TABLE OF ASTEROID GROUPS ----------------------= John Pazmino NYSkies Astronomy Inc www.nyskies.org nyskies@nyskies.org 2008 August 31

Introduction ---------- The NYSkies Seminar of 3 June 2008 highlighted asteroids. The theme was limited to asteroids inside of Jupiter's orbit, omitting those in the outer solar system. This was merely a simplicity to help organize the discussion. As part of that session I hacked a table from Internet of asteroid groups. The original was incomplete. I filled in missing data from various websites and print references. There are still holes in the table, but it's fleshed out to be useful in understanding and discussing asteroids of the inner solar system. For the Earth-threat asteroids I give the perihelion 'q' or aphelion 'Q' as well. The dashes are data that are not applicable in the group definition or have yet to be narrowed in my inquiries. Readers are invited to fill in the missing data.

Groups and families ----------------- When the number of known asteroid grew large enough, it became evident that they bunched around certain combinations of orbital elements. An 'asteroid group' is a set of asteroids sharing common elements. Groups were revealed several decades ago, when asteroids were still mere pinpoints on photographs with almost no physical characteristics. They were studied for their dynamical behavior under gravity of the planets, notably Jupiter. In the late 1990s we started to examine asteroids via radar, spectrometry, and colorimetry. These methods gave us physical properties of asteroids, leading to the establishment of families. An 'asteroid family' is a set of asteroid sharing common physical properties, as sussed out by their spectrum, color, and radargram. Some groups later were found also to be families, so they show up in dialog about both categories. There could be situations where pieces from one larger asteroid broke off, forming a family of the fragments. These may continue to travel in similar orbits, forming a group. In this article I discuss only groups, the dynamical category of asteroid.

Elements ------ The orbital elements of an asteroid are unlike those of the major planets. Asteroids, being infinitesimally small compared with a planet, are disturbed easily by gravity. This disturbance shows up in mutable orbit elements, such that a given set turns more and more stale within a decade. Hence, in a planetarium program, you must obtain current elements to faithfully plot the path of an asteroid. Using old elements will plot wrong paths, however neat and pretty they look. Asteroid elements are

Epoch - a given date, usually in Julian Day number at even 400th days, for which the elements are valid. The elements are valid for the next few hundred days, until the next epoch. Elements from a several epochs ago should be tagged for only historical use.

Mean anomaly - the angular revolution of the asteroid in its orbit completed at the epoch date. This is NOT the 'longitude' of the asteroid relative to the Sun! It's a uniform-flowing angular motion from which, by computation, the actual longitude is derived. It starts at 0 at the perihelion of the orbit and runs thru 360 degrees back at perihelion after one lap around the orbit.

Semimajor axis - the mean between the perihelion and aphelion distance of the asteroid from the Sun. Because asteroid orbits can be strongly excentric, it is not wise to equate semimajor axis with a 'mean orbit radius' like for a major planet.

Excentricity - the degree of ellipse in the shape of the orbit, like for a periodic comet. 0 is a circular orbit. 1 is a parabola, for which no asteroid so far qualifies. Perhaps a dead comet in a parabolic orbit could be called an asteroid? Asteroids have closed orbits, unlike most comets. Comets can have parabolic orbits, allowing only a single round of the Sun. Excentricities between 0 and 1 are ellipses, the shape for an asteroid orbit.

Inclination - the tilt of the orbit plane relative to the Earth's orbit, or ecliptic. Once in a while this is stated relative to Jupiter's orbit or to a combined plane of all the planets. For statistical purposes, it doesn't matter much which inclination is used, as long as you know which it is and don't mix them among asteroids. Inclinations from 0 to 90 degrees are direct or prograde orbits. 90 to 180 are for retrograde asteroids, of which there are apparently very few.

Longitude of ascending node - the heliocentric longitude of the south- to-north intercept of the asteroid orbit with the Earth's orbit. The other intercept is the descending node, 180 degrees away.

Argument of perihelion - The angular distance around the orbit from the ascending node to the perihelion point. This is measured along the orbit, not in the ecliptic plane. For low inclination orbits, the error is small. For inclinations greater than about 5 to 8 degrees, the error is too severe to ignore and you must use the proper measurement method.

longitude of perihelion - not a usual parameter, this is the sum of the longitude of ascending node and argument of perihelion This element is used in some statistical studies but is not a basic dynamical property of the asteroid. The sum is a pure arithmetic one along two paths, the ecliptic and then the orbit.

Mean daily motion - Not really necessary, but handy in computations, it's the angular change of mean anomaly per Earth day. It can be calculated from the semimajor axis with Kepler's laws, so it's not really an independent element. Alternatively, it's 360 degrees divided by the orbit period in Earth days.

Perihelion date - The date, in either decimal or in Julian Day number, when the asteroid passes thru the perihelion point in its orbit. This is often missed out for an asteroid because it repeatedly rounds perihelion every few years. A comet may round perihelion once, for a parabola orbit, or only at long intervals. At the moment of perihelion pass the mean anomaly is 0.

Perihelion distance - Not an independent element because it can be calculated from the semimajor axis and excentricity. It is the distance from the Sun to the perihelion point on the orbit.

Aphelion distance - Not an independent element because it can be calculated from the semimajor axis and excentricity. It is the distance from the Sun to the aphelion point on the orbit.

The big three ----------- Dynamical studies of asteroids involve all of the orbital elements. however, for many purposes, examining just three can yield extremely good insight to the behavior of an asteroid. These are the semimajor axis, inclination, and excentricity. Their symbols are 'a', 'i', and 'e' in charts, tables, formulae. Sometimes the Greek letters alpha, iota, and epsilon are used for one or an other of them. Asteroid groups are defined principally by these three elements, with the action of any others offering supplemental qualification.

Interrelations ------------ Several parameters are interrelated, being computed from each other. Additional, somewhat trivial, interrelations are permutations of the ones here.

Semimajor axis = ((perihelion) + (aphelion)) / (2) = (perihelion) / ((1) - (excentricity))

Perihelion = (semimajor axis) * ((1) - (excentricity))

Aphelion = (semimajor axis) * ((1) + (excentricity))

Excentricity = (1) - ((perihelion) / (semimajor axis))

Osculating elements ----------------- The orbital elements of asteroids mutate quickly, typicly over decades. Osculating elements are the instantaneous values of the elements for a given moment. they are valid for only a short time, within the current lap of the orbit. It is wise to use in most planetarium programs the osculating elements valid for an epoch close to the period of simulation. The files of asteroid elements available from planetarium author and the Minor Planet Center wwbsites are the osculating elements and must be replaced every couple years.

Mean elements ----------- Mean elements are those elements smoothed out to simulate an asteroid for many decades. The are cleared of shortterm deviations from the planets circulating around the Sun and tugging at the asteroid from all directions. However, to use them properly, a true dynamical model of the solar system is required. It moves the asteroid under gravity from the planets and will yield at a given moment the osculating elements.

Proper elements ------------- For studies of asteroids over geologic or astronomic time, a refined set of elements is required. They account for the longterm changes in the planet orbits. For the larger asteroids, the proper elements are close to mean elements. However, they can be quite different from values of the osculating elements for a given moment. The intent of the proper elements is to give elements for an asteroid as if it was in a pure two-body dynamical situation just it and the Sun without the other planets. To this end they are artificial and do not directly relate to the current behavior of the asteroid. Because there is no generally agreed method for deriving proper elements, sources will offer different sets. Articles in astronomy journals present new or alternative approaches for obtaining proper elements, with comparison with existing sets. Proper elements can be used only by a high-powered dynamical model, not the level commonly run on home computers. hence, they should NOT be the input to a planetarium or dynamic model; the resulting trajectory and ephemeris will be entirely wrong. Proper elements are about as good a 'book' set of elements there is, without having to issue now ones like for mean or osculating elements. Thus, they are handy for statistical analyses of asteroids and the detection of groups. Usually only the semimajor axis, inclination, and excentricity are computed as proper elements. The proper elements remain constant in the sense that they can begin a simulation at any point of time, backward or forward many millions of years. They will modify into the osculating elements at any given moment during the simulation.

Names of groups ------------- A group is named for a prototype member in that group. This may be the most important, best studied, brightest, largest, or first discovered. There is no rule for picking a prototype, the name settling in by general consensus among astronomers. In rare cases, the group name doesn't come from an actual asteroid of that name, due to historical quirk.

Kirkwood gaps ----------- In 1874 Kirkwood discovered that there were certain distances from the Sun where there were no or very few asteroids. Asteroids seem to avoid these specific distances and bunch up away from them. It's a bit of amazement that he figured this out with the only 200ish asteroids known is his day. In fact, he recognized that certain other gaps were possible but there were not enough asteroids to show them up. Kirkwood explained the avoided distances as those having simple ratios of orbital period with Jupiter, like 2/5 or 1/3. For the 1/3 case, the asteroid period is 1/3 Jupiter year, so every third revolution of the asteroid finds it in conjunction with Jupiter at the same longitude around the Sun. Thus, every third lap, Jupiter's tug of gravity pulls it a bit farther away from the Sun, out of the orbit at the 1/3 distance. This tug is cumulative, where tugs in other directions tend to net out over a revolution of the asteroid and Jupiter. After a few rounds, the asteroid is deposited in a new orbit at such distance that disrupts the simple ratio. The 1/3 place is in time sweeped clear of asteroids, forming one of the Kirkwood gaps. In the table below, some groups appear to sit in or very close to a Kirkwood gap. However, this is misleading from just the semimajor axis. Semimajor axis for asteroids can be a distance from the Sun rarely occupied by the asteroid during its rounds. In fact, such asteroids 'in a Kirkwood gap' have high excentricity so they are carried in and out of the gap, messing up the chances of always being tugged by Jupiter at each conjunction. Any low excentricity asteroids that once were in the gap were, yes, eventually sweeped away.

Resonance ------- The ratio of one planet period to that of an other creates a resonance situation. When the one planet is extremely small compared to the other, like an asteroid against a Jupiter, the little body can suffer badly under the resonance mechanism. Resonance is cited in four[!] ways, which can be hideously confusing if not clearly explained. Two are based on rounds of the bodies's orbit and two are based on the bodies's periods. In all four cases, the resonance may be written as a fraction or a proportion. So, there are in all EIGHT ways you may see, in different articles, the one and same resonance case. To give the resonance by laps we state the laps of the one body versus the laps of the other. If an asteroid makes three rounds of its orbit while Jupiter makes two in its own, we say the asteroid is in a 3/2 or 3:2 resonance with Jupiter. The first number applies to the asteroid; the second, Jupiter. Some authors bank off of Jupiter and say the asteroid is in a 2/3 or 2:3 resonance. All four of these statements are valid, provided you are explicit about which you are using. Similar reasoning applies to the method of periods. The asteroid, by comparing period of revolution, is in a 2/3, 2:3, 3/2, 3:2 resonance with Jupiter. To sort out things, we have

-------------------- Jupiter asteroid viewpoint viewpoint --------- ---------- periods fraction 2/3 3/2 proportn 2:3 3:2

laps fraction 3/2 2/3 proportn 3:2 2:3 ------------------------------

Jupiter the sweeper ----------------- Jupiter, being the most massive of the planets, more than every other solar system body combined (except, of course, the Sun), exerts a strong gravity influence over a wide sector around him. Woe be to the comet who wanders within 2 AU of Jupiter. Asteroids are more resistent to Jupiter's gravity by running in more or less stable orbits around the Sun in the asteroid belt. The pull of Jupiter comes from the whole 360 degree circle around the asteroid and from varying distances. The pulls net out in the short term, leaving minor oscillations of the asteroid's orbital elements. A comet pulled by Jupiter has no chance to 'undo' the distortion by a future pass within a few months or years, so its orbit is permanently changed. There are exceptions for short-period comets, like Oterma and Helin-Roman- Crockett, whose orbits are radicly altered several times by Jupiter within decades. When asteroids are caught in a resonance situation, in or near the Kirkwood gaps, they can suffer major orbit distortion. On each occurrence of conjunction with Jupiter the asteroid suffers an unbalanced extra pull, both in direction and strength. The accumulation of these tugs draws the asteroid out of the resonant orbit into one where the resonance is broken, there no longer being a simple ratio of the two periods or laps. The result is that the Kirkwood gaps are sweeped clean of asteroids, save for those that pass thru in severely excentric orbits and do not repeatedly suffer the resonant pulls. Beyond 4.05 AU Jupiter's gravity is so strong that no stable asteroid circular or low-excentricity orbit is possible. This region of the solar system (plus a similar one beyond Jupiter) is devoid of permanent residents. Thule, the only known asteroid to inhabit this 'forbidden zone', is in an unstable orbit that will decay within a few decades or a century. The asteroid will eventually drop lower into the main belt or be veered outward beyond Jupiter, depending on the exact dynamical simulation you rely on.

Earth & Mars resonance ------------------- Besides Jupiter, both Earth and Mars can create resonance with asteroids in the inner region of the asteroid belt. These are far weaker than Jupiter resonances, but given enough time, they are effective in shifting asteroids away from certain radii from the Sun.

Earth coorbiters -------------- Several asteroids have orbits almost coincident with Earth's but I found no generally agreed name for this group. Here I call it the Cruithne group, after perhaps the most famous of its members. There is likely no danger of collision, except if the orbit is disturbed by an other planet, because a Cruithne asteroid never actually gets too close to Earth. A Cruithne asteroid may start in an orbit just larger than Earth's, say with a period of 366.25 days versus Earth's 365.25. If we start with the two bodies in superior conjunction, the asteroid seems from Earth's eye to plod slowly thru the stars, falling behind in the stars by about 1 degree of heliocentric longitude per year. Eventually, after some 180 Earth years, the asteroid is a few million kilometers ahead of earth, with Earth still gaining on it. Earth's gravity than pulls the asteroid back, slowing it and making it fall into a lower heliocentric orbit, within that of Earth's. The asteroid now has a period of, say, 364.25 days. It pulls ahead of earth, 1 degree per year, to migrate away, round the Sun to superior conjunction, continue round to chase Earth from behind. When it gets within hailing distance of Earth, Earth's gravity pulls the asteroid forward, speeding it and shifting it to a slightly higher orbit. This has the 366.25 day orbit, and the see-saw motion repeats. A solar system plot of such an asteroid resembles a C-shape toilet seat, with the cutout at Earth! The asteroid runs faster than Earth along the inner edge of the toilet seat, loops around the cutout, runs slower than Earth along the outer edge, loops around the opposite side of the cutout, and so on. Some folk, for politeness sake, call this a horseshoe orbit, altho it hardly looks like any shoe a respectable horde will kindly wear. The usual solar system diagram and descriptions are really tricky to interpret correctly. You really should simulate a Cruithne asteroid with a solar system dynamics program. No, you can not use a 'trolley track' model, even if using current elements. The resulting path over even a few years will be totally wrong. In fact, the asteroid makes this round and back motion ONLY as viewed from Earth. it DOES NOT shift back and forth relative to the Sun. What the Sun sees is a normal prograde asteroid that indexes between one, faster-than-Earth, orbit and an other, slower=than=Earth, orbit, but always moving prograde thru the zodiac.

Earth=threats ----------- If an asteroid's orbit can graze or cross that of Earth, that asteroid can be a threat to Earth by collision. This is true even if there is a substantial inclination so the asteroid usually passes north or south of Earth when crossing Earth's orbit. That's because the inclination can be shifted by perturbations from other planets, including Earth itself, such that on a future round, we could suffer the collision. There are four cases of threat: Apohele, Aten, Apollo, and Amor. The last three are named for actual members of their groups. The Apohele asteroid group is named for an asteroid that was not confirmed after discovery. For several years, there were no other members of the Apohele group. By now, 2008, there are five or six members, but none are actually named Apohele.

Apohele ----- The Apohele asteroid orbits entirely within Earth's orbit with its aphelion at Earth's perihelion. That is, when an Apohele is farthest from the Sun, it COULD meet Earth when Earth is at perihelion. The two orbits are tangent, altho it will be some long while before both bodies are simultaneously at the common point. It is extremely tough to find and track an Apohele because it, in Earth's sky, are for the most part of their orbit in daylight or strong twilight. Their greatest elongation from the Sun is 90 degrees, at which point the asteroid is REALLY close to us. They also are seen only in partial phase, no greater than half- full. This, with the highly textures surface, makes an Apohele among the faintest of asteroids.

Aten -- An Aten asteroid has a semimajor axis less than Earth's but an aphelion greater than Earth's. This may be hard to visualize at first: How can an asteroid with a smaller orbit than Earth's cross Earth's orbit? The trick is in the excentricity, shown by a made-up example. Suppose an Aten has an SMA of 0.8 AU, against Earth's 1.0 AU. Also allow that its excentricity is such that the perihelion distance of the asteroid is 0.3 AU. Such an asteroid would have an excentricity of 0.625. Adding twice the SMA to the perihelion gives the aphelion, so we have ((2)*(0.8))+(0.3) = 1.1 AU. This example asteroid can ride 0.1 AU, about 15 million kilometers, beyond Earth, It could strike us on the outward rise to aphelion or on the inward fall from there.

Apollo ---- An Apollo asteroid has SMA greater than Earth's and crosses her orbit. This is the easy situation to visualize. The meet can be at either the inward or outward segment of the Apollo trajectory.

Amor -- an Amor asteroid has SMA greater than Earth's and its perihelion at Earth's aphelion. The orbits are tangent This is the inversion of the Apohele case, but there are a lot of members in the Amor group. The Earth collision can occur when an Amor rounds its perihelion.

SMA and period ------------ An equivalent definition of the Earth-threat asteroids is based on orbital period. Since the period is a direct relation to the semimajor axis, the two definitions are commutable. For period in Earth year's and semimajor axis in AU, the two are related by

(period) = (semimajor axis)^(3/2)

(smeimajor axis) = (period)^(2/3)

Thus the Atens and Apoheles have periods less than Earth's, less than one year, while the Apollos and Amors have periods more than one year.

Parameter limits -------------- The range of the orbit elements defining a group differs among authors because they are not truly hard and sharp frontiers. The best agreed boundaries are those of semimajor axis. Plots of asteroids generally show firm delimitation between groups along SMA, making vertical banding in the graphs (SMA on horizontal axis). The borders for excentricity and inclination are more diffuse. Authors may take as a limit the extreme values, a value enclosing some greater percent of group members, a statistical average, and so on. You may find else where limits for excentricity and inclination that appear to disagree with the ones in the table here.

Hilda --- This group of asteroids hovers near 4 AU from the Sun in strongly excentric orbits. Any that were in a low-excentricity orbit were sweeped away long ago. The Hilda asteroid survives because its orbit adjusted to put the aphelion far from Jupiter, where it would otherwise put it within a fractional AU from the planet for a certain death. The aphelia of the Hilda group are at 60, 180, and 300 degree longitudinal separation around the Sun from Jupiter. The perihelia are at 0, 120, 240 degrees. The 0 degree perihelion is close to Jupiter but the asteroid is moving fast enough to get away to a safe distance quickly. The Hilda group is a good example of an asteroid orbit that is hardly static at all. Old elements of a Hilda will produce utterly wrong trajectories. The entire orbit migrates in step with Jupiter to keep the line of apsides, joining perihelion and aphelion, at the stated separations from Jupiter. A given Hilda has an orbit aligned to ONE of the three longitude displacements and that orbit slews around the Sun pacing Jupiter. Collectively, the Hilda group sorted itself out into the three sets of orbit. The 60-degree and 300-degree aphelia of a Hilda touches the inner edge of the Trojans, but there is no chance of either group losing members to the other. A Trojan moves too quickly to drop down into a Hilda orbit and a Hilda moves too slowly to be caught up with the Trojans. The members of both groups, altho mixed together at about 4.8 AU from the Sun, bypass each other without interaction.

Trojan ----- Stricta ments the Trojan group is uniquely associated with Jupiter. However, the name is applied to similar dynamical groups attached to other planets. Only the Jupiter Trojans have members named for participants of the Trojan War. A loose descriptions of a Trojan is that they sit 60 degrees ahead and behind Jupiter in the planet's orbit. These points are also defined as the Lagrange libration points L4 (ahead) and L5 (behind). This is false in the general case because no planet has a true circular orbit. The definition is the equilateral triangle configurations of asteroid, planet, Sun, regardless of where the asteroid point of the triangle is relative to the orbit of the planet. There are two triangles, one with its asteroids lagging the planet; the other, leading. The L4 and L5 points are at the points of the triangles In the case of a strongly excentric planet orbit, the Trojans may be an unstable group because the gravity regime around them is in constant flux. Among the planets in 2007, Jupiter, Mars, and Neptune have Trojan asteroids. Earth is suspected to have Trojans but none as yet have been found.

Table of asteroid groups ---------------------- The table here is arranged in ascending order of semimajor axis. Mind well that the limits for the parameters may differ from those cited by other authors.

------------------------------------------------------------ group | SMA (AU) | exc | inc | prototype | remarks ---------+----------+-------+-------+--------------+------------ Vulanoid | Q<0.4 | N/A | --- | none yet | within Mercury Apohele | <1, Q=.98| --- | ~0 | 1998-DK36 | Earth grazer Aten | <1, Q>.98| --- | ~0 | 2062 Aten | Earth crosser Cruithne | 1 | 0 | ~0 | 3753 Cruithne| Earth coorbiter Earth Tj | 1 | ~0 | ~0 | none yet | At L4 & L5 Amor |>1, q=1.02| --- | ~0 | 1221 Amor | Earth grazer Apollo |>1, q<1.02| --- | ~0 | 1862 Apollo | Earth crosser Mars Tj | 1.52 | ~0 | ~0 | 5261 Eureka | At L4 & L5 Hungaria | 1.78-2 | <.18 | 16-34 | 434 Hungaria | 9:2 resonance Kirkwood | 1.9 | --- | --- | gap | 9:2 resonance Kirkwood | 2.06 | --- | --- | gap | 4:1 resonance Flora | 2.15-2.35|.03-.23|1.5-8.0| 8 Flora | 4:9 Mars res Kirkwood | 2.25 | --- | --- | gap | 7:2 resonance Phocaea | 2.25-2.5 | >.10 | 18-32 | 25 Phocaea | Nysa | 2.41-2.5 |.12-.21|1.5-4.3| 44 Nysa | AKA Hertha Kirkwood | 2.50 | --- | --- | gap | 3:1 resonance Alinda | 2.5 | .4-.65| --- | 887 Alinda | 1:4 Earth res Maria | 2.5-2.71 | --- | 12-17 | 170 Maria | Eunomia | 2.53-2.72|.08-.22| 11-16 | 15 Eunomia | family? Adeona | 2.66-2.69| <.18 | 0-21 | 145 Adonia | family? Pallas | 2.7-2.82 | --- | 33-38 | 2 Pallas | Kirkwood | 2.71 | --- | --- | gap | 8:3 resonance Dora | 2.76-2.81| <.2 | 0-14 | 668 Dora | family? Kirkwood | 2.82 | --- | --- | gap | 5:2 resonance Koronis | 2.83-2.91| 0-.11 | 0-3.5 | 158 Koronis | family? Kirkwood | 2.95 | --- | --- | gap | 7:3 resonance Eos | 2.99-3.03|.01-.13| 8-12 | 221 Eos | family? Themis | 3.08-3.24|.09-.22| 0-3 | 24 Themis | family? Griqua | 3.1-3.27 | >.35 | >20 | 1362 Griqua | Hygiea | 3.06-3.24|.09-.19|3.5-6.8| 10 Hygiea | family? Kirkwood | 3.27 | --- | --- | gap | 2:1 resonance Cybele | 3.27-3.7 | <.3 | 0-25 | 55 Cybele | 7:4 resonance Kirkwood | 3.7 | --- | --- | gap | 5:3 resonance Hilda | 3.7-4.2 | >.07 | 0-20 | 153 Hilda | 3:2 resonance forbidden| 4.05-5.0 | --- | --- | gap | empty Thule | 4.2 | --- | --- | 279 Thule | unstable 4:3 res Trojan | 5.05-5.4 | ~0 | --- | 588 Achilles | At L4 & L5 ---------+----------+-------+-------+--------------+------------