### J_{2}

(geopotential coefficient regarding a planet's oblateness)

**J**_{2} is a coefficient reflecting the
gravitational effect of a body's (planet's or moon's) **oblateness**
(extra width making it flatter than a perfect sphere),
generally due to the body's rotation.
J_{2} is of interest for space flight navigation and is
measured by observing the flight of spacecraft near a mass. Its
measure aids in modeling the composition of a body, thus measuring
that of solar system planets is of interest. Cassini has been
used for this with Saturn and Juno with Jupiter.
A spherically symmetric planet has a J_{2} of 0 and the
larger the J_{2}, the more oblate.

J_{2} is one coefficient in a standard
type of model of gravitational potential of such a body:
a **zonal harmonic gravity model** and in the case of Earth,
it is often termed a **geopotential model**
(which term is also sometimes borrowed for other bodies).
The word *zonal* indicates these are the coefficients that do not affect
gravity's symmetry about the axis of rotation.
The coefficients are called **gravitational moments**,
and indicate gravitational the arrangement of mass
other than that of a uniformly-dense sphere.
The set of coefficients generally used presume the body is somewhat
close to spherical, close to hydrostatic equilibrium and rotation is
the major factor producing the moments.
They are representations of the gravitational potential's
spherical harmonic coefficients (those that are axis-symmetric
around the axis of rotation), and are generally cited scaled to the
body's size and radius so as to be dimensionless.

**J**_{3} is a similar coefficient but reflects asymmetry across
the equator, thus is typically far less significant than J_{2}.
Further coefficients (**J coefficients**, J_{4}, J_{5},
J_{6}, etc.) are symmetric across the equator if they are
even-numbered. There are spherical harmonics corresponding to
J_{0} and J_{1} but gravitational potential models are
coordinated and scaled so J_{0} is taken as 1 (the coefficients
are scaled to the total mass of the body) and J_{1}
is 0 (the coefficients are based on the body's center of mass).
The moments corresponding to other (non-axis-symmetric) spherical
harmonics can be significant and of interest but the terms used
vary. Also, various other names are also used for the coefficients
and models described above.

All the solar system planets and the Sun are oblate from to
rotation, and have a significant J_{2} coefficient.
Jupiter and Saturn, with rotation periods of less
than half an Earth-day, are the most oblate.
It is also useful in modeling the behavior of rings.
Values are improving with continuing measurement and analysis but
here are some example determined values:

Body | J_{2} | J_{3} |

Earth | 0.001082 | -0.0000025 |

Mars | 0.001964 | 0.000036 |

Jupiter | 0.01475 | -0.00058 |

Saturn | 0.1656 | -0.001 |

Determinations of the Sun's J_{2} put it in the general
area of 0.00000022.

(*gravity,coefficient,model,measure*)
**Further reading:**

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

https://www.mathworks.com/help/aeroblks/zonalharmonicgravitymodel.html

**Referenced by pages:**

gravitational potential model

Legendre polynomials

Love number

Laplace radius (r_{L})

theory of figures (TOF)

Index