Fortran APEX Code

Fortran code to convert between geographic and magnetic quasi-dipole, magnetic apex, and modified magnetic apex coordinate systems based on Van Zandt et al. (1972) and Richmond [1995]. The geomagnetic main field is the International Geomagnetic Reference Field (IGRF) Version 13. The test program includes:

  • Magnetic apex parameters for a given location
  • Base vectors for scaling electric field and ion velocity components
  • Magnetic footpoints in both hemispheres for a given geographic point
  • Conversion between quasi-dipole and geographic coordinates
  • Subsolar point, solar zenith angle, magnetic local time


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***In case you need the updated coefficient file with IGRF13 in an older format please download here.

References for apex coordinate systems:

Emmert, J.T., Richmond, A.D., & Drob, D.P. (2010). A computationally compact representation of Magnetic-Apex and Quasi-Dipole coordinates with smooth base vectors. Journal of Geophysical Research, 115, A08322.

Laundal, K.M. & Richmond, A.D. (2017). Magnetic coordinate systems. Space Science Reviews, 206, 27-59.

Richmond, A.D. (1995). Ionospheric electrodynamics using Magnetic Apex Coordinates. Journal of Geomagnetism and Geoelectricity, 47, 191-212.

Thébault, E., Finlay, C.C., Beggan, C.D. et al. (2015). International Geomagnetic Reference Field: the 12th generation. Earth, Planets and Space 67, 79.

VanZandt, T.E., Clark, W.L.,and Warnock, J.M. (1972). Magnetic Apex Coordinates: A magnetic coordinate system for the ionospheric F2 layer, Journal of Geophysical Research, 77, 2406-2411.


Other resources for apex code:


Graph showing Quasi-Dipole coordinates
Quasi-Dipole coordinates at 250 km altitude, derived from the International Geomagnetic Reference Field (IGRF), Version 12 (Thébault et al., 2015), for epoch 2020.0. The dashed grid represents geographic coordinates, while lines of constant apex latitude and longitude are drawn at 10◦ intervals. The thick contours represent the magnetic equator and magnetic longitudes of 0◦ (Americas) and 180◦ (Asia).