© ESO, 2017

1. Introduction

One of the main challenges of astronomy and cosmology is to model, and reach an understanding, of the evolution of galaxies and large-scale structure. The star-formation rate (SFR) is a crucial parameter in these models. In order to measure the SFR in distant galaxies, several possible tracers have been and are being studied: ultraviolet (UV) radiation, infrared (IR) radiation, emission from polycyclic aromatic hydrocarbons (PAHs), atomic and molecular lines (e.g., Kennicutt 1998; Kennicutt & Evans 2012). With the advent of the Atacama Large (sub)Millimeter Array (ALMA), it has become popular to use the [C ii] 158 μm line as an indicator of the SFR over cosmic history (e.g., Herrera-Camus et al. 2015; Vallini et al. 2015; Pentericci et al. 2016). However, the origin of [C ii] emission on a galactic scale is still unclear.

Intuitively, the SFR is expected to depend on the local conditions in the interstellar medium (ISM), the gas and dust that form the environment of stars. The ISM comes in different phases, diffuse gas being the most prevalent. These phases are the cold neutral medium (CNM) with moderate gas densities, n ~ 30 cm^-3, and moderate gas temperatures, T ~ 100 K, the warm neutral and warm ionized medium (WNM and WIM) with low densities and high temperatures, n ~ 0.3 cm^-3 and T ~ 8000 K, and the hot ionized medium (HIM) with very low densities and very high temperatures, n ~ 3 × 10^-3 cm^-3 and T ~ 10⁶ K. Most of the gas of the ISM is in the neutral phase. Other ubiquitous components of the ISM are H ii regions around massive stars with densities ranging from n ~ 1 cm^-3 to n ~ 10⁵ cm^-3 and T ~ 10⁴ K, and molecular clouds with high density and low temperatures, n ~ 10³ − 10⁸ cm^-3 and T ~ 10 − 30 K (Hollenbach & Tielens 1999). These phases are not isolated from each other, but there is an exchange of matter between them, particularly driven by ionization, winds, and explosions of massive stars. Molecular clouds especially are the birthplaces of new (massive) stars and thereby of vital interest. Meyer et al. (2008) provide a review of star formation in L1630. At the interface between an H ii region, ionized by a massive star, and a parental molecular cloud, a photodissociation region (PDR) is formed, where intense stellar UV radiation impinges on the surface of the dense cloud. At the surface of these clouds, the gas is atomic; deeper inside the cloud, the molecular fraction increases. The study of PDRs reveals much about the interplay between stars (including hosts of newly formed stars) and the ISM, thereby yielding valuable insight into the process of star formation (see Hollenbach & Tielens 1999, for a review of PDRs).

The ISM is mainly heated by stellar radiation, specifically by far-ultraviolet (FUV) radiation with energies between 6 and 13.6 eV. The characteristics of the gas cooling allow us to infer the amount and, possibly, the source of the heating. One of the main coolants of the cold neutral medium is the [C ii] ${}^{\mathrm{2}}\mathit{P}_{\mathrm{3}\mathit{/}\mathrm{2}}{\mathrm{-}}^{\mathrm{2}}{\mathit{P}}_{\mathrm{1}\mathit{/}\mathrm{2}}$ fine-structure line at λ ≃ 158 μm, that is, ΔE/k_B ≃ 91.2 K. The [C ii] line is also one of the brightest lines in PDRs, carrying up to 5% of the total far-infrared (FIR) luminosity, the other 95% mainly stemming from UV irradiated dust. Carbon has an ionization potential of 11.3 eV, hence C⁺ traces the transition from H⁺ to H and H₂. Another important coolant is the [O i] line at λ ≃ 63 μm (ΔE/k_B ≃ 228 K). The ratio of those two main coolants depends on the temperature and density of the gas. For T = 100 K, the [O i] cooling efficiency overtakes the [C ii] cooling efficiency at n ≃ 3 × 10⁴ cm^-3; at n = 3 × 10³ cm^-3, the [O i] contribution to the total cooling is about 5% (cf. Tielens 2010).

The [C ii] line has been studied in a variety of environments. Important contributions may come from diffuse clouds (CNM), dense PDRs, surfaces of molecular clouds, and (low-density) ionized gas including the WIM (e.g., Wolfire et al. 1995; Ossenkopf et al. 2013; Gerin et al. 2015). Langer et al. (2010) identify warm CO-dark molecular gas in Galactic diffuse clouds by means of [C ii] emission. Jaffe et al. (1994) conducted an earlier study observing the extended [C ii] emission from the Orion B molecular cloud (L1630). This study is preceded by a [C ii] survey of the Orion Molecular Cloud 1 (OMC1) in Orion A by Stacey et al. (1989). Goicoechea et al. (2015) present a velocity-resolved [C ii] map toward OMC1, observed by the Heterodyne Instrument for the Far-Infrared (HIFI) onboard the Herschel satellite in 2012. Velocity-resolved [C ii] and [¹³C ii] emission from the star-forming region NGC 2024 in L1630 was observed in 2011 using the GREAT (German Receiver for Astronomy at Terahertz Frequencies) instrument onboard the airborne Stratospheric Observatory for Infrared Astronomy (SOFIA) and analyzed by Graf et al. (2012); the neighboring reflection nebula NGC 2023 was observed in 2013/14 using the same instrument. Sandell et al. (2015) discussed the physical conditions, morphology, and kinematics of that region. A theoretical study on collisional excitation of the [C ii] fine-structure transition was performed by Goldsmith et al. (2012). The GOT C⁺ survey (Galactic Observations of Terahertz C⁺) survey (Pineda et al. 2014), also a Herschel/HIFI study, investigated specifically the relationship between [C ii] luminosity and SFR. This study found a good correlation on Galactic scales. This was also established by Stacey et al. (2010) and Herrera-Camus et al. (2015) at low and high redshift.

On December 11, 2015, a part of the Orion B molecular cloud, including the Horsehead Nebula, was observed in [C ii] with the upGREAT instrument, the first multi-pixel extension of GREAT, onboard SOFIA, as presented and described in Risacher et al. (2016). The survey was conducted “to demonstrate the unique and important scientific capabilities of SOFIA, and to provide a publicly available high-value SOFIA data set”¹. It allows us to study [C ii] emission and its correlations with other astrophysical tracers under moderate conditions (intermediate density and moderate UV-radiation field), as opposed to the high density and intense UV-radiation field in OMC1.

In the present study, we analyze the [C ii] emission from a 12′ × 17′ area of the L1630 molecular cloud in Orion B that is illuminated by the nearby star system, σ Ori. Our distance to the star system is approximately 334 pc, which we also assume to be the distance to the molecular cloud. The projected distance between the star system and the molecular cloud is 3.2 pc (Ochsendorf et al. 2014, and references therein). Part of the mapped area, in which star formation is low, is the well-known Horsehead Nebula. The star-forming regions NGC 2023 and NGC 2024 are adjacent to the mapped area, but not included. We compare the velocity-resolved [C ii] SOFIA/upGREAT observations with new CO (1 − 0) observations of the molecular gas obtained with the 30 m telescope (Pety et al. 2017) at the Institut de Radioastronomie Millimétrique (IRAM), with Spitzer/Infrared Array Camera (IRAC) studies of the PAH emission from the PDR surfaces, Hα observations of the ionized gas, and with existing far-infrared continuum studies using Herschel/Photoconductor Array Camera and Spectrometer (PACS) and Spectral and Photometric Imaging Receiver (SPIRE) data to determine dust properties and trace the radiation field. This wealth of data allows us to separate emission from the ionized gas, neutral PDR, and molecular cloud, in order to derive global heating efficiencies and their dependence on the local conditions, and to make detailed comparisons to PDR models.

This paper is organized as follows. In Sect. 2, we present the observations. In Sect. 3, we divide the surveyed area into regions with specific characteristics. Furthermore, we study the kinematics of the gas as revealed by the SOFIA/upGREAT observations of [C ii] emission and the correlation of the various data sets with each other. Section 4 contains a discussion of the results obtained in Sect. 3 and we derive column densities and other gas properties. We conclude with a summary of our results and an outlook for the future in Sect. 5.

2. Observations

2.1. [C II] observations

The [C ii] emission in Orion B (L1630) was observed on December 11, 2015 using the upGREAT instrument onboard SOFIA. The region was observed using the upGREAT optimized on-the-fly mapping mode. The region was split into four tiles, each covering an area of 363″ × 508.2″. In this mode, the array is rotated 19.1° on the sky and an on-the-fly (OTF) scan is undertaken. By performing a second scan separated by 5.5″ perpendicular to the scan direction, it is possible to fully sample a region 72.6″ wide along the scan direction (cf. Risacher et al. 2016, for details). By combining OTF scans in the RA and Dec direction, it is possible to cover the map region with multiple pixels. Each tile was made up of 10 x-direction OTF scans and 14 y-direction OTF scans. A spectrum was recorded every 6″. Scans in the x-direction had an integration time of 0.4 s, while those in the y-direction had an integration time of 0.3 s. Since the y-scan length is longer than the x-scan length, the integration time was reduced. This is due to Allan variance stability time limits of 30 s. A reference position at about 12′ to the west of the map was observed, which was verified to be free of ¹²CO (2 − 1) and ¹³CO (2 − 1) emission with the James Clerk Maxwell Telescope (JCMT; G. Sandell, priv. comm.). The reference position was checked to be free of [C ii] emission to a 1 K level. A supplementary off contamination check was undertaken whereby the reference position was calibrated using the internal hot reference measurement; these spectra also showed no evidence of off emission to a 1 K level. The [C ii] map itself, showing no “absorption” features anywhere, confirms that there cannot be notable [C ii] emission at the reference position. An off measurement is ideally taken after 30 s of on source integration to avoid drift problems in the calibrated data. For this observing run, each tile was observed twice in the x- and y-directions, resulting in a total integration time per map pixel of 1.4 s. For a spectral resolution of 0.19 km s^-1, this results in a noise rms in the final data cube of 2 K in the velocity channels free of emission.

The data cube provided by the SOFIA Science Center was processed using the Grenoble Image and Line Data Analysis Software²/Continuum and Line Analysis Single-dish Software (GILDAS/CLASS). We subtract a baseline of order one from the spectra. The spectral data were integrated over the velocity range (with respect to the local standard of rest, LSR) v_LSR = 6 − 20 km s^-1 to obtain the line-integrated intensity, which is shown in Fig. 1. Channel maps are shown in Fig. 4. The spatial resolution of our final maps is 15.9″. For comparison with other tracers, we use a Gaussian kernel for convolution. At the rim of the map, the [C ii] signal suffers from noise and we ignore an outer rim of 45″ in our analysis.

2.2. Dust SED analysis

In this study we make use of the dust temperature and dust optical depth maps released by Lombardi et al. (2014). Lombardi et al. (2014) fit a spectral energy distribution (SED) to Herschel/PACS and SPIRE observations of the Orion molecular cloud complex in the PACS 100 μm and 160 μm, and SPIRE 250 μm, 350 μm, and 500 μm bands. The photometric data, convolved to the SPIRE 500 μm36″ resolution, are modeled as a modified blackbody, $\begin{array}{ccc}\mathit{I}\mathrm{\left(}\mathit{\lambda }\mathrm{\right)}\mathrm{=}\mathit{B}\mathrm{\left(}\mathit{\lambda ,}{\mathit{T}}_{\mathrm{d}}\mathrm{\right)}\hspace{0.17em}{\mathit{\tau }}_{\mathrm{0}}{\left(\frac{{\mathit{\lambda }}_{\mathrm{0}}}{\mathit{\lambda }}\right)}^{\mathit{\beta }}\mathit{,}& & \end{array}$(1)with T_d the effective dust temperature, τ₀ the dust optical depth at the reference wavelength λ₀, and β the grain-emissivity index. Lombardi et al. (2014) use the all-sky β map with 35′ resolution by the Planck collaboration, interpolated to the grid on which the SED is performed; only the effective dust temperature and τ₀ are free parameters in this fit. The β map shows a smooth increase of about 3% from the north-east to the south-west in the area surveyed in [C ii], with a mean of 1.56. Lombardi et al. present their dust optical depth map at λ₀ = 850 μm, following the Planck standard, but for our analysis we convert τ₈₅₀ to τ₁₆₀ using the β data. We integrate Eq. (1) from λ_min = 20 μm to λ_max = 1000 μm to obtain the far-infrared intensity I_FIR.

We notice that the Horsehead PDR has comparatively low dust temperature in the SED fit, T_d ≃ 20 − 22 K. This could be due to beam dilution. In the models of Habart et al. (2005), the dust temperature is T_d ≃ 30 K at the edge, dropping to T_d ≃ 22 K for a hydrogen nucleus gas density of n_H = 2 × 10⁴ cm^-3 within 12″, and to T_d ≃ 13.5 K for n_H = 2 × 10⁵ cm^-3. Throughout this paper, by “gas density” we mean the hydrogen nucleus gas density: n_H = n_{H i} + 2 n_H2.

The derived effective dust temperature and dust optical depth can depend significantly on the choice of β: the temperature can be up to 3 − 4 K lower if β = 2 instead of β = 1.5; τ₁₆₀ then increases by a factor of two. The FIR intensity is less sensitive to β: it only decreases by 10% for β = 2.

Furthermore, we employ Spitzer/IRAC observations in the 8 μm band, which is dominated by PAHs but which can be influenced by very small grains. We use a super mosaic image retrieved from the Spitzer Heritage Archive, created October 22, 2012. We also make use of the 850 μm observations from the Submillimetre Common-User Bolometer Array 2 (SCUBA-2) around NGC 2023/2024 first presented by Kirk et al. (2016) as part of the JCMT Gould Belt Survey (GBS). These trace dense regions within the molecular cloud. However, we do not use the map reduced by the GBS group, but we retrieved the data from the Canadian Astronomy Data Centre (CADC) archive, processed on October 1, 2015.

2.3. CO (1–0) observations

In this work we make use of part of the ¹²CO (1 − 0) large-scale map at 115.271 GHz obtained by Pety et al. (2017) with the Eight Mixer Receiver (EMIR) 090 at the IRAM 30 m telescope. The fully sampled on-the-fly line maps were taken with a channel spacing of 195 kHz (a velocity resolution of ~0.5 km s^-1). CO-emission contamination from the reference position was eliminated by adding dedicated frequency-switched line observations of the reference position itself (see Pety et al. 2017, for details). The median noise levels range from 100 to 180 mK (in the T_mb scale) per resolution channel. Here we use the CO (1 − 0) line-integrated intensity map in the v_LSR = 9 − 16 km s^-1 range³, convolved to the 36″ angular resolution of SPIRE 500 μm. The resulting map is shown in Fig. 3.

2.4. Hα observations

In this study we use the Hα image of the Horsehead Nebula and its environs in L1630 and the H ii region IC 434 taken by the Mosaic 1 wide field imager on Kitt Peak National Observatory (KPNO). For calibration of the KPNO image, we use Hα data of the Horsehead Nebula collected by the Hubble Space Telescope as part of the Hubble Heritage program. We obtained the image from the archive of the National Optical Astronomy Observatory (NOAO), but it was taken as part of the program presented in Reipurth et al. (1998).

The bright star at 05h41′02.70″, − 02°18′17.77″ in the Hα image is a foreground star; it is visible in the IRAC 8 μm image, as well. We masked it before convolution, such that it does not show in the convolved images.

3. Analysis

3.1. Kinematics: velocity channel maps

Perusing the [C ii] channel maps from 8.0 km s^-1 to 16.0 km s^-1 shown in Fig. 4, we recognize several continuous structures in space-velocity. From 10.5 − 11.5 km s^-1, we observe a [C ii] front that runs from the south-east to the north-west of the map. From 12.5 − 14.0 km s^-1, a front runs from the north to the south. The Horsehead mane is visible from 10.0 − 11.5 km s^-1. From 12.5 − 13.0 km s^-1, an intermediate [C ii] front lights up. Based on the kinematic behavior and assuming that [C ii] emission is related to PDR surfaces, we divide the [C ii] fronts into four groups: PDR1 in the north-west of the molecular cloud, PDR2 in the south-west, PDR3 in the south-east, and the Horsehead PDR. The intermediate PDR front we do not discuss in detail.

In the luminous north, an almost circular cavity forms in the center of the region of highest intensity. Its boundary lights up in the 14.0 km s^-1 map. In the 12.5 km s^-1 and 13.0 km s^-1 channels, we see bright emission where the rim of the cavity is. This cavity appears quite clearly in the unconvolved IRAC 8 μm image (see Fig. 15). Comparison with the 8 μm map reveals a (proto-)star at the northern edge of the bubble. This star is visible in the Hα image as well, but it is not identified as a pre-main-sequence (PMS) object in Mookerjea et al. (2009). Here it is listed as MIR-29, a more evolved star in the vicinity of NGC 2023, identified by the Two Micron All Sky Survey (2MASS). In addition to the main emission in the velocity range v_LSR = 6 − 20 km s^-1, we see a faint [C ii] component at v_LSR ≃ 5 km s^-1, that has also been detected in CO observations by Pety et al. (2017). Due to its faintness, however, we will ignore it in our analysis.

3.2. Global morphology

Apart from the four PDR surfaces discussed in Sect. 3.1, we singled out other specific regions that stand out in their morphology in the respective quantities [C ii], CO, and IRAC 8 μm emission (see Fig. 5). The 8 μm emission is a tracer of UV-pumped polycyclic aromatic hydrocarbons (PAHs) and therefore of PDR surfaces. Ionized gas is traced by Hα emission, and CO traces the molecular hydrogen gas. The regions are indicated in Figs. 2 ([C ii] map) and 3 (CO (1 − 0) map). We outline the boundary between the H ii region IC 434 and the molecular cloud L1630 by the onset of significant [C ii] emission at the molecular cloud surface where we also have been guided by the Hα contour of highest emission (see Fig. 18).

The four PDR regions, among them the Horsehead PDR⁴, are distinct in the IRAC 8 μm map. The Horsehead PDR is not the most luminous part of the region in all maps. The brightest part is region PDR1. We define the neck of the Horsehead Nebula that is traced by the CO (1 − 0) line; in itself, it has little [C ii] emission, but part of it is covered by the [C ii]-emitting molecular cloud. Another part of the cloud, where there is little CO emission, we call CO-dark cloud. Deeper inside the molecular cloud, CO emission is high and we define a region of CO cores or clumps. The Horsehead PDR is likely to suffer from beam dilution in all images, since its scale length is found to be less than 10″ (Habart et al. 2005), which is smaller than the beam sizes in question. To the north-east of the map, we recognize the reflection nebula NGC 2023, which has been studied with SOFIA/GREAT in [C ii] emission by Sandell et al. (2015). This region will not be discussed here.

Figure 5a shows the FIR intensity with [C ii] contours in the mapped area. The FIR intensity peaks close to [C ii] in the most luminous part (PDR1), but slightly deeper into the cloud. The Horsehead mane is bright in both [C ii] and FIR; the emission overlaps very well. PDR2 is more pronounced in [C ii] emission than in the FIR. PDR3 can only be surmised in I_FIR, but it cannot be distinguished very clearly in the integrated [C ii] map either.

In Fig. 5b, we compare the 8 μm emission with [C ii] in contours. The 8 μm emission behaves in a similar way as I_FIR, but structures stand out more decidedly. The 8 μm emission, too, peaks slightly deeper into the cloud than [C ii]. The bright regions in the Horsehead mane overlap; in both maps it is a thin filament. PDR2 is more pronounced in [C ii], but is distinguishable in I_{8 μm} as well. PDR3 is more distinct in I_{8 μm}.

The CO (1 − 0) emission in the mapped area does not resemble the pattern of [C ii] emission (Fig. 5c). In CO, the entire Horsehead and its neck light up with nearly equal intensity while the surroundings remain dark. PDR1 and 2 are not very bright in CO. Interestingly, there is a CO spot in PDR1, right where the cavity is observed in (unconvolved) [C ii] and 8 μm emission (see Figs. 4 and 15, respectively). A “finger” of CO emission, the “CO clumps”, points towards PDR1. PDR3 can be inferred as shadow in CO emission, that is, a ridge of low CO emission.

The τ₁₆₀ map (Fig. 5d) resembles the CO (1 − 0) map. The Horsehead and its neck have higher dust optical depth than their surroundings; the material directly behind the mane is a peak in τ₁₆₀, that overlaps partially with the [C ii] peak, but it peaks slightly deeper into the Horsehead. The [C ii] peak in PDR1 does not correspond to a peak in dust optical depth, although the onset of the molecular cloud is traced by an increase in τ₁₆₀. PDR3 corresponds to an optically thin region compared to its environment. The region of highest dust optical depth at the eastern border of the map (not containing NGC 2023) corresponds to a region with little [C ii] emission, but high CO emission. The CO “finger” relates to enhanced dust optical depth.

PDR1 and PDR2 border on the H ii region, as can be seen from Fig. 5e. PDR2 overlaps with a region of significant Hα emission, tracing the ionized gas at the surface of the molecular cloud. The Hα map and the logarithmic [C ii] cooling efficiency I_{[ C ii ]}/I_FIR map (Fig. 5f) resemble each other. High [C ii] over FIR intensity ratios are found near the boundary with high Hα emission. However, I_{[ C ii ]}/I_FIR is a misleading measure in the H ii region, since I_FIR does not trace the radiation field well here and I_{[ C ii ]} is sufficiently low to be significantly affected by noise. Variations in I_{[ C ii ]}/I_FIR across the map span a range from 3 × 10^-3 up to 3 × 10^-2. Table 1 lists the mean values of the [C ii] cooling efficiency η = I_{[ C ii ]}/I_FIR, [C ii] intensity I_{[ C ii ]}, CO(1 − 0) intensity I_{[ C0 (1 − 0) ]}, FIR intensity I_FIR, and dust optical depth τ₁₆₀ for the several regions defined above.

3.3. Kinematics: velocity-resolved line spectra

Figure 6 displays spectra extracted towards different positions in the map, as indicated in Fig. 1. The positions are chosen as representative single points for the different regions we identified earlier. Details on the spectra, that is, peak position, peak temperature, and line width, are given in Table 2. Point A represents the Horsehead mane, B lies in the most luminous part of the map, C just north of it, whereas D is displaced to the east; all three represent PDR1. We chose additional points in PDR1, because B might be affected by the bubble structure discussed later. Point E lies in PDR2 behind the Horsehead and F in the southern part of PDR2. Point G is located in the intermediate PDR front which we do not discuss in detail. Point I represents PDR3, whereas H is chosen in the CO-dark cloud, where there is little [C ii] emission (and little emission in other tracers).

The spectrum taken towards the Horsehead PDR (point A) shows a narrow line. Opposed to this is the line width of the spectrum extracted towards the most luminous part of the molecular cloud, point B: here, the line is broadened. It peaks at a slightly higher velocity than the Horsehead PDR. From comparison with the dust optical depth, we conclude that the broadening of the line is not due to a high column density (if dust density and gas density are related). The same holds for point C in PDR1. Here there appears a small side peak at higher velocity, which could also be inferred for point B (as a shoulder). Point D evidently has a spectrum with two peaks. From the distinctly different morphology of the channel maps at the two peak velocities (cf. Fig. 4 at 10.5 km s^-1 and 13.0 km s^-1), we surmise that the two peaks correspond to two distinct emitting components, rather than to one emission component with foreground absorption. The same goes for point E in PDR2, which also has two peaks. The southern part of PDR2, point F, has only one rather narrow peak. The intermediate PDR, point G, exhibits a strong narrow line, as well. The spectrum taken in the western PDR, point I, shows a somewhat broader line with somewhat lower intensity. At point H, where the intensity is low in all tracers, the [C ii] line is also broader.

Strikingly, the peak velocity of point D is shifted towards lower velocity by 1 km s^-1 with respect to points B and C (all PDR1). However, one component of this spectrum lies at about 11 km s^-1, which is also the velocity of PDR3 (point I). This is further evidence that PDR3 and a part of PDR1 are spatially connected, as concluded from the channel maps. Point D in PDR1 has a component at about 13 km s^-1, which is the velocity of PDR2. In B and C (PDR1) this component might be hidden beneath the strong side peak at 14 km s^-1. The affiliation of the second component at point E in PDR2 is unclear; there might be another layer of gas behind or in front of the main component, or it could originate from the gas of PDR1 and PDR3 at 11 km s^-1.

3.4. Edge-on PDR models

We supplement the correlation plots in the following section with model runs that are based on the PDR models of Tielens & Hollenbach (1985), with updates like those found in Wolfire et al. (2010) and Hollenbach et al. (2012). We include the most recent computations on fine-structure excitations of C⁺ by collisions with H by Barinovs et al. (2005) and with H₂ by Wiesenfeld & Goldsmith (2014), and adopt a fractional gas-phase carbon abundance of 1.6 × 10^-4 (Sofia et al. 2004). The line intensities are calculated for an edge-on case by storing the run of level populations with molecular cloud depth for the excited level of CO and C⁺ as calculated in the face-on model. For each line of sight, the intensity is found from integrating Eq. (B.14) in Tielens & Hollenbach (1985) through the layer of length z = N_H/n_H with the gas column density N_H and the gas density n_H, where we replace the factor (2π)^-1 for a semi-infinite slab with (4π)^-1. The cooling rate is given by Eq. (B.1) in Tielens & Hollenbach (1985) in the limit of no background radiation. For the escape probability we take the line-of-sight formulation $\begin{array}{ccc}\mathit{\beta }\mathrm{\left(}\mathit{\tau }\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}\mathrm{-}{\mathrm{e}}^{\mathrm{-}\mathit{\tau }}}{\mathit{\tau }}\mathit{,}& & \end{array}$(2)where τ is the line optical depth calculated as in Eq. (B.8) in Tielens & Hollenbach (1985).

The FIR continuum intensity in the edge-on case is calculated from the run of dust temperature with depth into the molecular cloud. We find the dust temperature, T_d, from the prescription given in Hollenbach et al. (1991). We integrate the dust absorption efficiency, Q_abs, through the layer of length z$\begin{array}{ccc}{\mathit{I}}_{\mathrm{FIR}}\mathrm{=}\frac{\mathrm{1}}{\mathrm{4}\mathit{\pi }}\mathrm{\int }\mathrm{4}\mathit{\pi }{\mathit{a}}^{\mathrm{2}}\hspace{0.17em}\frac{{\mathit{n}}_{\mathrm{d}}}{{\mathit{n}}_{\mathrm{H}}}{\mathit{Q}}_{\mathrm{abs}}\hspace{0.17em}\mathit{\sigma }{\mathit{T}}_{\mathrm{d}}^{\mathrm{4}}\hspace{0.17em}{\mathit{n}}_{\mathrm{H}}\hspace{0.17em}\mathrm{d}\mathit{z,}& & \end{array}$(3)where we take the grain size a = 0.1 μm, and n_d/n_H = 6.36 × 10^-12, which gives a grain cross section per hydrogen atom of 2.0 × 10^-21 cm². For Q_abs we use the average silicate and graphite value Q_abs = 1.0 × 10^-6(a/ 0.1 μm)(T_d/ K)² from Draine (2011).

In Fig. 7, we present the results of the models with an incident FUV intensity of G₀ = 100 appropriate for σ Ori (Abergel et al. 2003 and references therein) and a Doppler line width of Δv = 1.5 km s^-1 for different densities on a physical scale. The x-axes share the same range of visual extinction, A_V = 0.0 − 9.3. We computed models for gas densities n_H = 3.0 × 10³ cm^-3,1.6 × 10⁴ cm^-3, and 4.0 × 10⁴ cm^-3; those densities we estimate from the line cuts in Sect. 4.7 for different parts of the molecular cloud. We integrate along a length A_V,los of the line of sight estimated in Sect. 4.4 for each density, where we assumed N_H = 2.0 × 10²¹ cm^-2A_V. A_V,los = 2.5and5.0 with n_H = 3.0 × 10³ cm^-3 correspond to PDR1 and PDR2, respectively; A_V,los = 0.5and2.5 with n_H = 1.6 × 10⁴ cm^-3and4.0 × 10⁴ cm^-3 correspond to potentially dense cloud surfaces in PDR1 and PDR2. The Horsehead PDR should be matched with A_V,los = 2.5or5.0 with n_H = 4.0 × 10⁴ cm^-3.

The three models substantially show the same result, with a luke-warm surface layer where the gas cools through the [C ii] line. The colder gas deeper in the cloud emits mainly through CO. Not surprisingly, the FIR dust emission also peaks at the surface. The line-of-sight depth of the molecular cloud (A_V,los = 0.5,2.5,or5.0) only slightly affects the ratios of FIR, [C ii]-, and CO-line intensities.

3.5. Correlation diagrams

Figure 8 shows correlation diagrams between several quantities. The different colors indicate the selected regions assigned in Sect. 3.2 and shown in Figs. 1 and 3. Gray points represent points that do not lie in either of the defined regions.

Figure 8a shows that the [C ii] cooling efficiency I_{[ C ii ]}/I_FIR decreases with increasing I_FIR. The different PDRs lie on distinct curves, with similar slopes. Figure 8b shows increasing I_{[ C ii ]} with increasing I_FIR, but again in distinct branches for the different regions. The relation between I_{[ C ii ]} and I_{8 μm} resembles that, as can be seen from Fig. 8f, but simple fits reveal a tighter relation of I_{[ C ii ]} with I_{8 μm} than with I_FIR. Over-plotted is a least-square fit ${\mathit{I}}_{\mathrm{[}\mathrm{C}\hspace{0.17em}{ii}\mathrm{]}}\mathrm{\simeq }\mathrm{2.2}\mathrm{\times }{\mathrm{10}}^{-2}{\mathit{I}}_{\mathrm{8}\hspace{0.17em}\mathit{\mu }\mathrm{m}}^{\mathrm{0.79}}$ (ρ = 0.85).

As Fig. 8c shows, I_{[ C ii ]} is roughly constant for higher τ₁₆₀, that is, in PDR3 and the Horsehead PDR. This might reflect the fact that there is colder, non-PDR material located behind the PDR surfaces. PDR1 and PDR2 at the onset of the molecular cloud, where we do not expect a huge amount of colder material along the line of sight, show a nice correlation: PDR1 lies at twice as high τ₁₆₀ and has twice as high I_{[ C ii ]}. For small τ₁₆₀ (i.e., in the H ii region and parts of the CO-dark cloud), the data show a steep rising slope. In Fig. 8d, there is no obvious relation between I_{[ C ii ]}/I_FIR and the dust temperature T_d. Figure 8e shows I_{[ C ii ]} versus I_{CO (1 − 0)}. Here we notice that the various regions populate distinct areas in the plot. In the diagram of I_{[ C ii ]}/I_FIR versus I_{CO (1 − 0)}/I_FIR (Fig. 9), we observe no obvious correlation, only the Horsehead PDR exhibits a significant slope.

The relation between I_{[ C ii ]} and I_{8 μm} resembles the relation of I_{[ C ii ]} with I_FIR, as can be seen from Fig. 8f, but simple fits reveal a tighter relation of I_{[ C ii ]} with I_{8 μm} than with I_FIR. Over-plotted is a least-square fit, I_{[ C ii ]} ≃ 2.2 × 10^-2 (I_{8 μm} [ erg s^-1 cm^-2 sr^-1 ] )^0.79 erg s^-1 cm^-2 sr^-1 (ρ = 0.85).

Figure 10 shows that the Horsehead PDR lies at the high end of the τ₁₆₀ distribution. The general trend does not exactly fit a slope of ≃ − 1, as does OMC1 in a first approximation (from I_{[ C ii ]}/I_FIR ≃ C/ (1 − e^{− τ₁₆₀}), Goicoechea et al. 2015), indicating that the emission cannot be modeled by a homogeneous face-on slab of dust with [C ii] foreground emission. Of course, dust temperature differences should be taken into account, yet here we assume a constant pre-factor. Moreover, this simple model is derived from a face-on geometry, whereas here we are likely to deal with PDRs viewed edge-on.

FIR intensity increases with increasing τ₁₆₀, as Fig. 11a shows, but temperature differences play a role. The dust is comparatively hotter in PDR1 and PDR2, and in the CO-dark cloud. The FIR intensity scatters a lot when related to T_d, as can be seen from Fig. 11b. Opposed to that, I_FIR seems to be well-correlated to I_{8 μm} (Fig. 11c).

4. Discussion

4.1. [C II] emission from the PDR

The total [C ii] luminosity from the mapped area of ≃ 210 sq. arcmin is L_total ≃ 14 L_⊙. The luminosity stemming from the molecular cloud is L_cloud ≃ 13 L_⊙ and that from the H ii region is L_{H ii} ≃ 1 L_⊙. Thus, about 5% of the total [C ii] luminosity of the surveyed area originates from the H ii region; 95% stems from the irradiated molecular cloud. This compares to 20% and 80%, respectively, of the area. For comparison, the total FIR luminosity from the mapped area is L_FIR ≃ 1245 L_⊙, of which 1210 L_⊙ belong to the molecular cloud and 35 L_⊙ to the H ii region. However, a small part of the luminosity may be attributed to NGC 2023: 0.2 L_⊙ in [C ii] and 35 L_⊙ in FIR luminosity. Since the 8 μm intensity as a cloud surface tracer is very well correlated to the [C ii] intensity, we may conclude that in the studied region of the Orion molecular cloud complex most of the [C ii] emission originates from PDR surfaces. This is in agreement with the distribution of [C ii] emission in OMC1: here, Goicoechea et al. (2015) find that 85% of the [C ii] emission arises from the irradiated surface of the molecular cloud. On Galactic scales, however, Pineda et al. (2014) find that ionized gas contributes about 20% and dense PDRs about 30% to the total [C ii] luminosity (the remainder stemming in equal amounts from cold H i gas and CO-dark H₂ gas).

The [C ii] line-integrated intensity I_{[ C ii ]} ranges from 10^-5 erg s^-1 cm^-2 sr^-1 in the H ii region up to a maximum of 7 × 10^-4 erg s^-1 cm^-2 sr^-1 in PDR1, with an average over the mapped area of . The [C ii] cooling efficiency η = I_{[ C ii ]}/I_FIR takes its highest values at the edge of the molecular cloud, bordering on the Hα emitting region. Its peak value is about 3 × 10^-2, ranging down to 3 × 10^-3. The separation of molecular cloud and H ii region emission is not trivial, since we think that the surface of the cloud is not straight, but warped. However, [C ii] emission from the region exclusively associated with the ionized gas in IC 434 is very weak and has a much wider line profile (cf. Figs. 6 and 12; see also Sect. 4.2). Hence, we assume that at the edge of the molecular cloud the [C ii] and FIR emission from ionized gas is minor compared to emission stemming from the molecular cloud itself.

Considering the average [C ii] cooling efficiencies, where beam-dilution and column-length effects should divide out, we note that PDR2 has twice as high [C ii] cooling efficiency as the Horsehead PDR and PDR1 have (see Table 1). We remark that PDR2 lies in a region where there still is significant Hα emission, indicating a corrugated edge structure. Since the average [C ii] emission in the H ii region is quite low, we do not expect [C ii] emission from the ionized gas to be responsible for the enhanced [C ii] cooling efficiency in PDR2. However, I_FIR is unexpectedly low in PDR2, which may account for the mismatch in I_{[ C ii ]}/I_FIR.

4.2. [C II] emission from the H II region

From the Hα emission in the studied region, originating from the ionized gas to the west of the molecular cloud, we can estimate the density of the H ii region (Ochsendorf et al. 2014). The radiated intensity of the Hα line can be calculated by $\begin{array}{ccc}{\mathit{I}}_{\mathrm{H}\mathit{\alpha }}\mathrm{=}\underset{\mathrm{0}}{\overset{\mathit{d}}{\mathrm{\int }}}{\mathit{j}}_{\mathrm{H}\mathit{\alpha }}\hspace{0.17em}\mathrm{d}\mathit{z}\mathrm{=}\underset{\mathrm{0}}{\overset{\mathit{d}}{\mathrm{\int }}}\frac{\mathrm{4}\mathit{\pi }{\mathit{j}}_{\mathrm{H}\mathit{\beta }}}{{\mathit{n}}_{\mathrm{p}}{\mathit{n}}_{\mathrm{e}}}\frac{{\mathit{j}}_{\mathrm{H}\mathit{\alpha }}}{{\mathit{j}}_{\mathrm{H}\mathit{\beta }}}\frac{{\mathit{n}}_{\mathrm{p}}{\mathit{n}}_{\mathrm{e}}}{\mathrm{4}\mathit{\pi }}\hspace{0.17em}\mathrm{d}\mathit{z}\mathit{.}& & \end{array}$(4)Assuming a gas temperature of T ≃ 10⁴ K, we use 4πj_Hβ/n_pn_e = 8.30 × 10²⁶ erg cm³s^-1 and j_Hα/j_Hβ = 2.86 (Osterbrock 1989). Further, we assume a homogeneous gas distribution along the line of sight, which we take to be d ~ 1 pc. Hence, we obtain $\begin{array}{ccc}{\mathit{I}}_{\mathrm{H}\mathit{\alpha }}\mathrm{\simeq }\mathrm{7.0}\mathrm{\times }{\mathrm{10}}^{-8}\hspace{0.17em}{\mathit{n}}_{\mathrm{e}}^{\mathrm{2}}\mathrm{erg}\hspace{0.17em}\mathrm{c}{\mathrm{m}}^{\mathrm{4}}\hspace{0.17em}{\mathrm{s}}^{-1}\hspace{0.17em}\mathrm{s}{\mathrm{r}}^{-1}\mathit{,}& & \end{array}$(5)where n_p = n_e. When a molecular cloud is photoevaporated into a cavity, as the surface of L1630 is, we expect an exponential density profile as a function of distance from the surface. Fitting an exponential to the observed Hα emission along a line cut, we obtain a density law with n_e,0 = 95 cm^-2 at the ionization front and a scale length of 1.2 pc, which is in good agreement with Ochsendorf et al. (2014). In the surveyed area, the density varies between 60 and 100 cm^-3.

Applying again T ≃ 10⁴ K for the gas temperature to the cooling law of [C ii] (Eq. (2.36) in Tielens 2010), we obtain $\begin{array}{ccc}{\mathit{n}}^{\mathrm{2}}\mathrm{\Lambda }\mathrm{\simeq }\mathrm{2.7}\mathrm{\times }{\mathrm{10}}^{-24}\frac{{\mathit{n}}_{\mathrm{e}}}{\mathrm{1}\mathrm{+}\frac{{\mathit{n}}_{\mathrm{cr}}}{\mathrm{3}{\mathit{n}}_{\mathrm{e}}}}\mathrm{erg}\hspace{0.17em}{\mathrm{s}}^{-1}\hspace{0.17em}\mathrm{c}{\mathrm{m}}^{-3}\mathit{,}& & \end{array}$(6)where we assumed an ionization fraction of x = 1 and, hence, consider collisions with electrons only; the critical density scales with T and is, at T = 10⁴ K, n_cr ≃ 50 cm^-3 (Goldsmith et al. 2012). Neglecting $\frac{{\mathit{n}}_{\mathrm{cr}}}{\mathrm{3}{\mathit{n}}_{\mathrm{e}}}$ and assuming again d ~ 1 pc for the length of the line of sight, the above yields $\begin{array}{ccc}{\mathit{I}}_{\mathrm{\left[}\mathrm{C}\hspace{0.17em}{ii}\mathrm{\right]}}\mathrm{\simeq }\mathrm{7}\mathrm{\times }{\mathrm{10}}^{-5}\left(\frac{{\mathit{n}}_{\mathrm{e}}}{{\mathrm{10}}^{\mathrm{2}}\hspace{0.17em}{\mathrm{cm}}^{-2}}\right)\mathrm{erg}\hspace{0.17em}{\mathrm{s}}^{-1}\hspace{0.17em}\mathrm{c}{\mathrm{m}}^{-2}\hspace{0.17em}\mathrm{s}{\mathrm{r}}^{-1}\mathit{.}& & \end{array}$(7)The observed intensity values lie in the range of 10^-5 − 10^-4 erg s^-1 cm^-2 sr^-1, which is, given the range of densities, in good agreement with the values derived from Hα emission. However, it is difficult to recognize a declining trend in the [C ii] intensity away from the molecular cloud, since the signal is very noisy in the H ii region due to the low intensity.

The [C ii] spectra extracted from the H ii region show a very weak and very broad feature (Fig. 12) with a peak main-beam temperature of ~ 1 K and an FWHM of 5 − 10 km s^-1, as compared to 10 − 20 K and 2 − 4 km s^-1 for the PDR regions in the molecular cloud. This is distinct from the spectra taken towards the cloud. We cannot distinguish a broad feature in the spectra taken towards the molecular cloud, although there is some Hα emission and we should expect ionized gas in front of the multiple PDR surfaces. Most likely the intensity is simply too low, approximately ten times lower than the intensity towards the H ii region, assuming that the H ii column in front of the molecular cloud as seen along the line of sight is ~ 0.1 pc, which renders the signal undetectable.

4.3. FIR emission and beam-dilution effects

We expect that beam dilution affects all maps to some extent when convolved to the SPIRE 500 μm36″ resolution, since the unconvolved IRAC 8 μm map at 1.98″ resolution reveals features and delicate structures (see Fig. 15) that disappear upon convolution. The upGREAT beam has an FWHM of 15.9″, thus beam dilution might be noticeable in [C ii] observations towards thin filaments. From the IRAC 8 μm emission, we infer a dilution factor of ~ 2.5 for the Horsehead PDR going from the native resolution of the 8 μm image to 36″ resolution, but only for the narrowest (densest) part of the PDR. The average 8 μm emission is not significantly beam-diluted when convolved to 36″. PDR3 possibly suffers significantly from beam dilution as well, since it shows as a rather sharp filament in 8 μm, where it reaches high peak values (higher than the Horsehead PDR peak values). Values of quantities we observe and compute from that are taken to be beam-averaged values in the respective resolution.

Due to the edge-on geometry of the PDRs in L1630 with respect to the illuminating star system σ Ori (and the low dust optical depth), we expect that I_FIR depends on the re-radiating column along the line of sight, which might explain the excess intensity in 8 μm, [C ii], and FIR in PDR1 as compared to other PDR surfaces here. The commonly expected value of incident FUV radiation, re-emitted in I_FIR, is G₀ ≃ 100, calculated from properties of σ Ori (Abergel et al. 2003, and references therein). Given the edge-on geometry of the cloud-star complex, the formula by Hollenbach & Tielens (1999), $\begin{array}{ccc}{\mathit{G}}_{\mathrm{0}}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}\frac{{\mathit{I}}_{\mathrm{FIR}}}{\mathrm{1.3}\mathrm{\times }{\mathrm{10}}^{-4}\hspace{0.17em}\mathrm{erg}\hspace{0.17em}{\mathrm{s}}^{-1}\hspace{0.17em}\mathrm{c}{\mathrm{m}}^{-2}\hspace{0.17em}\mathrm{s}{\mathrm{r}}^{-1}}\mathit{,}& & \end{array}$(8)cannot be used to infer the intensity of the incident UV radiation. The FIR intensity varies substantially across the mapped area; for a face-on geometry with a single UV-illuminating source, one would expect less divergent values. This realization corroborates the assumption of an edge-on geometry and has been the rationale for building edge-on models with different molecular-cloud depths along the line of sight.

The dust optical depth τ₁₆₀ does not trace the PDR surface column, but the total gas and dust column. The FIR intensity does not increase with increasing τ₁₆₀ for PDR1, PDR2, and PDR3 (in comparison to τ₁₆₀ ≃ 2 × 10^-3, 1 × 10^-3, 3 × 10^-3, respectively⁵). Especially in the case of PDR3, τ₁₆₀ likely traces not only the PDR but the molecular cloud interior, as well. A fraction of I_FIR might stem from deeper, cooler parts of the molecular cloud and not from the PDR surface, although I_FIR is biased towards the hot PDR surface, as is τ₁₆₀. In PDR2, I_FIR is lower than we would expect, leading to significantly higher I_{[ C ii ]}/I_FIR than in PDR1 and PDR3. The dust temperature T_d in PDR2 is determined to be considerably lower than in PDR1, although the environments seem similar.

4.4. Column densities, gas temperature, and mass

Since we do not detect the [¹³C ii] line in single spectra, we cannot determine the [C ii] optical depth by means of it. The noise rms of the spectra is too high to put a significant constraint on τ_{[ C ii ]}. In averaged spectra, we can detect the [¹³C ii] F = 2 − 1 line just above the noise level (see Sect. 4.5). However, knowing the C⁺ column density and the intrinsic line width, we can estimate the [C ii] optical depth and the excitation temperature (see Appendix A) from single spectra. We compute the C⁺ column density of the PDR surface from the dust optical depth, assuming standard dust properties and that all carbon in the PDR surface is singly ionized. Additionally, we expect beam dilution to be insignificant for the [C ii] observations. From the native IRAC 8 μm map, we infer a dilution factor of 1.5 when going to 15.9″ resolution, but only towards the thinnest filament in the Horsehead mane. This equally yields a peak temperature of T_P ≃ 20 K there, as does the brightest part of the Horsehead PDR.

The gas column density can be computed from the dust optical depth τ₁₆₀, assuming a theoretical absorption coefficient. This yields $\begin{array}{ccc}{\mathit{N}}_{\mathrm{H}}\mathrm{\simeq }\frac{\mathrm{100}\hspace{0.17em}{\mathit{\tau }}_{\mathrm{160}}}{{\mathit{\kappa }}_{\mathrm{160}}{\mathit{m}}_{\mathrm{H}}}\mathrm{\simeq }\mathrm{5}\mathrm{\times }{\mathrm{10}}^{\mathrm{24}}\hspace{0.17em}{\mathrm{cm}}^{-2}{\mathit{\tau }}_{\mathrm{160}}\mathit{,}& & \end{array}$(9)where we have used a gas-to-dust mass ratio of 100 and assumed κ_abs = 2.92 × 10⁵(λ [ μm ] )^-2 cm² / g (Li & Draine 2001). With the fractional gas-phase carbon abundance [ C / H ] = 1.6 × 10^-4 (Sofia et al. 2004), we can estimate the C⁺ column density in the PDR surface from the dust optical depth, under the assumption that all carbon in the line of sight is ionized: $\begin{array}{ccc}{\mathit{N}}_{{\mathrm{C}}^{\mathrm{+}}}\mathrm{\simeq }\mathrm{8}\mathrm{\times }{\mathrm{10}}^{\mathrm{20}}\hspace{0.17em}{\mathrm{cm}}^{-2}{\mathit{\tau }}_{\mathrm{160}}\mathit{.}& & \end{array}$(10)Later studies have reported somewhat differing values for [ C / H ], varying by a factor of two for different sight lines (see e.g. Sofia & Parvathi 2009; Sofia et al. 2011). However, the average is not found to deviate substantially from the earlier value of [ C / H ] = 1.6 × 10^-4; the general uncertainty seems to be quite large. We discuss the effect of the uncertainty in the column density on the derived gas properties in the following. For PDR1 and PDR2 we have τ₁₆₀ ≃ 2 × 10^-3 and τ₁₆₀ ≃ 10^-3, respectively, from the τ₁₆₀ map, where we assume that all the dust actually is in the PDR surface. However, these values for the dust optical depth may be affected by significant uncertainties, up to a factor of two, since τ₁₆₀ depends on the assumed dust properties in the SED fit (see discussion in Sect. 2.2). For PDR3, we suppose that there is a significant amount of cold material located along the line of sight, which renders τ₁₆₀ an inaccurate measure for the depth of the PDR along the line of sight here.

Inferring the PDR dust optical depth of the Horsehead PDR requires further effort. According to Habart et al. (2005), there is a large density gradient from the surface to the bulk of the Horsehead PDR. In the surface the gas density might be as low as n_H ~ 10⁴ cm^-3, whereas in the bulk it assumes n_H ~ 2 × 10⁵ cm^-3. Abergel et al. (2003) find n_H ~ 2 × 10⁴ cm^-3 as a lower limit for the density of the gas directly behind the illuminated filament. The dust optical depth is likely to be beam diluted in the SPIRE 500 μm resolution; the filament has an extent of only 5 − 10″. Assuming that the maximum τ₁₆₀ ≃ 10^-2 occurs in the densest (inner) part of the PDR and that the length along the line of sight remains approximately the same, we conclude that the dust optical depth in the Horsehead PDR surface must be significantly lower than the maximum value, by a factor of approximately ten, due to the decreased density.

From deep integration of the Horsehead PDR with SOFIA/upGREAT, however, we are able to infer a [C ii] optical depth of τ_{[ C ii ]} ≃ 2 from the brightest [¹³C ii] line, which can be detected in these data (C. Guevara, priv. comm.; Guevara et al., in prep.). According to Eq. (A.5), this translates into a C⁺ column density of N_C⁺ ≃ 7 × 10¹⁷ cm^-2, that is τ₁₆₀ ≃ 10^-3, which is ten times lower than the maximum value in the Horsehead bulk. However, from our models, this [C ii] optical depth corresponds to a twice as large C⁺ column density of N_C⁺ ≃ 1.6 × 10¹⁸ cm^-2, if we assume that all carbon is ionized within our beam, which might not be the case.

We calculate the [C ii] optical depth τ_{[ C ii ]} and excitation temperature T_ex for PDR1 and PDR2, and T_ex for the Horsehead PDR, using the formulas of Appendix A⁶. The results are shown in Table 3. In principle, the values we infer for N_C⁺ are upper limits, but the general uncertainty in the C⁺ column density is potentially larger than the deviation from the upper limit. The dust optical depth, from which we calculate the C⁺ column density, is not well-determined (cf. Sect. 2.2) and the carbon fractional abundance may deviate. If we assume an error margin of the C⁺ column density of ±50%, this results in ranges of [C ii] optical depth and excitation temperature of τ_{[ C ii ]} ≃ 0.6 − 2.7 and T_ex ≃ 60 − 90 K, respectively, for PDR1, and τ_{[ C ii ]} ≃ 0.4 − 2.2 and T_ex ≃ 60 − 100 K, respectively, for PDR2. Certainly, these values are subject to uncertainties in the inferred gas density and the spectral parameters, as well; however, the uncertainty in N_C⁺ seems to be the most significant, leading to considerable deviations, so we will not discuss the influence of the other uncertainties here. In the Horsehead PDR, the precise value of τ_{[ C ii ]} does not overly influence the excitation temperature which we calculate: it is T_ex ≃ 60 ± 2 K.

If the density of the gas is known, one can compute the gas temperature from the excitation temperature: $\begin{array}{ccc}\mathit{T}\mathrm{=}\frac{{\mathit{T}}_{\mathrm{ex}}}{\mathrm{1}\mathrm{-}\frac{{\mathit{T}}_{\mathrm{ex}}}{\mathrm{91.2}\mathrm{K}}\ln \mathrm{(}\mathrm{1}\mathrm{+}{\frac{{\mathit{n}}_{\mathrm{cr}}}{\mathit{n}}}^{\mathrm{)}}}\mathit{,}& & \end{array}$(11)with the critical density ${\mathit{n}}_{\mathrm{cr}}\mathrm{=}\mathit{\beta }\mathrm{\left(}{\mathit{\tau }}_{\mathrm{[}\mathrm{C}\hspace{0.17em}{ii}\mathrm{]}}\mathrm{\right)}\frac{\mathit{A}}{{\mathit{\gamma }}_{\mathrm{ul}}}$, where β(τ_{[ C ii ]}) is the [C ii] 158 μm photon escape probabilty, A ≃ 2.3 × 10^-6 s^-1 is the Einstein coefficient for spontaneous radiative de-excitation of C⁺, and γ_ul is the collisional de-excitation rate coefficient, which is γ_ul ≃ 7.6 × 10^-10 cm³ s^-1 for C⁺–H collisions (Goldsmith et al. 2012) and γ_ul ≃ 5.1 × 10^-10 cm³ s^-1 for C⁺–H₂ collisions (Wiesenfeld & Goldsmith 2014) at gas temperatures of ≃ 100 K; n is the collision partner density. At T ~ 100 K,γ_ul, and thereby n_cr, is only weakly dependent on temperature; for C⁺–H collisions, n_cr ≃ β(τ_{[ C ii ]}) × 3.0 × 10³ cm^-3, and for C⁺–H₂ collisions, n_cr ≃ β(τ_{[ C ii ]}) × 4.5 × 10³ cm^-3. At the cloud edge, collisions with H dominate the excitation of C⁺, while deeper into the cloud H₂ excitation dominates. We expect that excitation caused by collisions involving both H and H₂ contribute within our beam. Since the points we chose in PDR1 and PDR2 lie close to the surface, we consider collisions with H; in the Horsehead PDR the choice of collision partner does not affect the derived gas temperature significantly due to the higher gas density. We take the photon escape probability to be β(τ) = (1 − e^{− τ}) /τ, as used in our edge-on PDR models. The densities are discussed in Sect. 4.7.

Using n_H ≃ 3 × 10³ cm^-3 for PDR1 and PDR2, we obtain T ≃ 86 K and T ≃ 93 K, respectively, for C⁺–H collisions. In the Horsehead PDR, we compute T ≃ 60 K. From models (see Sects. 3.4 and 4.9) we compute a gas temperature of about T ≃ 100 − 140 K in the top layers of a PDR (cf. Fig. 7). This is in reasonable agreement with the values derived from our observations in PDR1 and PDR2. In the Horsehead PDR, however, the results likely are affected by beam dilution, since the gas temperature drops quickly on the physical scale (within the beam size of 15.9″ of the [C ii] observations). Disregarding this, with a low C⁺ column density of N_C⁺ = 1.6 × 10¹⁷ cm^-2 we can match the gas temperature measured by Habart et al. (2011) from H₂ observations, T ≃ 264 K; this results in a [C ii] optical depth of τ_{[ C ii ]} ≃ 0.1. We can fit a gas temperature of T ≃ 120 K, as predicted by our models for conditions appropriate for the Horsehead PDR, with a column density of N_C⁺ ≃ 2.0 × 10¹⁷ cm^-2. This would yield a [C ii] optical depth of τ_{[ C ii ]} ≃ 0.3. Both values are significantly lower than what is observed in [¹³C ii].

According to Goldsmith et al. (2012), the [C ii] line is effectively optically thin, meaning the peak temperature is linearly proportional to the C⁺ column density, if T_P<J(T) / 3, where J(T) is the brightness temperature of the gas. Hence, even though our derived [C ii] optical depth for PDR1 and PDR2 is >1, the line is still effectively optically thin in these regions. It is optically thick in the Horsehead PDR.

From the H column densities, that is, from the dust optical depth, we can estimate the mass of the gas: M_gas = N_Hm_HA, where A is the surface that is integrated over. The total gas mass of the molecular cloud (not including the H ii region IC 434 and the north-eastern corner of gas and dust associated with NGC 2023) in the [C ii] mapped area is M_gas,total ≃ 280 M_⊙. The Horsehead Nebula and its shadow add M_gas ≃ 33 M_⊙ to the total mass, the Horsehead PDR surface contributing M_gas ≃ 3 M_⊙. The mass of CO-emitting gas (also computed from N_H) is M_gas ≃ 250 M_⊙. The assumption that CO-emitting gas contributes the bulk of the mass and that [C ii] traces PDR surfaces yields a gas mass of PDR surfaces of M_gas ≃ 20 M_⊙. Additionally, ≃ 12 M_⊙ are located in gas that emits neither strongly in [C ii] nor in CO. The H ii region comprises M_gas ≃ 10 M_⊙ as derived from the dust optical depth; from Hα emission, we estimate M_gas ≃ 5 M_⊙.

Table 4 compares the masses computed from the dust optical depth and from CO (1 − 0) emission, respectively, for the several regions we defined. Since the uncertainties (in the dust optical depth, but also in the conversion factors, especially in the X_CO factor Bolatto et al. cf. 2013) are substantial, we cannot draw clear-cut conclusions. We note, moreover, that PDR regions might overlap along the line of sight with CO-emitting regions whose emission stems from a different layer of the molecular cloud, as certainly is the case in PDR3. It seems that there is a fair amount of gas mass unaccounted for by CO emission, but within the error margin of 30%, this mass can be between 10 and 100 M_⊙. In a larger area mapped in CO, comprising the area mapped in [C ii], Pety et al. (2017) find that the CO-traced mass is in fact greater than the dust-traced mass; this especially influences the deeper layers of the Orion B molecular cloud (see Table 4 in Pety et al. 2017), thus these findings might not apply to our study field in L1630. However, they conclude that CO tends to overestimate the gas mass, whereas the dust optical depth underestimates it. In this way, with an area of ~2 pc², we find a gas mass surface density of 100 − 150 M_⊙ pc^-2 in the [C ii] mapped area.

4.5. Excitation properties from [¹³C II]

If we average over a large number of spectra in [C ii]-bright areas, we are able to identify the [¹³C ii] F = 2 − 1 line above the 5σ level (see Fig. 13); we cannot detect the other two (weaker) [¹³C ii] lines. From the [¹³C ii] F = 2 − 1 line, we can compute average values of the [C ii] optical depth, the excitation temperature, and the C⁺ column density. From an average spectrum of the [C ii]-bright regions of PDR1 and PDR2 (an area of 50 square arcmin, which is a third of the cloud area and corresponds to 3140 spectra), we obtain τ_{[ C ii ]} ≃ 1.5 (with an uncertainty of 20%). To obtain this result, we used [¹³C ii] line parameters established by Ossenkopf et al. (2013) and the ¹²C/¹³C isotopic ratio of 67 for Orion (Langer & Penzias 1990; Milam et al. 2005). This yields a C⁺ column density of N_C⁺ ≃ 3 × 10¹⁷ cm^-2. From an average over the Horsehead PDR (an area of 3 square arcmin, corresponding to 180 spectra), we obtain τ_{[ C ii ]} ≃ 5 (also with 20% uncertainty) and N_C⁺ ≃ 1 × 10¹⁸ cm^-2. We note that τ_{[ C ii ]} in PDR1 and PDR2 does match the value calculated in Sect. 4.4 from the dust optical depth and single spectra, and N_C⁺ does not, whereas in the Horsehead PDR τ_{[ C ii ]} does not agree, but N_C⁺ does. Excitation temperatures from the averaged spectra are T_ex ≃ 40 K, which is lower than what we infer from single spectra. Regions with lower-excitation [C ii] contribute to the averaged spectra, but we have to include them to obtain a sufficient signal-to-noise ratio. We stress that this spectral averaging over a large area with varying conditions will weigh the emission differently for the [¹²C ii] and [¹³C ii] lines in accordance with the excitation temperatures and optical depths involved. Hence, the averaged spectrum will not be the same as the spectrum of the average. In our analysis, we have elected to rely on the analysis based on the dust column density rather than this somewhat ill-defined average.

4.6. Photoelectric heating and energy balance

The most substantial heating source of PDRs is photoelectric heating by PAHs, clusters of PAHs, and very small grains. The photoelectric heating rate is deeply built into PDR models and controls the detailed structure and emission characteristics to a large extent. The heating rate drops with increasing ionization of these species. It can be parametrized by the ionization parameter γ = G₀T^0.5/n_e, where G₀ is the incident radiation, T is the gas temperature, and n_e is the electron density; the corresponding theoretical curve as derived by Bakes & Tielens (1994) is shown in Fig. 14. Okada et al. (2013) confirm in a study of six PDRs, which represent a variety of environments, the dependence of the photoelectric heating rate on PAH ionization and conclude on the dominance of photoelectric heating by PAHs.

The [C ii] 158 μm cooling rate increases with gas density and temperature. In the high-density limit (n_H ≫ n_cr), it scales with n_H, whereas for low densities it scales with ${\mathit{n}}_{\mathrm{H}}^{\mathrm{2}}$; the temperature dependence is largely captured in a factor exp( − ΔE/k_BT), where ΔE is the energy level spacing.⁷ The gas density of the Horsehead PDR, n_H ≃ 4 × 10⁴ cm^-3, lies above the critical density for C⁺, n_cr = β(τ_{[ C ii ]}) × 3.0 × 10³ cm^-3; the densities of PDR1 and PDR2 are close to n_cr, that is, in the intermediate density regime. The similar gas temperatures and densities of PDR1 and PDR2, however, do not reflect the difference in the [C ii] cooling efficiencies I_{[ C ii ]}/I_FIR calculated in Table 1 and visualized in Fig. 8a (cf. Sect. 4.3). In the case of the Horsehead PDR, gas cooling through the [O i] 63 μm line becomes important; the [O i] surface brightness is comparable to the surface brightness of the [C ii] line (Goicoechea et al. 2009).

We emphasize that we can directly measure the temperature and the density of the emitting gas in the PDR from our observations (see Sects. 4.4 and 4.7, respectively). Hence, we can test the theory in a rather direct way. Specifically, we assume that all electrons come from C ionization, hence n_e = 1.6 × 10^-4n_H, where we have adopted the gas-phase abundance of carbon estimated by Sofia et al. (2004). For the Horsehead PDR with n_H ≃ 4 × 10⁴ cm^-3 and T ≃ 60 K, we compute an ionization parameter of γ ≃ 1 × 10² K^{1 / 2} cm³; for PDR1 and PDR2 with n_H ≃ 3 × 10³ cm^-3 and T ≃ 100 K, we obtain γ ≃ 2 × 10³ K^{1 / 2} cm³. From the cooling lines, that is, assuming that all the heating is converted into [C ii] and [O i] emission, we arrive at a heating efficiency of 1.7 ± 0.4 × 10^-2 (with I_{[ O i ]} ≃ 1.04 ± 0.14 erg s^-1 cm^-2 sr^-1 from Goicoechea et al. 2009, at similar spatial resolution) for the Horsehead PDR, and 1.1 ± 0.3 × 10^-2 for PDR1. From their [O i] study of the Horsehead Nebula, Goicoechea et al. (2009) find a heating efficiency of 1 − 2 × 10^-2, which is consistent with our findings. For PDR2, we calculate an average [C ii] cooling efficiency of 2.2 ± 0.4 × 10^-2. However, I_FIR is unexpectedly low in PDR2, so we wonder whether the mismatch in I_{[ C ii ]}/I_FIR between PDR1 and PDR2 really is due to an erroneous determination of I_FIR and is thereby deceptive.

The general behavior of the observationally obtained heating efficiency is indeed very similar to the theoretical curve for photoelectric heating by PAHs, clusters of PAHs, and very small grains, as shown in Fig. 14, except that theoretical values seems to be offset to higher efficiency by about a factor of two. Such a shift might reflect a somewhat different abundance of PAHs and related species in the studied regions. We note that these differences between theory and observations can lead to considerable differences in the derived physical conditions. For example, adopting the theoretical relationship and solving the energy balance for the gas density and FUV field appropriate for the Horsehead PDR would result in a derived gas temperature of T ≃ 100 K, while the temperature as measured from the pure rotational H₂ lines by Habart et al. (2011) is T ≃ 264 K, and we determine it to be T ≃ 60 K, which is beam-averaged. For PDR1 and PDR2, the theoretical relationship would imply a temperature of T ≃ 125 K, while we measure T ≃ 90 K from the peak [C ii] intensity. From a theoretical perspective, we expect the temperature to decrease with increasing density (cf. Fig. 9.4 in Tielens 2010). This is what we see in [C ii] observations. However, from studies by Habart et al. (2011) and Habart et al. (2005), the observational temperature lies in the regime T ≃ 200 − 300 K, albeit in a very narrow surface gas layer. Here, the observational temperature in the Horsehead PDR appears to be higher than the theoretical temperature, despite the heating efficiency being underestimated. This may seem inconsistent; it certainly suggests that we have to be very careful with the assumptions we make. However, the discrepancy may be due to various reasons, not all of them implying inconsistency, that we will not discuss in this paper. Calculating the ionization parameter with either differing observational temperatures or the theoretical temperature does not shift the data points in Fig. 14 significantly towards the theoretical curve. Clearly, further validation of the theoretical relationship in a variety of environments is important as photoelectric heating is at the core of all PDR research, including studies on the phase structure of the ISM (Wolfire et al. 1995, 2003; Hollenbach & Tielens 1999). The significance of PAH photoelectric heating is reflected in the tight correlation between PAH and [C ii] emission from the PDRs (see Figs. 8f and 19d). In a future study, we intend to return to this issue of the importance of PAHs to the heating of interstellar gas.

4.7. Line cuts

The edge-on nature of the PDR in the Orion B molecular cloud (L1630) is well illustrated by line cuts taken from the surface of the molecular cloud into the bulk (cf. Figs. 15 and 16). In addition to previously employed tracers (8 μm, Hα, [C ii], CO (1 − 0)), we compare with SCUBA 850 μm observations, that trace dense clumps, and compare PACS 160 μm data as a measure for I_FIR.

Along line cut C, the Horsehead PDR is clearly distinguishable; Hα drops immediately and the other four tracers peak, which indicates high density. Assuming A_V = 2 for the transition from C⁺/C to CO and N_H = 2 × 10²¹ cm^-2A_V, which is consistent with our models (see Fig. 7), we obtain from the physical position of the transition, d ≃ 0.03 pc, n_H ≃ 4 × 10⁴ cm^-3. Since there are no further indications of dense clumps in 850 μm emission, we assume that the rest of the gas located along cut C is relatively diffuse. Having established that, we tentatively assign the CO peak between 0.6 and 0.8 pc to the PDR surface at 0.3 pc. We infer a gas density of n_H ≃ 3 × 10³ cm^-3. Identifying the CO peak at 1.1 pc with the PDR surface at 0.6 pc yields about the same density.

If we perform the same procedure for line cut A, we derive about the same densities, n_H ≃ 3 × 10³ cm^-3. Here, we assign the broad prominent CO feature at 0.8 pc to the broad PDR feature at 0.4 pc. The small CO peak at 0.5 pc possibly relates to the PDR feature at 0.4 pc, hence n_H ≃ 10⁴ cm^-3. The broad CO feature between 0.2 and 0.4 pc could originate from a dense surface, located at 0.2 pc.

Line cut B is difficult to interpret. Due to their respective shapes, the PAH peaks around 0.6 pc might correspond to the CO peaks at 0.7 and 0.85 pc, yielding a density of n_H ≃ 5 × 10³ − 10⁴ cm^-3. There is no indication of dense clumps in the SCUBA map at this point. The gas at the surface is likely to be relatively diffuse, since there is no distinct CO peak that could be related.

The gas at the surface cut by line D is denser again. Here, we estimate n_H ≃ 3 × 10⁴ cm^-3. PDR3 at 1.0 pc might be relatively dense too, since there is CO emission peaking directly behind the PDR front. However, this CO emission could certainly originate from deeper, less dense layers of the molecular cloud, as well. There is no 850 μm peak corresponding to the CO peak, rendering the latter hypothesis more plausible.

There are a number of CO peaks we cannot relate to a specific structure, for example, the one at 0.4 pc in line cut C. This might indicate that there are layers of gas with higher density stacked along the line of sight. Overall, this analysis suffers from numerous uncertainties and unknowns. Nevertheless, the calculated densities seem to be reasonable for a molecular cloud.

4.8. Geometry of the L1630 molecular cloud

Our data suggest that the L1630 molecular cloud as we see it consists of stacked layers of PDRs along the line of sight, which are offset against each other. Figure 17 shows a schematic illustration, which we inferred from our data, of the edge-on geometry of the studied region of the Orion B molecular cloud. The surface of the molecular cloud is inclined with respect to the incident radiation and the observer, but with steps where strong PDR surfaces can be discerned in the IRAC 8 μm image, for instance. This implies that we generally cannot compare our data with face-on PDRs; in the correlation plots, we need to correlate along line cuts and not along the line of sight. Also, the incident radiation is not easily estimated (see discussion in Sect. 4.3).

The CO emission in the Horsehead neck stems from a different distance along the line of sight than the overlapping [C ii] emission. The CO emission stems from the shadow of the Horsehead Nebula, whereas I_FIR and I_{[ C ii ]} originate from the surface of the bulk molecular cloud. They are spatially not coincident and are likely to correspond to different densities. Hence, the [C ii] and the CO emission from this part cannot be correlated.

We do not see sharp Hα edges along the line cuts, as we would expect for multiple PDR fronts, but only on the primary surface of the molecular cloud. We clearly notice the onset of the bulk cloud behind the Horsehead Nebula in line cut C, and in line cut D there is a distinct shoulder to the primary Hα peak. Apart from that, the Hα emission across the molecular cloud is somewhat diffuse. We do notice coincidence, however, between the Hα emission contours and the PDR surfaces as traced by [C ii] emission. In Fig. 18, PDR1 and PDR2 are lined-out by contours of high I_Hα, whereas PDR3 and the intermediate PDR fronts are traced by contours of lower I_Hα.

From the dust optical depth and the inferred densities, we can estimate the length of the PDR along the line of sight. At the surface of the Horsehead Nebula, τ₁₆₀ ≃ 10^-3 together with n_H ≃ 4 × 10⁴ cm^-3 yields an estimate of about l_PDR ≃ 0.05 pc. In the bulk, however, we would expect something closer to the projected extent, about ≃ 0.2 pc. Assuming a density of n_H ≃ 2 × 10⁴ cm^-3, Habart et al. (2005) arrive at l_PDR ≃ 0.5 pc, but comment that this seems implausibly high; they conclude that the density must actually be higher. If we adopt n_H ~ 2 × 10⁵ cm^-3 (Habart et al. 2005) for the bulk and a dust optical depth of τ₁₆₀ ≃ 10^-2, we obtain l_Horsehead ≃ 0.17 pc.

For PDR1 and PDR2, we estimate l_PDR1 ≃ 1 pc and l_PDR2 ≃ 0.5 pc, respectively. PDR3 yields l_PDR3 ≃ 1.5 pc, but this is most likely not the length of the PDR but of the whole PDR+molecular cloud interior column. From the G₀ values of Sect. 4.3, assuming that G₀ ∝ l_PDR, we infer that l_PDR3 lies in between l_PDR1 and l_PDR2.

4.9. Comparison with models

As described in Sect. 3.4, we ran PDR models for an edge-on geometry with varying length along the line of sight and varying density (see also Fig. 7) at G₀ = 100. The respective model relations between tracers are plotted in Figs. 8a, b, e, and 9: I_{[ C ii ]}/I_FIR versus I_FIR, I_{[ C ii ]} versus I_FIR, I_{[ C ii ]} versus I_{CO (1 − 0)}, and I_{[ C ii ]}/I_FIR versus I_{CO (1 − 0)}/I_FIR, respectively.

We have constructed models for selected depths of the molecular cloud along the line of sight A_V,los, which we calculated from the gas column density using N_H = 2.0 × 10²¹ cm^-2A_V, and gas densities n_H, that we inferred from τ₁₆₀ in Sect. 4.4 and the line cuts in Sect. 4.7, respectively. For PDR1, we estimated A_V,los ≃ 5.0, whereas for PDR2 we got A_V,los ≃ 2.5. The Horsehead PDR ranges from A_V,los ≃ 2.5 at the edge to A_V,los ≳ 5.0 deeper in the Horsehead Nebula. For the gas in PDR1 and PDR2, we assume n_H = 3.0 × 10³ cm^-3; for the Horsehead PDR we take n_H = 4.0 × 10⁴ cm^-3, but we note that the density increases with depth into the PDR according to a number of studies on that region (cf., e.g., Habart et al. 2005). Further, we ran models for n_H = 1.6 × 10⁴ cm^-3 to probe a density region that might either be occupied by the Horsehead PDR or by a denser (surface) structure overlaid on PDR1 and PDR2.

Figure 8a shows that the model predictions for the Horsehead with A_V,los = 0.5 and n_H = 1.6 × 10⁴ cm^-3 or n_H = 4.0 × 10⁴ cm^-3 are consistent with the data, whereas we estimated A_V,los = 2.5. The A_V,los = 2.5 line with n_H = 4.0 × 10⁴ cm^-3 could fit the data within the extent of beam dilution. The A_V,los = 5.0 model lines cannot explain the data, even when taking into account beam-dilution effects. Another reason for the discrepancy might be dust depletion in this region decreasing the FIR emission. PDR1 is matched quite well by the curve for A_V,los = 5.0 and n_H = 3.0 × 10³ cm^-3, but the data points lie close to the A_V,los = 2.5 with n_H = 3.0 × 10³ cm^-3 line, as well. The latter also fits PDR2 quite well. The rest of the data points lie between these two curves and the curves for A_V,los = 0.5. Similar conclusions can be drawn for Fig. 8b. The PDR data points are matched by the same model curves as before, with the remaining points lying in between these and the A_V,los = 0.5 curves.

The comparison of the data with the model predictions for the relation between [C ii] and CO (1 − 0) is less clear-cut. In Fig. 8e, PDR1 and PDR2 lie close to the A_V,los = 5.0 line. However, also the higher density lines agree with the data points in PDR1. The Horsehead points lie close to the A_V,los = 2.5 lines instead of the A_V,los = 5.0or0.5 lines. This might reflect the fact that A_V,los = 2.5 in deeper layers of the Horsehead PDR, whereas it might be lower in the [C ii] emitting surface layers.

In Fig. 9, model lines group according to their respective densities and there is comparatively little variation with A_V,los. PDR1 lies on the model lines with n_H = 1.6 × 10⁴ cm^-3. This behavior might imply that the density deeper in the cloud in this region is increased. PDR2 lies on the n_H = 3.0 × 10³ cm^-3 lines. All the A_V,los lines for n_H = 1.6 × 10⁴ cm^-3 run through the Horsehead data points, though some data points lie closer to the model lines for higher or lower density.

The gas temperatures in the top layers of the PDRs are predicted by the models as T ≃ 130 K for PDR1 and PDR2, and T ≃ 115 K for the Horsehead PDR. The latter again is a deviation from the value we infer, T ≃ 60 K, and the temperature estimated by Habart et al. (2011), which is T ≃ 264 K; it lies close to the value assumed by Goicoechea et al. (2009) for modeling [O i] emission from the Horsehead PDR, T ≃ 100 K, and close to the value deduced from equating the photoelectric heating efficiency with the [C ii]+[O i] cooling efficiency, T ≃ 100 K (cf. Sect. 4.6). For PDR1 and PDR2, the gas temperatures computed in Sect. 4.4 assuming excitation by C⁺–H collisions, T ≃ 90 K, are in good enough agreement with the value predicted by the model, especially considering the uncertainties. The derived temperatures would be substantially higher if we considered collisions with H₂ to dominate the excitation of C⁺ (T ≃ 160 − 200 K); the contribution of C⁺–H₂ collisions may be reflected in the underestimation of the gas temperature. Other reasons for slight discrepancies may be the uncertainties in gas density, the error margins of the spectral parameters that we used in the analysis, the uncertainty in the C⁺ column, and, of course, uncertainties within the models.

In conclusion, we are quite capable of reproducing the observed data with model data within the given uncertainties. Our A_V,los and the gas densities can only be estimates, as is the geometry of the molecular cloud that probably has a corrugated surface and density gradients running through it.

4.10. Comparison with OMC1 in the Orion A molecular cloud

One of the aims of the present study is to establish correlations of astrophysical tracers under moderate conditions (intermediate density and moderate UV-radiation field) and to compare those to correlations found under harsher conditions (higher density and strong UV-radiation field). An example of the latter conditions is OMC1 in the Orion A complex. Whereas L1630 has edge-on geometry, OMC1 can be approximated as a face-on PDR with respect to its UV-illuminating sources, the Trapezium cluster. In this section, we compare the correlations found in L1630 to those in OMC1. Data on OMC1 are from Goicoechea et al. (2015).

The OMC1 data seem to continue the trend found in our L1630 data very well in Figs. 19a, b, and d. Both in OMC1 and in L1630, the 8 μm emission correlates well with I_{[ C ii ]}. From a linear fit, we find I_{[ C ii ]} ≃ 4.9 × 10^-2I_{8 μm} + 9.0 × 10^-5 erg s^-1 cm^-2 sr^-1 (ρ = 0.79), which is similar to the result of Goicoechea et al. (2015): I_{[ C ii ]} ≃ 2.6 × 10^-2I_{8 μm} + 1.6 × 10^-3 erg s^-1 cm^-2 sr^-1 (ρ = 0.91). However, the slope of Goicoechea et al. (2015) should be divided by 2.9, since they use a bandwidth of the IRAC 8 μm band of 1 μm, whereas it is really 2.9 μm. Thus, there is some discrepancy, which expresses itself as a flattening of the correlation curve at high 8 μm and [C ii] intensity, comparable to the I_FIR − I_{[ C ii ]} dependency (Fig. 19a).

In Fig. 19c, the I_{[ C ii ]}/I_FIR versus I_CO/I_FIR relation in L1630 and OMC1 is shown. CO-line intensities in L1630 are from the CO (1 − 0) transition, whereas in OMC1 line intensities are from the CO (2 − 1) transition; we divide the CO (2 − 1) intensity in OMC1 by 8, the frequency ratio of the two lines cubed, which for optically thick thermalized CO emission as in OMC1 (cf. Goicoechea et al. 2015) gives a good estimate of the CO (1 − 0) intensity. The data points form two patches with different slopes. Yet, the L1630 data in their entirety seem to show a continuation of the trend set by the OMC1 data. We note that L1630 is characterized by a much higher I_{[ C ii ]}/I_FIR which reflects the decrease of photoelectric heating efficiency with increasing G₀ and the importance of [O i] cooling for high G₀ and high density. The difference in behavior of the I_{[ C ii ]}/I_FIR versus I_CO/I_FIR relation reflects the difference in geometry. As demonstrated in the study of Goicoechea et al. (2015), OMC1 is well-modeled as a face-on PDR, while L1630 has edge-on geometry (cf. Fig. 9).

Goicoechea et al. (2015) find a decrease of I_{[ C ii ]}/I_FIR with dust temperature, which is notably different from our observation shown in Fig. 8d; we find no significant slope. Also, we obtain a less good correlation of I_FIR with τ₁₆₀ than do Goicoechea et al. (2015) for OMC1 (cf. Sect. 3.5, Fig. 11a). The figure corresponding to Fig. 10 in our paper, I_{[ C ii ]}/I_FIR versus τ₁₆₀, in Goicoechea et al. (2015), reveals that OMC1 matches a slab of constant foreground [C ii] emission, that is, a face-on PDR geometry, much better than the studied region of L1630.

5. Conclusion

We have analyzed the velocity-resolved [C ii] map towards the Orion B molecular cloud L1630 observed by upGREAT onboard SOFIA. We compared the observations with FIR photometry, IRAM 30 m CO (1 − 0), IRAC 8 μm, SCUBA 850 μm, and Hα observations.

About 5% of the total [C ii] luminosity, 1 L_⊙, of the surveyed area stems from the H ii region IC 434; the molecular cloud (not including the north-eastern corner with possible contamination from NGC 2023) accounts for 95%, that is, 13 L_⊙. The bulk of the [C ii] emission originates from PDR surfaces. The total FIR luminosity of the mapped area (without NGC 2023) is 1210 L_⊙, of which 1175 L_⊙ stem from the molecular cloud and 35 L_⊙ from the H ii region. This yields an average [C ii] cooling efficiency in the molecular cloud within the mapped area of 1%. From the dust optical depth, we derive a total gas mass of M_gas ≃ 280 M_⊙. Most of the gas mass, M_gas ≃ 250 M_⊙, is contained in the CO-emitting molecular cloud. The [C ii]-bright gas contributes M_gas ≃ 20 M_⊙, which is only about 8% of the total gas mass in the mapped area. This is in close agreement with the PDR mass fraction traced by [C ii] found by Goicoechea et al. (2015) in OMC1, which also is 8% (within a factor of approximately two). The mass of the H ii region accounts for an additional M_gas ≃ 10 M_⊙.

The [C ii] cooling efficiency is found to decrease with increasing I_FIR, in continuation of the results from OMC1 (Goicoechea et al. 2015). Its peak value is about 3 × 10^-2, ranging down to 3 × 10^-3. Highest values are obtained at the edge of the molecular cloud towards the H ii region. The overall [C ii] cooling efficiency of the mapped area, calculated from the total luminosities L_{[ C ii ]}/L_FIR, is ~10^-2; this compares to an average single-pixel [C ii] cooling efficiency in PDR regions of 1.1 ± 0.3 × 10^-2. We note that due to the edge-on geometry of the molecular cloud, I_FIR does not trace the incident UV radiation in general, for there may be deeper and colder cloud layers located along the line of sight. The [C ii] intensity increases with FIR intensity. [C ii] intensity and PAH 8 μm intensity are closely related, reflecting the predominance of gas heating through the photoelectric effect on (clusters of) PAHs and very small grains.

We derive gas densities of the molecular cloud in the range n_H ≃ 10³ − 10⁴ cm^-3, with the Horsehead PDR having a slightly higher density, n_H ≃ 4 × 10⁴ cm^-3. Dust temperatures lie in the range T_d ≃ 18 − 32 K. From the column densities, we derive an extent of the cloud along the line of sight of l ≃ 0.5 − 1 pc at the edge of the cloud, and l ≃ 1.5 pc at the eastern border of the studied area. The Horsehead Nebula scores low with l ≃ 0.05 pc. As discussed, these values are afflicted by significant uncertainties.

We estimated the [C ii] optical depth and the excitation temperature towards three representative points in the mapped area. From our analysis we gather that the column density of the Horsehead PDR cannot be straightforwardly calculated from the dust optical depth. By deep observations of the brightest [¹³C ii] line, we can calculate the [C ii] optical depth directly, implying [C ii] emission to be optically thick: τ_{[ C ii ]} ≃ 2. The corresponding (beam-averaged) excitation temperature is T_ex ≃ 60 K, which is basically equal to the gas temperature. From other studies we observe higher temperatures, which would imply a lower C⁺ column density. Also our models for lower C⁺ column density match the Horsehead data in the correlation diagrams, as opposed to those for the C⁺ column density inferred from [¹³C ii]. Towards PDR1 and PDR2, we obtain [C ii] optical depths of τ_{[ C ii ]} ≃ 1.5 and excitation temperatures of T_ex ≃ 65 K, which gives a gas temperature of T ≃ 90 K.

The observed [C ii] intensity, in combination with the [O i] intensity where appropriate, provides a direct measure of the heating efficiency of the gas. We have compared the observed heating efficiency with models for the photoelectric effect on PAHs, clusters of PAHs, and very small grains by Bakes & Tielens (1994). Theory and observations show a very similar dependence on the ionization parameter γ, albeit that theory seems to be offset to a slightly higher efficiency. This may, for example, reflect a too high abundance of these species in the models, an issue we will revisit in a future study.

We have endeavored to establish the edge-on nature of the Orion B L1630 molecular cloud. The data suggest that there are multiple PDR fronts across the molecular cloud, implying that the cloud surface is warped and not a single edge-on bulk. The edge-on warped geometry makes it difficult to correlate different quantities with each other, since they may relate to offset layers of the molecular cloud and depend on the local inclination of the cloud surface as well as on the length of the emitting column along the line of sight. Our model predictions for edge-on PDRs are capable of replicating the observed correlations between I_{[ C ii ]} and I_FIR; we can also interpret the model correlations between I_{[ C ii ]} and I_{CO (1 − 0)}.

Velocity-resolved line observations are an excellent tool to study gas dynamics and to identify distinct gas components by their different kinematic behavior. The line profile allows inference of the line optical depth, especially when line observations of isotopes (or isotopologues in the case of molecular lines) are available. In the present study, the [C ii] line profile yields information on the origin of the emission, narrow in dense PDRs, broad in the H ii region. In combination with other tracers, we can form a picture of the physical conditions prevailing in a molecular cloud. Several more tracers could be included to render the picture more comprehensive such as [O i] and H i.

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Acknowledgments

J.R.G. and E.B. thank the ERC for funding support under grant ERC-2013-Syg-610256-NANOCOSMOS, and the Spanish MINECO under grant AYA2012-32032. J.P., F.L.P., E.R., and E.B. acknowledge support from the French program “Physique et Chimie du Milieu Interstellaire” (PCMI) funded by the Centre National de la Recherche Scientifique (CNRS) and Centre National d’Études Spatiales (CNES). Studies of the ISM at Leiden observatory are supported through the Spinoza Prize of the Dutch Science Foundation (NWO).

References

Appendix A: Calculating [C II] optical depth and excitation temperature

For optically thick [C ii] emission, we can employ the following two equations to get an estimate of the optical depth and the excitation temperature⁸: $\begin{array}{ccc}& & {\mathit{T}}_{\mathrm{P}}\mathrm{\approx }\mathit{J}\mathrm{\left(}{\mathit{T}}_{\mathrm{ex}}\mathrm{\right)}\mathrm{=}\frac{\mathrm{91.2}\hspace{0.17em}\mathrm{K}}{{\mathrm{e}}^{\mathrm{91.2}\hspace{0.17em}\mathrm{K}\mathit{/}{\mathit{T}}_{\mathrm{ex}}}\mathrm{-}\mathrm{1}}\\ & & {\mathit{\tau }}_{\mathrm{\left[}\mathrm{C}\hspace{0.17em}{ii}\mathrm{\right]}}\mathrm{=}\frac{\mathit{A}{\mathit{\lambda }}^{\mathrm{3}}}{\mathrm{4}\mathit{\pi b}}{\mathit{N}}_{{\mathrm{C}}^{\mathrm{+}}}\frac{{\mathrm{e}}^{\mathrm{91.2}\hspace{0.17em}\mathrm{K}\mathit{/}{\mathit{T}}_{\mathrm{ex}}}\mathrm{-}\mathrm{1}}{{\mathrm{e}}^{\mathrm{91.2}\hspace{0.17em}\mathrm{K}\mathit{/}{\mathit{T}}_{\mathrm{ex}}}\mathrm{+}\mathrm{2}}\\ & & \mathrm{\approx }\frac{\mathit{A}{\mathit{\lambda }}^{\mathrm{3}}}{\mathrm{4}\mathit{\pi b}}{\mathit{N}}_{{\mathrm{C}}^{\mathrm{+}}}\frac{\mathrm{91.2}\hspace{0.17em}\mathrm{K}\mathit{/}{\mathit{T}}_{\mathrm{P}}}{\mathrm{91.2}\hspace{0.17em}\mathrm{K}\mathit{/}{\mathit{T}}_{\mathrm{P}}\mathrm{+}\mathrm{3}}\mathit{,}\end{array}$where A = 2.3 × 10^-6s^-1 and b is the line width with ${\mathit{\tau }}_{\mathrm{[}\mathrm{C}\hspace{0.17em}{ii}\mathrm{]}}\mathit{b}\mathrm{\approx }{}^{\mathrm{\int }}\mathit{\tau }_{\mathrm{[}\mathrm{C}\hspace{0.17em}{ii}\mathrm{]}}\mathrm{d}\mathit{v}$ (cf. Eq. (5) in Gerin et al. 2015). For an optically thin emission, we take into account the dependence of the peak temperature on the optical depth. Here, we assume that $\begin{array}{c}{}_{\mathrm{˜}}\\ \mathit{f}\end{array}\mathrm{\left(}\mathit{\tau }\mathrm{\right)}\mathrm{=}\mathrm{1}\mathrm{-}{\mathrm{e}}^{\mathrm{-}\mathit{\tau }}$ accounts for the increase in the peak temperature with optical depth⁹. Hence, the results above become $\begin{array}{ccc}{\mathit{T}}_{\mathrm{P}}& \mathrm{\approx }& \end{array}$(A.4)$\begin{array}{ccc}{\mathit{\tau }}_{\mathrm{\left[}\mathrm{C}\hspace{0.17em}{ii}\mathrm{\right]}}& \mathrm{\approx }& \end{array}$(A.5)Equation (A.5) has the form τ = g(τ), which can be solved either graphically or by the fix-point method. In order to do so, we need to make some assumptions on b and N_C⁺. In the following, we assume that opacity broadening is insignificant, that is, the lines are Gaussian such that b ≈ Δv_FWHM. The C⁺ column density we take from the dust optical depth.

Appendix B: Face-on calculation

In order to compare our observations with face-on PDRs, we integrate our, presumably edge-on, observations along the depth into the molecular cloud from the surface with respect to the incident FUV radiation. We normalize the measured intensity to the pixel size (15″), assuming a line-of-sight length of l ≃ 1 pc and an average gas density of n_H ≃ 3 × 10³ cm^-3. We further assume that radiation is emitted isotropically and homogeneously along the line of sight. The correlation plots are shown in Fig. B.1.

All Tables

All Figures