EDP Sciences
Free Access
Issue
A&A
Volume 586, February 2016
Article Number A137
Number of page(s) 16
Section Interstellar and circumstellar matter
DOI https://doi.org/10.1051/0004-6361/201525616
Published online 09 February 2016

© ESO, 2016

1. Introduction

Stellar bubbles are formed from the combined action of ionization power and stellar winds from OB stars. They are a major source of turbulent energy injection into the interstellar medium (ISM), sweeping up the surrounding gas and dust and modifying the magnetic fields (e.g. Ferrière 2001). Bubbles are seen as agents of triggered star formation, as the swept-up material may become unstable and fragment. It is therefore important to understand the influence of star formation and feedback on the structure of the magnetic field, and how the ionized bubbles are shaped by the field. The all-sky polarization data from Planck1 open up new opportunities for studying the magnetic field structure of such objects and their connection with the large-scale Galactic field.

The interplay between the expansion of an ionized nebula and the action of the magnetic fields has been the subject of several studies, both observational (e.g. Pavel & Clemens 2012; Santos et al. 2012, 2014) and numerical (e.g. Bernstein & Kulsrud 1965; Giuliani 1982; Ferrière et al. 1991; Krumholz et al. 2007; Peters et al. 2011; Arthur et al. 2011). These observational studies rely partly on interstellar dust grains as tracers of the magnetic field. Non-spherical grains spin around their axes of maximal inertia, which are precessing around the magnetic field lines. The grains emit preferentially along their long axes, thus giving rise to an electric vector perpendicular to the magnetic field. At the same time, dust grains polarize the light from background stars and since extinction is higher along their longest axes, the transmitted electric vector is parallel to the magnetic field in the plane of the sky (see e.g. Martin 2007). Observations of dust emission or absorption allow us to retrieve the plane-of-the-sky orientation of the magnetic field that pervades interstellar matter. Results from high-resolution studies of H ii regions have revealed regions of well-ordered magnetic field along the edges of the nebulae and magnetic field strengths of tens to hundreds of μG. Santos et al. (2014) studied the star-forming region Sh2-29 in optical and near-infrared polarimetry and derived a field strength of around 400μG. The authors used the Chandrasekhar-Fermi method (Chandrasekhar & Fermi 1953), which relates the dispersion in magnetic field orientation with turbulent motions of the gas, under the assumption of equipartition between magnetic and turbulent/thermal pressures. Such high values of the field strength reflect the ordered structure of the magnetic field at sub-parsec scales, compressed by the expanding H ii region. The aforementioned dust polarization observations of ionized nebulae, both in emission and extinction, have covered small regions around the objects, generally tens of arcminutes. Planck data allow us to study the magnetic fields probed by polarized dust emission towards large H ii regions embedded in their parent molecular clouds, and the diffuse medium surrounding them.

Measurements of the magnetic fields towards H ii regions can also be performed through Faraday rotation of linearly polarized background sources, both extragalactic and Galactic (Heiles & Chu 1980). The plasma in the H ii regions rotates the plane of polarization of the background radio wave by an angle that is proportional to the square of the observing wavelength. The quantity derived from Faraday rotation observations, the rotation measure (RM), is directly related to the line-of-sight component of the magnetic field B|| weighted by the electron density of the ionized gas. Harvey-Smith et al. (2011) studied the line-of-sight magnetic field towards five large diameter Galactic H ii regions, tens of parsecs wide, using RM combined with Hα data to estimate the electron density. They found B|| values of 2–6 μG, consistent with those measured in the diffuse ISM through Zeeman splitting observations (Heiles & Crutcher 2005; Crutcher et al. 2010; Crutcher 2012). This indicates that the RM enhancement observed towards H ii regions may be the consequence of a local increase of the electron density. A similar study performed on the Rosette Nebula by Savage et al. (2013) attributed the high values of RM to an increase in the field strength, as will be discussed in Sect. 2.

The goal of this work is to study the structure of the magnetic field in and around an ionized bubble created by young stars. For this purpose we chose the Rosette Nebula, adjacent to the Monoceros (Mon) OB2 cloud, for its close to spherical shape and for its relatively large size on the sky, , relative to the Planck 353 GHz beam of . This paper is organized as follows. We start in Sect. 2 by describing the Rosette Nebula and its main features relevant for our study. In Sect. 3 we introduce the Planck and ancillary data used in this study, which are then analysed and discussed in Sect. 4. The interpretation of the radio and submillimetre polarization observations is presented in Sect. 5, in light of a 2D analytical model of the magnetic field in a spherical bubble-shell structure. The main results are summarized in Sect. 6. The detailed derivation of our magnetic field model is given in Appendix A.

thumbnail Fig. 1

The Rosette Nebula, its molecular cloud Mon OB2, and their surrounding medium as seen by Planck at 353 GHz. The dotted lines give the approximate outer radii of the Rosette H ii region and the Mon Loop supernova remnant (described in Sect. 2). The northern Monoceros cloud Mon OB1 is above the Galactic plane; part of the dust ring of λ Ori is visible at the bottom right corner of the map; the northern end of Orion B is seen at (l,b) = (204°, −12°) (Dame et al. 2001).

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2. The Rosette Nebula and the Mon OB2 molecular cloud

The Rosette bubble is located near the anti-centre of the Galactic disk, centred on (l,b) = (206°̣3, −2°̣1), and is part of a larger region called the northern Monoceros region. It is at (1.6 ± 0.1) kpc from the Sun and its age is estimated to be about (3 ± 1) Myr (Román-Zúñiga & Lada 2008). Figure 1 shows the position of the Rosette Nebula relative to its parent molecular cloud Mon OB2, as well as other known and nearby objects, marked on the Planck 353 GHz map (Sect. 3.1). The Rosette has been the object of extensive studies, as it has been considered an archetype of a triggered star formation site (Williams et al. 1994; Balog et al. 2007; Román-Zúñiga & Lada 2008; Schneider et al. 2010). However, based on the small age spread among the young stellar objects in the Rosette, Ybarra et al. (2013) and Cambrésy et al. (2013) suggest that the effect of the H ii region expansion on stimulating star formation is secondary relative to the original collapse of the cloud. The prominent OB association, NGC 2244, responsible for the ionized nebula and the evacuation of its central part, contains more than 100 stars, of which seven are O and 24 are B stars (Ogura & Ishida 1981; Marschall et al. 1982; Park & Sung 2002; Román-Zúñiga & Lada 2008). Cox et al. (1990) quote a luminosity of 22 × 105L for the brightest four O stars and 13 B stars of the central cluster.

The first study of the expansion of the H ii region in the Rosette was done by Minkowski (1949). The author also noted the existence of “elephant trunks” and dark globules, from the analysis of photographic data, in the north-western edge of the central cavity (Herbig 1974; Schneps et al. 1980; Carlqvist et al. 1998, 2002). Globules are also seen in the south-eastern region in the Herschel images of the Rosette (Schneider et al. 2010), where the ionized nebula is interacting with the molecular cloud. The Rosette H ii region is an ionization bounded Strömgren sphere (Menon 1962) and expands at a velocity of 15 km s-1 (Smith 1973; Fountain et al. 1979; Meaburn 1968). The mean electron temperature in the nebula, derived by Quireza et al. (2006), is 8500 K.

The Rosette H ii region has a projected optical extent of about . Celnik (1985) estimated the size of the nebula by fitting several shell models to observations at 1.4 and 4.7 GHz. The mean radial profile of the radio emission was reproduced by a model of a spherical shell with inner and outer radii of 7 and 19 pc, respectively, and constant electron density of 13.5 cm-3 (these values have been scaled from the distance of 1.42 kpc used by Celnik to the adopted distance of 1.6 kpc). Celnik also found that the ionized shell shows a radial density gradient out to an outer radius of 28 pc, with a mean electron density of about 5.7 cm-3. This model leads to a total mass of ionized matter of 1.2 × 104 M, similar to the value of 1.3 × 104 M obtained for the spherical shell of constant density. The total molecular mass derived from 12CO emission is 1.6 × 105 M (Heyer et al. 2006), of which 54% lies inside the Rosette H ii region.

The magnetic field in the Rosette Nebula has been recently studied by Savage et al. (2013) using RM observations of background galaxies. They fit the RM data with a simple stellar bubble model (Whiting et al. 2009) that consists of an inner low-density cavity of shocked stellar wind and a shell of shocked and photoionized ISM material. The authors assume that the ambient magnetic field B0 outside the bubble has a strength B0 = 4μG and that the field component in the shock plane is amplified by a factor of 4, appropriate for strong adiabatic shocks. Then by fitting the RM data to their model, they find that the angle between B0 and the line of sight is θ0 = 72°. The results of Savage et al. (2013) will be further discussed in Sect. 5.2.

Part of the northern Monoceros region has also been studied in polarization at radio frequencies by Xiao & Zhu (2012). These authors focused on the supernova remnant (SNR) Monoceros Loop, G205.7+0.1, north-west of the Rosette Nebula. The SNR and H ii region are thought to be interacting (Davies et al. 1978; Smith 1973; Jaffe et al. 1997; Fountain et al. 1979; Gosachinskii & Khersonskii 1982). The polarization data presented by Xiao & Zhu (2012) do not cover the interface region between the two objects and show that the magnetic field in the Rosette Nebula is largely parallel to the Galactic plane. However, at a frequency of 5 GHz the observed polarization vectors are the result of either highly rotated background polarized synchrotron emission, considering the high RMs of about 500 rad m-2 measured in the nebula (Sect. 4.2.2, Savage et al. 2013), or foreground emission.

3. Data

3.1. Planck data

Planck (Tauber et al. 2010; Planck Collaboration I 2011; Planck Collaboration I 2014) has mapped the polarization of the sky emission in seven frequency channels, from 30 to 353 GHz. In this paper we use the data from the High Frequency Instrument (HFI; Lamarre et al. 2010; Planck HFI Core Team 2011; Planck Collaboration VI 2014) at 353 GHz, where the dust polarized emission is brightest. A first description of the dust polarization sky is presented in a series of papers, namely Planck Collaboration Int. XIX (2015), Planck Collaboration Int. XX (2015), Planck Collaboration Int. XXI (2015), Planck Collaboration Int. XXX (2016), Planck Collaboration Int. XXXII (2016), and Planck Collaboration Int. XXXIII (2016).

3.1.1. Intensity and polarization

We use the full-mission temperature and polarization sky maps at 353 GHz from the 2015 release of Planck (Planck Collaboration I 2015). The map-making, calibration and correction of systematic effects is described in Planck Collaboration VIII (2015). We smooth the maps and their corresponding covariance from the initial angular resolution of to 6. This is a compromise between increasing the signal-to-noise ratio of the polarization data and preserving high resolution, as well as minimizing beam depolarization effects.

The 353 GHz intensity data are corrected for the cosmic microwave background (CMB) anisotropies using the Commander map, presented in Planck Collaboration IX (2015), although the contribution to the total signal in the region under study is small, at most 2%. We also subtract the Galactic zero level offset, 0.0885 ± 0.0067 MJy sr-1, from the intensity map and add the corresponding uncertainty to the intensity variance (Planck Collaboration VIII 2015). We do not attempt to correct for the polarized signal of the CMB anisotropies, as this is a negligible contribution 1% at 353 GHz in the region under study (Planck Collaboration Int. XXX 2016).

The linear polarization of dust emission is measured from the Stokes parameters Q, U, and I, delivered in the Planck data release. They are the result of a line-of-sight integration and are related as

(1)where p is the dust polarization fraction and ψ is the polarization angle. The two argument function arctan(X,Y) is used to compute arctan(X/Y), avoiding the π ambiguity. The Planck Stokes parameters are provided in the HEALPix (Górski et al. 2005) convention, such that the angle ψ = 0° is towards the north Galactic pole and positive towards the west (clockwise). In the commonly used IAU convention (Hamaker & Bregman 1996), the polarization angle is measured anticlockwise relative to the north Galactic pole. We adopt the IAU convention by taking the negative of the PlanckU Stokes map.

The Stokes Q and U maps in the region under study have a typical signal-to-noise ratio of 8 and 2, respectively. The main systematic effect concerning polarization data is signal leakage from total intensity. We used the corrections derived from the global template fit, described in Planck Collaboration VIII (2015), which accounts for all of the leakage sources, namely bandpass, monopole, and calibration mismatches. The 2015 Planck data release also includes polarization maps that were corrected only for dust bandpass leakage using a different method. We compare the two sets of maps in the region under study to quantify the systematic uncertainties associated with the leakage correction. The relative difference in the Q Stokes parameter is less than 10%, whereas it is typically 25% in U because the Rosette is 2° from the Galactic plane, where the magnetic fields are largely parallel to the plane and contribute more signal to Q than to U.

Because of the presence of noise in the data, the polarization intensity, P, calculated directly using Eq. (1) is positively biased. We debias this quantity according to the method proposed by Plaszczynski et al. (2014) by taking into account the diagonal and off-diagonal terms in the covariance matrix for Q and U. Since we only calculate the polarization intensity towards high (>10) signal-to-noise regions, namely the Mon OB2 cloud, the relative difference between the debiased and direct (Eq. (1)) estimates is less than 3%. A comparison between other debiasing methods as applied to the all-sky Planck data is presented in Planck Collaboration Int. XIX (2015).

3.1.2. Products

We use the Planck “Type 3” CO map from Planck Collaboration XIII (2014) to qualitatively inspect the molecular gas in the Monoceros region. This is a composite line map, where the line ratios between the first three CO rotational lines are assumed to be constant across the whole sky. The Type 3 CO map is the highest signal-to-noise map extracted from Planck, with an angular resolution of .

The dust emission from Planck wavelengths to about 100 μm is dominated by the contribution of large dust grains (radius larger than 0.05 μm). In order to trace their temperature, we use the map derived from the all-sky model of dust emission from Planck Collaboration XI (2014), at a resolution of 5.

3.2. RM data

We use the RM data presented by Savage et al. (2013) towards 23 extragalactic radio sources observed through the Rosette complex with the Karl G. Jansky Very Large Array2. The RM observations towards extragalactic sources are an integral of the line-of-sight magnetic field component B||, weighted by the electron density ne, and given by (2)where S is the path length from the source to the observer. The RM is positive when B points towards us, hence when B|| is also positive.

We consider 20 measurements (given in Table 3 of Savage et al. 2013); two of the sources are depolarized and we also exclude the only negative RM value detected. This measurement was also discarded in the analysis of Savage et al. (2013), who could not determine if its origin is Galactic or extragalactic. In any case this measurement lies outside the boundaries of the Rosette Nebula. Whenever two values are given for the same source, in the case of a double component extragalactic object, we take their mean and the dispersion as the uncertainty because we combine the RM data at a resolution of 128, with emission measure data at a lower resolution of 144 (Sect. 3.3). The 20 RM measurements are positive and have a typical uncertainty of 40%, or below 10% for half of the sample.

3.3. Emission measure data

The emission measure (EM) data are from the radio recombination line (RRL) survey of Alves et al. (2015) at 1.4 GHz and at a resolution of . These are a by-product of the H i Parkes All-Sky Survey (HIPASS, Staveley-Smith et al. 1996) and their analysis is presented by Alves et al. (2010, 2012). The emission measure is defined as (3)and derived from the integrated RRL emission as (4)where Te and TL are the electron and line temperatures, respectively. The overall calibration uncertainty in these data is 10%.

3.4. IRAS data

We use the IRIS (Improved Reprocessing of the IRAS Survey, Miville-Deschênes & Lagache 2005) data at 12 and 100 μm. The IRAS map at 12 μm traces the emission from dust particles that are smaller than those emitting at the longer Planck wavelengths.

4. Our observational perspective on the Rosette

In this section we start by describing the features of the Rosette Nebula and its parent molecular cloud from total intensity maps at different frequencies. We then study the polarized emission arising from the dust shell that surrounds the Rosette H ii region, along with the radio polarization signal created by the ionized gas.

thumbnail Fig. 2

The Rosette Nebula and its molecular cloud. a) Integrated RRL emission at resolution, showing the radio morphology of the H ii region. The green circles, with a diameter equivalent to the beam full width at half maximum (FWHM) of the RRL data, give the position of the 20 RM observations of Savage et al. (2013) used in this work. b) Planck 353 GHz emission at 6 resolution. The black star in panels a) and b) indicates the position of the central star cluster NGC 2244 and the black circles correspond to the inner and outer radii of the H ii region,  pc and  pc, respectively (see Sect. 4.2.2). c) Planck CO integrated intensity at resolution. d) Dust temperature derived from Planck at 5 resolution. e) Dust emission as seen by IRAS at 12 μm and at 100 μm (f), at 4 resolution. The contours in panels c) to f) show the Planck 353 GHz emission; these are at every 5% from 15 to 30%, at every 10% from 30 to 50%, and finally at 70 and 90% of the maximum intensity of 37 MJy sr-1.

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4.1. Intensity

The Rosette Nebula and its parent molecular cloud are presented in Fig. 2. The maps show the RRL thermal radio emission, dust emission at 353 GHz, integrated CO emission, dust temperature, and dust emission at 12 μm and 100 μm. The central cavity is seen both in radio and in dust emission, as a lower intensity region around the position of the ionizing source NGC 2244.

The expanding H ii region is interacting with the molecular cloud Mon OB2, creating photon-dominated regions at their interface (Cox et al. 1990; Schneider et al. 2010), which can be traced at 12 μm. The map of Fig. 2e shows bright 12 μm emission associated with the 353 GHz contours on the eastern side of the nebula’s centre. Within these structures elongated along the ionization front, there are dense molecular clumps with on-going star formation, as the result of the compression by the H ii region. The dust temperature is also higher at the interface of these regions with the nebula, as shown in Fig. 2d, compared to the lower temperature at the position of their maximum intensity.

Cox et al. (1990) studied the Rosette complex in all four IRAS bands and analysed the radial distribution of the dust emission. These authors showed that the 12 μm emission, which traces the photodissociation regions, comes from a shell surrounding the outer side of the ionization front, as well as from the molecular cloud. On the other hand, the longer wavelength emission at 60 and 100 μm mostly arises from the H ii region and thus is tracing ionized and neutral gas.

thumbnail Fig. 3

Planck polarization maps of the Rosette region: a) Q Stokes parameter; b) U Stokes parameter. The dashed circles, with radii 2° and 3°, define the region where we estimate the background level, the full circle centred on the star cluster represents the outer radius of the dust shell,  pc, and the rectangle encompasses the brightest part of the Mon OB2 cloud. c) Intensity map at 353 GHz with the plane-of-the-sky magnetic field orientation shown by headless vectors, obtained by rotating the polarization angle ψ by 90°. The vectors are plotted at every 6, from the average of Q and U within the same distance from the central pixel. d) Polarization fraction p, with the same intensity contours as in Fig. 2.

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thumbnail Fig. 4

Normalized distribution functions (DFs) of the a) Stokes Q and b) U maps measured within the background aperture (blue) and within the outer radius of the dust shell (red). The two regions are defined in Fig. 3 by the dashed and full circles, respectively.

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The western part of the Rosette complex has not been studied as thoroughly as the main molecular cloud. It is not as bright in dust emission, but still visible in the Planck and IRAS maps (Figs. 2b, e, and f) as a fragmented shell surrounding the central cavity. There is no significant counterpart in CO emission, except in the cloud located in the south-west, which appears elongated perpendicular to the ionization front in the Planck and CO maps. This cloud also has lower dust temperature than the remainder of the western dust shell.

The dust shell seen in the Planck 353 GHz map surrounding the H ii region is associated with the Rosette complex, as it correlates with the dust emission observed by IRAS (Cox et al. 1990) and identified in the dust temperature map. The map of Fig. 2d shows a clear enhancement of the dust temperature in the region occupied by the nebula as a result of dust heating by the central cluster. It also illustrates a decrease in the dust temperature towards the central cavity because the Rosette is not a filled H ii region but a shell, with an inner cavity created by the evacuation of material due to powerful stellar winds (Smith 1973; Fountain et al. 1979).

The origin of the dust shell as material swept up by the expanding H ii region will be further discussed in Sect. 5. The size of the shell can be estimated by fitting the Planck intensity map with a shell model, similar to that used by Celnik (1985) to derive the radius of the ionized nebula. We obtain  pc and  pc for the inner and outer radii of the dust shell, with a typical uncertainty of 2 pc. Further details are given in Appendix A.

4.2. Polarization

Table 1

The mean level of the Stokes I, Q, and U parameters at 353 GHz, in units of MJy sr-1.

4.2.1. Planck data

The maps of the Q and U Stokes parameters towards the Rosette Nebula and its parent molecular cloud are shown in Figs. 3a and b, respectively. There is significant Q emission towards Mon OB2 relative to the background, however the polarized emission surrounding the Rosette H ii region does not define the same shell as seen in intensity. Notably, the Q and U signal in some regions of the dust shell is at the same level as that outside the Rosette/Mon OB complex. Figure 4 compares the Q and U emission observed in the dust shell that surrounds the Rosette H ii region (in red) with that from an aperture that represents the local background (in blue). The first distribution is measured within 22 pc (the outer radius of the dust shell, Sect. 4.1) from the central cluster and the second within an aperture of inner and outer radii of 2° and 3°, respectively, centred at (l,b) = (207°, −2°) (see the circles in Fig. 3). As seen in Fig. 4a, the Q signal in the shell clearly deviates from that measured in the background, with a higher mean value and a broader distribution. The distribution of U emission, shown in Fig. 4b, is also broader in the dust shell, even if the mean level is similar to that of the background. We compare the histograms of Fig. 4 quantitatively by means of the Kolmogorov-Smirnov statistical test, which shows that the polarized signal from the shell and the background are significantly different. This test compares the cumulative distributions and thus takes into account the different number of points from each sample. The statistical difference applies when considering the dust shell as a whole, but it may not be valid in the case of individual structures located inside it. We will thus focus the analysis on the mean polarized quantities, without attempting to study the small-scale structure observed in the dust shell. This approach is also justified because our analytical magnetic field model is based on a uniform density distribution.

We estimate the mean value of the I, Q, and U Stokes parameters in the shell and background regions. These are listed in Table 1 along with the corresponding uncertainties, given by the standard error on the mean. The second to last column of Table 1 lists the mean of the Stokes parameters in the dust shell corrected for the background emission, required for comparison with the model results. We note that the choice of a constant value for the background follows from the absence of any clear pattern in the polarized emission outside the Rosette/Mon OB2 that we can use to model the background variations. As a result, the contribution of the background residuals to the polarized signal of the dust shell needs to be taken into account. The background distributions of Fig. 4 have about 8 times more points (Nback = 14 154) than those corresponding to the dust shell (Ndust = 1757). We extract Ndust points from the background distribution and assess the variation of its mean Stokes parameters by repeating this exercise 10 000 times. We take the standard deviation of the 10 000 mean values and add it in quadrature to the standard error on the mean Stokes parameters. The final uncertainty in the background-subtracted values is quoted in the second to last column of Table 1. We also computed these quantities using a different leakage correction (as explained in Sect. 3.1.1), which we will use in Sect. 5.3 to quantify the systematic uncertainties in our results. We note that the method used to compute the uncertainties does not take into account that the background fluctuations are spatially correlated. If we estimate the background in circular areas of the same radius as the Rosette dust shell, we find a larger uncertainty on the background-corrected values. Thus, we cannot exclude that the mean Q and U values in Table 1 for the dust shell have a significant contribution from local background fluctuations. However, we regard the possibility of having a second ISM structure in the same direction as the Rosette as unlikely.

We perform the same analysis in the rectangular region of Fig. 3 that delineates the brightest part of Mon OB2. The mean Stokes parameters, corrected for the background contribution are listed in the last column of Table 1. We use these values to estimate the polarization properties of the Rosette’s parent molecular cloud. The resulting debiased polarization fraction, derived according to the method introduced in Sect. 3.1.1, is p = (3.8 ± 0.9)% and the polarization angle is ψ = −3° ± 14°. The plane-of-the-sky magnetic field orientation in Mon OB2, along the Galactic plane, is consistent with the large-scale orientation seen in Fig. 3c.

The map of polarization fraction is shown in Fig. 3d, where the lowest values of p ≲ 3% are seen towards the densest regions surrounding the Rosette Nebula. Similar values are observed in the less bright part of the Mon OB2 cloud at (l,b) = (208°̣5, −2°̣5). From the all-sky analysis of Planck 353 GHz polarization data, Planck Collaboration Int. XIX (2015) find that higher density regions have lower polarization fraction relative to the maximum pmax = 19.8%, detected at 1° resolution. This decrease in p can be the result of several effects, namely depolarization in the beam or along the line of sight, variations in the intrinsic polarization fraction of dust grains, as well as changes in the magnetic field geometry. The latter was shown to generally explain the variations of p across the whole sky: Planck Collaboration Int. XIX (2015) find an anti-correlation between the fluctuations in the magnetic field orientation and the polarization fraction. Planck Collaboration Int. XXXIII (2016) used this result to model the variations of the Stokes parameters across three interstellar filaments with variations of the magnetic field orientation, for a fixed alignment efficiency of dust grains. In Sect. 5 we present an analytical solution for the magnetic field in the Rosette H ii region and associated dust shell, for constant dust properties. We will use this model to explain the present Planck polarization observations and the radio RM data consistently.

4.2.2. Radio data

The radio emission map of Fig. 2a shows the Rosette H ii region and its nearly circular shape. We fit the radial profile of the RRL emission with a shell model and find the same values as Celnik (1985) for the inner and outer radii,  pc and  pc, respectively, with an uncertainty of 1 pc. Also shown in Fig. 2a are the positions of the RM data from Savage et al. (2013; Sect. 3.2). The circles have a diameter equivalent to the beam FWHM of the radio survey, , in order to show the regions within which the electron density, ne, is estimated.

The RRL observations provide a measure of EM for a given electron temperature. We use Eq. (4) with Te = 8500 K (Sect. 2, Quireza et al. 2006) to calculate EM towards the 20 RM positions. The observed EM includes the contribution from the warm ionized gas from the background/foreground material in the Galaxy. We correct for this contribution by subtracting the average EM measured towards the RM positions located outside the H ii region. Of the 20 positions, 14 lie outside the radius  pc, with an average EM of (288 ± 124) cm-6 pc, where the uncertainty corresponds to the standard error on the mean. The EM values measured towards the shell are in the range 1500–5300 cm-6 pc. We can thus estimate the electron density in the Rosette towards the remaining six positions, which are listed in Table 2. In the general case when the electron density distribution is not uniform but concentrated in discrete clumps of ionized gas, the filling factor f is introduced to relate the true path length L of a given line of sight through the nebula to the effective path length Leff = fL, which is the total length occupied by the individual clumps. The true path length through a shell of inner and outer radii and , respectively, is given by (Savage et al. 2013)

where ξ is the linear distance between a given line of sight and the line of sight to the centre of the nebula, measured in the transverse plane through the nebula. For the Rosette, with  pc and  pc as derived above, L(ξ = 0) = 24 pc is the path length through the centre of the shell.

Table 2

Results from the analysis of the RM and EM data.

The average electron density along the line of sight is estimated from the EM observations using Eq. (3) as (5)Table 2 lists the average electron density at each of the six positions in the Rosette for f = 1. The derived values are between 8 and 13 cm-3 and are consistent with those obtained by Celnik (1985). The mean electron density across the entire H ii region is 12.3 cm-3, with a scatter of 4 cm-3.

Similarly to the EM data, the measured values of RM include the contribution from the large-scale magnetic field weighted by the Galactic warm ionized gas. This needs to be corrected for in order to study the local component of the field in the Rosette. The mean of the 14 measurements outside the radius  pc is (132 ± 20) rad m-2, where the uncertainty corresponds to the standard error on the mean. The background measurements vary between 50 and 230 rad m-2 (half of these have an uncertainty of more than 20%). The final six background-corrected RM values are listed in Table 2.

The line-of-sight component of the magnetic field can be obtained by combining the EM and RM observations, using Eqs. (2) and (3), as (6)This approximation holds if B|| is uniform across the nebula and if ne and B|| are uncorrelated along the line of sight. Beck et al. (2003) point out that the latter assumption can lead to uncertainties by a factor of 2–3 in the estimated value of B||. The derived B|| values in the nebula for f = 1 are listed in Table 2 and vary from + 1 to + 5 μG. If we consider a filling factor f = 0.1 (Herter et al. 1982; Kassim et al. 1989), the values increase by a factor of 3.2, varying between + 3 and + 16 μG. These results are in the range of the B|| values measured in the diffuse ISM (Crutcher 2012). As will be discussed in the next section, the hypothesis that the magnetic field is uniform throughout the Rosette H ii region is not satisfied. The field is confined to the ionized shell and thus its direction must vary across the nebula. We will assess this aspect by means of a 2D magnetic field model and compare its predictions with the measured B|| values.

5. The magnetic field in the Rosette

In this section we interpret the observations by comparing them with the results from an analytical model of an ionized nebula that has expanded in a uniform and magnetized medium and formed a neutral shell of swept-up matter. The model is presented in Sect. 5.1 and further detailed in Appendix A. We use it to study the RM distribution across the nebula (Sect. 5.2), as well as to reproduce the mean polarization of the dust shell observed by Planck (Sect. 5.3).

5.1. The magnetized Strömgren shell

The evolution of an expanding ionized nebula has been studied numerically, both in uniform and turbulent magnetized media (e.g. Krumholz et al. 2007; Arthur et al. 2011). As the H ii region expands, the surrounding ISM is swept up into a shell around the central stars. In accordance with the frozen-in condition, magnetic field lines are dragged with the expanding gas and concentrated in the dense shell. If the magnetic pressure is comparable to the thermal pressure in the H ii region, magnetic forces lead to departures from sphericity. Furthermore, because the swept-up magnetic flux increases from the magnetic poles (along the direction of the initial field B0) to the equator (90° from B0), magnetic pressure in the swept-up shell tends to make the shell thickness similarly increase from the poles to the equator. We will ignore magnetic effects and assume that the H ii region expands equally in all directions, creating a spherical neutral shell of swept-up ISM. We consider this assumption to be consistent with the radio observations of the Rosette H ii region, which show its nearly circular shape (e.g. Fig. 2a), despite a possible elongation along the line of sight.

Within this framework, we derive an analytical solution for the magnetic field in a spherical structure composed of a shell of swept-up gas formed around a shell of ionized gas. This configuration is shown in Fig. A.1 of Appendix A, where the details of the derivation are discussed. The model is an analytical description of the correspondence between the initial and present configurations of the radial distribution of matter, which characterizes the expansion of the H ii region. We consider that the initial gas density and the magnetic field B0 are uniform and that they evolve as the matter expands radially. This is translated in the form of an expansion law (Fig. A.2). The final magnetic field B (Eq. (A.8)) depends on the strength of the initial field, B0, and its direction, described by the polar and azimuthal angles (θ0,φ0), with respect to the line of sight.

We note that the Galactic magnetic field has uniform and random components, which are of the same order (see e.g. Beck 2001). In the present study, B0 corresponds to the initial local field, which is a sample of the total (uniform plus random) magnetic field.

In the following sections we describe how the combination of RM and dust polarization data allows us to fully describe the magnetic field in the Rosette and its parent molecular cloud. These data have different resolutions; however, this is not a concern since the model does not attempt to reproduce any fluctuations on the scale of the resolution of either of the observables.

thumbnail Fig. 5

Comparison between the RM observations and the predictions from the model. Panel a) shows the RM data as a function of the distance to the centre of the Rosette. The two vertical dashed lines show the inner and outer radii of the nebula,  pc and  pc, respectively (Sect. 4.2.2). The blue curve shows the radial profile of the modelled RM for the best fit to the six RM measurements within (filled circles), the reference fit. The red curve is the result of fitting the highest four RM data points. The open circles correspond to the RM observations outside the Rosette, used to estimate the background RM, and are not included in the fit. Panel b) presents the reduced χ2 from both fits. The stars indicate the best fit parameters B0 and θ0, which correspond to the minimum for each fit. The reference fit (blue) gives B0 = 7 μG and θ0 = 25° for χ2 = 271 and Ndof = 4. The second fit (red) gives B0 = 24 μG and θ0 = 70° for χ2 = 76 and Ndof = 2. The contours are at 10 and 30% above the corresponding minimum values of .

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thumbnail Fig. 6

Average of the line-of-sight component of the magnetic field as a function of the linear distance to the centre of the nebula. The curve shows the result of the reference fit, for which B0 = 7 μG, θ0 = 25°, and B0 | | = 6.3 μG, as indicated by the horizontal dashed line. The three vertical lines delineate the inner and outer radii of the H ii region and the outer radius of the dust shell:  pc;  pc; and  pc.

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5.2. Ionized shell: RM

The RM of the modelled H ii region is computed using Eq. (2). We consider a uniform nebula with constant electron density ne = 12.3 cm-3, the mean value derived from the EM data (Sect. 4.2.2), and integrate B|| along the line-of-sight depth of the H ii shell. Since the RM is derived solely from the line-of-sight component of the field, it does not depend on φ0.

We derive the B0 and θ0 values that best fit the RM data through a χ2-minimization procedure taking into account the corresponding uncertainties, as follows: (7)The reduced χ2 is computed as , where Ndof = NpointsNparams is the number of degrees of freedom, the difference between the number of points and the number of parameters in the fit. In the present case Nparams = 2, corresponding to B0 and θ0. The results are shown in Fig. 5. Figure 5a presents the best-fit RM curve as a function of the linear distance to the centre of the nebula, ξ, along with the 20 RM measurements discussed in Sect. 4.2.2. The blue line is the result of a fit to the six RM data points within the outer radius of the Rosette Nebula,  pc, and hereafter referred to as the reference fit. The best-fit parameters are B0 = 7 μG and θ0 = 25°. The shape of the RM curve is set by the radii of the H ii shell (thus by the expansion law used, Appendix A), and the scaling of the curve depends on a combination of the two free parameters. The model is unable to reproduce the scatter in the RMs observed close to the boundary of the H ii region, which could be due to the possible clumpiness of the medium and/or fluctuations in the magnetic field direction. These effects are not accounted for in our model, which also assumes that the expansion is perfectly spherical. We did not attempt to adjust the radii of the H ii shell, or equivalently its expansion law. Instead, we perform a second fit, excluding the two lowest RM data points, at ξ ~ 16 pc, to assess the departures of the observations from the simplified assumptions of the present model. The result is shown by the red curve in Fig. 5a. The best-fit parameters are B0 = 24 μG and θ0 = 70°, for χ2 = 76 with Ndof = 2, leading to a that is about 30% lower than that obtained in the reference fit.

The χ2 contours as a function of B0 and θ0 for the two fits are shown in Fig. 5b. The high χ2 values (given in the caption) are due both to the simplicity of the model and to the uncertainties in the RM observations, which do not reflect their true radial variation across the Rosette, as they correspond to individual line-of-sight measurements. The contours illustrate the degeneracy between the strength and the orientation of the initial field relative to the line of sight, as expected since RM ∝ B|| = Bcosθ (Eq. (2)). As a consequence, all the (B0,θ0) combinations that follow the minimum contour lead to approximately the same B0 | | value: B0 | | ≃ 6 μG for the reference fit and B0 | | ≃ 8 μG for the second fit. Furthermore, for a given angle θ0, the two different fits give B0 values that differ by less than about 4 μG.

In the rest of this section, we consider only the reference fit obtained when using all of the six RM measurements (blue curve in Fig. 5). Figure 6 shows the radial profile of the mean of B||, B||, measured along the line of sight. There is a significant difference between B|| in the H ii region and in the dust shell. We note that the exact shape of the curve is determined by the adopted expansion law (Fig. A.2), which characterizes the two distinct regimes in the evolution of the Rosette: the expansion of the ionized gas, leading to a decrease in B relative to B0, and the compression of the interstellar gas in the dust shell, accompanied by a compression of the field lines, and hence an increase in B (see Fig. A.3). The mean value of B|| across the projected surface of the ionized shell, which in 3D includes the central cavity, is 2.6 μG. This is comparable to the mean of the six values derived from the RM data listed in Table 2. Therefore, the present model indicates that we can use the RM data to recover the mean B|| in the H ii region, which is 62% lower than B0 | | in the molecular cloud.

The RM observations are from Savage et al. (2013), who fitted a different analytical model to the data, as introduced in Sect. 2. The authors found a value of θ0 = 72° for an assumed B0 = 4μG, under the assumption that a strong adiabatic shock produces an enhancement of the component of the field parallel to the expansion front, relative to the ambient medium. In addition, Savage et al. (2013) applied the shock boundary conditions to the whole thickness of the shell, although these only hold in the thin-shell approximation. The present magnetic field solution in Eq. (A.8) naturally explains the variations of the normal and tangential components of the field relative to the expansion front, throughout the nebula.

This analysis defines the loci of B0 and θ0 values that best fit the RM data towards the Rosette Nebula (blue curve in Fig. 5b). For all of these solutions the resulting line-of-sight field component in the ambient medium is B0 | | ≃ 6μG, which is at the low end of the range of values reported by Crutcher (2012) for molecular clouds of similar column density as Mon OB2 (around 3 × 1022 cm-3, Planck Collaboration XI 2014). In the following section we show that the degeneracy between B0 and θ0 can be alleviated by further comparing the predictions from our model with the Planck polarization observations towards the dust shell.

5.3. Neutral shell: dust polarized emission

We model the shell that surrounds the H ii region with constant intrinsic dust polarization fraction p0 and with inner and outer radii of  pc and  pc, respectively (see Appendix A). Since we will be comparing the ratios between the mean Stokes parameters, we do not specify the density or temperature of the gas and work with normalized quantities. The polarization fraction p can be written as p = p0sin2θ. We compute q = p0sin2θcos(2ψ) and u = p0sin2θsin(2ψ) (Eq. (1)) at every position in the 3D shell and integrate along the line-of-sight direction to obtain the normalized Stokes parameter maps.

thumbnail Fig. 7

Comparison between the Planck observations and the model. The ratios Q ⟩/⟨ I and U ⟩/⟨ I derived from the polarization data are given by the magenta point. The statistical and systematic uncertainties are shown by the solid and dashed error bars, respectively. The dashed grey lines show the solutions for φ0 = −5°, 0°, and 5°, with θ0 varying from 10° to 80°. The solid grey lines indicate how the ratios change with φ0, for a given θ0 angle. The model results are calculated here for p0 = 4%.

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There are three variables involved in modelling the dust polarization: the angles (θ0,φ0), which define the direction of B0 with respect to the line of sight, and the intrinsic polarization fraction p0. The Stokes parameters do not depend on the strength of the magnetic field. The polarization fraction relation given above indicates that p0 and θ are degenerate. We thus start by comparing the data with the model for a fixed p0 value of 4%, which corresponds to the observed value in the Mon OB2 cloud (Sect. 4.2.1). Figure 7 illustrates how the predicted ratios between the mean values of the Stokes parameters, Q ⟩/⟨ I and U ⟩/⟨ I, vary as a function of the initial magnetic field direction (θ0,φ0). The observed ratios, calculated using the values listed in the fifth column of Table 1, are Q ⟩/⟨ I ⟩ = (1.51 ± 0.07 (sta.) ± 0.17 (sys.)) × 10-2 and U ⟩/⟨ I ⟩ = (−0.06 ± 0.06 (sta.) ± 0.41 (sys.)) × 10-2. The systematic uncertainties correspond to the difference in the ratios when derived with the different leakage correction maps (Sect. 3.1.1). The observed U ⟩/⟨ I ratio constrains the sky projected orientation of the initial magnetic field, φ0 ≃ 0°, to within about 5°. This value is consistent with that measured towards the Rosette’s parent molecular cloud (Sect. 4.2.1) and corresponds to a magnetic field parallel to the Galactic plane. Fixing this parameter allows us to study how the Q ⟩/⟨ I ratio varies as a function of the angle between B0 and the line of sight θ0, and the intrinsic polarization fraction p0. This is shown in Fig. 8, for p0 ranging from 4 to 19.8%, the maximum polarization fraction observed across the sky (Planck Collaboration Int. XIX 2015). The comparison between the data and the different models gives an upper limit on θ0 of about 45° for p0 = 4%. A lower limit of θ0 ≃ 20°, implying a field that is nearly along the line of sight, is obtained for the maximum intrinsic polarization fraction p0 = 19.8%.

thumbnail Fig. 8

Variation of the modelled Q ⟩/⟨ I ratio as a function of p0 and θ0, for φ0 = 0° (grey lines). The observed Q ⟩/⟨ I ratio is shown by the vertical magenta line.

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The comparison between the present model and the Planck polarization observations leads to two main results. First, the inferred range of θ0, combined with the results from the RM study, restricts the range of the magnetic field strength in the Rosette’s parent molecular cloud to B0 ≃ 6.5–9 μG. Second, we find that the Mon OB2 cloud has a magnetic field structure distinct from that of the Perseus spiral arm; an azimuthal Galactic mean field with a pitch angle of −8° (Ferrière 2011), is expected to be oriented at about 60° from the line of sight at the position of Mon OB2. Within the uniform density and polarization fraction assumption of this model, such values of θ0 are only possible for a significantly low intrinsic polarization fraction p0. However, we cannot discard depolarization effects from a turbulent field and/or clumpy density distribution. While several observational studies indicate that molecular clouds preserve the large-scale field orientation (see Li et al. 2014), Planck observations question this general interpretation of almost no variation of the magnetic fields with interstellar structures. Modelling of the dust polarization data from Planck allows us to study the 3D geometry of the magnetic field in a variety of environments. The analysis of the magnetic field structure in nearby interstellar filaments by Planck Collaboration Int. XXXIII (2016) suggests that their evolution is coupled to the field, which is distinct from the field of the clouds in which they are embedded. This result agrees with our findings on the Rosette/Mon OB2 complex.

Finally, we can estimate the magnetic and thermal pressures in the H ii and dust shells. We do not include the possible contribution from a turbulent field at scales much smaller than the Rosette, whose quantification is difficult. In either case this is expected to be small owing to the fast expansion of the H ii region, which tends to order the magnetic field. For an initial field strength B0 = 9 μG, which corresponds to θ0 = 45°, the modelled magnetic field has a mean value BH ii = 3.2μG within the ionized nebula and Bdust = 21.4μG in the dust shell. The thermal pressure in the H ii region is  erg cm-3, where k is the Boltzmann constant, ne = 12.3 cm-3, and Te = 8500 K (Sects. 2 and 4.2.2). The magnetic pressure,  erg cm-3, is therefore smaller than the thermal pressure in the H ii region, supporting our initial assumption that the H ii region is roughly spherical. In the dust shell, the magnetic pressure is  erg cm-3, hence smaller than but of the same order as the ionized gas pressure. This implies that both the thickness and the density of the dense shell should not be constant (e.g. Ferrière et al. 1991). Modelling these effects is beyond the simplified nature of the present analysis, where we focus on the mean polarization properties of the Rosette, but will be taken into account in future work.

6. Conclusions

This work represents the first joint analysis and modelling of radio and submillimetre polarization observations towards a massive star-forming region (the Rosette Nebula) to study its 3D magnetic field geometry. We have developed an analytical solution for the magnetic field, assumed to evolve from an initially uniform configuration following the expansion of ionized gas and the consequent concentration of the surrounding ISM in a dense shell. The assumption of uniform density and temperature distributions for both the ionized and dust shells, along with constant intrinsic polarization fraction of dust grains, is clearly an approximation. Different parts of the fragmented, swept-up shell presumably have distinct properties and some may even be pre-existing dense clouds caught by the expanding H ii region. Nevertheless, the model is able to reproduce the mean observed quantities.

We use the Planck data at 353 GHz to trace the dust emission from the shell of swept-up ISM surrounding the Rosette H ii region. Even if the shell is clearly seen in intensity, the same pattern is not detected in dust polarized emission against the local background. When analysed as a whole, the polarized signal from the dust shell is significantly distinct from that of the background and can be reproduced by the current model. The correspondence between the model and the Planck observations constrains the direction of the magnetic field in the Rosette’s parent molecular cloud Mon OB2 to an angle in the plane of the sky φ0 ≃ 0° (roughly parallel to the Galactic plane) and an angle to the line of sight θ ≲ 45°. This result is crucial to removing the degeneracy between θ0 and B0 inherent in the RM modelling. We thus find that B0 is about 6.5–9 μG in Mon OB2.

The present magnetic field model provides a satisfactory fit to the observed RM distribution as a function of the distance from the centre of the Rosette H ii region. More data are needed to better understand the abrupt variations of RM close to the outer radius of the nebula. The RM modelling suggests a significant increase in the line-of-sight magnetic field from the H ii region to the dense shell, where B|| reaches nearly 4 times B0 | |.

The combination of RM and dust polarization data in this work is essential to constrain both the direction and the strength of the field in the Rosette region. The model presented here can be directly applied to other similar objects for which the expansion law can be derived.


1

Planck (http://www.esa.int/Planck) is a project of the European Space Agency (ESA) with instruments provided by two scientific consortia funded by ESA member states (in particular the lead countries France and Italy), with contributions from NASA (USA) and telescope reflectors provided by a collaboration between ESA and a scientific consortium led and funded by Denmark.

2

We checked for the presence of nearby pulsars in the Australia Telescope National Facility Pulsar Catalogue (Manchester et al. 2005). We did not find pulsars closer than from the centre of the Rosette with the required information (distance, dispersion measure, RM).

Acknowledgments

We thank the referee for the useful comments. We acknowledge the use of the HEALPix package and IRAS data. The Planck Collaboration acknowledges the support of: ESA; CNES and CNRS/INSU-IN2P3-INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MICINN, and JA (Spain); Tekes, AoF, and CSC (Finland); DLR and MPG (Germany); CSA (Canada); DTU Space (Denmark); SER/SSO (Switzerland); RCN (Norway); SFI (Ireland); FCT/MCTES (Portugal); and DEISA (EU). A detailed description of the Planck Collaboration and a list of its members can be found at http://www.rssd.esa.int/index.php?project=PLANCK&page=Planck_Collaboration. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement No. 267934.

References

Appendix A: Magnetic field model

thumbnail Fig. A.1

Sketch of the adopted spherical configuration. Left: the initial state, with uniform density and magnetic field; there are no separate ionized and dust shells. Right: present-day state with a cavity inside , an ionized shell between and , and a dust shell between and . The present-day radii , , and correspond to initial radii 0, , and , respectively.

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In this appendix we present the derivation of the analytical formula that describes the magnetic field structure in a spherical shell, following the expansion of an ionized nebula in a uniform medium with density n0 and magnetic field B0.

We start by deriving the expansion law, which will define how the initial uniform magnetic field is modified. Once the star cluster is formed, it ionizes the surrounding gas, which becomes overpressured and starts expanding at a velocity close to the ionized gas sound speed. The expansion of the ionized gas, in turn, creates a cavity and sweeps up the surrounding ISM into a thin and dense shell. The resulting structure, as observed at the present time, is composed of a cavity of radius , surrounded by a thick shell of ionized gas extending from to , itself surrounded by a thin dust shell extending from to (see Fig. A.1). For the Rosette,  pc and  pc, as derived from the radial distribution of the radio emission (Sect. 4.2.2);  pc and  pc, measured from the Planck 353 GHz latitude cut through the centre of the shell (Sect. 4.1). Thus, the inner radius of the dust shell is slightly smaller than the outer radius of the H ii region. This is not surprising, as the Rosette is an ionization bounded Strömgren sphere and thus the two shells are expected to overlap at the boundary, where the ionized and neutral gases are mixed. However, for the sake of simplicity, we take  pc.

We use a parameter grid of 1 pc resolution for all the radii. Therefore, we adopt an uncertainty of 1 pc in and . We ascribe a larger uncertainty of 2 pc to and ; this reflects the variation of the fitted radii when considering the longitude or radial profile of the dust emission, which are affected by the presence of the Mon OB2 cloud.

We now denote by r0 the initial radius of a particle currently at radius r. In our simplified model, the initial radius of a particle currently at is simply r0 = 0, while the initial radii of particles currently at and can be denoted by and , respectively. We emphasize that and are just two working quantities, which do not correspond to any physical boundaries. The value of can be obtained by noting that a particle currently at has just been reached by the expanding shell and has not yet moved from its initial radius , so that  pc. The value of can be inferred from the conservation of mass. In the initial state where MH ii is the mass inside the sphere with radius and Mdust is the mass inside the shell with inner and outer radii and , respectively, as shown in the left panel of Fig. A.1. From the previous equations we can write (A.3)We calculate the mass of ionized gas based on the size of the nebula and on the mean electron density of 12.3 cm-3, derived from the radio data (Sect. 4.2.2). Taking into account the contribution from ionized helium (Celnik 1985), we obtain MH ii = 1.2 × 104 M. For the mass of the dust shell we use the results of Heyer et al. (2006; Sect. 2), Mdust = 8.6 × 104 M. As a result,  pc. With the three known radii in the initial state, , and their corresponding values in the present state, , we can derive an expansion law as shown in Fig. A.2. With only three data points, we consider the simplest description of the expansion by writing r as a piecewise linear function of r0. This is in any case sufficient to describe the two clear regimes seen in Fig. A.2: the expansion of the H ii region (slope larger than 1) and the compression of the ISM within the outer shell (slope smaller than 1). The expansion law is hence given by r = αr0 + β, where α = 1.09, β = 7 pc for the H ii region () and α = 0.27, β = 16 pc for the dust shell ().

thumbnail Fig. A.2

Radial expansion law derived from the three known radii in the initial, r0, and final, r, states (red filled circles). The black curve is described by a piecewise linear function of the form r = αr0 + β, with α = 1.09, β = 7 pc for and α = 0.27, β = 16 pc for .

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Now consider a cartesian coordinate system with z-axis along the line of sight and x- and y-axes in the plane of the sky, along the trace of the Galactic plane and along the rotation axis, respectively. The initial uniform magnetic field can be written in these Cartesian coordinates as (A.4)with (A.5)where θ0 and φ0 are the polar and azimuthal angles, respectively, i.e. θ0 is the angle between B0 and the line of sight, while φ0 gives the plane-of-the-sky direction of B0 with respect to the Galactic plane (φ0 = 0° for B0 parallel to the Galactic plane).

The vector potential associated with B0 is (A.6)which satisfies the condition B0 = ∇ × A0. We use the frozen-in approximation and assume that the magnetic field evolves from the initially uniform configuration following the radial expansion of the gas. The vector potential in the final state is given by (see Eqs. (4) to (10) in Parker 1970) (A.7)and the resulting magnetic field works out to be (A.8)The previous equation, written in spherical coordinates, clearly shows the change in both the normal (radial) and tangential components of the magnetic field, relative to the expansion front.

We create a 3D cartesian grid of 1813 = 5 929 741 voxels, each equivalent to 0.25 pc ( at the distance of the Rosette), with the bubble-shell structure located at the origin. The resolution of the model is finer than that of the observations, which is needed to have the required sampling to compute the integrals along the line of sight. We use Eq. (A.8) along with the expansion law of Fig. A.2 to calculate the magnetic field strength in every pixel of the grid. Figure A.3 shows how the field strength in the shell, B, varies relative to the initial field strength, B0. The map corresponds to a vertical cut through the centre of the shell for an initial field with (θ0,φ0) = (90°,0°), therefore on the plane of the sky and along the Galactic plane. Figure A.3 illustrates that the largest compression of the field lines occurs towards the equator of the shell, or in the direction perpendicular to the initial field B0, where the ratio B/B0 is seen to increase from the centre to the outer radius of the dust shell. The change in expansion law at the boundary between the H ii and dust shells, r = 19 pc, results in a discontinuity in the tangential component of the magnetic field. On the other hand, close to the poles of the shell, or along B0, the field lines are little disturbed, with the ratio B/B0 continuously increasing from the centre to B/B0 = 1 at the boundary of the dust shell. Owing to the axial symmetry of the magnetic field model, the map of Fig. A.3 is reproduced in every plane about the direction of the initial field B0.

thumbnail Fig. A.3

Ratio of the field strength B/B0 for a vertical slice through the centre of the bubble-shell structure (xy plane). The initial field B0 is on the plane of the sky (θ0 = 90°) and along x (φ0 = 0°). The three black circles delineate the radii of the Rosette H ii and dust shells:  pc;  pc; and  pc.

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All Tables

Table 1

The mean level of the Stokes I, Q, and U parameters at 353 GHz, in units of MJy sr-1.

Table 2

Results from the analysis of the RM and EM data.

All Figures

thumbnail Fig. 1

The Rosette Nebula, its molecular cloud Mon OB2, and their surrounding medium as seen by Planck at 353 GHz. The dotted lines give the approximate outer radii of the Rosette H ii region and the Mon Loop supernova remnant (described in Sect. 2). The northern Monoceros cloud Mon OB1 is above the Galactic plane; part of the dust ring of λ Ori is visible at the bottom right corner of the map; the northern end of Orion B is seen at (l,b) = (204°, −12°) (Dame et al. 2001).

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In the text
thumbnail Fig. 2

The Rosette Nebula and its molecular cloud. a) Integrated RRL emission at resolution, showing the radio morphology of the H ii region. The green circles, with a diameter equivalent to the beam full width at half maximum (FWHM) of the RRL data, give the position of the 20 RM observations of Savage et al. (2013) used in this work. b) Planck 353 GHz emission at 6 resolution. The black star in panels a) and b) indicates the position of the central star cluster NGC 2244 and the black circles correspond to the inner and outer radii of the H ii region,  pc and  pc, respectively (see Sect. 4.2.2). c) Planck CO integrated intensity at resolution. d) Dust temperature derived from Planck at 5 resolution. e) Dust emission as seen by IRAS at 12 μm and at 100 μm (f), at 4 resolution. The contours in panels c) to f) show the Planck 353 GHz emission; these are at every 5% from 15 to 30%, at every 10% from 30 to 50%, and finally at 70 and 90% of the maximum intensity of 37 MJy sr-1.

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In the text
thumbnail Fig. 3

Planck polarization maps of the Rosette region: a) Q Stokes parameter; b) U Stokes parameter. The dashed circles, with radii 2° and 3°, define the region where we estimate the background level, the full circle centred on the star cluster represents the outer radius of the dust shell,  pc, and the rectangle encompasses the brightest part of the Mon OB2 cloud. c) Intensity map at 353 GHz with the plane-of-the-sky magnetic field orientation shown by headless vectors, obtained by rotating the polarization angle ψ by 90°. The vectors are plotted at every 6, from the average of Q and U within the same distance from the central pixel. d) Polarization fraction p, with the same intensity contours as in Fig. 2.

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In the text
thumbnail Fig. 4

Normalized distribution functions (DFs) of the a) Stokes Q and b) U maps measured within the background aperture (blue) and within the outer radius of the dust shell (red). The two regions are defined in Fig. 3 by the dashed and full circles, respectively.

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In the text
thumbnail Fig. 5

Comparison between the RM observations and the predictions from the model. Panel a) shows the RM data as a function of the distance to the centre of the Rosette. The two vertical dashed lines show the inner and outer radii of the nebula,  pc and  pc, respectively (Sect. 4.2.2). The blue curve shows the radial profile of the modelled RM for the best fit to the six RM measurements within (filled circles), the reference fit. The red curve is the result of fitting the highest four RM data points. The open circles correspond to the RM observations outside the Rosette, used to estimate the background RM, and are not included in the fit. Panel b) presents the reduced χ2 from both fits. The stars indicate the best fit parameters B0 and θ0, which correspond to the minimum for each fit. The reference fit (blue) gives B0 = 7 μG and θ0 = 25° for χ2 = 271 and Ndof = 4. The second fit (red) gives B0 = 24 μG and θ0 = 70° for χ2 = 76 and Ndof = 2. The contours are at 10 and 30% above the corresponding minimum values of .

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In the text
thumbnail Fig. 6

Average of the line-of-sight component of the magnetic field as a function of the linear distance to the centre of the nebula. The curve shows the result of the reference fit, for which B0 = 7 μG, θ0 = 25°, and B0 | | = 6.3 μG, as indicated by the horizontal dashed line. The three vertical lines delineate the inner and outer radii of the H ii region and the outer radius of the dust shell:  pc;  pc; and  pc.

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In the text
thumbnail Fig. 7

Comparison between the Planck observations and the model. The ratios Q ⟩/⟨ I and U ⟩/⟨ I derived from the polarization data are given by the magenta point. The statistical and systematic uncertainties are shown by the solid and dashed error bars, respectively. The dashed grey lines show the solutions for φ0 = −5°, 0°, and 5°, with θ0 varying from 10° to 80°. The solid grey lines indicate how the ratios change with φ0, for a given θ0 angle. The model results are calculated here for p0 = 4%.

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In the text
thumbnail Fig. 8

Variation of the modelled Q ⟩/⟨ I ratio as a function of p0 and θ0, for φ0 = 0° (grey lines). The observed Q ⟩/⟨ I ratio is shown by the vertical magenta line.

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In the text
thumbnail Fig. A.1

Sketch of the adopted spherical configuration. Left: the initial state, with uniform density and magnetic field; there are no separate ionized and dust shells. Right: present-day state with a cavity inside , an ionized shell between and , and a dust shell between and . The present-day radii , , and correspond to initial radii 0, , and , respectively.

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In the text
thumbnail Fig. A.2

Radial expansion law derived from the three known radii in the initial, r0, and final, r, states (red filled circles). The black curve is described by a piecewise linear function of the form r = αr0 + β, with α = 1.09, β = 7 pc for and α = 0.27, β = 16 pc for .

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In the text
thumbnail Fig. A.3

Ratio of the field strength B/B0 for a vertical slice through the centre of the bubble-shell structure (xy plane). The initial field B0 is on the plane of the sky (θ0 = 90°) and along x (φ0 = 0°). The three black circles delineate the radii of the Rosette H ii and dust shells:  pc;  pc; and  pc.

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Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.