Magnetic fields with random initial conditions in discs with Kepler rotation curve

Denis A. Grachev; Evgeny A. Mikhailov; Ekaterina N. Zhikhareva

doi:10.1515/astro-2022-0216

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Magnetic fields with random initial conditions in discs with Kepler rotation curve

Denis A. Grachev , Evgeny A. Mikhailov and Ekaterina N. Zhikhareva

Published/Copyright: February 18, 2023

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From the journal Open Astronomy Volume 32 Issue 1

Abstract

Magnetic fields of various astrophysical objects are described using the dynamo mechanism. Corresponding equations in the three-dimensional case are quite difficult to be solved. So, the two-dimensional models can be useful for such problems. For galactic and accretion discs, it is convenient to use the no- z approximation. The initial conditions for the magnetic field have a special interest. It seems that the seed field is generated by the Biermann mechanism, and after that they are transformed by the small-scale dynamo which gives a random structure of the field. Previously it has been shown that random initial conditions can lead to generation of large-scale magnetic field structures which correspond to the galaxies at the moment. In this work, we have studied generation of the magnetic field for the case of Kepler rotation curve, which is mostly suitable to the accretion discs. Here, we have studied the field generation in the thin disc for rapidly changing angular velocity in the case of simple model assumptions.

Keywords: magnetic field; accretion discs; dynamo; Kepler rotation

1 Introduction

Nowadays, it no doubt that a large variety of astrophysical objects have large-scale magnetic fields (Zeldovich et al. 1983, Bochkarev 2011). There is a wide range of observational data connected with the magnetic field of the Sun (Parker 1955, Obridko et al. 2017) and other stars (Katsova et al. 2022), fields of different planets (Busse and Simitev 2011, Cuartas-Restrepo 2018), and magnetic fields of galaxies (Beck et al. 1996, Arshakian et al. 2009). Today, it is very important to study magnetic fields of accretion discs surrounding massive compact astrophysical objects such as black holes, neutron stars, and white dwarfs (Shakura and Sunyaev 1973, Rüdiger and Shalybkov 2002, Moss et al. 2016). Most of the astrophysical objects have a difficult structure of the field, combining toroidal and poloidal components (Sokoloff 2015).

Generation of the magnetic fields in astrophysics is a result of different magnetohydrodynamic processes, which are connected with the so-called dynamo mechanism (Zeldovich et al. 1983). It describes transition of the energy of the turbulent motions to the magnetic field energy, and it is based on two main effects. First, most of the astrophysical objects rotate not as a solid body, so we can consider the differential rotation which transforms the poloidal component of the field to the toroidal one. The effectivity of this mechanism is proportional to the gradient of the angular velocity. Another important part of the dynamo is connected with the alpha-effect, which describes the helicity of the turbulent motions. It can describe the transformation of the toroidal component of the field to the poloidal one. Jointly these two effects lead to the generation of the magnetic field according to the exponential law. However, the magnetic field structures can be destroyed by turbulent diffusivity. So the magnetic field generation is a threshold process: the magnetic field can be generated only for some specific values of the parameters which correspond to the case when alpha-effect and differential rotation are more intensive than the dissipative processes (Arshakian et al. 2009).

The equations for evolution of the magnetic field are based on averaging the magnetohydrodynamics equations (Krause and Raedler 1980). This procedure allows us to find the regular part of the magnetic field, excluding random one which has the typical lengthscale comparable with the turbulent cells. These equations contain the terms connected with alpha-effect and differential rotation (Molchanov et al. 1985). Also there is a term connected with the turbulent diffusion. However, such equations are quite difficult to be solved: they are three-dimensional, and to have the realistic modeling, we should take a lot of grid nodes while solving them in simulations. Such problem requires large resources even for modern supercomputers. If we are taking the theoretical methods, they are very difficult to be realized, and they are principally connected with some rough assumptions or asymptotic methods. Also if we are taking full three-dimensional models, it is necessary to include the values of parameters that characterize the kinematics of the medium, which cannot be measured precisely for different important cases. So, it is necessary to take two-dimensional models for such problems, which take into account some specific features and symmetries of the astrophysical objects. This allows to solve the equations in much simpler case using both theoretical and computational methods. First, models were proposed by Parker for the magnetic field of the Sun (Parker 1955) and by Elsasser for the magnetic field of the Earth (Elsasser 1950). They gave the scientific community an opportunity to find the typical laws of the generation of the field in objects that have the shape of a sphere.

If we are speaking about the galaxies and accretion discs, it seems quite convenient to consider them as a thin disc. Such approaches were realized by Subramanian and Mestel (1993) and Moss (1995) while constructing the no- z approximation for galaxy dynamo. They assumed that the galaxy disc has a small half-thickness which can be described as a small parameter of the model. Also it is possible to use the following approximation: the magnetic field is mainly situated in the equatorial disc plane. This leads to the idea connected with neglecting the vertical part of the magnetic field. However, if we have vertical partial derivative of the magnetic field, we can find it from the solenoidality condition. After that, we can take the eldest mode of the magnetic field and replace the second vertical partial derivatives of the components of the field by algebraic expressions. Finally, we obtain a system of two connected parabolic equations which describe the field evolution. This model is widely used for modeling the magnetic fields in galaxy discs for different cases (Kleeorin et al. 2002, Moss et al. 2013, Mikhailov et al. 2014).

As for the accretion disc, it is strongly believed that they contain large-scale magnetic fields. They can describe the transition of the angular momentum between different parts of the disc, which is very important to understand their evolution (Shakura and Sunyaev 1973). As for the origin of such magnetic fields, there are different ideas. First, some researchers describe the field transport with the falling medium (Okuzumi et al. 2014). However, such processes are highly turbulent, and it is very difficult to carry regular structures of the field. Other approaches are connected with the influence of the central body. But different numerical works show that the field generation is based on the dynamo mechanism (Brandenburg and Donner 1997). Taking into account that the accretion discs have the similar shape with the galaxies, it is possible to use the same approaches and the no- z approximation.

The possibility of using no- z approximation was first showed by Moss et al. (2016). There are some differences between the dynamo mechanism in galaxies and in accretion discs. The main difference is connected with principally specific rotation laws. Most of galactic objects rotate according to the flat rotation curve (such law is based on the influence of dark matter in the galaxies), which leads us to the angular velocity proportional to r − 1 , where r is the distance from the centre. It gives us an opportunity to generate the magnetic field with quite moderate growth rate. As for the accretion discs, the rotation is connected with the Kepler rotation law and the angular velocity is proportional to r − 3 ∕ 2 . This can lead to much more intensive field generation (Moss et al. 2016). Also the field evolution is very different in various parts of the disc: as for galactic objects, we sometimes can assume nearly the same law for the field components. As for the accretion discs, the gradient of the field will be much larger. Another problem is connected with radial accretion flows. It makes the inner boundary condition very important, and the field evolution strongly depends on it (Boneva et al. 2021).

It is also necessary to discuss the initial condition. Most of the works connected with the dynamo (for both accretion and galactic discs) take some specific laws for the seed fields (Moss 1995, Phillips 2001, Andreasyan et al. 2020). It is usually connected with some typical modes multiplied on some initial field. Such approach does not take into account the origin of the seed field. Now both for the galactic and accretion discs, it is quite useful to suppose the following scheme. First, the magnetic field is generated based on the Biermann battery mechanism (Biermann and Schluter 1951, Mikhailov and Andreasyan 2021, Mikhailov and Andreasyan 2021). It is connected with flows of protons and electrons. They have the same charge and strongly different masses and interact with surrounding media according to different mechanisms. This produces circular currents which generate magnetic fields according to the Biot–Savart law. Such fields can be quite low, but they can be an initial condition for the small-scale dynamo. Small-scale dynamo model produces the random magnetic field which can be the initial condition for the large-scale dynamo and regular magnetic fields. So, it is necessary to study the dynamo mechanism using the random seed fields. Such works have been done for galactic magnetic fields (Moss et al. 2012, Mikhailov et al. 2021), and it is necessary to study the magnetic field generation in the accretion disc. Also the random initial conditions there can be connected with the transport of the field with the accreting medium.

Here, we describe the modeling of the magnetic field in the disc with the Kepler rotation law (which can be associated with accretion disc (Shakura and Sunyaev 1973) or galactic disc with specific rotation law). We solve the problem in the axisymmetric case: previously, it has been shown that the nonaxisymmetric structures cannot be stable, and all nonaxisymmetric fields (even random) will become axisymmetric. After that, we study the magnetic field growth and the structures which can be generated by the dynamo in no- z approximation.

2 Dynamo in thin discs

Large-scale magnetic field generation is based on the joint action of the alpha-effect and differential rotation. The evolution of the magnetic field is described by the Steenbeck–Krause–Raedler equation (Krause and Raedler 1980), which is obtained by averaging basic equations of magnetohydrodynamics at the distances associated with turbulence lengthscale (this gives us an opportunity to exclude the small-scale magnetic field which have zero average value and are not interesting from the point of view of observational tests based on Faraday rotation measurements):

∂ B ∂ t = ∇ ( α B ) + ∇ ( U × B ) + η Δ B .

Here, U = r Ω e φ − V ( r ) e r is the large-scale velocity connected with the rotation of the galaxy with angular velocity Ω ( r is the distance from centre) and radial velocity V ( r ) associated with accretion processes and η is the turbulent velocity coefficient.

The alpha-effect is described by a pseudo-scalar that charaterizes the helicity of the motions, and it can be constructed as shown by Molchanov et al. (1985):

α = − τ 3 ( v ⋅ ∇ v ) ,

where τ is a typical correlation time of the turbulence. As for discs, it can be shown that the alpha-effect coefficient can be obtained as follows (it is connected with the Coriolis force which differs for cases z > 0 and z < 0 and influences the turbulent motions) (Phillips 2001):

α = α 0 z h ,

where z is the distance from the disc plane, h is the half-thickness of the disc, and for α 0 , there is the following model:

α 0 = Ω l 2 h ,

where l is the lengthscale of the turbulence. Usually its value is 1–2 orders smaller than the typical half-thickness of the disc.

If the magnetic field enlarges, the turbulent motions that are the reason of the field generation become less intensive. There are different models based on complicated hydrodynamical models, but basic approaches based on energy conservation law lead us to the approximate law:

α 0 → α 0 1 − B 2 B ∗ 2 ,

where B ∗ is the so-called equipartition magnetic field which describes equal values of energy densities of the magnetic field and the turbulent motions. It can be shown that B ∼ ρ 1 ∕ 2 v , where ρ is the density of the medium and v is the typical value of turbulent velocities (Arshakian et al. 2009).

One of the main features of the no- z approximation is connected with using the main z -mode of the magnetic field. If we also take the axisymmetric case, this leads us to the approximation:

B ( r , z , t ) = B ˜ ( r , t ) cos π z 2 h ,

where B ˜ ( r , t ) is the amplitude value of the magnetic field. In the next equations, we are going to omit the tilde. Also we are going to assume that the z -component of the magnetic field is quite small, so we should take into account only its z -derivative which can be obtained using the solenoidality condition for the magnetic field. So the second z -derivative of the magnetic field components can be rewritten using the algebraic expression (Phillips 2001):

∂ 2 B r , φ ∂ z 2 = − π 2 4 h 2 B r , φ .

The angular velocity of the disc will be described by the Kepler law:

Ω = G M r 3 ,

where M is the mass of the central object.

As for the values of other parameters, it is useful to take the following models which are close to the ones for the accretion discs physics (Frank et al. 2002, Moss et al. 2016). We assume that B ∗ ∼ r − 21 ∕ 16 , h ∼ r 9 ∕ 8 and V ( r ) ∼ r − 1 (here, we omit the term Q = 1 − r c . o . r , where r c . o . is the radii of the central object, assuming that we are mainly studying the parts of the disc which are quite far from the centre of the disc).

The typical values of the parameters for the objects that are studied are the following: l ∼ 1 0 8 cm , h ∼ 1 0 9 cm , η ∼ 1 0 14 cm 2 /s and M ∼ 1 0 33 g , and B ∗ ∼ 1 0 − 2 G . As for the typical lengthscales, we can take that the disc has the outer radii of order of 1 0 8 cm and the inner radii is two orders smaller. Initial magnetic field has typical values which are two orders smaller than the equipartition one. Such parameters are quite normal for the accretion discs surrounding white dwarfs, neutron stars, and black holes which have masses comparable with the Sun. It also necessary to add that by describing the accretion discs situated near supermassive black holes, the models can be easily reduced using another numerical values and same model assumptions.

3 Numerical solutions of the equations

Previous assumptions lead us to a system of equations:

∂ B r ∂ t = − α 0 h B ϕ 1 − B r 2 + B ϕ 2 B ∗ 2 − η π 2 4 h 2 B r + V r ∂ B r ∂ r + + η ∂ 2 B r ∂ r 2 + 1 r ∂ B r ∂ r − B r r 2 ∂ B ϕ ∂ t = − Ω B r − η π 2 4 h 2 B ϕ + V r ∂ B ϕ ∂ r + η ∂ 2 B ϕ ∂ r 2 + 1 r ∂ B ϕ ∂ r − B ϕ r 2 .

As for initial conditions, we choose the following: the magnetic field within the rings [ n △ r , ( n + 1 ) △ r ] was assumed to be constant and random. The field has random direction, and its value is a random number B < B 0 , where B 0 that is two orders smaller than B ∗ . The equations are solved for r min < r < r max , where r min and r max are inner and outer radii, correspondingly.

We solve the equations for an accretion disc surrounding a white dwarf with mass M = 2 × 1 0 33 g . The value of the initial field is B 0 = 1 0 − 4 G and the equipartition one is B ∗ = 1.48 × 1 0 − 2 G . For outer radii, we use r max = 1.1 × 1 0 11 cm , and for the inner one, r min = 1.1 × 1 0 9 cm . However, to make the results more general, they are presented in dimensionless units: distances are measured in units of r max . So, the solutions can be taken for the accretion discs surrounding different objects, and even for the galactic objects where there is a rotation law that differs from classical flat rotation curve, being close to the Kepler formulae.

This problem was solved numerically using the parameter values described earlier and corresponding to the model assumptions for accretion discs. We used an explicit numerical scheme taking the finite difference method.

The results of the magnetic field evolution in time are shown on Figure 1. The magnetic field grows fastly and forms both separate small domains and merge within fairly large parts of the disc. The field enough quickly reaches the stationary value corresponding to the equipartition level. If in neighbouring areas the field takes values of equal magnitude and opposite directions, then there are sharp transition layers.

$Figure 1 Magnetic field for random initial conditions. Red line shows t = 1 0 5 s , t=1{0}^{5}\hspace{0.33em}\hspace{0.1em}\text{s}\hspace{0.1em}, black line – t = 1 0 6 s , t=1{0}^{6}\hspace{0.33em}\hspace{0.1em}\text{s}\hspace{0.1em}, blue line t = 1 0 7 s . t=1{0}^{7}\hspace{0.33em}\hspace{0.1em}\text{s}\hspace{0.1em}.$

Figure 1

Magnetic field for random initial conditions. Red line shows t = 1 0 5 s , black line – t = 1 0 6 s , blue line t = 1 0 7 s .

If we take different random realizations of the initial magnetic fields, the localization of the domains and their borders changes (Figure 2). However, nearly in every case, there are regions with magnetic fields of opposite directions.

$Figure 2 Magnetic field for t = 1 0 7 s t=1{0}^{7}\hspace{0.33em}\hspace{0.1em}\text{s} . Curves with different colour show different realizations of random seed fields.$

Figure 2

Magnetic field for t = 1 0 7 s . Curves with different colour show different realizations of random seed fields.

Such processes are quite well known for the galactic magnetic fields. They can be associated with magnetic field reversals that have been found in the Milky Way (Van Eck et al. 2011, Mikhailov and Khasaeva 2019, Andreasyan et al. 2020). Now it is a strong evidence that there are three regions with different directions of the magnetic field. They have been studied observationally using Faraday rotation measurements. From the theoretical point of view, they are described by the theory of contrast structures in mathematical physics (Bozhevol’nov and Nefedov 2011).

The contrast structures can take place in systems of parabolic systems which have two or more stable stationary solutions. The functions are quite close to these stationary values in different regions, which are connected with thin borders with large gradients. In our case, the field in different domains corresponds to one of these stationary solutions.

Such problem has been studied for the flat rotation curve in different works. Some of them are devoted to specific determined initial conditions, which lead to generation of magnetic field reversals. However, if we take random seed field, it is very difficult to find the solution with reversals (most of the simulations describe the same direction of the field in the whole object). As we can see, for the Kepler rotation law, the generation of contrast structures is much more often and realizes in majority of cases.

4 Conclusion

We have studied the magnetic field which is generated by the mean field dynamo mechanism in thin discs with Kepler rotation curve. The initial conditions for the large scale dynamo are connected with the result of the small scale dynamo which is connected with turbulence in the media, so the seed field can be associated random. We have studied the field evolution, and it was shown that it has the reversals that can be associated with cotrasts structures that are well known in mathematical physics.

The main feature of the model with the Kepler rotation law is that the reversals are connected with the majority of the cases. It is quite different from the flat rotation curve that was studied in previous works, where the reversals took place in a small part of random realizations. This result can be connected with much more intensive gradients of the angular velocity. We can conclude that for the Kepler rotation law, the influence of the initial conditions is higher than for smooth change of the velocity.

Funding information: The work of E.A.M. was supported by the Russian Science Foundation (Grant No. 19-72-30028).
Author contributions: D.A.G.: concept of the paper and editing the paper text. E.A.M.: writing the paper text and mathematical statement of the problem. E.N.Z.: numerical simulation and writing the paper text.
Conflict of interest: The authors state no conflicts of interests that could influence the work reported in the paper.

References

Andreasyan RR, Mikhailov EA, Andreasyan HR. 2020. Structure and features of the galactic magnetic-field reversals formation. Astron Rep. 64(3):189–198. 10.1134/S1063772920030014Search in Google Scholar

Arshakian T, Beck R, Krause M, Sokoloff D. 2009. Evolution of magnetic fields in galaxies and future observational tests with the square kilometre array. A&A 494(1):21–32. 10.1051/0004-6361:200810964Search in Google Scholar

Beck R, Brandenburg A, Moss D, Shukurova A, Sokoloff D. 1996. Galactic magnetism: recent developments and perspectives. ARA&A. 34:155–206. 10.1146/annurev.astro.34.1.155Search in Google Scholar

Biermann L, Schluter A. 1951. Cosmic radiation and cosmic magnetic fields. II. Origin of cosmic magnetic fields. Phys Rev. 82:863–868. 10.1103/PhysRev.82.863Search in Google Scholar

Bochkarev NG. 2011. Magnetic fields in space. Moscow: Nauka. (in Russian) Search in Google Scholar

Boneva D, Mikhailov E, Pashentseva M, Sokoloff D. 2021. Magnetic fields in the accretion disks for various inner boundary conditions. A&A. 652:A38. 10.1051/0004-6361/202038680Search in Google Scholar

Bozhevol’nov YuV, Nefedov NN. 2011. Front motion in a parabolic reaction-diffusion problem. Comput Math Math Phys. 50(2):264–273. 10.1134/S0965542510020089Search in Google Scholar

Brandenburg A, Donner KJ. 1997. The dependence of the dynamo alpha on vorticity. MNRAS. 288(2):L29–L33. 10.1093/mnras/288.2.L29Search in Google Scholar

Busse FH, Simitev RD. 2011. Remarks on some typical assumptions in dynamo theory. Geophys Astrophys Fluid Dyn. 105:234–247. 10.1080/03091929.2010.519891Search in Google Scholar

Cuartas-Restrepo P. 2018. Planetary Magnetic Fields and Habitability in super Earths. Open Astron. 27:183. 10.1515/astro-2018-0026Search in Google Scholar

Elsasser WM. 1950. The Earthas Interior and Geomagnetism. Rev Modern Phys. 22(1):1–35. 10.1103/RevModPhys.22.1Search in Google Scholar

Frank J, King A, Raine D. 2002. Accretion power in astrophysics. Cambridge: Cambridge University Press. 10.1017/CBO9781139164245Search in Google Scholar

Katsova MM, Obridko VN, Sokoloff DD, Livshits IM. 2022. Solar and stellar flares: frequency, active regions and stellar dynamo. ApJ. 936:49. 10.3847/1538-4357/ac85e3Search in Google Scholar

Kleeorin N, Moss D, Rogachevskii I, Sokoloff D. 2002. The role of magnetic helicity transport in nonlinear galactic dynamos. A&A. 387:453–462. 10.1051/0004-6361:20020383Search in Google Scholar

Krause F, Raedler K-H. 1980. Mean-field magnetohydrodynamics and dynamo theory. Oxford: Pergamon Press. 10.1515/9783112729694Search in Google Scholar

Mikhailov EA, Andreasyan RR. 2021. Initial galactic magnetic fields and the Biermann battery mechanism. Astron Rep. 65(9):715–722. 10.1134/S1063772921090055Search in Google Scholar

Mikhailov EA, Andreasyan RR. 2021. Biermann battery as a source of astrophysical magnetic fields. Open Astron. 30(1):127–131. 10.1515/astro-2021-0017Search in Google Scholar

Mikhailov E, Khasaeva T. 2019. Evolution of the magnetic field reversals in galaxies. Bulgarian Astronom J. 31:39–50. Search in Google Scholar

Mikhailov E, Kasparova A, Moss D, Beck R, Sokoloff D. 2014. Magnetic fields near the peripheries of galactic discs. A&A. 568:A66. 10.1051/0004-6361/201323341Search in Google Scholar

Mikhailov EA, Khasaeva TT, Teplyakov IO. 2021. The emergence of contrast structures for galactic magnetic field: theoretical estimates and modeling on GPU. Proc Instit Syst Program RAS. 33:253–264. 10.15514/ISPRAS-2021-33(6)-18Search in Google Scholar

Molchanov SA, Ruzmaikin AA, Sokolov DD. 1985. Kinematic dynamo in random flow. Soviet Phys Uspekhi. 28:307–327. 10.1070/PU1985v028n04ABEH003869Search in Google Scholar

Moss D, Stepanov R, Arshakian TG, Beck R, Krause M, Sokoloff D. 2012. Multiscale magnetic fields in spiral galaxies: evolution and reversals. A&A. 537:A68. 10.1051/0004-6361/201118122Search in Google Scholar

Moss D, Beck R, Sokoloff D, Stepanov R, Krause M, Arshakian TG. 2013. The relation between magnetic and material arms in models for spiral galaxies. A&A. 556:A147. 10.1051/0004-6361/201321296Search in Google Scholar

Moss D, Sokoloff D, Suleimanov V. 2016. Dynamo generated magnetic configurations in accretion discs and the nature of quasi-periodic oscillations in accreting binary systems. A&A. 588:A18. 10.1051/0004-6361/201525944Search in Google Scholar

Moss D. 1995. On the generation of bisymmetric magnetic field structures in spiral galaxies by tidal interactions. MNRAS. 275:191–194. 10.1093/mnras/275.1.191Search in Google Scholar

Obridko VN, Livshits IM, Sokoloff DD. 2017. Two populations of the solar magnetic field. MNRAS. 472(3):2575–2582. 10.1093/mnras/stx2015Search in Google Scholar

Okuzumi S, Takeuchi T, Muto T. 2014. Radial transport of large-scale magnetic fields in accretion disks. I. Steady solutions and an upper limit on the vertical field strength. ApJ. 785(2):127. 10.1088/0004-637X/785/2/127Search in Google Scholar

Parker EN. 1955. Hydromagnetic dynamo models. ApJ. 122:293–314. 10.1086/146087Search in Google Scholar

Parker EN. 1955. The formation of sunspots from the solar toroidal field. ApJ. 121:491–507. 10.1086/146010Search in Google Scholar

Phillips A. 2001. A comparison of the asymptotic and no-z approximations for galactic dynamos. Geophys Astrophys Fluid Dyn. 94(1):135. 10.1080/03091920108204133Search in Google Scholar

Rüdiger G, Shalybkov DA. 2002. The magnetic geometry of magnetic-dominated thin accretion discs. A&A. 393:L81–L84. 10.1051/0004-6361:20021254Search in Google Scholar

Shakura NI, Sunyaev RA. 1973. Black holes in binary systems. Observational appearance. A&A. 24:337–355. 10.1007/978-94-010-2585-0_13Search in Google Scholar

Sokoloff D. 2015. Problems of magnetic dynamo. Phys Uspekhi. 58:601–605. 10.3367/UFNe.0185.201506h.0643Search in Google Scholar

Subramanian K, Mestel L. 1993. Galactic dynamos and density wave theory - II. An alternative treatment for strong non-axisymmetry. MNRAS. 265:649–654. 10.1093/mnras/265.3.649Search in Google Scholar

Van Eck CL, Brown JC, Stil JM, Rae K, Mao SA, Gaensler BM, et al. 2011. Modeling the magnetic field in the galactic disk using new rotation measure observations from the very large array. ApJ. 728:97. 10.1088/0004-637X/728/2/97Search in Google Scholar

Zeldovich IaB, Ruzmaikin AA, Sokolov DD. 1983. Magnetic fields in astrophysics magnetic fields in space. New York: Gordon and Breach Science Publishers. Search in Google Scholar

Received: 2022-12-14

Revised: 2023-01-15

Accepted: 2023-01-18

Published Online: 2023-02-18

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Keywords for this article

magnetic field; accretion discs; dynamo; Kepler rotation

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