User:Iridia/J 2

The quadrupole gravitational coefficient $$J_{2}$$ is a dimensionless value used in planetary physics to measure the distortion of the gravitational field of a spinning, oblate spheroid planet. A stationary planet will be perfectly spherical, and have a spherical surrounding gravitational potential: as its rotation speed increases, the planet will deform under the centrifugal force, developing an equatorial bulge. This redistribution of mass alters the shape of the gravitational potential around the planet. The magnitude of $$J_{2}$$ provides information on the planet's internal density structure, including the size of its most dense region, the core. $$J_{2}$$ is best found by measuring the path of spacecraft in orbit around or flying by a planet. It can also be calculated, though with less precision, if the spin of a planet can be measured.

Theory
Planets and dwarf planets are in hydrostatic equilibrium, and most have rotational flattening, so can be considered as oblate spheroids: triaxial ellipsoids with two equal long axes and one short axis. The external gravitational potential of a body with an axis of symmetry can be written:
 * $$V_{gravity}(r,\theta) = -\frac{Gm}{r}[1 - \Sigma^{\infty}_{n=2} J_{n} (\frac{R}{r})^{n} P_{2}(\cos\theta)]$$
 * $$m$$: total mass
 * $$R$$: equatorial radius of the body
 * $$J_{n}$$: dimensionless constants, the quadrupole gravitational coefficients
 * $$P_{n}(\cos\theta)$$: Legendre polynomials of degree n. Choosing the origin of the coordinates ($$r, \theta$$) as the body's centre of mass means there is no $$n = 1$$ term.

The $$J_{n}$$ values are given by:
 * $$J_{n} = +\frac{1}{mR^{n}} \int^{R}_{0} \int^{+1}_{-1} r^{n} P_{n}(\mu) \rho(r, \mu) 2\pi r^{2} d\mu dr$$
 * $$\mu = \cos\theta$$
 * $$\rho(r, \mu)$$: the body's internal density distribution, its internal arrangement of mass.

Considering the magnitude of these $$J_{n}$$ values in turn, $$P_{2}(\mu)$$ is an odd function when n is odd, so
 * $$J_{(2n + 1)} = 0$$

if the body's northern and southern hemispheres are symmetric. The higher-order $$J_{n}$$ terms rapidly become small:
 * $$J_{n} \propto q^{n/2} $$

provided that q is small.

$$J_{2}$$ has a physical interpretation: it can be represented as a function of the three moments of inertia, A, B, and C, about the rotational axes of the body:
 * $$J_{2} = \frac{C - \frac{1}{2}(A + B)}{ma^{2}} \approx \frac{C - A}{ma^{2}} $$

which is valid under rotational distortion, when the two hemispheres are symmetric: $$A \approx B$$.

Effects and measurement
The modification of a planet's gravity field by its $$J_{2}$$ affects the orbits of objects travelling around or near the planet. There are several ways in which this happens; the most prominent is the rotation of the path of the orbit in the plane of the orbit, precession. Observing the orbits of satellites and some eccentric rings allows $$J_{2}$$ to be directly quantified. From this, the quantity (C-A) can be found, and through further relations, they can be determined independently. Once they are known, models of the interior of the planet can be constrained, limited to set ranges of values.

Repeated, close encounters of a large satellite or planet by a spacecraft allow measurements of $$J_{2}$$ to be made at each encounter. This produces gradual refinements in the value, and can mean the measurements of the moments will be extremely accurate. The rigidity of a satellite may also be found.