\documentclass[journal]{rmaa}

\onecolumn

\usepackage{multirow}
\usepackage{multicol}


\title{The Kinematics and Velocity Ellipsoid of the G III Stars}

\author{Richard L. Branham, Jr.
\affil{Instituto Argentino de Nivolog\'ia, Glaciolog\'ia y Ciencias Ambientales, Argentina}}


\fulladdresses{
\item Richard L. Branham, Jr.: Instituto Argentino de Nivolog\'ia, Glaciolog\'ia y 
Ciencias Ambientales, C.C. 330, 5500 Mendoza, Argentina (richardbranham\_1943@yahoo.com).}


\shortauthor{Branham}

\shorttitle{Kinematics and Velocity ellipsoid of G giants}


\listofauthors{R.~L.~Branham~Jr.}

\indexauthor{Branham, R.~L., Jr.}


\ReceivedDate{2010 October 27} 
\AcceptedDate{2011 March 10}
\SetYear{2011}


\resumen{Para estudiar la cinem\'{a}tica de las estrellas G gigante (clase de luminosidad III) se usan movimientos propios y paralajes 
de 3,075 estrellas, de las cuales 658 tienen velocidades radiales. Estas estrellas provienen de la reducci\'{o}n nueva hecha por van Leeuwen 
del cat\'{a}logo Hipparcos. La soluci\'{o}n da para la velocidad solar $16.72\pm 0.41$~km~s$^{-1}$; para las constantes de Oort, en 
unidades de km~s$^{-1}$~kpc$^{-1}$, $A=14.05\pm 3.28$ y $B=-9.30\pm 2.87$, valores que representan una velocidad local de rotaci\'{o}n de 
$198.48\pm 26.95$~km~s$^{-1}$ si suponemos una distancia al centro Gal\'{a}ctico de $8.2\pm 1.1$~kpc. Para las dispersiones de velocidades 
obtenemos, en unidades de km~s$^{-1}$: $\sigma _{x}=51.78\pm 0.55$, $\sigma _{y}=42.81\pm 0.32$, $\sigma _{z}=28.45\pm 0.22$ con una 
desviaci\'{o}n del v\'{e}rtice de $3.^{\circ }88\pm 6.^{\circ }62$. Una comparasi\'{o}n de esta dispersi\'on con las obtenidas de otras 
clases espectrales indica que la discontinuidad de Parengo existe tambi\'{e}n para las estrellas gigantes. }

\abstract{To study the kinematics of the G giant stars (luminosity class III) use is made of proper motions and parallaxes taken from van 
Leeuwen's new reduction of the Hipparcos catalog. 3,075 stars, of which 658 have radial velocities, were used in the final study. The solution 
gives: solar velocity of $16.72\pm 0.41$~km~s$^{-1};$ Oort's constant's, in units of km~s$^{-1}$~kpc$^{-1}$, $A=14.05\pm 3.28$ and $B=-9.30\pm 2.87$, 
implying a rotational velocity of $198.48\pm 26.95$~km~s$^{-1}$ if we take the distance to the Galactic center as $8.2\pm 1.1$~kpc; velocity 
dispersions, in units of km~s$^{-1}$, of: $\sigma _{x}=51.78\pm 0.55$, $\sigma _{y}=42.81\pm 0.32$, $\sigma _{z}=28.45\pm 0.22$ with a vertex 
deviation of $3.^{\circ }88\pm 6.^{\circ }62$. A comparison of the velocity dispersions with those given by other spectral types shows that 
Parenago's discontinuity also exists for the giant stars.}%\bigskip}


\addkeyword{Galaxy: kinematics and dynamics} 
\addkeyword{methods: numerical} 


\begin{document}


\maketitle


\section{Introduction}

This paper continues a series on the kinematics and velocity ellipsoids of
the giant stars (luminosity class III). Previously studied were the O-B5
giants (Branham 2006), the M giants (Branham 2008), the B6-9 and A giants 
(Branham 2009a), the K giants (Branham 2009b), and the F giants (Branham 
2010). The G giants fill the remaining lacuna and complete the investigation 
of all of the giant stars. A search of the ADS data base\footnote{\url{http://adswww.harvard.edu/}.} 
shows that the G giants as a group have not been studied since the research of Parenago
in 1951 (Delhaye 1965), which adds impetus to this current study.

The methodology remains similar to that for the previous studies, so similar
that I will eschew presentation of the mathematical development and refer
the reader to the relevant previous publications where the necessary
equations can be found. As in my investigation of the F giants the velocity
ellipsoid calculation uses the singular value decomposition (SVD) to include
stars for which only tangential velocities, but no radial velocities, are
available. The section on the equations of condition briefly discusses this
matter.

Little evidence exists to suggest that G stars form part of the Gould belt.
There appears to be a clean break between the O-B stars and the later
spectral types regarding participation in the Gould belt; O-B stars have
definite Gould belt members whereas the others do not. Nevertheless, a plane
will be fit to the G giants in a later section, but to examine the
randomness of the data, not to infer that some G giants actually belong to
the Gould belt.

To summarize briefly the mathematical procedure used, one solves for the
kinematics and velocity ellipsoid of the G giants by use of semi-definite
programming (SDP), which forces the solar motion calculated from the
velocity ellipsoid to be the same as that calculated from the kinematical
parameters. Nor is it necessary to use the same adjustment criterion for the
two sets of calculations: the kinematical parameters may be reduced by use
of a least squares criterion whereas the velocity ellipsoid may be
calculated with the robust $L_{1}$ criterion (minimize the sum of the
absolute values of the residuals), or with the same $L_{1}$ criterion
for both. For a readable discussion of SDP see Vandenberge \& Boyd (1996).

In their classical work {\it Statistical Astronomy}, Trumpler \& Weaver
(1962) refer to two incompleteness factors, $K_{1}$, which compensates for
the deficiency of proper motions in a parallax catalog compared with a
proper motion catalog, and $K_{2}$, which corrects for the absence of proper
motions nearly in the line of sight and thus not detectable in either a
proper motion or a parallax catalog. This study, however, shows that the
calculation of incompleteness factors for the G giants is unnecessary or
counterproductive.

\section{The Observational Data}

The proper motions and parallaxes used in this study were taken from van
Leeuwen's version of the Hipparcos catalog (van Leeuwen 2007), henceforth
called simply the Hipparcos catalog, the radial velocities from the
Wilson (Nagy 1991) and Strasbourg Data Centre (Barbier-Brossat \& Figon
2000) catalogs. van Leeuwen's catalog  (2007) omits a few stars contained in the
original catalog (ESA 1997). For those few stars the relevant data were
taken directly from the original catalog. The equinox of the 
Hipparcos catalog is J2000 and the catalog epoch is J1991.25. Stars listed
as spectral class G, luminosity class III were extracted from the catalog.
This resulted in a total of 3,075 G giants, of which 658 have radial
velocities. The G giants are skewed towards the later giants; over 94\% fall
between G5 and G9, and the G8 stars alone account for 64.9\% of the total.

The star's HD number determined if either of the two radial velocity
catalogs contained an entry for that particular star. Not all of the data
could be accepted. Negative parallaxes were excluded as were parallaxes
smaller than 1 mas because the Ogorodnikov-Milne (OM) model was used for the
equations of condition (Ogorodnikov 1965, pp. 61--63). This model, valid out
to about 1~kpc, should be adequate because the minimum parallax used in this
study, 1~mas, corresponds to a distance of 1~kpc. For a justification of
this distance limit see Smart (1968, p. 285). Parallaxes smaller than 1 mas
have such large mean errors that their inclusion seems unwarranted because
of the uncertainty in their distances. Known multiple stars, flagged in the
catalog, contaminate the proper motion by confusing orbital motion with
genuine proper motion and were also excluded. And some of the solutions for
the astrometric data in the catalog, also flagged, are substandard and were
likewise excluded. Smith \& Eichhorn (1996) have derived a procedure to
correct the observed parallaxes, and this procedure was used to transform
all of the parallaxes used in this study. In my study of the M giants
(Branham 2008) I show that the Smith-Eichhorn correction seems to leave
little residual parallax error.

What about the quality of the data? I have already commented on the high
quality of the Hipparcos proper motions (Branham 2009b). Tangential
velocities calculated from proper motions, therefore, should also be high
quality. The radial velocities, however, come from disparate sources
incorporated into the Wilson and the Strasbourg Data Center catalogs. My study of the
F giants showed lower homogeneity in the radial velocities. One could
perform a similar analysis with the G giants, but this seems otiose. The
data are what they are and must be used as they are. The radial velocities,
moreover, are not used alone but multiplied by the parallax in the equations
of condition. A runs test shows that the radial velocities are of somewhat
lower quality than the tangential velocities, but of acceptable quality. See
the \S~\ref{sec:results}. The runs test measures how often a variable,
distributed about the mean, changes sign from plus to negative or negative
to positive. The changes of sign, the runs, have a mean for $n$ data points
of $n/2+1$ and a variance of $n(n-2)/4(n-1)$. An advantage of the runs test
over other tests for randomness resides in its being nonparametric, making
no assumption about the normality of the data, although to actually
calculate probabilities for the observed runs one does assume approximate
normality. For a detailed description of the runs test see Wonnacott \&
Wonnacott (1972, pp. 409--411).


%%%%% FIGURA 1

\begin{figure}[!t]
\centering
\includegraphics[clip,width=0.7\textwidth]{rbranhamfig1.eps}
\caption{Space distribution of G giants.}
\label{Fig1}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%% FIGURA 2

\begin{figure}[!t]
\centering
\includegraphics[clip,width=0.7\textwidth]{rbranhamfig2.eps}
\caption{Distribution in $x-y$ plane.}
\label{Fig2}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%% FIGURA 3

\begin{figure}[!t]
\centering
\includegraphics[clip,width=0.7\textwidth]{rbranhamfig3.eps}
\caption{Distribution in $x-z$ plane.}
\label{Fig3}
\vspace{0.2cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%% FIGURA 4

\begin{figure}[!t]
\centering
\includegraphics[clip,width=0.7\textwidth]{rbranhamfig4.eps}
\caption{Distribution in $y-z$ plane.}
\label{Fig4}
\vspace{0.2cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{The space distributions}

Let $x, y, z$ be rectangular coordinates with origin at the Sun: $x$ points towards the Galactic 
centre, $y$ is perpendicular to $x$ in the direction if increasing $l$, and $z$ is positive for 
positive Galactic latitude. From $\varpi$, the star's parallax, $l$, its Galactic longitude, 
and $b$, its Galactic latitude, we calculate
\begin{equation}
\left( 
\begin{array}{c}
x \\ 
y \\ 
z
\end{array}
\right) =\frac{1}{\varpi }\left( 
\begin{array}{c}
\cos l\cos b \\ 
\sin l\cos b \\ 
\sin b
\end{array}
\right) .  
\label{1}
\end{equation}

Figure~1 shows the distribution of the G giants in space, Figures~2, 3, and 4
the distributions in the $x-y$, $x-z$, and $y-z$ planes. Define a moment matrix,
referred to the centroid of the distances, $\bar{x},\bar{y},\bar{z}$, from
the $x, y, z$:
\begin{equation}
\left( 
\begin{array}{ccc}
\sum_{i}(x_{i}-\bar{x})^{2} & \sum_{i}(x_{i}-\bar{x})(y_{i}-\bar{y}) & \sum_{i}(x_{i}-\bar{x})(z_{i}-\bar{z}) \\ 
\sum_{i}(y_{i}-\bar{y})(x_{i}-\bar{x}) & \sum_{i}(y_{i}-\bar{y})^{2} & \sum_{i}(y_{i}-\bar{y})(z_{i}-\bar{z}) \\ 
\sum_{i}(z_{i}-\bar{z})(x_{i}-\bar{x}) & \sum_{i}(z_{i}-\bar{z})(y_{i}-\bar{y}) & \sum_{i}(z_{i}-\bar{z})^{2}
\end{array}
\right) .  
\label{2}
\end{equation}

Before use of the moment matrix outliers should be eliminated from the
distances. The criterion selected for the cutoff was five times the median
of the distances to the stars. This cutoff results in only a sparse trim of
the data. Stigler (1977) has shown that modest trimming works better than
extreme trimming. Although the eigenvalues of the moment matrix indicate
little concentration towards the Galactic plane, 129.7, 94.0, and 80.9~pc$^{2}$, 
the normalized eigenvector associated with the $z$-axis points towards $b_{g}=55.^{\circ }671$, 
a significant tilt with respect to the Galactic plane. This tilt, however, seems 
to be a selection effect associated with the
giants rather than a tilt associated with the Gould belt. Look at the G
stars as a whole without discrimination as to luminosity class. There are 82
supergiants, 184 bright giants, 3,075 giants, 1,152 subgiants, 4,656 main
sequence, and 9,332 unspecified luminosity class G stars. With this
heterogeneous group the tilt becomes $79.^{\circ }494$. Further evidence
that the tilt seems not associated with the Gould belt arises from an
attempt to fit a plane to the G giants. Branham (2003) outlines the
procedure for doing this. 1,389 stars are classified as ``Gould belt'', but
the tilt of the plane of the remaining, supposedly ``Galactic belt'', stars
becomes even worse, $35.^{\circ }58$, and with high correlations, up to
40\%, among the $x, y, z$ coordinates.. Thus, the procedure that works so well
with the O-B5 stars to discriminate between Gould belt and Galactic belt
stars fails completely for the G giants.

That the tilt should not unduly bias the solution can be inferred from a
look at the randomness of the rectangular coordinates. The correlation
between $x$ and $y$ is $-$10.1\%, between $x$ and $z$ $-$12.9\%, and between $y$
and $z$ 11.3\%. Regarding the randomness in distance a runs test yields,
after elimination of discordant distances, 1,487 runs out of an expected
1531 implying an 11.3\% chance that the distances are random. The statistics
for the G giants, therefore, indicate \textit{relative} randomness.


\section{Equation of condition for kinematics and the velocity ellipsoid}

The equations of condition, given in detail in Branham (2009a) and which
come from Ogordnikov (1965, pp. 74--75), involve twelve unknowns for the
kinematical parameters: the components of the reflex solar motion $X, Y, Z$
and the components of the displacement tensor $u_{x}, u_{y}, u_{z}, v_{x}, v_{y}, v_{z}, w_{x}, w_{y}, w_{z}$. 
All of these quantities are referred to Galactic latitude and longitude, $l$ and $b$,
rather than right ascension $\alpha $ and declination $\delta $. Proper
motions in $\alpha $ and $\delta $ are converted to proper motions in $l$
and $b$ expressed in milli-arc-seconds (mas) per year; radial velocity is
expressed in km~s$^{-1}$. Parallax $\varpi $ is also expressed in mas. These
kinematical parameters are calculated from the least squares criterion
because there are fewer discordant observations, handled by a 2.5\% trim of
the data, than with the velocity ellipsoid calculation, for which the robust
$L_{1}$ criterion is indicated. As with linear programming, one does this by
minimization of an objective function. If $r_{i}$ is one of $m$ residuals
from a solution for the kinematical parameters and $r$ is the $m$-vector of
the residuals, then we impose the condition $r^{T}\cdot r-\tau =0$, where $\tau$ 
is an arbitrary parameter, and minimize $\tau $ in the objective function.

A sparse trim of the data seems indicated not only by what Stigler (1977)
has found, light trimming or even no trimming works better than extreme
trimming, but by what a General Colby reported to the Astronomer Royal, Sir
George Airy (Airy 1854). In a geodetic adjustment for the determination of the
scale of longitudes for England, inclusion of all data rather than just the
most concordant data gave results that went from, to use Airy's words,
``considerably in error'' to ``perfectly good.''

The equations as derived by Ogorodnikov actually use the distance $1/\varpi $
rather than the parallax $\varpi $ itself, but it is important to recast the
equations to remove the parallax error from the denominator and thus
ameliorate any possible Lutz-Kelker bias. The equations, therefore, are
multiplied by $\varpi$, which places the parallax in the numerator.

To calculate the velocity ellipsoid use $\dot{x}, \dot{y}, \dot{z}$, the space
velocities of a star, found from the proper motions and radial velocity and
expressed in km~s$^{-1}$ are:
\begin{equation}
\left( 
\begin{array}{c}
\dot{x} \\ 
\dot{y} \\ 
\dot{z}
\end{array}
\right) =\left( 
\begin{array}{ccc}
-\sin l & -\cos l\sin b & \cos l\cos b \\ 
\cos l & -\sin l\sin b & \sin l\cos b \\ 
0 & \cos b & \sin b
\end{array}
\right) \cdot \left( 
\begin{array}{c}
\kappa \mu _{l}\cos b/\varpi \\ 
\kappa \mu _{b}/\varpi \\ 
\dot{r}
\end{array}
\right) .  
\label{3}
\end{equation}

If radial velocities are unavailable, then we seem unable to calculate the
space velocities. This, however, can be done if we use the SVD to employ
only the tangential velocities. See Branham (2010) for details. There are
other possibilities. Fuchs et al. (2009) use a deprojection
formulation for the proper motions that assumes the lines of sight towards
the stars are statistically uncorrelated with the velocities of the stars.
The Section~\ref{sec:disc} shows that the SVD approach satisfies the
assumption of statistical randomness and seems, therefore, satisfactory.

One fits a quadric surface with ten coefficients, $a, b, c, d, e, f, g, h, k, l$, to
these velocities. To assure that the equation indeed corresponds to an
ellipsoid one must impose the condition that the matrix of the $a,b,c$
coefficients
\[
A=\left( 
\begin{array}{ccc}
a & c/2 & d/2 \\ 
c/2 & b & e/2 \\ 
d/2 & e/2 & c
\end{array}
\right) ,
\]
in fact be positive-definite and symmetric. To avoid the trivial solution $a=b=\cdots =q=0$ 
another condition must be imposed. The one I use is that
the volume of the ellipsoid must be a maximum. Because the volume is
proportional to the determinant of $A$, the condition becomes $\det (A)=\max$. An 
eigenvalue-eigenvector decomposition of the matrix $A$ yields the axes
of the velocity ellipsoid and their orientation with respect to the Galactic
coordinate system. Because of generally greater error in the data, the
coefficients $a,\ldots ,l$ are calculated by use of the robust $L_{1}$ criterion. 
SDP allows one to combine without difficulty the least squares
criterion for the kinematical parameters and the $L_{1}$ criterion for the
velocity ellipsoid. For the latter if there are $n$ space velocities with $n$
corresponding residuals $r_{sv,i}$ , let $\gamma _{i},i=1,\ldots ,n$, be $n$ 
\textit{positive} parameters. Then minimize the $\gamma $ in the objective
function subject to the conditions
\[
\mathrm{diag}(\gamma +r_{sv})>0; \qquad \mathrm{diag}(\gamma -r_{sv})>0; \qquad \mathrm{diag}(\gamma )>0. 
\]

The SDP formulation of the velocity ellipsoid calculation along with the $L_{1}$ criterion 
for the minimization of the residuals offers advantages over
competing methods. The ellipsoid calculated is unique and represents a
global minimum of the reduction criterion when that criterion is the $L_{1}$
(Calafiore 2002). Bochanski, Hawley, \& West (2011) prefer a geometric
simplex minimization with the least squares criterion. Although simplex
minimization can be used with the $L_{1}$ criterion, thus making the
procedure robust-see the algorithm in Branham (1990, pp. 191--197 ), the
method can converge to a local rather than to a global minimum, particularly
when many parameters are being fit, unless a good starting approximation is
available. Pourbaix (1998) , using the simplex algorithm to calculate orbits
of double stars, estimates that when $n$ parameters are being fit there are 
$\approx \mathrm{O}(e^{n})$ local minima. He implements a simulated annealing
modification of the simplex algorithm to reject the local minima. With SDP
and the $L_{1}$ criterion such a strategy becomes unnecessary because the
minimum \textit{is} global.

The solar velocity, $S_{0}=\sqrt{X^{2}+Y^{2}+Z^{2}}$, calculated from both
the solution for the kinematical parameters $X,Y,\ldots ,w_{y},w_{z}$ and
the coefficients of the velocity ellipsoid must be the same. This condition
can be imposed when one uses the SDP formulation of the problem.


\section{Some Corrections to the Observations and Covariance Matrices}

The total space motions needed in the velocity ellipsoid calculation should
be corrected for the effects of Galactic rotation by modifying the proper
motions and radial velocities used in the calculations to remove the
rotation. This was done by the same procedure used in Branham (2009a).

In theory one should also apply a correction for the incompleteness of the sample of 
the G giant stars taken from the Hipparcos catalogue. Trumpler \& Weaver (1962, p. 374) 
define a factor of incompleteness $K_{1}$ as
\begin{equation}
K_{1}=\frac{N(m,\mu)}{N_{\varpi }(m,\mu )},  
\label{4}
\end{equation}
where $N(m,\mu)$ is the number of stars in the sky for magnitude interval $m\pm \Delta m/2$ and 
proper motion interval $\mu \pm \Delta \mu /2$ and $N_{\varpi }(m,\mu)$ is the number of stars in 
the parallax catalogue for the same intervals. Equation~(4) is difficult to apply if there is 
insufficient overlap between the proper motion catalog and the parallax catalog. For the Hipparcos 
parallaxes a logical proper motion catalog would be the Tycho II catalog (H\o g et al. 2000). But 
for $K_{1}$ the sparse overlap between the two catalogs assures that the factor becomes large with 
large mean errors. One must question whether such corrections are realistic and should be applied. 
I feel they should not and that the randomness of the data is more important.

Trumpler \& Weaver (1962, p. 375) also define a second incompleteness factor, $K_{2}$, to correct for the 
absence of proper motions in the parallax catalog nearly along the line of sight and hence undetectable. 
$K_{2}$ depends on the velocity ellipsoid. Branham (2009b) shows how this incompleteness factor can 
be evaluated. For the G giants this factor becomes an insignificant $5.6 \cdot 10^{-7}$. Therefore, 
neither the $K_{1}$ nor the $K_{2}$ incompleteness factor need be applied.

The covariance matrix to calculate mean errors is given in equation~(25) of Branham (2006), and equation~(26) 
of that publication shows how to calculate the errors for quantities, such as the Oort constants, 
derived from the displacement tensor.


%%%%% TABLA 1

\begin{table}[!t]
\vspace{0.1cm}
\centering
\caption{Solution for Kinematic Parameters for the G III stars}
\label{Table1}
\small
\begin{tabular}{lcc}
\toprule
\multicolumn{1}{c}{Quantity} & Value & Mean Error \\ 
\midrule
$\sigma (1)$ (mean error of unit weight in mas~km~s$^{-1})$ & ~112.44   & $\cdots $ \\ 
$u_{x}$ (in mas~km~s$^{-1})$                                 & ~~$-$7.63 & 6.13 \\ 
$u_{y}$ (in mas~km~s$^{-1})$                                 & ~~23.35   & 4.06 \\ 
$u_{z}$ (in mas~km~s$^{-1})$                                 & ~~$-$3.59 & 5.23 \\ 
$v_{x}$ (in mas~km~s$^{-1})$                                 & ~~~4.75   & 4.63 \\ 
$v_{y}$ (in mas~km~s$^{-1})$                                 & ~~$-$8.08 & 6.00 \\ 
$v_{z}$ (in mas~km~s$^{-1})$                                 & ~~$-$6.87 & 5.21 \\ 
$w_{x}$ (in mas~km~s$^{-1})$                                 & ~~~1.57   & 4.15 \\ 
$w_{y}$ (in mas~km~s$^{-1})$                                 & ~~$-$7.68 & 3.74 \\ 
$w_{z}$ (in mas~km~s$^{-1})$                                 & ~$-$11.39 & 5.86 \\ 
$S_{0}$ (solar velocity in km~s$^{-1}$)                      & ~~16.72   & 0.41 \\ 
$A$ (Oort constant in km~s$^{-1}$~kpc$^{-1}$)                & ~~14.05   & 3.28 \\ 
$B$ (Oort constant in km~s$^{-1}$~kpc$^{-1}$)                & ~~$-$9.30 & 2.87 \\ 
$V0$ (Circular velocity in km s$^{-1})$                     & ~198.48   & 26.95 \\ 
$l_{1}$ (longitude displacement)                            & ~$-26.^{\circ}45$ & $7.^{\circ}05$ \\ 
$K$ (K term in km~s$^{-1}$)                                 & ~~~$-$7.85 & 4.99 \\ 
\bottomrule
\end{tabular}
\vspace{0.3cm}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%% TABLA 2

\begin{table}[!t]
\vspace{0.1cm}
\centering
\caption{Velocity dispersion and vertex deviation of the G III stars}
\label{Table2}
\small
\begin{tabular}{lcc}
\toprule
\multicolumn{1}{c}{Quantity} & Value & Mean Error \\ 
\midrule
Mean absolute deviation of residuals in km~s$^{-1}$      & ~~11.00            & $\cdots$ \\ 
$S_{0}$ (solar velocity in km~s$^{-1}$)                   & ~~16.72           & 1.05 \\ 
$\sigma _{1}$ (velocity dispersion in $x$ in km~s$^{-1})$ & ~~51.78           & 0.55 \\ 
$\sigma _{2}$ (velocity dispersion in $y$ in km~s$^{-1})$ & ~~42.81           & 0.32 \\ 
$\sigma _{3}$ (velocity dispersion in $z$ in km~s$^{-1})$ & ~~28.45           & 0.22 \\ 
$l_{1}$ (longitude of $\sigma _{1})$                      & ~~$3.^{\circ}88$   & $6.^{\circ}62$ \\ 
$b_{1}$ (latitude of $\sigma _{1})$                       & ~~$0.^{\circ}28$   & $0.^{\circ}61$ \\ 
$l_{2}$ (longitude of $\sigma _{2})$                      & ~$93.^{\circ}93$  & $1.^{\circ}90$ \\ 
$b_{2}$ (latitude of $\sigma _{2})$                       & ~~$9.^{\circ}86$   & $0.^{\circ}65$ \\ 
$l_{3}$ (longitude of $\sigma _{3})$                      & $-87.^{\circ}72$ & $1.^{\circ}62$ \\ 
$b_{3}$ (latitude of $\sigma _{3})$                       & ~$80.^{\circ}14$  & $0.^{\circ}49$ \\ 
\bottomrule
\end{tabular}
\vspace{0.3cm}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Results}
\label{sec:results}

After the equations of condition for the kinematical parameters had been
formed, I applied two checks for the adequacy of the reduction model. The
first check simply calculates the singular values of the matrix of the
equations of condition. An inadequate reduction model, for example one in
which some unknowns are strongly correlated, results in a high condition
number for the matrix because of small singular values. The condition number
of the matrix of the equations of condition for the G giants, however, is
low, 22.8. The second check calculates Eichhorn's efficiency (Eichhorn
1990), a parameter that varies from 0 to 1 with 0 indicating redundancy in
the parameters and 1 that all parameters are necessary. The efficiency of
0.94 indicates that \textit{all} of the variables in the model are necessary
and with little correlation among themselves.


%%%%% FIGURA 5

\begin{figure}[!t]
\centering
\includegraphics[clip,width=0.7\textwidth]{rbranhamfig5.eps}
\caption{'.'=rectangular velocity of star (upper); velocity ellipsoid (lower).}
\label{Fig5}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%% FIGURA 6

\begin{figure}[!t]
\centering
\includegraphics[clip,width=0.7\textwidth]{rbranhamfig6.eps}
\caption{Ellipsoid in $x-y$ plane.}
\label{Fig6}
\vspace{0.2cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


The first solution for the G giants was calculated from all of the equations
of condition. These solutions calculated residuals needed to find discordant
data. For the G stars the criterion was five times the mean absolute
deviation (MAD) of the residuals. This eliminated 168 of the 6,808 equations
of condition, a 2.5\% trim. I have used this elimination criterion in the
past, generally with good results, but a further reason exists to justify
its use. For the G giants there is a 19.0\% chance that the original
residuals are random, as calculated by a runs test, but a 78.7\% chance that
the trimmed residuals are random. Eliminating some of the residuals,
therefore, increases the randomness of the sample.

Table~1 shows the solution for the kinematical unknowns and Table~2 for the coefficients and 
orientation of the velocity ellipsoid for the G stars. For convenience the components for the 
displacement tensor are converted to the more familiar form of the solar motion, Oort constants, 
the deviation $l_{1}$ between the longitude of the geometric center and the kinematic center of 
the Galaxy, and $K$ term. Also shown is the circular velocity $V_{0}$, found from the relation 
$V_{0}=(A-B)R_{0}$, where $R_{0}$ is the distance to the centre of the Galaxy. Kerr \& Lynden-Bell 
(1986) determine a value of $8.5 \pm 1.1$~kpc for $R_{0}$. Perryman (2008, App. A), however, after 
a survey of recent determinations feels that 8.2~kpc is a better determination. The mean error 
for $V_{0}$ comes from the procedure given in Branham (2008) and uses 8.2~kpc for $R_{0}$ with the 
same mean error as given by Kerr \& Lynden-Bell (1986).

The orientation of the velocity ellipsoid in space and in the $x-y$, $x-z$, and
$y-z$ planes is shown in Figures~5--8 for the G giants. (Because of the density
of data points, it proved impossible to plot both the stars and the velocity
ellipsoid on one graph because the stars blotted out the ellipsoid;
therefore a subgraph was used for the stars and another for the ellipsoid.)


%%%%%% FIGURA 7

\begin{figure}[!t]
\centering
\includegraphics[clip,width=0.7\textwidth]{rbranhamfig7.eps}
\caption{Ellipsoid in $x-z$ plane.}
\label{Fig7}
%\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%% FIGURA 8

\begin{figure}[!t]
\centering
\includegraphics[clip,width=0.7\textwidth]{rbranhamfig8.eps}
\caption{Ellipsoid in $y-z$ plane.}
\label{Fig8}
\vspace{0.15cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Discussion}
\label{sec:disc}

The distribution of the residuals from the kinematical solution, after
eliminating discordant residuals, is seen in the histogram of Figure~9. As
mentioned in the previous section, 168 of the residuals were eliminated, a
2.5\% trim. The distribution is somewhat skewed, coefficient of skewness
0.09, more platykurtic, kurtosis of 1.42, than the normal distribution,
kurtosis of 3, and more lighter tailed, Hogg's Q factor of 0.36, than a
normal distribution, Q=2.58. The Q factor is defined as
\begin{equation}
Q=\frac{(U_{0.05}-L_{0.05})}{(U_{0.5}-L_{0.5})},  
\label{5}
\end{equation}
where $U_{\alpha }$ and $L_{\alpha }$ are averages of the respective upper
and lower 100$\alpha$ of the data (Stigler 1977). A runs test, however,
reveals 3,350 runs out of an expected 3,404. The residuals, therefore, can
be considered random. To be specific, there is a 19.0\% chance that the
residuals are taken randomly from a normal distribution. Because we
generally use a 5\% to 10\% limit before rejecting the null hypothesis that
a distribution is in fact not random, we infer that although the actual
distribution deviates from normality the residuals conform to the null
hypothesis of being random. If the runs test is applied separately to the
equations of condition arising from the tangential velocities and those from
the radial velocities, the former show 3,032 runs out of an expected 3,075
and the latter 303 runs out of an expected 329. This demonstrates that the
tangential velocities seem of higher quality than the radial velocities,
31.2\% probability of randomness versus 4.6\%, but because they are used
conjointly the residuals remain relatively random. All of this confirms,
along with the singular values and Eichhorn's efficiency, that the reduction
model suffers no serious defects and that inclusion of the $K_{1}$
incompleteness factor seems unnecessary.


%%%%% FIGURA 9

\begin{figure}[!t]
\centering
\includegraphics[clip,width=0.7\textwidth]{rbranhamfig9.eps}
\caption{Histogram of residuals from kinematical solution.}
\label{Fig9}
\vspace{0.2cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


For the residuals from the velocity ellipsoid the situation becomes
different, as Figure~10 shows. The residuals deviate even more from a normal
distribution, coefficient of skewness 3.62, platykurtic, kurtosis 0.59, and
light tailed, Q factor of 0.25. They are, however, even more random than the
residuals from the kinematical solution, 1,784 runs out of an expected
1,813. There is thus a 33.7\% probability that the residuals are random.
This is the principal justification for use of the SVD to calculate the
velocity ellipsoid.


%%%%% FIGURA 10

\begin{figure}[!t]
\centering
\includegraphics[clip,width=0.68\textwidth]{rbranhamfig10.eps}
\caption{Residuals in velocity for G giants.}
\label{Fig10}
\vspace{0.15cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%% TABLA 3

\begin{table}[!t]
%\vspace{0.1cm}
\centering
\caption{Recent determinations of the solar velocity}
\label{Table3}
\setlength{\tabnotewidth}{0.8\textwidth}
\tablecols{6}
\small
\begin{tabular}{lccccc}
\toprule
\multicolumn{1}{c}{Reference} & Class & $-X$ (km~s$^{-1}$) & $-Y$ (km s$^{-1}$) & $-Z$ (km s$^{-1}$) & $S_{0}$ (km s$^{-1}$) \\ 
\midrule
Branham (2008)              & M III             & $8.99\pm 0.42$      & $20.40\pm 0.40$       & $4.80\pm 0.39$      & $24.20\pm 0.70$\tabnotemark{a} \\ 
Yuan et al. (2008)          & O-B5              & $9.17\pm 0.40$      & $8.66\pm 0.38$        & $5.83\pm 0.34$      & $13.90\pm 0.38$\tabnotemark{b} \\ 
Yuan et al. (2008)          & K-M III           & $18.46\pm 0.32$     & $17.70\pm 0.32$       & $6.32\pm 0.32$      & $20.61\pm 0.32$\tabnotemark{c} \\ 
Aumer \& Binney (2009)      & mixture IV, V     & $9.96\pm 0.33$      & $5.25\pm 0.54$        & $7.07\pm 0.34$      & $13.29\pm 0.72$ \\ 
Branham (2009a)             & B69 III           & $9.58\pm 0.39$      & $11.94\pm 0.41$       & $6.03\pm 0.33$      & $16.40\pm 0.40$\tabnotemark{a} \\ 
Branham (2009a)             & A III             & $9.85\pm 0.59$      & $8.00\pm 0.58$        & $5.80\pm 0.52$      & $13.95\pm 0.58$\tabnotemark{a} \\ 
Branham (2009b)             & K III             & $7.53\pm 0.26$      & $19.11\pm 0.26$       & $7.41\pm 0.22$      & $21.83\pm 0.26$\tabnotemark{a} \\ 
Bobylev \& Bajkova (2010)   & Galactic masers   & $5.5\pm 2.2$        & $11.0\pm 1.7$         & $8.5\pm 1.2$        & $15.0\pm 3.0$ \\ 
Branham (2010)              & F III             & $10.84\pm 0.49$     & $12.62\pm 0.49$       & $8.14\pm 0.44$      & $16.72\pm 0.41$\tabnotemark{a} \\ 
Sch\"{o}nrich et al. (2010) & F-G V             & $11.1_{-0.75}^{+0.69}$ & $12.24_{-0.47}^{+0.47}$ & $7.25_{-0.36}^{+0.37}$ & $\cdots$ \\ 
Shen \& Zhang (2010)        & Galactic Cepheids & $12.58\pm 1.09$     & $14.52\pm 1.06$       & $8.98\pm 0.98$      & $21.21\pm 1.81$\tabnotemark{d} \\ 
\bottomrule
\tabnotetext{a}{Mean error calculated from equations~(25) and (26) in Branham (2006).} 
\tabnotetext{b}{For heliocentric distance 0.2--3~kpc.}
\tabnotetext{c}{For heliocentric distance 0.2--1~kpc.}
\tabnotetext{d}{For heliocentric distance 0.2--3~kpc.}
\end{tabular}
%\vspace{0.2cm}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Regarding recent determinations of the kinematical parameters Perryman (2008, p. 502) gives his Table~9.3 with pre-2007 
results while Table~3 shows some post-2008 values. If we look at the 33 values for the solar velocity in both tables, 
without discriminating among spectrum-luminosity classes nor weighting by number of stars, there is a range from a minimum 
of 11.8~km~s$^{-1}$ to a maximum of 24.6 with mean 18.29 and standard deviation 3.74. The value in Table~1 falls well within 
this range. With respect to the Oort constants and associated quantities such as $A-B$ and $-(A+B)$, the former equal to 
$V_{0}/R_{0}$ and the latter to ($dV/dR)$, Perryman's Table~9.3 shows determinations up to 2007 and Table 4 post-2008 
determinations, a total of 28. We see that A ranges from a minimum of 9.6~km~s$^{-1}$~kpc$^{-1}$ to a maximum of 19 with a 
mean of 14.52 and a standard deviation of 2.52, again without discriminating as to number of stars, spectrum or luminosity 
class, or other indicators. The value given in Table~1 coincides well with this mean. For B the corresponding values are 
minimum of $-$24~km~s$^{-1}$~kpc$^{-1}$, maximum of $-6.34$, mean $-12.63$, and standard deviation of 3.30. Once again, the 
value for B given in Table~1 shows no anomaly. The only quantity that shows a possibly discrepant value is the K term, 
putatively significant only for the early stars, with determinations falling near 5~km~s$^{-1}$, and close to 0 for later 
spectral types. The value in Table~1, large and moreover negative, seems discrepant. Its mean error, however, is also large 
and furthermore Branham (2009a) has shown that this term is sensitive to errors in the data; little credence, therefore, should 
be placed on its value. In a recent paper McMillan \& Binney (2010) find that the most probable range for $V_{0}/R_{0}$ falls 
between 29.9~km~s$^{-1}$~kpc$^{-1}$ and 31.6~km~s$^{-1}$~kpc$^{-1}$. Many of the values in Perryman's Table~9.3 and Table~3 fall 
outside of this range which, however, merely shows that quantities such as the distance to the centre of the Galaxy and the 
Sun's circular velocity are difficult to determine.


%%%%% TABLA 4

\begin{table}[!t]
\vspace{0.1cm}
\centering
\caption{Recent determintions of Oort constants and auxiliary quantities}
\label{Table4}
\setlength{\tabnotewidth}{0.8\textwidth}
\tablecols{6}
\setlength{\tabcolsep}{0.925\tabcolsep}
\footnotesize
\begin{tabular}{lccccc}
\toprule
\multicolumn{1}{c}{Reference}  & Class             & $A$ (km~s$^{-1}$~kpc$^{-1}$) & $B$ (km~s$^{-1}$~kpc$^{-1}$) & $A-B$ & $-(A+B)$ \\ 
\midrule
Branham (2008)            & M III             & $16.86\pm 2.78$ & $~-6.34\pm 2.56$ & $23.20\pm ~7.79$\tabnotemark{a} & $-10.52\pm 8.82$\tabnotemark{a} \\ 
Yuan et al. (2008)        & O-B5              & $15.33\pm 0.94$ & $-15.12\pm 0.71$ & $30.45\pm ~1.18$\tabnotemark{b} & $~-0.21\pm 1.18$\tabnotemark{b} \\ 
Yuan et al. (2008)        & K-M III           & $15.86\pm 1.30$ & $-14.57\pm 1.01$ & $30.44\pm ~1.65$\tabnotemark{c} & $~~1.29\pm 1.65$\tabnotemark{c} \\ 
Bobylev \& Bajkova (2010) & Galactic masers   & $17.8\pm 0.8$   & $-13.2\pm 1.5$   & $31.1~\pm ~1.7~$                & $~~4.6~\pm 1.7~$ \\ 
Branham (2009a)           & B69 III           & $11.77\pm 1.66$ & $~-9.05\pm 1.38$ & $20.82\pm ~3.83$\tabnotemark{a} & $~-2.72\pm 1.04$\tabnotemark{a} \\ 
Branham (2009a)           & A III             & $11.48\pm 5.45$ & $~-8.29\pm 4.23$ & $19.77\pm 12.09$\tabnotemark{a} & $~-3.19\pm 7.44$\tabnotemark{a} \\ 
Branham (2009b)           & K III             & $13.08\pm 1.72$ & $-10.21\pm 1.47$ & $23.29\pm ~2.20$\tabnotemark{a} & $~~2.86\pm 0.82$\tabnotemark{a} \\ 
Branham (2010)            & F III             & $14.85\pm 7.47$ & $-10.85\pm 6.83$ & $23.35\pm ~4.07$\tabnotemark{a} & $~-4.75\pm 4.63$\tabnotemark{a} \\ 
Shen \& Zhang (2010)      & Galactic Cepheids & $17.42\pm 1.17$ & $-12.46\pm 0.86$ & $29.88\pm ~1.45$                & $~-4.96\pm 1.45$\tabnotemark{d} \\ 
\bottomrule
\tabnotetext{a}{Mean error calculated from equations~(25) and (26) in Branham (2006).}
\tabnotetext{b}{For heliocentric distance 0.2--3~kpc.} 
\tabnotetext{c}{For heliocentric distance 0.2--1~kpc.} 
\tabnotetext{d}{For heliocentric distance 0.2--3~kpc.}
\end{tabular}
%\vspace{0.1cm}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%% TABLA 5

\begin{table} [!t]
\vspace{0.1cm}
\centering
\caption{Velocity dispersions for the giant stars}
\label{Table5}
%\setlength{\tabnotewidth}{0.8\textwidth}
\tablecols{5}
\setlength{\tabcolsep}{1.2\tabcolsep}
\small
\begin{tabular}{ccccc}
\toprule
Spectral type & $\sigma _{x}$ (km~s$^{-1}$) & $\sigma _{y}$ (km~s$^{-1}$) & $\sigma _{z}$ (km~s$^{-1}$) & Number of stars \\ 
\midrule
O-B5 & $32.44\pm 5.04$ & $26.16\pm 2.75$ & $18.71\pm 2.39$ & 107 total space motion \\ 
B6-9 & $39.25\pm 3.29$ & $10.83\pm 1.16$ & $14.07\pm 0.85$ & 147 total space motion \\ 
A    & $26.95\pm 4.26$ & $23.08\pm 2.14$ & $16.46\pm 0.55$ & 144 total space motion \\ 
\noalign{\smallskip}
\multirow{2}{*}{F} & \multirow{2}{*}{$36.89\pm 1.90$} & \multirow{2}{*}{$24.66\pm 1.16$} & \multirow{2}{*}{$17.97\pm 0.81$} & ~222 total space motion \\ 
                   &                                  &                                  &                                  & ~369 tangential velocity \\
\noalign{\smallskip}
\multirow{2}{*}{G} & \multirow{2}{*}{$51.78\pm 0.55$} & \multirow{2}{*}{$42.81\pm 0.32$} & \multirow{2}{*}{$28.45\pm 0.22$} & ~658 total space motion \\ 
                   &                                  &                                  &                                  & 2417 tangential velocity \\
\noalign{\smallskip}
K    & $50.58\pm 0.99$ & $42.42\pm 1.13$ & $32.92\pm 0.56$ & 880 total space motion \\ 
M    & $57.40\pm 1.67$ & $45.86\pm 1.63$ & $33.84\pm 1.02$ & 480 total space motion \\ 
\bottomrule
\end{tabular}
%\vspace{0.1cm}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


About the velocity ellipsoid little can be said because the only previous
study of all of the G giants is that of Parenago (Delhaye 1965, p. 64),
which used fewer stars, 345, and a different reduction method, the method of
moments (Trumpler \& Weaver 1962, pp. 283--286). The dispersions of the
velocity ellipsoid are higher than those Parenago found, but this is a
consequence of use of the SDP method; see Branham (2004).

Some insight, however, can be gained if we take these results not in
isolation but rather conjointly with my previous studies of the giant stars.
Because all of these studies use the same reduction method, variations
caused by differences in the calculation of the velocity ellipsoid, such as
use of the method of moments, will be minimized. Table~5 shows the velocity
dispersions for all of the giant stars from O to M.

Two conclusions follow from an examination of Table~5. There is a clear
break in all of the velocity dispersions between the F and the G giants.
Dehnen \& Binney (1998) confirmed this break, known as Parenago's
discontinuity, for the main sequence stars, but it appears as if the giant
stars also show the discontinuity. Use of the SVD to include stars for which
only tangential velocities are available, the second conclusion, seems
justified on more than just a statistical basis. Neither the F giants
compared with earlier spectral types that use only total space motions nor
the G giants with later spectral types exhibit glaring discrepancies,
indicating that the tangential velocities are integrated well with the total
space motions.

\section{Conclusions}

Semi-definite programming proves itself once again a useful tool for
problems of Galactic kinematics by allowing one to combine a solution for
the kinematical parameters such as the Oort constants with one for the
coefficients of the velocity ellipsoid. The singular value decomposition
allows one to incorporate stars for which only tangential velocities but no
radial velocities are available into the calculation of the velocity
ellipsoid. When applied to the G III stars the calculated solutions appear
concordant with what others have found. A comparison with giant stars of
other spectral types confirms that Parenago's discontinuity exists for the
giant stars as well as main sequence stars.

\vspace{-0.25cm}

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\end{document}
