Semimajor axis variations in co-orbital moons

Semimajor axis variations in co-orbital moons

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I've been playing with simulations of co-orbital bodies similar to Saturn's moons Janus & Epimetheus- horseshoe orbits where the two bodies are of comparable mass- and I'm seeing some very odd patterns that I can't explain.

What "should" happen is that the body with the slightly smaller orbit will eventually overtake the outer body; when their mutual interaction becomes significant compared to their interactions with the primary, the lower body is accelerated forward into a higher orbit, while the higher body is accelerated backward into a lower orbit, they switch positions around their average orbital radius, and then separate again.

What I'm seeing in simulation, however, is a little more complicated. When I simulate the Saturn+Janus+Epimetheus system, the moons' semimajor axis very slowly migrate farther away from their average up until they swap; right around the swap, the difference in semimajor axes is maximized, and they drift closer together for half a cycle and then farther apart again before the next swap. This creates "u" and "n" shapes on the plot of semimajor axis over time, with the plots for each moon in opposite phases.

For Janus and Epimetheus, the effect is very small, but increasing the masses of the moons amplifies it, to the point that the maximum separation in orbital radii that occurs during the moons' closest approach to each other is ten or more times larger than the "normal" separation during the middle of a cycle.

There is of course the possibility that this an artifact of errors in my simulation software, but I think that is unlikely given the following:

  1. All other aspects of the Janus+Epimetheus system are reproduced perfectly- individual orbital periods are correct, average semimajor axes are correct, the frequency of the swaps is correct, and my simulation conserves energy to within less than 1 part in 1 trillion over a petasecond of simulation time (approx. 31,689 years).
  2. The effect is perfectly regular, and simulated systems remain stable. If the effect was a result of simulation errors, I wouldn't expect them to behave so nicely.

My first thought was that perhaps the semimajor axis of each moon was being affected by the added gravity of the other moon when they are on opposite sides of the primary from each other, and when they approach more closely they won't be pulling each towards the primary as strongly. That, however, should result in a uniform expansion and contraction of both moons' orbits as they get respectively closer and farther away from each other. What I actually see is that the inner moon's orbit gets smaller while the outer moons orbit gets larger, and vice-versa.

So, is this likely to just be a simulation error after all (if it helps, I am using the 4th-order Hermite integration algorithm)? If not, what is the explanation for it? Has such an effect been documented before, and if so, where?

Epimetheus (moon)

Epimetheus (ep'-i-mee'-thee-us, Greek Επιμηθεύς) is a moon of Saturn. It is also known as Saturn XI. It is named after the mythological Epimetheus.

Epimetheus occupies essentially the same orbit as the moon Janus. Astronomers assumed that there was only one body in that orbit, and accordingly had a hard time figuring out their orbital characteristics it is obviously impossible to reconcile the observations of two distinct objects as a single object.

Audouin Dollfus observed a moon on December 15, 1966, which he proposed to be named "Janus"[3]. On December 18, Richard L. Walker made a similar observation which is now credited as the discovery of Epimetheus[4]. However, at the time, it was believed that there was only one moon, unofficially known as "Janus", in the given orbit.

Twelve years later, in October 1978, Stephen M. Larson and John W. Fountain realised that the 1966 observations were best explained by two distinct objects (Janus and Epimetheus) sharing very similar orbits. This was confirmed in 1980 by Voyager 1, and so Walker officially shares the discovery of Epimetheus with Larson and Fountain.

The 1980 Voyager discovery was designated S/1980 S 3, and it was officially named "Epimetheus" in 1983. The name Janus was officially approved by the IAU at the same time, although the name had been used informally since Dollfus proposed it a few days after the 1966 discovery.

Orbital relationship between Epimetheus and Janus

Epimetheus and Janus are "co-orbital". Janus' orbital radius from Saturn is 151,472 km and Epimetheus' orbital radius is 151,422 km, a separation of only 50 km. Since closer orbits have higher velocities the two moons must inevitably approach each other, and since Epimetheus' diameter is 115 km and Janus' is 178 km it would seem at first glance that a collision is also inevitable. But as the inner moon catches up with the outer moon their mutual gravitational attraction boosts the inner moon's momentum and raises its orbit, causing it to slow down. The outer moon loses an equal amount of momentum and drops into a lower orbit at the same time, speeding it up. The moons thus "trade" orbits and begin moving apart again, without actually approaching each other closely. The exchange takes place about once every four years the next closest approach is in Jan/Feb 2006. This arrangement is unique in the solar system, as far as is currently known.

Other unusual orbits include the retrograde orbit of Triton around Neptune, and the "horse-shoe" orbit of Cruithne and (potentially) dozens of other objects in similar orbits [5].

Physical characteristics

Epimetheus, as imaged by Voyager 1 (NASA)

There are several Epimethean craters larger than 30 km in diameter, as well as both large and small ridges and grooves. The extensive cratering indicates that Epimetheus must be quite old. Janus and Epimetheus may have formed from a disruption of a single parent to form co-orbital satellites, but if this is the case the disruption must have happened early in the history of the satellite system. From its very low density and relatively high albedo, it seems likely that Epimetheus is a very porous icy body. There is a lot of uncertainty in these values, however, and so this remains to be confirmed.

The Cassini orbiter is due to perform a flyby of Epimetheus on December 3, 2007.

List of geological features on Saturn's smaller moons

. | Pandora | Janus, Epimetheus | Mimas | .

Saturn's natural satellites

Pan | Daphnis | Atlas | Prometheus | S/2004 S 6 | S/2004 S 4 | S/2004 S 3 | Pandora | Epimetheus and Janus | Mimas | Methone | Pallene | Enceladus | Telesto, Tethys, and Calypso | Polydeuces, Dione, and Helene | Rhea | Titan | Hyperion | Iapetus | Kiviuq | Ijiraq | Phoebe | Paaliaq | Skathi | Albiorix | S/2004 S 11 | Erriapo | Siarnaq | S/2004 S 13 | Tarvos | Mundilfari | S/2004 S 17 | Narvi | S/2004 S 15 | S/2004 S 10 | Suttungr | S/2004 S 12 | S/2004 S 18 | S/2004 S 9 | S/2004 S 14 | S/2004 S 7 | Thrymr | S/2004 S 16 | Ymir | S/2004 S 8

2 Answers 2

There are many wrong points in your reasoning.

First of all, if a body is in the L3 point of the Earth-Sun system, it is orbiting the Sun along the same orbit of Earth, so it cannot be a second Moon.

Moreover, L3 is way more distant than the Moon from Earth, which is also one of your requirements.

Last but not least, to answer your question about stability

The Sun–Earth L3 is unstable and could not contain a natural object, large or small, for very long. This is because the gravitational forces of the other planets are stronger than that of Earth (Venus, for example, comes within 0.3 AU of this L3 every 20 months).

If instead by any chance your mean the L3 of the Earth-Moon system, same consideration will hold: the forces induced by the first Moon (which you place somewhere between L1 and L2) and the Sun would quickly perturb any body happening to be in L3.

A small asteroid could orbit Earth at the distance of the Moon in the L4 point, 60 degrees ahead of the Moon, or the L5 point, 60 degrees behind the Moon. Orbits in L4 or L5 points are called Trojan orbits.

The lunar L4 and L5 points are also considered to be stable points for artificial space habitats, so explaining the name of the L5 Society.

But I have the impression that in Trojan obits the main object, in this case the Earth, should have many times the mass of the secondary object, in this case the Moon, which in turn has to have many times the mass of the tertiary object in the L4 or l5 point.

As far as I know the only objects in the L4 and L5 points of the Moon are concentrations of interplanetary dust called the Kordylewski clouds, so faint that though they were first detected in 1956 they were not confirmed until 2018. Their mass must be minute compared to that of the Moon.

Perhaps the Earth could have two moons of equal mass in the same orbit, 60 degrees apart, one moon being in the L4 point 60 degrees ahead of the other, and the other moon being in the L5 position 6 degrees behind the other. But I don't know if such a situation would be stable, and I know of no examples of such a situation.

It has been claimed that two planets with similar mass could be stable in the same orbit if they were 60 degrees apart.

Two planets with similar masses can also share the same orbit if they orbit 60 degrees apart. This means that each is in the other’s L4/L5 Lagrange point. This kind of configuration comes out of our computer simulations, and we expect to find one of these setups among exoplanet systems.

If that is correct, two identical mass moons ought to also be able to share the same orbit, spaced 60 degrees apart.

Here is some additional information.

The PlanetPlanet blog has a set of posts called The Ultimate Solar System, designing solar systems which have as many habitable planets as possible.

The post The Ultimate Engineered Solar System designs a solar system which doesn't have single planets in each orbit, but rings of planets in each orbit.

It seems that a solar system could have a number of planets sharing the same orbit, as long as the planets have equal mass and are equally spaced in the orbit. The source is this paper:

Apparently such a ring of planets could be stable with seven to forty two planets in a single orbit.

And what is stable for a ring of planets around a star would also be stable for a ring of moons around a planet. Except that the gravity of the star would be a perturbing factor.

The Moon has a mass of 0.012300 of the Earth's mass, and the Moon's orbit with a semi-major axis of 384,399 kilometers would have a circumference of approximately 2,415,248.1 kilometers if it was circular.

So if there were 7 moons at the distance of the Moon and with the same mass as the Moon, they would have a total mass of 0.0861 Earth mass, and they would be spaced 51.4285 degrees, or about 345,035.44 kilometers, apart in their shared orbit.

So if there were 42 moons at the distance of the Moon and with the same mass as the Moon, they would have a total mass of 0.5166 Earth mass, and they would be spaced 8.5714 degrees, or about 57,505.904 kilometers, apart in their shared orbit.

In another post, Cohorts of Co-Orbital Planets, it was proposed that arcs of planets could share stable orbits and they didn't need to be complete rings.

1 Answer 1

The Hunt for Exomoons with Kepler Project has so far failed to find any exomoons. This is a negative finding (so far). Negative findings are always a bit trickier to explain than are positive findings. This negative finding might mean something very significant, or it might have very little significance:

Maybe exoplanets are much less likely to have exomoons that the abundance of moons in our solar system would suggest or

Maybe the close-in exoplanets that Kepler was predisposed to find are much less likely to have exomoons or

Maybe the group hasn't studied enough exoplanets and so far have just been a bit unlucky. From reading through their papers, their approach is exceeding CPU intensive, taking decades of CPU time per planet investigated or

Maybe their technique isn't as good at detecting exomoons as they think it is or


The earth-moon system is not an isolated two-body system, so that the calculation of the position of the moon requires a correction that goes beyond the great inequality , which is due in particular to the gravitational influences of the sun. In the context of a perturbation theory, one can calculate that the Kepler's orbit elements of the moon are subject to temporal changes due to the influence of the sun: The position of the perigee and the ascending node "move" linearly in time due to the perturbation (so-called secular perturbations ), all orbit elements and in particular the semi-major axis , numerical eccentricity and orbital inclination of periodic disturbances which depend on the ecliptical length of the moon λ m and the sun λ s . Some terms have periodic dependencies on the double angle between the sun and the moon , including a term that affects the semi-major axis. This term can be understood as the compression of the lunar orbit towards the sun. These perturbations lead to a change in the ecliptical length of the moon in a first approximation around the summand: 2 ( λ s - λ m ) < displaystyle 2 ( lambda _ - lambda _ )>

where μ = ω s / ω m ≈0.075, the ratio of the sidereal month to the sidereal year. This first approximation provides only a rough estimate with an amplitude of only about 0.44 degrees. Closer analysis shows that the total amplitude is 39.5 arc minutes, i.e. H. 0.66 degrees. The first links

do not depend on the numerical eccentricity in contrast to the large deviation and evection. The remaining 5 arc minutes, however, result from terms that depend on both the eccentricity of the lunar and earth orbits. The period of the disturbance results from

d. H. exactly one synodic month.

The calculation presented here is in principle also valid for the moons of other planets. Since it practically only depends on the frequency ratio μ, one quickly sees that it is much smaller for all other large moons of the solar system than for the earth's moon (μ≈1 / 13). In relation to μ, the Saturn moon Iapetus with μ≈1 / 135 is in second place before the Jupiter moon Callisto with μ≈1 / 260. However, due to the quadratic dependence of μ, the size of the effect at Iapetus is only 1% or 0.25% of the size at the Earth's moon. In addition, as in the case of evection, disturbances due to the flattening of the central planet and neighboring planets are far more relevant in the large moons of the gas planets .

Boccaletti, D. and Pucacco, G.: 1998, Theory of Orbits, Springer-Verlag, Vol. 2.

Fleming, H. J. and Hamilton, D. P.: 2000, 'On the origin of the Trojan asteroids: effects of Jupiter's mass accretion and radial migration', Icarus 148, 479-493.

Gomes, R. S.: 1998, 'Dynamical effects of planetary migration on primordial Trojan-type asteroids', Astron. J. 116, 2590-2597.

Henrard, J.: 1993, 'The adiabatic invariant in classical mechanics'. In: Jones, Kirchgraber and Walther (eds), Dynamics Reported, Springer-Verlag, pp. 117-235.

Jeans, J. H.: 1924, 'Cosmogonic problems associated with a secular decrease of mass', Mon. Not. R. Astron. Soc. 84, 2-11.

Lissauer, J. J., Goldreich, P. and Tremaine, S.: 1985, 'Evolution of the Janus-Epimetheus coorbital resonance due to torques from Saturn's rings', Icarus 64, 425-434.

Littlewood, J. E.: 1964, 'Adiabatic invariance II: elliptic motion about a slowly varying center of force', Ann. Phys. 26, 131-156.

Marzari, F. and Scholl, H.: 1998a, 'Capture of Trojans by a growing proto-Jupiter', Icarus 131, 41-51.

Marzari, F. and Scholl, H.: 1998b, 'The growth of Jupiter and Saturn and the capture of Trojans', Astron. Astrophys. 339, 278-285.

Marzari, F., Scholl, H., Murray, C. and Lagerkvist, C.: 2003, 'Origin and evolution of Trojan asteroids'. In: W. Bottke, A. Cellino, P. Paolichi and R. P. Binzel (eds), Asteroids III, University of Arizona Press, Tucson, USA, pp. 725-738.

Yoder, C. F., Colombo, G., Synnot, S. P. and Yoder, K. A.: 1983, 'Theory of motion of Saturn's coorbiting satellites', Icarus 53, 431-443.

Yoder, C. F., Synnot, S. P. and Salo, H.: 1989, 'Orbits and masses of Saturn's co-orbiting satellites, Janus and Epimetheus', Astron. J. 98, 1875-1889.

Uranus' Atmosphere

  • The reason is a lack of internal heat, unlike the other Jovian planets.
  • Clouds on Uranus are cold and don't billow up above the top haze layer.
  • Results in a generally uniform appearance.

More detail can be seen at mear infrared wavelengths, especially those infrared wavelengths that can peer through the methane haze layer. These infrared images reveal occasional clouds and cyclonic storms, as well as weak bands and zones.

Semimajor axis variations in co-orbital moons - Astronomy

Review One
For Test on 25-Sep-01

If you know the meanings of all the concepts outlined below, you should make at least 105 on the first test! You may also benefit from the computer assisted review, as described in the syllabus. Lesson 1 on Basic Astronomy and Lesson 2 on the Solar System should be the most useful lessons. You may also review the lecture notes on the web.

Definitions: Astronomy, universe, equatorial plane, right ascension, declination, astronomical unit, light-year, constellation, eclipse, ecliptic plane, geocentric, heliocentric.

Earth's motions: daily motions, annual motions of stars.

Kepler's laws of planetary motion:

1. Planets move in elliptical orbits, with the sun at one focus.

2. A radius line sweeps out equal areas in equal time intervals.

3. Period 2 (in years) = semimajor axis 3 (in AU).

1. An object remains at rest or in uniform motion unless acted upon by an outside force.

2. The net force = mass times acceleration.

3. For every action force there is an equal and opposite reaction force.

Newton's law of gravity: every object attracts every other object with a force which is proportional to their masses and inversely proportional to the square of the distance between them.

Law of gravity + Kepler's third law -> formula for mass vs period and radius of orbit of a satellite:

Light: Both electromagnetic wave and a stream of particles, moving at 186,000 mi/sec or 3x10 8 meters/sec.
inverse square law of intensity.

Wavelengths: (longest) radio - infrared - visible - ultraviolet - X-ray - Gamma-ray (shortest)

Telescopes: Reflecting (mirror), Refracting (lens).

Types of observation: Direct view, photographic, filters, spectra, photometers.

Planets: All revolve ccw near ecliptic plane, nearly circular orbits.

Terrestrial: denser, much less light elements, solid surface.

Mercury: hot, poor reflector, craters, scarps, maria, eccentric orbit, rotational period = 2/3 orbital period.

Venus: heavy cloud cover -> green house effect -> hot surface.
surface is mountainous, rocky. slow reverse rotation.
pressure = 90 x earth atmosphere.

Earth: Blue, white, has life, greatest erosion, plate tectonics, 23.5 degree tilt of axis produces seasons.

earth-moon may be called co-planets.

Moon: craters, maria, best early record of solar system,
causes tides, solar eclipses on earth, little atmosphere.

Mars: Red planet, seasons, dry, .01 of earth atmosphere.
Polar ice caps = H2O & CO2 ice.
Rocky, mountains, scarps. No life detected!

Giant: Less dense, more light elements, fast spin, more satellites, rings.

Jupiter: Largest. Red spot, bands, turbulent upper atmosphere,
mostly H and He, similar to original solar system stuff,
at least 16 moons, faint rings, huge magnetic field, strong radio emissions, 4 major distinctive satellites.

Saturn: Smallest density, prominent rings of small particles - 'shepherded' by small moonlets
At least 18 moons
Titan - moon with greatest atmosphere.

Uranus: Axis of rotation slightly below ecliptic plane, 5 major moons, at least 10 small moons and 11 faint rings.

Neptune: Physical twin of Uranus, except nearly normal rotation, 2 major moons with strange orbits.
several faint rings.
At least 6 minor moons, some "shepherd" moons.

Pluto: (not terrestrial, but certainly not a giant) Most eccentric orbit - sometimes inside Neptune's - usually beyond.
Satellite, Charon shows Pluto is smaller, less dense than previously thought.

Asteroids: mostly between orbits of Mars and Jupiter. small chunks, might have been a planet but didn't make it.
probably contributes some to meteoroids and satellites.

Comets: rocky snow-balls in greatly eccentric, random orbits.
when approaching sun, heat vaporizes, makes comma, tail(s).
Mostly in distant, spherical Oort cloud, 50,000-150,000 AU.

Meteoroids: small chunks of rocks. occasionally collide with earth or other planet.
In Earth's atmosphere = meteor.
On the ground = meteorite - used to study S. S. history.
Trails remaining from disintegrated comets - meteor showers.

Wednesday, 15 February 2012

21th Largest Asteroid, 45 Eugenia

45 Eugenia is a large main belt asteroid, with a diameter of 215 km. Eugenia is the 21th largest asteroid currently known.


Eugenia was discovered on June 27, 1857 by the Franco-German amateur astronomer Hermann Goldschmidt. His instrument of discovery was a 4-inch aperture telescope located in his sixth floor apartment in the Latin Quarter of Paris.

Eugenia was the forty-fifth asteroid to be discovered. The preliminary orbital elements were computed by Wilhelm Forster in Berlin, based on three observations in July, 1857.


The asteroid was named by its discoverer Hermann Goldschmidt after Empress Eugenia di Montijo, the wife of Napoleon III.

Eugenia was the first asteroid to be definitely named after a real person, rather than a figure from classical legend.


Diameter (mean): 214.6 km
Aphelion: 2.943 AU
Perihelion: 2.50 AU
Semi-major axis: 2.724 AU
Orbital Period: 4.49 years
Rotation period: 5.699 hrs
Date discovered: 1857.6.27
Class: F
Satellite: 2
Type: Main-belt Asteroid
(data from JPL Small-Body Database)

Physical characteristics

Eugenia is an F-type asteroid, which means that it is very dark in colouring (darker than soot) with a carbonaceous composition.

Eugenia's density appears to be unusually low, indicating that it may be a loosely-packed rubble pile, not a monolithic object. Eugenia appears to be almost anhydrous.

Satellite system

Eugenia is famed as one of the first asteroids to be found to have a moon orbiting it. It is also the second known triple asteroid, after 87 Sylvia.


As most of you will know, New Horizons didn't find any double moons or moonlets, not yet anyway. Except that Keberos is possibly a binary contact moon.

But there are lots of other Kuiper belt objects out there, and then the entire Oort cloud to explore as well, eventually. And even the gas giants are not explored so thoroughly you can rule out tiny moonlets of the moons.

What do you think, do you think there is a chance that some of the moons in our solar system, such as Saturn's Rhea, or Iapetus, or Uranus' Oberon, might have undiscovered moonlets or rings? Tiny moonlets or very faint rings? What about the Oort cloud and Kuiper belt - might they have moonlets or rings of moons?

Watch the video: 19- d orbitals (January 2023).