A 100 pc ELLIPTICAL AND TWISTED RING OF COLD AND DENSE MOLECULAR CLOUDS REVEALED BY HERSCHEL AROUND THE GALACTIC CENTER

A simplified toy model of the ∞-shaped feature is built assuming that the material is distributed along an elliptical orbit (projected onto the Galactic Plane) with semi-axes [a, b] and major-axis position angle θ_p. For simplicity, a constant orbital speed v_orb is assumed. An additional sinusoidal vertical oscillation component with a vertical frequency ν_z and a phase θ_z is added to the equations of the ellipse describing the orbit. Adopting a coordinate system with the x-axis oriented toward negative longitudes, the y-axis pointing away along the line of sight from the Sun, and the z-axes toward positive latitudes (see Figure 5), the position and radial velocity of each point along the orbit are described as a function of the polar angle θ_t (counterclockwise from the x-axes) as

$$\begin{displaymath} \left\lbrace \begin{array}{@{}l}x = a\,\cos \theta _t \, \cos \theta _p - b\, \sin \theta _t \, \sin \theta _p \\ y = a\, \cos \theta _t \, \sin \theta _p + b\, \sin \theta _t \, \cos \theta _p \\ z = z_0\, \sin \nu _z (\theta _p - \theta _z)\\ \hbox{$v$}_r = -\hbox{$v$}_{{\rm orb}} \cos (\theta _p +\theta _t). \end{array} \right. \end{displaymath}$$

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We let a, b, θ_p, and v_orb vary until they match the observed projected morphology of the ellipse and the distribution of radial velocities at 20 positions along the ∞-shaped feature. We keep z₀ fixed at 15 pc, which is half of the measured b-extent of the ∞-shaped feature; we also fix ν_z = 2 in units of the orbital frequency, as indicated by the projected twist shape (see Section 3.3). The fitting procedure minimizes a pseudo-χ² value ξ:

$$\begin{equation} \xi = \sum _{i = 1} ^{20} {{(\hbox{$v$}_r-\hbox{$v$}_{{\rm obs}})^2}\over {\Delta \hbox{$v$}_{{\rm obs}}}}, \end{equation} \tag{ 1 }$$

where Δv_obs is half of the velocity range toward each of the 20 test positions (see Figure 4).

The best fit is obtained for an ellipse with a = 100 pc, b = 60 pc, θ_p = −40°, and v_orb = 80 km s⁻¹; we obtain θ_z ∼ 0 implying that the ring crosses the mid-plane along the X-axes. These numbers should be taken with caution, as the velocities are derived by eye from a plot. However, the discrepancy between the model radial velocity and the observed ones is within ±25 km s⁻¹. The only exception is the 50 km s⁻¹ cloud (see Figure 3) located close in projection to Sgr A^⋆ (see Section 3.2).

The CMZ has been the subject of many studies in millimeter molecular tracers, and although the interpretation is difficult for the extreme confusion along the line of sight, several features were identified. Sofue (1995) analyzed the CO data in Bally et al. (1987) and identified two large-scale structures (the "Arms"). Tsuboi et al. (1999) recognized a large-scale feature in their CS position–velocity diagram that they call "Galactic Center Bow" which matches the [l⁻, b⁻]–[l⁺, b⁺] arms of the ∞-shaped feature. The cold, high column density dust features identified in the Herschel images, used as a proxy for the densest molecular gas, now indicate that the previously identified "Arms" and the "Galactic Center Bow" are parts of a single, twisted, and elliptical ring rotating around Sgr A^⋆. Figure 5 presents a sketch of the proposed arrangement of what we will call the 100 pc ring.

A rough estimate of the ring mass is obtained by integrating the N(H) map within a contour surrounding the 100 pc ring which encloses material with N(H) ≳  2 × 10²³ cm⁻² (for a corresponding solid angle of ∼4 × 10⁻⁵sr). The contribution of background or foreground gas is removed by subtracting a column density of N(H) = 2 × 10²³ cm⁻², the mean N(H) value in the interior of ∞-shaped feature. The resulting ring mass is ∼3 × 10⁷ M_☉ (it is ∼4 × 10⁷ M_☉ without background subtraction). Within the uncertainties, the derived ring mass agrees with 5.3 × 10⁷ M_☉ estimated from SCUBA maps (Pierce-Price et al. 2000) and 3 × 10⁷ M_☉ from CO data (Dahmen et al. 1998).

The 100 pc ring traces gas and dust moving about the nucleus on an elliptical orbit whose major axis (the straight red line in the right panel of Figure 5) is perpendicular to the major axis of the great Galactic bar, oriented 20°–45° with respect to our line of sight (see Figure 5). The 100 pc ring is likely located on the so-called x₂ orbit system, which precesses at the same rate as the solid-body angular velocity of the Galactic Bar, resulting in stable, non-intersecting trajectories (Binney et al. 1991). These x₂ orbits are enclosed within another larger system of stable orbits, called x₁, elongated along the Galactic Bar. The prominent Sgr B2 and Sgr C star-forming complexes are located at the two projected longitude extrema of the 100 pc ring where gas on x₁ and x₂ orbits may collide, producing strong shocks which may trigger the formation of massive molecular cloud formation as well as enhanced star formation rates by the shock focusing mechanism suggested by Kenney & Lord (1991) for the spiral-bar interface of M83.

It is tempting to speculate that the 100 pc ring constitutes the remnant of a more persistent dusty torus that may have played a role in past active galactic nucleus (AGN) phases of our Galaxy. The recent Fermi-LAT detection of a large gamma-ray bubble emanating from the Galactic center (Su et al. 2010; Crocker 2011; Zubovas et al. 2011) provides possible evidence for such past activity. Mid-IR interferometry of AGNs reveals compact dusty structures with a radius of a few parsecs (e.g., Jaffe et al. 2004; Radomski et al. 2008), but there is additional evidence from Spitzer imaging and spectroscopy suggesting that larger and more persistent structures play a role in shaping observed AGN properties (e.g., Shi et al. 2006; Quillen et al. 2006). Model predictions (e.g., Krolik 2007) for torus column density in excess of 10²⁴ cm⁻², and orbital speed ∼100 km s⁻¹, seem to be consistent with the parameters we estimate for the 100 pc ring.

Sgr A^⋆ is noticeably displaced ∼10' (24 pc in projection) toward the negative longitude side of the geometrical center of symmetry of the 100 pc ring. The ring displacement with respect to the supposed location of the Galactic center is consistent with the overall displacement of dense gas within the CMZ. Two-thirds to three-quarters of the cold interstellar gas and dust within the inner few hundred parsecs of the nucleus is at positive longitudes with respect to Sgr A^⋆ (Bally et al. 1988; Morris & Serabyn 1996). A similar asymmetry is present in the observed radial velocity on the 100 pc ring close to the location of Sgr B2, as opposed to Sgr C (see Figure 4). This may indicate an additional pattern motion of ∼10–20 km s⁻¹ of the 100 pc ring as a whole along its major axes and toward us, implying a "sloshing" motion back and forth that could be explained as the response of the gas to an m = 1 mode in the Galactic bar potential (Morris & Serabyn 1996).

The projected proximity of the 20 and 50 km s⁻¹ clouds to Sgr A^⋆has led to the suggestion (e.g., Herrnstein & Ho 2005) that they are physically interacting with Sgr A^⋆. If they are part of the 100 pc ring, then Sgr A^⋆ must be closer to the front of the ring than to its back; Sgr A^⋆ may then not be in the ring center, nor lie along the ellipse axes. Proximity to Sgr A^⋆ could be the source of the roughly 40 km s⁻¹ deviation of the radial velocity of the 50 km s⁻¹ cloud from the value predicted by the model of the rotating elliptical ring.

Given the extent of the 100 pc ring, its orbital motion is driven by the enclosed gas and bulge stars. If, however, ring material passes in proximity of Sgr A^⋆, the gravitational pull due to the strong mass concentration can be substantial. Indeed, the 3.6 × 10⁶ M_☉ black hole in Sgr A^⋆ dominates the gravitational potential only to a radius of a few parsecs. Beyond that distance the potential is dominated by the stellar mass in the bulge with a radial density profile of the form ρ(r) ≈ ρ₀(r/r_b)^−α, where α ≈ 1.9, r_b ∼ 0.34pc, and ρ₀ ∼ 2.1 × 10⁶ M_☉ pc⁻³ (Trippe et al. 2008). We then treat the gravitational pull from the Sgr A^⋆ mass concentration as an additional component over the general rotational motion. Using Newton's second law:

$$\begin{equation} {{{\it GM}_C(r)}\over {r^2}} = {\Delta {v}\over {t_0}}, \end{equation} \tag{ 2 }$$

where r is the distance between Sgr A^⋆ and the two molecular clouds (that is assumed similar for the two clouds if they belong to the 100 pc ring), M_C(r) is the mass enclosed in a sphere of radius r centered on Sgr A^⋆, and t₀ is the timescale for the cloud passing in front of Sgr A^⋆. The latter can be written as t₀ = Δx/v_orb, where Δx is the projected distance between the 50 and 20 km s⁻¹ clouds and v_orb is their orbital speed. Equation (2) can then be rewritten as

$$\begin{equation} r = \sqrt{{{\it GM}_C(r) \Delta x}\over {\Delta { {v}}\, { {v}}_{{\rm orb}}}}. \end{equation} \tag{ 3 }$$

Using this formulation to estimate M_C(r), the measured projected distance between the nearest edges of the two clouds of Δx ∼ 7pc, and assuming an orbital speed of 80 km s⁻¹ from our toy model fit, Equation (3) provides r ∼ 20 pc. In other words, to explain the difference in radial velocities between the 20 and 50 km s⁻¹ clouds would require that Sgr A^⋆ is closer than 20 pc to the foreground arm of the 100 pc ring.

Perhaps the most striking aspect of the 100 pc ring is its twist. Apparently, the gas in the ring oscillates twice about the mid-plane each time it orbits the nucleus once. Assuming the elliptical axes and the orbital speed derived from the toy model, the orbital period is P_o ∼ 3 × 10⁶ years; the period of the vertical oscillations is therefore P_z ∼ 1.5 × 10⁶ years. The coherence of the ring's morphology suggests stability over a few revolutions. Thus, the relationship between P_o and P_z can be used to infer constraints on the flattening of the gravitational potential in the CMZ. Assuming as a gross approximation that the matter (gas and stars) is distributed in a uniform-density ρ₀ slab of thickness h, P_o at a distance R can then be expressed as

$$\begin{equation} P_o = \sqrt{{4\pi R}\over {G \rho _0 h}}. \end{equation} \tag{ 4 }$$

A perturbation on a particle rotating in the flattened slab potential will result, in a first-order approximation (Binney & Tremaine 1987), in a harmonic oscillation both in radial and vertical directions. The vertical oscillation frequency can be expressed as

$$\begin{equation} \nu _z ^2 = \left({\partial ^2 \Phi }\over {\partial z^2} \right)_{R,0}. \end{equation} \tag{ 5 }$$

The gravitational field along the z-direction of the slab (approximated as infinite) is

$$\begin{equation} F _z = 2\pi \rho _0 G h, \end{equation} \tag{ 6 }$$

so that the period of the vertical oscillations is

$$\begin{equation} P _z = \sqrt{{2 \pi }\over {\rho _ 0 G} }\,. \end{equation} \tag{ 7 }$$

As noted above, the ∞ shape that the ring displays in projection implies that

$$\begin{equation} P _o = 2 P_z \end{equation} \tag{ 8 }$$

or, using Equations (4) and (7),

$$\begin{equation} h = {{R}\over {2}}. \end{equation} \tag{ 9 }$$

The shape of the 100 pc ring would therefore imply a flattened potential in the CMZ that is consistent with a x/z = 0.5 axial ratio bulge mass distribution, remarkably similar to the value of 0.5–0.6 recently derived (Rodriguez-Fernandez & Combes 2008), fitting the 2MASS data to a potential model that includes the bulge and the nuclear bar.