The warped young stellar disc in the Galactic centre
L. Subr^{1,2,3}  J. Schovancová^{1}  P. Kroupa^{3}
1  Faculty of Mathematics and Physics, Charles University,
V Holesovickách 2, 18000 Praha, Czech Republic
2 
Astronomical Institute, Academy of Sciences, Bocní II, 14131 Praha,
Czech Republic
3 
Argelander Institute for Astronomy (AIfA), Auf dem Hügel 71, 53121 Bonn,
Germany
Received 2 October 2008 / Accepted 25 November 2008
Abstract
Aims. Within the central parsec of the Galaxy, several dozen young stars orbiting a central supermassive black hole are observed. A subset of these stars forms a coherently rotating disc. Other observations reveal a massive molecular torus that lies at a radius 1.5 pc from the centre. In this paper we consider the gravitational influence of the molecular torus upon the stars of the stellar disc.
Methods. We derive an analytical formula for the rate of precession of individual stellar orbits and we show that it is highly sensitive to the orbital semimajor axis and inclination with respect to the plane of the torus, as well as to the mass of the torus.
Results. Assuming that both the stellar disc and the molecular torus are stable on the timescale 6 Myr, we constrain the mass of the torus and its inclination with respect to the young stellar disc. We further suggest that all young stars observed in the Galactic centre may have a common origin in a single coherently rotating structure with an opening angle
,
which was partially destroyed (warped) during its lifetime by the gravitational influence of the molecular torus.
Key words: stellar dynamics  Galaxy: nucleus
1 Introduction
Near infrared observations of the central parsec of the Galaxy that were made over the past decade have brought new views of the environment in the vicinity to a supermassive black hole. They revealed a numerous population of young massive stars that may be distributed into at least two different groups. Within a distance from the centre are found more than ten socalled Sstars orbiting the supermassive black hole on apparently randomly oriented and highly eccentric ( ) orbits. They appear to be standard OB main sequence stars (Ghez et al. 2002; Eisenhauer et al. 2005) which contradicts with strong tidal forces that prevent stellar formation at the place. Unless these stars mimic their age, it is likely that they have migrated to the centre from longer distances.Farther away, at , nearly one hundred young stars have been detected (see Paumard et al. 2006, for one of the most recent reviews). These stars are mainly classified as postmain sequence OB supergiants and WolfRayet stars. According to the evolutionary phase, their age is estimated to be . Levin & Beloborodov (2003) point out that a substantial fraction of these stars form a coherently rotating disc (usually referred to as a clockwise stellar disc or CWS). It is a flaring disc with an opening angle with a rather sharp inner edge at , and it extends up to radius of . The radial column density profile of the CWS decreases approximately as r^{2}; i.e., most of the stars are concentrated at the inner edge. The mean plane of the disc can be determined by two angles: inclination with respect to the plane of the sky and longitude of the ascending node (measured from the north; see Paumard et al. 2006, for a detailed description of the convention). Levin & Beloborodov (2003) suggest that this disclike pattern is a consequence of a stellar formation in a selfgravitating accretion disc.
Further analyses (Genzel et al. 2003; Paumard et al. 2006) indicate the presence of another coherent stellar system that is usually referred to as the counterclockwise stellar disc (CCWS). This structure is assumed to be formed by fewer (15) stars. It is narrower in the radial extent since concentrated around , and it has a larger opening angle . The existence of the CCWS disc is a matter of ongoing debate (e.g. Lu et al. 2007); nevertheless, even if it is accepted as an explanation of the origin of another subset of young stars in the Galactic centre, more than twenty stars not belonging to any of the two stellar discs would still remain without a satisfactory explanation of their origin.
The gravitational potential in the considered region is dominated by the supermassive black hole of mass (Ghez et al. 2002). It is surrounded by a roughly spherical cluster of latetype stars. The radial density profile is wellfitted with a broken powerlaw with index below and above the break radius (Schödel et al. 2007). Its mass within is comparable to the mass of the black hole.
The central region is surrounded by a molecular torus (circumnuclear disc; CND), which lies at the outer edge of the black hole's sphere of influence ( ). Its mass estimated from the radio observations of ionised molecular gas is (Christopher et al. 2005). This massive structure defines a nonspherical component of the gravitational field in the central parsec.
In this paper we investigate the influence of the CND upon the dynamical evolution of the disclike stellar structures. In the subsequent section we briefly review the dynamics in the perturbed Keplerian potential. In Sect. 3 we apply the results on the motion of the stellar discs  we present constraints on some parameters of the CND and the CWS determined from their gravitational interaction and we also give suggestions on the dynamics of the whole system of young stars over its lifetime. Conclusions and discussion of our results are given in Sect. 4.
2 Dynamics in the perturbed Keplerian potential
For the purpose of this study we introduce a simple model of the Galactic centre, which consists of three main constituents determining the gravitational field: (i) the central supermassive black hole of mass , which is treated as a source of the Keplerian potential; (ii) the massive molecular torus modelled as an infinitesimally thin ring of radius and mass ; and (iii) a spherical stellar cusp with a powerlaw density profile and mass within the radius . Both the ring and the cusp are centred on the black hole.
The stars are treated as test particles whose motions are determined by the composed smooth external potential. Their orbits can be represented by five orbital elements: the semimajor axis a, eccentricity e, inclination i, argument of pericentre , and longitude of the ascending node . Here we assume the angles to be measured in the frame in which the ring lies in the xy plane. For convenience, the results presented in the subsequent section will be transformed into coordinates with  representing the plane of the sky and pointing from the observer. The inclination and longitude of the ascending node in this observer's coordinate system will be denoted and , respectively.
If the gravity of the spherical cusp were ignored, the dynamics in the
field of the central body and the ring would be equivalent to the reduced
hierarchical threebody problem. In this case, the orbital elements
,
and
undergo secular evolution (Kozai 1962;
Lidov 1962) on a timescale of
The equations of motion for mean orbital elements read
The temporal evolution does not depend upon the angle , which is merely a consequence of the axial symmetry; furthermore, energy conservation implies a constant a in this order of approximation.
Including the gravity of the spherical cusp leads to an additional shift in the pericentre, which can be incorporated by an extra term in Eq. (4), dependent upon the global parameters of the cusp and the semimajor axis and eccentricity of the orbit (Ivanov et al. 2005). The overall influence of the cusp can be characterised by a decrease in the amplitude of the oscillations of eccentricity and inclination and shortening of their period (Karas & Subr 2007). This is also clearly seen in Fig. 1, which shows the evolution of an example orbit in the cases both with and without the potential of the spherical cusp. On the other hand, a generic influence of the cusp upon the evolution of lies in diminishing the variations of its first time derivative; the characteristic timescale of the change of generally becomes much longer than that of the mutually coupled elements and .
Figure 1: Evolution of orbital elements of two example orbits. The solid line represents a trajectory in the gravitational field of the central mass and a ring of radius and mass . The dotted line shows an orbit integrated in the field, including a spherical cusp of mass . In both cases the initial values of the orbital elements are: and . 

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Within the context of this paper we are interested in a system where
.
In this case the amplitude of the oscillations
of eccentricity and inclination can be considered negligible, and rotates with much higher frequency than .
This configuration allows us
to simplify Eq. (5) by averaging over one revolution of :
The change of over an interval can then be written as:
According to the underlying perturbation theory, Eqs. (1)(7) refer to the elements averaged over one orbital period, which cannot be easily mapped to the osculating elements defined by instant positions and velocities. We performed numerical tests of the error introduced by replacing averaged elements by the osculating ones in formula (6). For and , the difference between the analytical estimate and the numerically measured rate of change of was always less than a factor of 1.2. With increasing eccentricities, formula (6) overestimates the real precession rate by a somewhat larger factor which, however, still stays below 2.
The real source of the perturbing potential, CND, is a rather clumpy torus of a finite thickness. Hence, we performed several numerical integrations of a test particle orbit in a gravitational field of a central mass, spherical cusp, and a set of 10 discrete point masses with orbits confined in a torus of a toroidal and poloidal radii and , respectively. The difference in the orbital evolution with respect to that in the case of a ringlike perturbation was found to be negligible for orbits with .
3 Consequences of differential precession
3.1 Constraints on the CND
Let us now consider an ensemble of stars forming a thin disc, i.e. with inclinations and longitudes of ascending nodes lying in a narrow interval. Let us further assume that the stellar orbits evolve solely due to the external gravitational potential determined by fixed parameters and . The key feature of the orbital evolution will be precession around the symmetry axis of the ringlike component of the gravitational field.
If the semimajor axes of stars at the inner edge of the disc are smaller than those at the outer edge by a factor of 5, the two edges of the disc will precess at a rate that differs by a factor 10. After a certain period of time, their angular momenta will point to completely different directions, i.e. the disclike structure will be destroyed. Hence, the requirement of the stability of the disc over a given period of time transforms into the requirement of a sufficiently slow precession at its outer edge.
Let us consider the subset of the young stars in the Galactic centre that
form the clockwise stellar disc (CWS).
Inserting values
,
and e=0 into Eq. (7), we obtain
To be compatible with observations, has to be less than , which is the opening angle of the inner part of the CWS (Beloborodov et al. 2006). Hence, Eq. (8) poses a constraint upon the inclination of the disc with respect to the molecular torus, depending on its mass. Considering e.g. (Christopher et al. 2005) requires , i.e. . Simultaneously, this poses an upper limit on the initial opening angle of the stellar disc. For the sake of simplicity, we considered a common sense of precession of all disc stars, i.e. . Identical results would be obtained for due to the symmetry of the problem. The constraint would be tighter by a factor of 1.5 if we consider nonzero (0.5) eccentricities of the stellar orbits.
To conclude this analysis, we note that is in accord with estimates of the normal vectors^{} of the plane of the disc and torus ( , Paumard et al. 2006; , Jackson et al. 1993).
3.2 A common origin of young stars in the Galactic centre?
Formula (8) indicates that orbits of stars at radii and/or inclinations or were considerably affected by precession within the past , i.e. their current orbital parameters are different from their values at the time of the birth. We suggest a possibility that stars that are not considered to be members of the CWS nowadays have been its members at the time of its formation. During the of the dynamical evolution, their orbits were subject to precession due to the gravity of the CND and were detached from their parent stellar system. This model could represent a possible solution of the problem of the origin of all young stars in the Galactic centre.
The mapping between the initial and the current orientations of the stellar orbit is formally straightforward within our simple model. Unfortunately, the observational data do not provide us with accurate enough values of the parameters and . Furthermore, the high sensitivity of the precession rate upon the inclination and semimajor axis, together with a lack of robust determination of these orbital elements from the observational data, also stand as a severe obstacle to any attempt to track the orbits of the observed stars back in time, which could prove or discard the hypothesis of a common origin. In the rest of this section we describe a test that shows that our model is compatible with the publicly available observational data.
We took data from Table 2 of Paumard et al. (2006) from which we considered all stars with determined 3D velocity and index 15 (i.e. excluding the Sstars), which gives in total stars. Five free parameters of the model consist of the two angles, , determining the initial orientation, , of the stellar disc. Another two angles, , determine the orientation of the CND, and represents its mass. (The last parameter can be considered as a degenerate combination of and ; in the following, we will implicitly assume and .) For a given set of parameters we scan the neighbourhood of each star's velocity with sampling d_{v}. The and coordinates of the stars' positions are assumed to be determined exactly. On the other hand, the coordinate (along the line of sight) is unknown. Therefore, we scan it with sampling d_{z} in the full range allowed by the condition that the star is gravitationally bound to the black hole. In total, we consider pairs of position and velocity vectors, which represent states compatible with the observational data of a particular star. For each state we perform a rotation of the normal vector of the orbit around the axis of the CND according to formula (7), which gives its direction, at t=0, i.e. ago. We then calculate its angular distance to , and count the number of states, N_{j,5}, with . We consider the measured star's position and velocity to be compatible with the hypothesis that it was born in the disc with normal vector and thickness , provided N_{j,5}>0. Finally, we denote as the number of stars with N_{j,5}>0 for a given set of values of the parameters of the model.
Figure 2: Various tests of the compatibility of the hypothesis of a common origin of young stars in a single thin disc. N_{5}: number of stars that have at least one state in the neighbourhood of observed velocities, which corresponds to an orbit dragged from the parent disc with normal vector by the CND of mass . N_{e,5}: similar to the previous but excluding orbits with e>0.5. : volume of the subset of the neighbourhood occupied by the states originating in the parent disc. : mean value of the mean anomaly of orbits with . 

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As we discussed in Sect. 3.1, the inner part of the CWS must have only undergone negligible precession due to the gravity of the CND. Therefore, we assume that it conserves orientation of the putative single parent disc and we consider , which is the normal vector of the inner part of the CWS according to Beloborodov et al. (2006). We then set , which enables us to plot N_{5} as a function of and as shown in Fig. 2. We see that an extended region exists where the observational data of nearly all stars are compatible with the hypothesis of their origin in a parent disc of thickness . This region of high values of N_{5} extends along the set of perpendicular to the normal vector . This is a natural consequence of the assumption that 35 stars, identified as CWS nowadays, have not undergone large precession. The region of good compatibility also includes an approximate orientation of the CND as determined from observations, , e.g. by Jackson et al. (1993).
We performed an analogical test of compatibility also for and and neighbourhood of . In all cases we have obtained a picture similar in that there exists an extended region with . Enlarging the inspected neighbourhood of the observed velocities to leads to higher values of N_{5} with its maximum reaching 72. This means that the observational data are compatible with the hypothesis of the common origin of the young stars in a single thin disc; nevertheless, they do not pose strong constraints on the parameters of the model. We introduce three supplementary tests that may be applied to the observational data to verify validity of our hypothesis. First, the model of star formation is assumed to prefer low eccentricities of the stellar orbits. Hence, we introduce N_{e,5} in the same way as N_{5}, but now with an additional condition e<0.5 for all tested orbits. The middle panel of Fig. 2 shows that region with large N_{e,5} coincides with the region of large N_{5}. The maximum value of N_{e,5} is 55; i.e. the model requires about one third of the stellar orbits to have moderate to high eccentricities. Again, considering the neighbourhood of the measured velocity vectors weakens this constraint, giving for a wide range of the model parameters. Additional analysis reveals that most of the eccentric orbits do not belong to the CWS subset of stars.
A ratio
can be
considered as a measure of the
probability that the orbit of star j originated in a disc with opening
angle
and normal vector
.
Consequently,
we introduce
as a measure of the probability that all stars originated in a thin disc. (Adding a unity to N_{j,5} in (9) prevents from being zero everywhere, while it does not strongly affect its meaning.) Bottom panel of Fig. 2 shows that accentuates orientations of the CND nearly perpendicular to the normal vector , but it does not strongly differentiate among the models that fall into this region. Maximum of this function is at .
Following Beloborodov et al. (2006), the configurations compatible with the hypothesis of a single warped disc are expected to have equally distributed value of the mean anomaly, M, of the individual orbits. A full test of the distribution of the mean anomaly of all configurations that have fulfiled other criteria of compatibility is not possible as it would require analysis of combinations of orbits. (Other tests presented here require analysis of only individual orbits.) In the bottom panel of Fig. 2 we present a restricted test showing the mean value of the mean anomaly, , for all tested orbits with . This quantity is close to , which corresponds to the expected uniform distribution in the major part of the space. The analogical plot would show that is close to the expected value in the regions of a high value of N_{5} and . This indicates that our hypothesis does not require some preferred value of M and, therefore, some of the configurations with high N_{5} are also compatible with the assumption of random distribution of the orbital phases.
4 Conclusions
The massive molecular torus (CND) surrounding the central parsec of the Galactic centre causes precession of the orbits of young stars that move at distances around the supermassive black hole. The rate of the precession depends on the orbital parameters, as well as on the orientation and mass of the CND. This rate is comparable to the lifetime of the young stars for a wide range of parameters and, therefore, this process should be taken into consideration in attempts to determine the relation between initial and current values of their orbital parameters. We have shown that would destroy any coherently rotating disclike stellar structure within , provided the inclination of most of the orbits with respect to the CND deviates by more than from . In other words, the stability of the stellar disc within its lifetime poses constraints on its inclination with respect to the CND and on the mass of the CND.
We further suggest that most, if not all, young stars observed in the Galactic centre may have been formed in a single, initially coherently rotating structure, presumably via fragmentation of a thin selfgravitating gaseous disc. Within this hypothesis, the orientation of the stellar disc was nearly exactly perpendicular with respect to the CND. Its core, represented by the CWS nowadays, remained nearly untouched by the precession. On the other hand, stars that were formed at the outer parts of the disc and/or slightly off its mean plane, or that were scattered out of it via twobody encounters, have undergone a more rapid precession of their orbits; i.e., they apparently no longer belong to the stellar disc. We have shown that within the uncertainty of their current velocities, parameters of the stellar orbits exist that would have had their angular momenta collinear about ago. Due to the high sensitivity of the precession upon the orbit inclination with respect to the CND and the uncertainty in the observed parameters of the stellar orbits, the procedure described in the previous section cannot provide robust constraints on the parameters of the model. Therefore, the concept of a single warped disc of young stars in the Galactic centre may be considered as viable, but not proven yet. Our hypothesis, however,gives an explicit prediction of a specific pattern of the normal vectors of the stellar orbits, which may be determined from future, more accurate observations: all of them are assumed to be found close to the circumference perpendicular to the normal vector of the CND.
Let us emphasise that the gravitational influence of the CND leaves stronger imprints on the dynamics of stars more distant from the centre; hence, we suggest that these stars deserve further attention from the observational point of view. Improved measurements of their kinematical state may bring new light on the question of the formation and dynamical evolution of the population of the young stars in the Galactic centre. Besides a generic demand for better constraints on orbital parameters from the observational side, there is also room for improvement for the model itself. Its most important (and computationally rather expensive) modification will probably lie in an improved treatment of the evolution of the individual orbits, which would take gravity of the stellar disc itself into account.
As a final remark, let us note that the strict constraints on the mutual (perpendicular) orientation of the stellar disc and the CND raises a question about the dynamics of gas from which the young stars were formed. It is likely that the parent gaseous disc had to be nearly perpendicular to the CND, so that it would not be destroyed via differential precession before it gave birth to the numerous stellar population. Such an initial orientation is statistically not very probable, raising the question whether it can be a generic result of dissipative (hydro)dynamics in the resonant external potential.
Acknowledgements
We thank an anonymous referee for helpful comments. L.S. gratefully appreciates a fellowship from the Alexander von Humboldt Foundation and the hospitality of the host institute (AIfA). This work was also supported by the DFG Priority Program 1177, the Research Program MSM0021620860 of the Czech Ministry of Education, and the Czech Science Foundation (Ref. 205/07/0052). J.S. is grateful for utilisation of the Grid infrastructure of the Enabling Grids for EsciencE (EGEE II) project, a project cofunded by the European Commission under contract number INFSORI031688.
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Footnotes
 ... vectors^{}
 We follow the convention of Paumard et al. (2006), according to which angles and are related to the normal vector of the orbital plane as: .
All Figures
Figure 1: Evolution of orbital elements of two example orbits. The solid line represents a trajectory in the gravitational field of the central mass and a ring of radius and mass . The dotted line shows an orbit integrated in the field, including a spherical cusp of mass . In both cases the initial values of the orbital elements are: and . 

Open with DEXTER  
In the text 
Figure 2: Various tests of the compatibility of the hypothesis of a common origin of young stars in a single thin disc. N_{5}: number of stars that have at least one state in the neighbourhood of observed velocities, which corresponds to an orbit dragged from the parent disc with normal vector by the CND of mass . N_{e,5}: similar to the previous but excluding orbits with e>0.5. : volume of the subset of the neighbourhood occupied by the states originating in the parent disc. : mean value of the mean anomaly of orbits with . 

Open with DEXTER  
In the text 
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