WHAT DETERMINES THE DENSITY STRUCTURE OF MOLECULAR CLOUDS? A CASE STUDY OF ORION B WITH HERSCHEL^*

The column density map of Orion B (Figure 1) is dominated by the two active star-forming clumps NGC 2023/24 and NGC 2068/71 (Buckle et al. 2010) with very high local column densities N(H₂) up to a few 10²² cm⁻² and high dust temperatures of up to 35 K (Figure 2) due to the H ii-regions. These two dense ridges are outlined by a column density level of ∼3–4 × 10²¹ cm⁻² and stand out in an extended cloud with a typical column density of 1–2 × 10²¹ cm⁻². In contrast, the northern-eastern part of Orion B is colder and less active. A sharp cutoff in column density at the western border of NGC 2023/2024 and the Horsehead nebula is seen in column density cuts at constant declination (Figure 1) which was known from CO data (Wilson et al. 2005). From west to east, we first observe a strong increase of column density on a few pc scale for NGC 2023/2024, and a weaker but clearly visible increase for the southern region. East of the peaks, the column density decreases to a level of 1–2 × 10²¹ cm⁻², which is higher than the values at the western border. Such a profile was also seen in the Pipe nebula (Peretto et al. 2012) where the authors proposed a large-scale compression by the winds of the Sco OB2 association likely caused this sharp edge. A similar process may be at work for Orion B because the western part of the cloud is exposed to various OB aggregates (OB1b–d).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The distribution of number of pixels versus column density for Orion B, Aquila, and Polaris are displayed in Figure 4. We will use the term PDF and the notation p(η) though the pure pixel distribution is not strictly a PDF which is defined for a lognormal distribution as

$$\begin{equation} p(\eta)d\eta =\left(2\,\pi \,\sigma ^2_{\eta }\right)^{-0.5} \, \exp \left[-(\eta -\mu)^2\left/\left(2\sigma ^2_{\eta }\right)\right]\,d\eta,\right. \end{equation} \tag{ 2 }$$

with η = ln (N/〈N〉) and σ_η as the dimensionless dispersion of the logarithmic field, and μ the mean. The normalization allows a direct comparison between clouds of different column density, and σ_η is a measure for the density variation in a turbulent medium. We determine σ_η with a fit to the assumed lognormal low-density part of the PDF and the slope with the index s from a power-law fit with p(η) = p₀(η/η₀)^s to the high-density part. The results of the PDF fit are given together with the Mach number determination in Table 1.

Table 1. Temperature Regime, Mach Number, and PDF Fit Results for Orion B, Aquila, and Polaris

Cloud	Tex(CO)	Tdust	〈Tex〉(CO)	〈Tdust〉	Δv	$\mathcal {M}$	ση	α
	(K)	(K)	(K)	(K)	(km s−1)	(6)	(7)	(8)
	(1)	(2)	(3)	(4)	(5)
OrionB	5–70	5–45	20	16	∼3	∼8	0.45	1.99
Aquila	⋅⋅⋅	9–40	20	19	∼2.2	∼6	0.30	1.77
Polaris-quiet	⋅⋅⋅	12–15	10	13	∼1	∼3	0.22	⋅⋅⋅
Polaris-saxophone	∼10–15	11–14	12	13	∼2	∼7	0.27	⋅⋅⋅

Notes. (1) Observed excitation temperature range from ¹²CO 1→0 data (Orion B: Buckle et al. 2010; Aquila: Zeilik et al. 1978; Polaris: Bensch et al. 2003; Shimoikura et al. 2012). No large-scale CO data are available for Polaris and Aquila). (2) Observed dust temperature range from Herschel data. (3) and (4) Average temperature. (5) Line width from ¹²CO 1→0. (6) Sonic Mach number from average temperature and CO line width. (7) Dispersion of the PDF. (8) Exponent of the spherical density profile.

Download table as:  ASCIITypeset image

The Orion B and Aquila PDFs show a well-defined lognormal part for low column densities and a clear power-law tail at higher column densities, starting at an extinction¹³ A_V around 3 for Orion B and 6 for Aquila. To first order, the PDFs only differ in their width (σ_η = 0.45 for Orion B and 0.3 for Aquila). The PDFs of two subregions (Figure 3) in the Polaris cirrus cloud (Falgarone et al. 1998) are more narrow with σ = 0.22 and 0.27, respectively. The PDF of the quiescent region is almost perfectly lognormal; however, above A_V ∼ 1.5 we observe a slight excess which is most likely a resolution effect but may possibly result from a physical process (see Section 3.3). A clear deviation from the lognormal shape is found for the "saxophone" filament. Interestingly, the excess for A_V > 1 has not the form of a power-law tail. Note that all PDF features (shape, width, etc.) do not depend on the angular resolution (18'' versus 36'') or pixel number (our statistic here is high because the images are on a 6'' grid). To quantify better the deviation of the PDF from lognormal, we determined the higher moments¹⁴ skewness $\mathcal {S}$ and kurtosis $\mathcal {K}$. The skewness, describing the asymmetry of the distribution, is positive for Orion B, Aquila, and Polaris-saxophone ($\mathcal {S}$ = 1.17, 1.24, and 0.49, error typically 0.05), implying an excess at higher column densities, and is $\mathcal {S}$ = 0.19 ($\mathcal {K}$ = 3.0) for Polaris-quiet, confirming the nearly lognormal form of its PDF. The Orion B and Aquila PDFs have much higher values for the kurtosis ($\mathcal {K}$ = 6.8 and 7.9) that arise from pronounced wings. The Polaris-saxophone region has also a (high) value of $\mathcal {K}$ = 4.4, indicating an excess at high column densities.

Zoom In Zoom Out Reset image size

From theory, a purely lognormal distribution is only expected if the cloud structure is shaped by supersonic, isothermal turbulence, while deviations in the form of a power law for high column densities are predicted for self-gravitating clouds (Klessen et al. 2000; Ballesteros-Paredes et al. 2011). The concept of isothermality does not fully apply to all clouds, as can be seen in the temperature PDFs in Figure 2. While Polaris can be considered as a nearly isothermal gas phase, Orion B and Aquila show a more complex temperature distribution over a larger range. Froebrich & Rowles (2010) argued that the A_V-value where the transition of the PDF must take place is always around 6, and defined this value as a threshold for star formation. Kainulainen et al. (2011) found values between A_V = 2 and 5 and proposed a scenario in which this A_V-range marks a transition between dense clumps and cores and a more diffuse interclump medium. The PDFs of Orion B and Aquila shown in Figure 4 clearly show that the transition from lognormal to power law is not universal but varies between clouds (A_V ∼ 3 and 6, respectively). This behavior suggests that the A_V -transition value neither represents a universal threshold in star formation nor a phase transition (unless the density of the clumps and the interclump medium strongly varies from cloud to cloud). A similar result was obtained in the study of N. Schneider et al. (2013, in preparation), presenting a large sample of PDFs from low- to high-mass star-forming clouds. Deviations of the PDF from lognormal are more likely a function of cloud parameters, in particular the virial parameter, the dominant forcing mode,¹⁵ and the Mach number as shown in models (Federrath & Klessen 2013) and observations (N. Schneider et al. 2013, in preparation).

Zoom In Zoom Out Reset image size

From the slope of the power-law tail for Orion B and Aquila, the exponent of an equivalent spherical density distribution ρ(r)∝r^−α is determined to be α = 1.99 (1.77) for Orion B (Aquila), conforming with results typically obtained for individual collapsing cores. The high-density tail, however, cannot be explained by the core population alone because it does not provide sufficient mass (Könyves et al. 2010). Moreover, it is probably also caused by global gravitational collapse of larger spatial areas like filaments and ridges (see, e.g., Schneider et al. 2010; Hill et al. 2011; Palmeirim et al. 2013 for observational examples). It was shown that non-isothermal flows can also cause power-law tails (Passot & Vazquez-Semadeni 1998) so that this process may influence the shape of the PDF as well. Indeed, in regions with significant temperature variations, α can reach values larger than free fall (α = 2.4 for the high-mass star-forming cloud NGC 6334; Russeil et al. 2013), possibly pointing toward a scenario in which heating/cooling processes become important.

We compare our observational PDFs with those obtained from hydrodynamic simulations (Federrath & Klessen 2013), including gravity, magnetic fields, and different turbulent states ($\mathcal {M}$ = 2–50, star formation efficiencies from 0% to 20%, and different forcing modes). In addition, we determine the "b-parameter" ($\sigma _s^2=f^2\, \sigma _{\eta }^2= \ln {(1+b^2\,\mathcal {M}^2)}$, e.g., Federrath et al. 2010; Burkhart & Lazarian 2012), characterizing the link between density and velocity in a cloud. We find the following.

• 1.  
  The dispersion σ_η of the PDF for Aquila is 0.3 and for the two Polaris subregions 0.22 and 0.27, while σ_η = 0.45 for Orion B. At the same time, the Mach number for all regions is typically 6–8 (in view of the uncertainty of $\mathcal {M}$, they are basically the same), while only the Polaris-quiet subregion has a significant lower value of 3. Numerical models indicate that a larger width is caused by a higher Mach number and/or compressive forcing instead of solenoidal forcing (see also Federrath et al. 2010; Tremblin et al. 2012). Since Orion B and Aquila have similar values of Mach number, we conclude that the Orion B cloud is likely exposed to compressive modes—as seen also in the sharp cutoff of column density (Figure 1)—caused by the stellar winds from diverse OB aggregates.¹⁶ Aquila has been proposed to be located at an encounter of several superbubbles (Frisch 1998), but the impact on its density structure—more or less important than close-by OB-stars—cannot be inferred. Both clouds are exposed to relatively high magnetic fields (Crutcher et al. 1999; Sugitani et al. 2011), so that the more narrow PDF of Aquila is presumably not caused by magnetic fields alone.
• 2.  
  Polaris-quiet has a narrow (σ = 0.22), lognormal PDF, the gas is nearly isothermal (see temperature PDF in Figure 2), and has a low (∼3) Mach number. Only isothermal turbulence simulations without self-gravity reproduce this shape of the PDF. We computed the forcing parameter $b = 1/\mathcal {M} \times (\exp {((f\,\sigma _{\eta })^2)-1})^{0.5}$ using an average of 2.5 between solenoidal and compressive forcing (Federrath et al. 2010) for the factor f = σ_s/σ_η (estimation of the three-dimensional density fluctuation σ_s out of the two-dimensional column density fluctuation σ_η). The resulting value of b = 0.2 is lower than what was found by comparing with the purely solenoidal driven isothermal MHD simulations of Burkhart & Lazarian (2012). Our data point for Polaris-quiet fits on their model (Figure 3) with b = 1/3. In any case, these results show that the Polaris-quiet PDF is consistent with the view that the cloud's density distribution is mainly governed by solenoidal forcing.
• 3.  
  Power-law tails in the high-density PDF regime form under the presence of self-gravity, but can also be provoked by purely non-isothermal turbulence (Passot & Vazquez-Semadeni 1998). For Orion B and Aquila, gravity most likely dominates because a large number of prestellar and protostellar dense cores and supercritical filaments are present (André et al. 2010; V. Könyves et al., in preparation). The gas is not isothermal (Figure 2), but the temperature does not vary by several orders of magnitude either. The excess in the PDF for the Polaris-saxophone region is more difficult to interpret. In this gravitational and thermally subcritical filament, only one candidate prestellar core (Ward-Thompson et al. 2010; Shimoikura et al. 2012) was found. The gas can be considered as isothermal, so that here magnetic fields may play a role (the strength is not known), leading to a narrow PDF (e.g., Molina et al. 2012), or statistical density fluctuations and intermittency due to locally compressive turbulence. These effects may also explain the slight excess in the PDF of Polaris-quiet.