REVEALING THE PHYSICAL PROPERTIES OF MOLECULAR GAS IN ORION WITH A LARGE-SCALE SURVEY IN J = 2–1 LINES OF ¹²CO, ¹³CO, AND C¹⁸O

Observations of the J = 2–1 transitions of ¹²CO, ¹³CO, and C¹⁸ O were carried out with the 1.85 m telescope installed at Nobeyama Radio Observatory, which is enclosed in a radome that prevents telescope structure distortion due to outdoor conditions (e.g., precipitation, wind, and sunlight). At 230 GHz, the telescope has a beam size of 27 (HPBW) which was measured by continuum scans of Jupiter. We used a two sideband separating (2SB) superconductor-insulator-superconductor (SIS) mixer receiver to observe J = 2–1 lines of ¹²CO, ¹³CO, and C¹⁸O simultaneously. The typical noise temperature of the receiver T_RX was measured to be ∼100 K (single sideband) and the image rejection ratio was measured to be 10 dB or higher. A fast Fourier transform spectrometer with a 1 GHz bandwidth and 61 kHz frequency resolution was installed as the backend system. We used the spectrometer for the observations of the three lines by dividing the frequency band into three parts. Each part has a velocity coverage and a velocity resolution of ∼250 km s⁻¹ and 0.08 km s⁻¹, respectively. Further information on the telescope is provided by Onishi et al. (2013).

The intensity calibration was carried out by observing a standard source, Orion KL, as described in Onishi et al. (2013). They estimated the uncertainty of the calibration to be ∼10%. The other factor that may affect the calibration accuracy is the beam coupling to the sources with different extents. Figure 4 of Onishi et al. (2013) shows that there are no large-scale deformations affecting the strength of the error beam, and the main dish was made by monobloc casting, which has no small-scale fluctuation effect on the surface such as misalignments of panels sometimes seen in large telescopes (e.g., Greve et al. 1998 for the case of IRAM 30 m). Onishi et al. (2013) also showed that the beam has nearly circular symmetry without significant minor lobes observed. The typical antenna temperature toward Orion KL is ∼45 K in ¹²CO(J = 2–1) after correction for the effect of spillover to the image sideband. The brightness temperature of Orion KL is 63 K in ¹²CO(J = 2–1) (Onishi et al. 2013). Therefore, the typical scaling factor from the antenna temperature to $T_{\rm R}^{*}$ is 1/0.7. The moon efficiency was measured to be ∼70% with an error of ∼10%. All of these facts indicate that the calibration error due to the beam coupling to the sources with different extents is smaller than that for the intensity calibration to the $T_{\rm R}^{*}$ scale, which is ∼10% (Onishi et al. 2013). Therefore, the uncertainty in the calibration is estimated to be ∼10% here.

The observations were carried out from 2011 January to May. The ¹²CO, ¹³CO, and C¹⁸O lines were observed simultaneously. The system noise temperatures including atmospheric attenuation T_sys were in the range from 200 to 400 K for the three lines. We have covered 55 deg² around the Orion A and Orion B molecular clouds. The area was divided into 55 submaps of 1° × 1°. We observed each submap using the on-the-fly (OTF) mapping technique along the galactic coordinates. The scan data were obtained with a fully sampled grid of 1'. We selected 30 different OFF positions toward which we confirmed that the ¹²CO emission is absent at an rms noise level of ∼0.1 K at a velocity resolution of 0.08 km s⁻¹. In this paper, we use the calibrated $T^*_{\rm R}$ scale (Kutner & Ulich 1981). Before observing each submap, we observed the Orion KL ($\alpha _{\rm J2000} = 05^{\rm h}35^{\rm m}14 \buildrel{\mathrm{s}}\over{.}46, \delta _{\rm J2000} = -05 {^\circ}22 {^\prime}29 {\buildrel{\prime\prime}\over{.}} 6$) for intensity calibrations to the $T^*_{\rm R}$ scale by assuming that its peak temperature for ¹²CO(J = 2–1) is 63 K (Onishi et al. 2013). We applied each scale factor obtained by the ¹²CO observations for the intensity calibrations of ¹³CO and C¹⁸O. We subtracted a polynomial curve from each spectrum to ensure the linear baseline, and resampled the raw OTF data onto the 1' grid by convolving them with a Gaussian function. The rms noise of the resulting data, $\Delta T^*_{\rm R}$, is typically ∼0.45 K at a velocity resolution of 0.3 km s⁻¹ with an effective beam size of 34. In addition, we made a moment masked cube (e.g., Dame 2011) to suppress the noise effect in the velocity analyses. The moment masked cube has zero values at the emission free pixels, which is useful to avoid a large error arising from the random noise. The emission free pixels are determined by identifying significant emission from the smoothed data whose noise level is much lower than the original data.

The ¹²CO(J = 1–0) and ¹³CO(J = 1–0) data were taken with the two 4 m millimeter-wave telescopes at Nagoya University (Kawabata et al. 1985; Fukui et al. 1991). The beam sizes of the telescopes were 27 (HPBW) at 110 GHz. Each telescope was equipped with a 4 K cooled superconducting mixer receiver (Ogawa et al. 1990), which provided typical single sideband system noise temperatures of ∼400 K and ∼150 K for the ¹²CO and ¹³CO frequency bands, respectively, including atmospheric attenuation. The spectrometers were acousto-optical spectrometers with 40 MHz bandwidth and 40 kHz frequency resolution, corresponding to a velocity coverage and resolution of 100 and 0.1 km s⁻¹, respectively. The data were obtained using a frequency switching mode with a grid spacing of 4' and 2'. The rms noise level is better than 0.5 K on the $T^*_{\rm R}$ scale. The survey data were partially published by Nagahama et al. (1998) for ¹³CO(J = 1–0) data of the Orion A.

The C¹⁸O(J = 1–0) data were taken with the NANTEN 4 m telescope (Mizuno & Fukui 2004), which is equipped with the same receiver and spectrometer as the Nagoya University 4 m telescopes described above. The C¹⁸O(J = 1–0) observations were carried out toward the region where the ¹³CO(J = 1–0) line emission is strong. The data were obtained using the frequency switching mode at a grid spacing of 2'. The rms noise level is better than 0.1 K on the $T^*_{\rm R}$ scale. The survey data were partially published by Aoyama et al. (2001) for the observation of the Orion B.

3.1.1. ¹²CO(J = 2–1) and ¹²CO(J = 1–0)

Figure 1 shows the velocity-integrated intensity maps of ¹²CO(J = 2–1) and ¹²CO(J = 1–0) observed with the 1.85 m telescope and the 4 m telescopes, respectively. The intensities are calculated by integrating the spectra between v_LSR = 0 and 20 km s⁻¹ where the emission exists. The Orion A and B molecular clouds are fully covered with significantly improved sensitivity, and angular and frequency resolutions compared with previous ¹²CO(J = 2–1) observations carried out by Sakamoto et al. (1994). As pointed out by Sakamoto et al. (1994), we found that the two transitions of ¹²CO exhibit a similar spatial distribution on a large scale. However, small-scale differences are seen in the lower intensity regions. Actually, in the higher intensity regions (>100 K km s⁻¹), both images exhibit almost similar distributions, while in the lower intensity regions (<10 K km s⁻¹) the J = 1–0 emission is apparently more widely distributed than the J = 2–1 emission. In the following, we describe the spatial distribution for Orion A and B in more detail (see Figure 2).

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The Orion A molecular cloud is distributed almost parallel to the galactic plane at b = −195. The maximum intensities of ¹²CO(J = 2–1) and ¹²CO(J = 1–0) are both found at the position of Orion KL (l = 20898, b = −1936), whose peak temperatures are 62.6 K and 55.5 K, respectively. The Orion A molecular cloud can be divided into three physically different regions: the main ridge, the extended component (EC), and northern clumps. The main ridge is a major component of the Orion A cloud including the integral shaped filament (Bally et al. 1987), L1641, and other star forming sites. The main ridge consists of a number of clumps, filaments (Bally et al. 1987; Nagahama et al. 1998; Nakamura et al. 2012), and shells (Heyer et al. 1992; Nakamura et al. 2012) and most of the structures are also observed in the present survey. One notable feature in the main ridge is the existence of a gradient of some physical parameters including the line center velocity (e.g., Maddalena et al. 1986), volume density (Sakamoto et al. 1994), excitation temperature, and filament width (Nagahama et al. 1998), which we will discuss in the following subsections (Sections 3.2 and 3.3). Another striking feature is the well-defined boundary observed on the western side of the main ridge. This feature is observed by Wilson et al. (2005) with a 9' resolution and they suggested that the boundary is due to stellar wind and/or radiation from the Orion OB1 association, or ancient interactions with supernovae. The EC (Sakamoto et al. 1997) is located on the eastern side of the main ridge with a less intense emission typically of <15 K km s⁻¹ in the integrated intensity map of ¹²CO(J = 2–1). Molecular gas of the EC has been observed only at a coarse angular resolution (Wilson et al. 2005), or toward small regions (Sakamoto et al. 1997). Sakamoto et al. (1997) proposed that the EC is located in front of the main ridge, and it was formed as a result of the interaction between the galactic atomic gas and the dense molecular gas in the main ridge. In the present survey, we covered the entire extent of the EC at higher angular resolution. We detected a dozen clumps that have relatively high intensity and a well-defined boundary toward the EC region (hereafter "EC clumps"). The remarkable feature of the northern clumps is the lack of diffuse emission, probably due to their interaction with the surrounding OB associations.

The Orion B molecular cloud is located in the upper-right side of Figure 1. The maximum intensity of ¹²CO(J = 2–1) is observed toward NGC 2068 (l = 20537, b = −1433) with a peak temperature of 31.9 K, and that of ¹²CO(J = 1–0) is observed toward NGC 2023 (l = 20687, b = −1653) with a peak temperature of 35.4 K. The Orion B molecular cloud can be divided into three regions: the southern part, including NGC 2023 and NGC 2024 (hereafter, we call this part Southern cloud); the northern part, including NGC 2068 and NGC 2071 (hereafter, Northern cloud); and the central part, which has only diffuse extended emission (hereafter, 2nd component). The Southern cloud and the Northern cloud have clear boundaries in the direction of the Orion OB1 association, which may be due to the stellar wind and/or radiation from massive stars. The second component has a different velocity component from the Northern and Southern clouds, and thus seems to have no physical relation to the other clouds (Maddalena et al. 1986).

3.1.2. ¹³CO(J = 2–1) and ¹³CO(J = 1–0)

Figure 3 shows velocity-integrated intensity maps of ¹³CO(J = 2–1) and ¹³CO(J = 1–0). In general, both the J = 2–1 and J = 1–0 lines have similar spatial distributions, except for the lower intensity region around the main ridge. The ¹³CO emission is detected toward the region where ¹²CO emission is relatively strong. However, the ¹³CO(J = 2–1) emission is not detected in the regions with extended week ¹²CO emission. In the Orion A, the maximum intensity of ¹³CO(J = 2–1) is observed toward Orion KL with a peak temperature of 17.4 K, and that of ¹³CO(J = 1–0) is observed toward 10' north to Orion KL (l = 20880, b = −1927) with a peak temperature of 12.8 K. The main ridge exhibits greater filamentary shape than the ¹²CO distributions, which is considered to reflect the inner structure of the clouds due to its smaller optical depth. The main ridge has almost constant intensity (∼10 K km s⁻¹), expect for the local peaks around L1641N (l = 2101, b = −196). The helix-shaped structure is seen on the southern side of the main ridge between l = 211° and 213°, representing the possible influence of the magnetic field (Uchida et al. 1991). The main ridge has well-defined boundaries on both the western and eastern sides of the filament. The diffuse emission is not seen toward the EC region in the J = 2–1 emission, while some of the EC clumps are clearly detected. In the northern clumps region, ¹³CO is observed where ¹²CO is relatively strong. In Orion B, the maximum intensity of ¹³CO(J = 2–1) is observed toward NGC 2024 (l = 20657, b = −1637) with a peak temperature of 16.7 K, and that of ¹³CO(J = 1–0) is observed toward NGC 2023 (l = 20687, b = −1660) with a peak temperature of 14.4 K. In the J = 2–1 emission, the Southern cloud and the Northern cloud are clearly separated. The clouds have well-defined boundaries in the western direction while some diffuse components are extended in the opposite direction.

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3.1.3. C¹⁸O(J = 2–1) and C¹⁸O(J = 1–0)

Figure 4 shows velocity integrated intensity maps of C¹⁸O(J = 2–1) and C¹⁸O(J = 1–0). In Orion A, the maximum intensity of J = 2–1 is observed toward 14' north of Orion KL (l = 20878, b = −1923) with a peak temperature of 3.3 K, and that of J = 1–0 is observed toward L1641S (l = 21210, b = −1917) with a peak temperature of 2.8 K. In Orion B, the maximum intensities of J = 2–1 and J = 1–0 are observed toward NGC 2023 (l = 20687, b = −1657) with peak temperatures of 3.8 K and 3.6 K, respectively. The C¹⁸O emission is detected where the ¹³CO emission is strong, including the main ridge of Orion A and NGC 2023, NGC 2024, NGC 2068, and NGC 2071. The fact that most of the C¹⁸O(J = 2–1) emission has a higher intensity than C¹⁸O(J = 1–0) indicates that the region traced by C¹⁸O has a temperature and density high enough to excite the molecule to the J = 2 state, and also that the lines are optically thin. The distribution of C¹⁸O(J = 2–1) emission is similar to the distribution of CS (Lada et al. 1991; Tatematsu et al. 1993) emission, which implies that C¹⁸O(J = 2–1) traces a high-density region with n(H₂) ∼ 10⁴ in the cloud.

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Velocity structures are very complicated in the Orion molecular cloud complex, as seen in the velocity channel maps shown in Figures 5 and 6. In Orion A, the main ridge seems to consist of two giant filaments: one is located at l = 214°–211° in the velocity range v_LSR = 1.0–7.0 km s⁻¹, and the other is located at l = 212°–208° in the velocity range v_LSR = 7.0–13.0 km s⁻¹. Both of the filaments consist of many smaller filaments, clumps, and shell-like structures. The helical filament observed in the velocity range v_LSR = 10.0–11.5 km s⁻¹ is the Orion east filament (Wilson et al. 2005) which has no physical relation to the Orion main cloud. The EC is detected at the velocity v_LSR = 5.5–8.5 km s⁻¹. One of the striking features is that the EC consists of many small-scale structures (e.g., filaments and clumps) with weak intensities typically <5 K in the ¹²CO(J = 2–1) emission. The EC clumps clearly have a different velocity from the EC, which is v_LSR > 8.5 km s⁻¹ with relatively higher intensities typically >5 K in the ¹²CO(J = 2–1) and well-defined boundary. The Northern clumps are detected in the velocity v_LSR = 10.0–16.0 km s⁻¹. The Northern clumps consist of many small clumps. There are mainly two velocity components in the Orion B cloud: the lower velocity component corresponding to the second component is found in the velocity range v_LSR = 1.0–7.0 km s⁻¹, and the higher velocity component corresponding to the Northern cloud and the Southern cloud is found in the velocity range v_LSR = 7.0–16.0 km s⁻¹. The lower-velocity component seems to consist of shells, filaments, and clumps, as found for the EC in the Orion A. On the other hand, the higher-velocity component is not very filamentary in structure compared with the main ridge in Orion A. At velocity v_LSR > 13.0 km s⁻¹, both the Orion A and B clouds consist of many small clumps.

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Figure 7 shows the intensity-weighted mean velocity maps. The ¹²CO and ¹³CO maps exhibit quite similar velocity distributions. In Orion A, the main ridge has a velocity gradient while the EC has no velocity change. The Orion east filament is seen as the high-velocity components on the northeast side around l = 212°–216° of Orion A. In Orion B, it is clear that the cloud mainly consists of two different velocity components also as seen in Figure 5.

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The line width maps obtained by dividing the integrated intensity by the peak temperature are shown in Figure 8. In Orion A, the line width increases as it approaches the center of the main ridge and as it approaches the Orion KL. This tendency is more clearly seen in ¹³CO. The EC has a small line width typically of <2 km s⁻¹. In the case of the Orion B cloud, ¹²CO emission lines with very large line widths are widely seen, which is explained mainly being due to a mixture of some distinct velocity components. The ¹³CO map seems to trace the velocity structure of the main component of Orion B as the emission line is optically thinner. In ¹³CO, the largest velocity widths in the Southern cloud and the Northern cloud are seen toward NGC 2024 and NGC 2071, respectively.

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Figure 9 shows the longitude–velocity diagrams. The velocity gradient of Orion A is also clearly seen in this figure. It seems to have a velocity gradient along the longitude also observed in Orion B. The velocity gradients are calculated as 0.15 and −0.08 km s⁻¹ pc⁻¹ for Orion A and Orion B, respectively. The second component of Orion B is clearly seen around v_LSR = 5 km s⁻¹. The EC of Orion A cannot be identified clearly in the longitude–velocity diagram, because it has the same velocity as the main ridge.

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Figure 10 shows latitude–velocity diagrams. In the figure, both of the Orion A and B clouds have no velocity gradient. There is a clear boundary between the Orion A and B molecular clouds around b = −17°. In Orion A, the EC is clearly seen in the velocity around 5 km s⁻¹, and the EC clumps are seen with a velocity around 10 km s⁻¹. The EC of Orion B is also clearly seen in the velocity-latitude diagram around v_LSR = 5 km s⁻¹.

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In this subsection, we derive the intensity ratios of the observed molecular lines to investigate the physical properties of the molecular gas. We should note that we calculate the ratios without matching the angular resolutions. Both the J = 2–1 and J = 1–0 data were originally observed at an angular resolution of ∼27. However, because the observations by the 1.85 m telescope were carried out in the OTF mode, the resultant angular resolution for the J = 2–1 lines is lowered to ∼34, as stated in Section 2. On the other hand, the J = 1–0 observations by the 4 m telescopes were made with undersampling, which makes it very difficult to smooth the data exactly to the same angular resolution as that of the J = 2–1 data. We therefore decided not to attempt to standardize the angular resolutions but to use all of the data as they are. If we observe a point source, the observed intensity would differ by a factor of ∼1.5 due to the difference in the angular resolutions (34 or 27). This is the maximum estimate for the possible error in the following analyses arising from the difference of the angular resolutions. The actual errors should be much smaller, because the CO emission lines are spatially extended, as shown in the previous subsections. In this paper, we neglected all of the pixels where the intensities of each line are lower than the 3σ noise level when deriving the line ratios.

3.3.1. Intensity Ratio of ¹²CO(J = 2–1)/¹²CO(J = 1–0)

Figure 11 shows the distribution of the ¹²CO(J = 2–1)/¹²CO(J = 1–0) intensity ratio (hereafter, $R^{12}_{2-1/1-0}$). In general, the ratio approaches unity if the emission is quite optically thick, and it reflects the excitation temperature of the region if they are optically thin. The overall tendency in the figure is similar to that of Sakamoto et al. (1994); the ratio is approximately equal to unity along the main ridge of the clouds and decreases down to 0.5 in the peripheral regions. The present data reveal the ratio even in much lower intensity regions compared with Sakamoto et al. (1994) mainly because the present J = 2–1 observations are more sensitive.

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The maximum ratios of ${R^{12}_{2-1/1-0}}$ are observed toward the cloud boundary near NGC 1977 and the lower part of the main ridge around l = 209°–211° for Orion A. For Orion B, the maximum ratios are observed toward the western side of the main cloud. In these regions, the ratio becomes higher than 1.5. The high ratio indicates that the ¹²CO lines are optically thin, and that the gas is dense and warm enough to excite to the J = 2 level. This suggests the interaction of the molecular clouds with the stellar winds and the radiation from the surrounding massive stars. A relatively high ratio (∼1.3) is observed near Orion KL, indicating that the region is also affected by the star clusters in M42 including the Trapezium.

In the Orion A main ridge, a gradient of the ratio is observed which was previously discovered by Sakamoto et al. (1994). The ratio has local peaks near L1641N and L1641S that are well-known star forming regions associated with the shell-like structures (Heyer et al. 1992). The EC is observed as a low ratio (∼0.5) while the ratio of the EC clumps are relatively high (∼0.8). For the Northern clumps in Orion A, the ratio is relatively high, especially near the Orion KL. The ratio of Orion B is relatively high (∼0.8), except for the second component.

3.3.2. Intensity Ratio of ¹³CO(J = 2–1)/¹³CO(J = 1–0)

Figure 12 shows the distribution of the ¹³CO(J = 2–1)/¹³CO(J = 1–0) intensity ratio (hereafter, $R^{13}_{2-1/1-0}$). This ratio reflects both the kinematic temperature and density of the gas because of the small optical depth of the ¹³CO emission lines. The large-scale tendency is similar to that of the ratio of ¹²CO  but the dynamic range is larger. The maximum ratio in Orion A is observed toward the cloud boundary near Orion KL with a ratio of ∼2. The gradient seen in $R^{12}_{2-1/1-0}$ is also seen in $R^{13}_{2-1/1-0}$. We found that the ratio is ∼0.8 in the region near L1641N and is 0.3–0.5 in the region at l > 211°.5. Some of the EC clumps and the Northern clumps are detected with a ratio of ∼0.8. In the Orion B clouds, the maximum ratio (>1.5) is observed in the western side of NGC 2024. Other clouds in Orion B were observed to have a relatively high ratio of ∼0.9.

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3.3.3. Intensity Ratio of ¹³CO(J = 2–1)/¹²CO(J = 2–1)

Figure 13 shows the distribution of the ¹³CO(J = 2–1)/¹²CO(J = 2–1) intensity ratio (hereafter, $R^{13/12}_{2-1}$). The ratio roughly reflects the column density when the excitation temperatures are the same for both of the lines. Due to the photon trapping effect, the ratio is also sensitive to local density where the ¹³CO(J = 2–1) is sub-thermally excited and the ¹²CO(J = 2–1) is optically thick. The ratio is also affected by the abundance variation, which mainly reflects the intensity of the interstellar radiation field in the massive star forming region (e.g., Ripple et al. 2013). The distribution of $R^{13/12}_{2-1}$ is somewhat different from the intensity distributions of ¹³CO(J = 2–1) and ¹²CO(J = 2–1). In Orion A, the ratio is nearly constant from north to south all along b ∼ −195, although the intensity distributions of ¹³CO(J = 2–1) and ¹²CO(J = 2–1) are strongest at the northern edge and decrease toward the southern edge. In Orion B, the ratio is stronger around the Northern cloud than the Southern cloud, although this tendency is the opposite of the intensity distributions of ¹³CO(J = 2–1) and ¹²CO(J = 2–1).

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The column densities of the molecular gas are often derived by assuming the X-factor, which is a conversion factor from line intensities to column densities for optically thick lines, and by assuming local thermodynamic equilibrium (LTE) for optically thin lines. In this subsection, we derive the column densities and masses by using the above assumptions, and discuss the cause of their differences. In order to investigate the difference in terms of star formation activity depending on environment, we divide the observed area into four regions, i.e., Orion A-1, Orion A-2, Orion B-1, and Orion B-2 (see Figure 2). Orion A-1 is part of Orion A at l > 211° and includes no massive star formation site. Orion A-2 is the region at l < 211° where the massive star formation takes place. Orion B-1 is part of Orion B at b < 15° corresponding to the Southern cloud as introduced in Section 3.1.1. Orion B-2 is the region at b > 15° corresponding to the Northern cloud. Hereafter, we call these subregions simply A1, A2, B1, and B2.

4.1.1. Line Luminosities

We summarize the luminosities of the observed emission lines and their ratios in Table 1. To derive the intensity, we integrated the observed emission lines over the surface areas of each subregion. The ratio J = 2–1/J = 1–0 will be different depending on the isotopes. The ${R^{12}_{2-1/1-0}}$ is the highest ∼0.6–0.9 and the ${R^{18}_{2-1/1-0}}$ is the lowest ∼0.1–0.7. Especially in the A1 subregion, ${R^{18}_{2-1/1-0}}$ is very low compared with ${R^{12}_{2-1/1-0}}$ by a factor of three. The ratios of ¹³CO/¹²CO show a similar tendency both in J = 2–1 and J = 1–0. The A2 subregion is higher than the A1 subregion and the B2 subregion is higher than the B1 subregion.

Table 1. Observed Line Luminosities and Luminosity Ratios

Source		$L_{2-1}^{12}$	$L_{2-1}^{13}$	$L_{2-1}^{18}$	$L_{1-0}^{12}$	$L_{1-0}^{13}$	$L_{1-0}^{18}$	$R^{12}_{2-1/1-0}$	$R^{13}_{2-1/1-0}$	$R^{18}_{2-1/1-0}$	$R^{13/12}_{2-1}$	$R^{13/12}_{1-0}$
(1)		(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)
Orion		23200	2930	88	29500	4420	158	0.79	0.66	0.56	0.13	0.15
Orion A		14400	1830	48	19000	2890	102	0.76	0.63	0.48	0.13	0.15
⋅⋅⋅	A1	4200	413	3	6950	954	28	0.60	0.43	0.13	0.10	0.14
⋅⋅⋅	A2	10200	1420	45	12000	1940	74	0.85	0.73	0.61	0.14	0.16
Orion B		8790	1100	39	10500	1530	56	0.84	0.72	0.70	0.12	0.15
⋅⋅⋅	B1	5760	691	18	6350	830	27	0.91	0.83	0.67	0.12	0.13
⋅⋅⋅	B2	3030	408	20	3730	661	28	0.81	0.62	0.72	0.13	0.18

Notes. Column 1: source name. Columns 2–4: total luminosity of ¹²CO, ¹³CO, and C¹⁸O(J = 2–1), respectively, in K km s⁻¹ pc². Columns 5–7: total luminosity of ¹²CO, ¹³CO, and C¹⁸O(J = 1–0), respectively, in K km s⁻¹ pc². Columns 8–12: luminosity ratios of $R^{12}_{2-1/1-0} = L_{2-1}^{12} / L_{1-0}^{12}$, $R^{13}_{2-1/1-0}$ $= L_{2-1}^{13} / L_{1-0}^{13}$, $R^{18}_{2-1/1-0} = L_{2-1}^{18} / L_{1-0}^{18}$, $R^{13/12}_{2-1}$$= L_{2-1}^{13} / L_{2-1}^{13}$, and $R^{13/12}_{1-0} = L_{1-0}^{13} / L_{1-0}^{13}$, respectively.

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4.1.2. Column Densities

The X-factor, which converts from ¹²CO(J = 1–0) line intensities to the column densities of molecular hydrogen, has been derived by comparing the intensities with other tracers of mass, such as the virial masses (e.g., Solomon et al. 1987), proton masses from gamma-ray observations (e.g., Bloemen et al. 1986), and dust observations (e.g., Dame et al. 2001). For the Galactic clouds, the X-factor is derived to be approximately 1.8 × 10²⁰ cm⁻² K⁻¹ km⁻¹ s (Dame et al. 2001), and we use this value in this paper. The averaged column densities derived with the X-factor are $N^{1-0}_{\rm X}({\rm H_2}) = 18.9 \times 10^{20}$ cm⁻². We also derived the averaged column densities for each subregion and summarized them in Table 2.

Table 2. Averaged Column Densities and Column Density Ratios

Source		$N_{\rm X}^{1-0}$	$N_{\rm LTE}^{13,2-1}$	$N_{\rm LTE}^{13,1-0}$	$R^{N}_{\rm 13/12}$	$R^{N}_{\rm LTE,13}$	$N_{\rm LTE}^{18,2-1}$	$N_{\rm LTE}^{18,1-0}$	$R^{N}_{\rm 18/12}$	$R^{N}_{\rm LTE,18}$
(1)		(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
Orion		18.9	20.7	22.4	1.19	0.92	62.8	44.2	2.34	1.42
Orion A		15.6	19.4	20.8	1.34	0.93	61.8	44.6	2.87	1.38
⋅⋅⋅	A1	15.3	11.3	16.9	1.11	0.67	38.1	32.1	2.10	1.19
⋅⋅⋅	A2	15.8	24.1	23.2	1.47	1.04	64.9	54.4	3.45	1.19
Orion B		31.4	23.3	26.4	0.84	0.88	64.2	43.4	1.38	1.48
⋅⋅⋅	B1	37.3	25.6	28.4	0.76	0.90	74.1	47.6	1.28	1.56
⋅⋅⋅	B2	27.7	20.3	24.3	0.88	0.83	57.2	39.6	1.43	1.45

Notes. Column 1: source name. Column 2: Averaged column density of H₂ derived from ¹²CO(J = 1–0) in 10²⁰ cm⁻². Columns 3 and 4: averaged column density of H₂ derived from ¹³CO(J = 2–1) and ¹³CO(J = 1–0), respectively, in 10²⁰ cm⁻². Column 5: column density ratio of $R^{N}_{\rm 13/12} = N_{\rm LTE}^{13,1-0} / N_{\rm X}^{1-0}$. Column 6: column density ratio of $R^{N}_{\rm LTE,13} = N_{\rm LTE}^{13,2-1} / N_{\rm LTE}^{13,1-0}$. Columns 7 and 8: averaged column density of H₂ derived from C¹⁸O(J = 2–1) and C¹⁸O(J = 1–0), respectively, in 10²⁰ cm⁻². Column 9: column density ratio of $R^{N}_{\rm 18/12} = N_{\rm LTE}^{18,1-0} / N_{\rm X}^{1-0}$. Column 10: column density ratio of $R^{N}_{\rm LTE,18} = N_{\rm LTE}^{18,2-1} / N_{\rm LTE}^{18,1-0}$.

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The J = 1–0 transitions of ¹³CO and C¹⁸O have often been used to derive the column density under the assumption of LTE (e.g., Dickman 1978; Pineda et al. 2010) because the Einstein's A coefficient is small, and thus the critical density for the excitation is low. The J = 2–1 transitions have higher critical densities for the excitation and they can be sub-thermally excited in lower-density regions. In the analyses, we apply the LTE assumption for all of the transition lines and discuss the cause of the differences of the derived properties. Furthermore, we use the peak brightness temperature of each ¹²CO transition line to estimate the excitation temperature of the ¹³CO and C¹⁸O transitions. Assuming LTE, the excitation temperature T_ex is derived from the peak brightness temperature of the ¹²CO line, T_peak, as

$$\begin{equation} T_{\rm ex}^{1-0} = 5.53 \left\lbrace \ln \left[ 1+ \frac{5.53}{T_{\rm peak}^{12,1-0} + 0.84} \right] \right\rbrace ^{-1} \end{equation} \tag{ 1 }$$

$$\begin{equation} T_{\rm ex}^{2-1} = 11.06 \left\lbrace \ln \left[ 1+ \frac{11.06}{T_{\rm peak}^{12,2-1} + 0.19} \right] \right\rbrace ^{-1}. \end{equation} \tag{ 2 }$$

Using the excitation temperature, the optical depths of the ¹³CO and C¹⁸O emission lines are derived from the brightness temperature, T_mb(v),

$$\begin{eqnarray} &&\tau _{J=1}^{13}(v)\nonumber\\ &&\quad = - \ln \left\lbrace 1 - \frac{T_{\rm mb}^{13,1-0}(v)}{5.29} \left[ \frac{1}{\exp (5.29/T_{\rm ex}) - 1} - 0.17 \right]^{-1} \right\rbrace\nonumber\\ \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} &&\tau _{J=1}^{18}(v)\nonumber\\ &&\quad = - \ln \left\lbrace 1 - \frac{T_{\rm mb}^{18,1-0}(v)}{5.27} \left[ \frac{1}{\exp (5.27/T_{\rm ex}) - 1} - 0.17 \right]^{-1} \right\rbrace\nonumber\\ \end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray} &&\tau _{J=2}^{13}(v)\nonumber\\ &&\quad = - \ln \left\lbrace 1 - \frac{T_{\rm mb}^{13,2-1}(v)}{10.58} \left[ \frac{1}{\exp (10.58/T_{\rm ex}) - 1} - 0.02 \right]^{-1} \right\rbrace \nonumber\\ \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} &&\tau _{J=2}^{18}(v)\nonumber\\ &&\quad = - \ln \left\lbrace\! 1 - \frac{T_{\rm mb}^{18,2-1}(v)}{10.54} \left[ \frac{1}{\exp (10.54/T_{\rm ex}) - 1} - 0.02 \right]^{-1} \!\right\rbrace.\nonumber\\ \end{eqnarray} \tag{ 6 }$$

The column densities of ¹³CO and C¹⁸O in the upper state, N_u, are derived by the following equations:

$$\begin{equation} N_{J=1}^{13} = 1.98 \times 10^{16} \left[ \exp \left(\frac{5.29}{T_{\rm ex}} \right) - 1 \right]^{-1} \int \tau _{J=1}^{13}(v) dv \end{equation} \tag{ 7 }$$

$$\begin{equation} N_{J=1}^{18} = 1.97 \times 10^{16} \left[ \exp \left(\frac{5.27}{T_{\rm ex}} \right) - 1 \right]^{-1} \int \tau _{J=1}^{18}(v) dv \end{equation} \tag{ 8 }$$

$$\begin{equation} N_{J=2}^{13} = 1.65 \times 10^{16} \left[ \exp \left(\frac{10.58}{T_{\rm ex}} \right) - 1 \right]^{-1} \int \tau _{J=2}^{13}(v) dv \end{equation} \tag{ 9 }$$

$$\begin{equation} N_{J=2}^{18} = 1.64 \times 10^{16} \left[ \exp \left(\frac{10.54}{T_{\rm ex}} \right) - 1 \right]^{-1} \int \tau _{J=2}^{18}(v) dv. \end{equation} \tag{ 10 }$$

Assuming LTE, the column density of the rotational state of J is related to the total CO column density as

$$\begin{equation} N_{\rm total}({\rm CO}) = N_{J} \frac{Z}{2J+1} \exp \left[ \frac{h B_0 J (J+1)}{k T_{\rm ex}} \right] \protect,\protect \end{equation} \tag{ 11 }$$

where B₀ is the rotational constant of the CO isotopologues, B₀ = 5.51 × 10¹⁰ s⁻¹ and 5.49 × 10¹⁰ s⁻¹ for ¹³CO and C¹⁸O, respectively. Z is the partition function, which is given by

$$\begin{equation} Z = \sum _{J=0}^{\infty } (2J + 1) \exp \left[ - \frac{h B_0 J (J+1)}{k T_{\rm ex}} \right]. \end{equation} \tag{ 12 }$$

The column density of the molecular gas, N(H₂), is derived by

$$\begin{equation} N({\rm H_2}) = X N_{\rm total}({\rm CO}) \protect,\protect \end{equation} \tag{ 13 }$$

where X is the isotopic abundance ratio of the CO isotopologues relative to H₂. We adopt X[¹³CO] = 7.1 × 10⁵ and X[C¹⁸O] = 5.9 × 10⁶ (Frerking et al. 1982).

The derived averaged column densities over the whole observed area from ¹³CO and C¹⁸O of J = 2–1 and J = 1–0 are $N^{13,2-1}_{\rm LTE} = 20.7 \times 10^{20}$ cm⁻², $N^{13,1-0}_{\rm LTE} = 22.4 \times 10^{20}$ cm⁻², $N^{18,2-1}_{\rm LTE} = 62.8 \times 10^{20}$ cm⁻², and $N^{18,1-0}_{\rm LTE} = 44.2 \times 10^{20}$ cm⁻², respectively. The derived column densities are summarized in Table 2. We note here that the ¹³CO and C¹⁸O abundances can change depending on the surrounding environment, although we assume uniform distribution of the abundances throughout the clouds. The abundances seem to depend on self-shielding and the star formation activity, and the values for ¹³CO range mostly within [¹³CO]/[H₂] = 1–3.5 × 10⁻⁶ (e.g., Dickman 1978; Frerking et al. 1982; Lada et al. 1994; Harjunpää et al. 2004; Pineda et al. 2008, 2010; Shimoikura & Dobashi 2011; Ripple et al. 2013), which affect the estimates of the mass and column densities.

The column densities derived from ¹²CO are similar to those of ¹³CO  while C¹⁸O shows significantly higher averaged column densities. This indicates that C¹⁸O emission traces a higher column density region than ¹²CO and ¹³CO, probably due to the photodissociation and chemical fractionation of the species (e.g., Warin et al. 1996). Another possibility is that the abundance ratio of C¹⁸O in the Orion region is different from those in the other regions measured by Frerking et al. (1982).

4.1.3. Masses

The gas mass is calculated from the molecular gas column densities by

$$\begin{eqnarray} \left(\frac{M_{\rm gas}}{M_{\odot }} \right) &=& 4.05 \times 10^{-1} \mu _{\rm H_2} \left(\frac{m_{\rm H}}{\rm kg} \right) \left(\frac{d}{\rm pc} \right)^2 \left(\frac{\Delta l}{\rm arcmin} \right)\nonumber\\ &&\times \left(\frac{\Delta b}{\rm arcmin} \right) \left(\frac{N({\rm H_2})}{\rm cm^{-2}} \right) \protect,\protect \end{eqnarray} \tag{ 14 }$$

where $\mu _{\rm H_2} \sim 2.7$ is the mean molecular weight per H₂ molecule, m_H is the atomic hydrogen mass, d is the distance, and Δl and Δb are the pixel sizes along the galactic coordinates.

The derived gas masses are summarized in Table 3. The masses derived from J = 1–0 are larger than those derived from J = 2–1 in all of the CO isotopes. The total gas masses derived from ¹³CO(J = 1–0) for four regions are about 70–90% of those derived from the ¹²CO(J = 1–0) luminosities. The ratio of the total masses derived from the two molecular lines are almost uniform not depending on the regions. This implies that the optically thick ¹²CO(J = 1–0) line is well proportional to the total mass, and if we assume that the mass derived from ¹³CO(J = 1–0) traces the true total mass more reliably, then the X factor for the ¹²CO(J = 1–0) intensity is estimated to be 1.5 × 10²⁰ cm⁻² K⁻¹ km⁻¹ s. The mass derived from ¹³CO(J = 2–1) is lower than that from ¹³CO(J = 1–0) by a factor of about three, indicating that the J = 2–1 line is sub-thermally excited. Especially toward the A1 subregion, the ratio ¹³CO(J = 2–1)/¹³CO(J = 1–0) is lower than the other regions by a factor of 1.4. This indicates that the density of the Orion A1 region is lower than the other two regions, which is also discussed in the previous sub-subsection.

Table 3. Total Masses and Mass Ratios

Source		$M_{\rm X}^{1-0}$	$M_{\rm LTE}^{13,2-1}$	$M_{\rm LTE}^{13,1-0}$	$R^{M}_{\rm 13/12}$	$R^{M}_{\rm LTE,13}$	$M_{\rm LTE}^{18,2-1}$	$M_{\rm LTE}^{18,1-0}$	$R^{M}_{\rm 18/12}$	$R^{M}_{\rm LTE,18}$
(1)		(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
Orion		110.4	28.2	90.5	0.82	0.31	6.9	45.3	0.41	0.15
Orion A		70.6	17.5	59.8	0.85	0.29	3.9	28.1	0.40	0.14
⋅⋅⋅	A1	25.8	3.8	18.2	0.71	0.21	0.3	8.9	0.34	0.03
⋅⋅⋅	A2	44.8	13.7	41.6	0.93	0.33	3.6	19.2	0.43	0.19
Orion B		39.7	10.7	30.7	0.77	0.35	3.0	17.2	0.43	0.18
⋅⋅⋅	B1	24.0	6.7	18.2	0.76	0.37	1.4	9.0	0.37	0.16
⋅⋅⋅	B2	14.2	4.0	12.5	0.88	0.32	1.6	8.2	0.58	0.19

Notes. Column 1: source name. Column 2: total molecular cloud mass derived from ¹²CO(J = 1–0) in 10³ M_☉. Columns 3 and 4: total molecular cloud mass derived from ¹³CO(J = 2–1) and ¹³CO(J = 1–0), respectively, in 10³ M_☉. Column 5: mass ratio of $R^{M}_{\rm 13/12} = M_{\rm LTE}^{13,1-0} / M_{\rm X}^{1-0}$. Column 6: mass ratio of $R^{M}_{\rm LTE,13} = M_{\rm LTE}^{13,2-1} / M_{\rm LTE}^{13,1-0}$. Columns 7 and 8: total molecular cloud mass derived from C¹⁸O(J = 2–1) and C¹⁸O(J = 1–0), respectively, in 10³ M_☉. Column 9: mass ratio of $R^{M}_{\rm 18/12} = M_{\rm LTE}^{18,1-0} / M_{\rm X}^{1-0}$. Column 10: mass ratio of $R^{M}_{\rm LTE,18} = M_{\rm LTE}^{18,2-1} / M_{\rm LTE}^{18,1-0}$.

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Molecular lines with different critical densities for excitation can be used to estimate the density and temperature of the emitting region. For the optically thick molecular lines, we need to include the photon trapping effect. Photon trapping varies the excitation state and depends on the morphology of the cloud. For simplicity, we used the large velocity gradient (LVG) approximation method (e.g., Goldreich & Kwan 1974; Scoville & Solomon 1974), which assumes a spherically symmetric cloud of uniform density and temperature with a spherically symmetric velocity distribution proportional to the radius and uses a Castor escape probability formalism (Castor 1970). It solves the equations of statistical equilibrium for the fractional population of CO rotational levels at each density and temperature by incorporating the photon escape probability that is effective in the optically thick case. The widely used radiative transfer code RADEX also uses the same technique with the ability to choose three different formulations for the escape probability (van der Tak et al. 2007). In the non-LTE analyses, the intensities of a few lines are compared and used to determine the physical properties of the gas where lines are emitted. Therefore, the analyses are sensitive to the density similar to the critical densities of the used lines (e.g., Castets et al. 1990; Beuther et al. 2000; Zhu et al. 2003; Ward et al. 2003). The selection of lines for the non-LTE analyses are important. Recently, the combination of optically thin and thick lines with different transitions has been found to be a good tracer of the physical properties of the gas, and the derived physical properties well reflect the star formation activity and the surrounding environment (e.g., Martin et al. 2004; Nagai et al. 2007; Mizuno et al. 2010; Minamidani et al. 2011; Torii et al. 2011; Nagy et al. 2012; Peng et al. 2012; Fukui et al. 2014). In this paper, we use the ¹²CO(J = 2–1), ¹³CO(J = 2–1), and ¹³CO(J = 1–0) lines with single-component LVG analyses.

We use the intensity ratios, not the absolute intensities, for the analyses to minimize the effect of the beam filling factor when deriving the density and temperature. This is because the high spatial resolution observations revealed that molecular clouds contain many small-scale structures (see Ripple et al. 2013; Nakamura et al. 2012; Shimajiri et al. 2011 for the Orion case), and the beam filling factors are not unity. Sakamoto et al. (1994) discussed the effect of the unresolved clumps for their analyses based on the radiative transfer computations of clumps by Gierens et al. (1992), and suggested that the ratios reflect the physical conditions in constituent clumps and that the beam filling factors may change depending on the regions.

Our analyses include the lowest 40 rotational levels of the ground vibrational level and use the Einstein A coefficient and ortho/para H₂ impact rate coefficients of Schöier et al. (2005). The ratio of ortho- to para-H₂ molecules is calculated by assuming the thermal equilibrium state. We performed the calculations for a ¹²CO fractional abundance of X(¹²CO) = [¹²CO]/[H₂] = 1 × 10⁻⁴ and a ¹²CO/¹³CO abundance ratio of 71 (Frerking et al. 1982). Another parameter required for the calculation is the velocity gradient often described as dv/dr. Figure 14 shows contour plots of the LVG analyses by assuming dv/dr to be 1 km s⁻¹ pc⁻¹ which are derived from the typical line width and the size of the cloud. Solid and dashed lines show contours of ${R^{13}_{2-1/1-0}}$ and ${R^{13/12}_{2-1}}$ ratios, respectively. The figure indicates that the ${R^{13/12}_{2-1}}$ ratio basically depends on the density, and the ${R^{13}_{2-1/1-0}}$ ratio depends on both the density and temperature. The ${R^{13/12}_{2-1}}$ dependency comes from the facts that it reflects the optical depth of ¹³CO(J = 2–1) when ¹²CO(J = 2–1) is optically thick, and also that less H₂ density is needed for the collisional excitation of ¹²CO(J = 2–1) than the optically thin ¹³CO(J = 2–1) line due to the photon trapping effect of the ¹²CO(J = 2–1) line. When the ¹³CO lines are optically thin, ${R^{13}_{2-1/1-0}}$ does not depend on the column density or the change of the abundances, reflecting only the local physical properties of the density and temperature because of the absence of the photon trapping effect. This makes the ratio a good tracer of the physical properties. ${R^{13}_{2-1/1-0}}$ is dependent only on the temperature if both of the lines are optically thin and fully thermalized. This ratio also depends on the density in terms of the different critical density for the excitation. Roughly speaking, ${R^{13/12}_{2-1}}$ traces the density and ${R^{13}_{2-1/1-0}}$ is larger for higher temperature and density. Because these two ratios have different dependences on the density and temperature, we are able to estimate the density and temperature from the intersection in the figure.

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In the present paper, we assumed a uniform fractional abundance of CO and dv/dr for the analyses. The change of parameters results in the change of the derived n(H₂), and we estimate the effect here. The abundance fluctuations of [¹³CO]/[H₂] range mostly within a factor of two, as indicated in Section 4.1.2. Specifically, [¹³CO]/[H₂] is observed to be fairly constant in self-shielded clouds, where the ¹³CO(J = 1–0) intensity is relatively high (Ripple et al. 2013). The observed line width should reflect the ratio dv/dr. Figure 8(b) indicates that the line width ranges mainly between 1.5 and 3 km s⁻¹. We also calculated the change of the derived n(H₂) with the different assumptions of Xdr/dv. For a fixed ¹³CO(J = 2–1)/¹²CO(J = 2–1) ratio, the derived density is inversely proportional to square root of the assumed Xdr/dv (Figure 15). This means that even a factor of 10 variation in Xdr/dv only amounts to a factor of ∼3 in density.

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Finally, we briefly discuss the effect of density inhomogeneities on the above analyses. Here, we assumed uniform physical properties in a beam. However, this assumption may not be realistic because the density can vary along the line of sight. Because we used the ratios of the line intensities, the derived densities can vary depending on the density distributions. For example, in the peripheral area where the lower density gas is extended, the density contrast may not be as high. In this case, the derived density roughly represents the physical properties of the extended gas. On the other hand, toward dense core regions, the high-density gas is normally surrounded by the lower-density gas, and thus the derived density may not reflect the physical properties of the high-density gas. In this case, the derived density can thus vary depending on the density and optical depth of the lower density gas. Therefore, we need to estimate how much the lower-density gas can affect the line intensities, although this is not easy because it depends highly on the morphology of the clouds. In the present analyses, we mainly make use of the sub-thermality of the optically thin ¹³CO(J = 2–1) line to derive the density. In a sub-thermal density range, the line intensity depends highly on the density. With the column density fixed, the intensity is almost proportional to the density (e.g., see Figure A2 of Ginsburg et al. 2011), indicating that the higher density gas contributes to the line intensity more than the lower density gas, giving more weight to the higher density gas. This implies that the derived densities here represent those of the higher-density gas rather than those of the surrounding lower-density gas. For example, Snell et al. (1984) calculated the effect of the lower-density gas for CS lines and concluded that the effect is small when the optical depth of the foreground density gas is less than 0.5. Mundy et al. (1986) modeled spherical clouds with 1/R and 1/R² density dependence and suggested that the calculated line intensities for the optically thin line yield densities that are between the maximum and average densities along the line of sight.

4.2.1. Deriving Physical Parameters

First, we chose seven different points that have different environments for calculating the physical properties with the LVG analyses. Orion KL is an example of a region of high temperature (Figure 16). This region has a high ${R^{13}_{2-1/1-0}}$ and low ${R^{13/12}_{2-1}}$. The analyzed curves are well crossed, and thus the temperature and the density are well determined as 88 K and 1800 cm⁻³, respectively. The density is well consistent with the values estimated by Castets et al. (1990), who determined the density to be a few 10³ cm⁻³. The OMC-3 region is an example of a region of high density and moderate temperature (Figure 17). This region has a high ${R^{13/12}_{2-1}}$ with moderate ${R^{13}_{2-1/1-0}}$. The analyzed curves are also well crossed for this region, and the temperature and the density are determined to be 34 K and 2200 cm⁻³, respectively. L1641S is an example of a region of low density and low temperature (Figure 18). This region has low ${R^{13/12}_{2-1}}$ with low ${R^{13}_{2-1/1-0}}$. We determined the temperature and density to be 10 K and 1000 cm⁻³, respectively. From these analyses, the temperature and density are successfully derived for the different environments. We also analyzed some other regions. Our results are summarized in Table 4.

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Table 4. Results of LVG Analyses

Source	l	b	$T^{12}_{2-1}$	$T^{13}_{2-1}$	$T^{13}_{1-0}$	$R^{13/12}_{2-1}$	$R^{13}_{2-1/1-0}$	Tkin	n(H2)
	(deg)	(deg)	(K)	(K)	(K)			(K)	(cm−3)
Orion KL	209.00	−19.40	57.8	15.6	10.1	0.27	1.54	88	1800
OMC-3	208.60	−19.20	23.8	12.2	10.9	0.51	1.12	34	2200
L1641-N	210.07	−19.67	19.2	5.8	7.7	0.30	0.76	21	1300
L1641-S	212.00	−19.33	6.8	2.1	4.1	0.31	0.51	10	1000
NGC 2024	206.53	−16.33	20.2	13.6	14.7	0.67	0.93	30	2200
NGC 2023	206.87	−16.53	30.2	12.8	13.9	0.43	0.92	33	2000
NGC 2068	205.40	−14.33	25.6	12.5	14.2	0.49	0.88	26	1600
NGC 2071	205.13	−14.13	14.3	6.8	9.6	0.48	0.71	30	1400

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4.2.2. Spatial Distribution of Density and Temperature

As we described in the previous subsection, the temperature and density of the molecular gas can be well determined by the LVG analyses in various environment. Thus, we apply this method to all of the observed pixels. The procedures we used are as follows: (1) we first generated the integrated intensity ratio maps of $R^{13}_{2-1/1-0}$ and ${R^{13/12}_{2-1}}$, (2) then, we calculated the line intensity for each density and temperature by using LVG analyses assuming uniform sphere structure and a constant velocity gradient (dv/dr = 1 km s⁻¹ pc⁻¹), and (3) finally, we compared the observed line ratios and calculated intensity ratios to determine the physical properties of the molecular gas using χ² test. The results of the analyses are shown in Figures 19 and 20 for the kinematic temperature and the density of the molecular gas, respectively.

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The kinematic temperature is mostly in the range of 20 K to 50 K along the cloud ridge. The temperature tends to be high in the active star formation sites and decline to the peripheral regions. We found especially high temperatures in some regions. One region is the eastern part of the Orion KL region near the Trapezium cluster. This region is considered to interact with the stellar wind and radiation from the Trapezium cluster. The western part of this region has no significant high-temperature structures. Another region is the southern edge of the Orion B cloud which is located in front of the OB1b subgroup. This region seems to be influenced by the radiation of old OB stars. We also note some other high-temperature regions. One region is located in the vicinity of L1641N. Actually, this region has a high temperature (∼100 K) but not as large a spacial extent as that of Orion KL. This suggests that L1641N is more deeply embedded in the molecular gas. Another region is the south-west side of the main ridge of Orion A. In this region, molecular gas is probably heated by the OB1b or 1c subgroups located southeast of the Orion A molecular cloud.

The densities derived with the analyses show values in the range of 500 to 5000 cm⁻³. The lowest density we can probe is determined by the critical density of ¹³CO(J = 2–1) for excitation. The highest density we can identify is limited by the thermalization and/or by the high opacity of the line emission toward the dense gas region. The high-density regions (∼2000 cm⁻³) are located in the north of Orion KL for Orion A and in the south of the Southern cloud for Orion B. In Orion A, the main ridge has a density gradient decreasing toward the outer regions, as pointed out by Sakamoto et al. (1994). We can also find small-scale density variations. For instance, there are local peaks around the L1641N and L1641S regions.

In this subsection, we compare the derived physical parameters of the gas to the star formation activity. In the Orion region, a new catalog of young stellar objects (YSOs) was recently complied from the infrared survey using the Spitzer space telescope (Megeath et al. 2012) toward regions with high extinction, which we refer to as the Spitzer catalog. The cataloged YSOs have dusty disks or infalling envelopes and are considered to have formed recently in the currently existing molecular clouds. The catalog has unveiled the spatial distribution of thousands of YSOs, enabling us to carry out the direct comparison of the star formation efficiency (SFE) with gas temperature and density.

We calculated the surface number density of YSOs, N_*, using the Spitzer catalog in the same grid as our CO data set (Figure 21). We used both the "disked" and "protostar" objects in their catalog. We then derived the distribution of the SFE using the following equation:

$$\begin{equation} {\rm SFE} = \frac{M_*}{M_* + M_{\rm cloud}}, \end{equation} \tag{ 15 }$$

where M_* is the mass of YSOs estimated as M_* = m_*N_* assuming the mean stellar mass m_* = 0.5 M_☉ (Evans et al. 2009), and M_cloud is the mass of the molecular gas. We use the LTE mass derived from ¹³CO(J = 1–0) line emission for the total molecular gas mass. We calculated the averaged SFEs for each subregions introduced in Section 4.1. Our results are summarized in Table 5. The subregions in Orion A have higher SFE than those of the Orion B subregions.

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Table 5. Summary of Molecular Cloud Properties

Region/Subregion	$\langle N^{13}_{1-0}(\rm H_2) \rangle$	$M^{13}_{1-0}$	〈Tkin〉	〈n(H2)〉	SFE
	(1020 cm−2)	(103 M☉)	(K)	(cm−3)
The entire Orion region	22.4	90	28.8	1000	0.037
The entire Orion A region	20.8	59	25.4	1000	0.045
⋅⋅⋅      A1	16.9	18	14.9	870	0.025
⋅⋅⋅      A2	23.2	41	31.8	1100	0.054
The entire Orion B region	26.4	30	35.8	1000	0.020
⋅⋅⋅      B1	28.4	18	44.7	990	0.018
⋅⋅⋅      B2	24.3	12	25.5	1000	0.023

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In this subsection, we discuss the relationship between the cloud properties and the star formation activity in the Orion molecular cloud. Figure 21 clearly shows that there are more YSOs in regions where the gas column density is higher. Figure 22(a) shows that the number of YSOs are positively correlated with the gas column density, although the tendency is unclear in the case of Orion B2. This trend is also seen in the gas density as shown in Figure 22(b), indicating that the density of the gas is key to the star formation activity therein. Table 5 summarizes the SFEs of the subregions. It is very striking that the SFE of the Orion A2 subregion is much higher compared to the other subregions. However, the average column density, temperature, and density of Orion A2 are not significantly different from the other subregions, implying that the SFE is not necessarily determined by the overall properties of the molecular clouds. Figure 23(a) shows the relation of SFE to the gas column density, and Figure 23(b) the relation to the gas volume density. It is obvious that the SFE is well correlated with the gas density, i.e., more stars are formed in denser regions. The poorer correlation in Orion B1 may be a result of the gas dispersion due to the active star formation in NGC 2024 and the external disturbances, as discussed in the next subsection.

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The positive correlation between the gas number density and the SFE indicates that the timescale from gas to protostar, T_collapse, is shorter for denser gas if we assume steady star formation. In this case, the total mass of the formed stars M_* is proportional to the total gas mass M_cloud and inversely proportional to the timescale of star formation T_collapse (i.e., M_*∝M_cloud/T_collapse). Therefore, the SFE is inversely proportional to T_collapse. If T_collapse is described as αT_ff, where T_ff is the free fall timescale, then the SFE is proportional to (αT_ff)⁻¹, and then to α⁻¹n^1/2, where n is the volume density of the gas. α depends on the balance between self-gravity and the other forces, is unity for the self-gravity dominating case, and then T_collapse may depend on the volume density. The data in Figure 23(b) shows that the SFE is roughly explained as SFE∝n^1/2, although the scatter is large and supposes that the dynamics of the gas is different from region to region.

The gas temperature is highest toward the region around Orion KL, probably due to heating by massive stars forming therein. There is a slight temperature enhancement along the ridge of Orion A, and this may be due to the star formation inside. We see an enhancement of the gas temperature toward NGC 2023 in Orion B, although the gas temperature does not seem to be well correlated with the star formation activity in Orion B.

It should be noted that the Spitzer catalog of Megeath et al. (2012) does not cover the whole extent of the molecular gas. A recent study with the Akari and WISE cataloges indicates that there are YSOs identified outside the Spitzer area (Tóth et al. 2013). We are also interested in the SFE in somewhat isolated clumps like the EC clumps and the Northern clumps, and this is one subject in a forthcoming paper.

In this subsection, we discuss the effect of the surrounding environment on the physical properties of the molecular clouds. Figure 24 shows the intensity distribution of Hα (Gaustad et al. 2001) compared with the molecular gas distribution. There are some intense peaks that correspond to Orion KL, the southern side of the Orion B cloud, and the Bernard loop. The Bernard loop is considered to have formed through an interaction with an old supernovae, and the other H ii regions are considered to have formed through the Ori OB1 association. The Bernard loop seems to have no interaction with the molecular cloud, as suggested by Sakamoto et al. (1994).

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The Hα peak toward the Orion KL is clearly due to the current active massive star formation therein. The Hα enhancement toward the southern side of the Orion B consists of two parts: one is NGC 2024 and the other is along the southern edge of the Orion B cloud. The former seems to reflect ongoing star formation activity. The latter is ionized by the strong UV radiation from the OB1b subgroups. There is a clear gas density and temperature enhancement toward the southern edge of Orion B1, as shown in Figures 20 and 19, and this fact suggests that the strong stellar wind and UV radiation compress and heat the molecular gas. Another important feature of Orion B1 is that the gas temperature is higher than in other subgroups as a whole. Specifically, the temperature is higher toward the surrounding edge of Orion B1. This implies that the Orion B1 cloud is surrounded by an H ii region heating the outer edge of the molecular gas of Orion B.