DR21 Main: A Collapsing Cloud
The molecular cloud, DR21 Main, is an example of a large-scale gravitational collapse about an axis near the plane of the sky where the collapse is free of major disturbances due to rotation or other effects. Using flux maps, polarimetric maps, and measurements of the field inclination by comparing the line widths of ion and neutral species, we estimate the temperature, mass, magnetic field, and the turbulent kinetic, mean magnetic, and gravitational potential energies, and present a 3D model of the cloud and magnetic field.
Partially ionized material in a molecular cloud will respond to gravitational forces by moving, primarily, along magnetic field lines. If one assumes that the field is not significantly tangled by turbulence, one can expect a concentration of material in an oblate core. This model has been verified by the analysis of polarimetric and photometric data (Dotson et al., 2009) by Tassis et al. (2008) showing that the field is directed along the short axes of the cores. The neutral component of the material will move slowly across field lines toward the center of gravity thus increasing the gravitational field. Where a pinch in the field is indicated by a polarization map, one can infer that the gravitational field in the core is competing with the support provided by the field. Within a critical radius, one expects the cloud to collapse.
Examples of pinched fields have been found in Orion (Schleuning, 1998) and in NGC1333 (Girart et al., 2006). In the case of DR21 Main (DR21M), the polarization map extends to radii such that the field straightens out as the gravitational field diminishes leaving an hourglass field configuration. In this cloud one has an opportunity to estimate the mass within the collapsing region. We use 350 polarimetry to measure the field as projected on the sky (§3.1); line width measurements to estimate the inclination of the field to the line of sight (§4.1); photometric maps to estimate the mass distribution (§4.3); and the angular dispersion of the field vectors to estimate the field strength (§4.4).
The giant star forming complex, DR21 (Figure 1) is located in the Cygnus constellation 3kpc from Earth (Campbell et al., 1982). The southernmost component, DR21M, has a mass of (Richardson et al., 1989). It contains one of the most energetic star formation outflows detected (Garden et al., 1991; Garden & Carlstrom, 1992). Garden et al. (1991) measured the mass of the outflow to be . Observers using Spitzer found 5 near-infrared (NIR) sources (see Figure 2) inside DR21M, presumed to be protostars (Kumar et al., 2007; Davis et al., 2007) indicating that regions within the cloud are undergoing gravitational collapse. Glenn et al. (1999) at 1.3 mm and Minchin & Murray (1994) at 800 have previously measured the polarization of DR21 finding uniform position angles East of North (i.e. B field 65 °West of North). The galactic plane is oriented at 42 °East of North. Roberts et al. (1997) measured a line-of-sight magnetic field of using HI Zeeman measurements around the dust emission peak and star formation regions. The regions observed by Roberts et al. (1997) with detections are outlined in green in Figure 3.
Observations, made at the Caltech Submillimeter Observatory (CSO) with the polarimeter Hertz (Schleuning et al., 1997; Dowell et al., 1998), provided 350 polarization maps of the region. Hertz was later replaced by SHARP (Li et al., 2008), a module mounted ahead of the photometer SHARC-II (Dowell et al., 2003) to permit polarization measurements.
SHARP has a wire cross-grid to split the incoming signal into orthogonal components of polarization which are then detected on two 144 pixel subsets of the SHARC-II detector array. Standard infrared techniques of chopping the secondary mirror to remove sky emission and positional offsets to remove gradients in sky emission were used for all the polarimetry observations (Hildebrand et al., 2000). The chopping direction was azimuthal with a throw of 300″. To avoid chopping into the adjacent cloud, DR21OH, all observations were made when the chop throw was 30°from north-south on the sky A half-wave plate was rotated to four different angles separated by 22.5°. Each observation cycle consisted of a set of observations at the 4 half-wave plate angles. The signals from the two 144 pixel arrays were then combined to give the polarization and photometric signals as described by Kirby et al. (2005). The polarization and flux signals are given by
where and are the horizontal and vertical components of the signal. The Stokes parameter, , was calculated from the difference of the polarization signals of the 1st and 3rd angle in the cycle, while the parameter was calculated from the difference of the polarization signals of the 2nd and 4th angle. The source flux, , was determined by the average of the flux signals from the 4 half-wave plate positions. The , , and from each cycle were then combined after correcting for changes in attenuation and sky noise. Due to the lack of an instrument rotator, this combination had to be done a new way. The corrected , , and from all of the cycles were combined to form an irregularly sampled map. This map was then smoothed to a resolution of (for more detail see Houde & Vaillancourt (2007).
The observations were made in 7 fields spaced by 45″. Each field was observed with a 4-point dither on corners of a square 10″ wide in Right Ascension and Declination. A few additional observations were taken to fill in areas of low signal to noise in the above 7 fields. The data were corrected for instrument polarizations of (Li et al. (2008), Vaillancourt et al, in preparation)
The SHARP observations were taken on 2006 July 15-19 during intermediate weather ( corresponding to 1.7-2.0 mm zenith water vapor). The SHARP polarization data set shows 1270 sky positions with a signal to noise . The beam size of SHARP is the pixel size. Limiting the sample so there is no beam overlap leaves 78 measurements. The median value of the polarization is 2.4% ranging between 0.37% and 13%. Half of the points have a signal to noise better than 4.7. At this value, the accuracy of the position angle of the polarization vectors is . The direction of polarization shows a pinched or hourglass shape (see Figure 3). The axis of symmetry of the hourglass is inclined 15° East of North (determined by model see 4.5.
3.2 Heterodyne Measurements
The polarimetry measurements from SHARP and Hertz are supplemented with 345 GHz heterodyne measurements of ion and neutral molecular line widths from the CSO. The heterodyne measurements were taken on 2006 July 12-13 (). The inclination of the field at eleven points within the pinched region was determined from measurements of line widths of neutral and ionized molecules of comparable mass. The heterodyne measurements of the J4-3 transition in HCN and HCO+ were made using only on/off position differencing (Houde et al., 2000a, b). The data were reduced using the CLASS program as part of the GILDAS package of programs (http://www.iram.fr/IRAMFR/GILDAS). Each of the lines was smoothed to half of the natural resolution using the CLASS routine SMOOTH to improve the signal to noise. One of the positions was removed due to low signal to noise. Another position was unusable because bandwidth of the instrument was insufficient to image the entire line.
The spectrum at each position was then fit by a multi-Gaussian model (Figure 4) to determine the line widths (more properly the standard deviation ). Multiple Gaussians were used to account for the self-absorption seen in most of the positions, and for one-sided outflows, which can cause an artificial reduction in line widths if not accounted for properly. The effects from broad outflows were removed from two locations. The fits were done using the XGaussfit program from the FUSE package of software for IDL ( http://fuse.pha.jhu.edu/analysis/fuse_idl_tools.html). Results of the fits are given in Table 1.
4.1 Inclination Angle
Houde et al. (2000a, b) investigated the effect of a magnetic field on the velocities and ions in a partially ionized flow (not necessarily an outflow or jet but any situation where local mean velocity is not zero). For a significantly strong field, the ions were forced into gyromagnetic rotation about the magnetic field instead of following the flow. This rotation led to a reduction in line width and suppression of high velocity wings. The line widths were derived for both the neutrals and ions. The ratio of the linewidths depends only on the orientation of the neutral flow(s) and the inclination angle of the magnetic field with the line of sight (Houde et al., 2002, 2004). Assuming that the neutral component of the material is (mean molecular mass of 2.3), the square of the ratio of the ion to neutral line width is given in terms of the angle with the line of sight, , by
and and are the ion mass and the reduced mass respectively. The neutral flows are modeled as symmetrical and contained within a cone of width centered on magnetic field direction (symmetry axis, see Figure 5). All values of must satisfy the condition where is the maximum polarization seen by Hertz (10 % Houde et al. (2004)).
Although the cone half width, , can, in principle, be determined by finding the curve that best fits the above condition on , the data for DR21 do not provide significant constraints on this quantity. The values of presented in Table 1 are computed assuming . The inclination with the line of sight decreases toward the center of DR21 as would be expected for a magnetic field in the center of a gravitationally contracting region with an axis near the plane of the sky. Inclination angles for points located at the edge of the cloud, where the gravitational pinch has had only a small effect, should approximate the inclination of the cloud’s mean B field. These points for DR21 give an estimate of inclined to the line of sight or 20°to the plane of the sky.
Choosing the wrong value of leads to a systematic error in the calculation of each angle, but the difference between angles is not significantly affected. The overall trend of decreasing angle toward the center of the cloud remains. If, instead, we had assumed , the values shown in the table would decrease by but the overall trend toward decreasing inclinations toward the center would not change.
4.2 Temperature and Optical Depth
The spectral energy distribution at 350 and 850 was fitted to give estimates of the dust temperature and optical depth at 350 . The 850 map was acquired from the SCUBA online archive
where the exponent gives the atmospheric transmission at frequency as determined from the transmission at 350 , and is the Planck function at frequency and temperature . An example fit to the data for the center of DR21 is shown in Figure 7. This formula assumes the cloud is optically thin at the wavelengths used. The ratio of the the 350 flux to the 850 is constant throughout the majority of the cloud ( 15) with a discrete jump at the edges of the cloud ( 25) and a dip in the center ( 8). The constancy of the ratio implies optical thinness except at the center where the ratio drops.
Temperatures at the peak were 20 K and went up to 25 K in the cloud. The edges of the cloud where the ratio increased had higher temperature (30-50 K). The 25 K region of the cloud extended along a rough northeast-southwest direction similar to the direction of the outflow. The optical depth showed a similar trend with a value at the peak greater than 1. The optical depth quickly drops below 1 indicating that all but the bright peak areas of DR21 main are optically thin at 350 , as expected from the flux ratios. Harvey et al. (1986) found temperatures from 50 and 100 flux maps to be 50 K. These wavelengths may be coming from a different environment with a different temperature. Supporting a multi-environment hypothesis, Richardson et al. (1989) found a 350 to 800 ratio of 16 at DR21OH with a corresponding temperature of 25 K in regions given to be 37 K by Harvey et al. (1986). Throughout the rest of the paper, we used temperatures of 20-25 K from the SHARP and SCUBA fluxes.
The mass of cloud within a specified region of a cloud can be determined from its temperature and far infrared flux (Hildebrand, 1983)
where F is the flux, D is the distance to the cloud, and
where is the optical depth of the cloud at frequency . Within a radius of 1 pc (size of SHARP map in E-W direction), DR21 main has a mass of , comparing nicely to the found by Richardson et al. (1989).
4.4 Magnetic Field Strength
Chandrasekhar & Fermi (1953) described a method to estimate the magnetic field in the plane of the sky. This method related the dispersion of the direction of starlight polarization from a straight line to the strength of the magnetic field,
where is the density of the gas, is the observed line of sight velocity dispersion, and is the dispersion in polarization direction in the plane of the sky. The weaker the field, the greater the dispersion of the polarization vector.
The dispersion of the polarization vectors will be overestimated if it is taken with respect to the mean field. To determine the dispersion about the local magnetic field one must take into account the inclination of the mean field (§4.1) and the structure of the large scale field due to non-turbulent effects such as gravitational collapse, expanding HII regions, and differential rotation (Hildebrand et al. submitted to ApJ). A polarization map usually shows a smoothly-varying pattern of vectors, therefore at separations small compared to the cloud diameter, the 2-point angular correlation function () of position angles should increase almost linearly as shown in the bottom panel of Figure 8. The linear portion would have a zero-intercept if there were no measurement error and no turbulence. Both effects will cause the whole 2-point correlation to be displaced upward (except at values correlation length of turbulence which is expected to be much less than the SHARP or Hertz resolution of DR21). The y-intercept, b, of the best fit line is then the quadratic sum of the estimate of the dispersion, and the measurement error, . A cross correlation of repeat observations gives the estimate of the measurement error (value at 0 separation) and an estimate of the quadratic sum of measurement error and turbulent dispersion (intercept of best fit line to linear region at small non-zero separation)
Figure 8 shows the correlation function for two subsets of the Hertz data set. The subsets were constructed by taking every other raw data file starting with file 1 for subset 1 and file 2 for subset 2. The line best fit to the linear region of the correlation function has an intercept of 93. The value of the correlation function at zero separation is 76 giving an estimate of 53 for the turbulent dispersion. Using the Chandrasekhar and Fermi method formula with correction factor from Ostriker et al. (2001) gives a plane of the sky magnetic field of 3.1 mG where is the observed line of sight velocity dispersion of HCN measured to be and is the mean density calculated from the mass in §4.3 to be .
The method of data acquisition of SHARP prevents repeat observations of the same location (caused by lack of instrument rotator) preventing that sample from being split as done for Hertz. However, the mean of the individual errors on the angles give a good estimate for the measurement error (72 to 76 for Hertz). Using that method of estimating the mean error and fitting a line to the linear region of the correlation function gives a magnetic field estimate of 2.5 mG using SHARP data (dispersion of 66).
Using the measurement of 400 for the line of sight magnetic field by Roberts et al. (1997) and taking a 3 mG field in the plane of the sky gives a total magnetic field of 3 mG inclined to the plane of the sky at 10°. This estimate of the angle is probably low due to the Zeeman measurement being from HI, which exists at lower density than the dust we observe in polarization so the field strength is also likely lower there. Taking the scaling of Heiles & Crutcher (2005) and the density of the region observed in HI as , the angle with respect to the plane of the sky is . For a density of , the angle would be Roberts et al. (1997) assumed that the magnetic field was less than 1 mG because unless the field was very uniform, the field would have bent somewhere such that they detected a 1 mG field. The dust polarization measurements show a field that is very uniform implying the field could be larger than 1 mG and not detected in the Zeeman observations.
4.5 3-Dimensional Model
A three dimensional model of the flux and magnetic field was constructed and fit to the observed flux and polarization data. The flux was modeled as a Gaussian ellipsis of the form
where is coordinate along the symmetry axis of the magnetic field, and are the other two Cartesian coordinates, and and are widths of the Gaussians to be fit. The magnetic field was modeled as
This form was chosen to give a magnetic field with the shape of an hourglass that straightened at large distance from the center. The magnetic field strength in the x direction in the midplane was set proportional to the column density. Assuming flux freezing, the magnetic field strength in the x direction for locations outside the midplane was set to a value in the midplane that is found by tracing the field line back to the midplane. The field in the y-z plane was assumed radial and the strength was found from the strength in the x direction and the overall magnetic field direction. The flux and magnetic field strengths were then rotated and summed along the line of sight. The values of , , , and were then set to give the best fit to the flux and the magnetic field direction determined by the polarization measurements. The best fit values were (units for are SHARP pixels = 2375, 20°for the angle of the magnetic field with the plane of the sky, and 15°East of North for the angle of the projected minor axis on the plane of the sky. The fit for the magnetic field direction was quite good (chi-squared of 2). The flux fit was poor as DR21-Main is not an exact ellipse. Figure 9 shows the model vectors along with the SHARP vectors. The angle with the plane of the sky is somewhat higher than found in §4.4 but agrees well with the measured inclination angles farthest from the center (see §4.1).
4.6 Gravitational and Magnetic Potential Energies
Using the mass and magnetic field estimates discussed in §4.3, §4.4, and the 3D model presented in §4.5, one can estimate the dependence on radius of the gravitational and magnetic potential energies. The gravitational potential energy of a flattened centrally condensed spheroid at a radius R centered on the peak of the cloud is
and the magnetic energy of the uniform field is given by
where the values and and are calculated by setting the energies equal when the mass to magnetic flux ratio is the critical value (McKee et al., 1993). B is scaled according the model in §4.5 and normalized such that the energy inside 1 pc is the same as for the mean field calculated in §4.4. By comparing large scale polarization maps to with magnetohydrodynamic turbulence simulations, Novak et al. (2007) found that the ratios of the uniform to fluctuating magnetic field in the clouds NGC 6334 and G333.6-0.2 were in the range 0.6-2.0. Figure 10 shows the gravitational and magnetic potential energies (assuming a ratio between uniform to fluctuating field of 1) versus radius. Extrapolating the gravitational potential energy curve to find the radius at which the two energies are equal gives an estimate of 1.2 parsec. The dotted line in Figure 10 shows turbulent energy calculated from the average HCN line widths of 4.2 km/s and assuming the velocities are isotropic. The sum of these two energies is the dotted-dashed line and intersects the gravitational energy curve at 0.7 parsecs. One would then expect the critical radius to be between 0.7 and 1.2 parsecs. The critical radius should not be regarded as the radius which is supported against collapse since the virial theorem can not be applied in this way to magnetic clouds (Mouschovias, 1987, 1991; Dib et al., 2007). Figure 12 shows a plot of the mass to magnetic flux ratio versus radius. The ratio is in units of the critical value of the ratio defined by Mouschovias & Spitzer (1976)
where is taken to be 0.12 (Tomisaka et al., 1988). The ratio dips below the critical ratio at 1.2 parsec. The mass inside this region is (§4.3) which is approximately the entire mass of the cloud. Figure 11 shows the polarization vectors with circles at the critical radii with and without turbulent energy. The magnetic field configuration changes at around these radii from the pinched structure to the east-west mean direction (see Figure 3). This geometry implies that the cloud is relatively stable against collapse at these radii, implying that the magnetic field is providing a significant amount of support against collapse. However the presence of protostars are radii greater than 1.2 parsec in Figure 11 suggests that the true picture may be more complex.
We present a three-dimensional model of the molecular cloud DR21 (Main). Polarimetry at 350 has provided a map of the magnetic field as projected on the plane of the sky; line observations of ion and neutral molecules have provided measurements of the inclination of the field to the line of sight; photometry has provided column densities and color temperatures.
A 3-dimensional model of an hourglass configuration magnetic field inclined to the line of sight is fit to the observations. We find a pinched field with an axis of 10°-20° from the plane of the sky. The mean field strength is 3 mG. The gravitational potential energy is equivalent to energies from support mechanisms (magnetic field and turbulence) at a radius between , a region encompassing .
- Dissertation submitted to the Department of Astronomy and Astrophysics, University of Chicago, in partial fulfillment of the requirements for the Ph.D. degree.
- Position Offsets in arcseconds from , (2000).
- Position Offsets in arcseconds from , (2000).
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