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Table 2

Brief description of our four models of three-dimensional, X-shape + spiral magnetic fields in galactic halos.

  Modela Referenceb Labels ofc Referenced B r B ϕ B z Free parametersef
coordinate field lines field and functions

Ag r1 > 0 z1 (Eq. (13)), ϕ1 (Eq. (70)) Eq. (77) Eq. (73) Eq. (68) Eq. (74) p(r,z) m ϕ(z1)
B r1 > 0 z1 (Eq. (18)), ϕ1 (Eq. (70)) Eq. (77) Eq. (75) Eq. (68) Eq. (76) p(r,z) m ϕ(z1)
C z1 = 0 r1 (Eq. (27)), ϕ1 (Eq. (70)) Eq. (84) Eq. (80) Eq. (68) Eq. (81) p(r,z) m ϕ(r1)
D z1 =  | z1 |   sign  z r1 (Eq. (32)), ϕ1 (Eq. (70)) Eq. (84) Eq. (82) Eq. (68) Eq. (83) p(r,z) m ϕ(r1)

Notes.

(a)

The shape of field lines associated with the poloidal field is indicated in Table 1 (second column), while the spiraling of field lines is described 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 functions describing 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 azimuthal wavenumber and ϕ(z1) or ϕ(r1) the fiducial angle of the sinusoidal azimuthal modulation at r1 or z1, respectively (see Eqs. (77) and (84)). Note that ϕ becomes irrelevant in the axisymmetric case (m = 0).

(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 = 0.

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