At one astronomical unit from the sun ( the Earth's distance ) the dust orbits are probably nearly circular.

Hence S breaks up into uncountably many orbits under G. Using the axiom of choice, we could pick a single point from each orbit, obtaining an uncountable subset X of S with the property that all of its translates by G are disjoint from X.

The eccentricity ( orbit ) | eccentricities of the orbits are represented by segments ( extending from perihelion to aphelion ) with the inclination s represented on the vertical axis.

The ' hot ' and ' cold ' populations are strikingly different: more than 30 % of all cubewanos are in low inclination, near-circular orbits.

The parameters of the plutinos ’ orbits are more evenly distributed, with a local maximum in moderate eccentricities in 0. 15 – 0. 2 range and low inclinations 5 – 10 °.

When orbital inclinations are compared, ' hot ' cubewanos can be easily distinguished by their higher inclinations, as the plutinos typically keep orbits below 20 °.

Binaries are quite common on low-inclination orbits and are typically similar-brightness systems.

Binaries are less common on high-inclination orbits and their components typically differ in brightness.

Linear dynamical systems and systems that have two numbers describing a state are examples of dynamical systems where the possible classes of orbits are understood.

* Given a partition of A, G is a transformation group under composition, whose orbits are the cells of the partition ‡;

* Given a transformation group G over A, there exists an equivalence relation ~ over A, whose equivalence classes are the orbits of G.

In sum, given an equivalence relation ~ over A, there exists a transformation group G over A whose orbits are the equivalence classes of A under ~.

The equivalence classes of ~— also called the orbits of the action of H on G — are the right cosets of H in G. Interchanging a and b yields the left cosets.

These pulsar planets are believed to have formed from the unusual remnants of the supernova that produced the pulsar, in a second round of planet formation, or else to be the remaining rocky cores of gas giants that somehow survived the supernova and then decayed into their current orbits.

: When multiple planets are present, each one slightly perturbs the others ' orbits.

The radial-velocity method and the transit method ( which between them are responsible for the vast majority of detections ) are most sensitive to large planets in small orbits.

Thus many known exoplanets are " hot Jupiters ": planets of Jovian mass or larger in very small orbits with periods of only a few days.

It also appears that there are more planets in large orbits than in small orbits.

The meteoroids left by the comet are organized in trails in orbits similar to though different from that of the comet.

Most multiple stars are organized in a hierarchical manner, with smaller orbits nested inside larger orbits.

The lines traced out by orbits dominated by the gravity of a central source are conic section s: the shapes of the curves of intersection between a plane and a cone.

For discrete-time dynamical systems the orbits are sequences, for real dynamical systems the orbits are curves and for holomorphic dynamical systems the orbits are Riemann surfaces.

By 2004, robotic observations accounted for an overwhelming percentage of the published scientific information on asteroid orbits and discoveries, variable star studies, supernova light curves and discoveries, comet orbits and gravitational microlensing observations.

In dynamical systems, the integral curves for a differential equation that governs a system are referred to as trajectories or orbits.

The orbits of moving bodies are curves in spacetime ; the orbits of bodies moving at constant speed without changing direction correspond to straight lines.

In effect, the positions and velocities are perturbed directly, and no attempt is made to calculate the curves of the orbits or the orbital elements.

The integral curves or fibers respectively are certain pairwise linked great circles, the orbits in the space of unit norm quaternions under left multiplication by a given unit quaternion of unit norm.

Since the group action preserves the fibers of π: P → X and acts transitively, it follows that the orbits of the G-action are precisely these fibers and the orbit space P / G is homeomorphic to the base space X.

This action is clearly free and the orbits are just the fibers of π.

De Jager ( 1955 ) has calculated the times required for these particles to reach the atmosphere under the influence of the Poynting-Robertson effect, which in this case causes the orbits to become more and more eccentric without changing the semi-major axis.

In Thomson's model, electrons were free to rotate in rings which were further stabilized by interactions between the electrons, and spectra were to be accounted for by energy differences of different ring orbits.

Some sources, including Newton, refer to the entire force as a centripetal force, even for eccentric orbits, for which gravity is not aligned with the direction to the center of curvature.

The gravitational force acts on each object toward the other, which is toward the center of mass of the two objects ; for circular orbits, this center of gravity is the center of the circular orbits.

The form of ' biosphere ' which I envisaged consists of a loose collection or swarm of objects traveling on independent orbits around the star.

The actual speed with which the Earth orbits the Sun varies slightly during the year, so the speed with which the Sun seems to move along the ecliptic also varies.

The plane of the ecliptic intersects the celestial sphere along a great circle ( black ), the same circle on which the Sun seems to move as the Earth orbits it.

Comets may have orbits which take them out to more than 1000 AU.

More importantly, the incontrovertible discovery of celestial bodies orbiting something other than the Earth dealt a serious blow to the then-accepted Ptolemaic world system, or the geocentric theory in which everything orbits around the Earth.

Behold, therefore, four stars reserved for your illustrious name ... which ... make their journeys and orbits with a marvelous speed around the star of Jupiter ... like children of the same family ...

Hohmann demonstrated that the lowest energy route between any two orbits is an elliptical " orbit " which forms a tangent to the starting and destination orbits.

The Sun cannot be used in a gravitational slingshot because it is stationary compared to rest of the Solar System, which orbits the Sun.

Kepler's laws and his analysis of the observations on which they were based, the assertion that the Earth orbited the Sun, proof that the planets ' speeds varied, and use of elliptical orbits rather than circular orbits with epicycles — challenged the long-accepted geocentric models of Aristotle and Ptolemy, and generally supported the heliocentric theory of Nicolaus Copernicus ( although Kepler's ellipses likewise did away with Copernicus's circular orbits and epicycles ).

They proposed that, of all closed classical orbits traced by a mechanical system in its phase space, only the ones that enclosed an area which was a multiple of Planck's constant were actually allowed.

Ecliptic comets have relatively small orbits, below 10 AU, and follow the ecliptic plane, the same plane in which the planets lie.

Statistical models of the observed orbits of long-period comets argue that the galactic tide is the principal means by which their orbits are perturbed toward the inner Solar System.

Apart from long-period comets, only four known objects have orbits which suggest that they may belong to the Oort cloud: 90377 Sedna, 2000 CR < sub > 105 </ sub >, 2006 SQ < sub > 372 </ sub > and 2008 KV < sub > 42 </ sub >.