3 Results

3.1 CO morphology and kinematics

Figures 1 and 2 show the integrated $^{12}{\rm CO}(J = 2\to 1)$ and ${\rm C}^{18}{\rm O}(J\,=\,1\to 0)$ line intensities. The overall morphology of both images is very similar to the $^{12}{\rm CO}(J\,=\,1\to 0)$ distribution published by Shen & Lo (1995) and the $^{13}{\rm CO}(J\,=\,1\to 0)$ distribution published by Neininger et al. (1998). It shows a triple peak morphology of which the two outer lobes have been interpreted as the edge of a central molecular toroid (Nakai et al. 1987; Shen & Lo 1995) and a weaker central peak located 65 pc west of the M82's center (2.2 m$\mu$m peak; Dietz et al. 1986). The two outer lobes have a projected separation of 410 pc (26''). The separation of the central and the western molecular lobe is only about 130 pc (8''). More diffuse CO emission is detected in the $^{12}{\rm CO}(J = 2\to 1)$ intensity distribution east and west of the CO peaks and in the south-west of the galaxy. The eastern part of the CO distribution is significantly warped to the north. The total extent of the emission region is about 1 kpc from east to west. With respect to M82's center the distribution of the molecular gas is clearly displaced to the west. South of the central and western CO peak two CO spurs are detected (see Fig. 1). They extend about 100 pc below the main molecular disk and join just below the expanding molecular superbubble which is located between the central and western CO peak (Neininger et al. 1998; Weiß et al. 1999). At the same location hot gas emerges into the halo of M82 (e.g. Shopbell & Bland-Hawthorn 1998; Bregman et al. 1995) supporting the idea that the CO spurs indicate the walls of the superbubble.

Note that the chain of CO emission south of the eastern end of the $^{12}{\rm CO}(J = 2\to 1)$ distribution is most likely not real but an artifact from the primary beam correction. The kinematic of the central 400 pc is dominated by solid body rotation. The rotation amplitude is about 200 kms^-1 ranging from 115 kms^-1 at the western peak up to 320 kms^-1 at the eastern peak. A pv-diagram along the major axis of M82 in the $^{12}{\rm CO}(J = 2\to 1)$ transition is shown in Fig. 3. (For the corresponding diagram in the ${\rm C}^{18}{\rm O}(J\,=\,1\to 0)$ data see Weiß et al. 1999). The pv-diagram is centered on the brightest supernova remnant SNR 41.9+58. The intense, velocity crowded regions at 20'', 5'' and -7'' offset correspond to the western, central and eastern CO peak. Between the central and western CO peaks two velocity components at 100 kms^-1 and 190 kms^-1 are detected. These features have been interpreted as an expanding superbubble. The velocity of the CO spurs is about 140 kms^-1 (see Figs. 4 and 5) which is similar to the centroid velocity of the expanding superbubble. Outside the central 400 pc the CO rotation curve flattens. The dynamical center derived from the $^{12}{\rm CO}(J = 2\to 1)$ and ${\rm C}^{18}{\rm O}(J\,=\,1\to 0)$ data agrees very well with the value of ${\rm v}_{\rm sys}=225\pm10~\ifmmode{{\rm\thinspace km\thinspace s}^{-1}}\else{\thinspace km\thinspace s$^{-1}$ }\fi$ published by Shen & Lo (1995) for the $^{12}{\rm CO}(J\,=\,1\to 0)$, and Neininger et al. (1998) for the $^{13}{\rm CO}(J\,=\,1\to 0)$ transition. The channel maps of the $^{12}{\rm CO}(J = 2\to 1)$ and ${\rm C}^{18}{\rm O}(J\,=\,1\to 0)$ line emission are presented in Figs. 4 and 5.

Figure 1: Integrated $^{12}{\rm CO}(J = 2\to 1)$ line intensity derived from the short-spacing corrected data cube. The contours correspond to 200, 400, 600, 800, 1200, 1600, 2000 and 2400 K kms-1. The cross indicates the center of M82 (2.2  m$\mu$m peak), the star represents the location of the brightest radio continuum point source (SNR 41.9+58) in M82. The dotted line shows the 50% sensitivity level of the primary beam mosaic

Figure 2: Integrated ${\rm C}^{18}{\rm O}(J\,=\,1\to 0)$ line intensity. The contours correspond to 2, 8, 14, 20, 26 and 32 K kms-1. The cross indicates the center of M82 (2.2  m$\mu$m peak), the star marks the location of the brightest radio continuum point source (SNR 41.9+58) in M82

3.2 Location of the starburst with respect to CO

Most tracers of star formation in M82 indicate that the highest star-forming activity is not associated with the molecular peaks, which presumably indicate the location of the reservoirs for the "fuel'' for star formation, but rather takes place between the peaks. The high-resolution 12.4 m$\mu$m image of the central region of M82 published by Telesco & Gezari (1992) suggests that the young stellar clusters, which heat the dust, are located between the western molecular lobe and the 2.2 m$\mu$m nucleus (western mid infrared (MIR) peaks), at the central CO peak, and between the central CO peak and the eastern CO lobe (eastern MIR peak). A similar morphology is visible in the Ne II line emission (Achtermann & Lacy 1995). The radio continuum point sources, which are believed to be supernova remnants (SNR) and compact H II regions, are spread across a much wider region and seem to avoid MIR and Ne II peaks (Kronberg et al. 1985). Only the strongest SNR in M82, SNR41.9+58, appears to be related to features at other wavelengths: it is located near the center of the expanding molecular superbubble, between the central and western CO peak, from which hot X-ray emitting gas is released into the halo of M82 (Weiß et al. 1999). At the same location recent radio continuum studies by Wills et al. (1999) identified a blow-out in the form of a cone of missing 5-GHz continuum emission. In the same study three other chimneys were identified within the central 300pc of M82. All these observations indicate that the regions of violent star formation are confined by the molecular lobes. Since no indications for high activity have been found at the 2.2 m$\mu$m nucleus itself, it seems that the starburst is arranged in a toroidal topology around the nucleus.

Figure 3: A pv-diagram along the major axis of M82 in the $^{12}{\rm CO}(J = 2\to 1)$ transition. The contours correspond to 4, 8, 12, 14, 18, and 22 K. The pv-diagram is centered on the brightest radio continuum point source SNR 41.9+58 (see Figs. 1 and 2). The velocity crowded regimes at 20'', 5'' and -7'' offset correspond to the western, central and eastern CO peak

3.3 CO line ratios

To calculate the line ratios properly we used the short-spacing corrected $^{12}{\rm CO}(J\,=\,1\to 0)$, $^{12}{\rm CO}(J = 2\to 1)$ and $^{13}{\rm CO}(J\,=\,1\to 0)$ data cubes. Note that the missing flux in the pure interferometric maps can be as high as 60% (see Table 1). Therefore the short-spacing correction is vital to derive proper line ratios. The short-spacing correction is less crucial for the peak line intensities. Here the missing flux is 10%-30% only. The $^{12}{\rm CO}(J\,=\,1\to 0)$, $^{12}{\rm CO}(J = 2\to 1)$ and ${\rm C}^{18}{\rm O}(J\,=\,1\to 0)$ data were smoothed to the resolution of the $^{13}{\rm CO}(J\,=\,1\to 0)$ observations (4.2''). Since no single-dish data were obtained for the ${\rm C}^{18}{\rm O}(J\,=\,1\to 0)$ transition we applied the missing flux factors derived from the $^{13}{\rm CO}(J\,=\,1\to 0)$ peak intensity distribution to the ${\rm C}^{18}{\rm O}(J\,=\,1\to 0)$ observations. This procedure is justified because the frequency of both transitions is similar and the observations were carried out in the same configurations with the PdBI. This leads to similar uv-coverages for both observations. Furthermore the morphology in the interferometer maps is similar and both transitions are optically thin (see Sect. 3.4). To take the remaining uncertainties into account we assumed an error of 50% for the ${\rm C}^{18}{\rm O}(J\,=\,1\to 0)$ line intensities. The line ratios were calculated at 19 positions across the CO distribution of M82. The spacing between individual positions is about 4''. The analyzed positions are marked by the crosses in Fig. 8. The circles indicate the FWHM of 4.2'' used in the study. The positions include all molecular peaks, the 2.2 m$\mu$m nucleus, the MIR peaks, the CO spurs and the diffuse emission in the outer regions of M82. For clarity the positions have been labeled 1 to 16 from east to west. Positions 17 to 19 correspond to positions on the CO outflow (see Fig. 8). The line ratios at the analyzed positions are summarized in Table 2. Errors include 10% uncertainty of the flux calibrators, errors of the amplitude calibration (typically about 10%) and statistical errors. Our high-resolution line ratios for $^{12}{\rm CO}$ and ^13CO $^{13}{\rm CO}$ differ slightly from values derived from single dish observations by Mao et al. (2000). But our data confirms that $^{12}{\rm CO}(J = 2\to 1)$/ $^{12}{\rm CO}(J\,=\,1\to 0)$ ratios larger than 1.8 (e.g. Knapp et al. 1980; Olofsson & Rydbeck 1984; Loiseau et al. 1990) can firmly be rejected. $^{12}{\rm CO}(J\,=\,1\to 0)$/ $^{13}{\rm CO}(J\,=\,1\to 0)$ and $^{12}{\rm CO}(J\,=\,1\to 0)$/ ${\rm C}^{18}{\rm O}(J\,=\,1\to 0)$ line intensity ratios are about 10-20 and 40-60 respectively.

 

 

Table 2: CO line ratios in M82 at 4'' resolution. The offsets are given relative to $\alpha = 09^{\rm h}55^{\rm m}51^{\rm s}.94, \delta = 69^{\circ }40'47.14''$ (J2000.0). For Cols.8 & 9 we have corrected the C¹⁸O intensities with the missing flux factor determined from the ¹³CO data. The corresponding percentage is given in Col.10. Errors include 10% uncertainty of the flux calibrators, errors of the amplitude calibration (typically about 10%) and statistical errors. For the C¹⁸O intensities we assumed an error of 50% due to the unknown missing flux

	$\Delta \alpha$	$\Delta \delta$	$\frac{I{\rm (CO}(2-1))}{I{\rm (CO}(1-0))}$	$\frac{T{\rm (CO}(2-1))}{T{\rm (CO}(1-0))}$	$\frac{I{\rm (CO}(1-0))}{I{\rm (^{13}CO}(1-0))}$	$\frac{T{\rm (CO}(1-0))}{T{\rm (^{13}CO}(1-0))}$	$\frac{I{\rm (CO}(1-0))}{I{\rm (C^{18}O}(1-0))}$	$\frac{T{\rm (CO}(1-0))}{T{\rm (C^{18}O}(1-0))}$	MF $^{13}{\rm CO}^1$
	['']	['']							[%]
1	16.5	7.5	$1.16\pm0.3$	$1.02\pm0.3$	$15.2\pm4.9$	$19.2\pm5.7$	$44\pm24$	$66\pm35$	23.5
2	14.5	5.0	$1.17\pm0.3$	$0.96\pm0.3$	$12.7\pm3.9$	$15.4\pm4.9$	$38\pm21$	$45\pm24$	10.3
3	11.5	3.0	$1.12\pm0.3$	$0.98\pm0.3$	$11.5\pm3.4$	$12.2\pm3.7$	$40\pm21$	$44\pm24$	8.0
4	9.5	1.0	$1.19\pm0.3$	$1.09\pm0.3$	$12.0\pm3.6$	$13.0\pm4.1$	$39\pm21$	$43\pm23$	7.0
5	6.5	0.5	$1.34\pm0.4$	$1.03\pm0.3$	$14.2\pm4.4$	$12.8\pm4.2$	$37\pm20$	$40\pm22$	9.0
6	4.0	0.5	$1.42\pm0.4$	$1.28\pm0.4$	$15.0\pm4.8$	$12.6\pm4.1$	$33\pm18$	$37\pm20$	12.4
7	2.0	-0.5	$1.39\pm0.4$	$1.26\pm0.4$	$17.3\pm5.7$	$16.5\pm6.1$	$37\pm20$	$61\pm33$	10.3
8	-1.0	-1.5	$1.06\pm0.3$	$1.01\pm0.3$	$25.9\pm8.5$	$20.7\pm8.0$	$36\pm20$	$56\pm30$	13.0
9	-4.0	-2.0	$0.98\pm0.3$	$0.98\pm0.3$	$21.3\pm6.6$	$22.1\pm8.4$	$53\pm28$	$56\pm30$	15.6
10	-6.5	-3.0	$1.07\pm0.3$	$0.95\pm0.3$	$25.4\pm8.4$	$17.0\pm6.0$	$41\pm22$	$45\pm24$	12.9
11	-10.0	-4.0	$1.12\pm0.3$	$1.08\pm0.3$	$13.2\pm3.9$	$11.7\pm3.8$	$45\pm24$	$29\pm15$	14.0
12	-14.0	-4.0	$1.00\pm0.3$	$0.98\pm0.3$	$15.3\pm4.5$	$11.5\pm3.5$	$48\pm26$	$36\pm20$	4.3
13	-17.5	-4.0	$1.14\pm0.3$	$1.02\pm0.3$	$14.6\pm4.4$	$10.9\pm3.3$	$48\pm26$	$39\pm21$	6.6
14	-20.5	-4.5	$1.05\pm0.3$	$0.95\pm0.3$	$14.3\pm4.3$	$11.1\pm3.6$	$49\pm27$	$39\pm21$	4.1
15	-23.5	-4.5	$1.07\pm0.3$	$1.00\pm0.3$	$16.3\pm5.1$	$14.7\pm4.7$	$60\pm32$	$48\pm26$	5.6
16	-26.5	-4.0	$1.14\pm0.3$	$1.09\pm0.3$	$14.5\pm4.8$	$14.3\pm4.9$	$62\pm34$	$56\pm30$	6.2
17	-2.5	-5.5	$1.02\pm0.3$	$1.07\pm0.3$	$20.2\pm6.8$	$21.2\pm8.3$	$40\pm22$	$54\pm29$	19.0
18	-3.5	-9.0	$1.20\pm0.3$	$0.91\pm0.3$	$12.0\pm4.1$	$21.5\pm8.6$	$29\pm16$	$52\pm28$	27.4
19	-8.5	-7.0	$1.36\pm0.4$	$1.12\pm0.3$	$11.8\pm3.8$	$11.6\pm3.9$	$32\pm17$	$33\pm18$	10.8

¹Missing flux determined from the ¹³CO data.

Figure 4: $^{12}{\rm CO}(J = 2\to 1)$ velocity channel maps of the short-spacing corrected data cube. Offsets in RA, DEC are given relative to the 2.2  m$\mu$m nucleus indicated by the cross in each channel map. The velocity spacing between individual maps is 12.8 kms-1, the central velocity ( $V_{\rm lsr}$) of each map is given in the top right corner in units of kms-1. The intensities are corrected for attenuation of the primary beam. The contours correspond to 6, 12, 18, 24 and 30 K. The dotted line in each channel map shows the 50% sensitivity level of the primary beam mosaic

Figure 5: ${\rm C}^{18}{\rm O}(J\,=\,1\to 0)$ velocity channel maps of the PdBI data cube. The figure has the same layout and specifications as Fig. 4; contours correspond to 0.05, 0.15, 0.25, 0.35, 0.45 and 0.55 K

Figure 6: "Best'' LVG solution at position 3 (the eastern CO lobe). The top left diagram shows the observed $^{12}{\rm CO}(J = 2\to 1)$ line intensities across the major axis of M82 and the $^{12}{\rm CO}(J = 2\to 1)$ intensity as given by the LVG-model for position 3. The fixed LVG-parameters (abundances, velocity gradient, radiation field and beam filling) are given in the upper right box. The parameters below the LVG input parameters summarize the optical depth for each transition, the CO column density, the kinetic temperature and H2 density for the "best fit''. The black and the grey solid line in the lower left diagram are the $^{12}{\rm CO}(J = 2\to 1)$/^12 CO(J=10) $^{12}{\rm CO}(J\,=\,1\to 0)$ (R12) and $^{13}{\rm CO}(J\,=\,1\to 0)$/ $^{12}{\rm CO}(J\,=\,1\to 0)$(R12,13) line intensity ratio, respectively. The dashed lines indicate the observational errors for the corresponding line ratio. The ${\rm C}^{18}{\rm O}(J\,=\,1\to 0)$/ $^{12}{\rm CO}(J\,=\,1\to 0)$(R12,18) line intensity ratio is given by the dashed-dotted line. Its error is indicated by the dotted line. The contours in the lower right diagram are results of the $\chi ^2$ test comparing the predicted line ratios and the $^{12}{\rm CO}(J = 2\to 1)$ line intensity with the observed values. The star indicates the best solution in the kinetic temperature and H2 density plane

Figure 7: "Best'' LVG solution at position 9 (central MIR peak). The items and parameters are the same as in Fig. 6

   
3.4 Radiative transfer calculations

The excitation conditions of the CO-emitting volume were modeled using a spherical, isothermal one-component large velocity gradient (LVG) model (Goldreich & Kwan 1974; de Jong et al. 1975). LVG line intensities were calculated for a kinetic temperature and H₂ density range from 5K to 200K by 5K and ${\rm log}\,n({\rm H}_2)$ from 1.8 to 5.0 by 0.2 respectively. In addition, we varied the CO abundance relative to H₂, [CO], per velocity gradient and the fractional $^{13}{\rm CO}$ and ${\rm C}^{18}{\rm O}$ abundances ([CO]/grad(V): $1~10^{-5}\,{\rm to}\,2~10^{-4}\,{\rm by}\, 1~10^{-6}$; [CO]/[ $^{13}{\rm CO}$]: 30 to 100 by 5; [CO]/[ ${\rm C}^{18}{\rm O}$]: 100 to 300 by 20). For the comparison between the observed peak intensity ratios (Table 2, Cols. 3, 5, 7) and the predicted LVG ratios we used a $\chi ^2$ test. To account for the absolute intensities across the disk of M82 we also fitted the $^{12}{\rm CO}(J = 2\to 1)$ intensity at each position by varying the beam filling from 0.1 to 0.9 by 0.1. The "best'' solutions are shown for positions 3 and 9 in Figs. 6 and 7. Position 3 on the western CO lobe is an example for a solution with low kinetic temperatures and high H₂ densities; position 9 on the brightest MIR peak is representative for solutions with high kinetic temperatures and low H₂ densities.

The observed line ratios and $^{12}{\rm CO}(J = 2\to 1)$ intensities can be modeled within the errors at all positions. The fit agrees very well with the data at positions where $^{12}{\rm CO}(J = 2\to 1)$/ $^{12}{\rm CO}(J\,=\,1\to 0)$ is less than 1.2. At position 6 (eastern MIR peak) we do not find any intersection for all observed line ratios in the H₂ density and kinetic temperature plane. For a more detailed discussion see Sect. 4.1. The best agreement with the observed line ratios and absolute intensities is found for a beam filling of 0.4. Positions 6 and 7 at the eastern MIR peak (Telesco & Gezari 1992) and positions 18 and 19 at the CO outflow require a somewhat lower beam filling of 0.2 and 0.3 respectively.

The LVG parameters of the "best-fit'' across the major axis of M82 are shown in Figs. 8a-d. The CO abundance relative to H₂ per velocity gradient ([CO]/grad(V)) varies between 1 10^-5pc/kms^-1 and 7 10^-5pc/kms^-1. Assuming ${\rm grad}(V)\approx 1 {\rm\ifmmode{{\rm\thinspace km\thinspace s}^{-1}}\else{\thinspace km\thinspace s$^{-1}$ }\fi\,pc^{-1}}$, as suggested by comparing the linewidth with the linear extent of the region, this corresponds to CO abundances in the range of ${\rm [CO]}\approx 10^{-5}$-7 10^-5. Similar values have been determined in the Orion region (Blake et al. 1987) and were suggested by chemical models (Farquhar et al. 1994). [CO]/grad(V) increases towards the MIR peaks which indicates higher CO abundances at the active star-forming regions than in the more quiescent outer regions. The fractional $^{13}{\rm CO}$ abundance [ $^{12}{\rm CO}$]/[ $^{13}{\rm CO}$] across M82 does not show any significant spatial variation. The mean value of all positions is $70\pm20$. A low fractional $^{13}{\rm CO}$ abundance is consistent with recent radiative transfer calculations by Mao et al. (2000) and an independent chain of arguments based on CN and ¹³CN measurements (Henkel et al. 1998). In contrast, the fractional ${\rm C}^{18}{\rm O}$ abundance [ $^{12}{\rm CO}$]/[ ${\rm C}^{18}{\rm O}$] shows a trend towards higher ${\rm C}^{18}{\rm O}$ abundances at the MIR peaks and in the outflow. While the average [ $^{12}{\rm CO}$]/[ ${\rm C}^{18}{\rm O}$] ratio in the quiescent regions is about 270, it is only about 160 at position 6, 11, 17 and 19 (see Fig. 8d). Note that these values suggest ${\rm C}^{18}{\rm O}$ abundances 2-3 times higher than those used by Wild et al. (1992) for their LVG calculations of CO line ratios in M82.

The kinetic temperature is well correlated with the MIR emission and other tracers of high-level star formation. Within the prominent CO lobes with less signs of ongoing star formation, the kinetic temperature is about 50K. Towards the active star-forming regions we find two kinetic temperature peaks above 150K. These "hot-spots'' coincide with the location of MIR peaks (for a comparison between the MIR emission and the CO distribution see Telesco & Gezari 1992). Near the 2.2 m$\mu$m nucleus the LVG models suggest temperatures of about 75K. Along the CO outflow the temperature drops with increasing distance from the active regions. At position 17 and 19 we find temperatures above 100K. At position 18 (100pc distance from the plane) the kinetic temperature has dropped to 60K. The spatial variation of the kinetic temperature along the major axis of M82 is shown in Fig. 8a. The corresponding diagram of the H₂ density distribution is shown in Fig. 8b. Solutions are found between $n({\rm H}_2)= 10^{2.7}$ and $10^{4.2}~{\rm cm}^{-3}$. In general, the H₂ densities are high in regions with low kinetic temperatures and vice versa. The solutions for the outer CO lobes suggest an H₂ density about $n({\rm H}_2)=10^{4.0}~{\rm cm}^{-3}$ with a tendency towards somewhat lower values at the very edge of the CO distribution ( $n({\rm H}_2)=10^{3.5}~{\rm cm}^{-3}$). These values are in agreement with H₂ densities calculated by Wild et al. (1992) and Mao et al. (2000). At the "hot-spot'', low H₂ densities of $n({\rm H}_2)=10^{2.8-3.1}~{\rm cm}^{-3}$ are required to match the observed line ratios. H₂ densities in the CO outflow are about $n({\rm H}_2)=10^{3.0}~{\rm cm}^{-3}$.

Both the $^{12}{\rm CO}(J\,=\,1\to 0)$ and the $^{12}{\rm CO}(J = 2\to 1)$ transitions are optically thick. In the cold dense regions we find an optical depth of $\tau_{\ifmmode{^{12}{\rm CO}(J\,=\,1\to0)}\else{$^{12}{\rm CO}(J\,=\,1\to0)$ }\fi} = 2{-}5$ and $\tau_{\ifmmode{^{12}{\rm CO}(J = 2\to1)}\else{$^{12}{\rm CO}(J = 2\to1)$ }\fi} = 7{-}15$. At the "hot-spots'' the derived optical depths are somewhat lower and reach unity in the $^{12}{\rm CO}(J\,=\,1\to 0)$ transition at the eastern MIR peak (position 6 & 7). For the ground transitions of the rare isotopes $^{13}{\rm CO}$ and ${\rm C}^{18}{\rm O}$ we find optically thin emission at all positions. Typical optical depths are $\tau_{\ifmmode{^{13}{\rm CO}(J\,=\,1\to0)}\else{$^{13}{\rm CO}(J\,=\,1\to0)$ }\fi} = 5~10^{-2}$ and $\tau_{\ifmmode{{\rm C}^{18}{\rm O}(J\,=\,1\to0)}\else{${\rm C}^{18}{\rm O}(J\,=\,1\to0)$ }\fi} = 5~10^{-3}$.

Figure 8: LVG solutions for positions 1 to 16. Top: locations of the analyzed positions. The radii of the circles indicate the spatial resolution for which the line ratios have been determined. a) to d): spatial variations of the kinetic temperature, the H2 density, the CO abundance per velocity gradient and the fractional ${\rm C}^{18}{\rm O}$ abundance across the major axis of M82. The error bars in a) and b) correspond to the parameter range of kinetic temperatures and H2 densities for which the LVG line ratios and the $^{12}{\rm CO}(J = 2\to 1)$ intensity is consistent with the observations within the errors. This corresponds to the area within the $\chi ^2=1$ contour shown for position 3 and 9 in Figs. 6 and 7. The error bars in c) and d) correspond to the range for [CO]/grad(V) and fractional ${\rm C}^{18}{\rm O}$ abundance where the $\chi ^2$ of the corresponding fit is 50 times higher than for the "best'' solution

3.5 Column densities and mass distribution

For the determination of CO and H₂ column densities at each position we used three methods:
- LVG: The column densities were derived from the CO and H₂ densities, the velocity gradient and the observed line widths using $N{\rm (CO)}=3.08~10^{18} \,n({\rm CO})\,\frac{{\rm d}V}{{\rm grad}(V)}$ and $N(\ifmmode{{\rm H}_2}\else{H$_2$ }\fi)= 3.08~10^{18}\,n(\ifmmode{{\rm H}_2}\else{H$_2$ }\fi)\,\frac{{\rm d}V}{{\rm grad}(V)}$, where dV is the observed line width. Therefore $\frac{{\rm d}V}{{\rm grad}(V)}$ is an equivalent path length through the clouds;
- LVG$_{\rm PF}$ (PF=partition function): the $^{13}{\rm CO}$ and ${\rm C}^{18}{\rm O}$ column densities were calculated from the general relation between optical depth, excitation temperature and column density at rotation level J: $N_J=93.5\frac{g_J \nu^3}{g_{(J+1)} A_{J+1,J}}(1-{\rm exp}(-4.8\,10^{-2} \nu/T_{\rm ex}))^{-1} \int \tau {\rm d}v$ where g_J is the statistical weight of level J and A_J+1,J is the Einstein coefficient for the transition J+1 to J. $\int \tau {\rm d}v$ was approximated by $\int \tau {\rm d}v \approx 1.06\, \tau \,{\rm d}V$. $T_{\rm ex}$ and $\tau$ are given by the LVG code for each level. $^{13}{\rm CO}$ and ${\rm C}^{18}{\rm O}$ column densities were determined using the sum of the 6 lowest levels for each isotope. H₂ and CO column densities were derived from the relative abundances of the rare isotopes relative to H₂ and CO;
- LTE: $^{13}{\rm CO}$ and ${\rm C}^{18}{\rm O}$ column densities were derived using a standard LTE approach (e.g. Dickman 1978). As for the LVG$_{\rm PF}$ method, CO and H₂ column densities were derived from the abundances of the rare isotopes relative to CO and H₂ at each position.
Column densities calculated from $^{13}{\rm CO}$ and ${\rm C}^{18}{\rm O}$ via the LTE method match each other with less than 5% difference at each position. The same holds for the LVG$_{\rm PF}$ column densities calculated from $T_{\rm ex}$ and $\tau$ of the $^{13}{\rm CO}$ and ${\rm C}^{18}{\rm O}$ transition. For simplicity we therefore give in the following the average between the column densities calculated from $^{13}{\rm CO}$ and ${\rm C}^{18}{\rm O}$ via the LTE and LVG$_{\rm PF}$ method.

The spatial variations of the beam-averaged H₂ column density across the major axis of M82 as calculated with the three methods is shown in Fig. 9. The spatial distribution of the H₂ column densities is in good agreement for all three methods. This suggests that the low J levels are almost thermalized. The largest difference between the methods is apparent at the central CO peak. While the LTE solutions suggest a local H₂ column density maximum of about $N(\ifmmode{{\rm H}_2}\else{H$_2$ }\fi)_{4''}=1~10^{23}~{\rm cm^{-2}}$, the peak is less prominent ( $N(\ifmmode{{\rm H}_2}\else{H$_2$ }\fi)_{4''}=5~10^{22}~{\rm cm^{-2}}$) and displaced by 4'' in the LVG and LVG$_{\rm PF}$ solution (see Fig. 9).

Nevertheless, all methods clearly show that most of the molecular gas traced by CO is located in the outer CO lobes. The central 300 pc between the molecular lobes contain only about 20-30% of the molecular gas mass. Furthermore, the H₂ column density distribution is clearly asymmetric with respect to the 2.2 m$\mu$m nucleus. We find that the centroid of mass is located about 100 pc south-east of the nucleus. The location of the centroid of mass for each method is indicated by the vertical line in Fig. 9. The highest H₂ column density is found at the western CO lobe (position 12). Its beam-averaged LVG column densities are $N{\rm (CO)}_{4''}=2~10^{19}\, {\rm cm^{-2}}$ and $N(\ifmmode{{\rm H}_2}\else{H$_2$ }\fi)_{4''}=2.3~10^{23}\,{\rm cm^{-2}}$. The corresponding cloud-averaged LVG column densities are $N{\rm (CO)}_{\rm cloud}= 4~10^{19}\,{\rm cm^{-2}}$ and $N(\ifmmode{{\rm H}_2}\else{H$_2$ }\fi)_{\rm cloud}=6~10^{23}\, {\rm cm^{-2}}$, respectively. The corresponding values for the eastern CO lobe (position 3) are $N{\rm (CO)}_{4''}=1~10^{19}\,{\rm cm^{-2}}$, $N(\ifmmode{{\rm H}_2}\else{H$_2$ }\fi)_{4''}=1.5~10^{23}\,{\rm cm^{-2}}$, $N{\rm (CO)}_{\rm cloud}= 2.5~10^{19}\,{\rm cm^{-2}}$ and $N(\ifmmode{{\rm H}_2}\else{H$_2$ }\fi)_{\rm cloud}=3.8~10^{23}\, {\rm cm^{-2}}$. For an assumed line-of-sight of 350pc (for comparison with Mao et al. 2000) the mean molecular density in the CO lobes is $<n(\ifmmode{{\rm H}_2}\else{H$_2$ }\fi)>_{4''}= 140{-}210~{\rm cm}^{-3}$. This corresponds to a volume filling factor of $f_{{\rm v},4''}=~<n(\ifmmode{{\rm H}_2}\else{H$_2$ }\fi)>_{4''}\!\!\!/n(\ifmmode{{\rm H}_2}\else{H$_2$ }\fi)\approx 0.01$. With $f_{{\rm a},4''}=0.4$ and a linear resolution of 65pc we obtain characteristic cloud sizes of $r_{\rm cloud}= \frac{1}{2}\, 65\,{\rm pc}\,f_{{\rm v},4''}/f_{{\rm a},4''} \approx 1~{\rm pc}$. Volume filling factors and characteristic cloud sizes do not change significantly in the central star forming regions. These values are in good agreement with PDR models published by Wolfire et al. (1990).

H₂ column densities in the molecular outflow are in the range $N(\ifmmode{{\rm H}_2}\else{H$_2$ }\fi)_{4''}=1.5$- $3.0~10^{22}\,{\rm cm^{-2}}$. The total mass of the outflow is $7.2~10^5\,\ifmmode{ M_{\odot}}\else{$M_{\odot}$ }\fi$ (D=3.9 Mpc, Sakai & Madore 1999).

3.6 Conversion from I(CO) to N(H $_{\mathsfsl 2}$) and total mass

To derive the conversion factor from I(CO) to N(H₂), we have compared LVG, LTE, and LVG$_{\rm PF}$ H₂ column densities with the integrated $^{12}{\rm CO}(J\,=\,1\to 0)$ intensities at 4.2'' resolution at the analyzed positions across the central part of M82. The variation of the conversion factor $\ifmmode{X_{\rm CO}}\else{$X_{\rm CO}$ }\fi\,=\ifmmode{N({\rm H}_2)}\else{$N$ (H$_2$ )}\fi/I{\rm (CO)}$ with position is shown in Fig. 10. Note that $X_{\rm CO}$ is lower than the Galactic value of $1.6~10^{20}\,{\rm cm^{-2}\,(K \ifmmode{{\rm\thinspace km\thinspace s}^{-1}}\else{\thinspace km\thinspace s$^{-1}$ }\fi)^{-1}}$ (Hunter et al. 1997) at all positions and for all methods. We find that $X_{\rm CO}$ varies across the disk of M82 by about a factor of 5 if one considers the LTE solutions ( $\ifmmode{X_{\rm CO}}\else{$X_{\rm CO}$ }\fi\,=2.1$- $10.8~10^{19}\,{\rm cm^{-2} \,(K \ifmmode{{\rm\thinspace km\thinspace s}^{-1}}\else{\thinspace km\thinspace s$^{-1}$ }\fi)^{-1}}$) and by a factor of 8-9 for the LVG and LVG$_{\rm PF}$ solutions ( $\ifmmode{X_{\rm CO}}\else{$X_{\rm CO}$ }\fi\,=1.3$- $11.5~10^{19} \,{\rm cm^{-2} \,(K \ifmmode{{\rm\thinspace km\thinspace s}^{-1}}\else{\thinspace km\thinspace s$^{-1}$ }\fi)^{-1}}$ and $\ifmmode{X_{\rm CO}}\else{$X_{\rm CO}$ }\fi\,=1.5$- $12.2~10^{19}\, {\rm cm^{-2}\,(K \ifmmode{{\rm\thinspace km\thinspace s}^{-1}}\else{\thinspace km\thinspace s$^{-1}$ }\fi)^{-1}}$). All methods show that the lowest conversion factors are associated with the central star-forming regions where the gas is heated by UV photons from the newly formed stars and cosmic-rays from SNRs. The CO-emitting volumes at these positions have high kinetic temperatures. Towards the outer molecular lobes with higher H₂ densities and lower kinetic temperatures, the conversion factor rises. This is in agreement with simple theoretical arguments that suggest that the conversion factor $X_{\rm CO}$ should be proportional to $T_{\rm kin}^{-1}\,n(\ifmmode{{\rm H}_2}\else{H$_2$ }\fi)^{1/2}$ for virialized clouds (Maloney & Black 1988). The variation of $X_{\rm CO}$ with $T_{\rm kin}^{-1}\,\,n(\ifmmode{{\rm H}_2}\else{H$_2$ }\fi)^{1/2}$ is shown in Fig. 11. The linear correlation between $X_{\rm CO}$ and $T_{\rm kin}^{-1}\,n(\ifmmode{{\rm H}_2}\else{H$_2$ }\fi)^{1/2}$ for $T_{\rm kin}^{-1}\,n(\ifmmode{{\rm H}_2}\else{H$_2$ }\fi)^{1/2}>0.5$ is clearly visible. For $T_{\rm kin}^{-1}\,n(\ifmmode{{\rm H}_2}\else{H$_2$ }\fi)^{1/2}<0.5$ the scatter in the plot increases. This is in particular true for $X_{\rm CO}$ calculated under the assumption of LTE. This suggests that the gas is not close to LTE at the "hot spots''. The increased scatter of $X_{\rm CO}$ calculated with the LVG and LVG$_{\rm PF}$ method might suggest that either the clouds are not virialized or that more appropriate models (like PDR models) are required to calculate the physical gas properties in the center of the starburst. For a more detailed discussion see Sect. 4.1. Nevertheless, this result not only shows that the standard Galactic $X_{\rm CO}$ factor is not appropriate for a starburst system like M82, but that $X_{\rm CO}$ is a function of the intrinsic gas properties which strongly depend on environmental effects. This implies that spatial variations of ^12CO(J=10) $^{12}{\rm CO}(J\,=\,1\to 0)$ intensities can be due to variations of the excitation conditions of the gas rather than variations of column density. Similar results have been obtained by Wild et al. (1992) using low-resolution CO data (see also Sect. 4.3). Based on the analysis of $X_{\rm CO}$ we have calculated the "true'' H_2H₂ distribution in M82 by interpolating the changes of $X_{\rm CO}$ from the analyzed positions across the central CO distribution. Multiplication of this X_CO $X_{\rm CO}$-map with the integrated $^{12}{\rm CO}(J\,=\,1\to 0)$ intensity distribution thus results in an H₂ column density map. We show these maps in Fig. 12 for $X_{\rm CO}$ derived from the LVG$_{\rm PF}$ (top) and LTE solutions (middle) in comparison with the H₂ distribution one would derive assuming a constant, standard Galactic conversion (bottom) to illustrate the importance of detailed studies of $X_{\rm CO}$ to derive H₂ column density distributions. The H₂ column density maps in Fig. 12 (top and middle) indicate that the central star-forming region is surrounded by a double-lobed distribution of molecular gas, while H₂ seems to be depleted in the central starburst region itself (see also Fig. 9).

The total H₂ mass of the region shown in Fig. 12 is $2.3~10^8\,\ifmmode{ M_{\odot}}\else{$M_{\odot}$ }\fi$ for the LVG$_{\rm PF}$ and LVG and $2.7~10^8\,\ifmmode{ M_{\odot}}\else{$M_{\odot}$ }\fi$ for the LTE solution at a distance of D=3.9 Mpc (Sakai & Madore 1999). The corresponding values at D=3.25 Mpc (Tammann & Sandage 1968) are 1.6 and $1.9~10^8\,\ifmmode{ M_{\odot}}\else{$M_{\odot}$ }\fi$, respectively. These values are in good agreement with estimates from 450 m$\mu$m dust continuum measurements (Smith et al. 1991) and from C¹⁸O(2$\to$1) intensities (Wild et al. 1992). Therefore, the total molecular mass is 3 times lower than the mass one would derive using the standard Galactic conversion factor of $1.6~10^{20}\,{\rm cm^{-2}\,(K \ifmmode{{\rm\thinspace km\thinspace s}^{-1}}\else{\thinspace km\thinspace s$^{-1}$ }\fi)^{-1}}$ ( $4.9~10^8\,\ifmmode{ M_{\odot}}\else{$M_{\odot}$ }\fi$ D=3.25 Mpc; $7.1~10^8\,\ifmmode{ M_{\odot}}\else{$M_{\odot}$ }\fi$ D=3.9 Mpc).