﻿ UCLA IGPP Space Physics Exercises Help

# Potential Field

This module allows you to specify the coefficients that describe a dipole, quadrupole, or octupole magnetic field in a spherical harmonic expansion and to visualize the resulting magnetic field.

## Spherical harmonic coefficients of the internal field

The internal magnetic field is the field that is generated from the planet’s interior and in contrast to the external magnetic field, which is associated with the current system outside the planet. So, outside of the region where the internal magnetic field is generated, it can be represented as the gradient of a scalar potential.

Binternal = -∇V

The potential, V, can be expressed by the sum of spherical harmonics through the following expression:

V = a Σn=1–max Σm=0–n (a/r)n+1 (g(n, m) cos(mφ) + h(n, m) sin(mφ)) P(n, m, cosθ)

The expression for the magnetic field using the previous two equations, in spherical coordinates, is as follows:

Br = Σn=1—max Σm=0–n (n+1)(a/r)n+2 (g(n, m) cos(mφ) + h(n, m) sin(mφ)) P(n, m, cosθ)

Bθ = Σn=1–max Σm=0–n (a/r)n+2 (g(n, m) cos(mφ) + h(n, m) sin(mφ)) dP(n, m, cosθ) / dθ

Bφ = (1/sinθ) Σn=1–max Σm=0–n m (a/r)n+2 (g(n, m) sin(mφ) - h(n, m) cos(mφ)) P(n, m, cosθ)

where a is the equatorial radius of the Sun; r is the radial distance to the Sun’s center from the point of field evaluation, and angles θ and φ are colatitudes and longitude, respectively. P(n, m, cosθ) are Schmidt-normalized associated Legendre functions of degree n and order m, and g and h are the internal field parameters (Schmidt coefficients).

This model is a magnetic multipole representation of the internal field. This module uses the previous three equations to plot the magnetic field lines.

### Dipole coefficients

n = 1 in the above equations.

n = 2 in the above equations.

### Octupole coefficients

n = 3 in the above equations.

## Magnetic multipole configuration

The definition for the magnetic/electric multipole is that the nth multipole is composed of two opposite (n-1)th multipoles separated by a small distance dl. In the real world, the base magnetic multipole is a dipole, which by convenient comparison we think are composed of two “magnetic monopoles” separating by dl1. From that, any high-order multipoles can be assembled by using the above definition.

Spherical harmonic expansion decompose a magnetic scalar potential to a multipole forms starting from the magnetic dipole. For the nth multipole, it uses 2n+1 coefficients to describe the different forms of this multipole. The spherical harmonic coefficients are ordered as: g10, g11, h11, g20, g21, g22, h21, h22, etc.

## Magnetic dipole

There is only one separating vector dl1 for dipole, thus dipole is always linear. While magnetic monopoles do not exist in isolation, it is useful to think of magnetic dipole in analogy to electric dipole with separated charges along some axis.

There are three coefficients that describe a dipole magnetic field in the spherical harmonic expansion, g10, g11, and h11. These correspond to the projection of the dipole moment along the z-axis (rotation), the y-axis and the x-axis (in the rotational equator) respectively.

In this module, you can specify these three components and visualize the resulting magnetic field, by clicking Calculate. The module also allows you to view the magnetic field at different latitudes and longitudes. The module also displays the dipole tilt angle and dipole longitude at bottom left.

A quadrupole is composed of two anti-dipoles separating by a distance dl2. So the initial thoughts would be that we need six coefficients to describe them (dl1 and dl2).

In a linear type quadrupole dl2 is parallel to dl1, whereas in a planar type quadrupole dl2 is perpendicular to dl1. However, a planar type quadrupole can be composed of two linear quadrupoles by perpendicularly placing them, this relationship decrease one degree of freedom, so only five coefficients are needed. That these five configurations provide all possible quadrupole fields (separated dipole pairs) can be shown by expressing the dipoles as separated monopoles and comparing.

The five coefficients that describe a quadrupole magnetic field in a spherical harmonic expansion are:

g20: linear type quadrupole, along z-axis, corresponds to seperation along the z-axis (rotational). This is called zonal (rotationally symmetric).

g21: planar type quadrupole, in the y-z plane, g21 has the dipole moment along the z-axis and separated in the y-axis.

h21: planar type quadrupole, in the x-z plane, h21 has the dipole moment along z-axis and separated in the x-axis.

g22: planar type quadrupole, in the x-y plane, g22 has the dipole moment along a direction at 45° to the x- and y-axis and a separation along a line at 45° to the x- and y-directions.

h22: planar type quadrupole, in the x-y plane, h22 has the dipole moment along y-axis and separated in the x-axis.

In this module you can specify these five components and visualize the resulting magnetic field, by clicking Calculate. The module also allows you to view the magnetic field at different latitudes and longitudes. The module also allows you the flexibility of number of lines plotted. At the right bottom, there are two input boxes for longitude and latitude. The number of field lines plotted by the program along longitude and latitude are double the value specified by the user in these boxes.

For best viewing of the result you would need proper view angle and density of lines. Sample values of these variables for good viewing, when only single spherical harmonic is present are provided below.

Spherical harmonic Latitude Longitude Latitude steps Longitude steps
g20 any 5 5
g21 90° 5 5
h21 5 5
g22 90° 90° 7 7
h22 90° 90° 7 7

To obtain the best view for a combination of different spherical harmonics, you need to try different orientations and line densities. If you specify a large number of latitude and longitude lines, then the program will take more time to plot the lines. (The time the program takes to plot the field lines is proportional to the density of field lines coefficients for longitude and latitude you have specified).

## Magnetic octupole

An octupole is composed from two anti-quadrupoles separating by distance dl3. The initial thoughts would be that we need nine coefficients to describe them (dl1, dl2, and dl3).

In linear type octupole dl3 is parallel to dl1 and dl2, in planar type octupole dl3 is perpendicular to dl1 or dl2, and they are all in one plane, and in cubic type octupole dl3, dl1, and dl2 are perpendicular to each other. However, a planar type octupole can be composed of two linear octupoles by perpendicularly placing them, and similarly a cubic type octupole can be composed by two planar octupoles, thus reducing the degree of freedom by two, so we need only seven coefficients. That these seven octupole configurations provide all possible octupole fields (separated quadrupole pairs) can be seen by expressing the moments as distributed monopoles and comparing.

The seven coefficients that describe an octupole magnetic field in the spherical harmonic expansion are g30, g31, h31, g32, h32, g33, and h33. The g30 is a zonal moment created by separating two oppositely-directed zonal quadrupoles along the z-axis (rotational). Two planar octupoles g31 and h31 are constructed from linear quadrupoles; two planar types g33 and h33 are constructed from planar quadrupoles and there are two cubic types g32 and h32.

In this module you can specify these five components and visualize the resulting magnetic field, by clicking Calculate. The module also allows you to view the magnetic field at different latitudes and longitudes. The module also allows the user the flexibility of number of lines plotted. At the right bottom, there are two input boxes for longitude and latitude. The number of field lines plotted by the program along longitude and latitude are double the value specified by the user in these boxes.

For best viewing of the result you would need proper view angle and density of lines. Sample values of these variables for good viewing, when only single spherical harmonic is present are provided below.

Spherical harmonic Latitude Longitude Latitude steps Longitude steps
g30 any 5 5
g31 90° 5 5
h31 5 5
g32 90° 90° 11 11
h32 90° 90° 11 11
g33 90° 90° 11 11
h33 90° 90° 11 11

To obtain the best view for a combination of different spherical harmonics, you need to try with different values of rotation and density of lines, to obtain best view. If you provide a very large value for density of lines along latitude and longitude, then the program will take more time to do plotting. (The time the program takes to plot the field lines is proportional to the density of field lines coefficients for longitude and latitude the user has provided).