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Table 1

Brief description of our four models of poloidal, X-shape magnetic fields in galactic halos.

Model Shape of Referencea Label ofb Referencec Verticald B r B z Free parametersef
field lines coordinate field lines field parity

Ag Eq. (12) – Fig. 1a r1 > 0 z1 (Eq. (13)) Eq. (16) sym Eq. (14) Eq. (15) r 1 a B 1 H
Eq. (21) antisym

B Eq. (17) – Fig. 1b r1 > 0 z1 (Eq. (18)) Eq. (16) sym Eq. (19) Eq. (20) r 1 n B 1 H
Eq. (21) antisym

C Eq. (26) – Fig. 1c z1 = 0 r1 (Eq. (27)) Eq. (30) antisym Eq. (28) Eq. (29) a B 1 L

D Eq. (31) – Fig. 1d z1 =  | z1 |   sign  z r1 (Eq. (32)) Eq. (30) antisym Eq. (33) Eq. (34) z1 | n B 1 L
Eq. (35) sym



Prescribed radius or height at which field lines are labeled by their height or radius, respectively.


Height or radius of field lines at the reference coordinate.


Radial or vertical field at the reference coordinate as a function of field line label.


For each model, the first line refers to the original parity in Sect. 2.2, and, when present, the second line refers to the reversed parity.


In general, the free parameters can vary with azimuthal angle, ϕ. However, in the particular case considered in Sect. 2.4, only B1 depends on ϕ (through Eq. (36)).


r1 is the reference radius in models A and B and | z1 | the modulus of the reference height in models C and D; a is the parameter governing the opening of quadratic field lines in models A and C (a > 0); 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 (); B1 is the peak value of the reference field, taken as the normalization field (B1 ≷ 0); H is the exponential scale height (in models A and B) and L the exponential scale length (in models C and D) of the reference field (H > 0 and L > 0).


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 = 0 (see Sect. 2.4).

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