### spherical harmonics

(harmonic functions on the surface of a sphere)

**Spherical harmonics** are functions on the surface
of a sphere which fulfill the same role as the
sine function providing harmonics for periodic
functions.

**Laplace spherical harmonics** (a common type,
often what is meant by *spherical harmonics*)
effectively divide the sphere into portions,
portioning the sphere by number of meridian divisions
and number of latitude divisions. The harmonic's **mode**
is designated by two numbers,
*l* and *m*, *l* (the **degree**, **multipole moment**
or **multipole number**)
being a natural number indicating the number of latitude-like divisions,
and *m* (the **order** or **azimuthal number**)
being a natural number indicating the number of meridian-like divisions,
there being no more meridian divisions than
latitude divisions. For example, for *l*=1, *m*=1,
the sphere is divided along an equator-like line and
a meridian-like line, resulting in four portions.

Spherical harmonics are used for describing gravity fields of
planets
(and the letters *l* and *m* probably grew out of analysis Earth's
gravity field in terms of latitude-divided and meridian-divided
modes).
They are also used in describing seismology
(including asteroseismology), and are
of interest in the theory of
core collapse supernovae.
They are also used in characterizing the
distribution of the cosmic microwave background (CMB) variations
around the celestial sphere (CMB anisotropies).
They can be used in characterizing weather
around a world.
They are also used within a technique for solving some
types of differential equations.

(*mathematics*)
**Further reading:**

http://en.wikipedia.org/wiki/Spherical_harmonics

**Referenced by pages:**

angular power spectrum

CMB anisotropies

Goddard gravity model (GGM)

gravitational potential model

J_{2}

multipole expansion

theory of figures (TOF)

Index