Magnetic Field of Molecular Gas Measured with the Velocity Gradient Technique I. Orion A

Magnetic fields play an important role in the evolution of molecular clouds and star formation. Using the Velocity Gradient Technique (VGT) model, we measured the magnetic field in Orion A using the 12CO, 13CO, and C18O (1-0) emission lines at a scale of 0.07 pc. The measured B-field shows an east-west orientation that is perpendicular to the integral shaped filament of Orion A at large scale. The VGT magnetic fields obtained from 13CO and C18O are in agreement with the B-field that is measured from the Planck 353 GHz dust polarization at a scale of 0.55 pc. Removal of density effects by using a Velocity Decomposition Algorithm can significantly improve the accuracy of the VGT in tracing magnetic fields with the 12CO (1-0) line. The magnetic field strength of seven sub-clouds, OMC-1, OMC-2, OMC-3, OMC-4, OMC-5, L 1641-N, and NGC 1999 has also been estimated with the Davis-Chandrasekhar-Fermi (DCF) and MM2 technique, and these are found to be in agreement with previous results obtained from dust polarization at far-infrared and sub-millimeter wavelengths. At smaller scales, the VGT proves a good method to measure magnetic fields.


INTRODUCTION
Magnetic fields play a key role in regulating the formation of molecular clouds and their evolution (Larson 1981;Seifried & Walch 2015;McKee & Tan 2003;McKee & Ostriker 2007). However, their role in the star-formation process is not entirely understood (Li et al. 2015;Crutcher 2012). In addition, turbulence effects are considered to be another key factor affecting the dynamics of star formation processes in molecular clouds. Together with gas self-gravity, these processes appear at all physical scales and at different evolutionary stages (Parker 1965(Parker , 1979Jokipii 1966;Li & Henning 2011;Hull et al. 2013;Caprioli & Spitkovsky the plasma in the direction parallel to the magnetic field and produces a most significant acceleration, the velocity gradients in this location are dominated by the gravitational acceleration being parallel to the magnetic fields. This phenomenon of domination by gravitational acceleration has indeed been detected in the Serpens molecular clouds (Hu et al. 2019, G34.43+00.24 (Tang et al. 2019), and NGC 1333 . This reaction to self-gravity enables the VGT to reveal self-gravity-dominated regions, as well as quiescent areas supported by turbulence, thermal pressure, and magnetic fields. The velocity information of the molecular gas is obtained from Doppler-shifted spectral lines. Because of the effect of a velocity caustic (Lazarian & Pogosyan 2000), the observed intensity distribution in a given velocity channel is defined by both the emitter density and the velocity distributions. To separate the density and velocity contributions, Yuen et al. (2021) proposed a new method, i.e., the Velocity Decomposition Algorithm (VDA). The accuracy of the VGT is expected to be improved by eliminating the dependence on density.
In this work, we target the Orion A molecular cloud, which is the nearest high-mass star-forming region located at a distance of around 400 pc (Menten et al. 2007;Kounkel et al. 2017). The complex structure of the Orion A filament can be seen from H 2 column density ditribution in Orion A (see Figure 1). Abundant polarization observations at different wavelengths have been performed for this source, including near-infrared (NIR) polarimetry (Poidevin et al. 2011), far-infrared polarimetry (Schleuning 1998;Chuss et al. 2019;Harper et al. 2018), and sub-millimeter polarimetry (Schleuning 1998;Tang et al. 2010;Ward-Thompson et al. 2017;Pattle et al. 2017Pattle et al. , 2021Planck Collaboration et al. 2020a,b). Recently, the CARMA-NRO Orion Survey (Kong et al. 2018) provided high-resolution 12 CO, 13 CO, and C 18 O (1-0) spectral line data (beam size ∼ 6 -10 ), which is from CARMA observations combined with single-dish data from the Nobeyama telescope. In the following, these high-resolution CO data may make it possible to generate both a multi-scale (from 10 to 0.1 pc) and multi-wavelength view of the magnetic field as it interacts with the multi-phase gas in Orion A.
In this work, we aim to measure the magnetic field structure of Orion A with VGT method using 12 CO, 13 CO, and C 18 O (1-0) spectral line data. The paper is organized as follows. In Sect. 2, we provide the details of the observational data used in this work. In Sect. 3, we describe the details of the VGT method, and the VDA algorithms. In Sect. 4, the magnetic field measured with VGT and VGT-VDA method have been described.  Figure 1. Distribution of H2 column density derived from Herschel continuum data in Orion A (Poglitsch et al. 2010;Griffin et al. 2010;Roy et al. 2013;Polychroni et al. 2013). The white contour level represents 3 × 10 21 cm −2 . The orange contours show the region covered by the CARMA-NRO Orion Survey (Kong et al. 2018).
Sect. 5 will show the realm of applicability of magnetic field measurements with VGT and some physical parameters. A summary has been provided in Sect. 6.
2. ARCHIVAL DATA 2.1. The CO emission data Carbon monoxide is a ideal tracer of the kinematic characteristics of molecular clouds. There are three resolution spectral lines 12 CO (1-0), 13 CO (1-0) and C 18 O (1-0) from CARMA-NRO Orion Survey (Kong et al. 2018), where CARMA observations were combined with singledish data from the Nobeyama 45 m telescope to provide extended images at about 0.01 pc resolution. The final maps have an angular resolution of about 8 (from 6 to 10 ) and a pixel size of 2 . The velocity resolution is 0.25 km s −1 for 12 CO and 0.22 km s −1 for 13 CO and C 18 O.

The Polarization data
The Planck satellite 1 provides the 353 GHz thermal dust polarized emission (Planck Collaboration et al. 2020a,c) tracing the large scale magnetic field in Orion A. Based on the observations of the High-Frequency Instrument (HFI) at 353 GHz, the Stokes parameters I, Q, U and their dispersion value (σI, σQ, σU) maps have been obtained. The resolution of these maps is 5 and the pixel size is ∼ 1.71 . The polarization angle from the HFI Stokes maps may be calculated as: where ψ Planck varies from -90 • to 90 • with the HEALPix convention. One has to use ψ=0.5 × arctan(-U, Q) to convert the Planck measurement to this IAU convention. The magnetic field (hereafter, the B-field) orientation can be obtained by adding 90 • to the polarization angle: Furthermore, the B-field orientation in equatorial coordinates (FK5, J2000) is obtained using the following angle relation (Corradi et al. 1998): where ψ r is the angle relation for spherical triangles between Equatorial and Galactic coordinate systems. l and b show the pixel position information of the Galactic coordinates. The magnetic field orientation is then transformed from Galactic (θ GPA ) to Equatorial (θ EPA ) coordinates by: 3. METHOD

The Velocity Gradient Technique
The Velocity Gradient Technique (VGT; González-Casanova & Lazarian 2017;Hu et al. 2018) is the main analysis tool used in this work and has been developed on the basis of the anisotropy of magneto-hydrodynamic turbulence (Goldreich & Sridhar 1995) and fast turbulent reconnection theories (Lazarian & Vishniac 1999). We use the Velocity Channel Gradients (VChGs) as the main model of VGT (after here short as VGT). For extracting the velocity information from Position-Position-Velocity PPV cubes, thin velocity channels Ch(x,y) were employed. The gradient map ψ s g is then calculated by: where x Ch i (x,y) and y Ch i (x,y) are the x and y components of the gradient, respectively. This is done for all pixels with spectral line emission having a signal-tonoise ratio greater than 3. The orientation of the magnetic field is found to be perpendicular to the velocity gradient, as long as these gradients are statistically significant. A sub-block averaging method (Yuen & Lazarian 2017a) has been used to export the velocity gradients from the raw gradients within a sub-block of interest and then to plot the corresponding histogram. Gradients for each channel are then calculated by adaptive sub-block averaging, which results in eigen-gradient maps ψ i gs (x, y) with i = 1,2,...,n v . Pseudo-Stokes-parameters Q g and U g of the gradient-inferred magnetic field may then be constructed by: where ψ g is the pseudo polarization angle. This pseudo polarization angle is perpendicular to the POS orientation angle of the magnetic field: ψ B = ψ g + π/2 . For the dense region, the turbulent flow will be modified thoroughly by self-gravity and the Velocity gradient orientation will change from perpendicular to parallel to magnetic field (Yuen & Lazarian 2017b;Hu et al. 2020). Therefore the velocity gradient orientation angle in gravity-dominated region will be re-rotate 90 degrees. It is calculated by: where ψ S B is VGT orientation angle in the case of selfgravity and ψ B is the pseudo magnetic field angle measured by VGT in the case of turbulence dominated regions. The orientation of the turbulence velocity gradient will be parallel to the magnetic field orientation.

Velocity Decomposition Algorithm
The Velocity decomposition algorithm (VDA, Yuen et al. 2021) is a new method to separate velocity and density fluctuations from a PPV cube using its statistical properties (Lazarian & Pogosyan 2000). The sonic Mach number in star formation regions is usually greater than unity, and the supersonic version of the VDA algorithm makes it possible to only obtain the velocity flux structure from a PPV cube using the thin channel formulation of the VGT method. The VDA allows a separation of the pure velocity caustics from the PPV cube. In theory, the VGT orientation applied the VDA method (after here, VGT-VDA) would be closer to the plasma motion direction and better trace the local magnetic field.
The supersonic VDA algorithm to get the pure velocity caustics structures (Lazarian & Pogosyan 2000) for the each channel of PPV cube is based on the following expression: where Ch(X, v, ∆v) is the channel of the PPV cube, X means the position, v is the local velocity, and ∆v is the velocity channel width. c s can be calculated by assuming a uniform temperature ∼ 10 K, which results in a value c s ∼ 186 m s −1 . When using the pseudo PPV cube V(X, v, ∆v) with only the velocity contribution, the raw gradients ψ i g (see Eq. 6) may be re-applied with VGT to improve the accuracy to trace the magnetic field.
The use of this technique requires high-SNR spectral line data (Yuen et al. 2021). This technique has not yet been applied on self-gravity regions, and we will use the CO data in Orion A to see if applying VDA improves the accuracy of VGT to trace magnetic field.

Magnetic Fields measured with VGT
The VGT treats the regions dominated by turbulence or self-gravity differently and it is important to find out which region in Orion A favor the one or the other scenario. The column density probability function (N-DPFs) (Ballesteros-Paredes et al. 2011;Burkhart 2018;Körtgen et al. 2019) provides a simple way to make this distinction. The N-DPFs follow a log-normal (LN) distribution in the case of turbulence dominated regions, but they will follow a power-law (PL) distribution in self-gravity-dominated regions (Robertson & Kravtsov 2008;Burkhart 2018;Körtgen et al. 2019). The transition point of this LN-PL model is a key for distinguishing the VGT model that is dominated by turbulence or self-gravity. Recent studies shows that the critical column density at this transition point is ∼ 3× 10 21 cm −2 (Spilker et al. 2021), i.e. if the column density of one region is greater than this density, self-gravity could be dominating. This dense region in Orion A includes the large integral shaped filament (hereafter, ISF), dense clumps, L1641-N and NGC1999, and the gas around them which has high column density N(H 2 ) (> 3 × 10 21 cm −2 ). Assume that Orion A is a long cylinder, the volume density of N-DPFs' transition point (column density ∼ 3 × 10 21 cm −2 ; Spilker et al. 2021) is around × 10 3 cm −3 by estimating the effective radius of cloud (∼ 0.61 pc, the details see Appendix.A). This result is in agreement with that from , in which the VGT method reveals that the self-gravity is occurring at volume density n 0 ≥ 10 3 cm −3 . However, self-gravity could be more localized to the clumps and cores while the envelope may be more diffuse. There is a possibility of a global gravitational contraction (Larson 1981;Ballesteros-Paredes et al. 2011;Kauffmann et al. 2013;Vázquez-Semadeni et al. 2019) that could cause an inflow (Hu et al. 2020) and change the direction of the velocity gradient. This region of this work in Orion A is dominated by self-gravity where the gravitationally collapse could occur at the core's scale and gravitationally contraction could occur at the cloud's scale. The distribution of the H 2 column density in Fig. 1 suggests that the whole region covered by the CO lines (Kong et al. 2018) in Orion A has high column density (> 3 × 10 21 cm −2 ) and could dominated by self-gravity. One thing to note is that the power-law DPF model remains a statistical concept. When plotting N-DPFs of the overall region, it is possible that some small sub-regions are dominated by turbulence and are overwhelmed.
Several methods have been applied to measure the magnetic field morphology in Orion A. The magnetic field measured with VGT is called pseudo magnetic field after here. Six pseudo magnetic field line integral convolution (Cabral & Leedom 1993, LIC) maps are shown in Fig. 2. The top panels in this Fig. 2 show the pseudo magnetic field orientation LIC maps measured by the VGT method using the 12 CO, 13 CO and C 18 O (1-0) spectral lines. The velocity range of the three CO emissions has been set to [1, 15] km s −1 . Sub-block averaging was used to determine the pseudo beam of the VGT results (see Sect. 3.1) using a Sub-block size set as 20×20 pixels. The resolution of the pseudo B-field would be accessing 40 (∼ 0.07 pc). where the region is dominated by self-gravity, the VGT angle would be re-rotated by 90 • again (see Eq 10).
The VGT explores the anisotropy (the elongation) of the turbulent eddies resulting from magnetohydrodynamic (MHD) turbulence in the presence of magnetic fields under sub-Alfvénic and supersonic conditions (Lazarian 2006). However, under supersonic conditions, the presence of shocks and also of self-gravity, the MHD turbulent anisotropies will be affected by the local density (Yuen et al. 2021). This effect is most serious for higher density regions, where the density contribution to the thin velocity channels could degrade the accuracy of VGT. Under these conditions, the "velocity decomposition algorithm" (VDA) may be used to separate the velocity and density contributions from the PPV (position-position-velocity) cube (Yuen et al. 2021). The density contribution may then be removed from the PPV cube and the VGT-VDA methods may more accurately re-measure the turbulence related velocity fluctuations. The physical properties from the MHD turbulent theory (Goldreich & Sridhar 1995) and the statistics from PPV cubes (Lazarian & Pogosyan 2000) have been predicted well by velocity caustics (Lazarian & Pogosyan 2000). These velocity caustics are important for the VGT method, in particular for the supersonic case where there is density contamination. Therefore, the removal of density effects by the VDA method will enhance velocity caustics and improve the accuracy of the VGT method.
Considering that Orion A is a dense region, its magnetic fields should be studied using the combined VGT-VDA method. The result of this study using the same three molecular tracers ( 12 CO, 13 CO and C 18 O (J = 1-0)) has been shown in Fig. 2 in the bottom panels. The general orientation of the pseudo magnetic field measured by VGT-VDA is similar for tracers and is perpendicular to the ISF and reveals more details in the B-field LIC maps.

Magnetic Field Morphology
The large Integral Shaped Filament (ISF, Bally et al. 1987) is one of the most impressive structural features in Orion A, which includes regions OMC-1, 2, 3, 4, 5 (see Fig. 2). The pseudo magnetic field orientations from two measuring methods (VGT, and VGT-VDA) are generally similar. The pseudo magnetic field directions are almost perpendicular to the long axis of ISF and may vary in individual sub-regions. For instance, the pseudo magnetic field in the vicinity of OMC-1 is more scattered, while in the dense region L 1641N the magnetic field directions point towards the center of its density clumps. The Orion A generally can be separated into four subregions as shown in Fig. 2, i.e., the Orion Nebula Cluster components ONC-North, ONC-Central ONC-South, and L 1641N. In the following, the pseudo magnetic field morphology measured in those sub-regions with VGT and VGT-VDA is displayed in Fig. 3 and is described below in detail. ONC-North -This region includes sub-regions OMC-2 and OMC-3. The pseudo magnetic field orientation in this north-south ISF region is perpendicular to this filament. The pseudo B-field from 13 CO VGT is similar to that from C 18 O VGT. Comparing with pseudo B-field orientations derived from 13 CO and C 18 O, the result obtained from 12 CO has an offset. 12 CO traces diffuse gas and 13 CO and C 18 O probe dense gas, which could lead slightly different pseudo B-field derived from 12 CO, 13 CO, and C 18 O lines. Its pseudo B-field orientation tends to be close to a northeast-southwest direction. ONC-Central -This is the main sub-region OMC-1. The pseudo magnetic field orientation by VGT for 13 CO and C 18 O is perpendicular to the ISF filament. However, at the Orion-Bar the orientation of the B-field for 12 CO, 13 CO, and C 18 O is parallel to the structure. At east of OMC-1, the pseudo magnetic field shows a disturbance at a cavity structure in the nearby Pillars region (Kong et al. 2018). ONC-South -At ISF filament sub-regions OMC-4 and OMC-5, the pseudo magnetic field orientation is perpendicular to the filament direction. The pseudo B-field orientations from 13 CO and C 18 O are nearly the same, while the morphology from 12 CO shows a relatively complex structure. At other diffuse regions, the general orientation of the B-field is east-west direction and again there are some disturbances around the Pillars region (Kong et al. 2018). L1641 -At the dense region L 1641N, the pseudo Bfield orientations distribute along the east-west direction. This region shows two sub-filaments in the form of an inverted 'V' and the pseudo magnetic field orientations from the three CO tracers are perpendicular to these integrated intensity distributions.  Figure 2 shows that the pseudo magnetic field morphology measured with VGT from 13 CO is similar to that from C 18 O. The pseudo magnetic field directions are mainly east-west and are perpendicular to the shape of the ISF. In some diffuse regions, the distorted magnetic field follows the density structure. East of ONC-Central, the pseudo B-field orientations derived with VGT from 13 CO follow the shape of the gas range. This is not evident for VGT with C 18 O.
The pseudo magnetic field morphology measured from VGT for 12 CO is similar to that for 13 CO and C 18 O in dense regions and different in diffuse regions. At the ISF, the pseudo B-field orientations from the VGT for 12 CO are perpendicular to the ISF shape and similar to that from the VGT for 13 CO and C 18 O. At the east side of ONC-Central, the pseudo B-field orientations derived  from the VGT for 12 CO show a more obvious distribution along the dense structure than that from VGT for 13 CO. At the L1641 region, the pseudo B-field directions from the VGT for 12 CO are close to the northwestsoutheast direction rather than the southwest-northeast direction found from the VGT for 13 CO.
The 12 CO, 13 CO, and C 18 O have different optical depths as the regions traced by 12 CO are more diffuse and closer to the surface of the molecular structures than those traced by 13 CO, and C 18 O. Therefore, also the VGT for 12 CO is a good tracer for the magnetic field in more diffuse regions and the VGT for 13 CO and C 18 O would trace the magnetic field in dense regions.

Comparison of B-field Derived from VGT and Dust Polarization
In order to compare the magnetic field orientation obtained with different measuring methods, the offset angle θ r between the dust B-field polarization φ B and the pseudo magnetic field orientation of VGT ψ S B is defined as: The offset angle θ r is then quantified by the Alignment Measure (AM, González-Casanova & Lazarian 2017) defined as: AM = 2( cos 2 θ r − 1 2 ) .
The range of AM values would be from -1 to 1, where AM values close to 1 means that φ B is parallel toψ S B and an AM value close to -1 indicates that φ B is perpendicular to ψ S B . The uncertainty in the AM value, σ AM , may be given by a standard deviation divided by the square root of the sample size.
A detailed comparison between the magnetic fields from the Planck dust emission data with those from the VGT method may be achieved when setting a subblock size for the VGT method results of 150×150 pixels, which makes VGT pseudo beam of 5 the same as Planck 353 GHz dust polarization (see Fig. 3). The VGT pixel size of the pseudo stokes maps will be re-gridded to 1.71 and be equal to the pixel size of the Planck polarization. When considering the signal-to-noise ratio (SNR) greater than 3 times sigma for the B-field vectors for both the dust polarization and the spectral line data, the mean AM values are found to be about 0.34±0.01 for 12 CO, 0.64±0.01 for 13 CO and 0.70±0.01 for C 18 O. This means that 13 CO and C 18 O trace magnetic field well when using VGT. 12 CO, 13 CO, and C 18 O emission originate from different layers of cloud which has different critical density, ∼ 10 2 , 10 3 , and 10 4 cm −3 (Evans 1999;Shirley 2015), respectively. Dust continuum at sub-mm wavelength traces dense region. 13 CO, and C 18 O trace the dense gas whose origin of them is similar to that of dust emission at sub-mm wavelength. Consequently, the high AM values observed for dense tracers, 13 CO and C 18 O, are to be expected since dense molecular tracers probe the dense molecular gas (∼ 10 4 cm −3 ). 12 CO traces more diffuse gas so that the velocity gradients are less aligned with the Planck polarization.
In addition, the performance of the VDA methods for improving the VGT results may be tested. Using the same method (see Eq. 13) to compare the Planck and VGT-VDA B-field directions, the mean AM values from the three VGT-VDA rsults for 12 CO, 13 CO and C 18 O are 0.54±0.01, 0.66±0.01, and 0.71±0.01. The mean AM value from the VGT-VDA method for 12 CO (AM =0.34) has been greatly improved compared with the VGT-only method (AM = 0.54). All mean AM values from the three VGT-VDA results are above 0.5. It means that there is a smaller difference between the dust polarization results and the B-fields measured with VGT-VDA using the CO emissions. However, this improvement is insignificant for the 13 CO and C 18 O data (AM values improved by 0.01 ∼ 0.02), which may be explained by VDA only being effective in region with prevalent MHD turbulence (see Yuen et al. 2021). In the presence of dense gas, as for 13 CO and C 18 O, the VDA method does not seem to work well to improve the accuracy of the VGT method. Another possibility is that VDA relies on the high signal-to-noise ratio of the spectra (Yuen et al. 2021), while 13 CO and C 18 O data have a lower signal-to-noise ratio than 12 CO. It should be noted that the B-field measured by VGT for 12 CO in OMC-1 has distinct differences from the B-field measured by dust polarization, which we will discussed in detail in Zhao et al.(in prep).
Compared with VGT results from the different tracers, the AM (VGT-VDA) from 13 CO and C 18 O is above 0.65. The magnetic fields derived from the VGT-VDA method for 13 CO and C 18 O are more similar to the Bfields derived from the dust polarization emission than that from the 12 CO emission. The critical density for the optically thick tracer 12 CO (1-0) emission is around 10 2 cm −3 as it traces the diffuse regions of the cloud. The Planck 353 GHz dust polarized emission generally combined contributions from both the diffuse and the dense regions. The optical depth for 13 CO and C 18 O is generally lower than for 12 CO. The pseudo magnetic field measured with VGT therefore samples regions with different critial densities corresponding molecular tracer. The Planck 353 GHz dust polarization trace the magnetic field from optically thin and dense region.  sults would thus also be close to the local magnetic field inferred from the Planck 353 GHz dust polarization.

The VDA effect on VGT
To see whether the VDA method does improve the results of the VGT method in tracing the magnetic fields, the Planck polarization has been set as reference and the relative angle has been set as the absolute value of offset angle θ r (see Eq .12). Figure 4 shows three histograms of the relative alignment between the Planck polarization and the VGT results from 12 CO, 13 CO, and C 18 O.
The left panel of Fig. 4 shows that relative angles from the VGT-VDA method for 12 CO are closer to zero degrees than those for the VGT method. The columns for the VGT-VDA method between 60 • and 90 • show a significant decrease compared with those of the VGT method, which indicates that the VDA algorithm significantly improves the accuracy of VGT tracing by downshifting the B-field angles for 12 CO. The middle panel of Fig. 4 for 13 CO shows that between 60 • and 90 • the relative angle distribution is suppressed for both VGT and VGT-VDA methods. The VGT-VDA method results in a significant improvement from 0 • to 30 • over those for the VGT method. The mean AM value for the 13 CO VGT-VDA method (0.67) has not improved compared with the VGT method (0.66; see Sect.4). The right panel of Fig. 4 right panel shows that most of the relative angle for both VGT and VGT-VDA are in the 0 • ∼ 30 • range and that there is a small difference between the two methods. The middle and right panels suggest that the VGT-VDA method gives a small improvement over the VGT-only method for tracing the magnetic field when using the density tracers 13 CO and C 18 O.
The alignment of the pseudo magnetic field from the VGT method depends on the local column density. The column density distribution in Orion A has been derived using SED fitting (Roy et al. 2013;Polychroni et al. 2013;see Fig. 1). Fig. 5 shows that there is a correlation between column density and the AM value from both the VGT and VGT-VDA methods. The left panel of this Fig. 5 shows that the AM values from the 12 CO VGT-VDA procedure is always higher than the VGT values at different column densities. Removing the density contribution with VDA gives a more consistent relationship with the dust emission. The other two panels in Fig. 5 show that using the VDA method gives a small improvement of AM values for 13 CO and essentially no change for C 18 O. The VDA effect is less at dense regions. All AM values from 13 CO and C 18 O are above 0.5 for all column densities, which makes them good tracers of the magnetic field by using the VGT method.
In regions with different column densities, the AM values from the three CO emissions show a consistent trend: the AM values decline with increasing column density. Orion A is an active star-forming region and has many complex and interesting structures such as filaments, bipolar outflows, shells, bubbles, and photoeroded pillars (Kong et al. 2018;Tang et al. 2018;Li et al. 2020). These regions are not simply dominated by either turbulence or self-gravity. More diffuse regions has weak star-forming activity in comparison with the denser regions and the motion traced by turbulence and velocity caustics would be more aligned with the local magnetic field. This makes using VGT more conducive for tracing magnetic fields.
For all three CO lines, the AM values in Fig. 5 show a downward trend for a column density in the range 3×10 21 to 5×10 21 cm −2 . When the column density is above 5×10 21 cm −2 , the AM values do not drop drastically and remain near a stable value. The stable AM values from VGT-VDA for 12 CO, 13 CO and C 18 O are around 0.5, 0.7 and 0.7. In those dense regions the  Note-The location of these molecular clouds has been shown in Fig. 2). Details of parameters used for the calculated function have been shown in Table 2. The parameter T from cs is the kinetic temperature T k for OMC-1, 2, 3, 4, 5 (Friesen et al. 2017) and the dust temperature T d for other clouds. There are two models: the cylindrical model applied for OMC-1, 2, 3, 4, 5 and the spherical model applied for L 1641-N and NGC 1999. Specific calculation formulas have been shown in Appendix A. The magnetic field strength (DCF) is calculated from the VGT dispersion and the DCF method (see § 5.4.1).
353 GHz dust emission would be mostly optically thin and trace the inner projected magnetic field. 13 CO and C 18 O are also optically thin molecular tracers of the same inner structure in the molecular cloud, which would make their VGT results similar to the 353 GHz dust polarization (at 850µm). In relative terms, 12 CO is an optically thick tracer and traces the surface structure of clouds and is greatly affected by the density contribution in its velocity channel. The VDA removal of the density influence greatly improves the VGT accu-racy and results in magnetic fields similar to those of the inner clouds.

VGT dispersion with DCF method
The dust polarization results are more consistent with the VGT results from 13 CO and C 18 O than with those from 12 CO. Even in high-density areas, the AM values for VGT from 13 CO and C 18 O are above 0.5. Since the area covered by the 13 CO emission is much larger than that of C 18 O, 13 CO would be a better tracer of the magnetic field in Orion A using the VGT method. The VGT-VDA method from 13 CO may then be used to calculate the magnetic field correlation parameter in Orion A. The resolution of the magnetic field from 13 CO is close to 40 (FWHM ≈ 0.07 pc).
Earlier the Davis-Chandrasekhar-Fermi (DCF;Davis 1951;Chandrasekhar & Fermi 1953) method has been used in Orion A to estimate the magnetic field strength defined as (Crutcher et al. 2004;Pattle et al. 2017): where ∆v 1D is the line width (FWHM) of the molecular line in km s −1 , σ θ is the angular dispersion from dust polarization or VGT methods in degrees, and n(H 2 ) is the Hydrogen volume density in cm −3 . The parameters σ θ and B pos are the mean angular dispersion in Orion A and the mean plane-of-the-sky magnetic field strength at this region. However, the DCF method originally did not consider self-gravity and sub-regions of Orion A are the dense region, include OMC-1, OMC-2, OMC-3, OMC-4, OMC-5, L 1641-N, andNGC 1999, where is gravitationally collapse (Hacar et al. 2017). The magnetic field strength estimated with the VGT and DCF methods should be compared and the errors evaluated. OMC-1, OMC-2, OMC-3 OMC-4, and OMC-5 are located along the large Integral Shaped Filament (ISF). The shape of these filaments is roughly like a long cylinder. Two other components, L 1641-N and NGC 1999, are dense clumps in Orion A and have a morphological structure that is simply spherical. Using these two simple geometric models (cylinder and spherical, see Appendix.A), the physical parameters in these clouds have been calculated and are presented in Table 1. The functions to calculate these parameters has been shown in Table 2. Because of the 1D velocity dispersion σ v,1D that includes a turbulence velocity and a shear velocity, the magnetic field strength B pos could be overestimated.

Two Mach Numbers Analysis
A basic assumption of the classical DCF techniques is that the observed fluctuations in the molecular medium result from Alfvén waves (see § 5.4.1; Davis 1951;Chandrasekhar & Fermi 1953). Self-gravity in the region causes additional fluctuations in the magnetic field and the turbulence, which would make the classical DCF method incomplete. A new technique called Two Mach Numbers method (MM2), which B pos calculated by Alfvén and sonic Mach number, provides an alternative for measuring the magnetic field strength when external shear and self-gravity distorts the magnetic fields. . The relation between the magnetic field strength in the plane of the sky(Bpos) and the H2 column density (N(H2)). The black line is a critical condition for the mass to flux ratio(λ = 1). The green dashed line is an empirical relationship between the magnetic field strength and the column density (Liu et al. 2021). The magnetic field strength of the molecular clouds is distributed along this line. The red star is the OMC-1 magnetic field strength observed by JCMT . The Red square is Bpos observed by HAWC+ (Chuss et al. 2019 The associations of the MM2 method were analytically justified in Lazarian et al. (2020). The velocity gradients and also the magnetic fields are a function of the Alfvén Mach number M A within a sub-block ). The dispersion relation in the direction of the velocity gradient will show a powerlaw alignment with the M A , which is a so-called "topto-bottom" ratio of the distribution of the fine channel velocity gradients (VChGs). The Alfven Mach number M A may be calculated as: where T v denotes the maximum value of the fitted histogram of the velocity gradient orientation, while B v is the minimum value. And with the knowledge of two Mach number M S amd M A , this new technique, MM2 , may be used to calculate the POS magnetic field strength as: where the Ω is a geometrical factor (Ω = 1), c s is the sound speed, ρ 0 is the volume mass density in units of g cm −3 , and M S and M A are the sonic Mach number and the Alfvén Mach number. The B pos derived by 13 CO MM2 is similar to the B pos obtained from the VGT dispersion with DCF method in most sub-clouds (see Table 1), except for the case for NGC 1999. The multivelocity components of the 13 CO spectral line emission could further increase the velocity dispersion and also B pos−DCF . Figure 6 displays the magnetic field strength (measured by the MM2 mothed) distribution with H 2 column density. The different colors indicate the results for different molecular clouds. The circles are magnetic field strengths estimated by the 13 CO VGT method. The magnetic field strength at OMC-1 has been estimated from the dust polarization Chuss et al. 2019). The B-field strength B pos are 6.6 mG (submm;JCMT) and 0.9∼1 mG (Far-IR; HAWC+). The OMC-1 B pos measured by the MM2 method for 13 CO is around 1 mG, which is similar to the dust polarized values. In OMC-3, B pos measured by MM2 is 148 µG. The B pos values derived by HAWC+(154 µm and 214 µm) are 158.6 and 205.4 µG. The B pos value from MM2 and dust polarization is similar to B pos from the dust polarization. The B pos value from the VGT dispersion and the DCF is larger than obtained from MM2. In the self-gravity region, the distorted magnetic field and the turbulence cause the estimated magnetic field strength with DCF to become larger. The MM2 method may provide another possible method for measuring the magnetic field strength in self-gravity regions.

SUMMARY
The objective of this work is to measure the magnetic field in Orion A with the VGT and VGT-VDA methods by multiple CO spectra emission. For this purpose, the VGT and VGT-VDA methods have been applied to determine the magnetic field structures in the filament structure of Orion A using the 12 CO, 13 CO, and C 18 O (1-0) emission profiles with a spatial resolution of ∼ 0.07 pc. It has been found that the VGT-VDA method has a great accuracy in tracing the magnetic field at small scales and shows strong agreement with the larger scale field structures determined from the Planck dust polarization data. In addition, the MM2 method would be an alternative to to estimate the POS magnetic field strength in regions where self-gravity plays a role. Further results are the following: 1. The magnetic field morphology measured with the VGT method demonstrates east-west structural features in Orion A. The magnetic field orientation is mainly perpendicular to the direction of the Integral Shaped Filament.
2. On the whole, the B-field morphology measured with VGT for 13 CO is similar to that of C 18 O. In dense regions, the orientations of the magnetic field derived by VGT for 12 CO are comparable to those of 13 CO and C 18 O. In some relatively diffuse areas, the magnetic field orientations derived for 12 CO are different from those of 13 CO and C 18 O.
3. In dense regions, the B-fields measured using VGT method for 13 CO, C 18 O and the VGT-VDA method for 12 CO, 13 CO, C 18 O are in agreement with those derived from the Planck 353 GHz dust polarization at the same scale (∼ 0.55 pc). The AM values for these are 0.66, 0.70, 0.54, 0.66, and 0.71, respectively, and they are all over 0.5. This would indicate that the magnetic field measured with VGT is similar to that of dust polarization.
4. The VDA method can improve the accuracy of the VGT to trace magnetic fields by separating velocity and density contribution, specially for 12 CO (eg, AM values from VGT and VGT-VDA method for 12 CO are 0.33 and 0.54). In dense regions with N(H 2 ) > 3 × 10 22 cm −2 , the VDA method does not significantly improve the accuracy of VGT for 13 CO and C 18 O. Additional corrections may be needed for the VDA method for tracers of dense regions. The improved method, VGT-VDA, can provide a higher accuracy to trace magnetic field. 5. A new technique, MM2, has been applied for the 13 CO data in Orion A to measure the magnetic field strength. The POS B pos values for OMC-1, OMC-2, OMC-3, OMC-4, OMC-5, L 1641-N and NGC 1999are 1061, and 161 µG, repectively, which is consistent with previous results obtained from dust polarization at far-infrared and submillimeter wavelengths.
Plans are to continue the comparison of methods for measuring the magnetic field in more sources to test the VGT method in complex star formation regions. There are two models to estimate the physical parameter of the clouds in Orion A. A filament is assumed to be a long uniform cylinder and the clump is a uniform sphere (Fiege & Pudritz 2000). These computational formulas have been shown in table.2.