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\markboth{I. I. Nikiforov}%
{Systematic Error in $R_0$ from Solving for Stellar Orbit Around Sgr A*}
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\begin{document}
\sloppy %\large
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\title{On a Source of Systematic Error in Absolute Measurement of
Galactocentric
Distance from Solving for the Stellar Orbit Around Sgr A*}
%\end{center}
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%\large
\author{Igor' I. Nikiforov}
%\end{center}
%\it
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%\begin{center}
\affil{Sobolev Astronomical Institute, St.~Petersburg State
University, Universitetskij pr.~28, Staryj Peterhof, St.~Petersburg 198504,
Russia, nii@astro.spbu.ru}
%\end{center}
\rm
%\smallskip %\vskip 12pt
\begin{abstract}
Eisenhauer et al. (2003, 2005) derived absolute (geometrical) estimates
of the distance to the center of the Galaxy, $R_0$, from the star S2
orbit around Sgr~A* on the assumption that the intrinsic velocity of
Sgr~A* is negligible. This assumption produces the source of systematic
error in $R_0$ value owing to a probable motion of Sgr~A* relative to
the accepted velocity reference system which is arbitrary to some
extent. Eisenhauer et al.\ justify neglecting all three spatial
velocity components of Sgr~A* mainly by low limits of Sgr~A*'s proper
motion of 20--60 km/s. In this brief paper, a simple analysis in the
context of the Keplerian dynamics was used to demonstrate that neglect
of even low (perhaps, formal) radial velocity of Sgr~A* leads to a
substantial systematic error in $R_0$: the same limits of 20--60~km/s
result in $R_0$ errors of 1.3--5.6\%, i.e., (0.1--0.45)$\times
(R_0/8)$~kpc, for current S2 velocities. Similar values for Sgr~A*'s
tangential motion can multiply this systematic error in the case of S2
orbit by factor ${\approx}1.5$--$1.9$ in the limiting cases. \end{abstract}
\section{Introduction}
The distance from the Sun to the center of the Milky Way, $R_0$, is a
fundamental Galactic constant for solving many astronomical and
astrophysical problems \citep[see, e.g.,][]{R93}. That is why, in its
turn, the problem of determination of $R_0$ remains topical over many
years. Absolute (i.e., not using luminosity calibrations) estimates of
$R_0$ with a current 3\% formal uncertainty from modelling the star S2 orbit
around the compact concentration of dark mass, the so-called
``supermassive black hole'', associated with the radio source Sgr~A*
\citep{Eea03,Eea05,Tea06} present a major breakthrough in measuring
$R_0$! (For brevity, from here on the object in focus of S2 orbit will be
referred to as ``Sgr~A*''.)
However, even though to take no notice the issue on coincidence of Sgr~A*
with the dynamical and/or luminous center(s) of our Galaxy \citep[see
discussion in][]{Nishiyama_ea06}, taken alone the modelling the orbital motion
of a star near Sgr~A* can be plagued with various {\em systematic\/} sources
of error. Since \citeauthor{Eea05}\ solved for the {\em Keplerian\/} orbit of
the star S2, in the literature {\em relativistic\/} effects and {\em
non-Keplerian\/}
orbit modelling are primarily explored for this problem
\citep[e.g.,][]{Eea05,Mouawad_ea05,Weinberg_ea05}.
Meanwhile, \citeauthor{Eea05}\ also used another {\em assumption that the
intrinsic velocity of Sgr~A* is negligible\/}. This assumption can produce
the source of systematic error in $R_0$ value owing to a probable motion of
Sgr~A* relative to the accepted velocity reference system which is
arbitrary to some extent. Thus far, no consideration has been given to the
role of this factor in measuring $R_0$.
In this study, a simple analysis is used to evaluate the {\em impact of an
unaccounted motion of Sgr~A*\/} (i.e., the focus of S2 orbit) {\em on an $R_0$
value\/} found from the formal solution of orbit. The Keplerian dynamics only
is
taken into consideration because relativistic and non-Keplerian effects seem
to be insignificant for measuring $R_0$
\citep{Eea05,Mouawad_ea05,Weinberg_ea05}. Particular attention has been given
to the impact of a nonzero radial velocity of Sgr~A* relative to the
Local Standard of Rest.
\section{Structure of the Problem on Determination of Orbital Parameters,
Distance to and Mass at Orbital Focus (Sgr~A*)}%$
The completeness of solution of the problem in question is determined by the
type of available data on motion of an individual star (S2).
\subsection{Star's Proper Motions Alone are Available}%
In this case, {\em all six orbital parameters are solved, except that only the
absolute value of the inclination angle, $i$, is determined\/}, leaving the
questions of the direction of revolution (prograde, $i>0$, or retrograde,
$i<0$) and where along the line of sight the star is located behind the
central object unresolved
\citep[e.g.,][]{Ghez_ea03}. Besides, {\em the semimajor axis is derived in
angular units\/} (in arcsec), hereafter $a''$. The distance to the focus,
i.e., $R_0$, and the central mass, $M$, can not be solved.
With accepted $R_0$, however, the value of semimajor axis, $a$, is calculated
in linear units (in kpc) and the central mass is found from Kepler's third law
\be\label{M}
M=n^2a^3/G, \qquad n=2\pi/P,
\ee
where $G$ is the gravitational constant, $n$ is the mean motion, and $P$ is
the orbital period, as it has been done in \citet{Schoedel_ea02}.
\subsection{Proper Motions and at Least a Single Measurement of Radial
Velocity of Star are Available}%
In this case, {\em the problem is completely solved\/} if the value of star's
radial (line-of-sight) velocity, $V_r$, is significantly different from zero
(more exactly,
from the radial velocity of the focus).
A. {\em The sign of\/} $V_r$ {\em determines the sign of $i$.\/} Consequently,
this also breaks the ambiguity in the direction of rotation and in star's
location along the line of sight relative to the focus
\citep[e.g.,][]{Ghez_ea03}.
B. {\em The absolute value of\/} $V_r$ {\em determines values\/} $R_0$ {\em
and\/} $M$. To gain greater insight into the fact of the matter, the problem
can be symbolically divided into two subproblems: (1) the determination of
orbital parameters from the proper motions alone and (2) the determination,
knowing the orbit, of the distance to focus ($R_0$) and of the central mass
from
the measurement(s) of $V_r$. These subproblems are almost independent in the
case of modelling the motion of stars around Sgr~A*, since up to now proper
motion measurements are numerous, but $V_r$ ones are few or at all $V_r$
actually is single, for any S star with solved orbit. So, $V_r$ measurement(s)
contribute(s) almost nothing to the knowledge of orbit, and vice versa proper
motion measurements do not directly determine neither $R_0$ nor $M$. Thus,
such breaking the problem down seems to be quite realistic.
If so, the value of $|V_r|$ may be considered as determining $R_0$ and
$M$ from known orbital parameters as follows.
({\bf i}) The orbit elements enable to find the ratio between $|V_r|$ and the
total
space velocity, $V$, for the moment $t$: \be\label{Vr/V}
V_r^2/V^2=\frac{[e\sin v\sin u +(1+e\cos v)\cos u]^2\msin^2i}{1+2e\cos v+e^2},
\ee
where $e$ is the eccentricity, $v$ is the true anomaly, $u=v+\omega$\/ is
the argument of latitude, $\omega$ is the argument of pericenter. A
value of $v$ can be calculated from classical formalism: $$
\tan(v/2)=\sqrt{(1+e)/(1-e)}\tan(E/2),
$$
$$
E-e\sin E={\cal M},\qquad {\cal M}=n(t-t_0)+{\cal M}_0,
$$
where $E$ and $\cal M$ are the eccentric and mean anomalies, correspondingly
\citep[e.g.,][]{Subbotin68}. Consequently, the knowledge of $|V_r|$ determines
$V$.
({\bf ii}) The value of total velocity $V$ can be expressed as
\be\label{V}
%V^2=n^2a^2\frac{1+e\cos v+e^2}{1-e^2}.
V=na\left(\frac{1+2e\cos v+e^2}{1-e^2}\right)^{1/2}.
\ee
From this equation, the value of $a$ {\em in linear units\/} can be
calculated. Then the ratio between $a$ values in linear and angular units
gives $R_0$:
\be\label{R0}
R_0=\frac{a\text{ [kpc]}}{a''}.
\ee
({\bf iii}) Using Eq.~(\ref{M}) with $a$ in linear units determines the
central mass $M$.
\section{Systematic Error in $\mathbf{R_0}$ Owing to a Nonzero Motion of
Orbital Focus
(Sgr~A*)}%$
\subsection{Nonzero Radial Velocity of Sgr~A*}\label{Vr_ne_0}%$
\citet{Eea03,Eea05} assume that the radial velocity of Sgr~A*,
$V_r^*\equiv V_r(\text{Sgr A*})$, relative to the Local Standard of Rest (LSR)
is
zero. Neglect of a possible radial motion of Sgr~A* is equivalent to the
introducing a corresponding systematic error in all $V_r$ values. This
error is equal to a value of $V_r^*$ and is the same in
all measurements of $V_r$. From Eqs.~(\ref{Vr/V})--(\ref{R0}) follows
that the relative systematic error in $V_r$ velocity fully converts to
the relative systematic error in $R_0$, i.e., \be\label{delta}
\delta_{\text{sys}}\equiv\frac{\sigma_{\text{sys}}(V_r)}{|V_r|}=
\frac{\sigma_{\text{sys}}(R_0)}{R_0}.
\ee
These simple considerations make it possible readily to evaluate the
systematic error in $R_0$ knowing typical values of $V_r$ used for the
determination of distance to S2/Sgr~A*. The first S2 radial velocity
measurement of $V_r=-510\pm 40$~km/s by \cite{Ghez_ea03} was obtained just 30
days after the star's passage through the pericenter point when $V_r$ was
changing very rapidly. Therefore, this measurement contributes to the solution
for $R_0$ much less then subsequent ones, hence the evaluation of
$\sigma_{\text{sys}}(R_0)$ must lean upon these latter. Besides, the
subsequent radial velocities, having substantially higher absolute values, give
a {\em lower\/} limit for $\sigma_{\text{sys}}(R_0)$.
\citeauthor{Eea05}\ justify neglecting all three spatial velocity components
of Sgr~A* mainly by low limits of Sgr~A*'s proper motion of 20--60 km/s
\citep{Eea05}. Such values of radial velocities seem to be quite plausible for
massive objects in the Galactic center \cite[see][]{Blitz94}.
Table~\ref{tab_r0sys} presents values of systematical errors in $R_0$
calculated for possible Sgr~A*'s radial velocities of $V_r^*=20$ and 60~km/s
with $R_0=7.5$ and 8.0~kpc \citep{R93,N04,Tea06}. In Table~1, $\langle
V_r\rangle$ is the average of velocities $V_r$, used for estimation of $R_0$
in \cite{Eea05}, over the observational period.
\begin{table}[t]
\normalsize
\caption{Systematic error in $R_0$ because of
neglect of a possible radial motion of Sgr~A*}
\label{tab_r0sys}
\vskip 0.01\textheight
\begin{center}%\large%\scriptsize
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{lccccc}
%\hline
\hline
Observational & $\langle V_r\rangle$ & $V_r(\text{Sgr A*}) $ &
$\delta_{\text{sys}}$ & \multicolumn{2}{c}{$\sigma_{\text{sys}}(R_0)$
(kpc)}\\%$
\cline{5-6}
Period & (km/s) & (km/s) &
& $R_0=7.5$ kpc & $R_0=8$ kpc \\
%\hline
\hline
2003 April--June & $-1500$ & 20 & 0.013 & 0.10 & 0.11 \\
& & 60 & 0.040 & 0.30 & 0.32 \\
2004 July--August& $-1075$ & 20 & 0.019 & 0.14 & 0.15 \\
& & 60 & 0.056 & 0.42 & 0.45 \\
%\hline
\hline
\end{tabular}
\end{center}
\end{table}
Table~\ref{tab_r0sys} demonstrates that {\em neglect of even moderately
low radial velocity of the orbital focus (Sgr~A*) relative to the LSR
can lead to a substantial systematic error in $R_0$\/}: values of
$V_r^*=20$--60~km/s result in systematic $R_0$ errors of 1.3--5.6\%,
i.e., {\bf (0.1--0.45)$\mathbf{\times (R_0/8)}$~kpc}\/, for current
typical star's velocities. Notice that the value of
$\sigma_{\text{sys}}(R_0)$ can not be reduced statistically since {\em
all\/} $V_r$ values is biased coherently by any nonzero velocity of
Sgr~A*. Only solving for $V_r(\text{Sgr A*}) $ can correct this
systematic error in $R_0$!
It should be mentioned that \cite{Tea06} state that they already solved 3D
velocity of Sgr~A$*$, however, not presenting in their short paper any
details---no values of velocities and even no exact value of current point
estimate for $R_0$!
\subsection{Nonzero Proper Motion of Sgr~A*}%$
The reference frame for proper motions \citeauthor{Eea03}\ have
established by measuring the positions of nine astrometric reference
stars relative to typically 50--200 stars of the stellar cluster
surrounding Sgr A*; the uncertainty of the reference frame is
11.7~km/s \citep[see][]{Eea03}. The effect of nonzero proper motion Sgr
A$^*$ relative to this frame, $\vec\mu^{\,*}\equiv\vec\mu(\text{Sgr A*})$, can
be
approximately estimated if to imagine that the value of $R_0$ is
determined, also on the basis of $V_r$'s measurement at a moment
$t$, not from Eqs.~(\ref{V}) and (\ref{R0}) but from the ratio between
star's linear velocity on the sky, $V_\mu$, and star's proper motion,
$\mu$, measured for the same moment $t$:
\be\label{R0mu}
R_0=\frac{V_\mu}{\mu}.
\ee
The value of $V_\mu$ is a known function of $V_r$, orbital elements, and time:
\be\label{Vmu}
V_\mu^2=V^2-V_r^2=V_r^2(\Psi^{-2}-1),\qquad \Psi^2(t)\equiv \frac{V_r^2}{V^2},
\ee
where $\Psi^2(t)$ can be calculated from orbital elements [Eq.~(\ref{Vr/V})].
Any nonzero radial velocity $V_r^*$ and nonzero proper motion $\mu^*$ of
Sgr~A* are equivalent to the introducing systematic errors
$\varepsilon_{V_\mu}$ and $\varepsilon_{\mu}$ in $V_\mu$ and $\mu$,
correspondingly. Because values of $V_r^*$ and $\mu^*$ are independent and
unknown, their combined impact on an $R_0$ estimate can be described by the
formula of propagation of errors applied to Eq.~(\ref{R0mu}):
\bea
\varepsilon^2_{R_0}\equiv \sigma_{\text{sys}}^2(R_0)
&=&
\left(\frac{\varepsilon_{V_\mu}}{\mu}\right)^2+
\left(\frac{V_\mu}{\mu^2}\varepsilon_\mu\right)^2\nonumber\\
&=&(R_0/V_\mu)^2(\varepsilon^2_{V_\mu}+R_0^2\varepsilon^2_\mu).
\eea
%Here $\varepsilon_{V_\mu}$ and $\varepsilon_\mu$ present errors owing to
nonzero
%$V_r^*$ and $\vec\mu^{\,*}$, correspondingly. From Eq.~(\ref{Vmu}) follows
\be
\varepsilon_{V_\mu}=\varepsilon_{V_r}\sqrt{\Psi^{-2}-1},
\ee
if an uncertainty on orbit elements is ignored, as it was actually done in
section~\ref{Vr_ne_0} Then considering that $\varepsilon_{V_r}=|{V_r^*}|$ we
have
\be
\varepsilon^2_{R_0}=
\frac{R_0^2}{V_r^2}\left({V_r^*}^2+R_0^2\varepsilon^2_\mu\frac{\Psi^2}{1-\Psi^2}\right).
\ee
Value of $\varepsilon_\mu$ depends from the relative orientation of vectors
$\vec\mu$ and $\vec\mu^{\,*}$. In the general case $0 \le \varepsilon_\mu \le
\mu^*$. Hence, e.g., for equal radial and tangential components of
Sgr~A* motion, i.e., for $V_\mu^*=|V_r^*|$, or $\mu^*=|V_r^*|/R_0$,
\be
\max\varepsilon_{R_0}=\varepsilon_{R_0}(V_r^*)k_1,\qquad
k_1=\frac{1}{\sqrt{1-\Psi^2}},
\ee
\be
\varepsilon_{R_0}(V_r^*)\equiv R_0\left|\frac{V_r^*}{V_r}\right|.
\ee
Here $\varepsilon_{R_0}(V_r^*)$ is the systematic error in $R_0$ owing to
only the radial velocity of Sgr~A* [see Eq.~(\ref{delta})].
For ${V_\mu^*}^2=2{V_r^*}^2$, or $\mu^*=\sqrt{2}|V_r^*|/R_0$, i.e., for equal
all
three Cartesian components of Sgr~A* motion,
\be
\max\varepsilon_{R_0}=\varepsilon_{R_0}(V_r^*)k_2,\qquad
k_2=\sqrt{\frac{1+\Psi^2}{1-\Psi^2}}.
\ee
With the S2 orbit elements derived in \cite{Eea05}, $k_1\approx 1.4974$,
$k_2\approx 1.8666$.
Thus, for a given $V_r$ the effect of nonzero proper motion of Sgr A* on
$R_0$, being a function of the true anomaly, ranges from zero to values
comparable to the effect of nonzero radial velocity of Sgr~A*, in the latter
case increasing measurably the total systematic error in $R_0$.
\section{Conclusions}
Simple considerations show that {\em neglect of even low radial velocity
of Sgr~A* relative to the LSR leads to a substantial systematic error
in\/} $R_0$---up to 6\%, i.e., ${\sim}0.5$~kpc, for plausible values of
Sgr~A* velocity. It is too much to consider the distance to Sgr~A*, not
to mention the value of $R_0$, as being established reliable from the
present results on modelling the S2/Sgr~A* system.
A proper motion of Sgr~A* biases the distance value not so inevitably,
but in limiting cases can increase the systematic error in $R_0$ owing
to radial motion by factor up to ${\approx}1.5$--$1.9$ for similar
values of Sgr~A*'s tangential velocity.
\acknowledgments
I am grateful to Prof.~K.~V.~Kholshevnikov and to Prof.~S.~A.~Kutuzov
for valuable remarks and discussions. The work is partly supported by
the Russian Pre\-si\-dent Grant for State Support of Leading Scientific
Schools of Russia no.\ NSh-4929.2006.2.
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\end{document}