------------------------------------------------------------------------
From: Mark Reid reid@cfa.harvard.edu
X-Sender: reid@alpmjr
To: Galactic Center Newsletter
Subject: Reid's .tex file
MIME-Version: 1.0
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\def\masyr=09{mas y$^{-1}$}
\def\sgra=09{Sgr~A*}
\def\sgrab=09{Sgr~A*~}
\def\tnot=09{\ifmmode {\Theta_0}\else {$\Theta_0$} \fi}
\def\rnot=09{\ifmmode {R_0} \else {$R_0$} \fi}
\def\onot=09{\ifmmode {\Omega_0}\else {$\Omega_0$} \fi}
\def\vsun=09{\ifmmode {V_\odot} \else {$V_\odot$} \fi}
\def\peryr=09{y$^{-1}$}
\def\perkpc=09{kpc$^{-1}$}
\def\porm=09{\ifmmode {~\pm~} \else {$~\pm~$} \fi}
\def\kms=09{\ifmmode {{\rm km s}^{-1}} \else {km s$^{-1}$} \fi}
\def\msun=09{\ifmmode {{\rm M}_\odot} \else {${\rm M}_\odot$} \fi}
\def\lsun=09{\ifmmode {{\rm L}_\odot} \else {${\rm L}_\odot$} \fi}
\def\vmax=09{\ifmmode {V_{max}} \else {$V_{max}$} \fi}
\def\ie=09=09{i.e.,~}
\def\eg=09=09{eg,~}
\def\etal=09{et al.~}
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\begin{document}
\title{The Proper Motion of Sgr A*: I. First VLBA Results}
\author{M.~J.~Reid}
\affil{Harvard--Smithsonian Center for Astrophysics, Cambridge, MA 02138}
\authoremail{mreid@cfa.harvard.edu}
\author{A.~C.~S.~Readhead \& R.~C.~Vermeulen}
\affil{California Institute of Technology, Pasadena, CA 91109}
\authoremail{}
\author{R.~N.~Treuhaft}
\affil{Jet Propulsion Laboratory, Pasadena, CA 91109}
\authoremail{}
\begin{abstract}
=09We observed Sgr A* and two extragalactic radio sources nearby
in angle with the VLBA over a period of two years and measured=20
relative positions with an accuracy approaching 0.1 mas.
The apparent proper motion of Sgr A* relative to J1745--283 is
$5.90\pm0.4$~mas~/yr, almost entirely in the plane of the Galaxy. =20
The effects of the orbit of the Sun around the Galactic Center can
account for this motion, and any residual proper motion of Sgr A*,=20
with respect to extragalactic sources, is less than about 20 \kms. =20
Assuming that Sgr A* is at rest at the center of the Galaxy, we estimate th=
at
the circular rotation speed in the Galaxy at the position of the Sun,=20
{$\Theta_0$}, is $219\pm20$~{km/s,} scaled by ${$R_0$}/8.0$~kpc.
=09Current observations are consistent with Sgr A* containing all of the=20
nearly $2.6\times10^6~M_\odot$, deduced from stellar proper motions,=20
in the form of a massive black hole. =09
While the low luminosity of Sgr A*, for example, might possibly have come
from a contact binary containing of order 10 $M_\odot$, the lack of substan=
tial=20
motion
rules out a ``stellar'' origin for Sgr A*. =20
The very slow speed of Sgr A* yields a lower limit to the mass of Sgr A* of=
=20
about 1,000 $M_\odot$. Even for this mass, Sgr A* appears to be=20
radiating at less than $0.1\%$ of its Eddington limit. =20
\end{abstract}
\keywords{Individual Sources: \sgra; Black Holes; Galaxy: Center, Fundament=
al
Parameters, Structure; Astrometry}
\section{Introduction}
=09\sgrab is a compact radio source, similar to weak active nuclei=20
found in other galaxies. Since its discovery more than two decades ago
by Balick \& Brown (1974), the possibility that \sgrab is a super-massive=
=20
($\sim10^6~\msun$) black hole has been actively considered. However, its=
=20
radio luminosity of $\approx10^2~\lsun$ (Serabyn \etal 1997)
and its estimated total luminosity of $\lax10^5~\lsun$ are many orders of=
=20
magnitude below that possible from a $\sim10^6~\msun$ black hole.
Thus, on the basis of its spectral energy distribution
\sgrab could be an unusual contact binary containing=20
$\sim10$~\msun and radiating near its Eddington limit. =20
=09Recently, Eckart \& Genzel (1997) and Ghez \etal (1999)=20
measured proper motions of stars
near the position of \sgra, as determined by Menten \etal (1997). =20
Stellar speeds in excess of 1000 \kms\ at a distance of $0.015$~pc=20
from \sgrab indicate a central mass of $2.6\times10^6~\msun$.
While these dramatic results are consistent with the theory that
\sgrab is a super-massive black hole, it is still conceivable that most of=
=20
the central mass could come from a combination of stars and=20
perhaps some form of dark matter. Clearly, independent constraints
on the mass of \sgrab are needed to establish whether or not
it is a super-massive black hole nearly at rest at the dynamical=20
center of the Galaxy.
=09The {\it apparent} motion of \sgrab can be used to estimate the=20
mass and elucidate the nature of this unusual source.
An apparent motion of \sgrab can be attributed to
at least three possible components:=20
1) a secular motion induced by the orbit of the Sun about the=20
Galactic Center,=20
2) a yearly oscillation owing to the Earth's orbital motion around the=20
Sun (trigonometric parallax), and
3) a possible motion of \sgrab with respect to the dynamical center of=20
the Galaxy.
Measurement of, or limits for, these components of \sgra's apparent motion
can provide unique information on
the circular rotation speed (\tnot) of the Local Standard of Rest (LSR)=20
and the peculiar motion of the Sun (\vsun),=20
the distance to the center of the Galaxy (\rnot), and=20
the nature of \sgrab itself.
=09In 1991 we started a program with the Very Long Baseline Array (VLBA)
of the National Radio Astronomy Observatory=20
\footnote {The National Radio Astronomy Observatory is operated by
Associated Universities Inc., under cooperative agreement with the National
Science Foundation.}
(NRAO) to measure the apparent motion of \sgrab . In principle,
VLBA observations of \sgrab, phase-referenced to extragalactic radio source=
s,=20
can
achieve an accuracy sufficient to detect secular motions of $\lax1~\kms$ fo=
r
a source at the distance of the Galactic Center.
However, achieving this accuracy is quite challenging technically as it inv=
olves=20
observing at short wavelengths (7 mm), in order to minimize the effects
of interstellar scattering toward \sgra, phase-referencing to extragalactic
sources, and careful modeling of atmospheric effects (because of the
low source elevations). =20
=09In the early years of the project, we searched for strong, compact,=20
extragalactic sources nearby in angle to \sgra and worked toward an
optimum observing strategy. In this paper, we report results of=20
observations spanning a two year period from 1995 to 1997. Our VLBA images=
=20
clearly show movement of \sgrab with respect=20
to extragalactic sources over many synthesized beams. =20
While the current positional accuracy is inadequate to determine a=20
trigonometric parallax, the secular motion of \sgrab is easily measured. =
=20
This yields an accurate estimate of the angular rotation rate of the Sun
around the Galactic Center, $(\tnot+\vsun)/\rnot$,=20
and places interesting limits on the mass of the
black hole candidate responsible for the radio emission.
Our results, and those of Backer and Sramek (1999) from=20
observations with the Very Large Array (VLA) over a 15 year period,
indicate that the apparent proper motion of \sgrab is dominated by the=20
orbit of the Sun about the Galactic Center and that any peculiar
motion of \sgrab is very small.
\section{Observations}
=09Our successful observations using the VLBA were conducted
in the late-night, early-morning periods of 1995 March 4, 1996 March 20 and=
31,=20
and 1997 March 16 and 27. (Observations attempted during two August evenin=
gs
in 1996 experienced high water vapor turbulence in the atmosphere, and phas=
e
referencing was not successful.) The observing frequency was=20
43.2~GHz and we observed four 8 MHz bands, each at right and left circular=
=20
polarization. We employed 2-bit sampling at the Nyquist rate, which=20
required the maximum aggregate sampling rate supported by the VLBA of 256=
=20
megabits per second. Only the inner five VLBA stations=20
(Fort Davis, Los Alamos, Pie Town, Kitt Peak, and Owens Valley) were used,
as baselines longer than about 1500 km heavily resolve the scatter
broadened image of \sgrab at 43 GHz (\eg Bower \& Backer 1998).
=09The observing sequence involved rapid switching between compact=20
extragalactic sources and \sgra.
Two sources, J1745--283 and J1748--291, from the catalog of Zoonematkermani=
et=20
al.=20
(1990) were found to be strong enough ($>10$~mJy at 43 GHz) to serve as
reference background sources. These sources are two of the three used by
Backer \& Sramek (1999) in their program to measure the proper motion
of \sgrab with the VLA; their third background source, J1740--294, proved t=
o
have a steep spectrum and was too weak for inclusion in our 43 GHz program.
We switched among the sources repeating the following pattern:
\sgra, J1745--283, \sgra, J1748--291.
Sources were changed every 15 seconds, typically achieving 7 seconds
of on-source data, except for the earliest observation in 1995 when we were=
=20
experimenting with longer switching times. We used \sgrab as the=20
{\it phase-reference} source, because it is considerably stronger than=20
the background sources and could be detected on individual baselines
with signal-to-noise ratios typically between 10 and 20 in the 7 seconds=20
of available on-source time.
=09We edited and calibrated data using standard tasks in the
Astronomical Image Processing System (AIPS) designed for
VLBA data. This involved applying data flagging tables generated by
the on-line antenna and correlator systems, station gain curves, and system
temperature measurements. We solved for station-dependent,=20
intermediate-frequency band
delays and phases on a strong, compact source (NRAO~530).
After applying these corrections, the multi-band data for \sgrab could be=
=20
combined=20
coherently and interferometer phases as a function of time determined. =20
These phase solutions were examined by an AIPS task specially written for=
=20
our observations that looked for and flagged data when baseline-dependent p=
hases=20
on=20
adjacent \sgrab scans changed by more than one radian. =20
Under good weather conditions between 10 and 30\% of the data were=20
discarded by this process. =20
This provided relatively unambiguous ``phase connection'' for the
remaining data and allowed removal of most of the=20
effects of short-term atmospheric fluctuations from all sources.
(We note that during average-to-poor weather conditions,=20
our phase measurements on \sgrab every 30 seconds were not frequent=20
enough to provide unambiguous phase connection. Thus, our 15 second switc=
hing=20
time is probably an optimum trade-off between on-source duty cycle and=20
atmospheric coherence losses.) Data calibrated in this manner produced=20
high (\eg 50:1) dynamic range maps of all sources with little or no spatial=
=20
blurring. The images of the background sources appeared less resolved than
that of \sgra, with no signs of complex or multiple component structures.
=09We found that the differences in relative positions between a=20
background source and \sgrab for closely spaced ($\approx10$~d) epochs=20
were $\approx1$~mas. These differences exceeded the formal precision,=20
estimated by the least-squares fitting process, typically=20
by a large factor. Since the observational conditions and data analysis we=
re=20
nearly
identical for these epochs, small geometric errors (\eg in baselines, sourc=
e=20
coordinates,
or Earth's orientation parameters) are unlikely to yield position shifts of=
this=20
magnitude given the small angular separation of \sgrab and the background=
=20
sources.
Therefore, we evaluated the possibility that refractive scattering of the r=
adio
waves in the interstellar medium or modeling errors for the Earth's atmosph=
eric
propagation delay could be responsible.
=09Refractive scattering can cause changes in the apparent flux density and
position of a radio source. Gwinn \etal (1988) published VLBI observations=
that
limit refractive position wander for water vapor masers in Sgr~B2N, a star
forming region close to the Galactic Center. For these maser spots, their
22~GHz observations revealed a diffractive scattering size of 0.3~mas and a=
n
upper limit to a Gaussian component of refractive position wander of 0.018~=
mas.
Theoretical estimates of refractive position wander, based on the diffracti=
ve
scattering size, by Romani, Narayan \& Blandford (1986) agree with this lim=
it=20
for
a Kolmogorov electron density spectrum. Assuming that the refractive effec=
ts=20
scale as the diffractive scattering size, we expect any refractive wander
of \sgrab at 43~GHz to be $<0.04$~mas, a value about a factor of 25 smaller
than our observed position differences.
=09After careful study of the data,=20
we concluded that the most likely source of relative position error
is a small error in the atmospheric model used by the VLBA correlator. =20
The following simple analysis supports this view:
The phase-delay of the neutral atmosphere, $\tau$, can be approximated=20
by $\tau_0 \sec Z$, where $\tau_0$ is the vertical phase-delay and $Z$ is=
=20
the local source zenith angle.
When measuring the {\it difference} in position of two sources separated in=
=20
zenith angle by $\Delta Z$, a first-order Taylor expansion of $\tau$ yields
the expected {\it differenced} phase-delay error for a single antenna:
$$\Delta\tau~\approx~{\partial\tau \over \partial Z} \Delta Z~=3D~
=09\tau_0\sec Z \tan Z \Delta Z~~~~.~~\eqno(1)$$
The seasonally-averaged atmospheric model (Niell 1996) used by the VLBA=20
correlator
is likely to miss-estimate $\tau_0$ by about 0.1 nsec,=20
equivalent to a zenith phase-delay of $\approx3$~cm in path length.
This comes mostly from the highly variable contribution by water vapor
(\eg Treuhaft \& Lanyi 1987).
Based on Eq.~(1), this should result in an antenna-dependent error of=20
$\Delta\tau \approx 0.3$~cm=20
for typical source zenith angles of $\approx70^\circ$ and =20
for our typical source separations of $\Delta Z \approx 0.012~{\rm rad} (\a=
pprox=20
0.7^\circ)$. =20
Since atmospheric errors are largely uncorrelated for different antennas,
on an interferometer baseline at an observing wavelength of $0.7$~cm, we=20
would expect a relative position shift of roughly $\sqrt{2}\times0.3$~cm or
$\approx70$\%\ of a fringe spacing. This corresponds to=20
$\approx0.4$ and $\approx1.7$~mas in the easterly and northerly=20
directions, respectively, for our longer baselines. =20
This effect should only partially cancel among the different baselines and=
=20
can explain the position errors seen in the raw maps made from observations=
=20
made $\approx10$ days apart.
=09In order to improve our relative position measurements, we=20
modeled simultaneously our differenced-phase data for the ``J1745--283 minu=
s=20
\sgra'' and
``J1748--291 minus \sgra'' source pairs. The model allowed for a relative=
=20
position shift
for each source pair=20
and a single vertical atmospheric delay error in the correlator model for
each antenna. This approach significantly improved the accuracy of=20
the relative position measurements as evidenced by the smaller deviations
in relative positions for observations closely spaced in time. =20
The vertical atmospheric delay parameters typically indicated a=20
correlator model error of a few cm and these parameters were estimated with=
=20
uncertainties of about 1~cm. Using this approach, we would estimate=20
from Eq.~(1) and the above discussion that relative position errors should =
be=20
$\approx0.1$ and $\approx0.4$~mas in the East--West and=20
North--South coordinates, respectively, for one day's observation.
=09The differenced-phases often displayed post-fit residuals of $\sim 30$=
=20
degrees of phase, which were correlated over periods of hours. =20
Assuming equal and uncorrelated contributions from the
two antennas forming an interferometer pair, this suggests a delay change o=
f=20
about=20
0.1 nsec of time or about 3 cm of uncompensated path length. Since=20
typical source zenith angles were about 70 degrees, this corresponds to=20
a vertical delay change in the atmosphere above each antenna of about=20
0.03 nsec or about 1 cm of path length.
This behavior is consistent with expected large-scale changes in the=20
atmospheric delay, and it suggests that significant improvement can
be obtained by monitoring and correcting for =20
large scale atmospheric changes.
=09The data in Table~1 summarize our relative position measurements.
The data taken on 1995 March 4 were of poor quality, only the stronger of=
=20
the two background sources (J1745--283) was detected, and the positional=20
accuracy=20
was significantly worse than for the later epochs.
Positions of the strongest background source, J1745--283,=20
phase-referenced to \sgra, for epochs spanning 2 years are=20
plotted in Fig.~1 with open circles in the sense \sgrab relative to J1745--=
283.
They indicate a clear apparent motion for \sgrab relative to J1745--283,=20
consistent in
magnitude and direction with the reflex motion of the Sun around the=20
Galactic Center (see \S 3.1). The positions in the East-West direction hav=
e
typical uncertainties of about 0.1 mas, as estimated from the scatter
of the post-fit position residuals about a straight-line motion. =20
It is interesting to note that,=20
while it takes $\approx220$~My for the Sun to complete an orbit around the=
=20
Galactic Center, the East--West component of the parallax from only 10 days=
=20
motion=20
can be detected with the VLBA!
The position uncertainties in the North-South direction are larger,=20
about 0.4~mas, owing to the low declination of the Galactic Center.
=09
=09The apparent motions of \sgrab relative to J1745--283
over a 2 year time period and J1748--291 relative to J1745--283
over a 1 year time period are given in Table~2. =20
The uncertainties in Table 2 include estimates of the systematic
effects, dominated by errors in modeling of atmospheric effects,=20
as discussed above. Assuming that J1745--283 is sufficiently distant that
it has negligible intrinsic angular motion, \sgra's apparent motion is=20
$-3.33\pm0.1$ and $-4.94\pm0.4$~mas~\peryr\=20
in the easterly and northerly directions, respectively.
This motion is shown by the solid lines in Fig.~1.
=09As a check on the accuracy of our measurements,=20
we measured relative positions between the two calibration sources.
These positions are plotted in Fig.~1 (crosses) in the sense=20
``J1748--291 minus J1745--283,'' offset to fit the plotting scale
for the ``\sgrab minus J1745--283" data. =20
The best fit motions are $+0.17\pm0.14$=20
and $-0.22\pm0.56$~mas~\peryr\ in the easterly and northerly=20
directions, respectively, as indicated with the dashed lines in Fig.~1. =20
The {\it uncertainty} in the relative motion of J1748--291 with respect to
J1745--283 is $\approx$40\% larger than for=20
the motion of \sgrab with respect to J1745--283,because the angular separat=
ion=20
of the two background sources ($\approx1.0$~deg) is greater than between=20
\sgrab and either of the background sources ($\approx0.7$~deg). =20
Thus, the background sources display no statistically significant motion=20
relative to each other, as expected for extragalactic sources.
=09Finally, we have determined the position of \sgrab relative
to an extragalactic source with high accuracy and, therefore, can=20
derive an improved absolute position of \sgra. VLBI observations carried o=
ut
by the joint NASA/USNO/NRAO geodetic/astrometric array (Eubanks, private
communication) detected J1745--283 at 8.4 GHz and determined its
position in the U.S.~Navy 1997-1998 reference frame to be
\centerline{J1745--283~~~~~$\alpha$(2000)=3D17~45~52.4968,~~~$\delta$(2000)=
=3D--28~2
0~26.294~,}
with a uncertainty of about 12~mas. Assuming this result, we find
the position of \sgrab measured at 1996.25 to be
\centerline{\sgra~~~~~~~~~~$\alpha$(2000)=3D17~45~40.0409,~~~$\delta$(2000)=
=3D--29~0
0~28.118~.}
The uncertainty in this position is dominated by that of J1745--283.
Note that were one not to correct for the ``large" apparent proper motion o=
f=20
\sgra,=20
the position of \sgrab determined for observations made more than 2 years f=
rom=20
1996.25 would be shifted by an amount greater than the $\approx12$~mas=20
uncertainty.
\section{Discussion}
=09The apparent motion of \sgrab with respect to background
radio sources can be used to estimate the rotation of the Galaxy and
any peculiar motion of the super-massive black hole candidate \sgra.
In Fig.~2 we plot the change in apparent position on the plane of the sky o=
f
\sgrab relative to J1745--283. The dotted line is the variance-weighted=20
least-squares fit to the data, and the solid line denotes the orientation
of the Galactic Plane. Clearly the apparent motion of
\sgrab is almost entirely in the Galactic Plane.
Thus, it is natural to convert the apparent motion from
equatorial to galactic coordinates. For \sgrab relative to J1745--283,
this yields an apparent motion of
$-5.90\pm0.35$ and $+0.20\pm0.30$~mas~\peryr\ in=20
galactic longitude and latitude, respectively. =20
The apparent motion in the plane of the Galaxy should be dominated by=20
the effects of the orbit of the Sun around the Galactic
Center, while the motion out of the plane should contain only small terms
from the Z-component of the Solar Motion and a possible motion of \sgra.
In the following subsections, we investigate the various components of
the apparent motion of \sgra, place limits on any offset of \sgrab
from the dynamical center of the Galaxy, derive limits on the mass of \sgra=
,
and constrain the distribution of dark matter in the Galactic Center.
\subsection {Motion of \sgrab in the Plane of the Galaxy}
=09Assuming a distance of $8.0\pm0.5$~kpc (Reid 1993),=20
the apparent angular motion of \sgrab in the plane of the Galaxy
translates to $-223\pm19$~{\hbox \kms.} The uncertainty includes the effec=
ts of
measurement errors and the 0.5~kpc uncertainty in \rnot.
Provided that the peculiar motion of \sgrab is small (see \S 3.2),=20
this corresponds to the reflex of true orbital motion of the Sun around the=
=20
Galactic Center. This reflex motion can be parameterized
as a combination of a circular orbit (\ie of the LSR)
and the deviation of the Sun from that circular orbit (the Solar Motion).
The Solar Motion, determined from Hipparcos data=20
by Dehnen \& Binney (1998), is $5.25\pm0.62$~\kms\ in the direction of gala=
ctic=20
rotation.
%and 7.17 \porm 0.38 \kms\ towards positive galactic latitude. =20
Removing this component of the Solar Motion from the {\it reflex} of the=20
apparent=20
motion of \sgrab yields an estimate for \tnot\ of $218\pm19$~{\hbox \kms.} =
=20
This value is consistent with most recent estimates of about 220~\kms=20
(Kerr \& Lynden-Bell 1986) and can be scaled for different values of the=20
distance=20
to the Galactic Center by multiplying by $\rnot/8$~kpc.
=09The most straightforward comparison of our direct measurement of the=20
{\it angular} rotation rate of the LSR at the Sun (\tnot/\rnot)
can be made with Hipparcos measurements based on motions of Cepheids.
Feast \& Whitelock (1997)=20
conclude that the angular velocity of circular rotation at the Sun,
\tnot/\rnot (=3D Oort's A--B), is $27.19 \porm 0.87$~\kms\ \perkpc\=20
($218 \porm 7$~\kms\ for $\rnot=3D8.0$~kpc). =20
Our value of $\tnot/\rnot$, obtained by removing the Solar Motion in
longitude from the {\it reflex} of the motion of \sgrab in longitude,
is $27.2\pm1.7$~\kms\ \perkpc.
The VLBA and Hipparcos measurements are consistent within their joint error=
s,=20
and both measurements are insensitive to the value of \rnot, as it is only =
used=20
to remove the small contribution of the Solar Motion. =20
It is important to note that our value of $\tnot/\rnot$ is a=20
true ``global'' measure of the angular rotation rate of the Galaxy. =20
The consistency of the local (A--B) and global measures of $\tnot/\rnot$=20
suggests that=20
local variations in Galactic dynamics ($d\tnot/d\rnot$) are less than the j=
oint=20
uncertainties of about 2~\kms~\perkpc.
=09After removing the best estimate of the motion of the Sun around the
Galactic Center, our VLBA observations yield an estimate of
the peculiar motion of \sgrab of $0.0\pm\sqrt{0.87^2+1.7^2}$~\kms~\perkpc\ =
or
$0\porm15$~\kms\ towards positive galactic longitude.
This estimate of the ``in plane'' motion of \sgrab comes from differencing
two {\it angular} motions. Since this difference is negligible, the uncert=
ainty=20
in \rnot\ does not affect this component of the peculiar motion of \sgra.
Given the excellent agreement in the global and local measures of the angul=
ar=20
rotation rate of the Galaxy, and the lack of a detected peculiar motion for=
=20
\sgra,
it is likely that \sgrab is at the dynamical center of the Galaxy.
=09
\subsection=09{Motion of \sgrab out of the Plane of the Galaxy}
=09Whereas the orbital motion of the Sun (around the Galactic Center)
complicates estimates of the ``in plane" component of the=20
peculiar motion of \sgra, motions out of the plane are simpler to interpret=
. =20
One needs only to subtract the small Z-component of the Solar Motion from t=
he
observed motion of \sgrab to estimate the out-of-plane component of the pec=
uliar=20
motion
of \sgra. An implicit assumption in this procedure is that the=20
Solar Motion reflects the true peculiar motion of the Sun. Since most esti=
mates=20
of the=20
Solar Motion are relative to stars in the solar neighborhood, this
assumes that ``local'' and ``global'' estimates of the Solar Motion=20
are similar. This procedure could be compromised slightly were the=20
solar neighborhood to have a significant motion out of the=20
plane of the Galaxy, owing, for example, to a galactic bending or corrugati=
on=20
mode.
=09One way to limit the magnitude of a possible difference between a local=
=20
and global
estimate for the Solar Motion is to compare motions based on nearby stars=
=20
with those based on much more distant stars. Using stars within about=20
0.1~kpc Dehnen \& Binney (1998) find the Z-component for the Solar Motion
to be $7.17 \pm 0.38~\kms$, while Feast \& Whitelock (1997) determine a val=
ue
of $7.61 \pm 0.64~\kms$ for stars with distances out to a few kpc.
Since these values agree within their joint uncertainties of about 0.74 \km=
s,=20
it seems unlikely that local and globlal values for the Solar Motion could=
=20
differ by=20
more than about 1~\kms.
=09The Hipparcos measurements of large numbers of stars in the solar
neighborhood provide an excellent reference for determining the local solar=
=20
motion.
We adopt the value of Dehnen \& Binney (1988), which comes from the velocit=
ies=20
of=20
more than 10,000 stars within about 0.1~kpc of the Sun.
Removing $7.17$~\kms from our measured apparent motion of=20
\sgrab out of the plane of the Galaxy, we estimate the peculiar motion of=
=20
\sgrab to be $15\pm11$~\kms\ toward the north galactic pole (see Table 3). =
=20
The uncertainty is dominated by our proper motion measurements and can
be greatly improved by future measurements. Note, for example,=20
that increasing the weight (decreasing the estimated uncertainty) of the=20
1995 measurement would decrease the magnitude of our peculiar motion estima=
te.
We do not consider our estimate of the peculiar motion of \sgrab out of the
plane of the Galaxy to be statistically distinguishable from a null result.=
=20
\subsection {Limits on the Mass of \sgra}
=09Our estimates of a peculiar motion of \sgrab=20
provide an upper limit of about $20$~\kms each=20
for motions in and out of the galactic plane. =20
Since stars in the inner-most regions of the central
cluster move at speeds in excess of 1000~\kms\ (Eckart \& Genzel 1997,=20
Ghez \etal 1999), a central ``dark mass'' of approximately=20
$2.6\times10^6$~\msun contained within 0.015 pc of \sgrab seems required.
It is likely, but unproven, that most of this mass is contained in a=20
super-massive black hole: \sgra. =20
Given the fact that independent measurements (Backer \& Sramek 1999,=20
and this paper) show that \sgrab moves at least two orders of magnitude slo=
wer
than its surrounding stars, \sgrab must be much more massive than the=20
$\sim10~\msun$ stars observed in the central cluster. =20
(See also Gould \& Ram\'irez [1998] for discussion of the
implications of a lack of apparent {\it acceleration} of \sgra.)=20
In this section we derive a lower limit to the mass of \sgrab and constrain
possible distributions of dark matter, not in the form of a super-massive
black hole.
\subsubsection=09{Virial Theorem}
=09Unfortunately, the Virial theorem is of little help in relating the
masses and velocities of stars to that of a central massive black hole.
For Virial equilibrium,=20
$$T_{s} + T_{bh} =3D -{1\over2}~(U_{s} + U_{bh})~~~,\eqno(2)$$
where $T$ and $U$ correspond to the kinetic and potential energy terms
and the subscripts $s$ and $bh$ identify those associated=20
with the stars and a central, massive, black hole, respectively. =20
Essentially all the kinetic=20
energy can be tied up in the stars ($T_{s}$) and all the gravitational=20
potential energy found associated with the black hole ($U_{bh}$). =20
In this case, attempts to estimate the kinetic energy of the black hole=20
($T_{bh}$)
%or the potential energy of the stellar interactions ($U_{s}$),=20
require differencing two large and uncertain quantities and will be=20
essentially useless.
\subsubsection=09{Equipartition of Kinetic Energy}
=09Upper limits on the motion of \sgrab have been used to infer lower
limits on the mass of \sgrab by assuming equipartition of kinetic energy
(\eg Backer 1996, Genzel \etal 1997).
This is reasonable for stellar systems such as globular clusters, where
massive stars are found concentrated toward the cluster center and move
more slowly than lower mass stars.
Similar results might also hold for a system involving a central black hole
and a surrounding stellar cluster, provided the core mass of the cluster
{\it greatly exceeds} that of the black hole. =20
However, given the likely mass dominance of \sgrab over the stars=20
within 0.015 pc, where high stellar speeds have been measured,=20
equipartition of kinetic energy may be an unreliable approximation. =20
=09Indeed, our solar system may prove a better ``scale model'' (with=20
planets=20
corresponding to stars and the Sun corresponding to \sgra). =20
The Sun orbits the barycenter of the solar system, approximately
in a binary orbit with Jupiter. Neglecting small perturbations from
other planets, for a binary orbit in the center of mass frame, momentum=20
conservation
requires that $m_J v_J =3D M_\odot V_\odot$, where the subscripts $J$ and $=
\odot$=20
refer
to Jupiter and the Sun, respectively. The ratio of the kinetic energy of
Jupiter to the Sun is given by $m_J v_J^2 / M_\odot V_\odot^2 =3D M_\odot /=
m_J$.
Hence, the kinetic energy of Jupiter exceeds that of the Sun by a factor
equal to the inverse of the ratio of their masses and=20
equipartition of kinetic energy does not apply.
\subsubsection {Case 1: $M_{SgrA*} \sim 2.6\times10^6$~\msun}
=09In order to better evaluate how an upper limit to the motion of \sgrab
can be used to provide a lower limit to its mass, we carried out N-body=20
simulations of stars orbiting about a massive black hole. =20
We used a simple, direct integration code (NBODY0) of Aarseth (1985),=20
documented by Binney \& Tremaine (1987) and modified for our purposes.
Initial simulations used 255 stars orbiting a $2.6\times10^6~\msun$ black h=
ole.
The stellar masses were chosen randomly to represent the upper end of a=20
stellar mass function with a power law distribution from 20 down to 2 \msun=
. =20
The number of stars and their masses used in this simulation are comparable
to those observed within the central 0.5~arcsec, or 4000~AU (Genzel \etal 1=
997).
Stellar orbits were chosen by randomly assigning a distance from the black =
hole,
uniformly distributed in the range 10 to 10,000 AU, calculating a circular=
=20
orbital speed, and then adjusting the speed randomly by between $\porm20\%$=
=20
of the circular speed for each of the three Cartesian coordinates. =20
Before starting the N-body=20
integrations, the orbital orientations were randomized by rotating the=20
coordinates (and velocity components) through three Euler rotations with=20
angles chosen at random.
=09The N-body simulations show quasi-random motions of the massive black=20
hole. After relatively short periods of time ($\ll10,000$~years) a=20
``steady state'' condition appeared to be reached. The speed of=20
a typical star was about 700 \kms\ at an average distance of=20
6,000 AU. The motion of \sgrab changed completely in all three coordinates=
=20
on time scales $\ll100$~years and was typically $\lax0.1$~\kms\=20
in each coordinate. The rapid, but bounded, changes in the motion of \sgra=
b=20
suggests that close encounters with individual stars are responsible for mo=
st
of the observed motions. Assuming this is the case, one can make a=20
simple analytical estimate of the expected motion of \sgra, owing to close
encounters with stars in the dense central cluster.
=09For a two-body interaction conserving momentum and viewed in the center=
=20
of mass frame,
$$mv =3D MV~~~,\eqno(3)$$
where $m$ and $v$ are the mass and speed of the star
and $M$ and $V$ are the mass and speed of the black hole, respectively.
For the case of interest where $M \gg m$, the orbital speed of the star =09
at periastron, $v_p$, is given by the well known relation
$$v_p^2 =3D { GM \over a } \Bigl( {1+e \over 1-e} \Bigr)~~~,\eqno(4)$$
where $G$ is the gravitation constant, $a$ is the stellar semi-major axis,
and $e$ is the orbital eccentricity.
Defining $V_p$ as the speed of the black hole at periastron, combining
Eqs.~(3) and (4) yields
$$V_p =3D \Bigl( {m \over M} \Bigr) \Bigl( {GM (1+e) \over a (1-e)}=20
\Bigr)^{1/2}~~~.
\eqno(5)$$
%At periastron the distance between the star and the black hole, $r_p$,
%is $a(1-e)$.
=09Our observations are only sensitive to orbital periods longer=20
than of order 1 year. Such orbital periods occur for stellar=20
semimajor axes greater than about 50 AU (1000 Schwarzschild radii)
for a $2.6\times10^6 \msun$ black hole.
Thus, for our application reasonable values for the parameters in Eq.~(5) a=
re as=20
follows:
$m \sim 10~\msun$, $e \sim 0.5 $, and $a \sim 50$~AU, and the expected
orbital speed of \sgrab would be $\approx0.03$~\kms. =20
(We adopt the periastron speed, instead of the lower average orbital speed,
because the influence of many orbiting stars will likely increase the speed
of \sgra, compared to the single star result.)
This speed is
well below our current limit for the motion of \sgra.
Thus, the simplest interpretation consistent with the fast stellar motions
and the slow \sgrab motion is that \sgrab is a super-massive black hole.
\subsubsection {Case 2: $M_{SgrA*} \ll 2.6\times10^6$~\msun}
=09While the simplest interpretation is that \sgrab is a=20
$2.6\times10^6$~\msun black hole, our upper limit on any peculiar motion=20
for \sgrab currently is two to three orders of magnitude above its expected
motion for that mass. Thus, it seems reasonable to investigate the possib=
ility=20
that
the mass within a radius of 0.01 pc=20
%($2.6\times10^6$~\msun as indicated by the stellar proper motions)=20
is not dominated by=20
\sgra, but instead is in some form of ``dark'' matter. In this
case, \sgrab will react to the gravitational potential and orbit
the center of mass of this dark matter. Below we show that the upper
limits on the motion of \sgrab are complimentary to the stellar
proper motion results and strongly constrain both the mass of \sgrab=20
and any possible configuration of matter within the central 0.01 pc.
=09Fig.~3 displays the enclosed mass versus radius for=20
four mass models that are consistent with the stellar proper motions
(cf., Genzel \etal 1997, Ghez \etal 1999).
These models all yield flat ``enclosed-mass versus radius" relations at
distances $\gax0.01$~pc from the center of mass of the system where
measurements exist. The most centrally condensed
mass distribution, a point mass, is shown as the horizontal dash-dot
line labeled ``a''. The least centrally condensed mass distribution plotte=
d is=20
for
a Plummer density distribution, where density, $\rho$, is given by
$\rho=3D\rho_0~\bigl(1+(r/r_0)^2\bigr)^{-\alpha/2}$,
for $\rho_0=3D6\times10^{11}$~\msun pc$^{-3}$, $r_0=3D0.01$~pc, and $\alpha=
=3D5$. =20
This distribution is shown with a curved dash-dot line labeled ``d''.
It is difficult to make a physically reasonable mass
distribution that is significantly less centrally condensed than this
and consistent with the stellar motion data. These two ``extreme" model=20
distributions
approximately bound all allowed mass distributions; two particular
examples of intermediate models are shown in Fig.~3 with dashed curves
labeled ``b'' and ``c''.
=09Assuming that \sgrab has a mass $\ll10^6$~\msun, it will
orbit about the center of mass of the system. Orbital speed for
a body at a radius, r, from the center of mass is given by
% $$V =3D \sqrt{{GM_{encl}\over r}}~~~,~~\eqno(6)$$
$V =3D \sqrt{{GM_{encl}/r}},$\=20
where $M_{encl}$ is the enclosed=20
mass at that radius. Setting $V=3D20$~\kms, our limit for the motion
of \sgra,=20
% in Eq.~(6)=20
produces the sloping solid line in Fig.~3. =20
Only enclosed masses below that line are permitted by our observations. =20
For radii greater than about $3\times10^{-5}$~pc (6 AU), this limit rules=
=20
out all but the {\it least} centrally condensed mass models.
For radii less than about $3\times10^{-5}$~pc, the orbital motion
of \sgrab produces angular excursions less than 0.8~mas. In this
case, while the orbital speed might greatly exceed 20~\kms, we may
have failed to detect these excursions owing to our poor=20
temporal sampling and $\approx0.4$~mas errors in the North--South direction=
.
Thus, we do not extend the motion limit line below $3\times10^{-5}$~pc,
and at this radius we replace the motion limit with a vertical line=20
in Fig.~3.
=09The stellar proper motions and the limits on the proper motion=20
of \sgrab combine to exclude almost all of ``parameter space" for=20
models of the density distribution of material in the inner
0.1 pc of the Galactic Center. =20
The stellar motions exclude ``soft'' gravitational potentials
(\ie the least centrally condensed mass distributions) and the motion
limit for \sgrab excludes ``hard'' gravitational potentials.
Continued VLBA observations of
\sgrab over the next five years could reduce the uncertainty in
the peculiar motion of \sgrab to about 2~\kms (dotted line in Fig.~3)
out of the plane of the Galaxy. =20
Improved accuracy for positions in individual VLBA observations,=20
necessary for a trigonometric parallax of \sgra, could move the
small angular excursion limit to $<0.2$~mas. This would further and
drastically restrict the range of possible models for a dominant central
dark matter condensation. =20
=09The point-like mass distribution labeled ``a" in Fig.~3 essentially=20
requires a super-massive black hole. Since we have assumed in this section=
=20
that this is {\bf not} \sgra, we would be left with the question of
why a low-mass \sgrab radiates far more than a super-massive black hole in
essentially the same environment.
Other mass models, and especially those with the most centrally condensed=
=20
mass distributions (\eg labeled ``b" in Fig.~3) require exceedingly high
mass densities. For example, model ``b'' has $\approx5\times10^5$~\msun\=
=20
within a radius of 6~AU, resulting in a density of $10^{19}$~\msun~pc$^{-3}=
$.
Theoretical arguments suggest that such models are unlikely to be=20
stable for even $10^7$~y, regardless of the composition of the matter (Maoz=
=20
1998). =20
=09The hardest cases to exclude on either observational or theoretical
grounds are given by the mass distributions that are the least centrally=20
condensed,
but still consistent with the stellar motion data. For this case, one can =
ask=20
the question, at what mass will perturbations by stars in the central clust=
er=20
lead to detectable motion of \sgra? =20
In order to answer this question, we modified the N-body code described in
\S~3.3.3 to include
a fixed gravitational potential appropriate for a Plummer law mass distribu=
tion
with $\alpha=3D5$, $\rho_0=3D6\times10^{11}$~\msun~pc$^{-3}$, and $r_0=3D0=
.01$~pc
(model ``d").
\sgrab was assigned a mass ($\ll10^6$~\msun) and the entire system,
including 254 stars was allowed to evolve in time. =20
For the softer allowable gravitational potentials (\eg ``c'' and ``d'')
there is little enclosed mass within $\sim10^{-4}$~pc to bind \sgra.
We found that when \sgra's mass was less than $\approx3,000$~\msun, \sgrab=
=20
gradually
moved outward from the center of the gravitational potential and achieved
orbital speeds in excess of 20~\kms. Thus, we conclude that the
lack of detectable motion for \sgrab places a conservative lower limit of=
=20
about 1,000~\msun for the mass associated with \sgra.
=09
%While very centrally condensed
%mass distributions (``hard" gravitational potentials) are consistent=20
%with the observed stellar motions, they can be essentially ruled out by th=
e
%lack of motion of \sgra. In this case, while orbits with speeds=20
%approaching 0.1~c are possible and could have gone undetected, the=20
\section=09{Conclusions}
=09The {\it apparent} proper motion of \sgra, relative to extragalactic=20
sources,
is consistent with that expected from the Sun orbiting the center of the Ga=
laxy.
Thus, \sgrab must be very close to, and most likely at, the dynamical cente=
r of=20
the=20
Galaxy. In this case, the proper motion measurement gives the angular rota=
tion=20
speed at the Sun, $(\tnot+\vsun)/\rnot$, directly, from which we estimate
$\tnot=3D218\pm19$~\kms for $\rnot=3D8$~kpc.
=09Our lower limit for the peculiar motion of \sgrab of about 20~\kms=20
implies
a lower limit for mass of \sgrab of $\sim10^3$~\msun. This rules out the
possibility that \sgrab is any known multiple star system, such as a contac=
t
binary containing $\sim10$~\msun\ and radiating near its Eddington luminosi=
ty.
A mass of more than $\sim10^3~\msun$ and a luminosity=20
$\lax10^5~\lsun$ indicates that \sgrab is radiating at
$\lax0.1\%$ of its Eddington limit. =20
=09All observations are consistent with=20
\sgrab being a super-massive black hole. Since the lower limit for
the mass of \sgrab is only about 0.1\% of the gravitational mass inferred f=
rom=20
the
stellar motions, one cannot claim from our observations alone
that even a significant fraction of the dark mass must be in \sgra. =20
However, alternative models involving ``dark'' matter distributions are
severely restricted by observations.
=09Future VLBA observations should be able to reduce the uncertainty in=20
the measurement of the motion of \sgrab out of the Galactic plane=20
to $\sim0.2$~\kms, at which point=20
knowledge of the Solar Motion may become the limiting factor. =20
Should the peculiar motion of \sgrab be less than 0.2 \kms,=20
then its mass almost certainly exceeds $\sim10^5~\msun$. =20
Such a large mass tied {\it directly} to the
radio source, whose size is $\lax1$~AU from VLBI observations,
would be compelling evidence that \sgrab is a super-massive black hole.
\acknowledgments
We thank D. Backer for providing coordinates for sources J1745--283 and=20
J1748--291
early in our project, M. Eubanks for measuring the astrometric position of=
=20
J1745--283,
and V. Dhawan for helping with the VLBA setup.
\begin{references}
\reference{} Aarseth, S. 1985, in {\it Multiple Time Scales}, eds. J. U.=20
Brackbill
\& B. I. Cohen, (Academic Press: New York), pp. 378--418
\reference{} Backer, D. 1996, in {\it Unsolved Problems of the Milky Way},
=09=09eds. L. Blitz \& P. Teuben, (Kluwer: Dordrecht), pp. 193--198
\reference{} Backer, D. \& Sramek, R.A. 1999, {\it this volume}
\reference{} Balick, B. \& Brown, R. 1974, \apj, 194, 265
\reference{} Binney, J. \& Tremaine, S. 1987, {\it Galactic Dynamics},=20
(Princeton
U. Press; Princeton)
\reference{} Bower, G. C. \& Backer, D. C. 1998, \apj, 496, L97
\reference{} Dehnen, W. \& Binney, J. J. 1998, \mnras, 298, 387
\reference{} Eckart, A. \& Genzel, R. 1997, \mnras, 284, 576
\reference{} Feast, M. \& Whitelock, P. 1997, \mnras, 291, 683
\reference{} Genzel, R., Eckart, A. Ott, T. \& Eisenhauer, F. 1997, \mnras,=
291,=20
219
\reference{} Ghez, A. M., Klein, B. L., Morris, M. \& Becklin, E. E. 1999, =
to=20
appear in \apj
\reference{} Gould, A. \& Ram\'irez, S. V. 1998, \apj, 497, 713
\reference{} Gwinn, C. R., Moran, J. M., Reid, M. J. \& Schneps, M. H. 1988=
,=20
\apj, 330, 817
\reference{} Kerr, F. J. \& Lynden-Bell, D. 1986, \mnras, 221, 1023
\reference{} Maoz, E. 1998, \apj, 494, L181
\reference{} Menten, K. M., Reid, M. J., Eckart, A. \& Genzel, R. 1997, \ap=
j,=20
475, L111
\reference{} Niell, A. E. 1996, JGR, 101, 3227
\reference{} Reid, M. J. 1993, \araa, 31, 345
\reference{} Romani, R. W., Narayan, R. \& Blandford, R. 1986, \mnras, 220,=
19
\reference{} Serabyn, E., Carlstrom, J., Lay, O., Lis, D. C., Hunter, T. R.=
,
=09=09Lacy, J. H. \& Hills, R. E. 1997, \apj, 490, L77
\reference{} Treuhaft, R. N. \& Lanyi, G. E.. 1987, Rad. Sci., 22, 251.
\reference{} Zoonematkermani et al 1990, \apjs, 74, 181
\end{references}
\vfill\eject
\figcaption {Position residuals of \sgrab\ relative to J1745--283 (circles)=
and
J1748--291 relative to J1745--283 (crosses) versus time. Eastward
components are shown in the top panel and Northward components in the
bottom panel. The solid and dashed lines give the variance-weighted
best fit components of proper motion. The J1748--291---J1745--283
positions have been offset to fit the plot scale for the=20
\sgra---J1745--283 data.}
\figcaption {Position residuals of \sgrab\ relative to J1745--283 on the
plane of the sky. North is to the top and East to the left. Each
measurement is indicated with an ellipse, approximating the
apparent, scatter broadened size of \sgrab at 43 GHz, the
date of observation, and $1-\sigma$ error bars.
The dashed line is the variance-weighted best-fit proper motion,
and the solid line gives the orientation of the Galactic plane.}
\figcaption {Enclosed mass versus radius for various model distributions
of dark matter, {\it assuming the mass of} \sgrab $\ll 10^6$~\msun.
Models labelled ``a'' through ``d'' have decreasing central mass condensati=
ons
(progressively softer gravitational potentials) and approximately bound
mass distributions that are consistent with stellar proper motion data.
Model ``a'' is a point mass; model ``b'' through ``d'' have Plummer density
distributions with $\rho_0$ of $3.9\times10^{18}, 2.5\times10^{14}, {\rm=20
and}~6.0\times10^{11}$
\msun~pc$^{-3}$; $r_0$ of 0.00002, 0.001, and 0.01 pc; and $\alpha$ of 3, 4=
, and=20
5,
respectively. The sloping solid line indicates the upper limit for enclose=
d=20
mass based
on the proper motion of \sgra. The vertical solid line at $3\times10^{-5}$=
~pc=20
(6~AU)
indicates the upper limit in radius, where angular excursions of \sgrab of=
=20
$<0.8$~mas
could be missed owing to insufficient astrometric accuracy. The sloping do=
tted=20
line
indicates expected improvement in the measurement of the proper motion of \=
sgrab
within $\approx5$~years.}
\end{document}
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