Spherical harmonics

The Earth Gravitational Models (EGM) are a series of geopotential models of the Earth published by the National Geospatial-Intelligence Agency (NGA). They are used as the geoid reference in the World Geodetic System.

The NGA provides the models in two formats: as the series of numerical coefficients to the spherical harmonics which define the model, or a dataset giving the geoid height at each coordinate at a given resolution.

In quantum mechanics, an atomic orbital (/ˈɔːrbɪtəl/ ) is a function describing the location and wave-like behavior of an electron in an atom. This function describes an electron's charge distribution around the atom's nucleus, and can be used to calculate the probability of finding an electron in a specific region around the nucleus.

Each orbital in an atom is characterized by a set of values of three quantum numbers n, ℓ, and m_ℓ, which respectively correspond to an electron's energy, its orbital angular momentum, and its orbital angular momentum projected along a chosen axis (magnetic quantum number). The orbitals with a well-defined magnetic quantum number are generally complex-valued. Real-valued orbitals can be formed as linear combinations of m_ℓ and −m_ℓ orbitals, and are often labeled using associated harmonic polynomials (e.g., xy, x − y) which describe their angular structure.

In geophysics and physical geodesy, a geopotential model is the theoretical analysis of measuring and calculating the effects of Earth's gravitational field (the geopotential).The Earth is not exactly spherical, mainly because of its rotation around the polar axis that makes its shape slightly oblate. However, a spherical harmonics series expansion captures the actual field with increasing fidelity.

If Earth's shape were perfectly known together with the exact mass density ρ = ρ(x, y, z), it could be integrated numerically (when combined with a reciprocal distance kernel) to find an accurate model for Earth's gravitational field. However, the situation is in fact the opposite: by observing the orbits of spacecraft and the Moon, Earth's gravitational field can be determined quite accurately. The best estimate of Earth's mass is obtained by dividing the product GM as determined from the analysis of spacecraft orbit with a value for the gravitational constant G, determined to a lower relative accuracy using other physical methods.

In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors.