ohirsurv.tex to appear in the proceedings of the 4th ESO/CTIO workshop "The Galactic Center" \documentstyle[11pt,paspconf,epsf]{article} \begin{document} \title{OH/IR Stars - Dynamical Studies} \author{Anders Winnberg} \affil{Onsala Space Observatory, S - 439 92 Onsala, Sweden} \begin{abstract} Since 20 years surveys have been undertaken to find OH/IR stars in the central parts of our Galaxy using various radio telescopes around the world. The present paper gives a review of such surveys and of dynamical studies using their data. Presently, two major survey projects are going on: a deep search in the very Centre of the Galaxy using both VLA monitoring data and new AT data and a large--scale survey of the area ${|l| \le 45\deg}$, ${|b| \le 3\deg}$ also using both VLA and AT data. The fact that only three parameters per OH/IR star are known -- the two sky coordinates and the line--of--sight velocity -- implies that severely limiting and simplifying assumptions have to be made regarding the distribution function of the stars and the gravitational potential. Usually spherical symmetry and isotropic velocity dispersion have been assumed. Recently, a project has been initiated to determine the proper motions of the OH/IR stars close to the Galactic Centre using the VLA and the VLBA on the associated H$_2$O and SiO masers. It is pointed out that the analysis of $l-v$ diagrams so far has overlooked the fact that they are superpositions of many $l-v$ diagrams from concentric ring areas within the total area. The slopes of the regression lines of these 'inner' $l-v$ diagrams get increasingly larger for smaller areas. The '$l-v$ slope' as a function of galactocentric distance is that expected for differential rotation in an $r^{-2}$ mass density distribution, a fact which also is suggested by the surface density distribution of the stars. \end{abstract} \section{Introduction} OH/IR stars are variable (pulsating) Asymptotic Giant Branch (AGB) stars. They lose mass at a high rate (10${^{-7} - 10^{-4}}$ M$_{\sun}$/yr). Dust particles condense in the stellar wind to such an extent that the central stars become invisible and the spectral energy distribution of the objects is shifted into the far infrared. Water vapour is photodissociated by UV radiation penetrating the expanding envelope from the outside, thus forming a shell of hydroxyl (OH) molecules at a distance from the central star of 10$^{16}$ -- 10$^{17}$ cm. The OH molecules are pumped by IR radiation from the surrounding dust which results in a maser--amplified OH line at 1612 MHz. As the gas expands through the OH shell, the line profile consists of two Doppler shifted components originating at the back and front sides of the shell. Therefore, the line--of--sight velocity of the star is the average of the line--of--sight velocities of the two components and the envelope expansion velocity is half the velocity difference between them. (In this paper we reserve the word 'radial' for the radial direction from the Galactic Centre.) Habing (1996) has written a review on AGB stars. OH/IR stars are excellent objects to probe the gravitational potential in the central parts of the Galaxy because: they are insensitive to all other forces except gravitation, they probably form a dynamically relaxed system, their radio radiation is not absorbed in the interstellar dust, their OH line emission is maser amplified making them detectable throughout the Galaxy, their radial velocities can be measured accurately, their distances can be determined through a geometrical method (except for the very central parts of the Galaxy, where the interstellar scattering broadens the angular sizes of the OH shells), and their proper motions can be measured in the near future (see below). Despite these advantages, progress in dynamical studies using OH/IR stars has been slow. One reason for this is the fact that, whereas $l-b$ and $l-v$ diagrams of various {\sl inter}stellar gas components (as e.g. CO) delineate structures, the corresponding diagrams for OH/IR stars are scatter diagrams in which it is very difficult to see any structures, at least with today's modest number of stars. The data analysis so far has been limited to determinations of regression lines and dispersions from them. This very simplistic way of looking at the data might lead to wrong or, at least, incomplete conclusions as I will try to explain. \section{Surveys} Already in the 1970's, surveys were undertaken to find OH/IR stars in the direction of the Galactic Centre (Baud et al. 1979; Olnon et al. 1981; Habing et al. 1983). However, these surveys were either carried out with too low sensitivity or they had serious selection effects. It was shown by Habing et al. (1983) that the sky area around Sgr A had to be searched using a radio interferometer with sufficiently long baselines in order to avoid detecting strong interstellar absorption lines of OH (see below). Subsequent surveys in this part of the sky therefore were carried out using interferometers such as the Very Large Array (VLA) and the Australia Telescope (AT). One of the major surveys in the 1980's, however, was carried out using single--dish instruments -- the survey by te Lintel Hekkert et al. (1991; LCHHN). This survey was not aimed at any particular region of the sky -- it is an all--sky survey -- but it covers the Galactic Bulge fairly well. More than 3000 IRAS point sources were searched for 1612--MHz OH emission and about 700 were detected. The IR sources were selected according to their IR colours which were required to be similar to those of OH/IR stars with known IR counterparts. Radio telescopes on both hemispheres were used for the survey: the 25--m Dwingeloo telescope in the Netherlands, the 100--m Effelsberg telescope in Germany, and the 64--m Parkes telescope in Australia. Most of the observing time was spent on the Parkes radio telescope and therefore there is a slight bias toward southern sources. The detected sources are concentrated towards the Galactic plane and Centre but there is a distinct avoidance from the lowest Galactic latitudes due to saturation of the IRAS detectors. Therefore this survey is a good large--scale survey but it should not be used for, say, ${|b| \la 2\deg}$. Another word of warning: Some of the OH spectra ($\sim$10\%) are not typical of OH/IR stars. Lindqvist et al. (1992a; LHWM) used the VLA to survey a small area of typical size ${1.0\deg \times 1.5\deg}$ covering the Galactic Centre and they detected 134 OH/IR stars. This area covers a large part of the area surveyed by Habing et al. (1983) who detected 34 OH/IR stars only. One reason for this large discrepancy, apart from the difference in sensitivity between the two surveys, is that the single dish (the 100--m telescope) used by Habing et al. detected strong interstellar absorption lines of OH which prevented the detection of weak emission lines from OH/IR stars. LHWM, on the other hand, rejected VLA visibility data from baselines shorter than 3000 wavelengths (540~m) in order to resolve out structures with angular sizes larger than about 1'. This eliminated most of the continuum flux in the Sgr A region and the spectral baselines became straight, enabling the detection of weak, narrow emission lines. Van Langevelde et al. (1993) monitored, over a time period of 3 years, the 37 strongest OH/IR stars of LHWM. They derived periods for 13 of the stars and OH shell diameters for 3 of them. Recently Sjouwerman (Leiden/Onsala) has concatenated most of the data and searched for OH/IR stars stronger than 40 mJy. He has also made observations using the AT. These data are centred on Sgr A West and they cover a velocity range from --550 to +600 km s$^{-1}$ (relative to the LSR). Their {\sl rms} noise fluctuations are about 5 mJy -- at least 4 times better than LHWM. Taken together (the concatenated VLA data and the AT data), about 50 previously unknown OH/IR stars have been detected (Sjouwerman et al., these proceedings). Another survey "in the making" is the one by Sevenster (Leiden). This is a large-scale survey which is specially "designed" to complement the survey by LCHHN, i.e. to fill in the low--latitude strip in which the IRAS point source catalogue is incomplete. Another aim of this survey is to find dynamical evidence, in the stellar component, for a bar in the Centre of the Galaxy. The survey covers the area ${|l| \le 45\deg, |b| \le 3\deg}$ and uses data from both the VLA and the AT. A paper has been submitted which presents AT data (539 fields) covering the Galactic Bulge (${|l| \le 10\deg , |b| \le 3\deg}$). The {\sl rms} noise is less than 40 mJy for about 90\% of the survey area. From these data 245 OH/IR stars have been detected of which 145 are previously unknown. In this context another type of survey should be mentioned. Izumiura et al. (1994, 1995a, b) have searched for SiO ${J = 1-0, v = 1}$ and 2 maser emission from IRAS point sources toward the Galactic Bulge (${|l| < 15\deg, 3\deg < |b| < 15\deg}$). The sources were selected according to IR colour, using criteria similar to those used by LCHHN. As a matter of fact, many sources are in common with LCHHN (about 30\%) and these are certainly OH/IR stars. Probably most of the other sources are OH/IR stars as well. Izumiura et al. have detected 194 sources in total. They have tried to separate them into 'bulge' and 'disk stars' and claim that 134 of the OH/IR stars have a high probability of belonging to the Bulge and 60 to the Disk. \section{Boltzmann's equation} The relevant equation describing the relation between the OH/IR stars and the gravitational potential is the socalled {\sl collisionless Boltzmann equation}: \begin{equation} \frac{\partial f}{\partial t} + {\bf v} \cdot {\bf \nabla} f - {\bf \nabla} \Phi \cdot \frac{\partial f}{\partial {\bf v}} = 0 \end{equation} \noindent where ${f = f({\bf x}, {\bf v}, t)}$ is the {\sl distribution function} or {\sl phase-space density} and $\Phi$ the {\sl gravitational potential}. $f$ gives the distribution of the OH/IR stars in spatial position $\bf{x}$ and velocity $\bf{v}$ at time $t$. Thus, $f$ is a function of seven variables. We usually assume that the system is time independent and in that case $f$ is a function of six variables. However, each OH/IR star is characterized by three variables only; the sky coordinates and the line--of--sight velocity. Because of this we have to make several assumptions of symmetry. If we assume that the system is spherical, that there is no net streaming motion in the radial direction and in the direction perpendicular to the Galactic Disk, and integrate (1) over velocity, $\bf{v}$, we get, in spherical coordinates $(r, \theta, \phi)$: \begin{equation} \frac{{\rm d} \left(n \sigma_r^2 \right)}{{\rm d} r} + \frac{n}{r}\left[2\sigma_r^2 - \left(\sigma_\theta^2 + \left^2 + \sigma_\phi^2\right)\right] = - n \frac{{\rm d} \Phi}{{\rm d} r} \end{equation} \noindent where ${n = n(r)}$ is the number density of stars and \begin{equation} \sigma_r^2 = \left \end{equation} \begin{equation} \sigma_\theta^2 = \left - \left^2 \end{equation} \begin{equation} \sigma_\phi^2 = \left \end{equation} \noindent are the velocity dispersions in the galactocentric radial, azimuthal, and latitudinal directions, respectively. Eq. (2) is the {\sl Jeans equation}. In a spherical system the gravitational potential, $\Phi$, is related to the enclosed mass, $M(r)$, according to: \begin{equation} \frac{{\rm d} \Phi}{{\rm d} r} = \frac{GM(r)}{r^2} \end{equation} \noindent where $G$ is the {\sl gravitational constant}. We can then solve for the enclosed mass: \begin{equation} M(r) = \frac{r\sigma_r^2}{G}\left[-\frac{{\rm d} \ln n(r)}{{\rm d} \ln r} - \frac{{\rm d} \ln \sigma_r^2}{{\rm d} \ln r} + (\lambda - 2) + \frac{\left^2}{\sigma_r^2}\right] \end{equation} \noindent where: \begin{equation} \lambda = \frac{\sigma_\theta^2 + \sigma_\phi^2}{\sigma_r^2} \end{equation} \noindent i.e. ${\lambda = 2}$ for an isotropic velocity dispersion distribution. Since we know nothing about the anisotropy of the velocity dispersion we usually assume that it is isotropic. This latter assumption is most probably wrong, judging from observations of other galaxies. However, despite the fact that several adventurous assumptions have been made regarding both the gravitational potential (sphericity) and the distribution function (isotropic velocity dispersion), there are further problems in the application of eq. (7): How can we estimate the volume density, $n(r)$, the velocity dispersion, $\sigma_r(r)$, and the {\sl mean streaming motion} in galactocentric azimuth, $\left$? We know the projected counterparts of these quantities as functions of the projected galactocentric distance, $R$ -- the surface density, $N(R)$, the line-of-sight velocity dispersion, $\sigma_{los}(R)$, and the mean line--of--sight velocity, $\left$. Thanks to the symmetry assumptions we can 'deproject' these quantities by solving a set of Abel integral equations (Binney \& Tremaine 1987). If ${N(R) \propto R^{-\alpha}}$ one can show that ${n(r) \propto r^{-\alpha-1}}$. In particular, when ${\alpha = 1}$ the system is an {\sl isothermal sphere}, a dynamically relaxed, equilibrium configuration. The line-of-sight velocity dispersion for such a configuration is dominated by the velocity dispersion in the tangential point because of the high central density concentration, ${\sigma_r \approx \sigma_{los}}$. A similar effect is at work with regard to the mean streaming motion in galactocentric azimuth, $\left$; a good approximation is the mean line--of--sight velocity, $\left$, at a projected galactocentric distance corresponding to the tangential distance. However, it is not particularly difficult to solve the corresponding 'de-projection integrals' numerically. If we could determine the proper motions of the OH/IR stars close to the Galactic Centre we would know five parameters per star instead of three. This would give considerably more information on the distribution function especially the velocity dispersions. Sjouwerman and van Langevelde have started on such a project using the VLA and the VLBA on H$_2$O and SiO masers associated with OH/IR stars at the Galactic Centre. It is not possible to carry out such a project on the 18--cm OH masers because they are broadened in angle due to interstellar scattering (van Langevelde et al. 1992). The H$_2$O and SiO masers occur at 13 and 7 mm wavelength, respectively, where the broadening due to scattering is a factor of about 200 and 700 smaller, respectively. \section{Data analysis} Lindqvist et al. (1992b) analyzed their data following the procedure outlined above. In order to estimate the mean streaming motion of the stars they calculated the linear regression line through the $l-v$ diagram. In doing that, they applied a similar data analysis as did McGinn et al. (1989) on their IR data of the inner 5 parsecs around the Galactic Centre. In this way the two data sets are directly comparable and complementary. \begin{figure} \plotone{ohirinbl.ps} \caption{$l-v$ diagram of OH/IR stars in the central parts of the Galaxy. Open circles are stars from LCHHN and filled circles are stars from LHWM.} \end{figure} Te Lintel Hekkert (1990) took a completely different approach in his dynamical analysis of the data of LCHHN. Since these data cover the whole Galaxy and the overall gravitational potential of the Galactic Disk is known from optical work, he adopted a gravitational potential and tried to estimate the distribution function of the OH/IR stars. He did this in collaboration with Dejonghe (Ghent) by applying a new mathematical method called {\sl quadratic programming} invented by Dejonghe (1989). In this method the orbits of the stars are characterized by their binding energy, $E$, and their angular momentum, $L_z$. They were able to construct a probability density diagram in $E-L_z$ space. In this diagram one can see that most of the stars have nearly circular, prograde orbits except for stars in the Bulge, where there is a larger fraction of stars having elongated orbits and even stars with retrograde rotation. Sevenster et al. (1995) compared the data sets of LCHHN and LHWM. They used the quadratic programming technique described above and came to the conclusion that the two samples of OH/IR stars consist of two distinct populations, one that extends over the whole Galaxy and shows signs of evolution and another one which is only seen near the Galactic Centre, has high rotation and does not show signs of long evolution. However, one can ask oneself whether this approach is the most appropriate in this region of the Galaxy. Here the gravitational potential is not well known and this is precisely the reason why OH/IR stars are so valuable -- they can give us information on the gravitational potential which we do not have. It seems to me that, in this region, we have more knowledge about the distribution function of the OH/IR stars than we have about the potential and that, by assuming a potential, we lose information which is contained in the data. \section{Is there a kinematic connection between the OH/IR stars in the Galactic Centre and those in the Bulge?} When comparing the $l-v$ diagrams of the LCHHN stars within ${|l| < 30\deg}$ and of the LHWM stars it is obvious that their kinematics are very different (Figure 1). This was pointed out by Dejonghe (1993) already. However, it was noted by Lindqvist et al. (1992b) that the central 15 stars of the LHWM sample obey an even steeper slope in the $l-v$ diagram, 2000~km~s$^{-1}$~deg$^{-1}$ instead of 180~km~s$^{-1}$~deg$^{-1}$ for the whole sample. Moreover these 15 stars are in good agreement with the data of McGinn et al. (1989). The slope of the regression line for the LCHHN stars in Figure 1 is about 11~km~s$^{-1}$~deg$^{-1}$ only. It is rather strange that, so far, we have concluded that the OH/IR stars in the central regions of the Galaxy rotate like a solid body whereas their surface density indicate that they have an $r^{-2}$ volume density law. Why should the OH/IR stars have a solid-body rotation and yet show an isothermal distribution? The different $l-v$ slopes for samples of different sky extent mentioned above may give a hint of how to analyze the data. \begin{figure} \plottwo{slopemod.ps}{enclmass.ps} \caption{a. Regression--line slopes of $l-v$ diagrams for OH/IR stars plotted against galactocentric distance. b. Enclosed mass as a function of galactocentric distance.} \end{figure} In a still ongoing work I have divided the data of LHWM and that of LCHHN for ${|l| < 30\deg}$ into subsamples contained in concentric ellipses (Winn\-berg 1994). I have plotted $l-v$ diagrams of the stars within the central ellipse and within the surrounding elliptical annuli. Each $l-v$ diagram contains about 60 stars. I have then fitted linear regression lines to these $l-v$ diagrams and plotted the slopes of the regression lines as a function of the semi major axis of the outer ellipse in each annulus (Figure 2a). As you can see all the data points and their statistical errors can be described well by a straight line in this log--log plot, i.e. data points from the LHWM stars and from the LCHHN stars can be fitted well with the {\sl same} line. For comparison I have drawn three other lines, one for a system with constant density (dotted line), one for a system with an $r^{-2}$ density law (dashed line), and one for a system with a central point mass (dash--dotted line). These lines are valid only if the velocity dispersion is constant. The line--of--sight velocity dispersion for these stars is increasing slowly with $r$ due to an increasing fraction of older stars. Of course the enclosed mass can be calculated according to eq. (7) and it turns out to be somewhat higher at short galactocentric distances than previously believed and to follow a line with a slope of 1 in a log--log plot as expected for an $r^{-2}$ density law (Figure 2b). This is what Becklin \& Neugebauer (1968) claimed already a long time ago. Admittedly it is a bit bold to carry the comparison between the two data sets this far because the corresponding surveys were very different both in the methods used and in the sensitivities reached. Also, so far, I have made no attempt at separating 'disk stars' from 'bulge stars'. However, I believe that, as long as I only deal with the kinematics of the stars, the result is not too far from reality. In any case a large--scale, homogeneous survey is now carried out (Sevenster et al. 1996) and others will no doubt follow in the near future. Therefore my hypothesis will soon be tested. \begin{references} \reference Baud, B., Habing, H. J., Matthews, H. 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