Asteroids

The absolute magnitude of an object is a measure of how bright it is relative to the star Vega. Vega is defined to have an absolute magnitude of 0 in all colors; anything brighter than Vega is a negative absolute magnitude, and anything fainter is a positive magnitude. Magnitudes are on a logarithmic scale such that for every increase in 2.5 magnitudes, the object is 10 times fainter. So, a star that is 10^th magnitude means that it is 10,000 times fainter than Vega.

The absolute magnitude of asteroids is very faint, which is a result of them being very small and not reflecting a lot of sunlight. The absolute magnitude histogram is more a reflection of how good our detection technology is than what the actual distribution of the brightness of asteroids is.

The actual distribution of magnitudes is probably more of an exponential, considering that there are many more smaller asteroids than larger ones. There is also a slightly larger tail towards the bright end (lower numbers) of magnitudes. This is an artifact again of our detection methods: It is much easier to spot a bright object than a faint one, and so we know of many more bright asteroids than faint ones.

Asteroids are found throughout the solar system. However, in the interest of readability, I have isolated in the graph just the solar system out to Jupiter, since the vast majority of known asteroids lie within that region. This allows a certain structure to the location distribution of the asteroids to be discerned.

At about 5.2 AU, there is a marked increase in the number of asteroids. These asteroids are Jupiter's Trojans, the asteroids that orbit the Sun with Jupiter. They are not moons of Jupiter, they just share its orbit around the Sun. From about 3.2 AU in to about 1.8 AU is the realm of the main asteroid belt. Almost all known asteroids orbit within this region between Mars and Jupiter.

The next graph shows an expanded view of the main asteroid belt, from 1.5 to 3.5 AU. The data have been re-binned with intervals of 0.002 AU. It is clear from this chart that there are many gaps in the distribution of the asteroids, and in some cases almost no asteroids are found in the locations. These are called Kirkwood Gaps, and they arise because of resonances with Jupiter. (Martian resonances are inconsequential compared with those from Jupiter.)

At a distance of 2.49 AU from the Sun, Jupiter's gravitational pull on an object is only about 0.3% of the Sun's. Over time, though, this pull adds up and serves to nudge an object away from that region, but this will really only work if the pull is repeated at a regular interval. 2.49 AU happens to lie at a distance from the Sun such that an object's year there will be exactly 1/3 of Jupiter's. This means that once every three years, an asteroid in that location will feel a tug from Jupiter, and over time, this will remove it from that location. This is known as a 3:1 resonance.

There are many other resonances that happen, and based upon how often an asteroid feels Jupiter's tug, the relative strength of the resonance is established. The resonance is especially effective at nudging an asteroid if the tug occurs in the same location in the asteroid's orbit and only in that location, so the 2, 3, 4, and 5 to 1 resonances are the strongest. The major resonances are:

• 5:1 ~ 1.78 AU
• 9:2 ~ 1.91 AU
• 4:1 ~ 2.06 AU
• 7:2 ~ 2.26 AU
• 3:1 ~ 2.50 AU
• 8:3 ~ 2.70 AU
• 5:2 ~ 2.82 AU
• 7:3 ~ 2.96 AU
• 9:4 ~ 3.03 AU
• 2:1 ~ 3.28 AU

The asteroid belt is bounded by the 5:1 and 2:1 resonance. Asteroids that are pushed outside of these resonances will generally become planet-crossing asteroids.

The eccentricity of an orbit is how much it varies from a perfect circle. A stable orbit can have an eccentricity anywhere from a perfect circle with an eccentricity of 0, up to a highly elliptical orbit with an eccentricity up to (but not including) 1. If an orbit had an eccentricity of 1, it would be parabolic and escape from the system. If it were larger than 1, it would be hyperbolic and also escape from the system.

Earth's eccentricity is 0.017, while Jupiter's is 0.094. In our solar system, the planet with the largest eccentricity is Mercury with 0.205 (Pluto's is larger at 0.244, but it is not considered a planet any longer). The planet with the lowest eccentricity is Venus with 0.007. Unless there is some gravitational tugging (such as with the Galilean Satellites) that keeps an orbit eccentric, orbits will usually circularize with time.

As shown in this diagram, the average eccentricity is about 0.11. In general, asteroids' orbits are far from perfect circles. This is probably due to perturbations from the planets - especially Jupiter - and collisions. A collision between asteroids can dramatically alter their orbits, though over time they should average out to circularize the orbit. The eccentricity can also be an effect of sampling. As discussed in the section on semi-major axis vs. eccentricity, objects that are farther away from the Sun are more likely to be found by Earth-based searches if they have a large eccentricity that takes them much closer to Earth than they usually are.

When the solar system formed, most of the solar nebula evolved into a disk, and from this the planets, moons, asteroids, comets, and Sun formed. Because of this, the planets all generally orbit in the same plane. The largest difference is Mercury which orbits 7° from Earth. (Pluto's orbit is ~17° off from Earth's, but it is no longer considered a planet.) The rest of the planets orbit within 2.5° of Earth's orbital plane.

The orbital inclination histogram here shows that asteroids are another exception to this rule. The average orbital inclination is about 8.2°.

A probable mechanism for creating the - sometimes very large - orbital inclinations are collisions. When two asteroids hit each other, their orbits can be changed dramatically. And, since the inclination of orbits is much less subject to gravitational perturbations than other orbital elements, they really don't have any driving force to put them back into an orbit with the rest of the solar system.

Another factor is that when a rather large asteroid is fragmented into smaller pieces, the pieces tend to stay close to the original orbit, which creates a "family" of asteroids (Hirayama Family). This increases the raw number of asteroids with a given orbital inclination.

This semi-logarithmic scatter plot of eccentricity as a function of the semi-major axis shows mostly what is discussed in the other sections. Visible are the general asteroid trends, such as the large number of main belt asteroids, trojans, a sprinkling of Centaurs, and the TNOs. Most asteroids have an eccentricity relatively low, with higher eccentricities tapered off.

Two important new observations, however, can be made. The first is from the asteroids that orbit within Mars' orbit, about 1.5 AU from the Sun. There are many fewer asteroids in this region than in the main belt, but on average, their eccentricities are much higher. This is a strong indication that they did not form in these locations, but that they were pushed or pulled into this region. When that happens, the asteroid is likely to return to its original orbit distance for at least part of its orbit, until it evolves more. This will cause it to have a high eccentricity, for during part of its orbit it will be much closer to the Sun than during another part of it.

For the Kuiper Belt Objects beyond the orbit of Neptune, there is also a large increase in eccentricity. This is probably an artifact of how we detect them. Since we are limited by how much light we can detect from an asteroid, the closer it is to us, the more likely we are to detect it. So, if an object were to have a semi-major axis of 100 AU and lie on a circular orbit, then it might be too faint to detect. But if it were on an elliptical orbit and this brought it within 35 AU of the Sun, then we would receive about 8 times more light from it, and so we would be more likely to detect it. So, objects that lie far from the Sun are much more likely to be detected today if they have highly eccentric orbits.

The first graph is a semi-logarithmic scatter plot with orbital inclination plotted against semi-major axis. This is a much truer picture of what the asteroid distribution would actually "look like" in space than the eccentricity vs. semi-major axis.

Probably the most important feature of this plot is that the Hirayama Families become apparent. This is the name given to "families" of asteroids, which are asteroids that tend to share the same orbital characteristics, and they probably had a common parent body that was destroyed during a collision. Japaneses astronomer Kiyostugu Hirayama, in 1918, was the first astronomer to suggest that they existed.

The second graph is an enlarged section of the orbital inclination vs. semi-major axis. It has been made into a number density image by creating a 2-D histogram. The 2 AU range has been divided into 0.005 AU increments, and the inclination of 0°-50° has been divided into 0.25° increments. The number of asteroids per bin is based upon the color displayed in the color bar in the upper left corner. The reason the data are displayed this way is that, even with dots only one pixel large, over 400,000 asteroids would make the graph look like a lot of red dots.

With this chart, it now becomes clear that there are some obvious clusters of asteroids, as indicated by the larger number density in certain areas. These groupings are the Hirayama Families.

Each Hirayama family is named for the main asteroid in the group. Some of the major ones are:

• Hungarians (~1.9 AU at ~22°)
• Floras (~2.2 AU at ~6°)
• Phocaea (~2.4 AU at ~22°)
• Koronis (~2.9 AU at ~1°)
• Eos (~3 AU at ~11°)
• Themis (~3.1 AU at ~1°)
• Cybeles (~3.4 AU at ~ 4°)
• Hildas (~4.0 AU at ~8°)

It is also fairly apparent from this image that there is a resonance at 2 AU, 2.5 AU, 2.8 AU, 2.9 AU, and 3.3 AU, since these are distances at which the number density of asteroids is almost 0.