Three-dimensional magnetic fields of molecular clouds

To investigate the role of magnetic fields in the evolution of the interstellar medium, formation and evolution of molecular clouds, and ultimately the formation of stars, their three-dimensional (3D) magnetic fields must be probed. Observing only one component of magnetic fields (along the line of sight or parallel to the plane of the sky) is insufficient to identify these 3D vectors. In recent years, novel techniques for probing each of these two components and integrating them with additional data (from observations or models), such as Galactic magnetic fields or magnetic field inclination angles, have been developed, in order to infer 3D magnetic fields. We review and discuss these advancements, their applications, and their future direction.

Magnetic field orientation relative to density structures may indicate their role in the ISM evolution (e.g., Soler and Hennebelle, 2017). Observations of plane-of-sky magnetic fields (B POS ; e.g., Planck

3D MAGNETIC FIELDS
Several methods (e.g., Chen et al., 2019;Hu et al., 2021b,a;Tahani et al., 2019Tahani et al., , 2022a have examined the 3D magnetic fields of molecular clouds. Chen et al. (2019) and  use B POS (dust polarization) observations and their polarization fraction (p) to recover the mean inclination of the ordered 2 magnetic fields of molecular clouds, whereas Tahani et al. (2022a,b, scales of a few to ∼ 100 pc) incorporate B LOS and B POS observations along with Galactic magnetic field (GMF) models. 1 A number of recent studies have examined the 3D magnetic fields of the diffuse ISM (e.g., Ferrière, 2016;Van Eck et al., 2017;Alves et al., 2018;Panopoulou et al., 2019;Clark and Hensley, 2019;Hensley et al., 2019). 2 ordered: ignoring the random component due to turbulence or smaller-scale variations

3D fields
Numerous observatories, including the James Clark Maxwell Telescope (JCMT; e.g., Eswaraiah et al., 2021;Ngoc et al., 2021;Hwang et al., 2021;Kwon et al., 2022), Planck Space Observatory (e.g., Planck Collaboration et al., 2016Alina et al., 2019), Atacama Large Millimeter/sub-millimeter Array (ALMA; e.g., Pattle et al., 2021;Cortés et al., 2021), Sub-Millimeter Array (SMA; e.g., Zhang et al., 2014), and Stratospheric Observatory for Infrared Astronomy (SOFIA; e.g., Chuss et al., 2019) have observed B POS of numerous star-forming regions. However, the number of B LOS observations of molecular clouds are still limited. Although Zeeman splitting is a powerful technique for probing B LOS and the most accurate method for determining field strengths, it requires lengthy observing runs, making it challenging to observe. The observing technique of Tahani et al. (2018) can be used to map B LOS of numerous molecular clouds. Tahani et al. (2018) developed a new technique to probe B LOS associated with molecular clouds, using Faraday rotation. We provide a brief summary of the technique in this section.

Faraday rotation
Due to the lower abundance of electrons in molecular clouds (compared to ionized regions), it was previously believed that Faraday rotation 3 could not be used to investigate the magnetic fields of molecular clouds. Tahani et al. (2018) developed a technique to successfully determine B LOS of molecular clouds using Faraday rotation measures (RM), while previous attempts (Reich et al., 2002;Wolleben and Reich, 2004) were unable to provide a map of B LOS observations across the cloud.

Methodology and results
In this technique (Tahani et al., 2018), the non-cloud (background and foreground; Galactic) contribution to the RM (RM ref ) is subtracted from the observed RM of extra-galactic point sources (radio galaxies or quasars) using an on-off approach. Numerous catalogs (e.g., Taylor et al., 2009) provide observed RM point sources. Following the determination of the the cloud's RMs, the electron column densities associated with each RM point are calculated using a chemical evolution code and extinction maps. Any chemical evolution code (e.g., one used by Gibson et al., 2009) and extinction map (e.g., Kainulainen et al., 2009), or even Hydrogen column density map (Lombardi et al., 2014;Zari et al., 2016), can be utilized. To find electron column densities, the cloud is divided into sub-layers aligned along the line of sight using extinction values and the chemical code. The electron column density in each sub-layer is obtained separately. Calculating the average B LOS along the line of sight is made possible by adding the electron column density contributions of these sub-layers. Tahani et al. (2018) mapped B LOS of the Orion A, Orion B, California, and Perseus molecular clouds and found that their results were consistent with existing molecular Zeeman measurements. They found that the B LOS direction of the Orion A (see left panel of Figure 1) and California clouds reverses from one side to the other (along the short axis of the cloud). Their Perseus results suggested a weak indication of this reversal. The B LOS reversal across Orion A was previously observed via Zeeman splitting (Heiles, 1997), in the same directions as Tahani et al. (2018).  (Lombardi et al., 2014). The circle and square markers represent B LOS , with the square indicating non-detection points (with high uncertainties that may cause a change in B LOS direction) and blue (red) representing pointing toward (away from) us. The drapery lines represent the B POS observed by the Planck Space Observatory. The red vector depicts the modeled Galactic Magnetic field projected onto the plane of the sky. The same B LOS reversal throughout the cloud was previously detected using Zeeman measurements (Heiles, 1997, see their Figure 15). We note that in Zeeman measurements, the negative sign indicates magnetic field directed toward us, while in RM studies, it indicates magnetic field directed away from us. Right panel: From our vantage point, the inferred 3D ordered magnetic field of Orion A is semi-convex. The red vector, bent gray cylinder, and blue vectors represent the modeled GMF, cloud, and 3D magnetic field of the cloud, respectively. Without identifying the inclination angle of the cloud, rotations of up to 50 • along the black arrow may be possible, resulting in both B(1) and B(2) (see Section 2 of Tahani et al., 2022a).
along the line of sight, c) parameters taken to estimate electron densities (cloud's initial temperature and density and Ultra-Violet and cosmic ionization rates), and d) extinction maps.

Reconstructing the mean 3D magnetic fields of molecular clouds
Using B LOS observations, Tahani et al. (2019Tahani et al. ( , 2022a studied the 3D magnetic field morphologies of the Orion A and Perseus molecular clouds. Tahani et al. (2019) constrained models of the ordered, cloud-scale magnetic field, using B POS angles and B LOS estimates, whereas Tahani et al. (2022a,b)

3D fields
cloud-scale magnetic field vectors in 3D 4 , given a set of model assumptions. We discuss these techniques in this section.

Analytical models of the ordered magnetic field within clouds and comparison to synthetic observations
Tahani et al. (2019) constructed models that could explain the observed B LOS reversal discussed in Section 2.1.2, obtained synthetic observations from the models, and compared these synthetic observations with B LOS (direction and strengths) and B POS (angle and strength; using Planck 5 ) estimates of Orion A. They concluded that an arc-shaped morphology (see right panel of Figure 1) is the most probable magnetic morphology for Orion A, based on Monte-Carlo analysis, chi-square probability values, and examination of a range of systematic biases between B LOS and B POS observations. In the arc-shaped morphology, field lines bend around the filamentary cloud in response to environmental interaction (first proposed by Heiles, 1997), enabling mass to flow along the field lines and accumulate on the cloud .

Using
Galactic magnetic field models to reconstruct the cloud-scale ordered magnetic field 3D vector Tahani et al. (2022a,b) reconstructed the cloud-scale, ordered magnetic field vectors of the Orion A and Perseus clouds in 3D. Using B LOS and B POS observations, along with large-scale GMF models (Jansson and Farrar, 2012a,b), they inferred the approximate orientation and direction 6 of the 3D ordered magnetic field of these clouds (including their B POS direction). Although the B POS orientation of numerous molecular clouds had been observed previously, their B POS direction remained undetermined even in the 3D study by Tahani et al. (2019).
Moreover, by estimating M A values and/or comparing estimates of initial magnetic field vectors (using GMF models) with B POS maps, Tahani et al. (2022a,b) suggest that the magnetic fields of the Orion A and Perseus clouds retain a memory of the Galactic magnetic fields. Although some studies (e.g., Stephens et al., 2011) have suggested that the magnetic fields of molecular clouds are dissociated from larger Galactic scales, others (e.g., Han and Zhang, 2007) have concluded that they largely retain the large-scale Galactic magnetic fields.
We note that this technique relies on correctly identifying the ordered GMF vector at the cloud location. This vector provides an approximation of the initial magnetic fields prior to the cloud's evolution (allowing us to ignore the GMF random component caused by cloud-scale turbulence). Since GMF models vary (Jaffe, 2019), this technique is applied to clouds in a region of the Galaxy (pointing anti-Galactic and nearby) where there is less disagreement between the GMF models. For example, all models in Figure 2 from Jaffe (2019), except panel h (Fauvet et al., 2011), generate similar ordered GMF vectors at the locations of the Orion A and Perseus clouds. Moreover, the limited number of B LOS observations per cloud and the use of two tracers (dust emission and a Faraday-based technique) may increase the technique's uncertainties. Upcoming observations are required to advance these studies (see Section 3). 4 approximate 3D morphology at scales of a few to 100 pc (ignoring turbulence and smaller-scale variations) 5 http://www.esa.int/Planck 6 In this mini-review we distinguish between the terms direction and orientation. Knowing the direction reveals orientation, but not the other way around. For example, the direction of B LOS indicates either away from us or toward us, whereas the orientation of B LOS indicates only that the line is parallel to the line of sight without specifying its direction. Similarly for B POS , direction refers to the complete 2D vector, while orientation refers only to the line without specifying the vector's endpoint.

Inclination angle: statistical studies of polarization fraction
The 3D morphologies identified by Tahani et al. (2022a,b) can be improved by inferring γ at various points across the cloud and combining their method with studies that estimate γ (e.g., Chen et al., 2019;Sullivan et al., 2021;Hu et al., 2021a. In recent years, γ has been inferred in molecular clouds (e.g., Sullivan et al., 2021) and diffuse ISM 7 (e.g., Hensley et al., 2019), using the dependence of p and polarization angle dispersion (S) on γ (e.g., Falceta-Gonçalves et al., 2008;Hensley et al., 2019), under the assumption of homogeneous grain alignment efficiency. King et al. (2018) compared the p and S values of the Vela C cloud with their 3D, ideal magnetohydrodynamics (MHD) colliding flow simulations. The simulations were performed using the ATHENA code (Stone et al., 2008) and included gravity. Statistical comparisons (using relative orientation of column density and magnetic fields, average γ, and S) between these simulations and observations explored the effect of γ on p and S and were made possible by the high resolution and sensitivity of the Balloon-borne Large Aperture Sub-millimeter Telescope for Polarimetry (BLASTPol) observations of the Vela C (Fissel et al., 2016) cloud. These comparisons indicated that the Vela C observations and its high polarization angle dispersion were consistent with simulations of magnetic fields with high inclination angles. However, due to the degeneracy between disorder caused by turbulence and disorder caused by a large inclination angle (the field disorder seen in the plane of the sky), they were unable to infer a γ value for the Vela C cloud. Chen et al. (2019) extended the study of King et al. (2018) and determined γ for the Vela C cloud, assuming a small total S (applicable only to sub-Alfvénic regions). Using a statistical examination of the p values of the cloud and the maximum polarization fraction (associated with zero inclination), they calculated γ. They found an average γ value of ∼ 60 • for the Vela C cloud, with an estimated accuracy of ≤ 10 • − 30 • . Subsequently, Sullivan et al. (2021) analyzed the 3D magnetic field properties of nearby molecular clouds 8 and estimated their cloud-averaged γ values. This technique can be used to examine the relative alignment of magnetic field lines and the orientation of filamentary dense gas in 3D .
The technique's inherent uncertainty is dominated by the following assumptions: a) presence of a location within the cloud with zero γ, corresponding to the observed maximum p; b) homogeneous grain alignment efficiency across the cloud 9 ; c) neglecting depolarization effects along the line of sight; d) assuming uni-directional magnetic fields along the line of sight; and e) ordered field line, which was addressed by .  augmented the technique of Chen et al. (2019) by incorporating magnetic field fluctuations and dispersion (making the technique applicable to trans-and super-Alfvénic regions as well). They modified the equations of Chen et al. (2019) on the assumption that field fluctuations are perpendicular to the mean field. Additionally, we note that these studies still require both B LOS and B POS directions to infer 3D vectors (see Figure 2).

Other approaches
While this mini-review focuses on the techniques discussed in Sections 2.3 and 2.4 and their combination for recovering the 3D magnetic fields of molecular clouds, we note that other more theory-based techniques can also be used in clouds (e.g., Yan and Lazarian, 2005;Tritsis and Tassis, 2018;Hu et al., 2021a;

3D fields
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Q L P 8 A p v T u K 8 O O / O x 6 K 1 5 B Q z x / A H z u c P v r K P L Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " j 0 7 P 6 8 f Q 9 y h V + n 6 c Y e y 2 M y X 1 v w E = " > A A A B 6 n i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 l E 0 W O p F 4 8 V 7 Q e 0 o W y 2 m 3 b p Z h N 2 J 0 I J / Q l e P C j i 1 V / k z X / j t s 1 B W x 8 M P N 6 b Y W Z e k E h h 0 H W / n c L a + s b m V n G 7 t L O 7 t 3 9 Q P j x q m T j V j D d Z L G P d C a j h U i j e R I G S d x L N a R R I 3 g 7 G t z O / / c S 1 E b F 6 x E n C / Y g O l Q g F o 2 i l h 3 r f 6 5 c r b t W d g 6 w S L y c V y N H o l 7 9 6 g 5 i l E V f I J D W m 6 7 k J + h n V K J j k 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " q j T N Z n 0 X O a L Q V z R 2 j L u w 2 c a n r 3 I = " > A A A B 6 3 i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m K o s d S L x 4 r 2 A 9 o Q 9 l s p + 3 S 3 U 3 Y 3 Q g l 9 C 9 4 8 a C I V / + Q N / + N S Z u D t j 4 Y e L w 3 w 8 y 8 I B L c W N f 9 d g o b m 1 v b O 8 X d 0 t 7 + w e F R + f i k b c J Y M 2 y x U I S 6 G 1 C D g i t s W W 4 F d i O N V A Y C O 8 H 0 L v M 7 T 6 g N D 9 W j n U X o S z p W f M Q Z t Z n U G N R K g 3 L F r b o L k H X i 5 a Q C O Z q D 8 l d / G L J Y o r J M U G N 6 n h t Z P 6 H a c i Z w X u r H B i P K p n S M v Z Q q K t H 4 y e L W O b l I l S E Z h T o t Z c l C / T 2 R U G n M T A Z p p 6 R 2 Y l a 9 T P z P 6 8 V 2 d O s n X E W x R c W W i 0 a x I D Y k 2 e N k y D U y K 2 Y p o U z z 9 F b C J l R T Z t N 4 s h C 8 1 Z f X S b t W 9 a 6 r 7 s N V p d 7 I 4 y j C G Z z D J X h w A 3 W 4 h y a 0 g M E E n u E V 3 h z p v D j v z s e y t e D k M 6 f w B 8 7 n D / E W j Y M = < / l a t e x i t > B 2 Galac tic latitud e Ga la ct ic lo ng itu de Lin e of sigh t < l a t e x i t s h a 1 _ b a s e 6 4 = " Q c b F S F e a r v 1 L y 9 R X G a / M r l D Z / / s = " > A A A B 7 n i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 l E 0 W P R i 8 c K 9 g P a U C b b T b t 0 N w m 7 G 6 G E / g g v H h T x 6 u / x 5 r 9 x 0 + a g r Q 8 G H u / N M D M v S A T X x n W / n d L a + s b m V n m 7 s r O 7 t 3 9 Q P T x q 6 z h V l L V o L G L V D V A z w S P W M t w I 1 k 0 U Q x k I 1 g k m d 7 n f e W J K 8 z h 6 N N O E + R J H E Q 8 5 R W O l T n + E U m J l U K 2 5 d X c O s k q 8 g t S g Q H N Q / e o P Y 5 p K F h k q U O u e 5 y b G z 1 A Z T g W b V f q p Z g n S C Y 5 Y z 9 I I J d N + N j 9 3 R s 6 s M i R h r G x F h s z V 3 x M Z S q 2 n M r C d E s 1 Y L 3 u 5 + J / X S 0 1 4 4 2 c 8 S l L D I r p Y F K a C m J j k v 5 M h V 4 w a M b U E q e L 2 V k L H q J A a m 1 A e g r f 8 8 i p p X 9 S 9 q 7 r 7 c F l r 3 B Z x l O E E T u E c P L i G B t x D E 1 p A Y Q L P 8 A p v T u K 8 O O / O x 6 K 1 5 B Q z x / A H z u c P v r K P L Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Q c b F S F e a r v 1 L y 9 R X G a / M r l D Z / / s = " > A A A B 7 n i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 l E 0 W P R i 8 c K 9 g P a U C b b T b t 0 N w m 7 G 6 G E / g g v H h T x 6 u / x 5 r 9 x 0 + a g r Q 8 G H u / N M D M v S A T X x n W / n d L a + s b m V n m 7 s r O 7 t 3 9 Q P T x q 6 z h V l L V o L G L V D V A z w S P W M t w I 1 k 0 U Q x k I 1 g k m d 7 n f e W J K 8 z h 6 N N O E + R J H E Q 8 5 R W O l T n + E U m J l U K 2 5 d X c O s k q 8 g t S g Q H N Q / e o P Y 5 p K F h k q U O u e 5 y b G z 1 A Z T g W b V f q p Z g n S C Y 5 Y z 9 I I J d N + N j 9 3 R s 6 s M i R h r G x F h s z V 3 x M Z S q 2 n M r C d E s 1 Y L 3 u 5 + J / X S 0 1 4 4 2 c 8 S l L D I r p Y F K a C m J j k v 5 M h V 4 w a M b U E q e L 2 V k L H q J A a m 1 A e g r f 8 8 i p p X 9 S 9 q 7 r 7 c F l r 3 B Z x l O E E T u E c P L i G B t x D E 1 p A Y Q L P 8 A p v T u K 8 O O / O x 6 K 1 5 B Q z x / A H z u c P v r K P L Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " j 0 7 P 6 8 f Q 9 y h V + n 6 c Y e y 2 M y X 1 v w E = " > A A A B 6 n i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 l E 0 W O p F 4 8 V 7 Q e 0 o W y 2 m 3 b p Z h N 2 J 0 I J / Q l e P C j i 1 V / k z X / j t s 1 B W x 8 M P N 6 b Y W Z e k E h h 0 H W / n c L a + s b m V n G 7 t L O 7 t 3 9 Q P j x q m T j V j D d Z L G P d C a j h U i j e R I G S d x L N a R R I 3 g 7 G t z O / / c S 1 E b F 6 x E n C / Y g O l Q g F o 2 i l h 3 r f 6 5 c r b t W d g 6 w S L y c V y N H o l 7 9 6 g 5 i l E V f I J D W m 6 7 k J + h n V K J j k 0 1 I v N T y h b E y H v G u p o h E 3 f j Y / d U r O r D I g Y a x t K S R z 9 f d E R i N j J l F g O y O K I 7 P s z c T / v G 6 K 4 Y 2 f C Z W k y B V b L A p T S T A m s 7 / J Q G j O U E 4 s o U w L e y t h I 6 o p Q 5 t O y Y b g L b + 8 S l o X V e + q 6 t 5 f V m r 1 P I 4 i n M A p n I M H 1 1 C D O 2 h A E x g M 4 R l e 4 c 2 R z o v z 7 n w s W g t O P n M M f + B 8 / g C 6 r 4 1 u < / l a t e x i t > B 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " q j T N Z n 0 X O a L Q V z R 2 j L u w 2 c a n r 3 I = " > A A A B 6 3 i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m K o s d S L x 4 r 2 A 9 o Q 9 l s p + 3 S 3 U 3 Y 3 Q g l 9 C 9 4 8 a C I V / + Q N / + N S Z u D t j 4 Y e L w 3 w 8 y 8 I B L c W N f 9 d g o b m 1 v b O 8 X d 0 t 7 + w e F R + f i k b c J Y M 2 y x U I S 6 G 1 C D g i t s W W 4 F d i O N V A Y C O 8 H 0 L v M 7 T 6 g N D 9 W j n U X o S z p W f M Q Z t Z n U G N R K g 3 L F r b o L k H X i 5 a Q C O Z q D 8 l d / G L J Y o r J M U G N 6 n h t Z P 6 H a c i Z w X u r H B i P K p n S M v Z Q q K t H 4 y e L W O b l I l S E Z h T o t Z c l C / T 2 R U G n M T A Z p p 6 R 2 Y l a 9 T P z P 6 8 V 2 d O s n X E W x R c W W i 0 a x I D Y k 2 e N k y D U y K 2 Y p o U z z 9 F b C J l R T Z t N 4 s h C 8 1 Z f X S b t W 9 a 6 r 7 s N V p d 7 I 4 y j C G Z z D J X h w A 3 W 4 h y a 0 g M E E n u E V 3 h z p v D j v z s e y t e D k M 6 f w B 8 7 n D / E W j Y M = < / l a t e x i t > B 2 Figure 2. B POS direction required for 3D field determination. The 3D magnetic field vectors B 1 and B 2 have the same inclination angle (γ), run parallel to the Galactic longitude axis when projected onto the plane of the sky, and point toward us when projected along the line of sight. However, due to the difference in their B POS directions, they are two distinct 3D vectors. Since the projections of these two vectors onto the plane of the sky are parallel to the longitude axis, their inclination angle with respect to the plane of the sky is the angle between the 3D vector and the longitude axis. The left and right panels display two different viewing angles. Distinguishing between these two vectors is particularly important in studies of relative alignment of field lines and clouds, as a cloud aligned with B 1 may be approximately perpendicular to B 2 depending on the value of γ. Skalidis et al., 2021b) or within its high density regions (i.e., clumps or cores Houde et al., 2000a;Kandori et al., 2017Kandori et al., , 2020a. We briefly discuss these techniques here, excluding those applicable only to core scales (e.g., Kandori et al., 2017Kandori et al., , 2020a. Houde et al. (2000aHoude et al. ( ,b, 2002Houde et al. ( , 2004 proposed a method for measuring γ based on the ion-to-neutral line-width ratios. Their observations showed that, in the presence of strong magnetic fields, the line-width of ions is narrower than that of coexisting neutrals. They suggest that when the field lines are perpendicular to the line of sight, the difference in line-widths should be the greatest, enabling them to infer γ. Some studies found supporting (Li and Houde, 2008;Hezareh et al., 2010;Houde, 2011;Tang et al., 2018) or inconsistent (Pineda et al., 2021) observational evidence.

Atomic alignment
The atomic alignment (or ground state alignment) technique (Yan and Lazarian, 2005, 2006, 2007Yan et al., 2019) relies on the alignment of the angular momentum of atoms in their ground state with the photons' angular momentum from background anisotropic radiation, followed by their realignment with external magnetic fields. For best outcomes, absorption lines are used. Calculating the degree of alignment with magnetic field lines, Yan and Lazarian (2007) obtained the Stokes parameters of absorbed radiation and compared them with observations to infer γ and the 3D field lines. This method is most applicable to diffuse ISM (Yan and Lazarian, 2012), but may also be applied to molecular clouds and their envelopes.

Young stellar objects and position-position-velocity space techniques
Based on the observable anisotropy of turbulence eddies in the presence of magnetic fields, Hu et al. (2021b) estimate magnetic fields using structure function analysis (SFA). They demonstrate that for sub-Alfvénic regions, the ratio of perpendicular to parallel 10 velocity fluctuations has a power-law relation with M A , enabling determination of 3D field strengths. Hu et al. (2021a) extended the SFA analysis of Hu et al. (2021b) to infer 3D fields by incorporating Gaia observations of young stellar objects (for estimating 3D velocity fluctuations; assuming they inherit the velocity of their parent cloud).

Potential insights from 3D field mapping
This section briefly discusses the potential takeaways from the aforementioned 3D studies. Assuming a GMF model and given B LOS and B POS observations, Tahani et al. (2022b,a) inferred the 3D ordered magnetic field vectors of two molecular clouds. Including γ can enhance these studies. Inferring the 3D magnetic fields of numerous molecular clouds will enable us to compare them with models and numerical simulations to constrain cloud formation models (see Hennebelle and Inutsuka, 2019, and references therein), 3D structure and evolution of the ISM (e.g., Hacar et al., 2022), 3D GMF models (e.g., Jaffe, 2019), and the role of magnetic fields in cloud evolution (e.g., Fiege and Pudritz, 2000a).
For example, Tahani et al. (2022b) employed velocity information of the Perseus cloud along with GMF models to predict the cloud-averaged ordered line-of-sight and 3D magnetic field of this cloud based on the model of Inutsuka et al. (2015) 11 and found the predictions to be consistent with their inferred 3D field and B LOS data. The cloud-formation model of Inutsuka et al. (2015) requires multiple compressions caused by expanding interstellar bubbles to form filamentary molecular clouds. Using dynamics and bubble observations of the Orion A and Perseus clouds, Tahani et al. (2022a,b) proposed similar formation scenarios for their 3D fields: the field lines should have been initially bent on a large scale by recurrent supernovae shocks. This bending of field lines by bubbles has been detected in numerical simulations (Kim and Ostriker, 2015) and large-and small-scale observations (Soler et al., 2018;Bracco et al., 2020;Arzoumanian et al., 2021). Subsequently, interaction with a secondary bubble may have pushed the HI gas surrounding the clouds, causing a sharp field line bending (arc-shaped field) associated with the molecular cloud.

DISCUSSION
Observing the 3D magnetic fields of molecular clouds and their substructures is essential for understanding their formation mechanism and the role magnetic fields play in star formation. Observations of B LOS and B POS are necessary but insufficient for determining the 3D fields. While B LOS observing techniques provide both the strength and direction of this component, B POS observing techniques provide only the orientation and strength of this component, but not its direction. Knowing the strengths and complete directions of B LOS and B POS enables us to infer the ordered, line-of-sight-averaged 3D field vectors. 10 relative to the magnetic field 11 also see simulations by Inoue et al. (2018) However, due to systematic biases between the techniques for determining field strengths, additional observations, such as observing the magnetic field inclination angles are required. The B LOS strength and direction, γ, and B POS orientation (without its direction) do not fully infer the 3D fields, as they can lead to two different vectors depicted in Figure 2. Other techniques such as the use of GMF models (Tahani et al., 2022a,b) can help resolve this issue.
The studies of B LOS , B POS , γ, and GMF could enable us to infer the 3D ordered magnetic fields of molecular clouds with improved precision. Upcoming observations will 1) enhance the precision and accuracy of the inferred 3D magnetic field of each cloud, 2) result in 3D magnetic field maps of more regions, and 3) produce more accurate GMF models, thereby enhancing the technique's underlying assumptions.
The forthcoming Zeeman measurements (Robishaw et al., 2015, for the most accurate determination of field strengths) and Faraday rotation measure catalogs by the Square Kilometer Array (SKA) project (Heald et al., 2020) or the Australian Square Kilometer Array Pathfinder (ASKAP), such as the Polarisation Sky Survey of the Universe's Magnetism (POSSUM) rotation measure catalog (Gaensler et al., 2010), will provide the B LOS of numerous molecular clouds with lower uncertainties and greater source density than previous catalogs (e.g., Taylor et al., 2009). These observations will increase the number of B LOS detections per molecular cloud by a factor of ∼ 10. These B LOS maps and future B POS observations, such as those by the Fred Young Sub-millimeter Telescope (FYST; CCAT-Prime collaboration et al., 2021), will enable 3D magnetic field maps of many molecular clouds.
Finally, starlight polarization observations (e.g., Pereyra and Magalhães, 2007) combined with Gaiaobserved parallax distances allow us to differentiate between, and separate, various cloud components along the line of sight (e.g., Doi et al., 2021). This is made possible by existing and upcoming starlight polarization observations, including the Galactic Plane Infrared Polarization Survey (GPIPS; Clemens et al., 2020) and the upcoming optical polarimetry survey with the Polar-Areas Stellar Imaging Polarization High Accuracy Experiment (PASIPHAE; Tassis et al., 2018).