## Abstract

Geospace phenomena such as the aurora, plasma motion, ionospheric currents and associated magnetic field disturbances are highly organized by Earth’s main magnetic field. This is due to the fact that the charged particles that comprise space plasma can move almost freely along magnetic field lines, but not across them. For this reason it is sensible to present such phenomena relative to Earth’s magnetic field. A large variety of magnetic coordinate systems exist, designed for different purposes and regions, ranging from the magnetopause to the ionosphere. In this paper we review the most common magnetic coordinate systems and describe how they are defined, where they are used, and how to convert between them. The definitions are presented based on the spherical harmonic expansion coefficients of the International Geomagnetic Reference Field (IGRF) and, in some of the coordinate systems, the position of the Sun which we show how to calculate from the time and date. The most detailed coordinate systems take the full IGRF into account and define magnetic latitude and longitude such that they are constant along field lines. These coordinate systems, which are useful at ionospheric altitudes, are non-orthogonal. We show how to handle vectors and vector calculus in such coordinates, and discuss how systematic errors may appear if this is not done correctly.

## Keywords

Magnetic coordinates Ionospheric electrodynamics## 1 Introduction

The influence of Earth’s magnetic field extends to approximately 15 \(R_{E}\) (\(R_{E}\) is the mean Earth radius \(\approx6371.009~\mbox{km}\)) in the hemisphere facing the Sun, and up to several hundred \(R_{E}\) on the night side. At ionospheric altitudes, the Earth’s main field is orders of magnitudes stronger than even the strongest magnetic disturbances created by the interaction with the solar wind. To understand the dynamics of near Earth plasma, electromagnetic field disturbances, and even the neutral atmosphere at high altitudes, it is essential to work in a reference frame which takes the geometry of the Earth’s magnetic field into account. Exactly how this should be done differs by the application, region, and desired level of precision.

At large distances from the Earth, the dominating driver is the solar wind flowing roughly radially out from the Sun. The interaction of the solar wind with the magnetosphere in these regions is therefore often best understood in a coordinate system which has the Earth–Sun line along one of the axes. Examples of such coordinate systems include the geocentric solar ecliptic (GSE) and the geocentric solar magnetic (GSM) coordinate systems. The latter also contains the Earth’s magnetic dipole field in one of the coordinate planes, making it especially suitable for working with solar wind-magnetosphere interaction processes. Closer to the Earth the relative importance of the Earth’s field becomes stronger, so that a more precise alignment with the dipole axis is required. Solar magnetic (SM) coordinates is one such system, having one axis along the dipole axis, and the Earth–Sun line in a coordinate plane.

At ionospheric heights and on ground, the centered dipole (CD) coordinate system is probably the most commonly used magnetic coordinate system. This system, which is most used as spherical coordinates, represents a shift of the poles from the rotational axis to the dipole axis. At these altitudes however, the Earth’s field deviates significantly from a centered dipole. To achieve better accuracy with respect to the magnetic field, several alternative coordinate systems exist. One example is the eccentric dipole (ED) coordinate system, which is based on a dipole representation which not necessarily has its origin at the center of the Earth.

Even better accuracy can be achieved by taking the non-dipole features of the Earth’s field into account. Corrected geomagnetic (CGM) coordinates (Gustafsson et al. 1992; Baker and Wing 1989; Shepherd 2014, and references therein) and the Magnetic Apex Coordinate systems (Richmond 1995) are based on a more detailed model of the Earth’s field, and defined in terms of field line tracing. The improved accuracy comes at the expense of simplicity, as the result is a non-orthogonal coordinate system, in contrast to the aforementioned systems which can be defined in terms of Cartesian base vectors and converted between by means of rotation matrices.

In this review paper, we define all the above mentioned coordinate systems. The definitions of the orthogonal coordinate systems are given such that it is possible to implement coordinate conversion routines using only the most basic inputs: The first few Gauss coefficients in the International Geomagnetic Reference Field (IGRF) model, and time. The paper differs from other reviews on space physics coordinates (Matsushita and Campbell 1967; Russell 1971; Fraser-Smith 1987; Fränz and Harper 2002; Barton and Tarlowski 1991; Hapgood 1992) most notably by the inclusion of the non-orthogonal coordinate systems, which we also explain how to use with vector quantities.

In Sect. 2 we describe the geocentric and geodetic coordinate systems. Section 3 contains the definitions of the orthogonal magnetic coordinate systems, and Sect. 4 the non-orthogonal coordinate systems. Section 5 explains how to handle electrodynamic vector quantities in such systems. Section 6 is about magnetic local time, and the secular variation of magnetic coordinate systems is briefly discussed in Sect. 7. The appendices contain a list of abbreviations (Appendix A), notations (Appendix B), and code to calculate the position of the Sun in geocentric coordinates (Appendix C).

## 2 Geocentric and Geodetic Coordinates

The Earth is slightly oblate, and **r** is not precisely normal to its surface except at the poles and equator. The World Geodetic System 1984 (WGS84) datum defines a reference Earth surface as an ellipsoid with equatorial radius \(R_{eq} =6,378,137.0\) m and a reciprocal of flattening \(1/f=298.257223563\), giving a polar radius of \(R_{p} =6,356,752.3\) m. Geodetic (or geographic) coordinates are defined with respect to this surface. Altitude \(h\) is the distance of a point from the surface. The gradient of \(h\) gives a unit vertical (upward) vector \(\hat{\mathbf{k}}\). Geodetic longitude is identical with geocentric longitude. Geodetic colatitude is the angle between \(\hat{\mathbf{k}}\) and the Earth’s axis, and differs slightly from geocentric colatitude except at the poles and equator. Geodetic latitude, \(\lambda_{gd}\), is 90^{∘} minus geodetic colatitude.

Note that the ENU directions in (4) refer to geocentric coordinates, so that “north” is northward on a spherical Earth, and “up” is radially with respect to the center of the Earth. ENU commonly refers to the corresponding geodetic directions, in which case an additional rotation must be applied, about the east direction, to make the northward vector tangent, and the upward vector (which we then call \(\hat{\mathbf{k}}\)) normal to the ellipsoid.

Geocentric coordinates are used throughout Sect. 3. Geodetic coordinates are used in Sect. 4.

## 3 Orthogonal Magnetic Coordinate Systems

In this section we define the most common orthogonal magnetic coordinate systems in terms of their origin and Cartesian base vectors represented in geocentric coordinates. When these coordinate vectors are known, it is straightforward to construct matrices to rotate between coordinate systems, and to calculate corresponding spherical coordinates by use of (2). The information given here should be sufficient to write computer software for coordinate transformations between all the orthogonal magnetic coordinate systems and geocentric coordinates.

The vector \(\hat{\mathbf{m}}\) is needed in the definitions of all the orthogonal magnetic coordinate systems described below. Some of the coordinate systems also depend on the direction of the Sun in the geocentric system. A unit vector, \(\hat{\mathbf{s}}\), pointing in this direction can be expressed in the same way as the components of \(\hat {\mathbf{m}}\) in (13), using the geocentric colatitude and longitude of \(\hat{\mathbf{s}}\) instead of the pole location. These values of geocentric colatitude and longitude are also the values of geodetic colatitude and longitude of the point on Earth where the Sun is in zenith (the subsolar point), since \(\hat{\mathbf{k}}\) at the subsolar point is essentially parallel to \(\hat{\mathbf{s}}\), even if the line between the Earth center and the Sun does not precisely pass through the subsolar point owing to the Earth’s oblateness. The subsolar point depends only on the time and date. Python code which can be used to calculate its coordinates is given in Appendix C.

### 3.1 Centered Dipole (CD) Coordinates

The centered dipole coordinate system is arguably the most commonly used magnetic coordinate system. It is often called geomagnetic, geomagnetic dipole, or simply magnetic coordinates, and abbreviated MAG (e.g., Russell 1971). We use the term centered dipole, and abbreviation CD, which is more descriptive in terms of its definition, and to distinguish between this and the other magnetic coordinate systems. Earth-fixed magnetic coordinate systems such as centered dipole coordinates are often used for magnetically organized phenomena close to ground, for instance to represent ground magnetic field perturbations or to organize measurements of geomagnetic disturbances.

An ENU basis in centered dipole coordinates (\(\hat{\mathbf{e}}_{cd}\) pointing eastward along CD circles of latitude and \(\hat{\mathbf {n}}_{cd}\) pointing along CD meridians) can be found by applying \(\underset{geo\rightarrow cd}{\mathbf{R}}\) to \(\hat{\mathbf{e}}, \hat {\mathbf{n}}\), and \(\hat{\mathbf{u}}\) as defined in (4). A matrix whose rows are the resulting vectors is the change of coordinate matrix from geocentric ECEF to centered dipole ENU coordinates.

### 3.2 Local Magnetic Coordinates

Magnetic field measurements are often represented by so-called magnetic elements, three of which are \(H\), \(D\) and \(Z\), where \(H\) is the horizontal field strength, \(D\) is the declination angle (angle with geodetic north, positive eastward) and \(Z\) is the downward component of the field. Magnetic perturbations to the main field are often presented with an \(H\) component with a somewhat different meaning; the component of the perturbation in the direction of the main field, i.e. in the compass direction. This direction is often empirically determined using geomagnetic quiet days or other techniques to determine a baseline magnetic field. The other components of such a local magnetic coordinate system are often local magnetic east (\(90^{\circ}\) clockwise of the \(H\) direction, seen from above) and up.

**B**and \(\hat{\mathbf{k}}\) is a unit upward vector, then unit vectors in the local magnetic east (\(\hat{\mathbf{e}}_{m}\)) and north (\(\hat{\mathbf {n}}_{m}\)) directions are

**B**in the local magnetic-meridional plane, positive upward/poleward, is

Local coordinates are easy to determine from local measurements of the field, but are not generally conjugate to local coordinates at other places along a field line, like the opposite hemisphere. Interhemispheric differences between dip latitudes for high-latitude field lines can become quite large, and variations of dip latitude along a meridian are not always monotonic, as near South Africa in Fig. 2.

### 3.3 Eccentric Dipole (ED) Coordinates

The centered dipole upon which centered dipole coordinates are based depends on the three first coefficients in a spherical harmonic representation of the Earth’s magnetic field. These coefficients are enough to account for about \(95\%\) of the Earth’s magnetic field (Lowes 1994). It is possible to have a dipole description of a larger fraction of the field by treating its origin and orientation as free parameters. However, according to Lowes (1994), who described five different techniques to estimate such a dipole, “there is no optimum eccentric dipole with which to represent the geomagnetic field approximately”.

The base vectors in eccentric coordinates are the same as for centered dipole coordinates (16), since the orientation of the dipole axis is kept the same. That means that coordinate transformations to ED coordinates amounts to a translation from the CD system by a vector \(\mathbf{t}\), whose geocentric components are \((\Delta x, \Delta y, \Delta z)\), given in (23)–(25). The CD components of \(\mathbf{t}\) are \(\Delta x_{cd} = \mathbf{t}\cdot\hat{\mathbf{x}}_{cd}\), \(\Delta y_{cd} = \textbf {t}\cdot\hat{\textbf{y}}_{cd}\) and \(\Delta z_{cd} = \textbf{t}\cdot\hat {\textbf{z}}_{cd}\).

Fraser-Smith (1987) presents direct relationships between the angular coordinates in centered and eccentric dipole coordinates, based on work by Cole (1963). He also presents expressions for the eccentric dipole pole positions in terms of geocentric coordinates. Apart from an approximation in his expression for pole positions, his formulas give the same results as does the method described above, but computer implementation is more involved, since it requires extra checks to find the correct quadrant for the angles in some cases. This is avoided here if the atan2-function is used.

### 3.4 Geocentric Solar Magnetic (GSM) Coordinates

### 3.5 Solar Magnetic (SM) Coordinates

Solar magnetic coordinates are much used in regions which are more strongly dominated by the Earth’s magnetic field, such as in the inner magnetosphere. Like CD coordinates, the \(z\) axis in SM coordinates is the dipole axis. The other axes are defined such that the Sun–Earth line is contained in the \(xz\) plane. Thus the difference from CD coordinates is a rotation about the \(z\) axis. The \(y\) axis, being perpendicular to both the dipole axis and the Sun–Earth line, is the same as in GSM coordinates. The difference between SM and GSM coordinates is thus a rotation about the \(y\) axis by the dipole tilt angle (15). In SM coordinates, the dipole axis is static, while the Sun–Earth line moves in the \(xz\) plane, in contrast to GSM coordinates, where the Sun–Earth line is static and the dipole axis moves.

The spherical SM coordinates are similar to spherical CD coordinates apart from a different definition of longitude. The \(0^{\circ}\) longitude in SM coordinates will contain the subsolar point, and thus represents the centered dipole noon meridian (see Sect. 6 about magnetic local time).

## 4 Nonorthogonal Magnetic Coordinate Systems

The magnetic coordinate systems defined in the previous section take into account only the first few terms in the spherical harmonic representation of the Earth’s magnetic field (10). Since each term is proportional to \(1/r^{n+2}\), the dipole field (\(n=1\)) dominates at large distances. Taking into account higher degree terms at high altitudes is therefore often inexpedient. At ionospheric altitudes however, the higher degree terms can be significant. In this section we describe some of the coordinate systems which are defined using the IGRF at full resolution (to degree 13 in 2015). They have the common feature that they are nonorthogonal, and that they would reduce to CD coordinates if \(g_{n}^{m}\) and \(h_{n}^{m}\) in (10) were zero for \(n, m > 1\) (i.e. a pure dipole field), and the Earth were spherical. The inclusion of higher order terms means that the coordinate systems described here can be used to construct models and organize data with a higher precision with respect to the Earth’s magnetic field than with e.g. centered dipole coordinates.

### 4.1 Magnetic Apex Coordinates

QD coordinates are useful for magnetically organized phenomena that have a specific height distribution, such as ionospheric currents, which are confined to the conducting layer of the ionosphere. The three coordinates in QD coordinates are \(\phi_{qd}, \lambda_{qd}\) and \(h\), whereas in Modified Apex coordinates the third coordinate is the magnetic potential \(V\), which relates to the IGRF magnetic field (10) as \(\mathbf{B} = -\nabla V\). The third coordinate in Modified Apex coordinates is rarely explicitly used (in contrast to the third coordinate \(h\) in QD coordinates), but it is useful for reasons that we will return to in Sect. 5.

Direct calculation of apex coordinates by field line tracing is computationally expensive and complicated to implement. Emmert et al. (2010) published Fortran software for computationally compact conversion between apex and geodetic coordinates, using spherical harmonic analysis. Their code has been used to produce the figures in this paper.^{1}

Application of apex coordinates is only possible for planets like Earth whose magnetic field is not very distorted. On Mars for example, every point will not have unique MA or QD coordinates, since the same apex height (and consequently latitude), can appear multiple times along a given dipole longitude. This is because Mars lacks a dynamo in the core, so that small-scale magnetic patches in the crust dominate.

### 4.2 Corrected Geomagnetic Coordinates (CGM)

At lower latitudes however the differences can be significant. In some regions surrounding the QD equator (which is also the dip equator), the CGM coordinates are not even defined, since there are field lines which never cross the CD equatorial plane. The CGM equator (\(\lambda_{cgm} = 0\)) has been mapped to both hemispheres, and is shown as bold contours in Fig. 7. Between these contours, in the shaded areas, CGM coordinates are undefined. The CD equator, marked with bright red, lies at one of the boundaries and the other is traced by points which are magnetically conjugate (i.e., at the other field line intersection with \(h=0\)) to the CD equator. The CD equator is far south of its conjugate points in the Atlantic sector, but the orientation is reversed at some points along the dip equator.

Early implementations of CGM coordinate conversions were based on lookup tables (see e.g., Gustafsson et al. 1992, and references therein), and conversions were only computed for points at \(h = 0\). Thus CGM coordinates are sometimes described as undefined at \(h > 0\). Later implementations, using spherical harmonics (Baker and Wing 1989), allowed conversion of points at higher altitudes, and has become known as altitude-adjusted corrected geomagnetic (AACGM) coordinates. The definition of AACGM coordinates is mathematically the same as that described above for CGM coordinates. Despite the CGM/AACGM coordinates being undefined in certain regions, computer implementations prior to Shepherd (2014) have provided coordinates in these regions, using interpolation techniques. However Shepherd (2014) found that the interpolation introduced errors affecting also poleward latitudes. His code is more accurate than earlier software in the region where the coordinates are defined. The CGM coordinates in Fig. 7 were calculated by field line tracing using the software published by Shepherd (2014), which is available in the C and IDL languages.^{2} The software also contains code to calculate conversions using spherical harmonics, which is much faster than with field line tracing.

## 5 Vector Quantities in Nonorthogonal Coordinates

The fact that CGM and apex coordinates are nonorthogonal implies that conversion of vectors is more complicated than a multiplication by a rotation matrix. In this section we describe how to do such conversions, starting by briefly reviewing a text-book description on nonorthogonal coordinates. Then we describe the technique introduced by Richmond (1995), which leads to certain very convenient properties for electrodynamic quantities regarding mapping along magnetic field lines. This section is partly motivated by the fact that several authors use CGM or apex grids with vectors represented in e.g. CD or local magnetic coordinates. This practice is mathematically unsound, since the vectors, which should be invariant with respect to the coordinate system, are effectively changed. This can lead to systematic errors between longitudes and hemispheres (Gasda and Richmond 1998; Laundal and Gjerloev 2014). In Sect. 5.2 we look at such errors in a quantitative fashion.

This is a generalization of base vectors in Cartesian coordinate systems, where \(\pmb{\epsilon}^{i} = \pmb{\epsilon}_{i}\), and the base vectors are orthogonal and of unit length. For example, the spherical coordinate base vectors \(\hat{\mathbf{e}}, -\hat{\mathbf{n}}, \hat {\mathbf{u}}\) (4) can be derived by calculating \(\frac {\partial\mathbf{r}}{\partial u_{i}}/\|\frac{\partial\mathbf{r}}{\partial u_{i}}\|\), with \(\mathbf{r}\) given on the right hand side of (1), and \(u_{i} = r, \theta, \phi\). The same result would appear by calculating the gradient of the spherical coordinates (45). Notice the change in sign of \(\hat{\mathbf{n}}\), since \(\frac{\partial \mathbf{r}}{\partial\theta}\) is southward when \(\theta\) is colatitude. If \(\mathbf{r}\) was differentiated with respect to the latitude, the result would be northward.

In a non-Cartesian coordinate system, such as CGM and Magnetic Apex Coordinates, \(\pmb{\epsilon}^{i}\) is in general not equal to \(\pmb {\epsilon}_{i}\). \(\pmb{\epsilon}^{i}\) is perpendicular to surfaces of constant \(u_{i}\), and its length depends on the spacing between these surfaces. \(\pmb{\epsilon}_{i}\) is tangent to the contours defined by the intersection between the other two coordinate surfaces, and its length depends on the rate of change of \(u_{i}\) along these contours.

### 5.1 Apex Base Vectors

Since the field is not a dipole, \(\mathbf{d}_{1}\) in general does not point strictly eastward along contours of constant Modified Apex latitude. It points perpendicular to Modified Apex meridians in surfaces of constant \(V\). Likewise, \(\mathbf{d}_{2}\) points roughly equatorward and downward, exactly perpendicular to contours of constant latitude in surfaces of constant \(V\). Both \(\mathbf{d}_{1}\) and \(\mathbf {d}_{2}\) are perpendicular to the magnetic field. Because of the non-dipole terms, \(\mathbf{d}_{i}\) will in general not have unit length at \(r = R_{E} + h_{R}\), and the vectors will not be perpendicular to each other. The quantity \(D = \|\mathbf{d}_{1}\times\mathbf{d}_{2}\|\) can be seen as a measure of the deviation from a dipolar field at \(r = R_{E} + h_{R}\). A map of \(D\) can be found in the paper by Richmond (1995).

As is clear from the definitions of the base vectors, the gradients of the magnetic coordinates must be calculated in order to find their components. Computer code which provides all base vectors (except for \(\mathbf{g}_{i}\) and \(\mathbf{f}_{3}\) which can easily be found when the others are known using relations in Richmond 1995) was published by Emmert et al. (2010).

#### 5.1.1 Decomposing \(\mathbf{E}, \mathbf{j}, \mathbf{v}\) and \(\Delta\mathbf{B}\)

**J**in terms of scaled QD components:

**J**can be related to the curl of the magnetic-perturbation field \(\Delta\mathbf{B}\):

Although (84)–(89) can readily be used to calculate **J** from \(\Delta\mathbf{B}\) in QD coordinates, the inverse calculation to get \(\Delta\mathbf{B}\) from **J** would be considerably more difficult in QD coordinates than in geocentric spherical coordinates, because the Biot–Savart relation, or a spherical-harmonic representation of it, becomes very complicated in QD coordinates, owing to the complicated calculation of distances and the non-orthonormality of the base vectors.

*SML*index (Newell and Gjerloev 2011) yields less longitude-dependent variation when computed with \(\Delta B_{q\lambda}\) than when computed with the unscaled magnetic perturbations in the local magnetic-north direction. If satellite magnetic-perturbation data are binned with respect to magnetic latitude and magnetic local time, it can also be convenient to orient and scale them using the components \(\Delta B_{q\phi}\), \(\Delta B_{q\lambda}\), and \(\Delta B_{qr}\) as defined in (87)–(89). One reason for this is that scaled current-density components can be calculated from (84)–(86) directly in QD coordinates without the need for longitude-dependent scale factors. \(J_{f_{1}}\) and \(J_{f_{2}}\), as calculated from (84) and (85), represent the scaled horizontal current density components. For the vertical current density, on the other hand, \(J_{f_{3}}\) as computed from (86) does not itself make a convenient representation, because it depends not only on the QD coordinates, but also on the geographically varying scale factor \(F^{2}\). Instead, the appropriate scaled vertical current density component is a quantity we can call \(J_{qr}\), defined by

While the calculation of the divergence of **J** as expressed by (82) in QD coordinates is fairly straightforward (and results in identically zero divergence), it is not straightforward to calculate the divergence of \(\Delta\mathbf{B}\) as expressed by (90) in QD coordinates. (One might wish to calculate the divergence of an empirical model of \(\Delta\mathbf{B}\), for example, to ensure it is zero.) This difference in ease of calculation of the divergence is due to the fact that **J** in (82) is expressed in components related to covariant base vectors, while \(\Delta\mathbf{B}\) in (90) is expressed in components related to contravariant base vectors.

Richmond (1995) presented a way to relate magnetic perturbations to three-dimensional currents partially using Modified Magnetic Apex coordinates, but requiring an additional calculation of a three-dimensional magnetic potential by solving Poisson’s equation in geographic coordinates. That paper pointed out how the source term for the Poisson’s equation can be made relatively small by first solving for a suitable two-dimensional function, constant along field lines, that minimizes the source term in a region of interest. Apart from the component of the magnetic perturbation associated with the solution of the Poisson’s equation, the remaining portion of magnetic perturbations associated with field-aligned currents maps along geomagnetic-field lines. (\(\Delta B_{q\phi}\), \(\Delta B_{q\lambda}\), and \(\Delta B_{qr}\) do not map along field lines.) We are not aware of any attempt to implement the procedure outlined in that paper with numerical algorithms.

### 5.2 Applications/Examples

Modified Magnetic Apex and Quasi-Dipole coordinates can be useful for examining how plasma velocities vary along a geomagnetic field line in the ionosphere.

**v**with the local orthogonal unit vectors \(\hat{\mathbf{e}}_{m}\) and \(\hat{\mathbf{p}}\). Figure 9(a) shows the resultant magnetic-eastward velocity, and Fig. 9(b) shows the resultant magnetic-meridional velocity. By a similar procedure we can start from a unit velocity in the magnetic-meridional direction \(\hat {\mathbf{p}}_{m}^{conj}\) at the conjugate point, map it along the magnetic field to the local point, and calculate the local magnetic-eastward and magnetic-meridional components. These are shown in Fig. 9(c) and (d). For a purely dipolar geomagnetic field on a spherical Earth all the values in Fig. 9(a) and (d) would be 1, and all the values in Fig. 9(b) and (c) would be 0. In Fig. 9 values greater than these idealized dipolar values are plotted with red contours, while values less than the dipolar values are plotted with blue contours.

We can note some of the larger deviations from a purely dipolar mapping. A magnetic-eastward velocity of 1 m/s at 250 km altitude, 30^{∘}N QD latitude, corresponds at 30^{∘}S QD latitude to a magnetic-eastward velocity ranging from as small as 0.84 m/s in the middle South Atlantic Ocean to 1.51 m/s on the eastern coast of South America [Fig. 9(a)]. Owing to the twist of magnetic flux tubes, a purely horizontal magnetic-eastward velocity at 40^{∘}N QD latitude acquires a magnetic-meridional component that at 40^{∘}S QD latitude can be as large as 0.43 m/s downward/equatorward around \(-45^{\circ}\mbox{E}\) geographic longitude, or 0.43 m/s upward/poleward around \(+45^{\circ}\mbox{E}\) geographic longitude [Fig. 9(b)]. A magnetic-meridional velocity of 1 m/s at 250 km altitude, \(44^{\circ}\mbox{N}\) QD latitude, and \(60^{\circ}\mbox{E}\) QD longitude (the QD longitude line that grazes the coast of West Africa) corresponds at \(44^{\circ}\mbox{S}\) QD latitude to about 2.24 m/s in the local magnetic-meridional direction [Fig. 9(d)]. Above about \(70^{\circ}\) magnetic latitude the local magnetic-eastward and magnetic-meridional directions can deviate greatly from those of a tilted dipole, and the mapping factors can differ greatly from the dipole values of 0 or 1.

## 6 Magnetic Local Time

The CD, ED, CGM, QD, and MA coordinates are all fixed with respect to the Earth. It is often appropriate however to introduce a magnetic local time, instead of magnetic longitude, in order to organize data and models with respect to the position of the Sun (Vegard 1912, 1917).

One such definition of MLT is the hour angle (1 hour is \(15^{\circ}\) magnetic longitude) from the midnight magnetic meridian, positive in the magnetic eastward direction. The midnight magnetic meridian can be defined as the meridian that is \(180^{\circ}\) magnetic longitude away from the subsolar point. The MLT/magnetic latitude coordinate system will then rotate with respect to the Earth at the rate at which the subsolar point crosses magnetic meridians.

## 7 Secular Variations

^{3}and QD/MA poles, are shown in Fig. 12. Their positions in 1950 are marked with large circles. Later positions are marked at 5 years intervals, ending in 2015. The centered dipole poles mark the intersection of the centered dipole axis with the Earth’s surface. For all the other poles, the axes connecting them will be eccentric with respect to the center of the Earth. We see that the eccentric dipole poles and apex poles do not coincide at the surface. At very high altitudes, all the poles approach the centered dipole poles.

^{∘}latitude at Northern polar latitudes and 5

^{∘}latitude at mid and low latitudes.

## 8 Closing Remarks

Another magnetic coordinate system, the magnetospheric geomagnetic latitude, was defined by Papitashvili et al. (1997a,b). This system takes into account also the external, highly time varying magnetospheric field by means of the Tsyganenko (1989) model.

The terminology associated with magnetic coordinates can be confusing and sometimes ambiguous. Traditionally, the term “geomagnetic” has been used to signify the poles, \(z\) axis, and coordinate system associated with what we have called the centered dipole coordinate system (Vestine 1967; Chapman 1963). The term “magnetic” is often used to refer to the dip latitude, dip equator and dip poles, although it can also signify other systems. The terminology used in this paper is chosen because it is well established, and/or descriptive in terms of the definition of the coordinate system.

With at least six differently defined “magnetic latitudes”, and several different definitions of MLT, it is clearly important to be precise about which coordinate system is used, and how MLT was calculated. Since the coordinate systems change in time, the epoch of the geomagnetic field used in the coordinate conversions should also be specified.

## Footnotes

- 1.
A Python wrapper for this code is available at https://github.com/cmeeren/apexpy (by Christer van der Meeren and Karl M. Laundal, University in Bergen).

- 2.
A Python wrapper for this code is available at https://github.com/cmeeren/aacgmv2 (by Christer van der Meeren and Karl M. Laundal, University in Bergen).

- 3.
The dip pole locations were obtained from http://www.geomag.bgs.ac.uk/education/poles.html.

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