Magnetic field models for the Milky Way

At the moment, two models of the coherent component of Galactic magnetic field are available. Typically, the coherence length of the turbulent component is too small to induce large scale photon-ALP oscillations at gamma-ray energies and hence only the coherent components are considered here.

The three models are the ones described in detail in [Pshirkov2011], [Jansson2012] and [Unger2024]. For the Jansson & Farrar model [Jansson2012], also the updates from measurements with the Planck satellite are available [Planck2016].

Reference / API

class gammaALPs.bfields.gmf.GMF(model='jansson12')[source]

Bases: object

Class with analytical functions that describe the galactic magnetic field according to the model of Jannson & Farrar (2012)

Only the regular field components are implemented. The striated field component is absent.

Rsun

Assumed position of the sun along the x axis in kpc

Type:: float

bring

field strength in ring at 3 kpc < rho < 5 kpc

Type:: float

bring_unc

associated uncertainty of field strength in ring at 3 kpc < rho < 5 kpc

Type:: float

hdisk

disk/halo transition height

Type:: float hdisk_unc: float,

hdisk_unc

associated uncertainty disk/halo transition height

Type:: float

wdisk

transition width

Type:: float

wdisk_unc

associated uncertainty of transition width

Type:: float, wdisk_unc: floats,

b

arrays with field strengths of spiral arms at 5 kpc

Type:: ndarray

b_unc

associated uncertainties with field strengths of spiral arms at 5 kpc

Type:: ndarray

f_cov

relative cross-sectional areas of the spirals (for a fixed radius)

Type:: ndarray

rx

dividing lines of spiral arms, coordinates of neg. x-axes that intersect with arm

Type:: ndarray

idisk

spiral arms opening angle

Type:: float

bn, bn_unc

field strength northern halo

Type:: float

bn_unc

associated uncertainty of field strength northern halo

Type:: float

Bs

field strength southern halo

Type:: float

Bs_unc

associated uncertainty of field strength southern halo

Type:: float

rhon

transition radius in northern hemisphere

Type:: float

rhon_unc

associated uncertainty of transition radius in northern hemisphere

Type:: float

rhos

transition radius in southern heisphere

Type:: float

rhos_unc

associated uncertainty of transition radius in southern hemisphere

Type:: float

whalo

transition width

Type:: float

whalo_unc

associated uncertainty of transition width

Type:: float

z0

vertical scale height

Type:: float

z0_unc

associated uncertainty of vertical scale height

Type:: float

BX0

field strength at origin of X component

Type:: float

BX0_unc

associated uncertainty of field strength at origin of X component

Type:: float

ThetaX0

elev. angle at z = 0, rho > rhoXc

Type:: float

ThetaX0_unc

associated uncertainty of elev. angle at z = 0, rho > rhoXc

Type:: float

rhoXc

radius where thetaX = thetaX0

Type:: float

rhoXc_unc

associated uncertainty of radius where thetaX = thetaX0

Type:: float

rhoX

exponential scale length

Type:: float,

rhoX_unc

associated uncertainty of exponential scale length

Type:: float,

gamma

striation and / or rel. elec. number dens. rescaling

Type:: float

gamma_unc

associated uncertainty striation and / or rel. elec. number dens. rescaling

Type:: float

Notes

see http://adsabs.harvard.edu/abs/2012ApJ…757…14J Jansson & Farrar (2012)

BX(rho, z)[source]

X (out of plane) component of galactic magnetic field in galactocentric cylindrical coordinates (rho,phi,z)

BX is purely poloidal, i.e. phi component = 0

Parameters:

rho (array-like) – N-dim array with distance from origin in GC cylindrical coordinates, is in kpc
z (array-like) – N-dim array with height in kpc in GC cylindrical coordinates

Returns:

BX, BX_abs – tuple containing the magnetic field of the X component as a (3,N)-dim array with (rho,phi,z) components of disk field for each coordinate tuple and absolute value of the field as N-dim array

Return type:

tuple of ndarray

Bdisk(rho, phi, z)[source]

Disk component of galactic magnetic field in galactocentric cylindrical coordinates (rho,phi,z)

Parameters:

rho (array-like) – N-dim array with distance from origin in GC cylindrical coordinates, is in kpc
z (array-like) – N-dim array with height in kpc in GC cylindrical coordinates
phi (array-like) – N-dim array with polar angle in GC cylindircal coordinates, in radian

Returns:

Bdisk, Bdisk_abs – tuple containing the magnetic field of the disk as a (3,N)-dim array with (rho,phi,z) components of disk field for each coordinate tuple and absolute value of the field as N-dim array

Return type:

tuple of ndarray

Bhalo(rho, z)[source]

Halo component of galactic magnetic field in galactocentric cylindrical coordinates (rho,phi,z)

Bhalo is purely azimuthal (toroidal), i.e. has only a phi component

Parameters:

rho (array-like) – N-dim array with distance from origin in GC cylindrical coordinates, is in kpc
z (array-like) – N-dim array with height in kpc in GC cylindrical coordinates

Returns:

Bhalo, Bhalo_abs – tuple containing the magnetic field of the halo as a (3,N)-dim array with (rho,phi,z) components of disk field for each coordinate tuple and absolute value of the field as N-dim array

Return type:

tuple of ndarray

L(z, h, w)[source]

Transition function, see Jansson & Farrar Eq. 5

Parameters:

z (float or array-like) – array with positions (height above disk, z; distance from center, rho)
h (float) – height parameter
w (float) – width parameter

Returns:

L – array or float (depending on z input) with transition function values

Return type:

ndarray

__init__(model='jansson12')[source]

Init the GMF class, all B-field values are in muG

Parameters:: model (str) – either jansson12, jansson12b, or jansson12c, where jansson12 is the original model and the other two options are the modifications of the model with Planck data, see http://arxiv.org/abs/1601.00546

r_log_spiral(phi)[source]

return distance from center for angle phi of logarithmic spiral

Parameters:: phi (scalar or array-like) – polar angle values
Returns:: result – logarithmic spiral result with r(phi) = rx * exp(b * phi) as an array
Return type:: ndarray

Notes

see http://en.wikipedia.org/wiki/Logarithmic_spiral

class gammaALPs.bfields.gmf.GMFPshirkov(model='ASS')[source]

Bases: object

Class with analytical functions that describe the galactic magnetic field according to the model of Pshirkov et al. (2011)

Only the regular field components are implemented.

Rsun

position of the sun in kpc along x axis

Type:: float

p

pitch angle, dictionary with entries ‘ASS’ and ‘BSS’, in radian

Type:: dict

z0

height of disk in kpc

Type:: float

d

value if field reversal in kpc

Type:: float

B0

Value of B field at position of the sun, in muG

Type:: float

z0n

position of northern halo in kpc

Type:: float

Bn

northern halo field in muG

Type:: float

r0n

northern halo

Type:: float

z1n

scale height of halo toward galactic plane, |z| < z0n

Type:: float

z2n

scale height of northern halo away from galactic plane, |z| >= z0n

Type:: float

z0s

position of southern halo in kpc

Type:: float

Bs

southern halo field in muG

Type:: float

r0s

southern halo

Type:: float

z1s

scale height of southern halo toward galactic plane, |z| < z0s

Type:: float

z2s

scale height of southern halo away from galactic plane, |z| >= z0n

Type:: float

Notes

Paper: http://adsabs.harvard.edu/abs/2011ApJ…738..192P Pshirkov et al. (2011)

Bdisk(rho, phi, z)[source]

Disk component of galactic magnetic field in galactocentric cylindrical coordinates (rho,phi,z)

Parameters:

rho (array-like) – N-dim array with distance from origin in GC cylindrical coordinates, is in kpc
phi (array-like) – N-dim array with polar angle in GC cylindircal coordinates, in radian
z (array-like) – N-dim array with height in kpc in GC cylindrical coordinates

Returns:

Bdisk, Bdisk_abs – tuple containing the magnetic field of the disk as a (3,N)-dim array with (rho,phi,z) components of disk field for each coordinate tuple and absolute value of the field as N-dim array

Return type:

tuple of ndarray

Notes

See Pshirkov et al. Eq. (3) - (5)

Bhalo(rho, z)[source]

Halo component of galactic magnetic field in galactocentric cylindrical coordinates (rho,phi,z)

Bhalo is purely azimuthal (toroidal), i.e. has only a phi component

Parameters:

rho (array-like) – N-dim array with distance from origin in GC cylindrical coordinates, is in kpc
z (array-like) – N-dim array with height in kpc in GC cylindrical coordinates

Returns:

Bhalo, Bhalo_abs – tuple containing the magnetic field of the halo as a (3,N)-dim array with (rho,phi,z) components of disk field for each coordinate tuple and absolute value of the field as N-dim array

Return type:

tuple of ndarray

__init__(model='ASS')[source]

Init the GMF class, all B-field values are in muG

Parameters:: model (string,) – either ASS or BSS for axissymmetric or bisymmetric model, respectively.

class gammaALPs.bfields.gmf.UF23(model='base')[source]

Bases: object

Class with analytical functions that describe the galactic magnetic field according to the model of Unger & Farrar (2023/2024)

2 components are implemented for the spiral field: The spur_field is only used in the ‘spur’ model. The other 7 models use the spiral_field.

3 components are implemented for the halo field: The twisted_halo_field is only used for the ‘twistX’ model. The other 7 models use the sum of poloidal_halo_field and toroidal_halo_field.

Rsun

distance of sun from center of milky way in kpc

Type:: float

poloidal_A

coasting radius for poloidal halo (a_c)

Type:: float

r_ref

reference radius, for spiral field 5 kpc for spur field sun radius (8.2 kpc) (r_0)

Type:: float

r_inner

inner boundary (radius) of spiral field (r_1)

Type:: float

w_inner

inner boundary (transition width) of spiral field (w_1)

Type:: float

r_outer

outer boundary (radius) of spiral field (r_2)

Type:: float

w_outer

outer boundary (transition width) of spiral field (w_2)

Type:: float

disk_b1, disk_b2, disk_b3

magnetic field strength of mode 1, 2, 3 (B_m)

Type:: floats

disk_h

transition height (z_d)

Type:: float

disk_phase1, disk_phase2, disk_phase3

phase of mode 1, 2, 3 (phi_m)

Type:: floats

disk_pitch

pitch angle (alpha)

Type:: float

disk_w

vertical transition width (w_d)

Type:: float

poloidal_B

magnetic field strength normalization constant (B_p)

Type:: float

poloidal_P

field line exponent (p)

Type:: float

poloidal_R

radial scale or transition radius (r_p)

Type:: float

poloidal_W

transition width (w_p)

Type:: float

poloidal_Z

scale height (z_p)

Type:: float

toroidal_BN

northern magnetic field strength

Type:: float

toroidal_BS

southern magnetic field strength

Type:: float

toroidal_R

transition radius (r_t)

Type:: float

toroidal_W

radial transition width (w_t)

Type:: float

toroidal_Z

vertical scale height (z_t)

Type:: float

twisting_time

twisting time

Type:: float

spur_center

central azimuth in degree

Type:: float

spur_length

half angular length in degree

Type:: float

spur_width

gaussian width in degrees

Type:: float

Notes

see: Michael Unger and Glennys R. Farrar Astrophys.J. 970 (2024) 95 https://doi.org/10.3847/1538-4357/ad4a54 and https://doi.org/10.5281/zenodo.11321212

Bdisk(rho, phi, z)[source]

Returns disk component of milky way magnetic field depending on model in galactocentric cylindrical coordinates (rho, phi, z)

Parameters:

rho (array-like) – N-dim array with distance from origin in GC cylindrical coordinates, is in kpc
phi (array-like) – N-dim array with polar angle in GC cylindircal coordinates, in radian
z (array-like) – N-dim array with height in kpc in GC cylindrical coordinates

Returns:

Bdisk, Bdisk_abs – tuple containing the magnetic field of the disk as a (3,N)-dim array with (rho,phi,z) components of disk field for each coordinate tuple and absolute value of the field as N-dim array

Return type:

tuple of ndarray

Bhalo(rho, z)[source]

Returns halo component of milky way magnetic field depending on model in galactocentric cylindrical coordinates (rho, phi, z)

Parameters:

rho (array-like) – N-dim array with distance from origin in GC cylindrical coordinates, is in kpc
z (array-like) – N-dim array with height in kpc in GC cylindrical coordinates

Returns:

Bhalo, Bhalo_abs – tuple containing the magnetic field of the disk as a (3,N)-dim array with (rho,phi,z) components of disk field for each coordinate tuple and absolute value of the field as N-dim array

Return type:

tuple of ndarray

__init__(model='base')[source]

Initialize the UF23 class for 8 different models

Parameters:: model (str) – one of [‘base’, ‘expX’, ‘neCl’, ‘twistX’, ‘nebCor’, ‘cre10’, ‘synCG’, ‘spur’]

poloidal_halo_field(rho, z)[source]

calculates poloidal halo field component (phi-component = 0) in galactocentric cylindrical coordinates (rho, phi, z)

Parameters:

rho (array-like) – N-dim array with distance from origin in GC cylindrical coordinates, is in kpc
z (array-like) – N-dim array with height in kpc in GC cylindrical coordinates

Returns:

b_cyl, b_cyl_abs – tuple containing the magnetic field of the disk as a (3,N)-dim array with (rho,phi,z) components of disk field for each coordinate tuple and absolute value of the field as N-dim array

Return type:

tuple of ndarray

Notes

See Unger & Farrar (2024) Section 5.3.2, p. 11

spiral_field(rho, phi, z)[source]

calculates fourier-spiral disk component in galactocentric cylindrical coordinates (rho, phi, z)

Parameters:

rho (array-like) – N-dim array with distance from origin in GC cylindrical coordinates, is in kpc
phi (array-like) – N-dim array with polar angle in GC cylindircal coordinates, in radian
z (array-like) – N-dim array with height in kpc in GC cylindrical coordinates

Returns:

b_cyl, b_cyl_abs – tuple containing the magnetic field of the disk as a (3,N)-dim array with (rho,phi,z) components of disk field for each coordinate tuple and absolute value of the field as N-dim array

Return type:

tuple of ndarray

Notes

See Unger & Farrar (2024) Section 5.2.2, p. 9

spur_field(rho, phi, z)[source]

calculates spiral-spur disk component in galactocentric cylindrical coordinates (rho, phi, z)

Parameters:

rho (array-like) – N-dim array with distance from origin in GC cylindrical coordinates, is in kpc
phi (array-like) – N-dim array with polar angle in GC cylindircal coordinates, in radian
z (array-like) – N-dim array with height in kpc in GC cylindrical coordinates

Returns:

b_cyl, b_cyl_abs – tuple containing the magnetic field of the disk as a (3,N)-dim array with (rho,phi,z) components of disk field for each coordinate tuple and absolute value of the field as N-dim array

Return type:

tuple of ndarray

Notes

See Unger & Farrar (2024) Section 5.2.3, p. 9

toroidal_halo_field(rho, z)[source]

calculates toroidal halo field component (has only phi component) in galactocentric cylindrical coordinates (rho, phi, z)

Parameters:

rho (array-like) – N-dim array with distance from origin in GC cylindrical coordinates, is in kpc
z (array-like) – N-dim array with height in kpc in GC cylindrical coordinates

Returns:

b_cyl, b_cyl_abs – tuple containing the magnetic field of the disk as a (3,N)-dim array with (rho,phi,z) components of disk field for each coordinate tuple and absolute value of the field as N-dim array

Return type:

tuple of ndarray

Notes

See Unger & Farrar (2024) Section 5.3.1, p. 11

twisted_halo_field(rho, z)[source]

calculates twisted halo field for twistX model with twisting time not 0 in galactocentric cylindrical coordinates (rho, phi, z)

Parameters:

rho (array-like) – N-dim array with distance from origin in GC cylindrical coordinates, is in kpc
z (array-like) – N-dim array with height in kpc in GC cylindrical coordinates

Returns:

b_cyl_x, b_cyl_x_abs – tuple containing the magnetic field of the disk as a (3,N)-dim array with (rho,phi,z) components of disk field for each coordinate tuple and absolute value of the field as N-dim array

Return type:

tuple of ndarray

Notes

See Unger & Farrar (2024) Section 5.3.3, p. 12

gammaALPs.bfields.gmf.delta_phi(phi0, phi1)[source]

Calculates the angle between v0 = (cos(phi0), sin(phi0)) and v1 = (cos(phi1), sin(phi1)). Arguments: phi0 : float

First angle in radians

phi1float: Second angle in radians

Returns: The angle between v0 and v1 in radians.

gammaALPs.bfields.gmf.sigmoid(x, x0, w)[source]

Calculates the logistic sigmoid function. Arguments: x : float

The input value

x0float: The midpoint of the sigmoid
wfloat: The width of the sigmoid

Returns: The result of the sigmoid function

gammaALPs.bfields.gmf.signum(x)[source]: Return the sign of each entry of an array