{\odot}$.

\item The $\cal{M}_{\rm C}$ ~and $\cal{M}_{\rm B}$, the parallax, and the  absorption line RV amplitudes, $K_{\rm B}$ and $K_{\rm C}$, yield an inclination, $i_{\rm BC} \simeq 23\arcdeg$ and separation, $a_{\rm BC}\simeq 0.016$ AU, consistent with an observed $v\,sin\,i =11\pm2$ km s$^{-1}$, and plausible B and C radii.

\item Periodograms of the TRANS mode astrometric residuals provide extremely weak evidence for a $P_{\rm AD} \sim 25^{\rm d}$. The component A RV AD-BC orbit residuals exhibit a  strong peak in the periodogram at $P_{\rm AD} \sim 21^{\rm d}$.

\item The spectra show  H$\alpha$~  in emission with from two to four components.
The centrally located (in RV) components 2 and 3 are associated with the AD components, and may be caused by self absorption in a circumbinary hydrogen cloud. The amplitudes of the B and C RVs place emission further from the BC system barycenter.

\item  We see He\,\text{\small{I}}~ ($\lambda$ 5876.6\AA) in emission with from one to three components. We identify component B and C emission from
RV behavior, which in this case more closely tracks the absorption line RVs. Hence, we surmise a location for two of the He\,\text{\small{I}}~ components
closer to the surfaces of B and C than the H$\alpha$~ emission. The RV behavior of the central He\,\text{\small{I}}~ component supports association with the AD components of the system.

\item Only partially achieving our third goal, a qualitative model explaining all observed phenomena, we propose  hypotheses that will require future observations and theoretical interpretation for completion.

\end{enumerate}

Some future {\sl Gaia~} catalog could contain AD-BC orbital parameters and confirm the total system mass. However,  the very small magnitude difference between the AD and BC pairs could make that determination problematical. Additional low-cadence monitoring of the system with speckle techniques could further improve
the AD-BC orbit. {vA\,351~} remains a phenomena-rich laboratory for future magnetohydrodynamic model testing

\acknowledgments

Support for GFB, OGF, and BEM for this work was provided by NASA through grants GTO-4892, 5657;
and GO-6479, 6881 from the Space Telescope Science Institute,
which is operated by the Association of Universities for Research in
Astronomy, Inc., under NASA contract NAS5-26555.  GT acknowledges partial support from the NSF through grant AST-1509375.
Similarly, EPH acknowledges support from the NSF through grants AST-1517824 and AST-1909560 for the analysis of the speckle data.

This publication makes use of data products from the Two Micron All
Sky Survey, which is a joint project of the University of
Massachusetts and the Infrared Processing and Analysis
Center/California Institute of Technology, funded by NASA and the NSF.
This research has made use of the SIMBAD and Vizier databases and
Aladin, operated at CDS, Strasbourg, France, the NASA/IPAC
Extragalactic Database (NED) which is operated by JPL, California
Institute of Technology, under contract with the NASA, and NASA's
Astrophysics Data System Abstract Service. We acknowledge the essential contribution of the
WD cooling models Web site (\url{http://www.astro.umontreal.ca/~bergeron/CoolingModels}).

This work has made use of data from the European Space Agency (ESA) mission
{\it Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by the {\it Gaia}
Data Processing and Analysis Consortium (DPAC,
\url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding for the DPAC
has been provided by national institutions, in particular the institutions
participating in the {\it Gaia} Multilateral Agreement.

We thank Art Bradley and Linda Abramowicz-Reed for their unflagging and expert {\it FGS~}
instrumental support over the last 25 years.  Cassegrain Echelle (CE)
Spectrograph observing and data reduction assistants included
J.~Crawford, Aubra Anthony, Iskra Strateva, Tim Talley, Amber
Armstrong, Robert Hollingsworth, Casey Kyte, and Jacob Bean.  We thank Dave Doss,
John Booth, and many other support personnel at McDonald Observatory
for their cheerful assistance over many years. Thank you Ray Lucas for morale-building
violin playing during a particularly productive observing run!
GFB thanks Chick Woodward and Nancy Morrison
for the suggestion of self-absorption to explain the central H$\alpha$~ peaks.
We thank Team IGRINS, especially Greg Mace and Heeyoung Oh for getting an April 2019 DCT observation.
We thank Gerard van Belle, Catherine Clark, and Zachary Hartman at Lowell who battled difficult weather conditions to secure the February 2020
speckle observation.

The American Astronomical Society supported the preparation of this paper
as GFB carried out his duties as Society Secretary. For this he is sincerely grateful.
GFB fondly remembers Debbie Winegarten (R.I.P.), whose able assistance with Secretarial matters freed me to devote  time to this analysis.

We thank an anonymous referee for a careful and useful assessment,  improving the final paper.

redpajama_set_name: RedPajamaArXiv


text:
Q: Floor casting algorithm just doing weird things I'm actually using the algorithme of Toto Briac from Raycaster, searching for more effective floor/ceiling raycast
When using it, the floor casting is working well for east and west side, but for north and south it just does weird things (just look image).
    double pixelsToBottom;
double pixelsToMid;
double directDistFloor;
double realDistance;
double y;
t_point f_p;

pixelsToBottom = (double)data->s_height - wall[1].y;
pixelsToMid = (double)data->s_height / 2 - pixelsToBottom;
for (int i = pixelsToMid; i < data->s_height / 2; i += 1)
{
    directDistFloor = (data->dist_proj * (double)(data->s_height / 2)) / i;
    realDistance = directDistFloor / fabs(cos(angle));
    f_p.x = data->player.pos.x + cos(angle) * (realDistance) / (data->dist_proj / (64.0));
    f_p.y = data->player.pos.y + sin(angle) * (realDistance) / (data->dist_proj / (64.0));
    y = (wall->x + (i + data->s_height / 2) * data->s_width) / data->s_width;
    pixel_put(&data->obj, wall->x, y, f_pixel(data, f_p));
}

But i'm facing an issu, when i'm facing north and south side it's all ok, but when the ray is going into east or west side, the texture just do a weird thing like that :
I know that it refer to : realDistance = directDistFloor / fabs(cos(angle));
if i replace the cos(angle) by sin(angle) in this line, it just invert things. I didn't find a way to change the calcule in right moment. I you have any idea I will take it ! Thank you !!

A: It's okay I found the answer, for the people who are interest, in the line   realDistance = directDistFloor / fabs(cos(angle)); I was using the angle of the ray in the world, I changed it by the angle relative to my player dir (so 0° if it's the player dir ray) and it work properly ! Thank's @ZwergofPhoenix for the time you took !

redpajama_set_name: RedPajamaStackExchange


text:
Топірець — вид сокири, характерний для карпатських областей України.
 Топірець Степан Васильович — український кобзар
 Топірець (річка) — річка що протікає через селище Немішаєве, бере свій початок зі сторони села Микуличи, а кінець зі сторони села Козинці

redpajama_set_name: RedPajamaWikipedia


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redpajama_set_name: RedPajamaCommonCrawl


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While connections with others has received the greatest attention among mental health clinicians and researchers, there are a variety of forms in which connections exist. Psychiatrist and friend Dr. Ned Hallowell wrote in his book Connect: 12 Vital Ties that Open Your Heart, Lengthen Your Life, and Deepen Your Soul, "What is connection or what I call connectedness? It is a feeling of being a part of something larger than yourself. This something may be a friendship, a marriage, a team, a school, a company, an activity you love, a country, even a set of ideals, like the Bill of Rights, or a belief system, like a religion." Ned offers ample evidence of the healing power of connections throughout his wonderful book.
A thought-provoking report was recently released that provides further evidence of the significant role of connections in addressing and preventing an array of childhood problems. The report, "Hardwired to Connect: The New Scientific Case for Authoritative Communities," was prepared by the Commission on Children at Risk, a group comprised of 33 prominent children's doctors, research scientists, and mental health and youth service professionals. The Commission details the problems faced by children in the United States, casting the spotlight on the lack of connections in the lives of these youngsters.
"The first part is the deteriorating mental and behavioral health of U.S. children. We are witnessing high and rising rates of depression, anxiety, attention deficit, conduct disorders, thoughts of suicide, and other serious mental, emotional, and behavioral problems among U.S. children and adolescents.
To address this lack of connectedness, the Commission advocates the creation of "authoritative communities." The word "authoritative" is borrowed in part from the work of Dr. Diana Baumrind who more than 30 years ago described different styles of parenting, including authoritative, permissive, authoritarian, and neglectful. Parents who are defined as authoritative are warm, involved, and accepting, and establish clear-cut and reasonable guidelines, consequences, and expectations. Research has consistently demonstrated that children are more likely to experience healthy emotional development when they are reared by parents who practice an authoritative approach.
It is a social institution that includes children and youth.
It treats children as ends in themselves.
It is warm and nurturing.
It establishes clear limits and expectations.
The core of its work is performed largely by non-specialists.
It has a long-term focus.
It reflects and transmits a shared understanding of what it means to be a good person.
It encourages spiritual and religious development.
It is philosophically oriented to the equal dignity of all persons and to the principle of love of neighbor.
Knowledge of the 10 characteristics of an authoritative community together with an appreciation of this primary goal can guide our individual behaviors especially in terms of the responsibility each of us assumes to create environments in which children thrive and feel connected. When problems arise in our youth, we must certainly provide services to address these problems. However, if as the Commission contends, children are "hardwired" from birth to connect with others, the more we focus our time, attention, energy, and resources on forming strong bonds with our youth, the more likely we will prevent problems from arising. In past writings, I have referred to studies that indicate that when students feel a positive attachment to at least one adult in their school, they are less likely to engage in acts of violence or drop out. The power of relationships must never be underestimated in preventing the emergence of a wide spectrum of self-defeating, dangerous behaviors in our youth.
What are one or two things I've done this month to strengthen my connections with my family?
What are one or two things I've done this month to strengthen my connections with my community?
What activities have I engaged in this month that have contributed to the well-being of others?
What programs exist in my community that strengthen connections between adults and youth and adults with other adults and do I participate in these programs?
Do I treat others with respect and dignity, as I myself would like to be treated?
Do I teach children, by example, a set of moral values that are in accord with principles of honesty, decency, and compassion?
‹ Hazing: Rituals of Bonding or Humiliation?


text:
Q: Breadcrumb doesn't work on a component I created (Joomla!) I'm learning how to use Joomla! 3.2
I created a new component, then I added it to my main menu,
Here are the only parameters I changed in the main menu:

But now when I go to my new component the breadcrumb doesn't work correctly, there's only the home button and when I click on it I'm redirected to the same page:

Did i missed something very important? How can I fix it? The breadcrumb works on every other pages.

A: Ok I found the problem, I forgot to add this
$pathway = $app->getPathway();
$breadcrumb = $pathway->setPathway(array());
$pathway->addItem( JText::_( 'COM_PARTISPOLITIQUES' ),JRoute::_('index.php?option=com_partispolitiques&view=partispolitique'));
$pathway->addItem( "Current page name",'');

In the file view.html.php located in my component folder...

A: I am going to assume you are using the Helix template as it seems to be a common oversight.
Make sure that Enable Page Title is enabled from your menu Helix Page Title tab.
If not, please advise as to what template.


text:
Q: Creating a class with push and pop methods in Python I am trying to create a Python object from the class below, but when I call the method popit(), I am getting a None. Any help as to what I miss or do wrong is appreciated.
class Stack:
def __init__(self, *item):
    self.stack = []
def pushit(self, item):
    if len(self.stack) == 9:
        print("Stack is full: " + self.stack)
    else:
        print(self.stack.append(self.item))

def popit(self):
    if self.stack == []:
        print("Stack underflow: " + self.stack)
    else:
        print(self.stack.pop())

A: Your methods -- both of them -- don't return anything. What I would do to sort this is:
class Stack:
    def __init__(self):
        self.stack = []

    def pushit(self, item):
        if len(self.stack) == 9:
            print("Stack is full: " + self.stack)
            return None
        else:
            print(self.stack.push(self.item))
            return self.item

    def popit(self):
        if self.stack == []:
            print("Stack underflow: " + self.stack)
            return None
        else:
            ret = self.stack.pop()
            print(ret)
            return ret


text:
Signature Staff (formerly Occasional & Permanent Staff) has been successfully placing members of household staff with families based in the UK or overseas for over 60 years.
From nannies to chefs, Signature Staff acts as a matchmaker to source the candidate that is the best-fit for your household based on your requirements and their skill-set and personality.
The Signature Staff team know what our clients require from their household staff, having worked in private households as nannies, maternity nurses and housekeepers ourselves.
We are particular about who we represent both from a client and candidate perspective as we want to ensure roles are long-lasting and relationships enduring.
Tonya Illing
Head of Maternity & Temporary Nanny Desks
8D8B39A2-9AD3-42F2-8A37-B9F3D0D13847
Tonya leads the team dedicated to all temporary nanny and maternity nurse placements in the UK and abroad.
After leaving her family's farm in South Africa at eighteen, Tonya spent seven years as an international nanny before joining our ranks as a receptionist in 2002. She soon made herself indispensable and is now one of the best-known names in the business, as well as the cog that keeps our office turning.Tonya is a foodie, loves going on curious tours around the city and her specialist subject is London Public Transport.
tonya@signaturestaff.com
Sarah Hankinson
Head of Overseas Nanny & Live-In UK Nanny Desks
Sarah leads the team responsible for all overseas and live in positions.
Sarah grew up in Cheshire, before moving to London in 2015 to continue working as a Nanny whilst completing her degree. She joined the team in 2019 and has plenty of experience as a Nanny in London, so is able to offer plenty of advice to clients and candidates! Sarah enjoys travelling and trying to convince everyone in the office we need an office pet.
sarah@signaturestaff.com
Serena Zanchetti
Head of Household Staff & Live-Out UK Nanny Desks
Serena leads the team responsible for live-out nanny and domestic staffing roles.
Serena is originally from sunny Italy where she used to teach English as a foreign language to primary and secondary school children. She moved to the UK in 2009 to become a Nanny and joined the Signature Staff team in early 2020.
Serena can speak Italian, English and Spanish fluently and is currently challenging herself in learning French, Swedish and Romanian. Coffee enthusiast and Friends fanatic, after work you can find her at a pub quiz or a dance class.
serena@signaturestaff.com
Paula Costa
Head of Admin & Support
Paula Costa leads the administration and support team. She or one of her team are generally the first people you will speak to when calling in, and we are sure you will find them very enthusiastic, thoughtful, and extremely helpful.
Paula is originally from Brazil where she studied a Bachelor of Education before moving to London in 2014. After spending nearly a decade working as a nanny, Paula decided it was time for a career change and she joined Signature in 2019.
Not only is Paula a pro when it comes to recruitment and technology, but she can also advise our team about the best TV shows and series to watch.
admin@signaturestaff.com
Jana Kocevska
Recruitment Administrator
Jana provides administrative support to the recruitment consultants. She is the newest member of our team, having previously worked in childcare for 15 years.
Jana has an extensive experience as a Nanny and Personal Assistant, and is very passionate about her role as a Recruitment Administrator.
Before moving to the UK in 2004, she received a Bachelor Degree in Translation in English and French in North Macedonia, . Jana is a bright and fun individual who is an avid cyclist, enjoys travelling and learning new languages.
jana@signaturestaff.com
Sara Pasquini
Overseas Recruitment Consultant
Sara started her career as a Tutor in her native Italy and moved to London in 2009 to pursue new opportunities.
She joined Signature Staff in 2020 bringing a unique set of skills built up during her career in education and hospitality. Furthermore, Sara is multi-lingual which enables her to build relationships with Signature Staff's global client base.
Sara has a passion for discovering new cuisines, she is vegan and an animal lover. In her spare time, she enjoys new Netflix Series.
sara.p@signaturestaff.com
James Pawle
As Chairman, James oversees the strategic direction of the business.
James has a background in finance where he has more than 10 years experience investing in and providing operational oversight to companies across a diverse range of sectors including recruitment, luxury retail, professional services, school uniform, automotive, building materials, and film distribution.
James enjoys exploring new countries, and participating in and watching most sports. James is also a huge fan of great food and wine.
james@signaturestaff.com
I'm looking for staff
I'm looking for a job
Register as a candidate
Please do get in touch with one of our Signature Staff team members to discuss your requirements.
Work with Signature Staff
If you're interested in working with us, send us an email at admin@signaturestaff.com, and don't forget to send us your CV.


Located in Marin County, San Rafael is a community of just over 58,000 residents. San Rafael is located in the North Bay area of San Francisco, a short 30-minute drive from the city. Locals describe it as a country town with plenty of city vibe, providing residents with a plethora of activities to entertain their time. From visits to the magnificent parks and hiking trails, to nights at the art gallery, there is never a shortage of fun in San Rafael. Downtown San Rafael is a main highlight of the town, with many bars, restaurants, and shops available. The community is comprised of a variety of interesting neighborhoods, each with their own unique traits and characteristics. Glenwood is a desirable San Rafael neighborhood that sits upon China Camp State Park. Terra Linda is also a popular neighborhood with a mix of residential homes and businesses.
Match me with 3 top San Rafael Realtors!


Gary Barnes is the global Premier High-Performance Business and Sales Coach, also known as "The Traction Coach". He is an author, an international speaker and the founder of Gary Barnes International. Gary is dedicated to helping people create wealth in all areas of life.
People make a difference – UC Dominguez Hills California , public access TV.