Analytical models of X-shape magnetic fields in galactic halos
aa r X i v : . [ a s t r o - ph . GA ] D ec Astronomy&Astrophysicsmanuscript no. paper c (cid:13)
ESO 2018August 13, 2018
Analytical models of X-shape magnetic fields in galactic halos
Katia Ferri`ere and Philippe Terral IRAP, Universit´e de Toulouse, CNRS, 9 avenue du Colonel Roche, BP 44346, F-31028 Toulouse Cedex 4, FranceReceived ; accepted
ABSTRACT
Context.
External spiral galaxies seen edge-on exhibit X-shape magnetic fields in their halos. Whether the halo of our own Galaxyalso hosts an X-shape magnetic field is still an open question.
Aims.
We would like to provide the necessary analytical tools to test the hypothesis of an X-shape magnetic field in the Galactic halo.
Methods.
We propose a general method to derive analytical models of divergence-free magnetic fields whose field lines are assigneda specific shape. We then utilize our method to obtain four particular models of X-shape magnetic fields in galactic halos. In passing,we also derive two particular models of predominantly horizontal magnetic fields in galactic disks. All our field models have spiralingfield lines with spatially varying pitch angle.
Results.
Our four halo field models do indeed lead to X patterns in synthetic synchrotron polarization maps. Their precise topologiescan all be explained by the action of a wind blowing outward from the galactic disk or from the galactic center. In practice, our fieldmodels may be used for fitting purposes or as inputs to various theoretical problems.
Key words.
Galaxies: magnetic fields – galaxies: halos – galaxies: spirals – Galaxy: halo – Galaxy: disk – ISM: magnetic fields
1. Introduction
Low-frequency radio continuum observations provide a pow-erful tool to detect and measure magnetic fields in the disksand halos of external galaxies (see, e.g., the recent review byBeck & Wielebinski 2013). Polarization observations of nearbyedge-on spiral galaxies show that most galactic disks harborlarge-scale magnetic fields that are horizontal, i.e., parallel tothe disk plane (e.g., Wielebinski & Krause 1993; Dumke et al.1995). In the past few years, high-sensitivity polarization ob-servations have also revealed the presence in galactic halosof magnetic fields forming a general X pattern. These so-called X-shape magnetic fields are characterized by a verti-cal (i.e., perpendicular to the disk plane) component that in-creases with both galactic radius and height in the four quadrants(T¨ullmann et al. 2000; Soida 2005; Krause et al. 2006; Krause2009; Heesen et al. 2009; Braun et al. 2010; Soida et al. 2011;Haverkorn & Heesen 2012).At first thought, the denomination of X-shape magneticfields could be reminiscent of magnetic reconnection. However,in contrast to X-type reconnection configurations, which trulyform a complete X, the observed X-shape magnetic fields actu-ally have the central part of the X missing – in that sense, theterm ”X-shape” may be a little misleading. More importantly,X-shape magnetic fields appear on global galactic scales, whichare vastly larger than the resistive scales on which reconnectionpresumably takes place.Since X-shape magnetic fields are seen only at large dis-tances from the galactic center, it is hard to guess what theircomplete magnetic topology is and how their field lines con-nect together, i.e., horizontally near the rotation axis or verti-cally across the disk plane. This, in turn, makes it di ffi cult to pindown the exact nature of X-shape magnetic fields and to traceback to their origin. Various physical scenarios have been pro-posed in the literature, involving either the operation of a clas- Send o ff print requests to : Katia Ferri`ere sical galactic dynamo in the halo (T¨ullmann et al. 2000; Soida2005) or, more likely, the action of a large-scale galactic windfrom the disk (T¨ullmann et al. 2000; Beck 2008; Heesen et al.2009; Soida et al. 2011). Alternatively, it has been suggested thatthe X pattern does not pertain to the large-scale regular magneticfield, but rather to a small-scale anisotropic random field whichwould be associated with extremely elongated magnetic loopsproduced by a spiky wind (Michał Hanasz, private communica-tion). Only rotation measure (RM) studies can distinguish be-tween regular and anisotropic random fields. In the cases ofNGC 5775 (Soida et al. 2011) and NGC 4631 (Mora & Krause2013), the large values and the smooth distributions of the RMsmeasured toward their halos speak in favor of regular X-shapefields, but little information is available for other galaxies. In thepresent study, we will proceed on the premise that the X patternis really an attribute of the large-scale regular field.Now the question that naturally comes to mind is whetherthe halo of our own Galaxy also possesses an X-shape magneticfield. The most widely used description of the Galactic halo fieldrelies on the double-torus picture, originally sketched by Han(2002) and later modeled by Prouza & ˇSm´ıda (2003); Sun et al.(2008); Jansson & Farrar (2012a), where the halo field is purelyazimuthal, forms a torus on each side of the Galactic plane andhas opposite signs above and below the plane. For the first time,Jansson & Farrar (2012a,b) added to this double-torus compo-nent a simple X-shape component, which they chose to be ax-isymmetric and purely poloidal. They found that this additionimproved the overall fit to the RM and synchrotron data com-bined. Throughout this paper, we will follow the usual convention of em-ploying the term ”poloidal” to refer to magnetic fields with only radialand vertical components, i.e., with no azimuthal component, in galac-tocentric cylindrical coordinates. Strictly speaking, this terminology iscorrect only for axisymmetric magnetic fields (e.g., Mo ff att 1978). 1atia Ferri`ere: Models of X-shape magnetic fields In this paper, we will not attempt to settle the question ofwhether the Galactic halo possesses or not an X-shape magneticfield, but we will provide the analytical tools required to seri-ously address this question in view of the existing observations.More specifically, we will propose a general method to deriveanalytical models of divergence-free, X-shape magnetic fieldsthat can be applied to galactic halos. We will then derive fourparticular X-shape models having a small number of free pa-rameters, and we will verify that these models truly produce Xpatterns in synchrotron polarization maps.We emphasize that the method proposed here to obtain ana-lytical field models is very general, in the sense that it can be ap-plied to all sorts of magnetic configurations, with or without anX shape. Our four X-shape models, which represent a particularapplication of our general method to halo magnetic fields, havethemselves broad applicability. First of all, they can be applied toour own Galaxy, with di ff erent purposes in mind. For instance,they can be used to construct synthetic maps of RMs, of syn-chrotron total and polarized intensities or of any other relevantobservables, which can then be confronted to existing observa-tional maps with the aim of placing constraints on the propertiesof a putative X-shape magnetic field in the Galactic halo. In adi ff erent perspective, our X-shape models can serve as a frame-work to study theoretical problems, such as cosmic-ray propaga-tion and large-scale magnetohydrodynamic phenomena, in ourGalaxy. Similarly, they can be applied to external galaxies, bothto construct various synthetic maps which can help to constraintheir true magnetic morphologies and to study the same kind oftheoretical problems as in our own Galaxy.For pedagogical reasons, we will start by developing mod-els of purely poloidal magnetic fields, and since our intendedapplication is to galactic halos, we will restrict our attentionto poloidal fields with an overall X shape (section 2). We willthen extend our poloidal field models to three dimensions, by in-cluding an azimuthal field component that is directly linked tothe poloidal component, as expected from dynamo theory (sec-tion 3). Taking advantage of the analytical expressions obtainedfor magnetic fields in galactic halos, we will, with a little ex-tra work, derive three-dimensional models of magnetic fields ingalactic disks; these disk field models can be utilized either in-dependently or in combination with our halo field models (sec-tion 4). As a necessary check of the relevance of our halo fieldmodels, we will use them to simulate synchrotron polarizationmaps of a hypothetical external galaxy seen edge-on and makesure that these maps display the expected X patterns (section 5).Finally, we will conclude our study with a short summary (sec-tion 6).
2. Analytical models of poloidal, X-shape magneticfields
To model magnetic fields in galactic halos (in particular, in thehalo of our own Galaxy), we will consider various magnetic con-figurations characterized by field lines that form a general X pat-tern. For each considered configuration, we will adopt a simpleanalytical function to describe the shape of field lines, and wewill derive the analytical expression of the corresponding mag-netic field vector, B , as a function of galactocentric cylindricalcoordinates, ( r , ϕ, z ).A convenient way to proceed is to use the Euler potentials, α and β , defined such that B = ∇ α × ∇ β (1) (Northrop 1963; Stern 1966). A first great advantage of theEuler representation is that the magnetic field is automaticallydivergence-free. Another important advantage, particularly rel-evant in the present context, is that field lines can be directlyvisualized. Indeed, according to Eq. (1), any given field line isthe line of intersection between a surface of constant α and asurface of constant β . As an immediate consequence, each fieldline can be defined by, or labeled with, a given pair ( α, β ).In this section, we focus on the particular case when the mag-netic field is poloidal ( B ϕ = β , with galactic azimuthalangle: β = ϕ , (2)whereupon Eq. (1) yields B r = − r ∂α∂ z ! r (3) B ϕ = B z = r ∂α∂ r ! z , (5)where subscripts r and z indicate that the partial derivatives withrespect to z and r are to be taken at constant r and z , respectively.It appears that α represents the (poloidal) magnetic flux per unitazimuthal angle (taken with the appropriate sign) between theconsidered field line ( α, ϕ ) and a reference field line ( α , ϕ ). For a poloidal, X-shape magnetic field, the equation of field linesin a given meridional plane ϕ can generally be written in theform z = F z ( α, r ) (6)or r = F r ( α, z ) · (7)These two classes of field lines are successively studied in sec-tions 2.2.1 and 2.2.2. z as a function of r Each field line can be identified by a value of α or, equivalently,by a value of any monotonic function of α . A convenient suchfunction is z ( α ), where z denotes the height of the consideredfield line at a specified (and fixed) radius r (e.g., r = z , as z = f z ( z , r ) , (8)and, accordingly, the expressions for the radial and vertical fieldcomponents, Eqs. (3) and (5), can be transformed into B r = − r d α dz ∂ z ∂ z ! r = r r B r ( r , z ) ∂ z ∂ z ! r (9) B z = r d α dz ∂ z ∂ r ! z = − r r B r ( r , z ) ∂ z ∂ r ! z · (10)Clearly, Eq. (9) expresses magnetic flux conservation (in theradial direction) between ( r , z ) and ( r , z ), while the ratio ofEq. (10) to Eq. (9), B z B r = dzdr ! z , (11) Table 1.
Brief description of our four models of poloidal, X-shape magnetic fields in galactic halos.
Model Shape of Reference a Label of b Reference c Vertical d B r B z Free parameters ef field lines coordinate field lines field parityA g Eq. (12) – Fig. 1a r > z (Eq. 13) Eq. (16) sym Eq. (14) Eq. (15) r a B H Eq. (21) antisymB Eq. (17) – Fig. 1b r > z (Eq. 18) Eq. (16) sym Eq. (19) Eq. (20) r n B H Eq. (21) antisymC Eq. (26) – Fig. 1c z = r (Eq. 27) Eq. (30) antisym Eq. (28) Eq. (29) a B L D Eq. (31) – Fig. 1d z = | z | sign z r (Eq. 32) Eq. (30) antisym Eq. (33) Eq. (34) | z | n B L Eq. (35) sym a Prescribed radius or height at which field lines are labeled by their height or radius, respectively. b Height or radius of field lines at the reference coordinate. c Radial or vertical field at the reference coordinate as a function of field line label. d For each model, the first line refers to the original parity in section 2.2, and, when present, the second line refers to the reversed parity. e In general, the free parameters can vary with azimuthal angle, ϕ . However, in the particular case considered in section 2.4, only B dependson ϕ (through Eq. 36). f r is the reference radius in models A and B and | z | the modulus of the reference height in models C and D; a is the parameter governingthe opening of quadratic field lines in models A and C ( a > n is minus the power-law index of field lines asymptotic to the z -axis in model B( n ≥
1) or the r -axis in model D ( n ≥ ); B is the peak value of the reference field, taken as the normalization field ( B ≷ H is theexponential scale height (in models A and B) and L the exponential scale length (in models C and D) of the reference field ( H > L > g Model A can only be used in non-axisymmetric magnetic configurations, with the addition of an azimuthal field component near the z -axis,to prevent the magnetic field from becoming singular at r = simply states that the magnetic field is tangent to field lines. Model A.
To describe field lines with an X shape, we choosea function z = f z ( z , r ) such that | z | increases as a positive powerof r for r → ∞ . We discard the simple linear function of r , whichleads to a cusp at r =
0, and consider the quadratic function z = z + a r + a r , (12)where r and a are two strictly positive free parameters (seeFigure 1a). Let us emphasize that the values of r and a areunique and common to all field lines, in contrast to z , whichhas a di ff erent value on each field line. By inverting Eq. (12) toobtain z as a function of ( r , z ): z = z + a r + a r , (13)calculating the partial derivatives of z with respect to z and r ,and introducing these derivatives into Eqs. (9) and (10), we findfor the radial and vertical field components: B r = r r z z B r ( r , z ) (14) B z = a r z + a r B r ( r , z ) · (15)To complete the model, it remains to specify the z -dependenceof B r ( r , z ), which, for simplicity, we choose to be linear-exponential: B r ( r , z ) = B | z | H exp − | z | − HH ! , (16)with B ≷ H > | z | ,while the linear factor guarantees that the X-shape halo field remains weaker than the horizontal disk field (modeled in sec-tion 4) at low | z | . The peak value of B r ( r , z ), reached at | z | = H ,is [ B r ( r , z )] peak = B .It directly follows from Eq. (14) that B r → ∞ for r → z -axis. Nevertheless, in non-axisymmetric configurations (dis-cussed in section 2.4), a slight modification of the magnetictopology near the z -axis, involving the addition of a local az-imuthal field component, may be su ffi cient to remove the singu-larity. Model B.
An alternative way of avoiding the singularity in-herent in the parabolic form of Eq. (12) is to prevent field linesfrom reaching the z -axis. This can be done by replacing the con-stant term in the right-hand side of Eq. (12) by a negative powerof r . The second term can remain quadratic, but it can also bereplaced by a linear term without a cusp appearing at r = r , z ). Altogether, we write the equation of field lines as z = n + z " rr ! − n + n rr , (17)with r and n two strictly positive free parameters (seeFigure 1b). As before, we invert this equation in favor of z : z = ( n + z " rr ! − n + n rr − , (18)and we insert the partial derivatives of z into Eqs. (9) and (10),to obtain: B r = r r z z B r ( r , z ) (19) B z = − nn + r z r z " rr ! − n − rr B r ( r , z ) · (20) (a) Model A (b) Model B (c) Model C (d) Model D Fig. 1.
Small set of field lines for each of our four models of poloidal, X-shape magnetic fields in galactic halos, in a vertical planethrough the galactic center: (a) model A (described by Eq. 12) with r = a = / (10 kpc) ; (b) model B (Eq. 17) with r = n =
2; (c) model C (Eq. 26) with a = / (10 kpc) (and, as always, z = | z | = . n =
2. Field lines are separated by a fixed magnetic flux per unit azimuthal angle; thus, in models A and B, theirvertical spacing at r is determined by Eq. (16) or (21) with H = . z isdetermined by Eq. (30) or (35) with L =
10 kpc. Each panel is 30 kpc ×
30 kpc in size and is centered on the galactic center. Thetrace of the galactic plane is indicated by the horizontal, red, solid line, and the z -axis by the vertical, black, dot-dashed line.The z -dependence of B r ( r , z ) is again chosen to followEq. (16). Finally, the requirement that B r and B z remain finitein the limit r →
0, where B r ∝ r n − B r ( r , z ) ∝ r n − and B z ∝ r n − B r ( r , z ) ∝ r n − , leads to the constraint n ≥ B r and B z are, respectively, even and odd functions of z .The magnetic field is, therefore, symmetric with respect to thegalactic midplane. In principle, it is possible to obtain antisym-metric magnetic fields with the same families of field lines, bychoosing for B r ( r , z ) an odd function of z – for instance, byreplacing the linear factor in the right-hand side of Eq. (16) by Magnetic fields that are symmetric / antisymmetric with respect tothe midplane are sometimes also referred to as quadrupolar / dipolar. ( z / H ): B r ( r , z ) = B z H exp − | z | − HH ! · (21) r as a function of z Here, each field line is identified by a value of the function r ( α ),where r denotes the radius of the considered field line at a spec-ified (and fixed) height z (e.g., z = z with opposite signs and select the pos-itive / negative value of z for field lines lying above / below themidplane. The equation of field lines, Eq. (7), can be rewrittenin terms of r , as r = f r ( r , z ) , (22) and the expressions for the radial and vertical field components,Eqs. (3) and (5), can be transformed into B r = − r d α dr ∂ r ∂ z ! r = − r r B z ( r , z ) ∂ r ∂ z ! r (23) B z = r d α dr ∂ r ∂ r ! z = r r B z ( r , z ) ∂ r ∂ r ! z · (24)It is now Eq. (24) which expresses magnetic flux conservation(in the vertical direction) between ( r , z ) and ( r , z ), while theratio of Eq. (23) to Eq. (24), B r B z = drdz ! r , (25)corresponds again to the standard definition of field lines. Model C.
For the same reasons as in section 2.2.1, our firstchoice to describe X-shape field lines is the quadratic function r = r (1 + a z ) , (26)obtained for z = a > r = r + a z , (27)combines with Eqs. (23) and (24) to yield B r = a r zr B z ( r , z ) (28) B z = r r B z ( r , z ) · (29)For the r -dependence of B z ( r , z ), we assume a simple expo-nential: B z ( r , z ) = B exp (cid:18) − r L (cid:19) , (30)with B ≷ L > z -axis, thefield lines governed by Eq. (26) only cross the r -axis of their ownmeridional plane. Model D.
For completeness, we also examine the counter-part of Eq. (17), r = n + r " zz ! − n + n zz , (31)with | z | and n two strictly positive free parameters and z re-quired to take on the same sign as z , i.e., z = | z | sign z , suchthat the ratio ( z / z ) is always positive (see Figure 1d). With theinverse equation, r = ( n + r " zz ! − n + n zz − , (32)Eqs. (23) and (24) reduce to B r = − nn + r r z " zz ! − n − zz B z ( r , z ) (33) B z = r r B z ( r , z ) · (34) The r -dependence of B z ( r , z ) is again assumed to followEq. (30), while the exponent n must now satisfy the constraint n ≥ , as can be seen from the behavior B r ∝ z n − B z ( r , z ) ∝ z n − and B z ∝ z n B z ( r , z ) ∝ z n in the limit z → B r and B z are, respectively, odd and even functions of z ,so that the magnetic field is anti-symmetric with respect to themidplane (see footnote 2). Model D can also be made symmetric,by linking the sign of B z ( r , z ) to the sign of z – for instance,by multiplying the right-hand side of Eq. (30) by sign z : B z ( r , z ) = B sign z exp (cid:18) − r L (cid:19) · (35)On the other hand, model C cannot be made symmetric, becausefield lines actually cross the midplane, so that B z cannot changesign across the midplane. In section 2.2, we presented four di ff erent analytical models ofpoloidal, X-shape magnetic fields for application to galactic ha-los. Our models in their original form are either symmetric orantisymmetric with respect to the midplane, but their parity cangenerally be reversed, simply by flipping the parity of the func-tion chosen for B r ( r , z ) or B z ( r , z ). Thus, models A and B areoriginally symmetric, with B r ( r , z ) given by an even functionof z (Eq. 16), but they also have an antisymmetric version, ob-tained by turning B r ( r , z ) into an odd function of z (Eq. 21).Similarly, models C and D are originally antisymmetric, with B z ( r , z ) independent of the sign of z (Eq. 30), but model Dalso has a symmetric version, obtained by linking the sign of B z ( r , z ) to the sign of z (Eq. 35). For convenience, the de-scriptive equations and parameters of the four models, in theiroriginal and modified versions, are summarized in Table 1.Let us now discuss the possible physical origin of X-shapemagnetic fields, in connection with their vertical parity. Thenatural tendency of a conventional galactic dynamo is to gen-erate a symmetric, mainly horizontal, field in the disk andan antisymmetric, possibly dipole-like, field in the halo (e.g.,Sokolo ff & Shukurov 1990). In general, though, the disk andhalo fields are coupled together and do not evolve independently.What usually happens is that one field enslaves and imposes itsnatural parity to the other, so that the total (disk + halo) field endsup being either symmetric (if the disk dynamo dominates) or an-tisymmetric (if the halo dynamo dominates) (Moss & Sokolo ff a mere extension of the symmetric thick-disk field, plus an an-tisymmetric (almost purely poloidal dipole) component, whichwould become dominant in the upper halo (Braun et al. 2010).Regardless of vertical parity, conventional dynamo fieldsdo not naturally come with an X shape, but they can poten-tially acquire one under the action of a galactic wind (e.g.,Brandenburg et al. 1993; Moss et al. 2010). Quite naturally,symmetric X-shape fields would be related to the disk dynamo,while antisymmetric X-shape fields would be related to the halodynamo. More specifically, the former could be explained witha wind originating near the galactic plane and advecting the diskdynamo field into the halo, while the latter could be explainedwith a wind blowing from the base of the halo and stretching outthe halo dynamo field into an X shape.From a morphological point of view, in models A, B and D,where low- | z | field lines tend to follow the galactic plane, the X-shape field could conceivably be traced back to an initially hori-zontal disk field, assuming the wind has the following properties:in model A, the wind would blow obliquely outward from thedisk (possibly due to pressure gradients; see the hydrodynami-cal simulations of Dalla Vecchia & Schaye (2008)); in model B,the wind would have a weak oblique component emanating fromthe disk plus a strong bipolar-jet component emanating from thegalactic center (most likely driven by an AGN or by a nuclearstarburst); and in model D, the wind would take the form of achampagne flow originating in the central region (probably dueto strong stellar activity). In these three models, the X-shape fieldcould a priori be either symmetric or antisymmetric, but if it isindeed connected to the disk field and if the latter has evolved toand retained its natural parity (two non-trivial conditions), the X-shape field should be symmetric. In contrast, in model C, wherefield lines cross the galactic plane, the X-shape field is easily rec-onciled with an initially dipole-like halo field, which was blownopen either along the plane by an oblique wind from the disk oralong the z -axis by a champagne flow from the central region. Inthis model, the X-shape field would obvioulsy keep the antisym-metric character of the initial dipole-like field.Somewhat di ff erent conclusions on the vertical parity of thehalo field emerged from the galactic dynamo calculations ofMoss et al. (2010). They found that halo fields that are antisym-metric tend to come with an X shape – a tendency which is re-inforced by the presence of a modest wind, whereas halo fieldsthat have become symmetric, presumably under the influence ofthe disk, tend to lose all resemblance to an X pattern. The expressions derived in section 2.2 and discussed in sec-tion 2.3 are valid in any given meridional plane. They can begeneralized to full three-dimensional space by letting the freeparameters (listed in the last column of Table 1) be functions ofthe azimuthal angle, ϕ .Here, we consider the very simple case when the only freeparameter allowed to vary with ϕ is the normalization field, B ,defined as B = B r ( r , H ) in models A and B (see Eqs. (16)and (21)) and B = B z (0 , | z | ) in models C and D (see Eqs. (30)and (35)). We further assume that the variation of B with ϕ issinusoidal: B ( ϕ ) = B ⋆ cos (cid:0) m ( ϕ − ϕ ⋆ ) (cid:1) , (36)with m the azimuthal wavenumber (taken to be positive) and B ⋆ and ϕ ⋆ two free parameters (allowed to take on either sign).Clearly, m = m = m = r = R and semi-infinite length, centered on the z -axis and extending from z = z → + ∞ . The magnetic flux across the surface of this cylinderreads Φ m = R Z π d ϕ Z + ∞ dz B r ( R , ϕ, z ) − Z R dr r Z π d ϕ B z ( r , ϕ, + lim Z → + ∞ Z R dr r Z π d ϕ B z ( r , ϕ, Z ) , (37)where the three terms in the right-hand side represent, respec-tively, the flux through the side surface at r = R , the flux throughthe bottom surface at z = z → + ∞ . These terms can be calculated using Eqs. (14)and (15), in conjunction with Eq. (13) and Eq. (16) or (21). Thesecond and third terms are easily shown to vanish, leaving only Φ m = (exp 1) H r Z π d ϕ B ( ϕ ) · (38)Now, the divergence-free condition implies that Φ m must vanish,which, in turn, implies Z π d ϕ B ( ϕ ) = · (39)For the sinusoidally-varying field given by Eq. (36), this is pos-sible only if the azimuthal wavenumber m , m =
0) magnetic field inmodel A is intrinsically unphysical, in the sense that it automat-ically leads to a non-vanishing magnetic flux out of, or into, theconsidered cylinder – in contradiction with the divergence-freecondition. This occurs because, in the axisymmetric case, themagnetic field on the side surface of the cylinder points every-where outward or inward. It then follows that the magnetic fieldmust necessarily become singular along the axis of the cylinder.In contrast, a non-axisymmetric ( m ,
0) magnetic field hasno net magnetic flux across the side surface of the cylinder, be-cause the magnetic field points alternatively outward and inward.The singularity found at r = r →
0, so that field lines have no other choice but torun into each other at r =
0. This unacceptable situation couldbe avoided by simply introducing an azimuthal field componentwithin a small radius, which would enable field lines to remainseparate from each other, except for connecting two by two toensure field line continuity. In this manner, the magnetic fieldwould remain finite everywhere.Hence, model A can be retained to describe non-axisymmetric magnetic fields outside a small radius. Inside thisradius, an azimuthal component must be added to the poloidalfield so as to remove the singularity at r =
0. Note that, inpractice, it may not be necessary to specify the exact form ofthe azimuthal component; knowing that the model can be madephysical may be su ffi cient.Model B does not become singular at r =
0, even in the ax-isymmetric case, because all the magnetic flux entering / leavingthrough the side surface of the cylinder finds its way out / in through the top surface (see Figure 1b). This can be verifiedmathematically by examining the three terms in the right-handside of Eq. (37). The first term (flux through the side surface) hasthe same expression as in model A, because in both models z is a linear function of z (see Eq. (13) for model A and Eq. (18)for model B). The second term (flux through the bottom surface)vanishes, because, like in model A, B z = z = ff erence betweenthe two models comes from the third term (flux through the topsurface), which vanishes in model A, but is equal and oppositeto the first term (flux through the side surface) in model B.
3. Analytical models of three-dimensional magneticfields in galactic halos
In this section, we let the magnetic field have both poloidal andazimuthal components. Moreover, based on observations of ex-ternal face-on spiral galaxies, we assume that the magnetic fieldprojected onto the galactic plane forms a spiral pattern, in whicheach spiral line satisfies an equation of the type ϕ = ϕ + f ϕ ( r ) , (40)where ϕ is the azimuthal angle at which the line passes througha chosen (and fixed) radius r , and f ϕ ( r ) is a monotonic functionof r which vanishes at r . The pitch angle of the spiral, p , definedas the angle between the tangent to the spiral and the azimuthaldirection, is given bycot p = r d ϕ dr = r d f ϕ dr · (41)In most studies of galactic magnetic fields, p is supposed to beconstant, which implies a logarithmic spiral, with f ϕ ( r ) = cot p ln rr · (42)However, the assumption of constant pitch angle is generally notrealistic, either in external galaxies or in our own Galaxy (e.g.,Fletcher 2010). This unrealistic assumption is not made in ourformalism, which naturally allows the pitch angle to vary withgalactic radius.When the magnetic field has an azimuthal component ( B ϕ , ϕ is no longer constant along field lines and, therefore, mayno longer serve as one of the Euler potentials. Instead, one canuse ϕ and, accordingly, replace Eq. (2) by β = ϕ = ϕ − f ϕ ( r ) · (43)With this choice of β , Eq. (1) gives for the radial, azimuthal andvertical field components B r = − r ∂α∂ z ! r ,ϕ B ϕ = − d f ϕ dr ∂α∂ z ! r ,ϕ B z = r ∂α∂ r ! ϕ, z + r d f ϕ dr ∂α∂ϕ ! r , z or, equivalently, B r = − r ∂α∂ z ! r ,ϕ (44) B ϕ = r d f ϕ dr B r = cot p B r (45) B z = r ∂α∂ r ! ϕ , z · (46)Eqs. (44) and (46) are the direct counterparts of Eqs. (3) and (5)in section 2; the only di ff erence between both pairs of equationsis that the partial derivatives of α in Eqs. (3) and (5) are taken atconstant ϕ , i.e., in meridional planes, whereas those in Eqs. (44)and (46) are taken at constant ϕ , i.e., along the spiral surfacesdefined by Eq. (40). Eq. (45), for its part, states that the hori-zontal magnetic field has pitch angle p and, therefore, followsthe spiral lines defined in the galactic plane by Eq. (40). Let usemphasize again that p is not required to remain constant alongfield lines. The similarity between the pair of Eqs. (3) – (5) and the pair ofEqs. (44) – (46) indicates that the derivation presented in sec-tion 2.2 can be repeated here, with this only di ff erence that theworking space is now a spiral surface of constant ϕ rather thana meridional plane of constant ϕ . Likewise, the azimuthal varia-tion introduced in section 2.4 can be taken up here, with Eq. (36)written as a function of ϕ instead of ϕ . For future reference, werecall that ϕ = ϕ − f ϕ ( r ) (47)(see Eq. 40).In the same spirit as in section 2.2, we distinguish betweentwo classes of poloidal field lines, which we successively studyin sections 3.2.1 and 3.2.2. Note that we only need to worryabout the radial and vertical field components, as the azimuthalfield, given by Eq. (45), can easily be obtained from the radialfield once the shape of the spiral has been specified. z as a function of r As in section 2.2.1, we start by adopting a reference radius r .Each field line can then be identified by the coordinates ( ϕ , z )at which it intersects the cylinder of radius r . Since the refer-ence radius, r , and the normalization radius of the spiral, r ,both have unique values (common to all field lines), we may, forconvenience, let r = r , and hence ϕ = ϕ .Following the procedure of section 2.2.1, on a surface of con-stant ϕ rather than constant ϕ , we transform Eqs. (44) and (46)into B r = r r B r ( r , ϕ , z ) ∂ z ∂ z ! r (48) B z = − r r B r ( r , ϕ , z ) ∂ z ∂ r ! z , (49)with r the reference radius, z set by our adopted X shape(Eq. (13) in model A and Eq. (18) in model B), and ϕ set byour adopted spiral (Eq. 47). A comparison with Eqs. (9) – (10)shows that the expressions of B r and B z obtained in section 2.2.1remain valid here provided only that B r ( r , z ) be replaced by B r ( r , ϕ , z ). Thus, in model A , Eqs. (14) – (15) become B r = r r z z B r ( r , ϕ , z ) (50) B z = a r z + a r B r ( r , ϕ , z ) , (51)and in model B , Eqs. (19) – (20) become B r = r r z z B r ( r , ϕ , z ) (52) B z = − nn + r z r z " rr ! − n − rr B r ( r , ϕ , z ) · (53)We assume that B r ( r , ϕ , z ) has the same vertical variation as insection 2.2.1 (see Eqs. (16) and (21)) and that its azimuthal vari-ation is governed by Eq. (36) written as a function of ϕ insteadof ϕ (see first paragraph of section 3.2). The result is B r ( r , ϕ , z ) = B r ( r , z ) cos (cid:0) m ( ϕ − ϕ ⋆ ) (cid:1) , (54)with B r ( r , z ) given by Eq. (16) for symmetric magnetic fieldsand by Eq. (21) for antisymmetric magnetic fields, m the az-imuthal wavenumber and ϕ ⋆ a free parameter. r as a function of z Here, we adopt a reference height z – or two opposite-signedvalues of z if field lines do not cross the midplane. Each fieldline can then be identified by the coordinates ( r , ϕ ) at whichit intersects the horizontal plane – or one of the two horizon-tal planes – of height z . Since the value of r now dependson the considered field line, we may no longer equate r to r , which must remain a separate free parameter. According toEq. (40), the azimuthal angles associated with r and r are re-lated through ϕ = ϕ + f ϕ ( r ).In the same fashion as in section 2.2.2, we transformEqs. (44) and (46) into B r = − r r B z ( r , ϕ , z ) ∂ r ∂ z ! r (55) B z = r r B z ( r , ϕ , z ) ∂ r ∂ r ! z , (56)with z the reference height, r set by our adopted X shape(Eq. (27) in model C and Eq. (32) in model D), and ϕ set byour adopted spiral (Eq. 47). A comparison with Eqs. (23) – (24)shows that the results obtained in section 2.2.2 remain valid with B z ( r , z ) replaced by B z ( r , ϕ , z ).Thus, in model C , Eqs. (28) – (29) become B r = a r zr B z ( r , ϕ , z ) (57) B z = r r B z ( r , ϕ , z ) , (58)and in model D , Eqs. (33) – (34) become B r = − nn + r r z " zz ! − n − zz B z ( r , ϕ , z ) (59) B z = r r B z ( r , ϕ , z ) · (60)If we assume that B z ( r , ϕ , z ) has the same radial variation asin section 2.2.2 (see Eqs. (30) and (35)) and that its azimuthal variation is governed by Eq. (36) written as a function of ϕ , weobtain B z ( r , ϕ , z ) = B z ( r , z ) cos (cid:0) m ( ϕ − ϕ ⋆ ) (cid:1) , (61)with B z ( r , z ) given by Eq. (30) for antisymmetric magneticfields and by Eq. (35) for symmetric magnetic fields, m the az-imuthal wavenumber and ϕ ⋆ a free parameter. z -dependent pitchangles A remaining limitation of our formalism is that the magneticpitch angle depends only on galactic radius. Yet, because galac-tic di ff erential rotation probably decreases with distance fromthe midplane (e.g., Levine et al. 2008; Marasco & Fraternali2011; Jałocha et al. 2011, and references therein), we expectfield lines to become less tightly wound up, and hence to havea larger pitch angle, at higher altitudes. Note that this is just aqualitative argument, and one should not try to directly link thevertical variation of the pitch angle to that of the rotational ve-locity. For instance, while a vertical decrease in rotational veloc-ity does indeed reduce the generation of azimuthal field throughthe shearing of radial field, the vertical velocity gradient itselfgenerates azimuthal field through the shearing of vertical field.Furthermore, the pitch angle depends on other factors, such asthe intensity of helical turbulence and the scale height of the in-terstellar gas (e.g., Fletcher 2010).Mathematically, we can allow the pitch angle to vary withboth r and z by letting each field line obey an equation similar toEq. (40), with f ϕ ( r ) replaced by f ϕ ( r , z ): ϕ = ϕ + f ϕ ( r , z ) , (62)where ϕ is the azimuthal angle at which the field line passesthrough either the reference radius, r (in models A and B), orthe reference height, z (in models C and D). In both cases, onemust have f ϕ ( r , z ) = The pitch angle, p ( r , z ), associatedwith a given winding function, f ϕ ( r , z ), is easily obtained by dif-ferentiation: cot p = r d ϕ dr = r ∂ f ϕ ∂ r + ∂ f ϕ ∂ z dzdr ! , (63)where the factor ( dz / dr ) = ( B z / B r ) depends on the chosen modelfor the poloidal field. Conversely, the winding function associ-ated with a given pitch angle is obtained by integrating Eq. (63)along the poloidal field line through ( r , z ): f ϕ ( r , z ) = Z rr cot p ( r ′ , z ′ ) dr ′ r ′ , (64)where r is either the reference radius (in models A and B) or theradial label of the field line (in models C and D; see Eqs. (27) and(32), respectively); z is either the vertical label of the field line(in models A and B; see Eqs. (13) and (18), respectively) or thereference height (in models C and D); and z ′ is the height of thefield line at radius r ′ (given by the primed versions of Eqs. (12),(17), (26) and (31) for models A, B, C and D, respectively). A We may no longer take the surface r = r to be the normalizationsurface on which f ϕ vanishes, since not all field lines cross r in mod-els C and D. This was not an issue in section 3.1, where we considered,not individual field lines, but the spiral lines defined by the magneticfield projection onto the galactic plane. In model A, z ′ = z + a r ′ + a r ;8atia Ferri`ere: Models of X-shape magnetic fields Table 2.
Brief description of our four models of three-dimensional, X-shape + spiral magnetic fields in galactic halos. Model a Reference b Labels of c Reference d B r B ϕ B z Free parameters ef coordinate field lines field and functionsA g r > z (Eq. 13) , ϕ (Eq. 70) Eq. (77) Eq. (73) Eq. (68) Eq. (74) p ( r , z ) m ϕ ⋆ ( z )B r > z (Eq. 18) , ϕ (Eq. 70) Eq. (77) Eq. (75) Eq. (68) Eq. (76) p ( r , z ) m ϕ ⋆ ( z )C z = r (Eq. 27) , ϕ (Eq. 70) Eq. (84) Eq. (80) Eq. (68) Eq. (81) p ( r , z ) m ϕ ⋆ ( r )D z = | z | sign z r (Eq. 32) , ϕ (Eq. 70) Eq. (84) Eq. (82) Eq. (68) Eq. (83) p ( r , z ) m ϕ ⋆ ( r ) a The shape of field lines associated with the poloidal field is indicated in Table 1 (second column), while the spiraling of field lines isdescribed by Eq. (62) or, more conveniently, by the pitch angle, p ( r , z ). b Prescribed radius or height at which field lines are labeled by their height or radius, respectively, and by their azimuthal angle. c Height or radius of field lines and their azimuthal angle at the reference coordinate. d Radial or vertical field at the reference coordinate as a function of field line labels. e The free parameters of the poloidal field are listed in Table 1 (last column). Here, we only list the additional free parameters and functionsdescribing the azimuthal structure. f p ( r , z ) is the pitch angle of the magnetic field, needed to infer the azimuthal field from the radial field (see Eq. 68); m is the azimuthalwavenumber and ϕ ⋆ ( z ) or ϕ ⋆ ( r ) the fiducial angle of the sinusoidal azimuthal modulation at r or z , respectively (see Eqs. (77) and (84)).Note that ϕ ⋆ becomes irrelevant in the axisymmetric case ( m = g Model A can only be used in non-axisymmetric magnetic configurations, with a modification of the azimuthal field component near the z -axis, to avoid a singularity at r = possible simple choice for the pitch angle, which accounts forthe decrease in di ff erential rotation with height, is p ( r , z ) = p ( z ) = p ∞ + ( p − p ∞ ) + | z | H p ! − , (65)with p the pitch angle at midplane, p ∞ the pitch angle at infinityand H p the scale height.The three components of the magnetic field can now be de-rived in the same manner as in section 3.1. With β = ϕ = ϕ − f ϕ ( r , z ) (66)chosen for the second Euler potential, Eq. (1) yields B r = − r ∂α∂ z ! r ,ϕ − r ∂ f ϕ ∂ z ∂α∂ϕ ! r , z B ϕ = − ∂ f ϕ ∂ r ∂α∂ z ! r ,ϕ + ∂ f ϕ ∂ z ∂α∂ r ! ϕ, z B z = r ∂α∂ r ! ϕ, z + r ∂ f ϕ ∂ r ∂α∂ϕ ! r , z in model B, z ′ = z r ′ r ! − n + n r ′ r rr ! − n + n rr ;in model C, z ′ = sign z r a (cid:20) r ′ r (1 + a z ) − (cid:21) ;and in model D, z ′ is the solution of " z ′ z ! − n + n z ′ z = r ′ r " zz ! − n + n zz · or, equivalently, B r = − r ∂α∂ z ! r ,ϕ (67) B ϕ = r ∂ f ϕ ∂ r B r + ∂ f ϕ ∂ z B z ! = cot p B r (68) B z = r ∂α∂ r ! ϕ , z , (69)where Eq. (63) was used to write the second equality in Eq. (68).The expressions of B r and B z are the same as those obtained insection 3.1 (Eqs. (44) and (46)), with ϕ substituting for ϕ . Thefinal expression of B ϕ is also the same as in section 3.1 (Eq. 45),which is a direct consequence of the definition of the pitch angle.On the other hand, the intermediate expression of B ϕ is moregeneral, containing not only a term ∝ B r (as in Eq. 45), but alsoa term ∝ B z . This new term reflects the explicit z -dependenceof the azimuthal winding of field lines, as described by f ϕ (seeEq. 62). The first equality in Eq. (68) can also be viewed as themathematical statement that the magnetic field is tangent to thesurfaces defined by Eq. (62).While Eq. (68) provides a general expression of B ϕ , commonto the four di ff erent models, Eqs. (67) and (69) need to be specif-ically worked out for each model. The resulting expressions of B r and B z can be directly taken from sections 3.2.1 and 3.2.2,with ϕ replaced by ϕ , keeping in mind that ϕ = ϕ − f ϕ ( r , z ) (70)(see Eq. 62). The final equations are shown below, a summaryis presented in Table 2 and the four models are illustrated inFigure 2. z as a function of r As a reminder, r is the reference radius, z the vertical label offield lines (dependent on the chosen X-shape model) and ϕ theirazimuthal label (always set by Eq. (70)). For a general function (a) Model A (b) Model B (c) Model C (d) Model D Fig. 2.
Small set of field lines for each of our four models of three-dimensional, X-shape + spiral magnetic fields in galactic halos,as seen from an oblique angle. The parameters of the poloidal field take on the same values as in Figure 1, while the pitch angleis assigned a constant value of p = − ◦ , plausibly representative of galactic halos. The selected field lines correspond to thosedisplayed in Figure 1 (i.e., they have the same labels ( z , ϕ ) or ( r , ϕ )), except in model D, where only four field lines are drawnfor clarity. Each box is a (30 kpc) cube centered on the galactic center. The trace of the galactic plane is indicated by the red, solidcircle, and the z -axis by the vertical, black, dot-dashed line. z ( r , z ), B r = r r B r ( r , ϕ , z ) ∂ z ∂ z ! r (71) B z = − r r B r ( r , ϕ , z ) ∂ z ∂ r ! z · (72)In model A , where z is given by Eq. (13), B r = r r z z B r ( r , ϕ , z ) (73) B z = a r z + a r B r ( r , ϕ , z ) , (74)and in model B , where z is given by Eq. (18), B r = r r z z B r ( r , ϕ , z ) (75) B z = − nn + r z r z " rr ! − n − rr B r ( r , ϕ , z ) · (76) In the above equations, B r ( r , ϕ , z ) = B r ( r , z ) cos (cid:16) m (cid:0) ϕ − ϕ ⋆ ( z ) (cid:1)(cid:17) , (77)with B r ( r , z ) taken from Eq. (16) for symmetric magnetic fieldsand from Eq. (21) for antisymmetric magnetic fields, m the az-imuthal wavenumber and ϕ ⋆ ( z ) a smoothly varying function of z . When ϕ ⋆ ( z ) reduces to a constant, one recovers exactly theresults of section 3.2.1, where ϕ = ϕ . r as a function of z Here, z is the reference height, r the model-dependent radiallabel of field lines and ϕ their azimuthal label set by Eq. (70). For a general function r ( r , z ), B r = − r r B z ( r , ϕ , z ) ∂ r ∂ z ! r (78) B z = r r B z ( r , ϕ , z ) ∂ r ∂ r ! z · (79)In model C , where r is given by Eq. (27), B r = a r zr B z ( r , ϕ , z ) (80) B z = r r B z ( r , ϕ , z ) , (81)and in model D , where r is given by Eq. (32), B r = − nn + r r z " zz ! − n − zz B z ( r , ϕ , z ) (82) B z = r r B z ( r , ϕ , z ) · (83)In the above equations, B z ( r , ϕ , z ) = B z ( r , z ) cos (cid:16) m (cid:0) ϕ − ϕ ⋆ ( r ) (cid:1)(cid:17) , (84)with B z ( r , z ) taken from Eq. (30) for antisymmetric magneticfields and from Eq. (35) for symmetric magnetic fields, m the az-imuthal wavenumber and ϕ ⋆ ( r ) a smoothly varying function of r . To recover the results of section 3.2.2, where ϕ = ϕ + f ϕ ( r ),it su ffi ces to identify ϕ ⋆ ( r ) with f ϕ ( r ) to within a constant.
4. Analytical models of three-dimensional magneticfields in galactic disks
Magnetic fields in most galactic disks are approximately hor-izontal and follow a spiral pattern (e.g., Wielebinski & Krause1993; Dumke et al. 1995; Beck 2009; Braun et al. 2010). In sec-tion 3, we already derived the general expression of a magneticfield having a spiral horizontal component (Eqs. (67) – (69)), andwe applied this general expression to our four models of mag-netic fields with an X-shape poloidal component (sections 3.3.1and 3.3.2). We now use the results of section 3 to discuss andmodel magnetic fields in galactic disks.
In most studies of magnetic fields in galactic disks (in particular,in the disk of our own Galaxy), the field is assumed to be purelyhorizontal ( B z = r -dependence of B r , which are generallyoverlooked. These implications, which directly emerge from ourformalism, are discussed below.When B z =
0, Eq. (69) implies that α ( r , ϕ , z ) reduces to afunction of ϕ and z only, with the dependence on r dropped out.It then follows from Eq. (67) that B r can be written as B r = r fc( ϕ , z ) , (85)or, in the case of a sinusoidal variation with ϕ , B r = r fc( z ) cos (cid:16) m (cid:0) ϕ − ϕ ⋆ ( z ) (cid:1)(cid:17) , (86) where ϕ (given by Eq. 70) is the azimuthal angle at which thefield line passing through ( r , ϕ, z ) crosses a reference radius r , m is the azimuthal wavenumber, and fc( z ) and ϕ ⋆ ( z ) are two func-tion of z . The key point here is that, unless fc( z ) = B r ∝ (1 / r ) → ∞ for r →
0, regard-less of the azimuthal symmetry, i.e., in axisymmetric ( m = m =
1) and higher-order ( m ≥
2) con-figurations. The physical reason for this singularity is similar tothat put forward in model A (see below Eq. 16), namely, all fieldlines (with all values of ϕ ) in a given horizontal plane convergeto, or diverge from, the z -axis.Many existing studies completely ignore this (1 / r ) depen-dence of B r (e.g., Han & Qiao 1994; Sun et al. 2008). Otherscorrectly include the (1 / r ) factor in the expression of B r , buteither they choose to turn a blind eye to the singular behaviorof B r at r = / r ) by a constant or a regular func-tion of r (e.g., Stanev 1997; Harari et al. 1999; Jansson et al.2009; Pshirkov et al. 2011) – which leads to a violation of thedivergence-free condition.Like in model A, there are basically two ways of remov-ing the singularity at r =
0, while keeping the magnetic fielddivergence-free. The first way is to let the horizontal field atsmall radii deviate from a pure spiral (defined here by Eq. 62),in such a way that field lines connect two by two away fromthe z -axis and remain separate from all other field lines. This re-configuration is possible only for non-axisymmetric fields, be-cause connecting field lines must necessarily have opposite signsof B r for the field direction to remain continuous at their connec-tion point.The alternative approach is to allow for a departure from astrictly horizontal magnetic field. With an adequate vertical com-ponent, field lines can bend away from the midplane as they ap-proach the z -axis and continue into the halo without ever reach-ing the z -axis. The advantage of this approach is that it can be im-plemented in both axisymmetric and non-axisymmetric configu-rations. Two illustrative examples are provided in section 4.2. Two of the four models developed in sections 2 and 3 for mag-netic fields in galactic halos (models B and D) have specificproperties which should make them easily adapted to galacticdisks: unlike model A, they are regular at r =
0; unlike model C,they can be symmetric with respect to the midplane; and theyboth have nearly horizontal field lines at low | z | . In this section,we present disk versions of these two models, which we refer toas models Bd and Dd, respectively. The descriptive equations ofthese two models are listed in Table 3, and a few representativefield lines are plotted in Figure 3.The notation used for disk fields is the same as that usedfor halo fields, namely, ( r , ϕ , z ) denotes the point where thefield line passing by ( r , ϕ, z ) crosses the chosen reference coor-dinate, r (in model Bd) or z (in model Dd). In practice, z (inmodel Bd) and r (in model Dd) are related to ( r , z ) via the shapeof poloidal field lines (see Eqs. (88) and (32), respectively).Moreover, ϕ is related to ( r , ϕ, z ) via the winding function ofthe spiral, f ϕ ( r , z ) (see Eq. 70), which, in turn, can be expressedin terms of the pitch angle, p ( r , z ), through Eq. (64). The pitchangle also links the azimuthal field to the radial field throughEq. (68). Table 3.
Brief description of our two models of three-dimensional, X-shape + spiral magnetic fields in galactic disks. Model a Shape of Reference b Labels of c Reference d B r B ϕ B z poloidal field lines coordinate field lines fieldBd Eq. (87) r > z (Eq. 88) , ϕ (Eq. 70) Eq. (77), with Eq. (91) Eq. (89) Eq. (68) Eq. (90)Dd Eq. (31) z = | z | sign z r (Eq. 32) , ϕ (Eq. 70) Eq. (84), with Eq. (35) Eq. (82) Eq. (68) Eq. (83) a The free parameters and functions of models Bd and Dd are the same as those of models B and D, respectively (see Table 1 for the freeparameters of the poloidal field and Table 2 for the free parameters and functions describing the azimuthal structure). The only di ff erenceconcerns the power-law index of field lines asymptotic to the z -axis in model B (now restricted to n ≥
2) or the r -axis in model D (now set to n = ). b Prescribed radius or height at which field lines are labeled by their height or radius, respectively, and by their azimuthal angle. c Height or radius of field lines and their azimuthal angle at the reference coordinate. d Radial or vertical field at the reference coordinate as a function of field line labels. (a) Model Bd (b) Model Dd
Fig. 3.
Small set of field lines for our two models of three-dimensional, X-shape + spiral magnetic fields in galactic disks, as seenfrom an oblique angle: (a) model Bd (poloidal component described by Eq. 87) with r = n =
2; and (b) model Dd(Eq. 31) with | z | = . n = . The pitch angle is given by Eq. (65) with p = − ◦ , p ∞ = − ◦ and H p = ϕ or ϕ + π , and they are separated by a fixed magnetic flux per unit ϕ ; thus,in model Bd, their vertical spacing at r is determined by Eq. (91) with H = . z is determined by Eq. (35) with L =
10 kpc. Each box is a (30 kpc) cube centered on the galactic center. The trace of the galacticplane is indicated by the red, solid circle, and the z -axis by the vertical, black, dot-dashed line. Model Bd.
Model B is not immediately fit to represent diskmagnetic fields, for two reasons. First, low- | z | field lines outsidethe reference radius, r , rise up too steeply into the halo (seeFigure 1b). Second, the magnetic field strength peaks too highabove the midplane (the reference field B r ( r , z ) peaks at | z | = H ; see Eqs. (16) and (21)). As we now show, both shortcomingscan easily be overcome.First, to reduce the slope of low- | z | field lines outside r , wereplace the linear term ∝ ( r / r ) in Eq. (17) by a slower-risingterm ∝ √ r / r . Under our previous requirement that each fieldline reaches its minimum height at ( r , z ), the equation of fieldlines becomes z = n + z " rr ! − n + n r rr , (87)and, after inversion, the vertical label of field lines reads z = (2 n + z " rr ! − n + n r rr − · (88) In view of Eqs. (71) – (72), the poloidal field components arethen given by B r = r r z z B r ( r , ϕ , z ) (89) B z = − n n + r z r z " rr ! − n − r rr B r ( r , ϕ , z ) · (90)The value of the exponent n in Eq. (90) is a little more restrictedthan for halo magnetic fields, not because of di ff erences in theexpressions of B z itself (Eq. (90) for disk fields vs. Eq (76)for halo fields), but because of di ff erences in the expressionsof B r ( r , z ) (Eq. (91) for disk fields vs. Eq. (16) or (21) forhalo fields). Here, the behavior B r ∝ r n − B r ( r , z ) ∝ r n − and B z ∝ r n − B r ( r , z ) ∝ r n − in the limit r → n ≥ B r ( r , ϕ , z ) provided by Eq. (77), but we change the verticalprofile of the function B r ( r , z ) entering Eq. (77) from a linear- exponential (as in Eqs. (16) and (21)) to a pure exponential: B r ( r , z ) = B exp − | z | H ! · (91)With this simple choice, B r ( r , z ) is an even function of z , sothat the magnetic field is symmetric with respect to the midplane,as appropriate for galactic disks (e.g., Krause 2009; Braun et al.2010; Beck & Wielebinski 2013). Let us just mention for com-pleteness that model Bd could in principle be made antisym-metric, simply by mutliplying the right-hand side of Eq. (91) bysign z . Model Dd.
Model D is more straightforward to adapt togalactic disks. Indeed, low- | z | field lines are already closelyaligned with the disk plane, and the magnetic field is clearlymuch stronger in a narrow band around the plane than in thehalo (see Figure 1d).To obtain a disk version of model D, it su ffi ces to intro-duce two simple restrictions. First, in the equation of field lines(Eq. 31), the exponent n must be such that the magnetic fieldstrength peaks, while remaining finite, at the midplane. Since B r ∝ z n − and B z ∝ z n in the limit z → n = . Second, if diskmagnetic fields are indeed symmetric with respect to the mid-plane, one must select Eq. (35) rather than Eq. (30) for the radialprofile of the reference field B z ( r , z ).To conclude, we point out a very nice aspect of model Dd.Since its field lines are densely packed and quasi-horizontal inthe disk, while they are broadly spread out and X-shaped inthe halo, model Dd could potentially serve as a single modelfor complete galactic magnetic fields, including both the diskand the halo contributions. Similarly, although less strikingly,model Bd could potentially serve as a single, complete fieldmodel in the cases of galaxies harboring strong vertical fieldsnear their centers. Such a use of models Dd and Bd would, ofcourse, be conceivable only to the extent that the disk and halofields have the same vertical parity (namely, symmetric with re-spect to the midplane).
5. Synchrotron polarization maps
By construction, our models of magnetic fields in galactic halosare characterized by a poloidal component that has an X shapeat large r and | z | (see Figure 1). However, in the presence of anazimuthal field component, this X shape can be partly maskedby the spiraling of field lines (see Figure 2). How does this a ff ectsynchrotron polarization maps? Does the azimuthal field washout the X pattern formed by the poloidal field alone?To address this question, we simulate synchrotron polariza-tion maps of a hypothetical external galaxy seen edge-on, forour four models in their axisymmetric versions ( m = B , is set to an arbi-trary value (irrelevant here), and the pitch angle, p , is assumedto obey Eq. (65) with p = − ◦ , p ∞ = − ◦ and H p = E ∝ n e B ( γ + / ⊥ , where B ⊥ is the magnetic field perpendicular to the line of sight, n e isthe density of relativistic electrons and γ is the power-law in-dex of the relativistic-electron energy spectrum. Here, we adoptthe conservative value γ =
3, and we make the standard dou-ble assumption that (1) relativistic electrons represent a fixedfraction of the cosmic-ray population and (2) cosmic rays and magnetic fields are in (energy or pressure) equipartition, so that E ∝ B B ⊥ . We then compute the associated Stokes parameters U and Q , remembering that synchrotron emission is (partially)linearly polarized perpendicular to B ⊥ and assuming that the in-trinsic degree of linear polarization is uniform throughout thegalaxy (consistent with the power-law index, γ , being uniform).We define a grid of sightlines through the galaxy, integrate theStokes parameters along every sightline and rotate the resultingpolarized-intensity bar (or headless vector) by 90 ◦ to recover themagnetic field orientation in the plane of the sky. The polariza-tion maps obtained in this manner are displayed in Figure 4. Weemphasize that these maps do not include any Faraday rotation,Faraday depolarization or beam depolarization e ff ects.The synchrotron polarization patterns predicted by mod-els A, C and D are reminiscent of the run of poloidal field linesin Figure 1. By the same token, they are consistent with theX shapes observed in the halos of external edge-on galaxies.Model B leads to an X pattern as well, but because the polar-ization bars fade away (logarithmically) with decreasing polar-ized intensity, this X pattern is completely overshadowed by thestrong polarized emission arising from the strong vertical fieldalong the z -axis.The impact of the main model parameters on the polariza-tion maps can be understood by running a large number of sim-ulations with di ff erent parameter values. One of the most crit-ical parameters is the pitch angle, p , whose e ff ect is found toconform to our physical expectation. As a general rule, a smallpitch angle entails a strong azimuthal field in regions where thepoloidal field (shown in Figure 1) has a significant radial com-ponent (see Eq. 68). This strong azimuthal field, in turn, tendsto make the magnetic orientation bars (i.e., the polarization barsrotated by 90 ◦ ) more horizontal, especially at short projected dis-tances from the z -axis, where the azimuthal field along most ofthe line of sight lies close to the plane of the sky. This e ff ectis particularly pronounced toward the central galactic region inmodels B and D, where the magnetic orientation bars are almostperfectly horizontal (see Figures 4b and 4d), while they wouldhave been nearly vertical for a purely poloidal field.The scale height of the pitch angle, H p , is also found tohave the expected impact on the polarization maps. Choosing,somewhat arbitrarily, H p = H p , we find that the magneticorientation bars either remain nearly horizontal up to higher al-titudes (in models A, B and D) or start turning toward the hor-izontal closer to the midplane (in model C). As a result, the Xpattern either emerges higher up (in models A, B and D) or issqueezed toward the midplane (in model C).Another parameter a ff ecting the polarization maps is theazimuthal wavenumber, m . The results presented in Figure 4pertain to axisymmetric ( m =
0) configurations. Bisymmetric( m =
1) configurations are found to lead to a broader rangeof polarization arrangements, in which the X pattern can be ei-ther enhanced or weakened, depending on the viewing angle. Forhigher-order ( m ≥
2) modes, the distribution of magnetic orien-tation bars becomes more complex and does not always displaya recognizable X pattern.The variety of polarization structures produced by our fourmodels may be representative of the variety observed in thereal world of external galaxies (e.g., Soida 2005; Krause 2009).Model A could a priori be used to describe the halo fields of This double assumption, routinely made by observers, as well asthe other assumptions underlying our derivation, are not critical for thepresent discussion. 13atia Ferri`ere: Models of X-shape magnetic fields (a) Model A (b) Model B (c) Model C (d) Model D
Fig. 4.
Maps of the normalized synchrotron polarization bars (or headless vectors) rotated by 90 ◦ toward a hypothetical externalgalaxy seen edge-on, as predicted by our four models of three-dimensional, X-shape + spiral magnetic fields in galactic halos.The magnetic field is supposed to be axisymmetric ( m = p = − ◦ , p ∞ = − ◦ and H p = z -axis by the vertical, black, dot-dashed line. The rotated polarization barstrace the average magnetic field orientation in the plane of the sky, and their level of grey reflects the magnitude of the polarizedintensity (with a logarithmic scale).galaxies like NGC 891, where the X pattern appears to crossthe rotation axis almost horizontally (Krause 2009) – under thecondition that the magnetic configuration is not axisymmetric.Model D would be more appropriate to describe the halo fields ofgalaxies like NGC 4217, where the magnetic orientation bars arenearly horizontal near the midplane and gradually turn more ver-tical with increasing latitude away from the central region (Soida2005). Note, however, that both models A and D predict a strongpolarized intensity from the disk, which is not always observedin the polarization maps of external edge-on galaxies. This canbe because, at low radio frequencies, the (intrinsically polarized)emission from the disk is subject to strong Faraday depolariza-tion. Model C could be the best candidate to describe the halo fields of galaxies like the starburst galaxy NGC 4631, which ap-pears to host the superposition of an X-shape halo field crossingthe midplane at nearly right angles plus a horizontal disk field(Irwin et al. 2012; Mora & Krause 2013; Marita Krause, privatecommunication). Finally, model B could probably be used in thecase of galaxies with strong bipolar outflows from their centralregions. In that respect, it is noteworthy that many, if not all, ofthe highly inclined galaxies studied by Braun et al. (2010) showevidence for circumnuclear, bipolar-outflow fields.
6. Summary
In this paper, we proposed a general method, based on the Eulerpotentials, to derive the analytical expression (as a function ofgalactocentric cylindrical coordinates) of a magnetic field de-fined by the shape of its field lines and by the distribution of itsnormal component on a given reference surface (e.g., a verticalcylinder of radius r or a horizontal plane of height z ). Utilizingthe Euler formalism ensures that the calculated magnetic field isdivergence-free.The first and primary application of our method was to mag-netic fields in galactic halos, which are believed to come withan X shape. We derived four specific analytical models of X-shape magnetic fields, starting with purely poloidal fields (sec-tion 2) and generalizing to three-dimensional fields that have anX-shape poloidal component and a spiral horizontal component(section 3). The pitch angle of the spiral was allowed to varywith both galactic radius and height. The descriptive equationsand the free parameters of our four halo models (models A, B,C and D) are summarized in Table 2 (see also Table 1 for thepoloidal component alone) and the shapes of their field lines areshown in Figure 2 (see also Figure 1 for the poloidal componentalone).In models A and C, field lines have nice and simple X shapesin meridional planes (see Figures 1a and 1c), and synchrotronpolarization maps showcase the best-looking X patterns (seeFigures 4a and 4c). Unfortunately, model A is unphysical, in-sofar as field lines from all meridional planes converge to the ro-tation axis, thereby causing the radial field component to becomeinfinite. For non-axisymmetric fields, this di ffi culty can possiblybe circumvented with a slight re-organization of field lines atsmall radii, but for axisymmetric fields, model A must definitelybe ruled out (see section 2.4). Model C, for its part, is alwaysphysical, but because field lines cross the midplane vertically, itis restricted to antisymmetric fields.In models B and D, field lines have slightly more compli-cated shapes in meridional planes (see Figures 1b and 1d): theirouter parts form a well-defined X figure, while their inner partsconcentrate along the rotation axis (in model B) or along themidplane (in model D). This geometry is a direct consequenceof magnetic flux conservation in a setting where field lines tendto converge toward the rotation axis (in model B) or the mid-plane (in model D), but are not allowed to cross it. An e ff ect ofthis concentration of field lines is to overshadow the underlyingX pattern in synchrotron polarization maps (see Figures 4b and4d). Otherwise, the strong vertical field near the rotation axis inmodel B is not unrealistic, as it could be explained by strongbipolar outflows from the central region. The strong horizontalfield near the midplane in model D is also realistic, but it wouldbe more appropriate for a model of the disk field than a modelof the halo field.A secondary application of our general method was to mag-netic fields in galactic disks, which were assumed to be approx-imately horizontal near the midplane and to have a spiral hori-zontal component everywhere, again with spatially varying pitchangle (section 4). Instead of going through an entire new deriva-tion, we took up two of our halo models (models B and D) andmodified them slightly in such a way as to adapt them to galacticdisks. The descriptive equations of our two disk models (mod-els Bd and Dd) are summarized in Table 3 and the shapes of theirfield lines are shown in Figure 3.Models Bd and Dd were both designed to represent diskmagnetic fields, with a high density of nearly horizontal fieldlines near the midplane. However, field lines do not remain strictly confined to the disk; at some point, they leave the diskand enter the halo, where they continue along X-shaped lines.Therefore, models Bd and Dd could potentially represent thedisk and halo fields combined and, thus, provide complete mod-els of galactic magnetic fields. One necessary condition for thisto be possible would be that the disk and halo fields have a com-mon vertical parity (presumably, symmetric with respect to themidplane), which would be expected on theoretical grounds, butdoes not seem to be fully supported by current observations (seesection 2.3).Anyway, because galactic disks and halos are di ff erent struc-tures, formed at di ff erent times and shaped di ff erently (e.g.,Silk 2003; Agertz et al. 2011; Guo et al. 2011; Aumer & White2013), it may be warranted to appeal to separate models for thedisk and halo fields. This keeps open the possibility for thesefields to possess di ff erent vertical parities, azimuthal symme-tries, pitch angles... This also makes it possible to consider thedisk or halo field alone, as required in some specific problems.Our magnetic field models can serve a variety of purposes,related to either our Galaxy or external spiral galaxies. Most ob-viously, they can be used to test the hypothesis of X-shape mag-netic fields in galactic halos against existing RM and synchrotronobservations, and, if relevant, they can help to constrain their ob-servational characteristics. Equally important, they can providethe galactic field models needed in a number of theoretical prob-lems and numerical simulations.The interested readers can resort to one of the specific fieldmodels derived in this paper (see Table 2 for the halo fields andTable 3 for the disk fields), but they can also derive their ownfield models, relying on Eqs. (67) – (69), once they have settledon a given network of field lines. These new field models mayin principle apply to galactic disks or halos, come with an Xshape or not, have one or several lobes in each quadrant..., themain di ffi culty being to find a workable analytical function forthe first Euler potential, α . Acknowledgements.
Katia Ferri`ere would like to express her gratitude to themany colleagues with whom she had lively and enlightening discussions on X-shape magnetic fields: Robert Braun, Chris Chy˙zy, Ralf-J¨urgen Dettmar, AndrewFletcher, Michał Hanasz, Marijke Haverkorn, Anvar Shukurov, with specialthanks to Rainer Beck and Marita Krause. The authors also thank the referee,Marian Soida, for his careful reading of the paper and his valuable comments.
References
Agertz, O., Teyssier, R., & Moore, B. 2011, MNRAS, 410, 1391Aumer, M. & White, S. D. M. 2013, MNRAS, 428, 1055Beck, R. 2008, in American Institute of Physics Conference Series, Vol. 1085,American Institute of Physics Conference Series, ed. F. A. Aharonian,W. Hofmann, & F. Rieger, 83–96Beck, R. 2009, in IAU Symposium, Vol. 259, IAU Symposium, ed. K. G.Strassmeier, A. G. Kosovichev, & J. E. Beckman, 3–14Beck, R. & Wielebinski, R. 2013, Magnetic Fields in Galaxies, ed. T. D. Oswalt& G. Gilmore, 641Brandenburg, A., Donner, K. J., Moss, D., et al. 1993, A&A, 271, 36Braun, R., Heald, G., & Beck, R. 2010, A&A, 514, A42Dalla Vecchia, C. & Schaye, J. 2008, MNRAS, 387, 1431Dumke, M., Krause, M., Wielebinski, R., & Klein, U. 1995, A&A, 302, 691Fletcher, A. 2010, in Astronomical Society of the Pacific Conference Series, Vol.438, Astronomical Society of the Pacific Conference Series, ed. R. Kothes,T. L. Landecker, & A. G. Willis, 197Frick, P., Stepanov, R., Shukurov, A., & Sokolo ff , D. 2001, MNRAS, 325, 649Guo, Q., White, S., Boylan-Kolchin, M., et al. 2011, MNRAS, 413, 101Han, J. 2002, in American Institute of Physics Conference Series, Vol.609, Astrophysical Polarized Backgrounds, ed. S. Cecchini, S. Cortiglioni,R. Sault, & C. Sbarra, 96–101Han, J. L., Manchester, R. N., Berkhuijsen, E. M., & Beck, R. 1997, A&A, 322,98Han, J. L., Manchester, R. N., & Qiao, G. J. 1999, MNRAS, 306, 371 Han, J. L. & Qiao, G. J. 1994, A&A, 288, 759Harari, D., Mollerach, S., & Roulet, E. 1999, Journal of High Energy Physics, 8,22Haverkorn, M. & Heesen, V. 2012, Space Sci. Rev., 166, 133Heesen, V., Krause, M., Beck, R., & Dettmar, R.-J. 2009, A&A, 506, 1123Irwin, J., Beck, R., Benjamin, R. A., et al. 2012, AJ, 144, 44Jałocha, J., Bratek, Ł., Kutschera, M., & Skindzier, P. 2011, MNRAS, 412, 331Jansson, R. & Farrar, G. R. 2012a, ApJ, 757, 14Jansson, R. & Farrar, G. R. 2012b, ApJ, 761, L11Jansson, R., Farrar, G. R., Waelkens, A. H., & Enßlin, T. A. 2009, J. CosmologyAstropart. Phys., 7, 21Krause, M. 2009, in Revista Mexicana de Astronomia y Astrofisica, vol. 27,Vol. 36, Revista Mexicana de Astronomia y Astrofisica Conference Series,25–29Krause, M., Wielebinski, R., & Dumke, M. 2006, A&A, 448, 133Levine, E. S., Heiles, C., & Blitz, L. 2008, ApJ, 679, 1288Marasco, A. & Fraternali, F. 2011, A&A, 525, A134Mo ff att, H. K. 1978, Magnetic field generation in electrically conducting fluidsMoss, D. & Sokolo ff , D. 2008, A&A, 487, 197Moss, D., Sokolo ff , D., Beck, R., & Krause, M. 2010, A&A, 512, A61Northrop, T. G. 1963, Reviews of Geophysics and Space Physics, 1, 283Oren, A. L. & Wolfe, A. M. 1995, ApJ, 445, 624Prouza, M. & ˇSm´ıda, R. 2003, A&A, 410, 1Pshirkov, M. S., Tinyakov, P. G., Kronberg, P. P., & Newton-McGee, K. J. 2011,ApJ, 738, 192Rand, R. J. & Lyne, A. G. 1994, MNRAS, 268, 497Silk, J. 2003, Ap&SS, 284, 663Sofue, Y. & Fujimoto, M. 1983, ApJ, 265, 722Soida, M. 2005, in The Magnetized Plasma in Galaxy Evolution, ed. K. T.Chyzy, K. Otmianowska-Mazur, M. Soida, & R.-J. Dettmar, 185–190Soida, M., Krause, M., Dettmar, R.-J., & Urbanik, M. 2011, A&A, 531, A127Sokolo ff , D. & Shukurov, A. 1990, Nature, 347, 51Stanev, T. 1997, ApJ, 479, 290Stern, D. P. 1966, Space Sci. Rev., 6, 147Sun, X. H., Reich, W., Waelkens, A., & Enßlin, T. A. 2008, A&A, 477, 573T¨ullmann, R., Dettmar, R.-J., Soida, M., Urbanik, M., & Rossa, J. 2000, A&A,364, L36Wielebinski, R. & Krause, F. 1993, A&A Rev., 4, 449, D. & Shukurov, A. 1990, Nature, 347, 51Stanev, T. 1997, ApJ, 479, 290Stern, D. P. 1966, Space Sci. Rev., 6, 147Sun, X. H., Reich, W., Waelkens, A., & Enßlin, T. A. 2008, A&A, 477, 573T¨ullmann, R., Dettmar, R.-J., Soida, M., Urbanik, M., & Rossa, J. 2000, A&A,364, L36Wielebinski, R. & Krause, F. 1993, A&A Rev., 4, 449