------------------------------------------------------------------------ Nii_DoG.tex Dynamics of Galaxies, Proceedings of the International Conference held at Pulkovo Observatory, August 6-10, 2007, in press (fwd) Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; CHARSET=US-ASCII; FORMAT=flowed Content-ID: X-MailScanner-Information: Please contact the postmaster@aoc.nrao.edu for more information X-MailScanner: Found to be clean X-MailScanner-SpamCheck: not spam, SpamAssassin (not cached, score=0.68, required 5, autolearn=disabled, SARE_ADLTOBFU 0.68) X-MailScanner-From: nii@dyna.astro.spbu.ru X-Spam-Status: No %astro-ph/0803.0825 \documentclass[a4paper,12pt]{article} %\usepackage[koi8-r]{inputenc} %\usepackage[russian]{babel} %\usepackage{babel} %\usepackage{srcltx}% To use Kile package \usepackage{amsmath}% E.g. for \text command \usepackage{epsf} \usepackage{comment} \usepackage{natbib} \bibpunct[, ]{(}{)}{;}{u}{}{,} %\usepackage{../asp2004_DoG} \usepackage{asp2004_DoG}% For astroph %usepackage{/mnt/dosd/nii/tex/sty/art_nii} %\input /mnt/dosd/nii/tex/prof_lin %\input /mnt/dosd/nii/tex/p_phys \input prof_lin.inc% For astroph \input p_phys.inc% For astroph \DeclareMathOperator{\msin}{\smash[t]{\mathrm{sin}}} \pagestyle{myheadings} \markboth{I. I. Nikiforov}% {Systematic Error in $R_0$ from Solving for Stellar Orbit Around Sgr A*} \setcounter{page}{1} \voffset=-10mm%+ low \hoffset=-13.1mm%+ right \textheight 24cm \textwidth 16.5cm \binoppenalty=10000 \relpenalty=10000 \begin{document} \sloppy %\large %\begin{center} %\LARGE\bf %\large\bf \title{On a Source of Systematic Error in Absolute Measurement of Galactocentric Distance from Solving for the Stellar Orbit Around Sgr A*} %\end{center} \thispagestyle{empty} \sc %\vskip 12pt %\medskip %\begin{center} %\large \author{Igor' I. Nikiforov} %\end{center} %\it %\vskip 1pt %\smallskip %\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. \begin{thebibliography}{} \bibitem[{Blitz(1994)Blitz}]% {Blitz94} Blitz, L. 1994, in ASP Conf.\ Ser., Vol.~66, Physics of the Gaseous and Stellar Disks of the Galaxy, ed.\ I. R. King (San Francisco: ASP), 1 %\bibitem[{Duboshin(1976)Duboshin}]% % {Duboshin76} % Duboshin, G. 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