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Phoebe, the largest irregular satellite of Saturn.

Ninety-three irregular satellites have been discovered since 1997 orbiting all four giant planets. Before 1997, only ten had been known, including Phoebe, the largest irregular satellite of Saturn, and Himalia, the largest irregular satellite of Jupiter. It is currently thought that the irregular satellites were captured from heliocentric orbits near their current locations, early after the formation of their parent planet. An altenative theory, that they orginated further out in the Kuiper Belt, is not supported by current observations.

Population overview

planet	total	un- named	pro grade	retro grade
Jupiter	53	14	7	46
Saturn	35	21	7	28
Uranus	9	0	1	8
Neptune	6	4	3	3
Total	103	39	18	85

(core facts)

Most of the irregular satellites are retrograde (explained by the assymetry of thestability regions)

Mention Sheppard (ca 100 smaller rocks...)

Mention obesrvational bias: outer planets' populations are expected to be richer than discovered so far.

Divided into groups; each has a prominent member and a varying number of smaller ones

Given their distance and inclination, the orbits are highly perturbed by the Sun. Some of them are involved in complex secular and Kozai resonances. Their orbital elements change dramatically over short intervals. (Pasihae example: 1.5Gm in two years in a, 10 deg in inclination and 0.4 in eccentricity in 24 years; check figures; Carruba 2002!)

The interest of the irregulars

Understanding of the origin of should proving insights into the early epoch of the solar system,

Orbits as a memory of the planetary migration

History rich in collisions (retro hitting progrades, example from Holman 2004)

Tisserand's relation

${\frac {1}{2a}}+{\sqrt {a(1-e^{2})}}\cos i\approx const$

Derivation

2007 NC7 The relation is derived from the Jacobi constant selecting a suitable unit system and some approximations. The relation is an approximate constant of motion for the (third) small mass μ₃orbiting μ₁ but which orbit have been modified b the second mass μ₂. $G(\mu _{1}+\mu _{2})\approx 1\approx G(\mu _{1}+\mu _{3})$

These conditions are statisfied for example for the Sun - Jupiter system with a comet or a spacecraft being the third mass.

From the two-body (μ₁,μ₃) vis-viva equation

$({\dot {\xi }}^{2}+{\dot {\eta }}^{2}+{\dot {\zeta }}^{2})=v^{2}=\mu \left({{2 \over {r}}-{1 \over {a}}}\right)$

For the angular momentum h

$\mathbf {h} =\mathbf {r} \times \mathbf {\dot {r}}$

the $\zeta$ component is $\xi {\dot {\eta }}-\eta {\dot {\xi }}=h\cos I$

where I is the inclination of μ₃ orbit to μ₂ orbit.

substituting these into the Jacobi constant

$C_{J}=2\cdot ({\frac {\mu _{1}}{r_{1}}}+{\frac {\mu _{2}}{r_{2}}})+2n(\xi {\dot {\eta }}-\eta {\dot {\zeta }})-({\dot {\xi }}^{2}+{\dot {\eta }}^{2}+{\dot {\zeta }}^{2})$

and taking μ₂<<1 gives

${\frac {1}{2a}}+{\sqrt {a(1-e^{2})}}\cos i\approx const$

except for small r₂

Jacobi integral

Co-rotating system.

One of the suitable co-ordinates system used is so called synodic or co-rotating system. The line connecting the two masses is chosen as X axis, with the distant unit equal to their distance. The beginning of the system is the barycentre and the system co-rotates with m2, so the masses are stationary and positioned in (-μ₂,0) and (0,-μ₁).

In this co-ordinate system, the Jacobi constant is expressed as follows:

$C_{J}=n^{2}(x^{2}+y^{2})+2\cdot ({\frac {\mu _{1}}{r_{1}}}+{\frac {\mu _{2}}{r_{2}}})-({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2})$

where:

$x\,\!,y\,\!$ are co-ordinates in the co-rotating system

$n={\frac {2\pi }{T}}$ mean motion

$\mu _{1}=Gm_{1}\,\!,\mu _{2}=Gm_{2}\,\!$ are the two masses

$r_{1}\,\!,r_{2}\,\!$ are distances of the test particle from the two masses

Derivation

In the co-rotating system, the accelerations can be expressed as derivatives of a single scalar function $U(x,y,z)={\frac {n^{2}}{2}}(x^{2}+y^{2})+{\frac {\mu _{1}}{r_{1}}}+{\frac {\mu _{2}}{r_{2}}}$

[Eq.1] ${\ddot {x}}-2n{\dot {y}}={\frac {\delta U}{\delta x}}$

[Eq.2] ${\ddot {y}}-2n{\dot {x}}={\frac {\delta U}{\delta y}}$

[Eq.3] ${\ddot {z}}={\frac {\delta U}{\delta z}}$

Multiplying [Eq.1] , [Eq.2] and [Eq.3] par ${\dot {x}},{\dot {y}}$ and ${\dot {z}}$ respectively and adding all three yields

${\dot {x}}{\ddot {x}}+{\dot {y}}{\ddot {y}}+{\dot {z}}{\ddot {z}}={\frac {\delta U}{\delta x}}+{\frac {\delta U}{\delta y}}+{\frac {\delta U}{\delta z}}={\frac {dU}{dt}}$

Integrating yields

${\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2}=2U-C_{J}$

where C_J is the constant of integration.

The left side represents the square of the velocity $v\,\!^{2}$ of the test particle in the co-rotating system.