Slope of Magnetic Field-Density Relation as An Indicator of Magnetic Dominance
Mengke Zhao Guang-Xing Li Keping Qiu




0000-0003-0596-6608]Mengke Zhao
School of Astronomy and Space Science, Nanjing University, 163 Xianlin Avenue, Nanjing 210023, Jiangsu, People’s Republic of China

0000-0003-3144-1952]Guang-Xing Li
South-Western Institute for Astronomy Research, Yunnan University, Kunming 650091, People’s Republic of China

0000-0002-5093-5088]Keping Qiu
School of Astronomy and Space Science, Nanjing University, 163 Xianlin Avenue, Nanjing 210023, Jiangsu, People’s Republic of China
Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210023, Jiangsu, People’s Republic of China


§ ABSTRACT 
The electromagnetic field is a fundamental force in nature that regulates the formation of stars in the universe. 
Despite decades of efforts, a reliable assessment of the importance of the magnetic fields in star formation relations remains missing. 
In star-formation research, our acknowledgment of the importance of magnetic field is best summarized by the  
  B-ρ relation 

    log B(ρ) /G = {
        -5,    if ρ ≲  10^-20  g cm^-3
    2/3· logρ +  logρ_0,    if ρ ≳  10^-20  g cm^-3
        .
whose interpretation remains controversial <cit.>. The relation is either interpreted as proof of the importance of a magnetic field in the collapse <cit.>, or the result of self-similar collapse where the role of the magnetic is secondary to gravity <cit.>. 



Using simulations, we find a fundamental relation, M_ A-k_B-ρ(slope of B-ρ relation) relation:

    M_ A/ M_ A,c = k_B-ρ^ K≈ M_ A/7.5≈   k_B-ρ^1.7± 0.15.

This fundamental B-ρ-slope relation enables one to measure the Alfvénic Mach number, a direct indicator of the importance of the magnetic field, using the distribution of data in the B-ρ plane. It allows us to drive the following empirical  B-ρ relation

    B/B_c =  exp((γ/ K)^-1( ρ/ρ_c)^γ/ K) ≈B/10^-6.3 G≈ exp(9 (ρ/10^-16.1 g cm^-3)^0.11) ,

which offers an excellent fit to the Cruther et al. data, where we assume M_ A-ρ relation (M_ A/ M_ A,c = (ρ/ρ_c)^γ≈ M_ A/7.5≈(ρ/10^-16.1 g cm^-3)^ 0.19). 

The foundational M_ A- k_B-ρ relation provides an independent way to measure the importance of magnetic field against the kinematic motion using multiple magnetic field measurements. Our approach offers a new interpretation of the <cit.>, where a gradual decrease in the importance of B at higher densities is implied. 






















§ BACKGROUND




The magnetic field is a fundamental force that plays a crucial role in the evolution of interstellar gas and star formation <cit.> through magnetic force and it also changes the initial condition <cit.> by affecting the transport of cosmic rays.

Despite decades of research, constraints on the importance of magnetic fields remain sparse. There are two approaches. 
The first one involves the measurement of dimension numbers which reflects the importance of the magnetic fields. One example is the mass-to-flux ratio, which is the ratio between mass and magnetic flux <cit.>. 
The second approach is to derive scaling relations between the magnetic field and density<cit.>, which serve as bridges between theory and observations.  










The empirical magnetic field density relation (B-ρ relation) is an empirical relation describing the magnetic field strength in the interstellar constructed using a sample of regions with Zeeman observations <cit.>. The relation can be described using different functions, such as

    log B(ρ) /G = {
        -5,    if ρ ≲  10^-20  g cm^-3
    2/3· logρ +  logρ_0,    if ρ ≳  10^-20  g cm^-3
        .

 The classical B-ρ relation is an empirical relation, which does not explain how the B-field evolves with density growing. 
 explain a part of classical B-ρ relation with the 2/3 power-law exponent being caused by the collapse of star formation 
This exponent of B-ρ in star-forming can be linked to the density profile of the region, where B∝ρ^1/2 is related to the density profile of ρ∼ r^-2 <cit.>.
The B-ρ relation is affected transiently by galactic magnetic field morphology <cit.>.



















The magnetic field-density relation, although widely cited, is non-trivial to interpret because both the magnetic field and gravitational force are long-range forces where the scale information is critical <cit.>. For example,  the energy balance of a cloud reads B^2r^3  ≈ρ^2 r^5, where the scale information is an essential part. 
To address the energy balance, one should perform joint analyses of the magnetic field-density relation and density-scale relation <cit.>.

B-ρ relation derived from several observations targeting single regions can provide crucial insight into the importance of B field <cit.>. 
When targeting individual regions, one eliminates the uncertainties in the scale.  B-ρ relation constructed in this way can better reflect the interplay between turbulence, gravity, and the B-field. 


The scope of this paper is to offer a direct interpretation of the slope of the B-ρ relation, as a direct indicator of the importance of magnetic field. 
Compared to the mass-to-flux ratio, which is indirect diagnostic and can be hard to measure, the Alfenvi Mach number is a direct indicator of the importance and is directly measurable using the slope. 
Our approach also allows for re-interpreting the <cit.>.
Using numerical simulation, we establish a fundamental link between the slope, d( log B) / d( logρ) with the importance of magnetic field measured in terms of the Alfvén Mach number M_ A. 
This allows us to evaluate the importance of the magnetic field directly from observations even to approach the fundamental B-ρ relation.  


































§ ALFVÉNIC MACH NUMBER & FUNDAMENTAL B-Ρ RELATION





 §.§ Slope of B-ρ Relation: An Indicator of M_ A 








To establish ration between B-ρ and M_ A, we select a numerical magnetohydrodynamic (MHD) simulation <cit.>. 
The initial conditions are generated by a PPML code without self-gravity  <cit.>,  after which the main simulation was performed using AMR code Enzo<cit.> where self-gravity is included. 
In both simulations, the equation of state is isothermal. 


The relatively short simulation time (0.8  Myr) means the effect of self-gravity is only noticeable in the regions of the highest density 
[The simulations were terminated at around  0.76 Myr, The density after which self-gravity can have an effect should be around 2.6×10^-20 g cm^-3 (estimated using ρ_* = 1/(t_*^2· G), where t_* is the time at which the simulation terminates), which is far above the mean density (3.8×10^-21) of simulations and this density. ]. 
The observed B-ρ  relation mostly reflects the interplay between the magnetic field and turbulence cascade, which is the main focus of this work.

We choose three simulations of different initial degrees of magnetization β_0 = 8π c_s^2 ρ_0/B_0^2 = 0.2, 2, 20.  The slopes of the  B-ρ relation can be derived after some averages (Method <ref>).
The Alfvén Mach number M_ A in various simulations can be calculated by the total energy ratio between the magnetic energy E_B and kinetic energy E_k:

    M_ A = √(4πρ)σ_v/B = 0.5ρσ_v^2/B^2/8π =  ∑ e_ k/∑ e_B ,

where the M_ A in strong, medium, and weak magnetic field states can be estimated as 0.31, 0.49, and 1.29, respectively.
The  Alfvénic Mach numbers M_ A are measured at the time of the snapshots (Method <ref>), which is different from the Alfvénic Mach numbers inferred from the initial conditions, due to the dissipation of turbulence and the amplification of the magnetic field.


We then plot the Mach number against the measured slope of the B-ρ  (see Fig. <ref>), to reveal a surprisingly simple relation between the slope of the B-ρ relation (k_B-ρ = d (log_10 B) / d (log_10 ρ)) and the Alfvén Mach number M_ A,

    M_ A/ M_ A,c = k_B-ρ^ K ,

where the K = 1.7±0.15, and the M_ A,c is the characteristic M_ A (∼ 7.5±1.3, see Fig. <ref>). 
The degree of magnetization quantified using M_ A determines the power-law exponent between density and magnetic field.
The M_ A- k_B-ρ relation could be a fundamental relation in natural.
An intuitive explanation for the correlation between the stronger magnetic field and shallower magnetic field-density relation is that a strong field can regulate the way collapse occurs, such that the matter accumulates along the field lines, weakening the correlation between B and ρ. 


























 §.§ Determining the Importance of B-field using the M_ A- k_B-ρ relation


 





M_ A is a fundamental physical quantity that directly characterizes the importance of the magnetic field. 
The fact that this quantity is directly related to the slope of the B-ρ relation provides a new way to study the importance of the magnetic field (see Sect. <ref>). 
However, to construct this single-region B-ρ relations, we require multi-scales, multi-densities observations towards single regions (Methods <ref>). Another case is the outer region of the Taurus molecular cloud, where multiple constraints of the magnetic field strength using Zeeman splitting <cit.> are available. 





We reveal vastly different roles of the magnetic field (see Fig. <ref>) by applying our results to both clouds. 
Toward the Taurus outer region, using Zeeman observations <cit.>, we estimate a M_ A of 0.8, indicative of a relatively strong (dynamically significant) magnetic field. Towards the NGC 6334 center region, the M_ A is estimated as 3.0, implying a weak (dynamically weak) magnetic field. 


Our assessment of the importance of the magnetic field is consistent with the behavior of these regions as reported in the literature <cit.>: 
The strong magnetic field observed towards the outer region of the Taurus molecular cloud, where  M_ A (∼ 0.4) indicates that the B-field is strong enough to dominate the gas evolution. 
This is consistent with conclusions from previous studies <cit.>. 
At the Taurus outer region, people have observed striations which are taken as signs of the dominance MHD wave <cit.> – a phenomenon that occurs only if the B-field is strong enough. Towards the dense part where our observations target, it is found that denser gas has lost a significant amount of magnetic flux as compared to the diffuse envelope <cit.>, and this critical transition, and agrees well with our estimate where √(E_B / E_k)∝ M_ A. 
Towards the central region NGC 6334, we find that kinetic energy is far above the magnetic energy (M_ A≈1.6), dynamically insignificant magnetic field). 
This is consistent with NGC 6334 being one the most active star-forming regions with a high fraction of dense gas <cit.>. 

Using the fundamental M_ A- k_B-ρ relation (Eq. <ref>), we can use the slope of B-ρ relation to estimate the Alfvén Mach number M_ A, which determine the importance of B-field.






















 §.§ Fundamental B-ρ relation




In the past, a convenient way to study magnetic field was to plot the field strength against the gas density, forming the magnetic field-density relation <cit.>. 
This multi-region magnetic field-density relation is constructed from observations of different regions which have different sizes and geometries. 
The plural nature of this multi-region magnetic field-density relation makes it not particularly useful in estimating the M_ A. 
However, we can decipher this relation using a minimum of assumptions. 


We start from the fundamental M_ A- k_B-ρ relation (Eq. <ref>), and assume another relation between Alfvén Mach number M_ A and density ρ as a power low:

    M_ A/ M_ A,c = (ρ/ρ_c)^γ ,


The fundamental B-ρ relation can be derived by combining fundamental M_ A-k_B-ρ relation (Eq. <ref>) with the empirical M_ A-ρ relation (Eq.<ref>) :

    B/B_c =  exp((γ/ K)^-1( ρ/ρ_c)^γ/ K)   ,

and by fitting Eq.<ref> to the observational data from <cit.>, assuming K =1.7±0.15 (Sec.<ref>), we find γ = 0.19 ± 0.07, log_10ρ_ c / (g cm^-3) = 16.09 ± 1.02, and log_10 B_ c/  (G) = -6.29 ± 0.65. 
This relation (Eq. <ref>) perfectly crosses all Zeeman observations and three MHD simulations and agrees to the classical B-ρ relation <cit.> 
In the diffuse region, the B-field slowly grows with density increasing.
In high density region (ρ ∼ 10^-18 g cm^-3), the slope of B-ρ relation is also up to 2/3, which is caused by star formation <cit.>.

Our fitting implies

    M_ A/ M_ A,c = (ρ/ρ_c)^0.19 ± 0.07 ,

which is a general relation connecting the degrees of magnetization of clouds in the Milky Way with the density. 
At low density (10^-22 g cm^-3), the magnetic field is non-negligible. As the density increases, the importance of the magnetic field is gradually diminished. The high-density regions are mostly kinematically-dominated. 
This flat M_ A-ρ relation (M_ A ∝ ρ^0.19 ± 0.07) also agrees to actual observations (see the Fig.2 of ).

This gradual decrease in the importance of the magnetic field is consistent with observational data. At the low-density end, the outer region of Taurus <cit.> and Musca <cit.> molecular clouds have striations (MHD wave, the feature of strong magnetic field state), indicative of strong, dynamically-important magnetic field. At intermediate densities, the increase of the gas density can be accompanied by an increase of the mass-to-flux ratio, and a gradual decrease of the importance of the magnetic field, as observed in the L1544 region <cit.>, and most high-density regions appears to be kinematically-dominated, such as NGC 6334 <cit.>, Ser-emb 8 <cit.>, and W51 <cit.>.





















































 





§ CONCLUSION





  
We find that the slope of the magnetic field-density relation  k_B-ρ = d (log_10 B)/ d (log_10ρ) linked to the  Alfvén Mach number, M_ A using numerical simulation results, 

    M_ A/ M_ A,c = k_B-ρ^ K≈ M_ A/11.0≈ k_B-ρ^ 1.7 ± 0.15,

where the K = 1.7±0.15, and the M_ A,c is the characteristic M_ A (∼ 7.5±1.3). This fundamental relation results from the interplay between turbulence cascade and the implication of magnetic field in MHD turbulence. 
This fundamental relation allows us to estimate Ma using the slope of the B-rho relation. We apply this technique to several regions and find that the estimated M_ A values exhibit clear correlations between  M_ A with the observed star formation activities. 

We find that the empirical  B-ρ relation can be explained using our fundamental relation, together with a relation between Ma and gas density. 
The M_ A-ρ relation reads

    M_ A/ M_ A,c = (ρ/ρ_c)^γ≈ M_ A/7.5≈(ρ/10^-16.1 g cm^-3)^ 0.19± 0.07 ,

and the B-ρ relation takes the following form

    B/B_c =  exp((γ/ K)^-1( ρ/ρ_c)^γ/ K) ≈B/10^-6.3 G≈ exp(9  (ρ/10^-16.1 g cm^-3)^0.11) ,

which provides a good fit to existing observational data. 
This fitting result reveals a general trend where a general decrease in the importance of the magnetic field accompanies the increase in the gas density. 


The fundamental M_ A- k_B-ρ relation provides an independent way to estimate the importance of the magnetic field, and it can be widely applied to further observations. The decreasing importance of magnetic field at higher density appears consistent with most of the existing observations and will be tested against future results. 







































                                                                                                                        




































Method








§ SIMULATION


The numerical simulation of molecular cloud selected in this work is identified within the three β_0 simulations applied by the constrained transport MHD option in Enzo (MHDCT) code <cit.>. 
The simulation conducted in this study analyzed the impact of self-gravity and magnetic fields on supersonic turbulence in isothermal molecular clouds, using high-resolution simulations and adaptive mesh refinement techniques. which detail in shown in <cit.>. 
The Enzo simulation with three dimensionless parameters provides us with massive information on physical processes:

    M_s = v_ rms/c_s = 9


    α_vir = 5 v_ rms^2 / 3 G ρ_0 L_0^2 = 1


    β_0 = 8π c_s^2 ρ_0/B_0^2 = 0.2, 2, 20

where v_rms is the rms velocity fluctuation, c_s is sonic speed, ρ_0 is mean density, and B_0 is the mean magnetic field strength.
With the same sonic Mach number M_ s and viral parameter α_vir in initial conditions, the difference only exists in initial magnetic pressure β-0 with various B-field strength,
The β_0 as 0.2, 2, and 20 present the strong B-field state, medium B-field, and weak B-field state.
The clouds are in three different evolutionary stages started with self-gravity existing at 0.6t_ ff(∼ 0.76 Myr), where t_ ff represents 1.1((n_H / 10^3)^-1/2 Myr <cit.>. 
Due to the short timescale of evolution, the gravitational collapse could barely affect the B-ρ relation in the simulation
The size of these molecular clouds is around 4.6^3 pc (256^3 pixels), with one-pixel size of approximately 0.018 pc <cit.>.
Fig. <ref> show the distribution of B-field and density with three β in the timescale of 0.6 t_ ff (∼ 0.76 Myr).












In this Enzo simulation, the thermal energy is located at the low state (M_s = 9).
Their timescale of evolution stage is 0.6 t_ ff, around  0.76 Myr, which the timescale starts from when interstellar media collapse.
Due to the short evolutionary time (∼ 3.85 × 10^-21 g cm^-3, or 821 cm^-3), the gravitational energy less affects the system in simulation.
With the similar mean density ρ_0 in these simulations, the different β_0 (∼0.2, 2, 20) present the different magnetic field states, strong B-field, stronger B-field (weaker than strong B-field but stronger than weak B-field), and weak B-field.




The energy ratio between kinetic energy E_ k and magnetic energy E_ B E_ k/E_B in the evolutionary timescale of 0.6 t_ff (∼ 0.76 Myr) is calculated:

    E_ k/E_ B = 1/2ρ v_ rms^2/B_0^2/8π

The E_ k/E_B with stronger, medium, and weak B-field states are 0.09, 0.24, and 3.37, respectively.




§ RE-SCALING METHOD

The values obtained from numerical simulations are without dimensions, and the numerical values of the physical quantities are obtained during the interpretation phase, where the pre-factors are inserted to ensure the dimensionless parameters such as the Mach number stay unchanged. When applying the result of numerical simulations to reality, one is allowed to change the numerical values of the prefactors, as long as the dimensionless controlling parameters stay unchanged. 

Each simulation has three dimensionless controlling parameters, including the Mach number, the plasma β, and the virial parameter. We argue that the virial parameter has little influence on the slope of the magnetic-field-density relation. This is because gravity is not important in determining the relation. This assertion is supported by the fact that the magnetic field-density relation do not evolve significantly during the collapse phase. 










We thus rescale our simulations using the following transform:



    B' = λ_B · B,     ρ' = λ_ρ·ρ

where λ_B and λ_ρ are the rescalling factors, the B and ρ are the B-field and density before rescalling, the B' and ρ' are the B-field and density after rescalling. 
In this simulation, initial M_ A,0 is amount to the plasma β_0, M_ A_0 = (ρ_0 v_ rms^2/B_0^2)^0.5 = (81 c_s^2 ρ_0 /B_0^2)^0.5 = (81/8πβ_0)^0.5.
With evolving, the unchanged of M_ A means that plasma β is not changed.
We thus require that this rescaling ensure that the Alfvenic Mach number M_ A of simulations do not change.



The density after rescalling, ρ', is obey on the M_ A-ρ relation (See Eq. <ref>):

    ρ' = ( M_ A/ M_ A,0)^1/γ·ρ_c

where the γ is the slope of M_ A-ρ empirical relation, the M_ A,c is the characteristic M_ A (see Eq. <ref>).
The rescalling factor of density is calculated as:

    λ_ρ = ρ'/ρ = ( M_ A/ M_ A,0)^1/γ·ρ_c/ρ

where the ρ is density before rescalling.
The B-field strength after rescalling, B', obeys the B-ρ relation (See Eq. <ref>) and is calculated by rescalling density ρ':

    B' =  exp((γ/ K)^-1( ρ'/ρ_c)^γ/ K) · B_c

where the parameter, B_c, ρ_c, γ, and cal K, are the same as Eq. <ref>, the B is the B-field before scalling.
The B-field rescalling factor λ_B is defined as:

    λ_B = B'/B =  exp((γ/ K)^-1( ρ'/ρ_c)^γ/ K) ·B_c/B

where the B is B-field before rescalling.
The parameters before or after rescalling are shown in Tab. <ref>.
The rescalling factors are shown in tab. <ref>.








§ FITTING SLOPE OF B-Ρ DISTRIBUTION




Fig. <ref> show the distribution between B-field and density in simulation with M_ A = 0.31, 0.49, 1.29, respectively.
Fitting the main structure of B-ρ distribution at the main density range of simulation in log-log space, the slope of B-ρ relation, d (log_10 B)/ d (log_10ρ) in simulations (M_ A = 0.31, 0.49, 1.29) can be derived as 0.15, 0.21, 0.35, respectively. 

table, with all  parameters, energy ratios,












§ CALCULATED THE SLOPE OF B-Ρ RELATION IN OBSERVATION




Calculating Slope of B-ρ relation have two method: 

i) Single source with various scales 

ii) Single source with various densities 






 §.§ Single source with various scales


The B-field B and density ρ of a single source is observed at various scales, such as NGC 6334 <cit.>(d (log_10 B)/ d (log_10ρ) = 0.41).



 §.§ Single source with various densities


The region of a single source has sub-regions with various densities. 
For example, the diffuse region (N(H_2)  2× 10^22 cm^-2) in the Taurus could exist MHD wave <cit.>.
As Fig. <ref> shows, the Taurus out region has sub-areas with various densities, 3C132, 3C133, L1544 <cit.>.
Zeeman source 3C132 is located in the MHD Wave region and 3C133, L1544 is located at the boundary line of the MHD Wave region.
<cit.> found that L1544 exists as the transition point between strong and weak magnetic fields.
Fig. <ref> shows the distribution between the mean density and mean B-field of three sources, which mean ρ and B come from <cit.>. 
The slope of the B-ρ relation in the Taurus outer region is fitting as 0.17. 



§ ENERGY SPECTRUM  VELOCITY DISPERSION




The energy spectrum of density ρ in three simulations presents the density structure focused on the 6-pixel scale (see Fig. <ref>). 
We use the 8-pixel as the sub-block size to calculate the velocity dispersion σ_v in each pixel of simulation data, which is close to the 6-pixel scale and can divide the whole scale of simulation data (256 pixels) without remainder:

    v_ijk, m = ∑ v_ijk·ρ/∑ρ∑ v_ijk


    σ_v = (∑ (v_ijk - v_ijk, m)^2)^1/2

where the i,j,k show the axis of x,y, and z.
The velocity dispersion can be used to compute the kinetic energy density (e_k = 1/2ρσ_v^2).






























