Tarski–Grothendieck set theory

This diagram shows the orbits of Saturn's irregular satellites. At the centre, the orbit of Titan, a regular satellite, is highlighted in red for comparison.

In astronomy, an irregular moon is a natural satellite following a distant, inclined, and often eccentric and retrograde orbit. They are believed to have been captured by their parent planet, unlike regular satellites, which formed in situ.

113 irregular satellites have been discovered, orbiting all four of the giant planets (Jupiter, Saturn, Uranus and Neptune). Before 1997, when Uranian irregulars Caliban and Sycorax were discovered, only eleven were known (counting Triton), 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, shortly after the formation of their parent planet. An alternative theory, that they originated further out in the Kuiper belt, is not supported by current observations.

Definition

Planet	rH, 106 km	rmin, km	Number
Jupiter	51	1.5	59
Saturn	69	3	38
Uranus	73	7	9
Neptune	116	16	7 (counting Triton)

There is no widely accepted precise definition of an irregular satellite. Informally, satellites are considered irregular if they are far enough from the planet that the precession of their orbital plane is primarily controlled by the Sun.

In practice, the satellite's semi-major axis is compared with the planet's Hill sphere (that is, the sphere of its gravitational influence) $r_{H}$. Irregular satellites have semi-major axes greater than 0.05 $r_{H}$ with apoapses extending as far as to 0.65 $r_{H}$. The radius of the Hill sphere is given in the adjacent table. 50 year old Petroleum Engineer Kull from Dawson Creek, spends time with interests such as house brewing, property developers in singapore condo launch and camping. Discovers the beauty in planing a trip to places around the entire world, recently only coming back from .

Orbits

Current distribution

Irregular satellites of Jupiter (red), Saturn (yellow), Uranus (green) and Neptune (blue) (excluding Triton). The horizontal axis shows their distance from the planet (semi-major axis) expressed as a fraction of the planet’s Hill sphere's radius. The vertical axis shows their orbital inclination. Points or circles represent their relative sizes.

The orbits of the known irregular satellites are extremely diverse, but there are certain rules. Retrograde orbits are far more common (83%) than prograde orbits. No satellites are known with orbital inclinations higher than 55° (or smaller than 130° for retrograde satellites). In addition, some groupings can be identified, in which one large satellite shares a similar orbit with a few smaller ones.

Given their distance from the planet, the orbits of the outer satellites are highly perturbed by the Sun and their orbital elements change widely over short intervals. The semi-major axis of Pasiphae, for example, changes as much as 1.5 Gm in two years (single orbit), the inclination around 10°, and the eccentricity as much as 0.4 in 24 years (twice Jupiter’s orbit period). Consequently, mean orbital elements (averaged over time) are used to identify the groupings rather than osculating elements at the given date. (Similarly, the proper orbital elements are used to determine the families of asteroids.)

Origin

DTZ's public sale group in Singapore auctions all forms of residential, workplace and retail properties, outlets, homes, lodges, boarding homes, industrial buildings and development websites. Auctions are at present held as soon as a month.

We will not only get you a property at a rock-backside price but also in an space that you've got longed for. You simply must chill out back after giving us the accountability. We will assure you 100% satisfaction. Since we now have been working in the Singapore actual property market for a very long time, we know the place you may get the best property at the right price. You will also be extremely benefited by choosing us, as we may even let you know about the precise time to invest in the Singapore actual property market.

The Hexacube is offering new ec launch singapore business property for sale Singapore investors want to contemplate. Residents of the realm will likely appreciate that they'll customize the business area that they wish to purchase as properly. This venture represents one of the crucial expansive buildings offered in Singapore up to now. Many investors will possible want to try how they will customise the property that they do determine to buy by means of here. This location has offered folks the prospect that they should understand extra about how this course of can work as well.

Singapore has been beckoning to traders ever since the value of properties in Singapore started sky rocketing just a few years again. Many businesses have their places of work in Singapore and prefer to own their own workplace area within the country once they decide to have a everlasting office. Rentals in Singapore in the corporate sector can make sense for some time until a business has discovered a agency footing. Finding Commercial Property Singapore takes a variety of time and effort but might be very rewarding in the long term.

is changing into a rising pattern among Singaporeans as the standard of living is increasing over time and more Singaporeans have abundance of capital to invest on properties. Investing in the personal properties in Singapore I would like to applaud you for arising with such a book which covers the secrets and techniques and tips of among the profitable Singapore property buyers. I believe many novice investors will profit quite a bit from studying and making use of some of the tips shared by the gurus." – Woo Chee Hoe Special bonus for consumers of Secrets of Singapore Property Gurus Actually, I can't consider one other resource on the market that teaches you all the points above about Singapore property at such a low value. Can you? Condominium For Sale (D09) – Yong An Park For Lease

In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value. Irregular satellites are believed to have been captured from heliocentric orbits. (Indeed, it appears that the irregular moons of the gas giants, the Jovian and Neptunian trojans, and grey Kuiper belt objects have a similar origin.) For this to occur, one of three things needs to have happened:

• energy dissipation (e.g. in interaction with the primordial gas cloud)
• a substantial (40%) extension of the planet's Hill sphere in a brief period of time (thousands of years)
• a transfer of energy in a three-body interaction. This could involve:
  • a collision (or close encounter) of an incoming body and a satellite, resulting in the incoming body losing energy and being captured.
  • a close encounter between an incoming binary object and the planet (or possibly an existing moon), resulting in one component of the binary being captured. Such a route has been suggested as most likely for Triton.

After the capture, some of the satellites could break up leading to groupings of smaller moons following similar orbits. Resonances could further modify the orbits making these groupings less recognizable.

Long-term stability

Phoebe, Saturn's largest irregular satellite

Remarkably, the current orbits prove stable in numerical simulations, in spite of substantial perturbations near the apocenter. The cause of this stability in a number of irregulars is the fact that they orbit with a secular or Kozai resonance.

In addition, simulations indicate the following conclusions:

• Orbits with inclinations higher than 50° (or less than 130° for retrograde orbits) are very unstable: their eccentricity increases quickly resulting in the satellite being lost
• Retrograde orbits are more stable than prograde (stable retrograde orbits can be found further from the planet)

Increasing eccentricity results in smaller pericenters and large apocenters. The satellites enter the zone of the regular (larger) moons and are lost or ejected via collision and close encounters. Alternatively, the increasing perturbations by the Sun at the growing apocenters push them beyond the Hill sphere.

Retrograde satellites can be found further from the planet than prograde ones. Detailed numerical integrations have shown this asymmetry. The limits are a complicated function of the inclination and eccentricity, but in general, prograde orbits with semi-major axes up to 0.47 r_H (Hill sphere radius) can be stable, while for retrograde orbits stability can extend out to 0.67 r_H.

The boundary for the semimajor axis is surprisingly sharp for the prograde satellites. A satellite on a prograde, circular orbit (inclination=0°) placed at 0.5 r_H would leave Jupiter in as little as forty years. The effect can be explained by so-called evection resonance. The apocenter of the satellite, where the planet’s grip on the moon is at its weakest, gets locked in resonance with the position of the Sun. The effects of the perturbation accumulate at each passage pushing the satellite even further outwards.

The asymmetry between the prograde and retrograde satellites can be explained very intuitively by the Coriolis acceleration in the frame rotating with the planet. For the prograde satellites the acceleration points outward and for the retrograde it points inward, stabilising the satellite.

Physical characteristics

Size

Illustration of the power law. The number of objects depends on their size.

Given their greater distance from Earth, the known irregular satellites of Uranus and Neptune are larger than those of Jupiter and Saturn; smaller ones probably exist but have not yet been observed. However, with this observational bias in mind, the size distribution is similar for all four giant planets.

Typically, the relation expressing the number $N\,\!$ of objects of the diameter smaller or equal to $D\,\!$ is approximated by a power law:

${\frac {dN}{dD}}\sim D^{-q}$ with q defining the slope.

A shallow power law (q~2) is observed for sizes 10 to 100 km^† but steeper (q~3.5) for objects smaller than 10 km^‡ .

For comparison, the distribution of Kuiper belt objects is much steeper (q~4), i.e. for one object of 1000 km there are a thousand objects with a diameter of 100 km. The size distribution provides insights into the possible origin (capture, collision/break-up or accretion).

^†For every object of 100 km, ten objects of 10 km can be found.
^‡For one object of 10 km, some 140 objects of 1 km can be found.

Colours

This diagram illustrates the differences of colour in the irregular satellites of Jupiter (red labels), Saturn (yellow) and Uranus (green). Only irregulars with known colour indices are shown. For reference, the centaur Pholus and three classical Kuiper belt objects are also plotted (grey labels, size not to scale). For comparison, see also colours of centaurs and KBOs.

The colours of irregular satellites can be studied via colour indices: simple measures of differences of the apparent magnitude of an object through blue (B), visible i.e. green-yellow (V), and red (R) filters. The observed colours of the irregular satellites vary from neutral (greyish) to reddish (but not as red as the colours of some Kuiper belt objects).

albedo	neutral	reddish	red
low	C 3–8%	P 2–6%	D 2–5%
medium		M 10–18%	A 13–35%
high		E 25–60%

Each planet's system displays slightly different characteristics. Jupiter's irregulars are grey to slightly red, consistent with C, P and D-type asteroids. Some groups of satellites are observed to display similar colours (see later sections). Saturn's irregulars are slightly redder than those of Jupiter.

The large Uranian irregular satellites (Sycorax and Caliban) are light red, while smaller Prospero and Setebos are grey, as are Neptunian sateliites Nereid and Halimede.

Spectra

With the current resolution, the visible and near-infrared spectra of most satellites appear featureless. So far, water ice has been inferred on Phoebe and Nereid and features attributed to aqueous alteration were found on Himalia.

Rotation

Regular satellites are usually tidally locked (that is, their orbit is synchronous with their rotation so that they only show one face toward their parent planet). In contrast, tidal forces on the irregular satellites are negligible given their distance from the planet, and rotation periods in the range of only ten hours have been measured for the biggest moons Himalia, Phoebe and Nereid (to compare with their orbital periods of hundreds of days). Such rotation rates are in the same range that is typical for asteroids. 50 year old Petroleum Engineer Kull from Dawson Creek, spends time with interests such as house brewing, property developers in singapore condo launch and camping. Discovers the beauty in planing a trip to places around the entire world, recently only coming back from .

Families with a common origin

Some irregular satellites appear to orbit in 'groups', in which several satellites share similar orbits. The leading theory is that these objects constitute collisional families, parts of a larger body that broke up.

Dynamic groupings

Simple collision models can be used to estimate the possible dispersion of the orbital parameters given a velocity impulse δV. Applying these models to the known orbital parameters makes possible to estimate the δV necessary to create the observed dispersion. It is believed that δV of tens of meters per seconds (5–50 m/s) could result from a break-up. Dynamical groupings of irregular satellites can be identified using these criteria and the likelihood of the common origin from a break-up evaluated.

When the dispersion of the orbits is too wide (i.e. it would require δV in the order of hundreds of m/s)

• either more than one collision must be assumed, i.e. the cluster should be further subdivided into groups
• or significant post-collision changes, for example resulting from resonances, must be postulated.

Colour groupings

When the colours and spectra of the satellites are known, the homogeneity of these data for all the members of a given grouping is a substantial argument for a common origin. However, lack of precision in the available data often makes it difficult to draw statistically significant conclusions. In addition, the observed colours are not necessarily representative of the bulk composition of the satellite.

Observed groupings

Irregular satellites of Jupiter

The orbits of Jupiter's irregular satellites, showing how they cluster into groups. Satellites are represented by circles that indicate their relative sizes. An object's position on the horizontal axis shows its distance from Jupiter. Its position on the vertical axis indicates its orbital inclination. The yellow lines indicate its orbital eccentricity (i.e. the extent to which its distance from Jupiter varies during its orbit).

Typically, the following groupings are listed (dynamically tight groups displaying homogenous colours are listed in bold)

• Prograde satellites
  • The Himalia group shares an average inclination of 28°. They are confined dynamically (δV ≈ 150 m/s). They are homogenous at visible wavelengths (having neutral colours similar to those of C-type asteroids) and at near infrared wavelengths
  • Themisto is so far believed to be isolated.
  • Carpo is so far believed to be isolated.
• Retrograde satellites
  • The Carme group shares an average inclination of 165°. It is dynamically tight (5 < δV < 50 m/s). It is very homogenous in colour, each member displaying light red colouring consistent with a D-type asteroid progenitor.
  • The Ananke group shares an average inclination of 148°. It shows little dispersion of orbital parameters (15 < δV < 80 m/s). Ananke itself appears light red but the other group members are grey.
  • The Pasiphae group is very dispersed. Pasiphae itself appears to be grey while other members (Callirrhoe, Megaclite) are light red

Sinope, sometimes included into Pasiphae group, is red and given the difference in inclination, it could be captured independently. Pasiphae and Sinope are also trapped in a secular resonances with Jupiter. 50 year old Petroleum Engineer Kull from Dawson Creek, spends time with interests such as house brewing, property developers in singapore condo launch and camping. Discovers the beauty in planing a trip to places around the entire world, recently only coming back from .

Irregular satellites of Saturn

Irregular satellites of Saturn, showing how they cluster into groups. For explanation, see Jupiter diagram

The following groupings are commonly listed for Saturn's satellites:

• Prograde satellites
  • The Gallic group shares an average inclination of 34°. Their orbits are dynamically tight (δV ≈ 50 m/s), and they are light red in colour; the colouring is homogenous at both visible and near infra-red wavelengths.
  • The Inuit group shares an average inclination of 46°. Their orbits are widely dispersed (δV ≈ 350 m/s) but they are physically homogenous, sharing a light red colouring.
• Retrograde satellites
  • The Norse group is defined mostly for naming purposes; the orbital parameters are very widely dispersed. Sub-divisions have been investigated, including
    • The Phoebe group shares an average inclination of 174°; this sub-group too is widely dispersed, and may be further divided into at least two sub-sub-groups
    • The Skathi group is a possible sub-group of the Norse group

50 year old Petroleum Engineer Kull from Dawson Creek, spends time with interests such as house brewing, property developers in singapore condo launch and camping. Discovers the beauty in planing a trip to places around the entire world, recently only coming back from .

Irregulars of Uranus and Neptune

Irregular satellites of Uranus (green) and Neptune (blue) (excluding Triton). For explanation, see Jupiter diagram

According to current knowledge, the number of irregular satellites orbiting Uranus and Neptune is smaller than that of Jupiter and Saturn. However, it is believed this is simply a result of observational difficulties due to the greater distance of Uranus and Neptune. The table at left shows the minimum radius (r_min) of satellites that can be detected with current technology, assuming an albedo of 0.04; thus, there are almost certainly small Uranian and Neptunian moons that cannot yet be seen.

Due to the smaller numbers, statistically significant conclusions about the groupings are difficult. A single origin for the retrograde irregulars of Uranus seems unlikely given a dispersion of the orbital parameters that would require high impulse (δV ≈ 300 km) implying a large diameter of the impactor (395 km), which is incompatible in turn with the size distribution of the fragments. Instead, the existence of two groupings has been speculated:

• Caliban group
• Sycorax group

These two groups are distinct (with 3σ confidence) in their distance from Uranus and in their eccentricity. However, these groupings are not directly supported by the observed colours: Caliban and Sycorax appear light red while the smaller moons are grey.

For Neptune, a possible common origin of Psamathe and Neso has been noted. Given the similar (grey) colours, it was also suggested that Halimede could be a fragment of Nereid. The two satellites have had a very high probability (41%) of collision over the age of the solar system.

Exploration

Distant Cassini image of Himalia

To date, the only irregular satellite to have been visited by a spacecraft is Phoebe, the largest of Saturn's irregulars, which was photographed by the Cassini probe in 2004. Cassini also captured a distant, low resolution image of Jupiter's Himalia in 2000. There are no spacecraft planned to visit any irregular satellites in the future. 50 year old Petroleum Engineer Kull from Dawson Creek, spends time with interests such as house brewing, property developers in singapore condo launch and camping. Discovers the beauty in planing a trip to places around the entire world, recently only coming back from .

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

• David Jewitt's pages
• Scott Sheppard's pages
• Discovery circumstances from JPL
• Mean orbital elements from JPL
• MPC: Natural Satellites Ephemeris Service