Introduction

Maser emission from molecular species has provided critical insights into the kinetic properties of young stars and has even enabled monitoring of their accretion processes (Hunter et al. 2018, and references therein). Energetic events (e.g., outflows and collimated jets) and infrared radiation surrounding massive protostars provide the energy to overpopulate the upper-energy levels of specific molecular transitions and amplify the molecular emission intensities (Moscadelli et al. 2016;Sanna et al. 2016). Common species exhibiting maser emission include, but are not limited to, OH, SiO, H 2 O, H 2 CO, NH 3 , and CH 3 OH. An empirical taxonomy based on pumping mechanisms classifies molecular masers into two categories (Batrla et al. 1987;Cyganowski et al. 2009); H 2 O and Class I CH 3 OH masers are most often excited via collisions and tend to reside in shocked material associated with outflows and expanding H II regions (e.g., Baudry et al. 1974;Elitzur & Fuqua 1989;Voronkov et al. 2014;Towner et al. 2017), while OH and Class II CH 3 OH masers are excited via radiation and are generally close to young stellar objects and luminous infrared sources (e.g., Sobolev & Deguchi 1994;Sobolev et al. 1997;Cragg et al. 2002Cragg et al. , 2005)). The most extensively studied Class I CH 3 OH masers include the 4 -1 -3 0 E transition at 36 GHz and the 7 0 -6 1 A transition at 44 GHz, while Class II CH 3 OH masers include the 5 1 -6 0 A transition at 6.7 GHz and the 2 0 -3 -1 E transition at 12.2 GHz (e.g., Voronkov et al. 2014;Song et al. 2022).

By correlating the maser emission from various species with a sample of star-forming regions at different stages, statistical analyses have also suggested that masers may help to trace the evolutionary phases of massive stars (e.g., Figure 6 in Breen et al. 2010). For example, Forster & Caswell (1989) found that isolated H 2 O masers tend to appear in sources at earlier evolutionary stages than OH masers based on a survey toward 74 star-forming regions. In addition, Class II CH 3 OH masers, which appear exclusively in massive star-forming regions (Xu et al. 2008;Breen et al. 2013), typically have a significant overlapping time line with H 2 O masers and arise before the OH masers and ultracompact (UC) H II regions turn on (Walsh et al. 2003;Szymczak et al. 2005).

However, within the picture of massive star formation, the position for Class I CH 3 OH masers in an evolutionary sequence is less clear. Initial comparisons of the mid-infrared properties of the sample of CH 3 OH maser sources suggested that Class I masers are signposts of the earliest stage of massive star formation, even prior to H 2 O masers (Ellingsen 2006). In contrast, detailed mapping of multiple Class I CH 3 OH maser transitions toward a luminous young stellar object, IRAS 16547-4247, led Voronkov et al. (2006) to argue that Class I CH 3 OH masers can appear earlier than the Class II masers but also last long enough to coexist with the OH masers. Subsequent observations toward the massive protocluster G18.67+0.03 further suggested that the Class I CH 3 OH masers might associate with both young sites and relatively evolved UC H II regions within the same protocluster (Cyganowski et al. 2012). In any case, high angular resolution imaging is essential for determining which protostar(s) within a protocluster currently power these masers (Towner et al. 2017).

Recently, new maser species, such as HNCO and CH 2 NH, have been discovered to exhibit compelling roles in inferring fine-scale structures and gas dynamics in the maser region. In particular, the HNCO maser was found to be associated with accretion flows, which helped to build an arm-spiral-structure model of one particular high-mass young stellar object (Chen et al. 2020); and the CH 2 NH maser toward galactic nuclei helps to probe the inner nuclear processes in extragalactic environments (Gorski et al. 2021). The pumping mechanism of these new maser species, however, is less constrained, as is their relation to other common maser species. Observations of these species in maser-active regions that contain both known collisionally and radiatively pumped masers are therefore critical as a first step in understanding their pumping mechanisms.

The high-mass star-forming region Sgr B2 north (N) displays rich maser activity from a diverse set of molecules, ranging from common maser species to rare complex molecules (Morita et al. 1992;McGrath et al. 2004;Hoffman et al. 2007;Fujisawa et al. 2014;Qiao et al. 2018). The single-dish PRebiotic Interstellar MOlecular Survey (PRIMOS) project has identified numerous species exhibiting maser activity at centimeter wavelengths. In fact, the discovery of interstellar carbodiimide (HNCNH) was only achievable due to its maser emission (McGuire et al. 2012). Radiative transfer calculations have also revealed that most emission lines of both methyl formate (HCOOCH 3 ) and CH 2 NH detected in PRIMOS are weak masers amplifying the background continuum radiation of Sgr B2(N) at centimeter wavelengths (Faure et al. 2014(Faure et al. , 2018)). However, the large ~¢ ´¢ 2.5 2.5 field of view of PRIMOS at the C band (∼5 GHz) includes both the north and main components of Sgr B2, complicating the interpretation of the spectral profiles. Therefore, it is crucial to have interferometric observations to disentangle in detail the various possible spatial components of maser emission.

In this work, we present a rigorous imaging study of maser emission from CH 3 OH and CH 2 NH toward Sgr B2(N). We describe the interferometric observations at centimeter and millimeter wavelengths in Section 2. A description of the methodology employed to model maser emission is described in Section 3, which includes the implementation of a Bayesian inference of physical characteristics. Section 4 presents both the spectra and spatial distribution per maser species. Section 5 contains discussions on the metastable states of maser transitions (Section 5.1), pumping mechanisms inferred from the spatial correlations (Section 5.2), derived physical conditions of masing gas clumps (Section 5.3), and a caveat on modeling maser intensity (Section 5.4).

Observations

The interferometric data used here were acquired from two data sets involving the Karl G. Jansky Very Large Array (VLA) in the C-band (nominal frequency coverage 4-8 GHz) and the Atacama Large Millimeter/submillimeter Array (ALMA) in Band 3 (nominal frequency coverage 84-116 GHz).

VLA Observations.

The VLA observations were carried out on 2016 May 27 (project code: 16A-076, PI: B. McGuire) with the C-band receiver. The correlator was configured to deliver 24 subbands or spectral windows, with bandwidths of 4, 8, and 16 MHz, to target various spectral lines. Additionally, nine 128 MHz spectral windows were set up to help with calibration, if needed. The results presented here are based on three spectral windows that delivered channel widths between 0.485 and 2.31 km s -1 , depending on the corresponding bandwidth and observing frequency. Detailed observational parameters of each spectral window are presented in Table 1. The phase center of the observations was targeted toward Sgr B2 (N), with the field center at α J2000 = 17 h 47 m 19 80, d = - ¢  28 22 17. 00 J2000

. The bandpass and absolute flux density scale calibrator was 3C286, while the complex gain calibrator was J1744-3116. The array was in the B configuration with 27 antennas, which covered baselines from 250 m up to 11 km and resulted in the maximum recoverable scale of 37″ at 4 GHz.

The CASA software package version 5.6.1 (CASA Team et al. 2022) was used for data reduction and imaging analysis of the VLA observations. For each spectral window, the continuum emission was subtracted in the uv-plane before imaging the spectral cubes. We used the tclean task with the automasking algorithm for imaging with the natural weighting scheme and a cell size of 0 27. The resulting size of the synthesized beam (FWHM) of each image cube has a median value of ∼3 8 × 1 5. The median of the resultant typical rms noise levels is ∼0.9 mJy beam -1 .

ALMA Observations. The ALMA observational data were acquired from the ALMA Science Archive of the survey project entitled "Exploring Molecular Complexity with ALMA (EMoCA)" (project code: 2011.0.00017.S, PI: A. Belloche). A detailed description of the EMoCA data has been presented in Belloche et al. (2016). In this work, two measurement sets, spectral window 0 of setup S1 and spectral window 0 of setup S4 (Xue et al. 2019), were used for imaging the 84 and 95 GHz CH 3 OH transitions. Different from Xue et al. (2019), we used the tclean task with the automasking algorithm for the imaging analysis, consistent with the VLA imaging process. The resultant synthesized beam size is 2 23 × 1 67 at 84 GHz and 1 94 × 1 49 at 95 GHz with a median rms value of 4 mJy beam -1 .

Note.

a The synthesized beam size obtained with Natural weighting.

Analysis

In order to constrain the excitation conditions quantitatively, we used the molsim program to model the molecular spectra with non-LTE radiative transfer models and to explore the physical parameter space with the Markov Chain Monte Carlo (MCMC) approach (Lee et al. 2023).

Non-LTE Radiative Transfer Modeling

We implemented the RADEX non-LTE radiative transfer code (van der Tak et al. 2007) into molsim written in Python. The module was accelerated with Numba to provide the high performance required for MCMC analysis.

RADEX considers a gas clump consisting of a targeted molecular species and its collision partners, such as H 2 and e -. The gas clump is embedded in a local background radiation field and has a certain geometry (e.g., slab, uniform sphere, expanding sphere). With this physical picture, RADEX iteratively solves for the level population by incorporating an equilibrium with inelastic collisions and the local radiation field. Lastly, it computes the peak optical depth, excitation temperature, and peak intensity of each line based on the level population and the line properties, assuming the local radiation field as the only source of background continuum emission. The model can be described with physical parameters, including kinetic temperatures (T k ), densities of collision partners (n c ), background radiation field temperatures (T bg ), source velocity (V lsr ), line widths (dV ), and molecular column densities (N col ).

In our implementation, we construct a modification and extend it to support a continuum temperature (T cont ) different from the local radiation field temperature (T radfoot_1 ) and allow multiple components to align perfectly along the line of sight (LOS). In the case of a population inversion, T cont serves as background radiation that stimulates the maser gain and is amplified by the foreground gas clump. We assume the continuum source and the gas components to be spatially distant from each other such that the continuum radiation is not included in the excitation analysis and the embedded radiation field of each component is only dominated by the local radiation field. Detailed justifications for this assumption will be discussed in Section 5.3. With this assumption, the level population of each component is computed with a distinct excitation condition using RADEX. The frequency-dependent optical depth and source function of the ith component are reconstructed by summing over all lines as follows:

Here, t i j , peak and T ex,i,j are the optical depth and excitation temperature returned by RADEX for the jth line of the ith component, B ν is the Planck function, f i is the line profile for the ith component using the Gaussian function G, and τ i and S i are the total optical depth and source function with all lines being accounted for.

Using these definitions, for an n-component model where the continuum radiation is first processed by the first component, then the second component and so on, the final intensity with the continuum subtracted can be expressed as

In the case of two components, the continuum-subtracted intensity is

It is worthwhile to note that the expressions reduce back to that of RADEX in the case of one component when the continuum temperature is equal to the local radiation temperature (i.e., T cont = T rad = T bg ). The calculations in molsim reproduces the results returned by RADEX down to machine precision when the results are converged.

In this work, we assume that the local radiation is contributed by the cosmic microwave background only, i.e., T rad = 2.725 K (Mather et al. 1999) and a uniform spherical geometry. We only consider H 2 as the collision partner, assuming a kinetic temperature-dependent thermal ortho-topara ratio (OPR), which can be approximated by

We left all the rest as free parameters and allowed them to be fitted using the MCMC approach as described below.

MCMC Fitting

For line fitting, we used a Bayesian approach to compare models with different parameters and calculate their probability distributions. This approach provides an inference by conditioning the data on priors and sampling posterior distributions. In practice, we use the MCMC approach to explore parameter space.

We use a logarithmic likelihood distribution with the formula below:

where x is the observed spectral line intensity, μ is the model line intensity, and σ is the uncertainty of the observations including an estimated 10% systematic uncertainty with the standard baseline deviation. The distribution applies to the case where the data set has a nonuniform uncertainty, in which case we compute the uncertainties for each spectral chunk separately.

The resulting corner plots are shown in Appendix B. A corner plot shows the probability distribution for each pair of parameters and for each parameter individually. In the following, the fitted results are represented by the 50th, 16th, and 84th percentiles as the representation value and the range of uncertainty, also known as the 68% confidence interval. This choice is equivalent to choosing the mean and ±1σ for a normal distribution, which is the case for most of our posterior distributions. One of the advantages of the MCMC technique over a traditional least-squares fit approach is that a larger parameter space can be explored more efficiently. Rather than only returning the parameters that maximize the likelihood function (e.g., least-squares fitting),foot_2 it returns the actual probability of the parameters, which can be used to assess whether parameters are highly covariant with one another. The correlation between parameters manifests itself as nonseparable distributions in the corner plots.

Results

We investigate the maser activities of two transitions in this work. The detailed spectroscopic parameters for each maser transition along with the observational results of the corresponding emission features are summarized in Table 2. For CH 3 OH, we examined the molecular spectra extracted toward two distinct regions, maser spot 1 (MS1, α J2000 = 17 h 47 m 19 730, d = - ¢  28 22 04. 580 J2000

) and maser spot 2 (MS2, α J2000 = 17 h 47 m 20 393, d = - ¢  28 22 12. 410 J2000

). Toward both regions, we performed MCMC analyses with a two-chained-component model, as presented in Section 4.1.1 for MS1 and 4.1.2 for MS2, respectively. For CH 2 NH, as there is no detectable signal toward MS1, we examined its spectrum extracted toward MS2 only and performed the MCMC analysis with a one-component model as presented in Section 4.2.

Empirically, a brightness temperature (T B ) of an emission line much greater than the source physical (kinetic) temperature is considered as an indication of a maser emission rather than a thermally excited line (Voronkov et al. 2006;Gargaud et al. 2011). As elaborated in the following sections, our MCMC analyses suggest the T k for the masing clumps in Sgr B2(N) to be ∼29-37 K. In contrast, as summarized in Table 2, the observed T B values of the molecular emission investigated in this work (>120 K) are significantly higher than the physical temperatures. Furthermore, we also performed a spectral analysis assuming a single excitation temperature to equal T k for comparison and found that the observed spectral profiles deviate substantially from the thermal model. Additional evidence for maser activities, including the non-Gaussian shape of line profiles, the compact emitting regions, the negative opacity values, and the metastable upper-energy levels, will be discussed further in the following sections. All combine to support the observed emission of CH 3 OH at 84 GHz toward both MS1 and MS2 and CH 2 NH at 5.29 GHz toward MS2 to consist of astronomical (weak) masers.

As a reference for the relative positions, Figure 1 shows the continuum emission from Sgr B2(N) observed at 6.8 cm (left) and 3.5 mm (right). The free-free continuum emission concentrations at 6.8 cm include four cores, K1-K4, and the two bow-shaped H II regions, K5 and K6 (Corby et al. 2015, marked in green in Figure 1). At 3.5 mm, three main hot cores, N1-N3, were revealed by their molecular emission (Bonfand et al. 2017, marked in orange in Figure 1).

Class I CH 3 OH Masers

Class I CH 3 OH masers are prevalent in massive star formation regions. In particular, the 44 GHz maser line was observed toward Sgr B2 at 3″ resolution (Mehringer & Menten 1997). In contrast, there is limited information about the 5 -1 -4 0 E Class I CH 3 OH maser at 84 GHz, a member of the same J + 1 -1 -J 0 E-type transition family along with the 36 GHz line (Breen et al. 2019, and references therein). Interferometric observations of the 84 GHz line in the Milky Way are even rarer (Voronkov et al. 2006), as recent work has focused on extragalactic detections (McCarthy et al. 2018;Humire et al. 2022).

Here, we present the first interferometric ALMA observations of the 84 GHz CH 3 OH masers in a Milky Way object at a resolution of 1 5, as well as the first detection of the 84 GHz maser toward Sgr B2. The spread of 84 GHz maser spots is manifest in the channel images shown in Figure 2, with velocities ranging from 50-86 km s -1 . The brightest maser spot located ∼3 6 west of the K4 core with a V lsr of 58 km s -1 (hereafter referred to as MS1) has a peak T B of 389.2 ± 0.2 K (8.454 ± 0.004 Jy beam -1 , Figures 2 (q)-(t)). In addition to the brightest spot MS1, another maser knot associated with the K6 H II region is also identified with a T B of 75.4 ± 0.4 K (1.636 ± 0.008 Jy beam -1 ) at a V lsr of 83 km s -1 (hereafter referred to as MS2), as shown in Figures 2 (b)-(e).

The molecular data file for E-type CH 3 OH was taken from the Excitation of Molecules and Atoms for Astrophysics (EMAA) database,foot_3 with spectroscopy from the Cologne Database for Molecular Spectroscopy (CDMS) catalogfoot_4 (Müller et al. 2005) and collision rate coefficients reported by Rabli & Flower (2010) for the rotational excitation of CH 3 OH by para-and ortho-H 2 (hereafter p-H 2 and o-H 2 ). The calculations of Rabli & Flower (2010) were performed for rotational transitions within the three first torsional states of CH 3 OH, ν = 0, 1, 2, with a CH 3 OH basis including rotational states j 1 15 for p-H 2 ( j 2 = 0) and j 1 9 for o-H 2 ( j 2 = 1), where j 1 and j 2 are the rotational quantum numbers. The coupled-states approximation was used to generate rate coefficients at temperatures 10 T k 200 K. Because torsionally inelastic transitions were not considered by Rabli & Flower (2010), the EMAA data set for E-CH 3 OH includes rotational transitions within the ground torsional state (ν = 0) only. The first excited state (ν = 1) of E-CH 3 OH opening at 204.2 cm -1 , the highest level of E-CH 3 OH included in the EMAA data set is the rotational state 9 -6 ( j 1 = 9, K = -6) with an energy of 203.9 cm -1 .

Masers with Dark Continuum Background-MS1

Four uncontaminated E-type CH 3 OH transitions covered in the collision data were used to constrain the physical parameters quantitatively using a non-LTE model and MCMC fitting algorithm. The observed spectra extracted toward the brightest maser spot MS1 are shown in Figure 4. We first attempted with a single-clump model but found that the observed spectra could not be reproduced with physically realistic parameters even after relaxing the constraints on T cont and T rad . In order to match the bright maser emission at 84.5 GHz, strong background radiation is needed to stimulate the maser gain in the single-clump model but it will also introduce significant absorption features at both 85.5 and 108.9 GHz.

Moreover, as shown in Figure 2(r), there is no significant millimeter-wave continuum source at MS1. While this spot also exhibits a strong CH 3 OH emission at 44 GHz (T B = 157 K, Mehringer & Menten 1997), it shows no continuum emission at centimeter wavelengths either. The closest continuum source, K4, is offset by ∼3 6, which corresponds to ∼0.14 pc at the distance of Sgr B2 (8.2 kpc, Reid et al. 2019). In addition, as shown in Appendix A, there are no point sources associated with this maser spot. The nearest bright infrared source observed with both the Two Micron All Sky Survey (2MASS) and the Galactic Legacy Infrared Midplane Survey Extraordinaire (GLIMPSE) program is located at ∼5 2 to the west of the maser spot. Instead, the background radiation stimulating the maser gain may be weak and diffuse, causing it to be resolved out by interferometers, such as the free-free and dust emission in the Galactic plane.

As illustrated in Figure 3, the model we consider consists of two-chained gas clumps overlapping in velocity along the LOS, where the foreground clump exhibits maser activity but the

Notes.

a Peak brightness temperature measured at the brightest emission spot for each molecule. The brightest emission spot for 84 GHz CH 3 OH Class I maser is located near the K4 core at a V lsr of 58 km s -1 . The brightest emission spot for 5.29 GHz CH 2 NH maser is located in the K6 H II region at a V lsr of 83 km s -1 . The measured T B of CH 2 NH represents a lower limit to the T B, peak because of the limited angular resolution of the VLA observations. b Characteristics of the CH 3 OH foreground masing clump toward MS1. c Characteristics of the CH 2 NH masing gas toward MS2. background clump can either be also an inverted maser (i.e., the double-maser scenario) or noninverted emitting (i.e., the singlemaser scenario). In both scenarios, this model can account for the strong maser emission with an absence of a bright and compact continuum source. It is analogous to the "selfamplification" model of two masing clouds that has been used to interpret water megamasers observed in circumnuclear disks in active galactic nuclei (Kartje et al. 1999).

A chained two-component model has been constructed to iteratively generate synthetic spectra, each component being described with five parameters, T k , n c , N col , dV, and V lsr . We assumed the two components have coherent but not identical values of V lsr of ∼58 km s -1 , although spatially separated along the LOS. As such, two V lsr parameters have the same prior probability distributions but are fit independently. Including both V lsr values, the priors for the free parameters and the MCMC analysis are discussed in detail in Appendix B. Posterior probability distributions for each parameter and their covariances were generated with 100 walkers with 50,000 samples each. 10As revealed in the bimodal posterior distributions, the samples converge toward two distinct peaks, each corresponding to the double-maser and single-maser scenarios, respectively. The statistics of the resulting parameters for each scenario were summarized in Table 3, which were used to construct the synthetic spectra for the double-maser scenario shown with the green trace and the single-maser scenario shown with the purple trace in Figure 4. Both synthetic profiles well reproduce the observed spectra, capturing both the maser and the optically thick emission features. Our model, unfortunately, falls short in matching the low-frequency side of the emission plateau at 108.87 GHz, corresponding to the 0 0 -1 -1 transition at a velocity range of 70-90 km s -1 . This divergence from the observed spectrum implies the potential presence of another optically thick velocity component(s) along the LOS. Because neither of the two scenarios completely reproduces the observed data, it is noteworthy that the uncertainties derived from the MCMC results should be considered as a lower limit of the uncertainties.

In the double-maser scenario, the τ and T ex of the 5 -1 -4 0 maser transition are found to be -2.44 and -5.8 K for the foreground clump and -0.02 and -2027.3 K for the background clump. In the single-maser scenario, the corresponding τ and T ex values are -2.52 and -6.5 K for the foreground clump and 0.13 and 245.2 K for the background clump. As a T ex of 245.2 K is significantly higher than the T k value in this case, it is evident that this transition exhibits strong noninverted emission and is on the verge of a population inversion.

Masers with Bright Continuum Background-MS2

In contrast to MS1, MS2 is aligned with the K6 H II region along the LOS, which provides background radiation to be amplified by the stimulated emission (Figure 2(d)). The spectra extracted toward MS2 are shown in Figure 6 with two individual velocity components being identified for each of the spectral features. We also performed the spectral analysis for MS2 with a two-component model, where each component has a distinct V lsr . The adopted prior and resultant posterior probability distributions for each parameter are depicted, respectively, in Figure 5. The priors for V lsr were chosen to be Gaussian distributions centering on 83 and 63 km s -1 , respectively, based on the difference between the observed sky frequency and the rest frequency. Meanwhile, the T cont prior also has a Gaussian distribution, based on the measured continuum level of 6.67 ± 0.52 K. The remaining parameters all have uninformative uniformly distributed priors. Following 100 walkers with 10,000 samples each, all parameters converge to primarily Gaussian-like distributions in our model. The corresponding corner plot of the posterior distributions and the parameter covariances are shown in Appendix B, with the largest covariance observed between H 2 densities and CH 3 OH column densities. This covariance aligns with the fact that H 2 densities and CH 3 OH column densities are related through the maser emission measure, ξ ∝ n c × N col , which, together with temperature, determines the overall rate coefficient for a collisionally pumped maser transition (Leurini et al. 2016).

The posterior medians from the MCMC inference (Table 4) were used to construct the synthetic spectra shown in Figure 6. If we take the noise level measured in each passband into account, the synthetic spectral profiles fit satisfactorily with the observed spectra for both the maser and optically thick absorption features. However, the low-velocity component (64 km s -1 ) of the 6 -2 -7 -1 transition fails to match the observed absorption feature at 85.55 GHz. This absorption feature might be attributed to potential contamination from an unidentified transition. The excitation temperature and opacity of the 5 -1 -4 0 maser transition were found to be negative, i.e., -5.6 K of T ex and -2.52 of τ for the high-velocity component (83 km s -1 ), confirming its maser activity toward MS2.

CH 2 NH Masers

The maser activity of the 1 1,0 -1 1,1 CH 2 NH transitions at 5.29 GHz has been revealed toward both the Galactic Center and external galaxies as point source emission (Faure et al. 2018;Gorski et al. 2021). With the VLA observations, we resolve this emission spatially into three velocity components with a V lsr of 64, 73, and 83 km s -1 . The peak intensity map for each component is shown in Figure 7, respectively. The 83 and 64 km s -1 components are both associated with the K6 H II region, while the 73 km s -1 component is concentrated toward the K5 H II region. The brightest 83 km s -1 component displays a more compact distribution than the other two components, with a T B of 121.8 ± 8.6 K (12.1 ± 0.85 mJy beam -1 ). This measured T B represents a lower limit to the peak T B since the observed emission morphology of the 83 km s -1 component may result from point sources convolved with the coarse VLA beam. Nevertheless, this component of the CH 2 NH emission is coherent with that of the 84 GHz CH 3 OH maser emission both in the velocity domain and spatial position. Therefore, the significant brightness, highly compact morphology, and coherence with other maser emission suggest maser activity in the 83 km s -1 component.

The molecular data file for CH 2 NH was taken from the EMAA database. Extended from the calculations in Faure et al. (2018), we calculated the collisional data for rotational transitions of CH 2 NH among the first 49 rotational levels, i.e., up to the state 8 2,6 at 99.5 cm -1 , and for kinetic temperatures in the range of 10-150 K. The detailed 3.

.4 0.1 0.1 Background -+ 29.2 5.6 5.7 -+ 1.4 10 0.4 0.6 3 -+ 57.9 0.1 0.1 -+ 8.1 0.2 0.2 -+ 12.1 1.3 2.6 L Foreground -+ 31.9 4.4 5.8 -+ 5.3 10 1.6 2.0 4 -+ 58.5 0.1 0.1 -+ 6.2 0.2 0.2 -+ 1.3 0.2 0.2 L Double-maser scenario d L L L L L -+ 3.3 0.1 0.2 Background -+ 29.3 8.8 11.2 -+ 2.1 10 0.8 2.1 3 -+ 59.4 0.3 0.2 -+ 5.8 0.2 0.2 -+ 8.1 1.3 1.5 L Foreground -+ 32.2 4.8 5.9 -+ 5.1 10 2.0 2.5 4 -+ 57.3 0.2 0.2 -+ 8.0 0.3 0.3 -+ 1.5 0.2 0.4

L

Notes. The quoted uncertainties represent the 16th and 84th percentiles (1σ for a Gaussian distribution) uncertainties. a n c represents the number density of the collision partners, which we take to be n H2 as hydrogen is assumed to be the dominant partner. b Two velocity components have similar V lsr of ∼58 km s -1 along the LOS. c Continuum temperature for the 84-109 GHz band. d The MCMC analysis revealed two distinct peaks in the posterior distributions, each representing one of the two scenarios. Depending on the scenario, the background clump exhibits maser or noninverted emission, while the foreground clump always exhibits maser emission.

calculations of the collisional rate coefficients are presented in Appendix C. We performed the MCMC analysis with a one-component model using the prior probability distributions depicted in Figure B4. The VLA observations only cover two transitions of CH 2 NH, while the other transition at 4.44 GHz has no signal over the noise levels. As such, the data are insufficient to simultaneously constrain the physical characteristics so the CH 2 NH MCMC analysis requires a confined choice of priors on some parameters. We opted to use E-type CH 3 OH discussed in Section 4.1.2 for physical constraints, where the 83 km s -1 components between CH 3 OH and CH 2 NH are spatially coherent and are both associated with the K6 H II region. We therefore extracted the CH 2 NH spectrum toward the same position as MS2. The posteriors on T k and n H 2 obtained from CH 3 OH (Table 4; Figure 5; Figure B3) are adopted as MCMC priors for the CH 2 NH model. The priors for V lsr and dV are chosen to be Gaussian distributions, informed by the strongest emission feature (12σ) detected in the data. The Gaussian prior for T cont is based on the continuum level of 2687 ± 58 K measured by the VLA. Meanwhile, the N col of CH 2 NH has a noninformative uniformly distributed prior.

As shown in the posterior distributions in Figure B4, following 10,000 samples with each of the 100 walkers, the samplers converge at an N col of -+ -4.76 10 cm 0.64 1.09 14

2 . The synthetic line profile is shown in Figure 8, reproducing six hyperfine components of the 1 1,0 -1 1,1 transition within the noise level. The six hyperfine components have T ex in the range of -0.75 to -0.55 K, τʼs in the range of -0.03 to -0.006, and, in particular, the F = 2 -2 transition has a T ex of -0.65 K and a τ of -0.03.

Discussion

Metastable Energy Levels of Maser Transitions

The majority of the molecular lines observed toward the H II regions in Sgr B2(N) are in absorption (Corby et al. 2015, and references therein). In contrast, specific transitions of certain species toward these regions exhibit emission features (e.g., McGuire et al. 2012;Corby et al. 2015;Remijan et al. 2022). These transitions establish population inversions between energy levels (n up g low > n low g up ), a prerequisite for maser action to occur. The mechanism of population inversion is conceptualized by spontaneous decay and collisions: spontaneous decay populates the levels determined by the Einstein coefficients, meanwhile collisions redistribute molecules due to their specific geometry and rotational energy level patterns. Figures 9 and 10 show the energy level structure of spontaneous transitions connected to the two target maser transitions, with Einstein spontaneous emission coefficients (A ij ) shown in colors.

In the case of the CH 3 OH 5 -1 -4 0 E transition at 84 GHz, the upper-energy level 5 -1 E is fed by the other higher levels with strong landing transitions but spontaneously decays to the lower-energy level 4 0 E with a weak escaping transition .foot_6 5 -1 E has a total escaping -A log s ij 10 1 of -4.2139 (equivalent to a lifetime of ∼1.6 × 10 4 s), which is ∼3.0 orders of magnitude lower than its total landing A ij . As discussed in Lees (1973) and Menten (1991), CH 3 OH spontaneously decays more rapidly within the K = -1 stack as compared with decaying out of the K = -1 stack. Specifically, among the two escaping transitions, 5 -1 -4 -1 E is stronger than 5 -1 -4 0 E (Figure 9). In fact, the K = -1 levels are the endpoints for all spontaneous decay routes (Lees 1973). For example, the lowerenergy level of this maser transition (4 0 E) decays rapidly to the lower K = -1 levels.

In addition to the spontaneous decay, collisions redistribute molecules over the vertical stacks of the K = -1 levels because the ΔK = 0 collisions are preferred compared with |ΔK| = 1 and 2 (Menten 1991). In particular, at a T k of 30 K, the total rate coefficient of the ΔK = 0 collisions de-populating a K = -1 level (∼1 × 10 -9 cm 3 s -1 ) is almost two orders of magnitudes higher than that of the |ΔK | = 1 collisions (∼5 × 10 -11 cm 3 s -1 ) and the |ΔK | = 2 collisions (∼3 × 10 -11 cm 3 s -1 ).

For CH 2 NH, we sum the hyperfine components of the 14 N splittings and use the sum of A ij as the representative A ij for each transition in Figure 10. In the case of the CH 2 NH 1 1,0 -1 1,1 transition at 5.29 GHz, the upper-energy level 1 1,0 is coupled to the higher levels 2 1,1 and 2 2,1 with strong spontaneous transitions but to the lower-energy level 1 1,1 with a weak escaping transition. Compared with the upper-energy level 1 1,0 , which has a total escaping Meanwhile, collisions of CH 2 NH exhibit a propensity for the |ΔK a | = 0 transitions compared with the inter-stack transitions |ΔK a | = 1 and 2 (Faure et al. 2018). In particular, the total rate coefficient of the ΔK a = 0 collisions de-populating a K a = 1 level (∼10 -10 cm 3 s -1 ) is 1 order of magnitude higher than that of the |ΔK a | = 1 and 2 collisions at a T k of 30 K. However, the inter-stack transitions from the ground state 0 0,0 complicate this multilevel system. The collisions from 0 0,0 more efficiently populate the lower level 1 1,1 than the upper level 1 1,0 , suppressing the inversion. Therefore, while these energy structures allow the temporary trapping of the population in the metastable 1 1,0 level, it only results in a weak population inversion.

Spatial Origin and Pumping Mechanism

The maser activity of several simple species was reported previously toward Sgr B2(N) with precise locations, including CH 3 OH, H 2 O, OH, and NH 3 , which are indicated by + symbols in Figures 1 and 11. The Class II CH 3 OH maser spot at 6.7 GHz is spatially associated with the N3 millimeter-wave core (Caswell 1996;Caswell et al. 2010). A survey performed with the Australia Telescope Compact Array characterized 30 H 2 O maser spots with the majority of them located within the N1-K2-K3 region (Walsh et al. 2014). Six OH maser spots have been identified with a VLA survey of OH masers, with the brightest spot being associated with the K3 centimeter-wave core (Argon et al. 2000;Qiao et al. 2014). In a recent VLA observation, nine NH 3 (9, 6) maser spots were resolved in this region; most of these spots are concentrated in the N1-K2-K3 region, while the brightest spot is located toward the N3 hot molecular core (Yan et al. 2022). In contrast, the maser emission of the organic molecular species presented in this work is found to be located close to the K4 centimeter-wave core and the K6 H II region, being spatially offset from the spots of the radiative pumping maser species OH and Class II CH 3 OH. The 84 GHz CH 3 OH maser displays a bright but extended distribution, centered around MS1 (near the K4 core), and subtending an angular size of ∼15″. The emission tapers

.7 0.1 0.1b -+ 30.7 1.5 2.0 -+ 8.3 10 1.6 1.5 4 -+ 83.3 0.1 0.1 -+ 6.6 0.1 0.1 -+ 1.1 10 0.1 0.1 16 L -+ 37.3 5.6 6.4 -+ 6.6 10 1.5 2.1 3 -+ 63.6 0.2 0.1 -+ 11.4 0.3 0.3 -+ 1.2 10 0.2 0.2 16 L CH 2 NH L L L L L -+ 2687.6 62.4 62.4c -+ 31.3 4.3 4.3 -+ 6.2 10 2.4 3.7 4 -+ 83.5 0.3 0.3 -+ 7.7 0.5 0.5 -+ 4.8 10 0.6 1.1 14

L

Notes. Similar to Table 3. a Two velocity components are identified along the LOS at a V lsr of 83 and 64 km s -1 , respectively. b Continuum temperature for the 84-109 GHz band. c Continuum temperature at 5 GHz. ). See Table 4.

off rapidly from the center with non-Gaussian profiles, a morphology possibly attributed to maser beaming effects. We found that MS2, superimposed on the K6 H II region, harbors maser emission of both Class I CH 3 OH at 84 GHz and CH 2 NH at 5.29 GHz (Figure 11). The 83 km s -1 component of the CH 2 NH maser at K6 remarkably resembles that of the Class I CH 3 OH maser. Their consistent morphology serves as strong evidence to support a similar collisional pumping mechanism for the CH 2 NH maser with the Class I CH 3 OH maser as well as a common physical condition for triggering these masers. By understanding the conditions that are necessary for maser activities to occur, we can also better understand how they can be used to monitor astronomical objects. For example, since these masers are pumped by intense collisions, we can expect them to be good candidates for studying regions with time-dependent gas dynamics, such as episodic outflows from massive protostars. Future surveys of these new masers in addition to such objects are needed to determine how frequently they accompany Class I CH 3 OH masers and how they might fit into proposed evolutionary time lines of masers in massive star formation (e.g., Ellingsen et al. 2012).

Alongside our observations of the Class I 84 GHz CH 3 OH maser, we also examined the CH 3 OH 8 0 -7 1 A-type transition at 95 GHz, which exhibits maser activity toward several famous star-forming regions, e.g., Orion-KL (Plambeck & Wright 1988), DR 21 (Plambeck & Menten 1990), and W51 (Pratap & Menten 1993). A recent single-dish survey of ATLASGAL clumps shows that the 84 GHz line is generally weaker than the 95 GHz line (Yang et al. 2023). In contrast, we found that the 95 GHz line exhibits weaker emission toward the identified maser spots in Sgr B2(N) compared to the 84 GHz line. The spatial distribution for the 95 GHz emission is similar to that for other thermal molecular lines, concentrating toward the N1 millimeter-wave source. This distribution may be attributed to either a weak maser or thermal radiation at different emission spots. It is noteworthy that Mehringer & Menten (1997) identified both maser and quasi-thermal emission components toward Sgr B2 in the 44 GHz line, a consecutive transition (J = 6) of the (J + 1) 0 -J 1 A series with the 95 GHz transition (J = 7). A more detailed analysis of other A-type CH 3 OH transitions is needed to determine the nature of the 95 GHz emission.

Physical Conditions of Masing Clumps

The high spatial resolutions of interferometric observations enable us to resolve and segregate compact molecular emissions in diverse environments. In particular, the rich thermalized emission of CH 3 OH and CH 2 NH occurs in the hot core regions. In the N1 hot core, the ALMA observations reported an N col of 2.0 × 10 19 cm -2 for CH 3 OH and 9.0 × 10 17 cm -2 for CH 2 NH at a T ex of 230 K (Motiyenko et al. 2020;Margulès et al. 2022). In contrast, the bright Figure 8. Observed spectrum of the CH 2 NH 1 1,0 -1 1,1 masing transition extracted toward MS2 is shown in black with noise levels shaded in gray. The synthetic spectrum with hyperfine splitting is overlaid in red for the 83 km s -1 components. The F = 1 -0 and 2 -2 hyperfine components are blended. Each hyperfine component is denoted by a vertical bar, with the red bars corresponding to a V lsr of 83 km s -1 and the blue bars corresponding to 64 km s -1 in blue. stimulated maser emission of these two molecules concentrates in the less dense masing clumps. The N col of these two molecules in the masing clumps (Tables 3 and 4) are 2-3 orders of magnitudes less abundant than those in the hot core.

From the MCMC analyses on the CH 3 OH 84 GHz maser presented in Section 4.1, we found the T k for the masing clumps to be ∼29-37 K. This range of T k is consistent with the large-scale measurement of the Sgr B2 envelope from the OCS lines (30-37 K, Armijos-Abendaño et al. 2020) as well as the dust temperature of Sgr B2(N) obtained from Herschel observations (∼28 K, Etxaluze et al. 2013). It is also consistent with the value of T k (30 K) derived from the non-LTE modeling of the weak masers of methyl formate detected in PRIMOS (Faure et al. 2014).

We also constrain the n H 2 of the masing clumps to be (5-8) × 10 4 cm -3 , a level significantly lower than that derived for (Argon et al. 2000;Caswell et al. 2010;Walsh et al. 2014;Yan et al. 2022). The K4 core and arc-like K6 H II region are marked with a green circle marker and dotted line, respectively.

the hot core regions. The n H 2 of the Sgr B2(N) hot cores are >10 7 cm -3 , with the N1 core reaching ∼(1-2) × 10 9 cm -3 (Sánchez-Monge et al. 2017). In contrast, the n H 2 of the masing clumps is comparable to the n H 2 of the colder envelope of Sgr B2, which is estimated to be 10 3 -10 4 cm -3 based on the absorption molecular lines (Huettemeister et al. 1995;Schmiedeke et al. 2016). The slight enhancement in physical densities might be attributed to local energetic events occurring in the envelope necessary for a collisionally pumped maser.

This work presents a quantitative constraint on the physical conditions characterizing a masing clump of organic molecules. It serves as a reference for the physical conditions sufficient for exciting population inversion and collisionally pumped masers. These conditions include a T k of ∼30, > ´n 5 10 cm H 4 3 2 , and moderate molecular N col (>10 16 cm -2 for CH 3 OH and >10 14 cm -2 for CH 2 NH). The relatively low values of temperatures and densities indicate that the masing clumps are situated in the colder and low-density external envelope. The distance from Sgr B2(N) to the edge of the external envelope is suggested to be approximately 19 pc (Schmiedeke et al. 2016). The continuum emission from Sgr B2(N) has an angular size of ∼10″, or 0.4 pc at a distance of 8.34 kpc (Reid et al. 2014), which subtends a solid angle of 3.5 × 10 -4 sr to the envelope. This aligns with our assumption mentioned in Section 3.1, where background continuum emission sources are far from masing clumps, and therefore, do not significantly affect the local energy distribution of the molecules. Maser activities can manifest themselves in the presence of either a bright continuum background radiation or multiple chained masing clumps in the direction of a diffuse and weak background.

Caveat on Geometric Effects

Geometric effects related to the propagation of maser radiation become increasingly significant as the optical depth becomes lower than -1. Although RADEX and molsim support various geometries through escape probability approximation, other effects, such as maser beaming, are not treated. Such effects are particularly important in the case of unresolved clumps. The models also assume homogeneous physical conditions (i.e., one-zone models), which is inappropriate for very negative optical depths. For precisely modeling maser lines with strong amplification, a nonlocal treatment of radiation and level population is required.

Although our current treatment does not provide a precise solution, it is in principle appropriate for |τ| = 1 (the case of CH 2 NH) and still offers a reasonable approximation to the first order as |τ| is of the order of unity (the case of CH 3 OH). Incorporating corrections such as nonlocal treatments and other geometric effects would definitely improve the solution but it would also introduce significant computational costs. This becomes especially infeasible for MCMC calculations. The current solution represents the best that is achievable with our current resources, acknowledging that this approach is subject to future expansions and refinements.

Summary

We report interferometric observations of maser activity from organic molecules toward Sgr B2 using the ALMA and VLA. Enabled by the ALMA observations, we report the first detection of the Class I CH 3 OH maser at 84 GHz toward Sgr B2. Multiple 84 GHz CH 3 OH maser knots are discovered with the brightest spot MS1 located ∼3″ west of the K4 free-free continuum emission core. The peak brightness temperature of MS1 reaches 389.2 ± 0.2 K at a velocity of 58 km s -1 . In addition, another CH 3 OH maser spot MS2 coexists with the K6 UC H II region at a velocity of 83 km s -1 . With the VLA observations, we resolve the 5.29 GHz CH 2 NH maser emission into three velocity components. These three components display distinct fine-scale structures associated with the K5 and K6 bow-shaped H II regions. The 83 km s -1 component is cospatial with MS2 and its peak brightness temperature has a lower limit 121.8 ± 8.6 K.

We present a tool for modeling and fitting maser spectra with non-LTE calculations and a Bayesian approach. By leveraging the theoretical calculations on collisional rate coefficients and cross sections for both E-type CH 3 OH and CH 2 NH transitions, this fitting technique enables us to statistically constrain the physical conditions of the masing regions. In particular, our analysis suggests a two-chain-clump maser model for explaining the intense CH 3 OH maser activity observed toward a region with weak and diffuse background radiation. This model can reproduce both the maser and the optically thick emission features of the CH 3 OH spectra observed toward the brightest maser spot MS1.

Interferometric observations have not only enabled us to characterize the spatial origin and extent of maser emission but also helped to constrain the pumping mechanism through direct comparisons of their morphologies. We found a definite association between the maser emission from complex molecules and the K6 UC H II region. They are offset from the masing regions of the radiatively pumped maser species and the infrared sources. In contrast, the spatial correlation between the activities of the 84 GHz CH 3 OH Class I maser and 5.29 GHz CH 2 NH maser suggests that the CH 2 NH maser is pumped by intense collisions, analogous to the Class I CH 3 OH maser. By understanding the nature of the new maser species, which are pumped by intense collisions, we can better understand how they may serve as good candidates for studying and monitoring astronomical objects with timedependent gas dynamics.

Data Access and Code

The code used to perform the analysis is part of the molsim open-source package; an archival version of the code can be accessed in Lee et al. (2023). The spectroscopic and collisional data for E-CH 3 OH and CH 2 NH can be found in the EMAA database (EMAA 2021). Appendix B MCMC Fitting Details E-type CH 3 OH. There are 11 free parameters in total to be adjusted in the MCMC analyses for E-type CH 3 OH. For the spectra extracted toward MS1, the adopted prior and resultant posterior probability distributions for each parameter are depicted in Figure B1. The priors for V lsr of both components are chosen to be a Gaussian distribution centered at 58.3 km s -1 with a standard deviation of 2 km s -1 . Since the continuum is weak, as evidenced by the observations, we opted for a uniform prior distribution in the range of 2.725-15 K for T cont . Meanwhile, all the remaining parameters have uninformative uniformly distributed priors. We found that the convergence rate of the RADEX algorithm slows down tremendously at high N col > 10 17 cm -2 . Therefore, we capped N col at 10 17.5 cm -2 and used 50,000 samples per walker (five times the value used for MS2). At high N col , we observed deviations between the results obtained from our modified non-LTE algorithm and those from RADEX due to results not being converged in the latter.

A corner plot of the parameter covariances and their distribution for the MS1 MCMC fit is shown in Figure B2. As shown in Figure B1, we found that the optimal set of parameters converges to two peaks, each corresponding to one of the two scenarios (double maser and single maser). In particular, the peak at a lower velocity of the foreground component, corresponding to the higher velocity peak of the background component, corresponds to the double-maser scenario, while the other peak corresponds to the single-maser scenario. To better visualize the posteriors, we isolated the two peaks using V lsr = 58 km s -1 as the criterion and plotted the posteriors separately in Figure B2. The corner plots with green and purple backgrounds correspond to the double-maser and single-maser scenarios, respectively.

In each corner, the first five parameters belong to the foreground component, while the last five are those of the background component. The 2D posterior distribution among parameters for the foreground component is shown in blue, while that for the background component is shown in red. The cross terms between the two components are shown in brown. The middle parameter is the background continuum temperature and the distribution between continuum temperature and other parameters is shown in yellow.

A similar MCMC analysis to that for the spectra extracted toward MS1 was carried out for the spectra extracted toward MS2. In contrast to Figure B1, the optimal set of parameters of MS2 converges to a single peak as shown in Figure 5. The corresponding corner plot of the parameter covariances and their distribution for the MS2 MCMC fit are shown in Figure B3. Similar to Figure B2, the parameters of the lowvelocity component are shown in blue, while those of the highvelocity component are shown in red. We found that the MCMC analyses yield consistent results, regardless of whether the foreground or background velocity component is assigned as the high or low velocity along the LOS. Furthermore, the brown circular distribution, the cross terms between the two components, reveals that the two components are completely independent. This independence arises because the two components fit distinct portions of the spectrum because of their disparate central velocities compared with the line widths. In contrast, the strong degeneracy of the H 2 density and CH 3 OH column density is revealed for each individual velocity component.

CH 2 NH. An MCMC analysis with a one-clump model was carried out for the CH 2 NH spectra. The prior and posterior distributions are shown in Figure B4, while a corner plot of the parameter covariances for the CH 2 NH MCMC fit is shown in Figure B5. The prior distribution for each parameter is shown in gray, while the priors on T k and n H2 are from the MCMC posteriors of CH 3 OH in MS2 (Figure 5). The posterior distributions for the physical characteristics are shown in red, while the posterior for T cont is shown in yellow.