Other Applications of Spherical Harmonics

Supernova remnants

This idealized Type Ia supernova begins as an approximately spherical explosion. The simulation advances the explosion for 454 years, approximately the present age of Tycho's supernova remnant.

Check "Show only current (ℓ, m)" to see the contribution of a single spherical harmonic mode to the remnant's structure. Uncheck it to see the full superposition of all modes. Random superpositions of modes introduce large-scale asymmetry and increasingly finer structures. Click on the model and rotate it to explore its geometry.

Earth's magnetic field

Earth's magnetic field is often shown as a dipole, like the field around a bar magnet. That familiar pattern is the dominant spherical-harmonic mode with ℓ = 1.

Higher-order modes add fine structure near the planet; the field becomes increasingly dipolar farther from Earth. Click "Show only current (ℓ, m)" to see the contributions of different values, "BUILD FIELD" to see a random superposition of modes that simulate a real field, and "EARTH-LIKE FIELD" to see a magnetic field resembling Earth's. Click and rotate the model to explore its geometry.

Visible gaps or endpoints indicate limits of the model, not breaks in the field. Also missing is the distortion of the field by the solar wind.