From vicente@oan.es Thu Oct 17 11:00:01 1996 From: Pablo de Vicente Date: Thu, 17 Oct 1996 16:51:50 +0200 To: vicente@oan.es, gcnews@astro.umd.edu Subject:CH3CN Sgr B2 article % laa.dem version 1.1 as of 25-Feb-91 % % This is LAA.DEM, the demonstration file of the % LaTeX style file from Springer-Verlag for the % Astronomy and Astrophysics Main Journal % % It is for use with LaTeX version 2.09 % % Please report all errors via e-mail to SPRINGER@DHDSPRI6.bitnet % or to the address mentioned on page 2 of the documentation % %NO%%%%\documentstyle{laamt} % LaTeX A&A Monotype Times Fonts %\documentstyle{laa} % LaTeX A&A Standard Fonts \documentstyle[referee]{l-aa} % LaTeX A&A Standard Fonts \begin{document} % \thesaurus{ 09 % A&A Section 9: ISM 09.03.1; % Clouds, 09.08.1; % HII regions, 09.11.1; % Kinematics and dynamics, 09.13.2; % Molecules, } % \title{ A Hot Ring in the Sgr B2 molecular cloud} % \subtitle{ } % \author{ P. de Vicente\inst{1} \and J. Mart\'\i n-Pintado\inst{1} \and T. L. Wilson\inst{2}} % \institute{ Centro Astron\'omico de Yebes, Apartado 148, 19080 Guadalajara, Spain \and Max-Planck Institut f\"ur Radioastronomie, Auf dem H\"ugel 69, 53121 Bonn 1, Germany} \offprints{P. de Vicente} % \date{ Received , 1995; accepted, 1996} % \maketitle % \begin{abstract} We present high angular resolution ($13-26''$) large scale mapping ($4'\times 7'$) of the J=5--4, J=8--7, and J=12--11 lines of ${\rm CH_3CN}$ and ${\rm CH_3^{13}CN}$ and of the J=11--10 line of ${\rm HC_3N}$ towards the Sgr B2 molecular cloud. All the K components of all ${\rm CH_3CN}$ lines are observed in emission except towards Sgr B2M where we have detected the J=5--4, K=4 and J=6--5, K=5 lines in absorption. ${\rm CH_3CN}$ and ${\rm HC_3N}$ show a ridge of strong emission along a north-south direction which contains the star forming regions Sgr B2M, Sgr B2N and Sgr B2S. The kinematics of the molecular gas shows four major molecular clouds with radial velocities of 44-54, 55-66, 67-78 and 90-120 ${\rm km\, s^{-1}}$ and sizes of a few parsecs. The main molecular cloud with a radial velocity of 55-66 ${\rm km\, s^{-1}}$ is observed over the whole region. Maps of the kinetic temperature and density derived from an LVG analysis of the ${\rm CH_3CN}$ data are presented for the molecular clouds at 44-54, 55-66 and 67-78 ${\rm km\, s^{-1}}$. The kinetic temperature for the three clouds ranges between 40-400 K, while the density is $\sim 10^5\ {\rm cm^{-3}}$ for all clouds. The total mass in these clouds is $3\cdot 10^6\ M_\odot$, with 70\% of the mass in the 55-66 ${\rm km\, s^{-1}}$ molecular cloud. This cloud reveals the presence of four different components: the hot cores, the warm envelope, the very hot component and the hot ring. The largest kinetic temperatures (200-400 K) are found towards the hot cores associated to the star forming regions Sgr B2M and Sgr B2N, with sizes of 0.5 and 0.7 pc respectively. Two new cores close to Sgr B2N with sizes of 0.3 pc have been found. ${\rm H_2}$ densities for the hot cores are $10^6-10^7\ {\rm cm^{-3}}$. The mass in the cores is typically $10^3-10^4\ {\rm M_\odot}$. The warm envelope extends over the whole region; this has a uniform kinetic temperature, between 40-80 K. The kinetic temperatures are higher than the dust temperatures at distances larger than 1 pc. The density in the warm envelope decreases with distance as ${\rm n(H_2)}= 2.96\cdot 10^5\, {\rm cm^{-3}} \, (r/{\rm pc}) ^{-0.87}$. The analysis of the absorption lines in the J=K components of the J=5--4 and J=6--5 lines shows the presence of a hotter and more diffuse envelope probably surrounding the warm envelope. An analysis of our data gives a kinetic temperature of 300 K and a density of $\sim 10^3\ {\rm cm^{-3}}$. The kinetic temperature maps reveal, for the first time, the presence of a ring of hot gas (100-120 K) surrounding Sgr B2M and Sgr B2N with a radius of 2 pc and a thickness of 1.4 pc. Our data suggest that the density in the hot ring is similar to that in the warm envelope. The high temperature of the hot cores and the kinetic temperature distribution for distances smaller than 1 pc can be accounted for by gas-dust collisional heating. This temperature is consistent with the total luminosity of the central sources Sgr B2M and Sgr B2N. In contrast, the dust temperatures in the warm envelope are too low (10-20 K) to heat the warm envelope molecular gas by this mechanism. Heating by dissipation of turbulent motions in the envelope of Sgr B2 can explain the high gas kinetic temperatures. The presence of the hot ring suggests the existence of another heating mechanism. The morphology of the hot ring which surrounds the 50$\mu$m and the radio continuum emission of Sgr B2M and Sgr B2N suggests that this feature might be associated to the interface between the warm envelope and the ionized bubble created by the OB stars recently formed in the Sgr B2 core. In this interface, heating by UV photons and or shock fronts produced by the expansion of the ionized gas could explain the hot ring. % \keywords{ interstellar medium: clouds Sgr B2 -- interstellar medium: HII regions: Sgr B2 -- interstellar medium: Kinematics and dynamics -- interstellar medium: Molecules -- } \end{abstract} % \section{Introduction} The Sagittarius B2 molecular cloud is one of the most active regions of massive star formation in the Galaxy. The most luminous regions in the molecular cloud, Sgr B2M and Sgr B2N, are very strong IR emitters (Thronson \& Harper 1986; Goldsmith et al. 1987a; Goldsmith et al. 1990) and contain all the signposts of recent massive star formation. Both regions contain several compact and ultracompact HII regions (Martin \& Downes 1972; Benson \& Johnston 1984; Carlstrom \& Vogel 1989; Gaume \& Claussen 1990; Mehringer et al. 1993), hot cores (Vogel, Genzel \& Palmer 1987; Goldsmith et al. 1987b) and maser emission in OH, ${\rm H_2O}$, ${\rm H_2CO}$ and SiO (Genzel \& Downes 1977; Forster et al. 1978; Hasegawa et al. 1985; Gardner et al. 1986; Elmegreen et al. 1980; Kobayashi et al. 1989; Gaume \& Mutel 1987; Mehringer et al. 1993). A third region with less luminosity than the two main cores and located $30''$ to the south of Sgr B2M also shows compact HII regions and maser emission. These regions, which contain $10^5\ {\rm M_\odot}$, are embedded in a giant molecular cloud with a size of $\sim$45 pc (Scoville et al. 1975) and a total mass of $\sim7\cdot 10^6\ {\rm M_\odot}$ (Lis \& Goldsmith 1989). In spite of the large mass in the ``envelope'' surrounding the main star forming regions most of the molecular studies have been directed towards studying the physical properties of the star forming cores Sgr B2M and Sgr B2N (Vogel, Genzel \& Palmer 1987; Lis \& Goldsmith 1991) and little is known about the properties of this massive envelope in Sgr B2. The properties of the molecular envelope have beEn derived from observations of ${\rm ^{12}CO}$ and ${\rm C^{18}O}$ (Scoville et al. 1975; Lis \& Goldsmith 1989). These authors have proposed that the Sgr B2 cloud is composed by two components. The first has a uniform ${\rm H_2}$ density between 1800 and 3500 $\rm{ cm^{-3}}$ and the second shows a density law varying as ${\rm r^{-2}}$, truncated at a value of approximately $7\cdot 10^5\ {\rm cm^{-3}}$ for $r\, \leq \, 1.25 \ {\rm pc}$. Gordon et al (1993) show maps of the dust emission at 1.3 mm and $870\ \mu{\rm m}$; they obtained dust grain temperatures in the range of 15-20 K for the envelope. On the other hand, several molecules present deep absorption lines towards the continuum sources in Sgr B2M and Sgr B2N (Wilson et al. 1982, Henkel et al. 1983, Greaves et al. 1992). From the analysis of these data it is inferred the presence of a warm ($T_k \geq 100$ K) and moderate density envelope ($10^3-10^4\ {\rm cm^{-3}}$) around the sources (Wilson et al. 1982, Huttemeister et al. 1993). The presence of this envelope suggests that turbulent heating is operating in the molecular cloud (Wilson et al. 1982). Unfortunately, the absorption lines only sample a very specific region along the line of sight which provide limited information on the extension of the hot material traced by these lines and therefore of the thermal structure in the Sgr B2 molecular cloud. The goal of the study presented in this paper is to determine the kinetic temperature and density structure of the molecular gas in the Sgr B2 cloud at scales of several parsecs with high angular resolution. To carry out this study we have mapped the J=5--4, J=8--7 and J=12--11 lines of the symmetric top rotor ${\rm CH_3CN}$. These maps, when combined with a model for the excitation of this molecule provides, for the first time, {\it maps} of kinetic temperature and ${\rm H_2}$ density in the Sgr B2 molecular cloud. From the kinetic temperature and density maps one can make the study of the thermal balance in this prototypical molecular cloud in the Galactic Center and estimate the influence of the central sources, Sgr B2M and Sgr B2N, in the heating of the molecular cloud. \section{Observations} The observations were carried out with the IRAM 30-m telescope at Pico de Veleta (Spain). Table 1 summarizes the observed molecules, transitions and frequencies. For ${\rm CH_3CN}$, Table 1 only gives the frequency of the $K=0$ component. The other $K$ transitions were also observed simultaneously in the 512x1 MHz contiguous filter backends. This Table also gives the typical single sideband system temperature, the half power beam width (HPBW) and the beam efficiency of the radiotelescope. We mapped the ${\rm CH_3CN}$ J=5--4 line in a region of $4'\times 7'$, with HPBW spacing. We also observed the J=5--4, J=8--7 and J=12--11 transitions of ${\rm CH_3CN}$ simultaneously with a 6 arcsec spacing in the region around the Sgr B2M and Sgr B2N sources. Spectra of the J=8--7 and J=6--5 lines were also taken at selected positions out of the central region. The two 512 $\times 1$ MHz channel filter bank and the 864 channel acustoptic spectrometer were used as backends. The velocity resolution provided by these spectrometers was 3.2, 2.7, 2.0, 0.8 ${\rm km\, s^{-1}}$ for the 3.3, 2.7, 2 and 1.4 mm lines respectively. The spectra were taken in position switching and we measured 5 on source positions for one reference spectrum. Pointing was checked frequently on the continuum emission maxima, Sgr B2M and Sgr B2N. Within the spectrometer we also observed the H41$\alpha$ (92.034 GHz), and H35$\alpha$ (147.046 GHz) radio recombination lines simultaneously. These lines, which trace the ionized material, were also used to check for pointing errors. The ${\rm HC_3N}$ J=11--10 transition was observed in the image band when observing the ${\rm CH_3CN}$ J=5--4 line. Most of the observations were made with the receiver tuned to SSB mode with an image band rejection of 7 to 9 dB. The line intensity calibration was done measuring cold and ambient temperature loads. The intensity scales are given in antenna temperature ${\rm T_a^*}$, and for the small sized sources the line intensities have been converted to main beam brightness temperatures by using the beam efficiencies in Table 1 and a forward efficiency of 0.9. \begin{table*} \caption{List of molecules, transitions and observing parameters} %\label{table1} \begin{flushleft} \begin{tabular}{llllll} \hline Molecule & Transition & Frequency (MHz) & $T_{\rm sys}$ (K) & HPBW ($''$) & Beam Efficiency \\ \hline ${\rm CH_3CN}$ & $J$ = 5--4, $K$ = 0 & 91987.094 & 300 & 26 & 0.6 \\ ${\rm CH_3^{13}CN}$ & $J$ = 5--4, $K$ = 0 & 91941.580 & 300 & 26 & 0.6 \\ ${\rm CH_3CN}$ & $J$ = 6--5, $K$ = 0 & 110383.523 & 300 & 21 & 0.6 \\ ${\rm CH_3^{13}CN}$ & $J$ = 6--5, $K$ = 0 & 110328.891 & 300 & 21 & 0.6 \\ ${\rm CH_3CN}$ & $J$ = 8--7, $K$ = 0 & 147174.594 & 400 & 17 & 0.6 \\ ${\rm CH_3^{13}CN}$ & $J$ = 8--7, $K$ = 0 & 147101.766 & 400 & 17 & 0.6 \\ ${\rm CH_3CN}$ & $J$ = 12--11, $K$ = 0 & 220747.266 & 700 & 13 & 0.45 \\ ${\rm CH_3^{13}CN}$ & $J$ = 12--11, $K$ = 0 & 220637.984 & 700 & 13 & 0.45 \\ ${\rm HC_3N}$ & $J$ = 11--10, $K$ = 0 & 100076.388 & 300 & 24 & 0.6 \\ \hline \end{tabular} \end{flushleft} \end{table*} \section{Results} \subsection{Morphology of the region} Fig. 1 shows our collection of profiles of ${\rm CH_3CN}$ lines towards Sgr B2M and Fig. 2 shows the integrated line intensity maps of the ${\rm HC_3N}$ J=11--10 line, ${\rm CH_3CN}$ J=5--4, J=8--7, J=12--11 lines and ${\rm CH_3^{13}CN}$ J=5--4 line. In these maps the positions of the ionized gas (HII regions) have been obtained from the recombination line emission and are shown by filled triangles. The angular resolution of our data only allows us to separate the two main strong complexes of HII regions: Sgr B2M and Sgr B2N. The offsets in the maps are measured relative to the Sgr B2M position ($\alpha_{1950}$=17:44:10.6, $\delta_{1950}$=-28:22:05). \begin{figure}[htbp] \picplace{0.5 cm} \caption{${\rm CH_3CN}$ J=5--4, J=6--5, J=8--7 and J=12--11 transitions towards Sgr B2M. The vertical lines indicate the position of the main isotope K components. The K=0 ${\rm CH_3^{13}CN}$ component is also marked. The J=5--4 K=4 and J=6--5 K=5 lines are seen in absorption. Radial velocities are computed from the main isotope K=0 components.} %\label{fig1} \end{figure} \begin{figure}[htbp] \picplace{0.5 cm} \caption{Integrated line intensity maps towards Sgr B2 of a) ${\rm HC_3N}$ J=11--10, b) ${\rm CH_3CN}$ J=5--4, c) ${\rm CH_3^{13}CN}$ J=5--4, d) ${\rm CH_3CN}$ J=8--7 e) ${\rm CH_3CN}$ J=12--11. a) Velocity interval: 45 ${\rm km\, s^{-1}}$. Contour levels of 5.5 K${\rm km\, s^{-1}}$. Lowest level: 2 K${\rm km\, s^{-1}}$. b) Velocity interval: 93 ${\rm km\, s^{-1}}$ comprising transitions K=0,1,2,3. Contour levels of 10 K${\rm km\, s^{-1}}$. Lowest level: 7 K${\rm km\, s^{-1}}$. c) Velocity interval: 93 ${\rm km\, s^{-1}}$ comprising transitions K=0,1,2,3. Contour levels of 14 K${\rm km\, s^{-1}}$. Lowest level: 5 K${\rm km\, s^{-1}}$. d) Velocity interval: 80 ${\rm km\, s^{-1}}$. Contour levels of 33 K${\rm km\, s^{-1}}$. Lowest level: 43 K${\rm km\, s^{-1}}$. e) Velocity interval: 100 ${\rm km\, s^{-1}}$. Contour levels of 47 K${\rm km\, s^{-1}}$. Lowest level: 20 K${\rm km\, s^{-1}}$. The lowest contours correspond approximatedly to 5$\sigma$. The filled triangles show the position of the Sgr B2M and Sgr B2N HII regions derived from the ${\rm H41}\alpha$ line maps. The HPBW of the telescope is shown in the lower right corner.} %\label{fig2} \end{figure} The map of emission from the ${\rm HC_3N}$ J=11--10 line is almost identical to that from the ${\rm CH_3CN}$ J=5--4. Both arise from a ridge which has a total extent of $6'-7'$ in the North-South direction and $3'-4'$ wide in the East-West direction. This ridge is centered on Sgr B2M. The most intense emission arises from a 4' long ridge. Toward the north, the ridge bends to the east. This structure is similar to that seen in the dust emission at 1.3 mm by Gordon et al. (1993). Although the J=8--7 ${\rm CH_3CN}$ line map is less complete, the bow-like shape is less pronounced in the J=8--7 than in the J=5--4 line map. The maximum of the molecular emission for the J=5--4 ${\rm CH_3CN}$ and J=11--10 ${\rm HC_3N}$ transitions is $36''$ west of the peak of Sgr B2M and $12''$ west of the peak of Sgr B2N. These offsets between the molecular and ionized material cannot be due to pointing errors because both, molecular and line emission were observed simultaneously. The offset between the HII regions and the molecular emission changes with the observed molecular line. The molecular emission of the J=8--7 line peaks closer to Sgr B2M and extends from $4''$ to $24''$ west of the HII regions. The maximum emission from more highly excited ${\rm CH_3CN}$ peaks towards the HII regions: the ${\rm CH_3CN}$ J=12--11 transition only shows two maxima which coincide with the position of the HII regions. As it will be discussed in section 8, the different locations of the molecular maxima for each transition are caused by a combination of optical depth and temperature effects. Because of sensitivity, the ${\rm CH_3^{13}CN}$ J=5--4 line is less extended than the main isotope but shows a spatial distribution similar to the high J lines of ${\rm CH_3CN}$; this emission peaks very close to Sgr B2N. \subsection{The kinematics in the Sgr B2 molecular cloud} It is well known that Sgr B2 shows a very complex kinematic structure with several molecular clouds at different radial velocities along the line of sight (see for example, Mart\'\i n-Pintado et al. 1990). In fact, most of the line profiles do not show a gaussian shape, indicating the presence of several velocity components along the line of sight. This is illustrated in Fig. 3 where we show the spatial distribution of the ${\rm HC_3N}$ J=11--10 line for different radial velocity intervals. The kinematic structure of the molecular gas has been derived from ${\rm HC_3N}$. The multiple K component structure for ${\rm CH_3CN}$, which overlaps in velocity in Sgr B2, makes the spatial structure derived from this molecule very uncertain. Low radial velocities are mainly found towards the south, while high velocities (67-78 ${\rm km\, s^{-1}}$) are towards the north. This structure is confirmed by gaussian fits to the profiles of the ${\rm HC_3N}$ 11--10 lines measured towards Sgr B2. We find that the peak velocities obtained from the gaussian fits mostly fall into four intervals: 44-54 ${\rm km\, s^{-1}}$, 55-66 ${\rm km\, s^{-1}}$, 67-78 ${\rm km\, s^{-1}}$ and 90-120 ${\rm km\, s^{-1}}$, which correspond to four different clouds (de Vicente, 1994). These velocity groups overlap along the line of sight, but can be separated in position. \begin{figure}[htbp] \picplace{0.5 cm} \caption{Integrated intensity maps of ${\rm HC_3N}$ J=11--10 for the radial velocity intervals given in the lower left corner of the panels. Contour levels correspond to an integrated intensity of 1.3 K${\rm km\, s^{-1}}$ being the lowest level 1 K${\rm km\, s^{-1}}$.} %\label{fig3} \end{figure} This velocity structure is consistent with that described by Mart\'\i n-Pintado et al. (1990) who proposed the existence of 3 molecular clouds with radial velocities of 55 ${\rm km\, s^{-1}}$, 65 ${\rm km\, s^{-1}}$ and 80 ${\rm km\, s^{-1}}$, in front of the HII regions. The lowest radial velocity material (44-54 ${\rm km\, s^{-1}}$) is located to the south of Sgr B2M. This shows a velocity gradient from 45 ${\rm km\, s^{-1}}$ to the south, to 50 ${\rm km\, s^{-1}}$ to the north of this cloud. The bulk of the emission (55-66 ${\rm km\, s^{-1}}$) has an elongated structure that overlaps in the south with the cloud at 44-54 ${\rm kms^{-1}}$ and in the north with the 67-78 ${\rm km\, s^{-1}}$ cloud. The 67-78 ${\rm km\, s^{-1}}$ material is located in a cloud with a nearly triangular shape, with its lower vertex on Sgr B2M and the upper vertices offset ($-100'',80''$) and ($80'', 120''$) from Sgr B2M. For velocities higher than 90 ${\rm km\, s^{-1}}$ the molecular gas is concentrated in a cloud of $\sim 100''$ diameter placed to the southwest of Sgr B2M. \subsection{The absorption line in ${\rm CH_3CN}$} Our data also show absorption lines for ${\rm CH_3CN}$ towards Sgr B2M. Surprisingly, the absorption lines are only observed for the J=5--4 and J=6--5 lines of ${\rm CH_3CN}$ in the J=K component (see Fig. 1). In agreement with the absorption lines observed in other species like ${\rm NH_3}$ (Wilson et al. 1982; Vogel et al. 1987; H\"uttemeister et al. 1993) and ${\rm H_2CO}$ (Rogstad et al. 1974, Henkel et al. 1983; Mart\'\i n-Pintado et al. 1990, Mehringer et al. 1993), the absorption in ${\rm CH_3CN}$ towards Sgr B2M also occurs at a radial velocity of 65 ${\rm kms^{-1}}$. The absorption lines in ${\rm CH_3CN}$ were checked using different reference positions and shifting the central frequency of the receiver. These tests eliminate the possibility that the absorption line comes from emission in the reference position or from the image band. Therefore the absorptions are due to ${\rm CH_3CN}$ and arise from gas located in front of the continuum sources in Sgr B2M. \section{Excitation models and physical properties derived from ${\rm CH_3CN}$} ${\rm CH_3CN}$ is a symmetrical top molecule which has been successfully used to determine the kinetic temperature of molecular clouds (Cummins et al. 1983; Loren \& Mundy 1984; Churchwell et al. 1991; Olmi et al. 1993). As in the case of other symmetrical top molecules, transitions, between J levels in different K ladders are radiatively forbidden. Within a J ladder, collisional excitation is thought to dominate radiative excitation. Thus the relative population of the K components of a given J depends mainly on the kinetic temperature of the molecular gas. Since different K components of a given rotational line can be observed simultaneously, the calibration should do not affect the kinetic temperature determinations. Furthermore, ${\rm CH_3CN}$ is also a molecule with a relatively high dipole moment, which can also be used to estimate hydrogen densities from measurements of different rotational transitions. Therefore ${\rm CH_3CN}$ is one of the few molecules which one can use to determine both the kinetic temperature and the density structure in molecular clouds. Different methods have been used to derive the kinetic temperatures in molecular clouds from ${\rm CH_3CN}$ (Churchwell \& Hollis 1983, Cummins et al. 1983, Loren \& Mundy 1984). In the first approach, the kinetic temperature was derived from rotational diagrams in which a single excitation temperature was used to describe the population distribution for all levels. A second approach (Churchwell \& Hollis 1983) considered the previous method too approximate and two temperatures were obtained, a rotational temperature connecting different rotational levels in the same K ladder and the kinetic temperature that describes the relative population between states in different K ladders. Cummins et al. (1983), compared the results obtained from this method with those obtained for a statistical equilibrium model for which two sets of collisional coefficients were derived from an analogous molecular species such as OCS. They concluded that when observing high rotational transitions ($J>7$) the two temperature approach gives reliable estimates for the kinetic temperatures but less accurate estimates for hydrogen densities. To derive the kinetic temperature and density in Sgr B2, we have made a multitransition analysis of the ${\rm CH_3CN}$ data by solving the statistical equilibrium equation under the assumption that the Large Velocity Gradient approximation (LVG) is valid. Due to the complexity of the line profiles in Sgr B2 (see 3.2), with several broad velocity components, our model also incorporates a more complete treatment than previous LVG analyses. Our model considers both species, A and E (ortho and para) separately with blending between the K=0 and K=1 lines. The model also incorporates the most recent collisional cross sections derived by Green (1986). The blending between the K=0 and K=1 components was taken into account by considering total overlapping to calculate both the opacity and the source function for the $K=0$ and $K=1$ lines. If $\tau_0$ is the opacity for the $K=0$ line and $\tau_1$ for the $K=1$ line, $v_d$ is the velocity difference between both lines and $\Delta v$ the linewidth, the final opacities $\tau_0^f$ and $\tau_1^f$ for lines $K=0$ and $K=1$ were estimated by, \begin{eqnarray} \tau_0^f &= \tau_0 + \tau_1 e^{-4 \log 2 (v_d/\Delta v)^2} = \tau_0 + \tau_1 K \\ \tau_1^f &= \tau_1 + \tau_0 e^{-4 \log 2 (v_d/\Delta v)^2} = \tau_1 + \tau_0 K \end{eqnarray} and the final source function $S_0^f$ and $S_1^f$ for each line are, \begin{eqnarray} S_0^f =& \frac{S_0\tau_0 + S_1 K \tau_1} {\tau_0^f} \\ S_1^f =& \frac{S_1\tau_1 + S_0 K \tau_0} {\tau_1^f} \end{eqnarray} where $S_0$ and $S_1$ are the source functions for the $K=0$ and $K=1$ lines respectively. The complex velocity field of the molecular gas in Sgr B2 has been taken into account by the following: we have derived densities and kinetic temperatures by simultaneously fitting intensities and line profiles of all the K components for all rotational lines, as well as the corresponding lines of ${\rm CH_3^{13}CN}$. Since the observed line profiles are not single gaussians, we have also considered that the observed profiles could also be caused by two velocity components whose radial velocities and line widths were derived from our ${\rm HC_3N}$ data. The free parameters used in the fits were ${\rm CH_3CN}$ column density, hydrogen density and kinetic temperature. A/E ratios used in the fits were always within 20\% of the expected LTE value of 3 to 1. For the ${\rm ^{13}C}$ isotope line we have used the typical ${\rm ^{12}C/^{13}C}$ ratio of 20 for the Galactic Center (see for example, Wilson \& Rood 1994) Fig. 7 shows a sample of profiles of the J=5--4 and J=8--7 line with the fits superimposed. In order to improve the signal to noise ratio and to obtain the same angular resolution for all transitions, we have averaged several spectra with similar profiles, radial velocity and linewidth. This averaging provided an effective resolution of $40''$. Since the emission in the envelope is more extended than the beam we have used the ${\rm T_a^*}$ scale to determine the physical conditions for this component. For the hot cores, where the intensities are larger, we were able to obtain an effective angular resolution of $17''$. In this case, the sources have smaller sizes and we used the main beam temperature scale. \section{The kinetic temperature structure of the Sgr B2 molecular cloud} The kinetic temperature distribution has been determined for the three lower velocity (44-54 ${\rm km\, s^{-1}}$, 55-66 ${\rm km\, s^{-1}}$, 67-78 ${\rm km\, s^{-1}}$) molecular clouds in Sgr B2. The lines arising in the highest velocity molecular cloud ($>90\ {\rm kms^{-1}}$) are very weak and prevented a reliable determination of the kinetic temperature. The results are presented in Fig. 4, panels a, b and c. The gray scale has been selected in each case to stress the lack of uniformity on a large scale. In the hot core region there is saturation. Fig. 4 also shows, in panels d) and e), the ${\rm H_2}$ density and ${\rm CH_3CN}$ column density for the 55-66 ${\rm km\, s^{-1}}$ cloud. High temperatures ($\geq 40$ K) are found for the three molecular clouds. The molecular clouds at 44-54 ${\rm km\, s^{-1}}$ and 67-78 ${\rm km\, s^{-1}}$ show kinetic temperatures which range between 40 K and 120 K, and a mean ${\rm H_2}$ density of $10^5\ {\rm cm^{-3}}$. The 44-54 ${\rm km\, s^{-1}}$ cloud shows a high temperature region (120 K) bow-shaped $150''$ to the south of Sgr B2M while the 67-78 ${\rm km\, s^{-1}}$ has several hot clumps (${\rm T_k}=$100 K) close to star formation regions. For the largest molecular cloud with velocities of 55-66 ${\rm km\, s^{-1}}$, the kinetic temperature reaches 400 K in the central sources associated with the recent star formation regions. Farther from the center, where the ${\rm CH_3CN}$ intensity is lower, the kinetic temperature is $\sim$ 40 K. In general the kinetic temperature of the molecular gas in Sgr B2 is {\it at least} a factor of two larger than the dust temperature derived by Gordon et al. (1993). The kinetic temperature map for this molecular cloud shows three different components: the hot cores, a warm envelope and a hot ring. These, plus a very hot component which presumably originates the ${\rm CH_3CN}$ absorption line, will be discussed in detail in the following sections. \begin{figure}[htbp] \picplace{0.5 cm} \caption{Upper panels (a,b,c) contain the kinetic temperature distribution for molecular gas in the velocity ranges a) 44-54 ${\rm km\, s^{-1}}$, b) 55-66 ${\rm km\, s^{-1}}$ and c) 67-78 ${\rm km\, s^{-1}}$. Contours represent kinetic temperatures of 40, 80, 100 and 120 K. The letters show the positions where we have taken the spectra shown in Fig. 7. The rectangle in the central part of map b) is enlarged in Fig. 5. Lower panels contain (d,e): maps of the ${\rm H_2}$ density and ${\rm CH_3CN}$ column density distributions for the 55-66 ${\rm km\, s^{-1}}$ molecular gas. The contours in d) are ${\rm n(H_2)}=1.2\cdot 10^5$, $1.6\cdot 10^5$, $4.4\cdot 10^5$ and $1.2\cdot 10^6\ {\rm cm^{-3}}$. In e) the column density/linewidth ratios are $1.9\cdot 10^{11}$, $5.3\cdot 10^{11}$, $8.8\cdot 10^{11}$, $1.4\cdot 10^{12}$ and $3.9\cdot 10^{12}\ {\rm cm^{-2} km^{-1}s}$. In figures a) to c) the lowest contour coincides with the map boundary} %\label{fig4} \end{figure} \subsection{The hot cores} The highest temperatures in Sgr B2 are found towards the continuum sources Sgr B2M and Sgr B2N. Our complete set of data in the J=5--4, J=6--5, J=8--7 and J=12--11 transitions of ${\rm CH_3CN}$ allows a determination of the thermal structure on a scale of 1 pc (corresponding to $28''$ at 7.5 kpc) in these regions. As seen in Fig. 5, four hot cores were detected. Two of these are extended, with deconvolved FWHP sizes of $15''\times 10''$ and $20''\times 20''$. Both have a ${\rm T_k}$ of 300 K, and are associated with Sgr B2M and Sgr B2N respectively. The other hot cores, with kinetic temperatures of 200 K and angular sizes of $\sim 7''\times 7''$, are newly found hot spots. We have labelled these as ${\rm CH_3CN(NW)}$ and ${\rm CH_3CN(NS)}$, as an indication of their positions relative to Sgr B2N ($20''$ to the west and $25''$ south of Sgr B2N respectively). These two hot cores have different positions than the clumps HNO(NW) and HNO(E) discovered by Kuan \& Snyder (1994), and should not be confused with them. The line profiles towards Sgr B2N and Sgr B2M are very complex because more than one region is seen along the line of sight. Towards Sgr B2M, the J=5--4 and J=6--5 transitions peak at 56 ${\rm km\, s^{-1}}$, while the J=8--7 and J=12--11 peak at 64 ${\rm km\, s^{-1}}$. The 64 ${\rm km\, s^{-1}}$ feature towards Sgr B2M probably arises from denser and hotter gas than that at 56 ${\rm km\, s^{-1}}$. Towards Sgr B2N the J=5--4, J=6--5 and J=8--7 transitions peak at 64 ${\rm km\, s^{-1}}$, while the J=12--11 peaks at 68 ${\rm km\, s^{-1}}$. In this source the gas at 68 ${\rm km\, s^{-1}}$ is denser than that at 64 ${\rm km\, s^{-1}}$. This behaviour clearly shows that several components with different physical conditions are present even in the hot cores themselves. Though it is very likely that the material close to these sources shows density and temperature gradients, we have used, as a first approximation, a two component model, with different physical conditions and radial velocities to fit the ${\rm CH_3CN}$ profiles. Two different cases have been considered: a) the J=12--11 transition was smoothed to the angular resolution of the J=8--7 line and b) the J=12--11, J=8--7 and J=6--5 lines were smoothed to the angular resolution of the J=5--4 line. The kinetic temperatures and densities are different for both cases and this is an indication that there is a kinetic temperature and hydrogen density gradient in the hot cores. In Sgr B2M the densest (${\rm n(H_2)}>10^6\ {\rm cm^{-1}}$) and hottest (${\rm T_k} > 300$ K) component produces the emission observed in the J=12--11 and J=8--7 lines at 64 ${\rm km\, s^{-1}}$ (Fig. 6). The J=5--4 and J=6--5 lines at 54 ${\rm km\, s^{-1}}$ arise from colder (${\rm T_k}=100$ K) and lower density gas (${\rm n(H_2)}\sim 10^5\ {\rm cm^{-1}}$). The spectra towards Sgr B2N show that most lines (K=0, K=1, K=2 and K=3) have large opacities ($>3$) making the analysis more uncertain. However the two velocity component model used previously for Sgr B2M with higher kinetic temperature (400 K) and hydrogen density ($10^7\ {\rm cm^{-3}}$) is consistent with all of the emission observed in J=12--11 profile and some of the J=8--7 and the ${\rm CH_3^{13}CN}$ J=5--4 and J=6--5 profiles. The values of ${\rm T_K}$ we obtain are a factor 2 higher than those obtained by Vogel et al. (1987) using ${\rm NH_3}$ towards the hot cores Sgr B2M and Sgr B2N while the densities are similar for Sgr B2N and one order of magnitude lower for Sgr B2M. Similar differences are also found for the hot core in Orion A where Loren \& Mundy (1984) derived a kinetic temperature of 275 K using ${\rm CH_3CN}$ while Hermsen et al. (1988) obtained a temperature of 165 K using several transitions of ${\rm NH_3}$. Further observations and models are needed to establish if ${\rm CH_3CN}$ comes from the hottest gas in the cores. \begin{figure}[htbp] \picplace{0.5 cm} \caption{The kinetic temperature and column density distribution for the nearby gas to the hot cores. In the left panel: the contours represent kinetic temperature values of 120, 200 and 300 K. The right panel: contours represent column density/linewidth ratios of $1.5\cdot 10^{12}$, $3.9\cdot 10^{12}$, $8.0\cdot 10^{12}$ and $1.1\cdot 10^{13}\ {\rm cm^{-2}km^{-1}s}$. The thick line on the left panel and the white line on the right panel represent the 5 mJy/beam contour of the continuum emission intensity at 3.6 cm from Mehringer et al. (1993).} %\label{fig5} \end{figure} \begin{table*} \caption{Physical properties derived for the hot cores using a 3 component model. We used continuum main beam temperatures of 3.1 and 1.9 K at 3.2 mm towards Sgr B2M and Sgr B2N respectively and a linewidth of 15 ${\rm kms^{-1}}$. a) the ${\rm CH_3CN}$ 12--11 transition was smoothed to the 8--7 angular resolution, b) the 12--11, 8--7 and 6--5 transitions were smoothed to the 5--4 angular resolution} %\label{table2} \begin{flushleft} \begin{tabular}{lllllllll} \hline \multicolumn{1}{l}{} & \multicolumn{4}{c}{a)} & \multicolumn{4}{c}{b)} \\ \cline{2-9} \\ Source & $T_k$ & ${\rm N_{CH_3CN}}$ & ${\rm n(H_2)}$ & $V_{lsr}$ & $T_k$ & ${\rm N_{CH_3CN}}$ & ${\rm n(H_2)}$ & $V_{lsr}$ \\ & (K) & ${\rm (cm^{-2})}$ & ${\rm cm^{-3}}$ & ${\rm km\, s^{-1}}$ & (K) & ${\rm (cm^{-2})}$ & ${\rm cm^{-3}}$ & ${\rm km\, s^{-1}}$ \\ \hline Sgr B2M & 300 &$5\cdot 10^{13}$ & $\leq 10^3$ & 64 & 300 &$5\cdot 10^{13}$ & $\leq 10^3$ & 64 \\ Sgr B2M & 100 &$3\cdot 10^{13}$ & $2\cdot 10^5$ & 54 & 100 &$2.5\cdot 10^{13}$ & $2\cdot 10^5$ & 54 \\ Sgr B2M & 300 &$1.7\cdot 10^{14}$ & $3\cdot 10^6$ & 64 & 200 &$1.2\cdot 10^{14}$ & $1.5\cdot 10^6$ & 64 \\ Sgr B2N & 300 &$5\cdot 10^{13}$ & $\leq 10^3$ & 64 & 300 &$5\cdot 10^{13}$ & $\leq 10^3$ & 64 \\ Sgr B2N & 400 &$4\cdot 10^{14}$ & $5\cdot 10^5$ & 64 & 400 &$3.1\cdot 10^{14}$ & $ 10^5$ & 64 \\ Sgr B2N & 400 &$10^{14}$ & $ 10^7$ & 68 & 400 &$1.6\cdot 10^{14}$ & $ 10^7$ & 68 \\ ${\rm CH_3CN(NS)}$ & 200 & $10^{14}$ & $10^6$ & 64 & & & & \\ ${\rm CH_3CN(NW)}$ & 200 & $1.4\cdot 10^{14}$ & $7\cdot 10^5$ & 68 & & & & \\ \hline \end{tabular} \end{flushleft} \end{table*} \begin{figure}[htbp] \picplace{0.5 cm} \caption{${\rm CH_3CN}$ J=12--11, J=8--7, J=6--5 and J=5--4 spectra taken towards Sgr B2M. The continous line represents the profile predicted by a 3 layer model. The physical conditions for the hot cores are given in Table 2. The 12--11 spectrum is smoothed to the angular resolution used to take the 8--7 data.} %\label{fig6} \end{figure} \subsection{The warm envelope} Our results show a warm envelope ($7\times 14$ pc) around the star forming region. Apart from the hot ring which will be discussed in the next section, the kinetic temperature for $r>$ 1 pc is rather uniform, between 40-60 K, independently of the radius. This envelope is relatively dense with densities of $2\cdot 10^5 \ {\rm cm^{-3}}$. The density shows, however, a systematic change with radius, ${\rm n(H_2)}= 2.96\cdot 10^5\, {\rm cm^{-3}} \, (r/{\rm pc}) ^{-0.87}$ up to a distance of 8 pc, where $r$ is the distance to Sgr B2M. This dependence was obtained by averaging densities from 8 radial cuts and using a least square mean analysis to the averaged curve. The ${\rm n(H_2)}$ dependence on the radius we obtain is different from that derived by Lis \& Goldsmith (1989). However Lis \& Goldsmith (1989) result was obtained from the integrated intensity of ${\rm C^{18}O}$ 1--0. Thus, this must be an average for all three molecular clouds, while our relation has been obtained only for the 55-66 ${\rm km\, s^{-1}}$ cloud. \subsection{The diffuse and hot envelope} In addition to the dense and warm envelope the ${\rm CH_3CN}$ absorption lines indicate the presence of another component with lower density. We have estimated the physical conditions of this component by considering the presence of a third layer of molecular gas surrounding the hot cores. However there are two possible scenarios: a) all three velocity components absorb the continuum of Sgr B2M b) only the layer of lower density gas absorbs the continuum radiation while the other two, of higher density, only contribute to the emission (see H\"uttemeister et al. 1993). For both cases our model generates similar profiles for the ${\rm CH_3CN}$ lines. For this analysis we considered a continuum source with a size of $13''\times 13''$ (3.2 mm) and $13''\times 7''$ (1.2 mm) and a main beam temperature of 3.1 K (3.2 mm) and 4.4 K (1.2 mm) (Mart\'\i n-Pintado et al. 1990). Table 2 contains the results of the fits. Our ${\rm CH_3CN}$ profiles can only be explained by a diffuse and hot envelope with a density of $\sim 10^3\ {\rm cm^{-3}}$ and a kinetic temperature of 200-300 K. If only this layer were in front of the continuum sources all the K components of the J=5--4 and J=6--5 lines would be observed in absorption. However, the denser hot cores cause that all K components except the 5(4)-4(4) and 6(5)-5(5) are seen in emission. Larger kinetic temperatures of the hot and diffuse envelope could also explain the ${\rm CH_3CN}$ absorption. Although there is no absorption observed towards Sgr B2N we have also performed fits to the observed profiles using both, a two component and a three component model. In the latter case we considered the same physical conditions for the hot and low density envelope as those determined for Sgr B2M. The results for a three component model are given in Table 2 and show that the data are consistent with both, the diffuse and hot density envelope and the warm envelope. This diffuse and very hot envelope probably comes from low density material located in the outer parts of the molecular cloud, and might be heated by additional mechanisms, as for example shocks, or UV radiation. In summary, we find a warm (40-80 K) dense ($2\cdot 10^5\ {\rm cm^{-3}}$) envelope surrounding the active star forming region in Sgr B2 and probably surrounded by hotter (300 K) and more diffuse ($10^3\ {\rm cm^{-3}}$) molecular material. \subsection{The hot ring} Fig. 4 shows that the kinetic temperature decreases abruptly from 300-400 K at the positions of Sgr B2M and Sgr B2N to 60-80 K at a distance of 1.3 pc ($40''$), in all directions. At a distance of $\sim$ 2 pc from the central sources, the kinetic temperature increases again up to 100-120 K in a relatively thin region approximately 1.4 pc wide and decreases again at larger distances to 40-60 K. In ${\rm T_K}$ there is a ring like structure with a radius of 2 pc and a width of 1.4 pc. In the following we will refer to this remarkable feature as the hot ring. The presence of such a hot ring with ${\rm T_K}$ values of 100-120 K, surrounded by material with kinetic temperature of approximately 60 K, is clearly illustrated by the spectra in Fig. 7; Here we show four spectra taken towards the hot ring and four outside it. Superimposed on the observed lines we show the expected line profiles for two kinetic temperatures of 80 and 100-130 K. It is clear that the spectra from the hot ring cannot be explained by lower kinetic temperatures, because the observed intensities of the high K transitions are larger than would be predicted by a model with ${\rm T_k}=$80 K. We stress that the model also fits the ${\rm CH_3^{13}CN}$ intensity and thus takes opacity effects into account. The opposite occurs for the spectra taken outside the hot ring. In this case, the observed intensities for the high K transitions are smaller than would be predicted by a model that uses a kinetic temperature of 120 K. Therefore, we conclude that the bulk of the molecular gas at 55-65 ${\rm km\, s^{-1}}$ shows a hot ring with a radius of 2 pc, a thickness of 1.4 pc and a kinetic temperature of 100-120 K around Sgr B2M and Sgr B2N. \begin{figure}[htbp] \picplace{0.5 cm} \caption{Spectra from transitions ${\rm CH_3CN}$ J=5--4 and J=8--7 on the positions marked in Fig. 4. A, B, C and D were taken towards the hot ring. E, F, G and H towards the warm envelope. The thick line represents the profile predicted by the model we considered correct. The thin line is the profile by a model using a lower or higher kinetic temperature. The kinetic temperature used for each case is specified in each box.} %\label{fig7} \end{figure} \section{Masses} {}From the sizes and ${\rm H_2}$ densities obtained from ${\rm CH_3CN}$, we have estimated the masses of the three molecular clouds, assuming that the size along the line of sight is equal to that in the plane of the sky. The results are summarized in Tables 3 and 4. The total mass for the hot cores plus the Sgr B2 Ridge ($14\times7$ pc), taking into account the three different clouds, is $3\cdot 10^6\ M_\odot$. The 55-66 ${\rm kms^{-1}}$ cloud has the largest contribution ($70\%$) to the total mass. The previous total mass estimate does not include the mass of the lower density hot gas seen in absorption towards Sgr B2M. If we include the mass of the diffuse envelope with a mean density of $10^3\ {\rm cm^{-3}}$ (estimated in section 5.4) and a diameter of 27.5 pc, as proposed by Lis \& Goldsmith (1989), the total estimated mass for Sgr B2 is $7\cdot 10^6\ M_\odot$ in excellent agreement with previous estimates from ${\rm ^{13}CO}$ data (Lis \& Goldsmith, 1989) It is therefore very likely that a large fraction of the molecular gas surrounding the Sgr B2 ridge is hot and diffuse. This diffuse envelope would then contribute to more than half of the mass of the Sgr B2 molecular cloud. Our mass estimates for the dense envelope are, however, 5 times larger than those derived for the 1.3 mm dust emission by Gordon et al. (1993). This discrepancy is probably due to the extended dust emission, which is not recorded because beam switched observations were used to take the dust data. From our maps (see Fig. 4) the hydrogen density is still high ($>10^4\ {\rm cm^{-3}}$) at distances larger than the full extent of the dust maps of Gordon et al. (1993). \begin{table*} \caption{${\rm H_2}$ densities and masses for the components of the Sgr B2 Ridge.} %\label{table3} \begin{flushleft} \begin{tabular}{rrr} \hline Gas (${\rm V_{lsr}}$) & ${\rm n(H_2)}$ & Mass \\ (${\rm km\, s^{-1}}$) & ${\rm cm^{-3}}$ & ($M_\odot$) \\ \hline 44-54 & $\sim 10^5$ & $4.8\cdot 10^5$ \\ \hline 55-66 & $10^5\ -\ 4\cdot 10^5$ & $2.0\cdot 10^6$ \\ \hline 66-78 & $\sim 10^5$ & $4.9\cdot 10^5$ \\ \hline \end{tabular} \end{flushleft} \end{table*} \begin{table*} \caption{Sizes, ${\rm H_2}$ densities and masses for the Sgr B2 hot cores. The sizes come from the kinetic temperature maps once deconvolved with a beam of $17''$. The assumed distance to Sgr B2 is 7.1 Kpc. The density was estimated from the ${\rm CH_3CN}$ analysis.} %\label{table4} \begin{flushleft} \begin{tabular}{lrrr} \hline Cores & Radius & ${\rm n(H_2)}$ & Mass \\ & (pc) & ${\rm cm^{-3}}$ & ($M_\odot$) \\ \hline Sgr B2M & 0.27 & $3\cdot 10^6$ & $1.3\cdot 10^4$ \\ ${\rm CH_3CN(NS)}$ & 0.17 & $10^6$ & $1.1\cdot 10^3$ \\ ${\rm CH_3CN(NW)}$ & 0.17 & $10^6$ & $1.1\cdot 10^3$ \\ Sgr B2N & 0.34 & $3\cdot 10^6$ & $2.5\cdot 10^4$ \\ \hline \end{tabular} \end{flushleft} \end{table*} The sizes of the hot cores in Table 4 have been obtained by deconvolving the central hot regions with a $17''$ beam. The sizes and masses for the cores Sgr B2M and Sgr B2N are larger than those obtained for the millimeter dust emission by Mart\'\i n-Pintado et al. (1990) and Carlstrom \& Vogel (1989) or from ${\rm NH_3}$ observations by Vogel at al. (1987). Our mass estimates are similar to those derived by Lis \& Goldsmith (1990) from observations in the continuum. The difference between the masses from Vogel et al. (1987), Carlstrom \& Vogel (1989) and Mart\'\i n -Pintado et al. (1990) and ours is due to the different size and hydrogen density estimates for the hot cores. The ${\rm CH_3CN}$ emission also traces hot gas in both the very high density cores and from more diffuse molecular gas surrounding the cores. \section{${\rm CH_3CN}$ Abundances} Our estimates for the abundance of ${\rm CH_3CN}$ in the hot cores is $\sim 7\cdot 10^{-11}$, while in the warm envelope is $\sim 2-5\cdot 10^{-11}$. These values have been obtained from the ${\rm CH_3CN}$ column density, the hydrogen density and assuming that the depth of the cloud along the line of sight is similar to that obtained in the plane of the sky. The fractional abundances of ${\rm CH_3CN}$ are similar to those found by Cummins et al. (1983) with much lower angular resolution towards Sgr B2. Loren and Mundy (1984) also derived similar values for the hot cores and the ridge in Orion, although they reported that ${\rm CH_3CN}$ is enhanced a factor 10 in the hot cores. Such a large enhancement of ${\rm CH_3CN}$ abundances is not seen from our data towards the hot cores in Sgr B2. \section{Morphology of the ${\rm CH_3CN}$ emission} The different positions of the emission maxima for the different rotational transitions of ${\rm CH_3CN}$ can be explained by the combination of the spatial distribution of the hydrogen density, column density of ${\rm CH_3CN}$, and the overlap of two molecular clouds, with distinct radial velocities and physical conditions, along the line of sight. The absorption seen in the ${\rm CH_3CN}$ J=5--4 transition does not have an important influence in the spatial shift of the maxima since the intensity of the J=5--4, J=8--7 and J=12--11 lines only decreases by 20\% due to the hot diffuse envelope in front of the continuum source. In the lower panels in Fig. 8, we show the derived physical conditions along a right ascension cut at the Sgr B2M declination. The hydrogen density, the ${\rm CH_3CN}$ column density and the kinetic temperature have a maximum towards Sgr B2M, decrease abruptly towards the east but smoothly towards the west. Under these conditions the intensity maxima for the ${\rm CH_3CN}$ lines should peak at different positions for the different rotational lines. The higher J lines will peak where the column density and hydrogen density are highest. The second panel of Fig. 8 shows the expected line intensity for the three lines along the cut, while the upper panel shows the observed ${\rm T_a^*}$. One also must consider that the integrated intensity maps (Fig. 2) include the emission from other molecular clouds with different radial velocities. The lower density gas at 67-78 ${\rm km s^{-1}}$ in the northwest of Sgr B2M excites the ${\rm CH_3CN}$ J=5--4 transitions but not the ${\rm CH_3CN}$ J=8--7 and J=12--11 lines. This makes the integrated intensity emission to shift to the northwest of Sgr B2M. In Fig 8. we also find a high hydrogen density at the interface between the hot cores and the ambient cloud towards positive right ascentions. This should be further investigated with higher resolution. \begin{figure}[htbp] \picplace{0.5 cm} \caption{Physical conditions, and predicted and observed line intensities along a right ascension strip at the declination of Sgr B2M for the ${\rm V_{lsr}}=$55--65 ${\rm km\, s^{-1}}$ component. Panel a): observed ${\rm T_a^*}$ of the ${\rm CH_3CN}$ transitions J=12--11, J=8--7 and J=5--4. Panel b): Predicted ${\rm T_a^*}$ for the ${\rm CH_3CN}$ transitions J=12--11, J=8--7 and J=5--4 K=0,1 as a function of the offset in right ascension along a constant declination strip. Panels c), d) and e): values of ${\rm T_K}$, ${\rm CH_3CN}$ column density and n(${\rm H_2}$), along the strip.} %\label{fig8} \end{figure} \section{The kinetic temperature distribution of the Sgr B2 molecular cloud} The kinetic temperature structure of the main molecular clouds in Sgr B2 can be characterized by four components: the hot cores, the warm and dense envelope, and the hot ring. The first 3 components can be easily recognized in Fig. 9, where we show the averaged kinetic temperature as a function of the distance to Sgr B2M. As discussed in section 5.3 our data also show the presence of an additional component, a hot and diffuse envelope. In order to explain the overall thermal distribution of the molecular clouds in Sgr B2 we have constructed a simple model which determines ${\rm T_K}$ by balancing heating and cooling mechanisms as a function of distance from the star forming region. The main heating mechanisms in the interstellar medium have been reviewed recently by Black (1987) and Genzel (1991). The photoelectric effect, the photoionization of C atoms and the photodissociation and deexcitation of molecular hydrogen are the most relevant agents in Photo Dissociation Regions. However these processes are generated by the ultraviolet radiation from OB stars which are attenuated at visual extinctions of less than 10 magnitudes (i.e. 0.02 pc at densities of $10^5\ {\rm cm^{-3}}$) and therefore should have little importance in dense molecular cloud cores. As main heating mechanisms to explain the thermal distribution we have considered in our model the heating of the dust by the radiation from the stars, gas dust collisions and the dissipation of turbulent motions. Heating by cosmic rays and Alfven wave dissipation may also be important, so we included them in our analysis. \begin{figure}[htbp] \picplace{0.5 cm} \caption{Kinetic temperature as a function of distance to Sgr B2M as derived from ${\rm CH_3CN}$ and predicted by models. The thick solid line represents the averaged kinetic gas temperature obtained from our ${\rm CH_3CN}$ maps. In all cases ${\rm T_K}$ was obtained by considering gas-dust coupling as the main heating mechanism. The last case also includes the dissipation of turbulent motions. The luminosity of the infrarred sources is $10^7\ {\rm L_\odot}$. For Scoville \& Kwan model see Scoville \& Kwan (1976)} %\label{fig9} \end{figure} \subsection{The hot cores} The high kinetic temperature of the gas in dense cores is mainly due to the presence of newly formed stars. As previously mentioned heating agents associated with PDRs can be ignored because these are not effective for visual extinctions higher than 10 magnitudes, and the column density near the hot cores causes much larger extinctions. Most of the gas is heated by collisions with dust grains, which have been heated by the absorption of stellar radiation. The amount of energy transferred by the dust grains to the gas through collisions considered in our model is given by (Lis \& Goldsmith 1991), \begin{eqnarray} \Gamma_{gr} = 2.1\cdot 10^{-33} \ {\rm n(H_2)} \, T_k^{1/2} (T_{\rm gr} - T_k) \ {\rm erg \ s^{-1}\ cm^{-3}} \end{eqnarray} where $T_{gr}$ is the grain temperature and $T_k$ the kinetic temperature of the gas. The grain temperature which decreases with the distance to the star depends on the luminosity of the heating sources. The radial distribution of dust temperatures has been obtained by two different methods: \begin{enumerate} \item we use the Scoville and Kwan (1976) approximation which applies for the low opacity limit, \begin{eqnarray} T_{\rm gr} = 49 \, Q^{-1/5}_{50} \left( {2\cdot 10^{17} \ {\rm cm}} \over r \right)^{2/5} \left( L_\star \over {10^5\, L_\odot} \right) ^{1/5} \end{eqnarray} where $Q_{50}$ is the grain emissivity at 50 $\mu$m and $L_\star$ the source luminosity in $L_\odot$. This approximation was obtained assuming that the emissivity depends on the frequency as $Q\sim \nu$. \item we solve the radiation transfer equation numerically, for a spherical cloud with a 3 pc radius and two density regimes. Within the internal 0.07 pc radius we used a constant density of $3\cdot 10^6 \ {\rm cm^{-3}}$ and up to a distance of 3 pc the density law derived in section 5. Dust opacities were obtained from the Hildebrand relation (1983), assuming a mass density of 3.3 gr ${\rm cm^{-3}}$ and a radius of 0.1 $\mu$m for the dust grains. We used 1.1 for the spectral index for the dust emissivity law (Mart\'\i n-Pintado et al. 1990) with an emissivity value at 1300 $\mu$m of $3 \cdot 10^{-5}$ (Righini-Cohen \& Simon 1977; Goldsmith et al. 1987a). The radial dust distribution was evaluated for three different bolometric luminosities of the central source, $5\cdot 10^6$, $7.5\cdot 10^6$ and $10^7\ L_\odot$. The dust temperature distribution derived from our model is in very good agreement with that obtained by Lis \& Goldsmith (1990) for radii between $10^{-2}$ pc and 10 pc. \end{enumerate} The cooling has been estimated using the analytical expression given by Lis and Goldsmith (1990) that takes into account several molecular species (Goldsmith \& Langer 1978). Fig. 9 shows the kinetic temperature distribution predicted when heating is by gas-dust collisions. We used one and two infrared sources, and two methods to estimate dust temperatures. In order to compare the results from these models with our data, we have convolved our results of the calculations with a $17''$ beam. The closest distance to the star forming region considered for the convolution was 0.001 pc. The convolved kinetic temperature near the hot cores is strongly dependent on this distance. Our model for one heating source predicts a kinetic temperature for the hot cores which is substantially lower than that derived from ${\rm CH_3CN}$ for all positions even with the highest luminosity for the central source. However when considering the heating by a second IR source (Sgr B2N) at a distance of 0.1 pc from Sgr B2M, the predicted kinetic temperatures agree quite well with the observed kinetic temperature distribution within a region of 0.5 pc around Sgr B2M. Thus, the spatial distribution of the molecular gas close to the Sgr B2M hot core ($\leq 0.5$ pc) can be explained by the heating of dust by radiation from the young stars in the star forming regions, Sgr B2M and Sgr B2N. For Sgr B2N there is a discrepancy between the size of the observed ${\rm CH_3CN}$ hot core (0.34 pc) and the size of the region at 300 K derived from the previous model and the assumed luminosities. This discrepancy can be due to the fact that Sgr B2N is a very complex region with multiple HII regions which could heat the gas at larger distances than expected for a point-like source as considered in our model. Interferometric mesurments of J=12--11 ${\rm CH_3CN}$ are needed to confirm the large extent of the hot material in Sgr B2N. \subsection{The warm envelope} Fig. 9 shows that for distances greater than 1 pc the dust grain temperatures ($\sim 15-20$ K) agree with those obtained by Lis \& Goldsmith (1990) and Gordon et al. (1993) but are systematically lower than the predicted kinetic gas temperature observed from ${\rm CH_3CN}$, $\sim 40-60$ K. Based on absorption lines Wilson et al. (1982) and Huttemeister et al. (1995) showed that the dust is cooler than the gas and a heating mechanism acting only on the gas is required. The Scoville \& Kwan solution, which is valid in the low opacity limit, approximately fits the data in the 0.5 to 1.5 pc. This may indicate a clumpy structure which allows a deeper penetration of radiation that heats the dust farther inside the cloud. Heating by cosmic rays and the dissipation of Alfven waves will not significantly affect the total heating rate for the typical densities we have obtained and the expected cosmic ray flux in the Sgr B2 molecular cloud. The cosmic rays flux would have to be 3 orders of magnitude higher than that of the solar system ($10^{-17}\ {\rm s^{-1}}$) in order to account for the heating in the molecular cloud. This is unrealistic since then a large fraction of molecules would be dissociated. It has been proposed that turbulent heating could account for the high kinetic temperature in the Galactic Center (Wilson et al., 1982). Heating by turbulent motions can be estimated according to Black (1987) by, \begin{eqnarray} \Gamma_{\rm turb} = 3.5\cdot 10^{-28} \, v_t^3 \, F \, {\rm n(H_2)} \left( {1\ {\rm pc}} \over R_c \right) \ {\rm erg \ s^{-1}\ cm^{-3}} \end{eqnarray} where $v_t$ is the turbulent velocity. We have assumed that $v_t$ ranged from 17 to 13 ${\rm km s^{-1}}$ from the nearest high density regions to the farthest lower density regions mapped. $R_c$ was estimated as 8 pc. Since the turbulence may extend farther than mapped, we used a scale factor $F$ to adjust the observed kinetic temperatures to the model prediction. The best value was $F=1.5$ which corresponds to a region of a radius of $\sim 12$ pc. The dissipation of turbulent motions can explain the observed kinetic temperature distribution of the gas in the warm envelope (Fig. 9) but does not heat the dust. Under these conditions dust-gas collisions act as a cooling agent. The large turbulent state in the Galactic Center is probably due to the velocity shear caused by the differential rotation, which is stronger at the Galactic Center molecular clouds (Wilson et al. 1982). \subsection{The diffuse and very hot envelope} In addition to the warm envelope, which requires ``moderate'' energy input to be heated, there is a very hot (500 K) and low density ($\sim 10^3\ {\rm cm^{-3}}$) component, seen from the absorption of some ${\rm CH_3CN}$ lines. Though the kinetic temperature of the hot component is similar to that derived by Flower et al. (1995) our derived densities for this component are one order of magnitude smaller than those inferred by Flower et al. (1995). Based on the ${\rm NH_3}$ absorption data of H\"uttemeister et al. (1995) these authors proposed that the envelope of Sgr B2 is filled by a very hot (500 K) and relatively dense ($10^4\ {\rm cm^{-3}}$) material. We have searched for this extended component in a position offset from the main continuum sources. Fig. 10 shows, superimposed on ${\rm CH_3CN}$ J=5-4 and J=8-7 spectra taken at a position $100''$ west from Sgr B2M, the profiles predicted for these ${\rm CH_3CN}$ lines for ${\rm T_K=}$500 K, ${\rm N(CH_3CN)}=5\cdot 10^{13}\ {\rm cm^{-2}}$ and two hydrogen density regimes, ${\rm n(H_2)}=10^3\ {\rm cm^{-2}}$ and ${\rm n(H_2)}=10^4\ {\rm cm^{-2}}$. From the fits we conclude that the ${\rm CH_3CN}$ data do not support a very hot (500 K) and relatively dense ($10^4\ {\rm cm^{-2}}$) envelope around Sgr B2. The ${\rm CH_3CN}$ data is in better agreement with a two component model in which the star forming region cores are surrounded by a dense ($10^5\ {\rm cm^{-3}}$) and warm (80 K) envelope and a more diffuse ($10^3\ {\rm cm^{-3}}$) and hotter component. The column density of ${\rm CH_3CN}$ in the diffuse and hot envelope is $\sim 5\cdot 10^{13} \ {\rm cm^{-3}}$. When compared with the column density of other molecules like ${\rm NH_3}$ or ${\rm HC_3N}$ we obtain that ${\rm N(CH_3CN)/N(HC_3N)}\simeq 2$ and ${\rm N(CH_3CN)/N(NH_3)}\simeq 10^{-3}$. We used the ${\rm N(HC_3N)}$ value obtained by Huttemeister et al. (1995). In spite of the different physical conditions the previous abundance ratios are similar to those found in the hot core in Orion (Irvine, 1987), though the origin of the diffuse and hot envelope in Sgr B2 is unknown. A similar chemistry between these two regions might support the idea that, like for the hot core, the chemistry of the envelope is strongly affected by the evaporation of grain mantles. These data support the suggestion by Flower et al. (1995) that C shocks might heat and evaporate grain mantles in the diffuse and hot envelope. \begin{figure}[htbp] \picplace{0.5 cm} \caption{Profiles of the ${\rm CH_3CN}$ J=5-4, J=8-7 lines towards the position (-100,0) from Sgr B2M. The thin and thick continuous lines superimposed on the observed lines represent the predicted profile for physical conditions ${\rm T_K}$=500 K, N(${\rm CH_3CN}$)=$5\cdot 10^{13}\ {\rm cm^{-2}}$ and ${\rm n(H_2)}=10^3$ and $10^4\ {\rm cm^{-3}}$ respectively. The grey line represents the line profile for a two component model: first component: ${\rm T_K}$=80 K, N(${\rm CH_3CN}$)=$1.8\cdot 10^{13}\ {\rm cm^{-2}}$, ${\rm n(H_2)}=4.7\cdot 10^4\ {\rm cm^{-3}}$, second component: ${\rm T_K}$=400 K, N(${\rm CH_3CN}$)= $5\cdot 10^{13}\ {\rm cm^{-2}}$, ${\rm n(H_2)}= 10^3\ {\rm cm^{-3}}$.} %\label{fig10} \end{figure} \subsection{The hot ring} The most remarkable feature of the kinetic temperature maps, the hot ring is clearly seen in the plots of averaged kinetic temperature as a function of distance to Sgr B2M (Fig. 9). As shown in Fig. 9 none of the above mechanisms previously discussed explains this structure. The origin of the hot ring is unclear. In order to gain insight, we have compared the morphology of the hot ring with the morphology of other types of emission. We find that the morphology of the hot ring resembles the morphology of the ``hole'' of $^{13}{\rm CO}$ emission at radial velocities of 40-50 ${\rm km\, s^{-1}}$ reported by Hasegawa et al. (1994). From the spatial distribution of $^{13}{\rm CO}$ at 20-40 ${\rm km\, s^{-1}}$, 40-50 ${\rm km\, s^{-1}}$ and 70-80 ${\rm km\, s^{-1}}$, the authors proposed that cloud-cloud collisions are responsible for this morphology, and that during the course of the collision, dense and massive cores may have formed in the interface between the colliding clouds. However the hot ring appears at different radial velocities than the ${\rm ^{13}CO}$ ``hole'', and does not match exactly with it; the HII regions are at the edge of the ``hole'' but in the center of the hot ring. We also do not observe any density enhancement associated to the ${\rm ^{13}CO}$ morphology. Therefore it is not clear that the hot ring is associated with the $^{13}{\rm CO}$ ``hole'' reported by Hasegawa et al (1994). Fig. 11 shows the comparison between the kinetic temperature (grey contours) and the radio continuum emission at 43 GHz from Akabane et al. (1988), shown as a solid line, and the continuum at 50 $\mu$m from Goldsmith et al. (1992), shown as a white line. The radio continuum at 7 mm traces the ionized gas with a small contribution from dust and molecular lines (Mart\'\i n-Pintado et al. 1990) while the continuum at 50 $\mu$m samples mainly hot dust heated by the radiation from the OB stars. The hot ring surrounds both the hot dust and the free-free emission. This morphological relation between the hot ring and both tracers of activity from OB stars suggests that the hot ring may be related to the massive star formation activity in Sgr B2. Very likely the hot ring is a thin interface between the molecular cloud and the ionized ``bubble'' created by the young OB stars in the core of Sgr B2. In this picture two possibilities can be considered to explain the increase of ${\rm T_K}$ in the hot ring. \begin{enumerate} \item Photoelectric heating from the UV radiation of the OB stars. This heating mechanism would be more effective in thin regions with low visual extinctions ($A_v\leq 4$). Howe et al. (1991) have estimated that for a molecular hydrogen density of $3\cdot 10^4\ {\rm cm^{-3}}$ the ${\rm C^+}$ layer would be 0.03 pc thick. Therefore the UV radiation would not heat a region with more than 10 mag. of visual extinction. However clumpiness would allow the UV photons to penetrate deeper in the cloud. Howe et al. 1991 show that the UV depth of penetration probed by CII can be 10-100 times greater than for uniform gas with densities larger than $10^4\ {\rm cm^{-3}}$. In any case one, expects the kinetic temperature to decrease monotonically from the OB star forming region, which is not the case for the hot ring which appears separated from the Sgr B2M and Sgr B2N cores. \item Heating is produced by shock fronts associated to the expansion of the ionized bubble. This mechanism would better match the morphology of the hot ring. It has been proposed that the low density high temperature material of the hot envelope in Sgr B2 observed in absorption lines is due to the presence of C shocks with a shock velocity of 25 ${\rm kms^{-1}}$ (Flower et al. 1995). This model explains the absorption lines of ${\rm NH_3}$ towards Sgr B2, and the abundance ratios of ${\rm CH_3CN}$ found in the hot and diffuse envelope (see section 9.3). It is then very likely that the absorption lines in Sgr B2 might arise from the hot ring located in front of the HII region. High angular resolution observations of ${\rm NH_3}$ will help to elucidate the origin of this remarkable feature. \end{enumerate} \begin{figure}[htbp] \picplace{0.5 cm} \caption{In gray scale, ${\rm T_K}$ map. The white line represents a level $>430$ Jy in the 50 $\mu$m emission (Goldsmith et al. 1992), which we take as the boundary of the warm dust emission region. The black solid lines correspond to levels 0.2 and 0.4 Jy at 43 GHz; the maximum intensity is 0.8 Jy/beam (Akabane et al. 1988).} %\label{fig11} \end{figure} \section{Conclusions} We have used the 30m telescope to map the Sgr B2 molecular cloud in three transitions (J=5--4, 8--7, 12--11) of ${\rm CH_3CN}$ and ${\rm CH_3^{13}CN}$ and the J=11--10 ${\rm HC_3N}$ lines. In certain positions we have also observed the J=6--5 lines of ${\rm CH_3CN}$. All these data have been combined to produce the first large scale map of the kinetic temperature distribution in the Sgr B2 molecular cloud. The main results derived from this work are the following: \begin{enumerate} \item The molecular emission in ${\rm HC_3N}$ and ${\rm CH_3CN}$ in Sgr B2 comes from an elongated ridge of $\sim 4'\times 7'$ (7 $\times$ 14 pc) which contains the main three star forming regions, Sgr B2M, Sgr B2N and Sgr B2S. Basically the ridge consists of four molecular clouds with radial velocities of 44-54, 55-66, 67-78 and 90-120 ${\rm km\, s^{-1}}$ as observed from the ${\rm HC_3N}$ J=11--10 line, which overlap partially along the line of sight. The bulk of the molecular gas at 55-66 ${\rm km\, s^{-1}}$ extends throughout the ridge, while the low velocity gas (44-54 ${\rm km\, s^{-1}}$) and the high velocity gas (67-78 ${\rm km\, s^{-1}}$) appear at different locations in the ridge. \item In addition to the emission lines we have detected an absorption feature at 65 ${\rm km\, s^{-1}}$ towards the Sgr B2M continuum source in the 5(4)--4(4) and 6(5)--5(5) lines of ${\rm CH_3CN}$. \item To determine the physical properties of the Sgr B2 molecular cloud we have developed a LVG model which fits simultaneously the A and E species, several velocity components and the main isotope as well as the ${\rm ^{13}C}$ lines of ${\rm CH_3CN}$. From the multitransition analysis we have obtained the ${\rm H_2}$ density, the kinetic temperature and the ${\rm CH_3CN}$ column density maps of the molecular clouds in Sgr B2. \item The 44-54 and the 67-78 ${\rm kms{-1}}$ clouds, which contain $10^6\ {\rm M_\odot}$, are warm with typical kinetic temperatures of 60-80 K and ${\rm H_2}$ densities of $10^5\ {\rm cm^{-3}}$. \item The bulk of the mass ($2\cdot 10^6\ {\rm M_\odot}$) of the Sgr B2 molecular cloud is in the 55-66 ${\rm kms^{-1}}$ cloud. The kinetic temperature map of this cloud decreases with distance to Sgr B2M from 300 K to 40 K. {}From the kinetic temperature distribution we have identified three main features: the hot cores, the warm envelope and the hot ring. The absorption lines also show the presence of a very hot and diffuse component. \item Our ${\rm T_K}$ maps show two hot cores associated with Sgr B2M and Sgr B2N. In addition we have discovered another two hot cores near Sgr B2N; these have been labelled ${\rm CH_3CN(NS)}$ and ${\rm CH_3CN(NW)}$. The kinetic temperatures of the hot cores Sgr B2M and Sgr B2N are 300-400 K and the densities $3\cdot 10^6$ to $10^7\ {\rm cm^{-3}}$. For ${\rm CH_3CN(NS)}$ and ${\rm CH_3CN(NW)}$, ${\rm T_K=}$200 K and ${\rm n(H_2)}\sim 10^6\ {\rm cm^{-3}}$. \item Surrounding the hot cores there is a cloud of warm envelope (40-80 K) and dense ($2\cdot 10^5\ {\rm cm^{-3}}$) gas. The density decreases with distance from Sgr B2M as ${\rm n(H_2)}= 2.96\cdot 10^5\, {\rm cm^{-3}} \, (r/{\rm pc}) ^{-0.87}$ from 1 to 8 pc. \item The most remarkable feature found in the kinetic temperature map is the hot ring. This feature appears in the ${\rm T_K}$ map as a ring like structure of higher temperature (120 K versus 60-80 K) surrounding the star forming region Sgr B2M and Sgr B2N. The radius of the ring is $\sim$ 2 pc and its thickness is 1.4 pc. \item The very hot component contains gas at very high temperature (500 K) and low density ($10^3\ {\rm cm^{-3}}$). This component is seen towards the continuum source Sgr B2M from ${\rm CH_3CN}$ absorption lines. More observations are needed to determine the extent and physical properties of this envelope. \item The kinetic temperature distribution found in the Sgr B2 molecular cloud cannot be explained only by the heating from the OB stars. While the high kinetic temperature distribution found in the hot cores up to a distance of 1 pc can be accounted by heating through grain-gas collisions when the total luminosity of Sgr B2M plus Sgr B2N is considered, the high temperature (40-60 K) at larger distances requires a different heating agent. Turbulent heating given a turbulent velocity of 13 ${\rm kms^{-1}}$ can account for the observed temperatures in the warm envelope. However an additional heating mechanism is needed to explain the presence of the hot ring in Sgr B2. The hot ring, which surrounds the hot dust emission and the ionized gas, very likely represents the interface between the ionized gas and the molecular cloud. The increase of the temperature is probably related to the presence of shocks originated by the expansion of the ionized gas into the molecular cloud. \end{enumerate} \acknowledgements We would like to greet the IRAM staff, and operators for their help during observations. This work has been partially supported by the Spanish CICYT under grant number PB93-0048 \begin{thebibliography}{} \bibitem{} Akabane K., Sofue Y., Hirabayashi H., Morimoto M., Inoue M., 1988, PASJ 40, 459. \bibitem{} Benson J.M., Johnston K.J., 1984, ApJ 277, 181. \bibitem{} Black J.H., 1987, in: Interstellar Processes, 561-609, eds. D.J. Hollenbach and H.A. 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