Published 2021‑02‑12 at 03:10 UT by Pierre Paquette · Updated 2021‑02‑16 at 02:20 UT by Pierre Paquette

In August 2008, astronomers announced the discovery of a second satellite around minor planet 87 Sylvia, making it the first known *trinary* minor planet. Soon after the discovery, the two moons of Sylvia were named Remus and Romulus, the names of her twin children. The list has now grown to include at least sixteen triple minor planets including dwarf planets Pluto and Haumea. *EcliptiQc* has decided to take a look at these very special minor planets and see if any of them has anything, well, even *more* special. Let’s explore them one by one.

Table 1. Minor Planets with Two Satellites | |||
---|---|---|---|

45 Eugenia Main-Belt Asteroid |
130 Elektra Main-Belt Asteroid |
3749 Balam Main-Belt Asteroid |
134340 Pluto Dwarf Planet, Trans-Neptunian Object, Plutino ⚠️ Updated 2021‑02‑16 ⚠️With proper values for orbital precession of the satellites |

87 Sylvia Outer Main-Belt Asteroid |
216 Kleopatra Main-Belt Asteroid |
4666 Dietz Main-Belt Asteroid ⚠️ Updated 2021‑02‑16 ⚠️Added extract from IAU Telegram #4346 about discovery |
136108 Haumea Dwarf Planet, Trans-Neptunian Object, Haumea Family |

93 Minerva Main-Belt Asteroid |
2577 Litva Mars-Crossing Asteroid, Hungaria |
6186 Zenon Main-Belt Asteroid |
(136617) 1994 CC Near-Earth Asteroid, Apollo Family (Potentially Hazardous Asteroid) |

107 Camilla Outer Main-Belt Asteroid |
3122 Florence Near-Earth Asteroid, Amor Family (Potentially Hazardous Asteroid) ⚠️ Updated 2021‑02‑17 ⚠️Added rotation periods and distances to main body of the satellites. |
47171 Lempo Trans-Neptunian Object, Plutino |
(153591) 2001 SN₂₆₃ Near-Earth Asteroid, Amor Family |

Clicking on a name will bring you to the section about it. |

Note: In addition to these systems, “tantalizing but inconclusive” evidence exists for a second moon to near-Earth asteroid 2002 CE₂₆.

**⚠️ UPDATE 2021‑02‑20 ⚠️** Note: Other possible trinary systems include 1830 Pogson, 2006 Polonskaya, 8306 Shoko, and 16635 1993 QO..

A *minor planet* is a body orbiting the Sun that is neither a planet nor exclusively classified as a comet.

An *asteroid* is a minor planet of the inner Solar System—up to and including those coorbital with Jupiter. This includes, but is not limited to, the “main belt” between the orbits of Mars and Jupiter.

Jupiter coorbital bodies include “Greeks” and “Trojans” and orbit 60° ahead of and behind Jupiter, respectively; they are estimated to outnumber main-belt asteroids.

*Centaurs* orbit between Jupiter and Neptune. We know of around 30 *Neptune trojans*, but there may be more than Jupiter coorbital bodies. Finally, there are the *transneptunian objects*, which are at least as distant as that planet is from the Sun; this category includes, but is not limited to, the *Kuiper belt* inside approx. 55 AU from the Sun.

In the following orbital animations, all bodies are represented to relative scale—which means they may or may not be large enough to be visible—and in such a way that the main body has a radius of approximately pixels. The only exceptions are the system of 136108 Haumea, which is represented at one-third of the scale of other systems, while those of 47171 Lempo and 134340 Pluto are represented shrunk by factors of 2.5 and 2.25, respectively, in order to make these systems fit within their respective windows.

All trajectories are shown centred on the main body, which in practice is almost always the case. A notable exception is the Pluto–Charon pair, where the barycentre of Charon’s orbit is actually *outside* Pluto, which means that Pluto itself should be considered an orbiting body here, but I chose to keep Pluto fixed and show Charon’s path with respect to it rather than with respect to the barycentre.

Please note that viewing this document on a larger screen increases resolution, as animations are scaled to a maximum size of 33% width or 66% height of your screen size, whichever value is lower. In general, satellite orbital elements are taken from the *Asteroids with Satellites* page. Time is greatly accelerated, but uniform for all animations, at approximately one hour of real time per second of simulation—the exact rate depends on the viewer’s device.

A rather large asteroid (estimates vary between 232 × 193 × 161 km and 305 × 220 × 145 km) discovered in 1857, 45 Eugenia was “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.” It was named after Eugénie de Montijo [1826–1920], the wife of Napoleon III, making it “the first asteroid to be definitely named after a real person.”

Its first satellite, officially named (45) Eugenia I Petit-Prince, was the first satellite of a minor planet to be discovered with a telescope—before it, only Dactyl, satellite of 243 Ida, had been known as a minor planet’s satellite, and it had been discovered by the automated space probe *Galileo*. Its name honours both Louis-Napoléon [1856–1879] and child novella *The Little Prince* by Antoine de Saint-Exupéry. It was discovered in November 1998 by a “large collaboration” of astronomers (Merline et al.) at the Canada-France-Hawaii Telescope on Mauna Kea, Hawaii, and measures approximately 13 km in diameter.

The second satellite (provisional designation S/2004 (45) 1), is about 6 km in diameter and was found in February 2004 at the Very Large Telescope at Cerro Paranal, Chile.

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We have briefly spoken about this one in the introduction. Its name comes from that of the woman (Rhea Silvia) who supposedly gave birth to Remus and Romulus, legendary founders of the city of Rome, Italy.

According to Wikipedia, “[t]he orbital planes of both satellites and the equatorial plane of the primary asteroid are all well-aligned; all planes are aligned within about 1 degree of each other, suggestive of satellite formation in or near the equatorial plane of the primary.”

I notice that the periods are *almost* in an 8:3 resonance, and believe this warrants a more careful study of the trio.

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This one was discovered in 1867 and named after the Roman equivalent of the goddess Athena. Minerva “is the Roman goddess of wisdom and strategic warfare, justice, law, victory, and the sponsor of arts, trade, and strategy.”

Her two satellites are Aegis (or Ægis?) and Gorgoneion, two attributes of the goddess in either culture. They were discovered at the W.M. Keck Observatory in Hawaii, in 2009. While Minerva itself measures around 150 km in diameter, the satellites are much smaller, at 4 and 3 km, respectively.

These two satellites are on very inclined orbits, basically perpendicular to the J2000 equatorial frame. This is consistent with Minerva’s axis of rotation also being tilted by a similar amount (its pole points to λ = 21 ± 10°, β = 21 ± 10° in the ecliptic frame). Despite this great angle, I have left them seen from above the equatorial frame, for consistency.

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There is insufficient information in the *Johnston Archive* about this system to build a simulation. However, a 2018 paper provided the required data. Funnily enough, one of its authors had published another paper, back in 1987, “that predicted there were no moons around asteroids because constant collisions would prevent satellites from remaining in orbit.”

Considering the mass of the outermost satellite (estimated at 1.12 × 10^{19} kg) and the eccentricity of the innermost one, it is likely the orbit of the latter precesses significantly. However, I was not able to find any numeric data about this. Maybe this would be worth computing, to add to the simulation…

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I wasn’t able to find orbital elements for the satellites of this minor planet. The first one was detected in 2003 and given the provisional designation S/2003 (130) 1. It’s about 4 km in diameter and orbits Elektra at around 1,170 km. The second one is about half the size and one-third the distance to the main body; it was discovered in December 2014 and bears provisional designation S/2014 (130) 1. Elektra itself measures 215 × 155 ± 12 km.

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Size and Orbital Elements of the Satellites* | ||
---|---|---|

Alexhelios | ||

Diameter | d | 678 km |

Semimajor axis | a | 289 km |

Eccentricity | e | 0 (?) |

Period | P | 2.32 days |

Mean motion | n | 155.1724°/d |

Orbit pole α | 74 ± 2° | |

Orbit pole δ | 16 ± 1° | |

Epoch | 2454728.5 (2008.715947981) | |

Cleoselene | ||

Diameter | d | 454 km |

Semimajor axis | a | 20 km |

Eccentricity | e | 0 (?) |

Period | P | 1.24 days |

Mean motion | n | 290.3226°/d |

Orbit pole α | 79 ± 2° | |

Orbit pole δ | 16 ± 1° | |

Epoch | 2454728.5 (2008.715947981) |

The primary was discovered on April 10, 1880, and named after the famous Egyptian queen. This nonfamily (“background”) asteroid is known from radar observations to be dog-bone-shaped and measure 276 × 94 × 78 km. It reflects between 10% and 20% of received light.

Once again, I wasn’t able to find orbital elements for the satellites of 216 Kleopatra. One source does mention the data from the table on the right; *n* is inferred. This data is, however, insufficient to create a model, as we don’t know and can’t imply the position of the moons along their orbits at any given time.

The discovery of both moons was done by a team led by Franck Marchis and Pascal Descamps and announced in September 2008.

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This minor planet was discovered in 1975 by a Ukrainian astronomer and named for Lithuania; its name is that of the country in Ukrainian (Литва). At the time, both countries were members of the Union of Soviet Socialist Republics.

I wasn’t able to find much, if any, information about its moons. They were discovered in 2009 and 2012 and thus bear provisional designations S/2009 (2577) 1 and S/2012 (2577) 1.

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Discovered March 2, 1981, this Amor-group, near-Earth, and potentially hazardous asteroid, is named after Florence Nightingale, the founder of modern nursing.

No information was found on these satellites, other than they were discovered on August 29 (JPL says 30–31), 2017 during radar observations from Arecibo and Goldstone. The inner moon of Florence has the shortest orbital period (0.3 d) of any of the moons of the 60 near-Earth asteroids known to have moons. They measure around 200 m (inner) and 300 m (outer), while Florence itself is approximately 4.4 km in diameter.

**⚠️ UPDATE 2021‑02‑16 ⚠️** Florence’s moons orbit at approximately 4.6 and 9.8 km of the main body in 0.3 d and 1.02 ± 0.1 d, respectively.

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Size and Orbital Elements of the Satellites* | ||
---|---|---|

S/2002 (3749) 1 | ||

Diameter | d | 1.84 km |

Semimajor axis | a | 289 km |

Eccentricity | e | 0.9 |

Period | P | 61 days |

Mean motion | n | 5.9016°/d |

Epoch | 2454728.5 (2008.715947981) | |

S/2007 (3749) 1 | ||

Diameter | d | 1.66 km |

Semimajor axis | a | 20 km |

Period | P | 1.391 days |

Mean motion | n | 258.8066°/d |

Epoch | 2454728.5 (2008.715947981) |

The primary was discovered on January 24, 1982, and named after Canadian astronomer David Balam, who is credited with the discovery or codiscovery of more than 600 asteroids—one of which was named Tsawout after the First Nation of the same name, who live on Vancouver Island, Canada. It makes a pair with 312497 (2009 BR₆₀) in that they have very similar orbital elements (see table below), and may be former parts of the same binary asteroid.

Comparison of Orbital Elements of the Balam–2009 BR₆₀ Asteroid Pair | ||
---|---|---|

Parameter | Balam | 2009 BR₆₀ |

a | 2.237802375865213 au | 2.23752665653593 au |

e | 0.1089878757626575 | 0.108969679253321 |

i | 5.382059856664287° | 5.389474131903992° |

Ω | 295.6883575156554° | 296.3409350068008° |

ω | 174.1727672938047° | 173.2470711018107° |

M | 188.3503046057333° | 162.9699453483741° |

Tₚ | 2459583.504033032550 | 2458447.078823951383 |

P | 1222.731280761296 days | 1222.505308887556 days |

Epoch: 2459000.5 · Source: JPL Small-Body Database Browser |

Hardly any information was found on its satellites, other than they were discovered February 8, 2002 and July 15, 2007.

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Both of this minor planet’s moons were discovered on September 2, 2015 by an international team. While the first was almost immediately announced (October 2015), the second wasn’t until July 20, 2018. I wasn’t able to find more data.

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No data was found about these satellites discovered January 1, 2017 and announced two weeks later.

Extract from *IAU Telegram #4346*:

“[. . .] with an orbital period of 14.392 ± 0.004 hr. Mutual eclipse/occultation events that are 0.08- to 0.16-magnitude deep indicate a secondary-to-primary mean-diameter ratio of 0.28 ± 0.03. The eclipsing/occulting secondary has a synchronous rotation with an observed lightcurve amplitude of 0.04 mag. Superimposed to the eclipse/occultation lightcurve are two rotational lightcurves with periods 2.6832 ± 0.0002 hr and 3.2981 ± 0.0002 hr, each with an amplitude of 0.08 mag; their behavior in the mutual events—they are present with unchanged shapes during the events—indicates that none of them belongs to the eclipsing/occulting secondary. This suggests that, while one of the short periods belongs to the primary, the other belongs to a third body in the system (compare with other cases of this kind in Pravec et al. 2016, *Icarus* 267, 267).”

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Lempo was the Finnish god of love and fertility before the advent of Christianity in Finland in the 11th or 12th century; it’s been seen as a devil since. Paha and Hiisi, this minor planet’s satellites, were discovered on December 8, 2001 and October 2007, respectively, and named October 5, 2017, from mythological Finnish demons, cohorts of Lempo. Together, they brought down the hero Väinämöinen, an old wise man who may be seen as a source for J.R.R. Tolkien’s Gandalf. It is to be noted that “hiisi” and “lempo” are considered (very mild) swear words in the Finnish language.

This is a triple system as far as the size of each body is concerned (see comparative sizes below). For this reason, there is a very fast apsidal precession, estimated to be of a full 360° in approximately 50 years. Because this is *apsidal* rather than *nodal* precession, the effects are barely seen in our animation, even when left to run for very long timespans.

Hiisi orbits so close to Lempo that the Hubble Space Telescope, used for the discovery, wasn’t able to resolve the two apart.

Like for the Pluto/Charon duet, it must be stated that orbits in this system are around a common barycentre, which is between Lempo and Hiisi, most likely outside either body.

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Discovered in 1930 by Clyde Tombaugh, Pluto was originally thought to have a mass similar to Earth’s, but the astronomical community soon realized such isn’t the case. This was a surprise, as Pluto had been found following what was perceived as irregularity in the orbit of Neptune—a planet which itself had been discovered very close to a computed position after Uranus’ orbit turned out to be different from calculations. More observations of Neptune and a better knowledge of the mass of the various gas giant planets allowed astronomers to sleep better at night considering Pluto’s lower mass.

The story wasn’t over, though, for what was considered to be the “ninth planet,” as the discovery of similarly sized trans-Neptunian bodies (TNOs) in the late 1990s and early 2000s brought astronomers to wonder whether those deserved to also be called “planets,” or if Pluto had to be called something else. In 2006, the International Astronomical Union, through a vote which brought controversy as not all delegates were present, decided on a new definition of the term “planet,” and Pluto became instead a “dwarf planet,” along with Ceres and Eris, the first TNO discovered to be comparable in size to Pluto. Eventually, the IAU added Haumea and Makemake to the list of dwarf planets. (Additionally, Quaoar, Sedna, Orcus, and Gonggong are likely dwarf planets, the latter being recognized as such by JPL and NASA, though the IAU still has to officially decide on either of these four bodies.)

A first satellite to Pluto was discovered in 1978 by James Christy, who named it “Charon” partly after his wife Charlene, but also from the Greek ferryman of the dead. Two pronounciations are accepted for it, plus variants due to local accents: [UK]

Nix and Hydra were discovered in 2005, Kerberos in 2011, and Styx in 2012, all four during search programmes that were run in prevision of the arrival of the probe New Horizons, launched in 2006 and who flew through the Pluto system in 2015. “Nix was named after Nyx, the Greek goddess of darkness and night and mother of Charon.” Hydra got its name from the mythological nine-headed monster, which makes a reference to Pluto formerly being classified as the “ninth planet.” Kerberos was named after Cerberus, Pluto’s watchdog in the underworld, while Styx bears the name of a goddess of the river of the same name in the underworld; both were named following a 2013 internet poll launched by Mark Showalter of the SETI Institute—the names were second and third after Vulcan, made popular following a tweet by Canadian–American actor William Shatner of *Star Trek* fame, but the IAU discounted Vulcan, as it’s not an underworld deity, and the name had already been given to an hypothetical planet inside the orbit of Mercury and to the hypothetical category of vulcanoid minor planets.

Styx, Nix, and Hydra are in a 18:22:33 resonance when considering the precessing orbits, with Hydra and Nix being in a simple 3:2 resonance. Additionally, “Styx, Nix, Kerberos, and Hydra are in a 1:3:4:5:6 sequence of near resonances” with Charon.

For this system, I have exceptionally used orbital elements from the JPL HORIZONS Web-Interface, for March 1, 2021, at 0 h UT, except for the rate of change of the nodes and apsides, which is from another JPL page.

Due to an erroneous conversion algorithm, this page wrongly stated that Charon and Styx had extreme orbital precession, which is *not* the case! Our most sincere apologies for this mistake. Consequently, Charon’s and Styx’s orbits are now drawn accurately and do not result in strange shapes.

For more information about the Pluto system, see for example Marina Brozović et al 2014.

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Haumea is the matron goddess—of fertility and childbirth—of the island of Hawaiʻi. She’s the mother of Pele, Kāne Milohai, Kāmohoaliʻi, Nāmaka, Kapo, and Hiʻiaka, among many others.

Haumea is currently the third-largest known transneptunian object, after 136199 Eris and 134340 Pluto; its mass is estimated to be one-third that of Pluto. It is most likely ellipsoid in shape, with its major axis twice as long as its minor: 2,100 × 1,680 × 1,074 km. A ring of 2,287 km in diameter was discovered around it—it is not present in our model.

Both of its satellites were discovered in 2005: January 26 for Hiʻiaka, and June 30 for Namaka. Note that the special character in the former’s name is *not* an apostrophe, but an ʻokina, which is considered a letter in the Hawaiian alphabet, marking a glottal stop.

“On timescales longer than a few weeks, no Keplerian orbit is sufficient to describe the motion of the inner, fainter satellite Namaka.”

The orbit is very eccentric, and its argument of periastron has a precession period of about 55 years.

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The main body was discovered by the Spacewatch project, founded in 1980 to search of potentially hazardous asteroids, of which it is part of the Apollo group. Its Earth minimum orbit intersection distance (MOID) is 0.0157 AU or 6.1 lunar distances—the maximum MOID for an object to be considered “potentially hazardous” is 0.05. It reflects slightly less than 40% of the light that reaches it.

Both satellites (informal names given here, from Brozović 2011) in this system were discovered on June 12, 2009, during Goldstone and Arecibo planetary radar observations, and announced a week later. They have very highly inclined orbits, basically perpendicular to the J2000 equatorial frame.

The orbital elements we use here are from .

A funny but meaningless peculiarity (*trivia*) is that, seen from above the J2000 equatorial frame, the outermost satellite never passes directly above or below the zone covered by the innermost satellite’s orbit.

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This is another near-Earth asteroid, discovered by the Lincoln Near-Earth Asteroid Research (LINEAR) project on September 20, 2001 (with precovery observations going back to September 25, 1990). Both of its moons were discovered on February 12, 2008 from radar observations made at Arecibo—the announcement was made the same day. It is an unusual carbonaceous near-Earth asteroid of a C- or somewhat brighter B-type. Its MOID is of 0.0520 AU or 20.3 lunar distances, putting it just outside the limit for being potentially hazardous to Earth. Its size being known from radar observations, we now know that it reflects slightly less than 5% of the light that reaches it.

This triple asteroid system is the target for the planned ASTER mission of the Brazilian Space Agency. After originally being planned for launch in 2015 with asteroid rendez-vous in 2019, it is currently scheduled for launch in February 2021 with rendez-vous in April 2022. The probe includes a multispectral imaging camera with wide and narrow bands, a laser rangefinder to map the surface and texture of the asteroid, an infrared spectrometer and a mass spectrometer to determine surface composition, and a synthetic aperture radar.

The orbital elements used here are also from .

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- “The J2000 (aka EME2000) frame definition is based on the earth’s equator and equinox, determined from observations of planetary motions, plus other data.”

- Benecchi, Susan D., Keith S. Noll, Will M. Grundy, and Harold F. Levison. “(47171) 1999 TC₃₆, A transneptunian triple.”
*Icarus,*Vol. 207, Issue 2 (June 2010): 978–991, PDF. - Brozović, Marina, Lance A. M. Benner, Patrick A. Taylor, Michael C. Nolan, Ellen S. Howell, Christopher Magri, Daniel J. Scheeres, Jon D. Giorgini, Joseph T. Pollock, Petr Pravec, Adrián Galád, Julia Fang, Jean-Luc Margot, Michael W. Busch, Michael K. Shepard, Daniel E. Reichart, Kevin M. Ivarsen, Joshua B. Haislip, Aaron P. LaCluyze, Joseph Jao, Martin A. Slade, Kenneth J. Lawrence, and Michael D. Hicks. “Radar and optical observations and physical modeling of triple near-Earth Asteroid (136617) 1994 CC.”
*Icarus,*Vol. 216, Issue 1 (November 2011): 241–256, PDF. - Brozović, Marina, Mark R. Showalter, Robert A. Jacobson, and Marc W. Buie. “The orbits and masses of satellites of Pluto.”
*Icarus,*Vol. 246 (January 2015): 317–329, PDF. - Dailey, Jeanne. “AFRL’s Starfire Optical Range discovers asteroid satellite.”
*Air Force Life Cycle Management Center. News,*January 20, 2021, link. - Fang, Julia Anna, Jean-Luc Margot, Marina Brozovic, Michael C. Nolan, Lance A. M. Benner, and Patrick A. Taylor. “Orbits of Near-Earth Asteroid Triples 2001 SN263 and 1994 CC: Properties, Origin, and Evolution.”
*The Astronomical Journal,*Vol. 141, Issue 5 (May 2011), Article id. 154, 15 p., PDF. - Fang, Julia Anna, Jean-Luc Margot, and Patricio Rojo. “Orbits, Masses, and Evolution of Main Belt Triple (87) Sylvia.”
*The Astronomical Journal,*Vol. 144, Issue 2 (August 2012), Article id. 70, 13 p., PDF. - Green, Daniel W. E.
*International Astronomical Union Circular No. 8582,*August 11, 2005, link. - Hirabayashi, Masatoshi, and Daniel J. Scheeres. “Analysis of Asteroid (216) Kleopatra using dynamical and structural constraints.”
*The Astrophysical Journal,*Vol. 780, Issue 2 (January 2014), Article id. 160, 11 p., PDF. - Jet Propulsion Laboratory. “HORIZONS Web-Interface.”
*Solar System Dynamics,*revised November 5, 2015, except for Charon: January 26, 2018, link. - Jet Propulsion Laboratory. “Planetary Satellite Mean Orbital Parameters.”
*Solar System Dynamics,*link. - Jet Propulsion Laboratory. “Small-Body Database Browser.”
*Solar System Dynamics,*link. - Johnston, William Robert.
*Asteroids with Satellites.*Updated December 6, 2020, link. - Johnston, William Robert. “(45) Eugenia, Petit-Prince,
*and*S/2004 (45) 1.”*Asteroids with Satellites Database—Johnston’s Archive.*Updated September 21, 2014, link. - Johnston, William Robert. “(87) Sylvia, Romulus,
*and*Remus.”*Asteroids with Satellites Database—Johnston’s Archive.*Updated September 21, 2014, link. - Johnston, William Robert. “(3749) Balam, S/2002 (3749) 1,
*and third component*.”*Asteroids with Satellites Database–Johnston’s Archive.*Updated September 21, 2014, link. - Marchis, Franck, Pascal Descamps, Daniel Hestroffer, and Jérôme Berthier. “Discovery of the Triple Asteroidal System (87) Sylvia.”
*Nature,*Vol. 436, Issue 7052 (August 2005): 822–824 (PDF). - Marchis, Franck, Frédéric Vachier, Josef Ďurech, Juan Emilio Enriquez, Alan W. Harris, Paul A. Dalba, Jérôme Berthier, Joshua P. Emery, Hervé Bouy, J. Melbourne, Alan Stockton, Christopher D. Fassnacht, Trent J. Dupuy, and Jean Strajnic. “Characteristics and Large Bulk Density of the C‑type Main-Belt Triple Asteroid (93) Minerva.”
*Icarus,*Vol. 224, Issue 1 (May 2013): 178–191 (PDF). - Meeus, Jean.
*Astronomical Algorithms,*Second Edition. Richmond: Willmann-Bell, 1998. - Navigation and Ancillary Information Facility.
*An Overview of Reference Frames and Coordinate Systems in the SPICE Context.*January 2020, PDF. - Pajuelo, Myriam Virginia, Benoît Carry, Frédéric Vachier, Michael Marsset, Jérôme Berthier, Pascal Descamps, William J. Merline, Peter M. Tamblyn, J. Grice, Al Conrad, Alexander Storrs, Bradley Timerson, David W. Dunham, Steve Preston, Arthur Vigan, Bin Yang, Pierre Vernazza, Stéphane Fauvaud, Laurent Bernasconi, David Romeuf, Raoul Behrend, Christophe Dumas, Jack D. Drummond, Jean-Luc Margot, Pierre Kervella, Franck Marchis, and Julien H. Girard. “Physical, spectral, and dynamical properties of asteroid (107) Camilla and its satellites.”
*Icarus,*Vol. 309 (July 2018): 134–161, PDF. - Pravec, P., P. Scheirich, P. Kušnirák, K. Hornoch, A. Galád, S.P. Naidu, D.P. Pray, J. Világi, Š. Gajdoš, L. Kornoš, Yu.N. Krugly, W.R. Cooney, J. Gross, D. Terrell, N. Gaftonyuk, J. Pollock, M. Husárik, V. Chiorny, R.D. Stephens, R. Durkee, V. Reddy, R. Dyvig, J. Vraštil, J. Žižka, S. Mottola, S. Hellmich, J. Oey, V. Benishek, A. Kryszczyńska, D. Higgins, J. Ries, F. Marchis, M. Baek, B. Macomber, R. Inasaridze, O. Kvaratskhelia, V. Ayvazian, V. Rumyantsev, G. Masi, F. Colas, J. Lecacheux, R. Montaigut, A. Leroy, P. Brown, Z. Krzeminski, I. Molotov, D. Reichart, J. Haislip, and A. LaCluyze. “Binary asteroid populations. 3. Secondary rotations and elongations.”
*Icarus,*Vol. 267 (March 2016): 267–295, PDF. - Ragozzine, Darin Alan, and Michael E. Brown. “Orbits and Masses of the Satellites of the Dwarf Planet Haumea (2003 EL61).”
*The Astronomical Journal,*Vol. 137, Issue 6 (June 2009): 4766–4776, PDF. - Royal Astronomical Society of Canada. “Asteroid (3748) Tatum.”
*Asteroids with a Canadian Connection,*posted May 3, 2011, link. - Shepard, Michael K., Jean-Luc Margot, Christopher Magri, Michael C. Nolan, Joshua Schlieder, Benjamin Estes, Schelte J. Bus, Eric L. Volquardsen, Andrew S. Rivkin, Lance A.M. Benner, Jon D. Giorgini, Steven J. Ostro, and Michael W. Busch. “Radar and infrared observations of binary near-Earth Asteroid 2002 CE26.”
*Icarus,*Vol. 184, Issue 1 (September 2006): 198–210, PDF. - Tatum, Jeremy B. “Celestial Mechanics.”
*Physics Topics,*last updated 12 July 2020, link. - Wikipedia, s.v. “45 Eugenia,” last modified December 30, 2020, 21:19, link.
- Wikipedia, s.v. “87 Sylvia,” last modified January 15, 2021, 20:09, link.
- Wikipedia, s.v. “Julian day,” last modified February 2, 2021, 12:51, link.
- Wikipedia, s.v. “Minerva,” last modified January 28, 2021, 10:30, link.
- Wikipedia, s.v. “Minor-planet moon,” last modified January 29, 2021, 01:40, link.
- Wikipedia, s.v. “Moons of Pluto,” last modified January 26, 2021, 01:20, link.
- Wikipedia, s.v. “Nix (moon),” last modified December 25, 2020, 11:23, link.
- Wikipedia, s.v. “(136617) 1994 CC,” last modified January 2, 2021, 02:09, link.
- Wikipedia, s.v. “(153591) 2001 SN₂₆₃,” last modified January 23, 2021, 19:19, link.

Planetary and satellite orbits are defined by a minimum of six parameters, explained in the table below. Following the table is an interactive diagram.

Symbol | Parameter | Definition |
---|---|---|

Epoch | The moment for which the values below are given. | |

a | Semimajor axis | Half the maximum dimension of the orbital ellipse. |

e | Eccentricity | A measure of the “stretching” of the orbital ellipse. It is defined as $ \displaystyle e = \sqrt { 1 - \frac {b^2}{a^2} } $, where b is half the minimum dimension of the orbital ellipse (also known as the semi-minor axis). |

i | Inclination | Tilt of the orbital ellipse with respect to a defined reference plane. In this document, this plane is that of the Earth’s equator at epoch J2000.0. Other reference planes used in astronomical literature include the ecliptic plane, defined as that of the Earth’s orbit around the Sun); the invariable plane (usually that of the Solar System, but may also be of a given planet/satellites or minor planet / satellites systems), defined as passing through the system’s barycentre and perpendicular to its angular momentum vector; or a main body’s equatorial plane. |

Ω | Longitude of the ascending node | The angle, measured along the reference plane, between the reference point (usually, the vernal equinox; symbol ♈︎) and the ascending node. The ascending node is the point where the object crosses the reference plane from “below” to “above”—from south to north, usually. The symbol for the ascending node is ☊, which must not be confused with Ω; notice the shape of the bottom. |

ω | Argument of the periastron | The angle, measured along the plane of the body’s orbit, between the ascending node and the periastron. The periastron (most often called “periapsis,” although this term’s etymology is wrong) is the point along the body’s orbit which is the closest to the main body. In the case of a body orbiting the Sun, this point is the perihelion; if orbiting the Earth, it’s the perigee.Sometimes, the (symbol: ϖ or π) is given. It is equal to Ω + ω, even though these two angles are measured on two different planes (for historical reasons). Care must be taken not to confuse ω, ϖ, and π—and then, the latter with the mathematical constant π = 3.14159265…longitude of periastron |

And at least one of the following, which can be calculated from each other: | ||

Tₚ | Time of periastron passage | The moment when the body is at periastron. It may be a calendar date with decimal day or day and hours, or a Julian Day (see Appendix II). |

P | Orbital period | The time it takes the body to complete one full orbit. Its unit of measurement may depend on the context: for example, measuring the orbital period of a star around its galaxy’s centre in days is pointless. |

ν | True anomaly | This is the angle, measured along the body’s orbit, between the periastron and the body’s current position. It is sometimes noted θ or f. |

M | Mean anomaly | The fraction of the orbit’s period that has elapsed since the passing of the periastron, expressed as an angle. This angle is that of a fictitious “mean body,” that would travel at uniform speed along a fictitious circular orbit whose radius is equal to the semimajor axis of the ellipse. |

n | Mean motion | The average angular speed of the body—also that of the mean fictitious body used in calculating M. Its value is $ \displaystyle n = \frac{360°}{P} $. |

L | Mean longitude | The longitude at which the body would be located if its orbit were circular and free of perturbations. It does not correspond to any real angle. Its value is $ \displaystyle L = Ω + ω + M $. |

Because orbits are subject to gravitational perturbations from other bodies in the system or the “parent” system (for planetary satellites, the system is that of the main planet and its satellites, whereas the “parent” system is the Solar System with the Sun, the other planets and their satellites, the minor planets, etc.), the first five elements will change slowly over long periods of time (whereas ν, *M*, and *L* change on generally much shorter timescales). A rate of change is sometimes given for these elements; for example, we have given ΔΩ and Δω for several of the minor planet satellites mentioned in the present document. These are sometimes noted instead by $ \displaystyle \dot{Ω} $ and $ \displaystyle \dot{ω} $. The **epoch** of the orbit must be known in order to derive accurate positions for the bodies being studied.

*Because of the time-dependent nature and evolution of orbital parameters, we do not recommend using the starting values or the results of the present document to compute positions for practical purposes such as celestial observing or navigation. We recommend using accurate and up-to-date orbital elements from the JPL HORIZONS Web-Interface.*

Converting observed positions into orbital elements and vice versa is a rather complex procedure, which is detailed in Jeremy Tatum’s Course Notes from when he was teaching at the University of Victoria, Canada. There, he was “the chief founder and is the driving force behind the [University of Victoria’s] Climenhaga Observatory’s program of astrometry of minor planets and comets, which was the only one of its kind in Canada.” Consequently, an asteroid was named in his honor: (3748) Tatum (1981 JQ)—its number is incidently just one less than that of Balam from our trinary minor planets section above.

*For the following, we use the notation found on Wikipedia, not that found on Pr. Tatum’s website; the rest of the procedure is left unaltered.*

Parts of the below procedures are from the author of the present document.

An ellipse can described in various ways. An orbit generally being described by its semimajor axis *a* and its eccentricity *e*, we can find its semiminor axis *b* through $ \displaystyle e = \sqrt{1 - \frac{b^2}{a^2}} $. If it is not centered on the origin, and that it is rotated (as is usually the case for orbits), its angle of rotation may be called $ \theta $ and its centre point being $ \displaystyle \left( x_0, y_0 \right) $, we have for general form of the ellipse:

Another way to describe an ellipse mathematically is by its parametric form:

$$ \displaystyle Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 $$Conversion back and forth between these two forms is possible:

$$ \begin{array}{cl} A & = & a^2\text{ sin}^2\ \theta + b^2\text{ cos}^2\ \theta \\ \\ B & = & 2\ (b^2 - a^2)\text{ sin }\theta\text{ cos }\theta \\ \\ C & = & a^2\text{ cos}^2\ \theta + b^2\text{ sin}^2\ \theta \\ \\ D & = & -2Ax_0 - By_0 \\ \\ E & = & -Bx_0 - 2Cy_0 \\ \\ F & = & Ax_0^2 + Bx_0y_0 + Cy_0^2 - a^2b^2 \\ \\ a, b & = & \displaystyle \frac {-\sqrt{2 \left( AE^2 + CD^2 - BDE + \left( B^2 - 4AC \right) F \right) \left( \left( A + C \right) ± \sqrt{\left( A - C \right)^2 + B^2 } \right)}}{B^2 - 4AC} \\ \\ x_0 & = & \displaystyle \frac{2CD - BE}{B^2 - 4AC} \\ \\ y_0 & = & \displaystyle \frac{2AE - BD}{B^2 - 4AC} \\ \\ \theta & = & \begin{cases} \begin{array}{ll} \displaystyle \text{arctan }\left( \frac{1}{B} \left( C- A - \sqrt{\left( A - C \right)^2 + B^2} \right) \right) & \text{for }B \neq 0 \\ 0 & \text{for }B = 0, A \lt C \\ 90° & \text{for }B = 0, A \gt C \\ \end{array} \end{cases} \end{array} $$When the equation of the ellipse is unknown, it is possible to find it with at least five points; while there is a way to solve for five equations with five unknowns with a matrix, Tatum presents “a better way.”

Let us first designate our five points as A, B, C, D, and E. Between them are imaginary straight lines AB, whose equation we’ll call α = 0; CD, to be β = 0; AC, to be γ = 0; and BD, to be δ = 0 (we’ll ignore point E for the time being). With αβ = 0 being lines AB and CD, and γδ = 0 being lines AC and BD, we have αβ + λγδ = 0, where λ is an arbitrary constant, to define any conic section that passes through A, B, C, and D. Adding E in the lot allows us to “find the value of λ that forces the equation to go through all five points.”

Let us now define points A = (1, 8); B = (4, 9); C = (5, 2); D = (7, 6); and E = (8, 4). Finding α thus boils down to finding the slope and origin of AB, and so on. We thus have $ \displaystyle \mathrm{_{slope}AB} = \frac {B_y - A_y} { B_x - A_x } = \frac {9 - 8} {4 - 1} = \frac {1}{3} $, so $ \displaystyle y = \frac {x}{3} + d $. Inputting values in there, we have $ \displaystyle A_y = \frac {A_x}{3} + d \Rightarrow 8 = \frac {1}{3} + d \Rightarrow d = \frac {23}{3} $, hence $ \displaystyle y = \frac {x}{3} + \frac {23}{3} = \frac {x + 23}{3} $, which can be rewritten to $ \displaystyle \alpha = x - 3 y + 23 = 0 $. In a similar way, we find the other lines:

$$ \begin{array}{lr} \alpha = 0: & x - 3y + 23 = 0\\ \\ \beta = 0: & 2x - y - 8 = 0\\ \\ \gamma = 0: & 3x + 2y - 19 = 0\\ \\ \delta = 0: & x + y - 13 = 0 \end{array} $$We then find the pairs of lines by multiplying two equations together. For example:

$$ \alpha\beta = (x - 3y + 23) (2x - y - 8) = 0 \Rightarrow 2x^2 - 7xy + 3y^2 + 38x + y - 184 = 0 $$Thus:

$$ \begin{array}{lr} \alpha\beta = 0: & 2x^2 - 7xy + 3y^2 + 38x + y - 184 = 0\\ \\ \gamma\delta = 0: & 3x^2 + 5xy + 2y^2 - 58x - 45y + 247 = 0\\ \\ \alpha\beta + \lambda\gamma\delta = 0: & (2 + 3 \lambda)x^2 - (7 - 5 \lambda) xy + (3 + 2 \lambda) y^2 + (38 - 58 \lambda) x + (1 - 45 \lambda) y - 184 + 246 \lambda = 0 \end{array} $$Now we substitute the coordinates of E in lieu of *x* and *y* to find the value of λ:

And we get the final equation of the ellipse by putting λ back in our equation:

$$ \displaystyle 508x^2 + 578xy + 382y^2 - 7827x - 6814y + 32760 = 0 $$Coming back to our parametric equation $ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 $, we thus have the following values:

$$ \begin{array}{cr} A & = & 508 \\ B & = & 578 \\ C & = & 382 \\ D & = & -7828 \\ E & = & -6814 \\ F & = & 32760 \end{array} $$An additional test is needed here to know if the conic is a circle, an ellipse, a parabola, or a hyperbola, but we’ll skip it by assuming all observed object positions are indeed elliptical in nature.

We then calculate the following cofactors:

$$ \begin{array}{ccr} \bar C & = & \displaystyle AC - \left( \frac{B}{2} \right)^2 & = & 110\,535 \\ \\ \bar D & = & \displaystyle \frac{DB}{4} - \frac{AE}{2} & = & 599\,610 \\ \\ \bar E & = & \displaystyle \frac{BE}{4} - \frac{CD}{2} & = & 510\,525 \\ \end{array} $$The centre of the ellipse is at $ \displaystyle \left( \frac {\bar E}{\bar C}, \frac {\bar D}{\bar C} \right) = \left( \frac {510\,525}{110\,535}, \frac {599\,610}{110\,535} \right) = \left( 4.61867, 5.42462 \right) $, and its major axis is inclined at an angle $ \displaystyle \text{tan }2\theta = \frac{B}{A - C} = \frac {578} {508 - 382} = \frac {578}{126} = 4.58730 \Rightarrow 2\theta = \text{atan } 4.58730 = 77.70231° \Rightarrow \theta = 38.85115° = 38°\ 51^′\ 4.15^″ $ to the *y* axis (or 128° 51′ 4.15″ to the *x* axis).

Another, less precise way is to draw the observations and the fitted ellipse, and to measure its semimajor and semiminor axes and tilt on the resulting graph with a ruler and a protractor.

It is common practice, when dealing with orbits, to set the position of the primary body as (0, 0), with North up and East corresponding to 90°.

We first need to establish the apparent orbit of the body. This is done by measuring, at various moments, its separation and position angle from the primary. *In the case of minor planets and other Solar System objects, compensation must be made for the relative movement of the system and the Earth around the Sun; this is the subject for a possible future article.* Once the apparent orbit’s equation is known (see above procedure), we can move to finding the actual orbital elements.

For this example, we’ll be using the values from Tatum’s §17.4 (while still using usual equation parameters, and not those used by Tatum):

$$ \displaystyle 14x^2 - 23xy + 18y^2 - 3x - 31y - 100 = 0 $$So we have:

$$ \begin{array}{} A = 14\text{, }& B = -23\text{, }& C = 18\text{, }& D = -3\text{, }& E = -31\text{, }& F = -100 \end{array}$$This gives us a centre at:

$$ \begin{array}{ccrcrcr} \bar C & = & \displaystyle 14 \cdot 18 - \left(\frac{-23}{2}\right)^2 & = & 252 - 132.25 & = & 119.75 \\ \\ \bar D & = & \displaystyle \frac{(-3)\cdot(-23)}{4} - \frac{14 \cdot (-31)}{2} & = & 17.25 - (-217) & = & 234.25 \\ \\ \bar E & = & \displaystyle \frac{(-23) \cdot (-31)}{4} - \frac{18 \cdot (-3)}{2} & = & 178.25 - (-27) & = & 205.25 \\ \\ x & = & & & \displaystyle \frac {205.25}{119.75} & = & 1.71399 \\ \\ y & = & & & \displaystyle \frac {234.25}{119.75} & = & 1.95616 \end{array} $$The slope of the major axis is $ \displaystyle m = \frac {\bar D}{\bar E} = \frac {234.25}{205.25} = 1.14129 $; any point on the major axis of the apparent ellipse will be along the line $ \displaystyle y = 1.14129\ x $ (there is no + *d* here as it passes by the origin). The apparent ellipse has two points in common with the actual ellipse: periastron and apoastron. If we replace *y* in the parametric equation of our ellipse, we get the coordinates of these two points, by solving the following two equations:

and

$ (C + Bn + An^2)\ y^2 + (E + Dn)\ y + F = 0 $,

where $ \displaystyle n = \frac {1}{m} = 0.87620 $. This gives us:

$$ \begin{array}{rcl} (14 + (-23) \cdot (1.14129) + 18 \cdot (1.14129)^2)\ x^2 + ((-3) + (-31) \cdot (1.14129))\ x + (-100) & = & 0 \\ \\ 11.19612\ x^2 + (-38.38002)\ x + (-100) & = & 0 \\ \\ x & = & 5.15919\text{ or }-1.73121 \end{array}$$ $$ \begin{array}{rcl} (18 + (-23) \cdot (0.87620) + 14 \cdot (0.87620)^2)\ y^2 + ((-31) + (-3) \cdot (0.87620))\ y + (-100) & = & 0 \\ \\ 8.59557\ y^2 + (-33.62860)\ y + (-100) & = & 0 \\ \\ y & = & 5.88814\text{ or }-1.97582 \end{array}$$“If *m* is positive the larger solution for *y* corresponds to the larger solution for *x*; If *m* is negative the larger solution for *y* corresponds to the smaller solution for *x*.”

There are two ways, not detailed here, for knowing which is the periastron and which is the apoastron: Either draw the graph, or calculate the distance between each point and the origin. Here we find the periastron to be located at (-1.73121, -1.97582).

The **eccentricity** of the actual orbit is calculated from the ratio of distances focus–centre (or origin–centre) to periastron–centre (semimajor axis). This gives us:

We then need to calculate an auxiliary ellipse. The first step is calculating a lengthening factor $ \displaystyle k = \frac {1}{\sqrt{1 - e^2}} = \frac {1}{\sqrt{1 - 0.49750^2}} = 1.15279 $, then the slope of the projected latus rectum—a chord perpendicular to the major axis and passing by the focus—of the actual orbit $ \displaystyle m = \frac{-\frac{D}{2}}{\frac{E}{2}} = \frac{-\frac{-3}{2}}{\frac{-31}{2}} = -0.09677 $. We find the two points where the latus rectum intersects the apparent ellipse the same way we found the position of the apsides, except with different values of *m* and $ \displaystyle n = \frac{1}{m} = -10.33333 $:

Again, “[i]f *m* is positive the larger solution for *y* corresponds to the larger solution for *x*; If *m* is negative the larger solution for *y* corresponds to the smaller solution for *x*,” so we have *M* = (-2.46975, 0.23901) and *N* = (2.46975, -0.23901).

We also need to find (at least one of) the point(s) on the apparent ellipse where a line parallel to the latus rectum crosses the centre. This is a slightly more complex set of equations, but nothing dramatic:

$$ \begin{array}{rcl} (A + Bm + Cm^2)\ x^2 + (Bd + 2Cmd + D + Em)\ x + Cd^2 + Ed + F & = & 0 \\ \\ (C + Bn + An^2)\ y^2 + (Be + 2Ane + E + Dn)\ y + Ae^2 + De + F & = & 0 \end{array} $$where $ \displaystyle d = \bar y - m \bar x $ and $ \displaystyle e = \frac{-d}{m} $ (not to be confused with the eccentricity!), where (*x̄*, *ȳ*) are the coordinates of the centre point. We thus have:

Because, once again, “[i]f *m* is positive the larger solution for *y* corresponds to the larger solution for *x*; If *m* is negative the larger solution for *y* corresponds to the smaller solution for *x*,” so we have *K* = (-1.13310, 2.23168).

Points on the auxiliary ellipse are found by multiplying segment lengths MF, FN, and KC (centre) by *k* to get points and segments M′F, FN′, and K′C. For M′F and FN′, this is easy enough, as MF (hence M′N′) passes by the origin, so we have $ M^′(x\text{, }y) = M(x \cdot k\text{, }y \cdot k) = (-2.84710, 0.27552) $ and $ N^′(x\text{, }y) = N(x \cdot k\text{, }y \cdot k) = (2.84710, -0.27552) $. For K′, we replace *x* and *y* by *x̄* + *k*(*x* - *x̄*) and *ȳ* + *k*(*y* - *ȳ*), so we get $ x = 1.71399 + 1.15279\ ((-1.13310) - 1.71399) = -1.56810 $ and $ y = 1.95616 + 1.15279\ (2.23168 - 1.95616) = 2.27378 $.

We now have five points on the auxiliary ellipse, which means we can derive its parametric formula from the procedure in the previous section.

$$ \begin{array}{rcl} A & = & (+5.15919, +5.88814) \\ P & = & (-1.73121, -1.97582) \\ M^\prime & = & (-2.84709, +0.27552) \\ N^\prime & = & (+2.84709, -0.27552) \\ K^\prime & = & (-1.56810, +2.27378) \\ \end{array} $$ $$ 10.5518\ x^2 - 16.9575\ xy + 15.3528\ y^2 - 3\ x - 31\ y - 100 = 0 $$ $$ \begin{array}{rcccccl} \text{tan }2\theta & = & \displaystyle \frac{B}{A-C} & = & \displaystyle \frac{-16.9575}{10.5518-15.3528} & = & 3.53208 \\ \\ & & \theta & = & 37.09607 \\ \\ & & m & = & \text{tan }\theta & = & 0.75619 \\ \\ & & n & = & \displaystyle \frac{1}{m} & = & 1.32243 \end{array}$$The **semimajor axis** of the auxiliary ellipse is its only axis that has not been foreshortened by projection. Its length is thus equal to the semimajor axis of the true orbit:

Considering the original observations must have been in arcseconds, the answer is also in arcseconds. Once the distance to the object is known, it is simple to convert angular dimensions to physical dimensions.

The **inclination** of the true orbit to the plane of the sky is found by:

The **longitude of the node** is given by:

from the *x* axis or 127.09607° from the *y* axis (the position angle used in astronomy). In strict mathematical terms, it is impossible to know which of the nodes this is, the ascending or the descending one, but observational data may allow answering this question—whether it’s drawing the apparent orbit and verifying the direction of motion, or using spectroscopically derived radial velocities.

Finally, we know the position of the node and of the periastron, but the longitude of the latter is *not* the apparent angle between the two, as it’s affected by perspective. Let’s instead call the measured angle λ. While it can be measured on a graph, it can also be calculated. Let’s first introduce angle θ:

Deciding which value is good is done by looking at the sign of $ P_x $: if it is negative, then the second angle is good—looking at the graph would also give the answer. Angle λ is then given by subtracting the lower values of θ and Ω (the one measured from the *x* axis):

The penultimate piece of our puzzle is the **argument of periastron**, given by:

The last bit of information that is required cannot be obtained by mathematical devices but by observation alone: Determining the orbital period *P* and the moment of periastron passage *Tₚ*.

Orbital Elements of Iapetus | ||
---|---|---|

Semimajor axis | a | 3564524.725648914 km |

Eccentricity | e | 0.02908967274787569 |

Ascending node | Ω | 253.4568570764956° |

Perichron | ω | 349.4625601591317° |

Inclination | i | 15.75897128219843° |

Mean anomaly | M | 3.149688132606785° |

True anomaly | ν | 3.339731721713938° |

Period | P | 6865675.917013768 seconds |

Mean motion | n | 0.00005243474995781366°/d |

Time of perichron | Tₚ | 2459273.804760225117 |

Epoch | 2459274.5 TDB (2021‑03‑01 00:00:00 TDB) |

While it is easy to find various apparent positions of a body knowing its orbital elements, the author of the present document wanted a quick and efficient way of drawing the apparent ellipse with graphic-design software such as Adobe Illustrator, or merely coding it in SVG. The easiest solution found was to determine five random apparent points from the orbital elements, then to use the above-mentioned procedure to find an ellipse’s parametric equation from five points. We’ll use the orbital elements of Iapetus, a high-inclination moon of Saturn, for our example (table to the right). These elements, taken from the JPL Horizons Web‑Interface, are *relative to Saturn’s equator and node of date* for their epoch. (Note that they differ from those given, e.g., on Wikipedia, as these are for epoch J2000.0, while those we obtained from JPL Horizons are for March 1, 2021.)

The five points may be taken at random, but the author of the present document normally uses the apsides, the nodes, and the current position of the object. One way or another, we will end up with points A, B, C, D, and E and can determine the apparent ellipse from them. The procedure here is usable for any orbit seen from any position and is the same one used in the interactive diagram above. However, we do not discuss here the determination of the viewing angle, which will be left to the viewer.

At perichron (yes, that *is* the word for “closest point of the orbit to Saturn”), true anomaly ν = 0°. At apochron, ν = 180°. At the ascending and descending nodes, ν = (−ω) and ν = (−ω) + 180°, respectively. Finally, let’s use a different value of ν than the one in the table, just to get a more precise ellipse. We’ll use instead ν = 45°. From here, and *for each of the five points*, we compute the following:

Where *r* is the radius vector of the object—its distance to the main body—and is found from:

Each of the $ (x, y, z) $ values (measured in the same units as *a* and *r*) must then be multiplied successively by each of the following matrices:

where $ \gamma $ is the roll angle, i.e. the left–right angle between the reference point and the observer. It corresponds to the observer’s longitude measured in the same reference frame as the orbital elements.

$$ \begin{vmatrix} 1 & 0 & 0 \\ 0 & \text{cos }\beta & \text{sin }\beta \\ 0 & -\text{sin }\beta & \text{cos }\beta \end{vmatrix} $$where $ \beta $ is the yaw angle, i.e. the up–down angle between the reference point and the observer. It corresponds to the observer’s latitude measured in the same reference frame as the orbital elements.

$$ \begin{vmatrix} \text{cos }\alpha & 0 & \text{sin }\alpha \\ 0 & 1 & 0 \\ -\text{sin }\alpha & 0 & \text{cos }\alpha \end{vmatrix} $$where $ \alpha $ is the pitch angle, i.e. the apparent tilt of the reference frame’s *z* axis from the point of view of the observer. It corresponds to the observer-centric (geocentric) position angle of the North pole of the main body, when the reference frame is the main body’s equator, such as is the case in our example.

As a reminder, the procedure to multiply a 1 × 3 matrix (the point’s coordinates) by a 3 × 3 matrix (the transformation one) is the following:

$$ (x, y, z) \begin{vmatrix}a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = (ax + by + cz, dx + ey + fz, gx + hy + iz) $$(Proceeding with three successive matrices allows us to avoid long, complicated equations. For example, after just the first matrix transformation, *x* is replaced by $ x \text{ cos }\gamma + y\ (-\text{sin }\gamma) $.)

After transformations using $ \beta = 71.47459, \alpha = 92.32287 $ (respectively, 90° minus Earth’s *saturnicentric* declination and 90° plus Earth’s *saturnicentric* right ascension, we now have the following values:

The *z* values are useless to us from now on, but may be used after the procedure is completed to know which of the nodes is the ascending and which is the descending. Using the (*x*, *y*) values to find the parametric equation of the apparent ellipse (procedure not shown), we have:

With its centre at (90564.31908, −11077.09827) and tilt angle $ \theta = 84.95072355° $ from the *y* axis. We have also seen how to find its dimensions, which are (432437.7422, 3563145.572). As the distance of Saturn on that day is 1,616,001,830.5 km, we can find angular distances from:

to get an apparent ellipse of 0.126369013 × 0.007845615° centered at (0.003210982, −0.000392741°) relative to Saturn. This is illustrated in the image to the right. Note that Saturn’s angular dimension at the time is 15.38509″. The respective sizes of the orbit ellipse and Saturn’s globe (rings not shown) are proportional.

**Note:** We have not attempted to correct for Earth’s tilt relative to Saturn, so the orientation of the image does not match the actual orientation of Iapetus’s orbit with respect to the celestial meridian.

The Julian day system is used by astronomers to measure time in a locale- and culture-independent way. Its unit is the day, and its era (beginning moment) is noon on January 1, 4713 BCE (−4712) of the proleptic Julian calendar. This system was created by Joseph Scaliger in 1583 and is based on the calendrical solar cycle (28 years), lunar cycle (19 years), and indiction cycle (15 years). The current Julian day is .

For dates in the Gregorian calendar (used by most countries in the world), the Julian day can be calculated with the formula:

$$ \displaystyle JD = \frac{ 1461 \times \left( Y + 4800 + \frac { \left( M - 14 \right) } { 12 } \right) } { 4 } + \frac { 367 \times \left( M - 2 - 12 \times \frac {M - 14}{12} \right) } { 12 } - \frac { 3 \times \frac { Y + 4900 + \frac { M - 14 } { 12 } } { 100 } } { 4 } + D - 32075 $$Meeus instead offers the following algorithm, valid for any date of any calendar:

Let *Y* be the year, *M* the month number (1 for January, 2 for February, etc., to 12 for December), and *D* the day of the month (with decimals, if any) of the given calendar date.

- If
*M*> 2, leave*Y*and*M*unchanged.

If*M*= 1 or 2, replace*Y*by*Y*− 1, and*M*by*M*+ 12.

In other words, if the date is in January or February, it is considered to be in the 13th or 14th month of the preceding year. - In the
*Gregorian*calendar, calculate $$ \begin{array}{llllllllllll} \displaystyle A = \text{INT }\left( \frac { Y } { 100 } \right) & & & & & & & & & & & \displaystyle B = 2 - A + \text{INT }\left( \frac { A } { 4 } \right) \end{array} $$

In the*Julian*calendar, take*B*= 0. - The required Julian Day is then

$$ \displaystyle JD = \text{INT }\left( 365.25\ \left( Y + 4716 \right) \right) + \text{INT }\left( 30.6001\ \left( M + 1 \right) \right) + D + B - 1524.5 $$

Some computer software have built-in or add-in functions to compute the Julian day: In PHP, one may use the function `gregoriantojd (int $month, int $day, int $year)`

; in JavaScript, one may use the NPM package `julian`

; in SQLite, there is a built-in `julianday()`

function; etc.