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Ananke group


The group is believed to have been formed when an asteroid was captured by Jupiter and subsequently fragmented. Based on the sizes of the satellites, the original asteroid may have been about 28 km in diameter. Since this value is near the approximate diameter of Ananke itself, it is likely the parent body was not heavily disrupted.

More recent calculations of the velocity impulse δV necessary to create the observed dispersion of the orbital parameters in the group yield 15 < δV < 80 m/s, a value considered compatible with a single collision event.

Available photometric studies add further credibility to the common origin thesis: three of the moons of the family (Harpalyke, Praxidike and Iocaste) display similar grey colours (average colour indices: B−V = 0.77 and V−R = 0.42) while Ananke itself appears on the limit between grey and light red.

The diagram illustrates the Ananke group in relation to other irregular satellites of Jupiter. The eccentricity of selected orbits is represented by the yellow segments (extending from the pericentre to the apocentre). The outermost regular satellite Callisto is located for reference.

The eight core members are (from the largest to the smallest): S/2003 J 16, Mneme, Euanthe, Harpalyke, Praxidike, Thyone, Ananke, Iocaste.



Orbit parameters
The diagram illustrates the orbits of the irregular satellites of the giants planets discovered so far. The semi-major axes are expressed as a fraction of the planet’s Hill sphere’s radius and inclination represented on Y axis. The satellites above the X axis are prograde, the satellites beneath are retrograde.

Colours
Observed colours vary from neutral to reddish but not as red as the colours of some Kuiper Belt objects (KBO). Each planet's system displays slightly different characteristics. Jupiter's irregulars are grey to slightly red, consistent with C, P and D-type asteroids. Groupings with similar colours can be identified (see later sections). Saturn's irregulars slightly redder than that of Jupiter. The very red colours typical for classical KBOs are rare among the irregulars.

Colour indices are simple measures of differences of the apparent magnitude of an object through blue (B), visible (V) i.e. green-yellow and red (R) filters. The diagram illustrates these differences (in slightly enhanced colour) for the irregulars with known colour indices. For reference, the Centaur Pholus and three classical Kuiper Belt objects are plotted (grey labels, size not to scale). For comparison, see colours of centaurs and KBOs.

Irregular satellites of Jupiter
Typically, the following groupings are listed (dynamically tight groups with displaying homogenous colours are listed in bold)
 * Prograde satellites
 * Himalia group (inclination 28° cluster): confined dynamically (δV~150m/s); very homogenous: neutral colours similar to C-type asteroids.
 * Themisto (isolated so far)
 * Carpo (isolated so far)
 * Retrograde satellites
 * Carme group (165° cluster): dunaically tight (5<δV<50m/s)very homogenous, light-red colour consistent with a D-type progenitor
 * Ananke group (148° cluster): little dispersion of orbital parameters (15<δV<80m/s); Ananke itself appears light-red while the satellites following similar orbits are grey
 * Pasiphae group: dispersed; Pasiphae appears to be grey while other members are light-red 1.

1 Sinope, sometimes included into Pasiphae group, is thought to be independent, trapped in a secular resonance with Pasiphae

Irregular satellites of Saturn
Typically, the following groupings are listed (dynamically tight groups with displaying homogenous colours are listed in bold)
 * Prograde satellites
 * Gallic group (inclination 34° cluster): tight dynamically (δV~50m/s), homogenous ((light-red colours)
 * Inuit group: (34° cluster) dispersed (δV~350 m/s) but homogenous (light-red colours)
 * Retrograde satellites
 * Norse group is defined mostly for naming purposes; the orbital parameter’s dispersion is large and different sub-divisions have been investigated, including
 * Phoebe group (174° cluster); large dispersion suggesting at least two sub-groupings
 * Skathi sub-group

Jacobi integral
$$C_J=n^2(x^2+y^2)+2\cdot (\frac{\mu_1}{r_1}+\frac{\mu_2}{r_2})-(\dot x^2+\dot y^2+\dot z^2)$$

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 CJ is the constant of integration.

The left represents the square of the velocity $$v^2$$ of the test particle in the co-rotating system.

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 masses

$$r_1\,\!,r_2\,\!$$ are distances of the test particle from the two masses

Irregulars of Uranus and Neptune
It is believed that the relatively poorer (known) populations of irregulars for Uranus and Neptune are explained by the varying observational limits (table on the left, the albedo of 0.04 is assumed).

Statistically significant conclusions about the groupings are difficult. Single origin for the retrograde irregulars of Uranus seems unlikely given the dispersion of the orbital parameters that would require high impulse (~300 km) implying a large diameter of the impactor (395km), incompatible in turn with the size distribution of the fragments. Instead, the existence of two groupings is speculated


 * Caliban group
 * Sycorax group

For Neptune, the possible common origin of Psamathe and S/2002 N4 was noted.

Sizes
The table gives the maximal apparent magnitude [m], the absolute magnitude and the the minimal size [D} assuming for the the irregulars of all giant planets.

HPD draft
H, P -> D

Early draft only; the formula may be wrong, the data are unchecked, there's no real error bars, colour spheres could be used to illustrate colors etc...
 * Diameter D[km] on the vertical axis (linear)
 * Absolute magnitude H[] on the horizontal axis (log)
 * Albedo p[] grey curves

Orbit of Ceres
Ceres follows an orbit between Mars and Jupiter, inside the Main asteroid Belt, completing it in 4.6 years. Its orbit is moderately inclined (i=10.6° to be compared with 7° for Mercury and 17° for Pluto) and moderately eccentric (e=0.08 to compare with 0.09 for Mars).

The diagram illustrates the orbits of Ceres (yellow) and Mars (red) as seen from the ecliptic pole (top) and from the ascending node (below). The segments of orbits below the ecliptic are plotted in darker colours. The periphelia (q) and aphelia (Q) are labelled with the date of the nearest passage. Ceres is currently approaching its aphelion (October 2006) while Mars passed its perihelion in June. Interestingly, the perihelia of Ceres and Mars are on the opposite side of the Sun, minimising the perturbation of Ceres Orbit by Mars. Ceres, like most asteroids, is highly perturbed by Jupiter.