You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'Thermal images of cold dust in the Central Molecular Zone of the Milky Way, obtained with the far-infrared cameras on-board the Herschel satellite, reveal a $\sim 3\times 10^7$ [M$_{\odot}$]{}  ring of dense and cold clouds orbiting the Galactic Center. Using a simple toy-model, an elliptical shape having semi-major axes of 100 and 60 parsecs is deduced. The major axis of this 100-pc ring is inclined by about 40[$^{\circ}$]{} with respect to the plane-of-the-sky and is oriented perpendicular to the major axes of the Galactic Bar. The 100-pc ring appears to trace the system of stable $x_2$ orbits predicted for the barred Galactic potential. [Sgr A$^{\star}$]{} is displaced with respect to the geometrical center of symmetry of the ring. The ring is twisted and its morphology suggests a flattening-ratio of 2 for the Galactic potential, which is in good agreement with the bulge flattening ratio derived from the 2MASS data.'
author:
- 'S. Molinari, J. Bally, A. Noriega-Crespo, M. Compiègne, J. P. Bernard, D. Paradis, P. Martin, L. Testi , M. Barlow, T. Moore, R. Plume, B. Swinyard, A. Zavagno, L. Calzoletti, A. M. Di Giorgio, D. Elia, F. Faustini, P. Natoli, M. Pestalozzi, S. Pezzuto, F. Piacentini, G. Polenta, D. Polychroni, E. Schisano, A. Traficante, M. Veneziani, C. Battersby, M. Burton, S. Carey, Y. Fukui, J. Z. Li, S. D. Lord, L. Morgan, F. Motte, F. Schuller, G. S. Stringfellow, J. C. Tan, M. A. Thompson, D. Ward-Thompson, G. White, G. Umana'
title: 'A 100-parsec elliptical and twisted ring of cold and dense molecular clouds revealed by *Herschel*around the Galactic Center.'
---

Introduction
============

Observations of the central regions of our Milky Way Galaxy provide the nearest template for the study of the conditions in galactic nuclei in normal galaxies. The $3.6\times 10^6$ [M$_{\odot}$]{} black hole at the center dominates the ÔCentral Molecular ZoneÕ (CMZ) which extends from Galactic longitude l = -1[$^{\circ}$]{} to about +1.5[$^{\circ}$]{} and hosts the densest and most massive molecular clouds in the Milky Way [@morser96]. The CMZ may be a Galactic analog of the nuclear star-forming rings observed in the centers of star forming barred galaxies (@kk04, @kc04). The CMZ contains tens of millions of solar masses of cold interstellar matter (ISM) [@ppetal00; @bally10b] and harbors several of its most active sites of star formation such as Sgr A, Sgr B2, and Sgr C (@yusef08,@yusef09). The gas kinematics and distribution of near-infrared light indicate that the gravitational potential in the inner Galaxy is dominated by a bar with a major axis inclined by about 20 to 45[$^{\circ}$]{} with respect to our line-of sight (@binney91, @ben03, @minchevetal07).

Despite the active star formation and bar dynamics, warm dust has been found to be a relative minor constituent in the CMZ. @sod94 used COBE/DIRBE data to constrain dust properties in the CMZ and @rodfer04 presented ISO observations. These studies found that 15 to 30% of the far-IR emission comes from molecular clouds with dust temperatures of about 19 K, 70 to 75% arises from the HI phase with dust temperatures of 17 to 22 K, and less than 10% is emitted by warm dust with T $>$ 29 K associated with the extended HII and warm (100 to 300 K) molecular phases traced by H$_3^+$ absorption (@oka05, @goto08). With Herschel we can now extend this type of analysis into a new domain of spatial resolution and sensitivity.

In this Letter, we present the first high-angular resolution maps of the CMZ far-IR dust emission, temperature, and column density based on data obtained with the Herschel satellite [@pilbratt10].

Observations and results
========================

The Galactic Center was observed with Herschel as part of the Hi-GAL Key-Project (the Herschel infrared Galactic Plane Survey, @moli10a). Observing strategy and basic data reduction are described in @moli10b and @trafi11. The central 2[$^{\circ}$]{}x2[$^{\circ}$]{} region of the Galaxy was imaged in Parallel Mode by acquiring simultaneously PACS [@poglitsch10] and SPIRE [@griffin10] images in 5 photometric bands centered at 70, 160, 250, 350 and 500 [$\mu$m]{}. SPIRE was used in “bright-source mode”, to minimize non-linearities and saturation in this very bright region; preliminary SPIRE calibration checks indicate an accuracy within 15% compared to standard SPIRE setting (M. Pohlen, priv. comm.). We corrected the zero-level in our SPIRE and PACS images using offset values derived from the comparison with the Planck/IRAS data using the same procedure as in @bernard10.

Figure \[gc70\] shows the PACS 70 [$\mu$m]{}  image. [Sgr A$^{\star}$]{} and [Sgr B2]{} are the brightest spots, and the entire area is filled with intense far-IR emission punctuated by several InfraRed Dark Clouds (IRDCs) seen in silhouette. A large bubble dominates at 70 [$\mu$m]{}  from 0[$^{\circ}$]{}${\;\lower.6ex\hbox{$\sim$}\kern-7.75pt\raise.65ex\hbox{$<$}\;}l {\;\lower.6ex\hbox{$\sim$}\kern-7.75pt\raise.65ex\hbox{$<$}\;}$ +0.20[$^{\circ}$]{}  but disappears at longer wavelengths (see SPIRE 250 [$\mu$m]{}  in fig. \[gc250\]). At 250 [$\mu$m]{}  [Sgr B2]{} is the brightest feature (it is saturated at 250 [$\mu$m]{}). Two bent chains of emission extend above and below the mid-plane from [Sgr B2]{} to [Sgr A$^{\star}$]{}, where they appear to cross, and continue on to [Sgr C]{}. This structure can also be identified in the SCUBA map [@ppetal00], but only the unique multi-band far-IR/submm capabilities of the Herschel PACS and SPIRE cameras make possible the determination of its temperature and column density.

Opacity and dust temperature are estimated from a pixel-to-pixel fit to the 70-350 [$\mu$m]{} data (the resolution is matched to the 350 [$\mu$m]{}  images – $\sim$25[$^{\prime \prime}$]{}) using the DustEM model [@comp11] following the method described in @bernard10. Assuming constant dust properties, the opacity is converted into a hydrogen column density ($N_H\,=\,N_{HI}+N_{H_2}$) using $\tau_{250}/N_H\,=\,8.8\,10^{-26}$ cm$^{2}$/H. This value is estimated from latitudes b$>$0, and it could be higher by a factor 2 - 4 in the Galactic plane [@bernard10] and could change our mass estimates accordingly. The temperature and column density maps (Figs. \[gc\_tmap\] and \[gc\_nmap\]) represent the integrated contribution of dust along the line of sight. High opacity, cold dust appears in silhouette against the warmer background even at 70 [$\mu$m]{}  (fig. \[gc70\]). Although the derived temperature might be underestimated, a temperature map without the 70 [$\mu$m]{} data provides similar results.

The detailed study of the great bubble visible in the 70 [$\mu$m]{} and the temperature maps (figs. \[gc70\] and \[gc\_tmap\]) will be presented in a forthcoming paper. This Letter is dedicated to the analysis of the large-scale filaments visible in the far-IR (fig.\[gc250\]), that trace a continuous chain of cold and dense clumps (T$_{dust}\leq$20 K; $N(H) >  2\times 10^{23}$ [cm$^{-2}$]{}) organized along an $\infty$-shaped feature that dominates the image between the [Sgr C]{} complex at $l$ = 359.4[$^{\circ}$]{}  and [Sgr B2]{} at 0.7[$^{\circ}$]{}. The total projected extent is $\sim$1.4[$^{\circ}$]{}, or about 180 pc, for a solar Galactocentric distance of 8.4 kpc [@reid09]. Warmer dust with relatively lower column density (T$_{dust}\geq$25 K, $ N(H) <  2 \times 10^{23}$ [cm$^{-2}$]{}) fills the interior of the $\infty$-shaped feature.

A 100-parsec elliptical ring of cold and dense molecular clouds: the $x_2$ orbits
=================================================================================

The $\infty$-shaped feature in the temperature map in fig. \[gc\_tmap\] coincides with high-column density material ($N(H)\geq 2\times 10^{23}$ [cm$^{-2}$]{}) organized in a continuous chain of irregular clumps (see fig. \[gc\_nmap\]). Most of the cold, high column density clumps in the [\[$l^-,b^-$\]]{} and [\[$l^+,b^+$\]]{}  arms of this feature are seen in silhouette against the 70 [$\mu$m]{}  background (fig. \[gc70\]); the same is not true for clumps along the [\[$l^+,b^-$\]]{} and [\[$l^-,b^+$\]]{} arms. We conclude that the [\[$l^-,b^-$\]]{} and [\[$l^+,b^+$\]]{} arms are in front of the bulk of the warmer dust emission (i.e. between us and the Galactic Center), while the [\[$l^-,b^-$\]]{} and [\[$l^+,b^+$\]]{} arms are located in the background.

We use the CS survey by @tsuboi99 (their fig. 4) to extract the radial velocity of CS condensations positionally coincident with the cold and dense peaks on varius positions along the $\infty$-shaped feature in figs. \[gc\_tmap\] and \[gc\_nmap\]. The position-velocity information is averaged in latitude bins of 1.5, so the information can be extracted with some degree of approximation but should be adequate to represent the mean velocity of the various clouds and clumps. The full span of velocities found toward each position is reported in fig. \[gc\_nmap\].

The [\[$l^-,b^+$\]]{} arm shows steadily increasing (in absolute values) negative velocities from the center of symmetry toward [Sgr C]{} indicating approaching material with increasing relative speed. Following the [\[$l^-,b^-$\]]{}  arm from [Sgr C]{} back to the centre we find negative velocities decreasing and changing to positive (receding) values at $l\sim$359.8[$^{\circ}$]{}. Continuing along the [\[$l^+,b^+$\]]{} arm the radial velocities are still positive and slowly increasing, indicating material receding at higher and higher speeds up to [Sgr B2]{}. Finally along the [\[$l^+,b^-$\]]{} arm we see velocities decrease again going back toward the center of symmetry of the $\infty$-shaped feature. The distribution of radial velocities along the $\infty$-shaped feature seems to indicate an ordered, rotating pattern around its projected center of symmetry.

A simple model for the $\infty$-shaped feature: a 100-pc elliptical twisted ring
--------------------------------------------------------------------------------

A simplified toy model of the $\infty$-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 $\theta_p$. For simplicity, a constant orbital speed $_{orb}$ is assumed. An additional sinusoidal vertical oscillation component with a vertical frequency $\nu_z$ and a phase $\theta _z$ is added to the equations of the ellipse describing the orbit. Adopting a coordinate system with the $x$ axis oriented toward negative longitudes, $y$ axis pointing away along the line of sight from the Sun, and $z$ axes toward positive latitudes (see Fig. \[ring\]), the position and radial velocity of each point along the orbit is described as a function of the polar angle $\theta_t$ (counter-clockwise from the $x$ axes) as

$$\left\{ \begin{array}{l}
x=a\,\cos \theta_t \, \cos \theta_p - b\, \sin \theta_t \, \sin \theta_p \\
y=a\, \cos \theta_t \, \sin \theta_p + b\, \sin \theta_t \, \cos \theta_p \\
z=z_0\, \sin \nu_z (\theta_p - \theta_z)\\
\textrm{v}_r=-\textrm{v}_{orb} \, \cos(\theta_p +\theta_t)
\end{array} \right.
\label{ringmodel}$$

We let $a, b, \theta_p$ and $_{orb}$ vary until they match the observed projected morphology of the ellipse and the distribution of radial velocities at 20 positions along the $\infty$-shaped feature. We keep $z_0$ fixed at 15 pc, which is half of the measured $b$-extent of the $\infty$-shaped feature; we also fix $\nu_z$=2 in units of the orbital frequency, as indicated by the projected twist shape (see §\[twist\_par\]). The fitting procedure minimizes a pseudo-$\chi^2$ value $\xi$

$$\xi = \sum _{i=1} ^{20} {{(\textrm{v}_r-\textrm{v}_{obs})^2}\over{\Delta \textrm{v}_{obs}}}$$

where $\Delta \textrm{v}_{obs}$ is half of the velocity range toward each of the 20 test positions (see fig. \[gc\_nmap\]).

The best fit is obtained for an ellipse with $a=100$pc, $b=$60pc, $\theta_p=-$40[$^{\circ}$]{}, and $_{orb}$=80 [kms$^{-1}$]{}; $\theta _z \sim$0 implying that the ring crosses the midplane 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 $\pm$25 [kms$^{-1}$]{}. The only exception is the 50 [kms$^{-1}$]{} cloud (see fig. \[gc\_tmap\]) located close in projection to [Sgr A$^{\star}$]{} (see §\[sgradist\]).

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. @sofue95 analysed the CO data in @bally87 and identified two large-scale structures (the “Arms”). @tsuboi99 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 $\infty$-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$^{\star}$]{}. Figure \[ring\] presents a sketch of the proposed arrangement of what we will call the100-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)${\;\lower.6ex\hbox{$\sim$}\kern-7.75pt\raise.65ex\hbox{$>$}\;}2\times 10^{23}$ [cm$^{-2}$]{}  (for a corresponding solid angle of $\sim 4\times 10^{-5} sr$). The contribution of background or foreground gas is removed by subtracting a column density of N(H) = $2\times 10^{23}$ [cm$^{-2}$]{}, the mean N(H) value in the interior of $\infty$-shaped feature. The resulting ring mass is $\sim 3\times 10^7$ [M$_{\odot}$]{} (it is $\sim 4\times 10^7$ [M$_{\odot}$]{}  without background subtraction). Within the uncertainties, the derived ring mass agrees with 5.3$\times 10^7$ [M$_{\odot}$]{}  estimated from SCUBA maps [@ppetal00] and 3$\times 10^7$ [M$_{\odot}$]{}  from CO data [@dahmen98].

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 fig. \[ring\]) is perpendicular to the major axis of the great Galactic bar, oriented 20 to 45[$^{\circ}$]{}  with respect to our line-of-sight (see fig. \[ring\]). The 100-pc ring is likely located on the so-called ‘$x_2$’ orbit system, which precesses at the same rate as the solid-body angular velocity of the Galactic Bar, resulting in stable, non-intersecting trajectories [@binney91]. These $x_2$ orbits are enclosed within another larger system of stable orbits, called $x_1$, 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 $x_1$ and $x_2$ 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 @kl91 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 AGN phases of our Galaxy. The recent Fermi-LAT detection of a large gamma-ray bubble emanating from the Galactic Center (@suetal10, @croker11, @zubo11) 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. @jaffeetal04, @radom08), 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. @shietal06, @quill06). Model predictions (e.g. @krolik07) for torus column density in excess of 10$^{24}$ [cm$^{-2}$]{}, and orbital speed $\sim$100 [kms$^{-1}$]{}, seem to be consistent with the parameters we estimate for the 100-pc ring.

The relation between the 100-pc ring and [Sgr A$^{\star}$]{} {#sgradist}
------------------------------------------------------------

[Sgr A$^{\star}$]{}is noticeably displaced $\sim$10[$^{\prime}$]{} (24 pc in projection) towards the negative longitude side of the geometrical center of symmetry of the 100-pc ring. The ring displacement with respect to the supposed location of the Galactic Center is consistent with the overall displacement of dense gas within the CMZ. Two-thirds to three-quarters of the cold interstellar gas and dust within the inner few hundred parsecs of the nucleus is at positive longitudes with respect to [Sgr A$^{\star}$]{}(@bally88, @morser96). 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 fig. \[gc\_nmap\]). This may indicate an additional pattern motion of $\sim 10-20$ [kms$^{-1}$]{}  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 [@morser96].

The projected proximity of the 20 and 50 [kms$^{-1}$]{}  clouds to [Sgr A$^{\star}$]{}, has led to the suggestion (e.g. @herrho05) that they are physically interacting with [Sgr A$^{\star}$]{}. If they are part of the 100-pc ring, then [Sgr A$^{\star}$]{} must be closer to the front of the ring than to its back; [Sgr A$^{\star}$]{}  may then not be in the ring center, nor lie along the ellipses axes. Proximity to [Sgr A$^{\star}$]{}  could be the source of the roughly 40 [kms$^{-1}$]{} deviation of the radial velocity of the 50 [kms$^{-1}$]{} 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$^{\star}$]{} the gravitational pull due to the strong mass concentration can be substantial. Indeed, the $3.6\times 10^6$ [M$_{\odot}$]{} black hole in [Sgr A$^{\star}$]{} 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 $\rho (r) \approx \rho_0 (r/r_b)^{-\alpha}$ where $\alpha \approx -1.9$, $r_b\sim 0.34$pc and $\rho_0 \sim 2.1\times 10^6$ [M$_{\odot}$]{}/pc$^3$ [@trippe08]. We then treat the gravitational pull from the [Sgr A$^{\star}$]{} mass concentration as an additional component over the general rotational motion. Using Newton’s 2$^{nd}$ law:

$${{GM_C(r)}\over{r^2}}={{\Delta \textrm{v}}\over{t_0}}
\label{fma}$$

where $r$ is the distance between [Sgr A$^{\star}$]{} 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$^{\star}$]{} and t$_0$ is the timescale for the cloud passing in front of [Sgr A$^{\star}$]{}. The latter can be written as $t_0=\Delta x / \textrm{v}_{orb}$, where $\Delta x$ is the projected distance between the 50 and 20 [kms$^{-1}$]{} clouds and $v_{orb}$ is their orbital speed. Eq. \[fma\] can then be rewritten

$$r=\sqrt{{GM_C(r) \Delta x}\over{\Delta \textrm{v}\, \textrm{v}_{orb}}}
\label{rdiff}$$

Using this formulation to estimate M$_C(r)$, the measured projected distance between the nearest edges of the two clouds of $\Delta x \sim 7$pc, and assuming an orbital speed of 80[kms$^{-1}$]{} from our toy model fit, Eq. \[rdiff\] provides $r\sim $20 pc. In other words, to explain the difference in radial velocities between the 20 and 50 [kms$^{-1}$]{}clouds would require that [Sgr A$^{\star}$]{} is closer than 20pc to the foregreound arm of the 100-pc ring.

The ring’s twist: constraints on the flattening of the Galactic potential {#twist_par}
-------------------------------------------------------------------------

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$_o \sim 3 \times 10^6$ years; the period of the vertical oscillations is therefore P$_z \sim 1.5 \times 10^6$ 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 $\rho _0$ slab of thickness $h$, P$_o$ at a distance R can then be expressed as

$$P_o=\sqrt{{4\pi R}\over{G \rho_0 h}}
\label{per_orb}$$

A perturbation on a particle rotating in the flattened slab potential will result, in a 1$^{st}$-order approximation [@bt], in a harmonic oscillation both in radial and vertical directions. The vertical oscillation frequency can be expressed as

$$\nu _z ^2 = \left({\partial ^2 \Phi}\over{\partial z^2} \right)_{R,0}
\label{nu_z}$$

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

$$F _z = 2\pi \rho_0 G h 
\label{f_z}$$

so that the period of the vertical oscillations is

$$P _z  = \sqrt{{2 \pi}\over{\rho_ 0 G} }
\label{per_ver}$$

As noted above, the $\infty$ shape that the ring displays in projection implies that

$$P _o = 2 P_z$$

or, using Eqs. \[per\_orb\] and \[per\_ver\],

$$h={{R}\over{2}}
\label{hr}$$

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 [@rc08] fitting the 2MASS data to a potential model that includes the bulge and the nuclear bar.

Conclusions
===========

The far-infrared multi-band images provided by the Hi-GAL survey using the Herschel PACS and SPIRE cameras provide evidence for a 100x60 parsecs elliptical ring of cold and dense clouds orbiting around the Galactic Center on $x_2$ orbits elongated along the minor axis of the Galactic center Bar. The ring is twisted with a vertical frequency that is twice the orbit frequency, indicating a flattening-ratio of 2 for the Galactic Center gravitational potential, consistent with the value derived from bulge star 2MASS number counts. Data processing has been possible thanks to ASI support via contract I/038/080/0. We thank the referee, M. Morris, whose comments led to a more focused paper. We thank L. Spinoglio, N. Sacchi and A. Marconi for useful discussions on AGNs, and M. Pohlen and the SPIRE ICC for the preliminary analysis of the SPIRE “bright source mode” calibration.

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