The Collision between Earth and a planet-size body

[dougettinger]

Previous discussions about the Earth-Moon system bring up more questions.

If a Mars-size planet struck the Earth to form the Moon would not the potential to knock Earth into a different orbit be possible?

Likewise, could this collision by a planet-size body at the right latitude cause the condition of the Earth's axis to be tilted?

The current hypothesis claims that both bodies were already differentiated with iron cores of varying sizes at their centers. I can certainly accept this idea. What is troubling me is how the iron core of the impactor became the iron core of the Moon. Certainly, the iron core of the Earth was not broached because it was comparatively too deep. So how was the Moon's iron core pulled from the magician's hat ?

The current hypothesis also claims that asteroid type bodies were created by this collision that either fell back to Earth or eventually collided with the new Moon, or were ejected farther into the solar system. So where are these asteroids that were ejected and should have various eliptical and inclined orbits ? Another claim is that these collisional bodies settled into stable orbits in the Asteroid Belt. So why do a sizeable population of meteorites and asteroids not have matching mineralogy as Earth or not the same dating as Earth's oldest rocks ?

This collision which I believed occurred during the Later Bombardment Period, about 3.9 billion years ago, when the Earth more than likely was hotter, more molten and softer, but with a developing crust. This should have affected the coefficient of restitution of impact making a partial inelastic collision. My contention is that some the impactor was absorbed by Earth. Is this scenario at least credible ?

Doug Ettinger
Pittsburgh, PA

[jaksichj]

I read your post with some interest ---and although I have not seen the previous posts---there has been work or currently work being done by a former-Princeton mathematician in conjunction with NASA. The individual's name is Dr. Edward Belbruno

Here are a few links:


http://science.nasa.gov/science-news/sc ... apr_theia/

http://arxiv.org/abs/astro-ph/0405372

http://arxiv.org/abs/0808.3268

My point being is ---that there would be a follow-up from the NASA --mission ---in the near future

possibly regarding your questions:

The current hypothesis also claims that asteroid type bodies were created by this collision that either fell back to Earth or eventually collided with the new Moon, or were ejected farther into the solar system. So where are these asteroids that were ejected and should have various eliptical and inclined orbits ? Another claim is that these collisional bodies settled into stable orbits in the Asteroid Belt. So why do a sizeable population of meteorites and asteroids not have matching mineralogy as Earth or not the same dating as Earth's oldest rocks ?

This collision which I believed occurred during the Later Bombardment Period, about 3.9 billion years ago, when the Earth more than likely was hotter, more molten and softer, but with a developing crust. This should have affected the coefficient of restitution of impact making a partial inelastic collision. My contention is that some the impactor was absorbed by Earth. Is this scenario at least credible ?

Doug Ettinger
Pittsburgh, PA

I hope this is of assistance

John

[dougettinger]

Thanks, John. I will pursue these links.

Doug Ettinger

[Chris Peterson]

If a Mars-size planet struck the Earth to form the Moon would not the potential to knock Earth into a different orbit be possible?

In general, probably not. Or more accurately, not from the collision itself. The energy released in a planetary collision is small compared with the energy necessary to change a planetary orbit- even though that may be enough energy to vaporize the planet.

However, when two large bodies pass close together (including a collision) they can transfer orbital angular momentum, with the result that the two may be left with very different orbits. The degree to which Earth's orbit was modified by the collision that created the Moon depends on the original orbit of the colliding body. If it was in a similar orbit to the Earth, there would probably be little change; if it were in an eccentric orbit coming from some other part of the system, there was the potential to shift Earth's orbit significantly.

Likewise, could this collision by a planet-size body at the right latitude cause the condition of the Earth's axis to be tilted?

That's more likely, as the energy required is on the same order as the collisional kinetic energy. Planetary axes could get tilted by collisions, or more often by the transfer of momentum during near misses.

The current hypothesis claims that both bodies were already differentiated with iron cores of varying sizes at their centers. I can certainly accept this idea. What is troubling me is how the iron core of the impactor became the iron core of the Moon. Certainly, the iron core of the Earth was not broached because it was comparatively too deep. So how was the Moon's iron core pulled from the magician's hat?

I don't think the theory excludes the possibility of iron from the Earth's core. In addition, with a much smaller impactor, access to its iron core would have been easier. Given that the energy released in the collision essentially melted both bodies, the presence of iron seems entirely reasonable. It would be harder to explain its absence.

The current hypothesis also claims that asteroid type bodies were created by this collision that either fell back to Earth or eventually collided with the new Moon, or were ejected farther into the solar system. So where are these asteroids that were ejected and should have various eliptical and inclined orbits ?

You'd expect most material to be gone, because the orbits they would end up in are not stable over long periods. The (current) planets have done a fine job of sweeping most other material out of the Solar System.

Another claim is that these collisional bodies settled into stable orbits in the Asteroid Belt.

I've never heard that suggested.

This collision which I believed occurred during the Later Bombardment Period, about 3.9 billion years ago, when the Earth more than likely was hotter, more molten and softer, but with a developing crust. This should have affected the coefficient of restitution of impact making a partial inelastic collision.

I don't think so. Given the energies involved, the consistency of the Earth makes no difference to the nature of the collision.

My contention is that some the impactor was absorbed by Earth. Is this scenario at least credible ?

Probably. They had to have mixed, because both were melted and some of that material re-coalesced into the Earth. But most of the impactor may simply have been destroyed- broken into small bodies in unstable orbits, resulting in ejection from the Solar System or ending up in the Sun.

[dougettinger]

If a Mars-size planet struck the Earth to form the Moon would not the potential to knock Earth into a different orbit be possible?

In general, probably not. Or more accurately, not from the collision itself. The energy released in a planetary collision is small compared with the energy necessary to change a planetary orbit- even though that may be enough energy to vaporize the planet.

However, when two large bodies pass close together (including a collision) they can transfer orbital angular momentum, with the result that the two may be left with very different orbits. The degree to which Earth's orbit was modified by the collision that created the Moon depends on the original orbit of the colliding body. If it was in a similar orbit to the Earth, there would probably be little change; if it were in an eccentric orbit coming from some other part of the system, there was the potential to shift Earth's orbit significantly.

You mentioned that the energy to move a planet into another orbit would also be enough energy to vaporize the planet. I have made some rudimentary calculations using energy and momentum conservation equations. However, I am at a lost to determine how much energy is converted to heat and how much heat is needed to vaporize a certain quantity of matter.

Assume that most of the impactor's material becomes absorbed including materials that are vaporized and fall back to the impacted body. The ejecta that goes beyond escape velocity is of insignificant mass that it can be neglected; I am using the estimated mass of the Asteroid Belt to help with this postulation. The kinetic energy of the impactor must be divided into the additional kinetic energy added to the impacted body, the energy to tilt the axis of the impacted body, and heat energy caused by impact. The angle and speed of impact, possibly added angular momentum from another close body involved in the collision scenario, and potential energy because the vectors involved caused the impacted planet to move toward an inner orbit. The vectors chosen are more additive than substractive.

How does a physicist estimate the amount of heat energy generated and whether that heat energy can vaporize both bodies ?
A datum point for the maximum temperature that was attained after impact could possibly be that temperature which would not boil-off Earth's already differentiated water or steam into space.

Doug Ettinger
Pittsburgh, PA

[jaksichj]

A possible manner in which to look at the situation would be to ask-- what would be the effect of the planets in the vicinity of the "struck-body" --this would turn into a very complex question of a multi-body gravity question---striking the earth would in effect produce a gravitational force on nearby bodies. In effect by producing the moon---a gravitational displacement would tug at nearby planets to a certain degree --by a proportion equivalent to 1/r*r---(or in words --by inverse square law of Newton).

I hope I am not barking up a wrong tree?

If a Mars-size planet struck the Earth to form the Moon would not the potential to knock Earth into a different orbit be possible?

In general, probably not. Or more accurately, not from the collision itself. The energy released in a planetary collision is small compared with the energy necessary to change a planetary orbit- even though that may be enough energy to vaporize the planet.

However, when two large bodies pass close together (including a collision) they can transfer orbital angular momentum, with the result that the two may be left with very different orbits. The degree to which Earth's orbit was modified by the collision that created the Moon depends on the original orbit of the colliding body. If it was in a similar orbit to the Earth, there would probably be little change; if it were in an eccentric orbit coming from some other part of the system, there was the potential to shift Earth's orbit significantly.

Likewise, could this collision by a planet-size body at the right latitude cause the condition of the Earth's axis to be tilted?

That's more likely, as the energy required is on the same order as the collisional kinetic energy. Planetary axes could get tilted by collisions, or more often by the transfer of momentum during near misses.

The current hypothesis claims that both bodies were already differentiated with iron cores of varying sizes at their centers. I can certainly accept this idea. What is troubling me is how the iron core of the impactor became the iron core of the Moon. Certainly, the iron core of the Earth was not broached because it was comparatively too deep. So how was the Moon's iron core pulled from the magician's hat?

I don't think the theory excludes the possibility of iron from the Earth's core. In addition, with a much smaller impactor, access to its iron core would have been easier. Given that the energy released in the collision essentially melted both bodies, the presence of iron seems entirely reasonable. It would be harder to explain its absence.

The current hypothesis also claims that asteroid type bodies were created by this collision that either fell back to Earth or eventually collided with the new Moon, or were ejected farther into the solar system. So where are these asteroids that were ejected and should have various eliptical and inclined orbits ?

You'd expect most material to be gone, because the orbits they would end up in are not stable over long periods. The (current) planets have done a fine job of sweeping most other material out of the Solar System.

Another claim is that these collisional bodies settled into stable orbits in the Asteroid Belt.

I've never heard that suggested.

This collision which I believed occurred during the Later Bombardment Period, about 3.9 billion years ago, when the Earth more than likely was hotter, more molten and softer, but with a developing crust. This should have affected the coefficient of restitution of impact making a partial inelastic collision.

I don't think so. Given the energies involved, the consistency of the Earth makes no difference to the nature of the collision.

My contention is that some the impactor was absorbed by Earth. Is this scenario at least credible ?

Probably. They had to have mixed, because both were melted and some of that material re-coalesced into the Earth. But most of the impactor may simply have been destroyed- broken into small bodies in unstable orbits, resulting in ejection from the Solar System or ending up in the Sun.

[dougettinger]

A n-body problem is appropriate to use in determining the possibility of a collision assuming certain trajectories and massres prior to collision.

Conservation laws are more appropriate to studying what happens after the collision occurs.

Doug Ettinger
Pittsburgh, PA

[Chris Peterson]

Assume that most of the impactor's material becomes absorbed including materials that are vaporized and fall back to the impacted body. The ejecta that goes beyond escape velocity is of insignificant mass that it can be neglected; I am using the estimated mass of the Asteroid Belt to help with this postulation.

I don't understand the basis of this assumption, nor what the Asteroid Belt has to do with it.

[Chris Peterson]

A possible manner in which to look at the situation would be to ask-- what would be the effect of the planets in the vicinity of the "struck-body" --this would turn into a very complex question of a multi-body gravity question---striking the earth would in effect produce a gravitational force on nearby bodies. In effect by producing the moon---a gravitational displacement would tug at nearby planets to a certain degree --by a proportion equivalent to 1/r*r---(or in words --by inverse square law of Newton).

I don't think this is a productive way of looking at the problem. The masses of terrestrial planets have only very tiny perturbing influences on other planets. Adding another planet in Earth's orbit would affect the rest of the planets only in very subtle ways, which would require fine measurements to detect. There would be no evidence left in the orbits of planets today that could tell us anything about a collision between two terrestrial planets early in the history of the Solar System.

N-body simulations are very valuable for assessing the physicality of different models- figuring out, for instance, how two planets might form in the same orbit, or a planet might move from one orbit to another resulting in a collision. The analysis of a collision itself, however, is probably more about energy calculations than dynamics.

However, such collisions can be modeled in extreme detail by finite element methods that include n-body calculations as part of the process of understanding where all the pieces actually go.

[jaksichj]

Chris, Thanks for the clarification

[dougettinger]

Assume that most of the impactor's material becomes absorbed including materials that are vaporized and fall back to the impacted body. The ejecta that goes beyond escape velocity is of insignificant mass that it can be neglected; I am using the estimated mass of the Asteroid Belt to help with this postulation.

I don't understand the basis of this assumption, nor what the Asteroid Belt has to do with it.

I probably used a rash assumption. My present thinking is that the asteroid belt was the resulting debris from a collision whether it happened in the asteroid belt's orbital region or whether the collisional debris came from elsewhere and was trapped by the resonance of Jupiter's gravity. Anyway, the total estimated mass of all the asteroids in the belt region including Jupiter's Trojan asteroids amounts to only a very small fraction of the solar system's largest moons. And perhaps this resulting debris came from the Earth's collision with a sizable impactor or another similar collision.

I hope this is the clarification you are looking for. And, Chris, you did a much better job of explaining the use of n-body investigations. Thanks.

Doug Ettinger
Pittsburgh, PA

[Chris Peterson]

My present thinking is that the asteroid belt was the resulting debris from a collision whether it happened in the asteroid belt's orbital region or whether the collisional debris came from elsewhere and was trapped by the resonance of Jupiter's gravity.

There is no evidence that the asteroid belt is the product of some sort of collision; indeed, there is evidence that it is not. Most theorists believe that the asteroids are made up of the accreted clumps that were present very early in the formation of the Solar System, and because of their location were prevented from reaching the planetesimal stage. That is, the Asteroid belt is made up of material that never had the chance to become a planet, not a planet or planetoid that was destroyed or partly destroyed.

[dougettinger]

How does the resonance from Jupiter's gravity prevent a planet from being formed in the Asteroid belt? Resonance for me is when, for example, 5 orbits of an asteroid matches 8 orbits of Jupiter for each span of time. Is the accumulative tug of gravity from Jupiter every 5 orbits enough to hold it from being attracted to another asteroid. Do I have the correct concept ?
Why does not any resonance created between Jupiter and Saturn's orbital belt prevent Saturn from being accreted ?

Doug Ettinger
Pittsburgh, PA

[neufer]

How does the resonance from Jupiter's gravity prevent a planet from being formed in the Asteroid belt? Resonance for me is when, for example, 5 orbits of an asteroid matches 8 orbits of Jupiter for each span of time. Is the accumulative tug of gravity from Jupiter every 5 orbits enough to hold it from being attracted to another asteroid. Do I have the correct concept ?

Why does not any resonance created between Jupiter and Saturn's orbital belt prevent Saturn from being accreted ?

Saturn & Jupiter orbital periods are in a ratio of ~77:31 or (154/155)ths shy of a narrow 5:2 'Kirkwood gap' resonance.

Uranus & Saturn orbital periods are in a ratio of ~20:7 or (20/21)ths shy of a narrow 3:1 'Kirkwood gap' resonance.

Neptune & Uranus orbital periods are in a ratio of ~43:22 or (43/44)ths shy of a 2:1 'Kirkwood gap' resonance.

(Pluto & Neptune are in a stable 3:2 resonance.)

They all accreted long before they landed in these stable orbital relationships.

[img3="This graph was created in June 2007 using all asteroids with "well-determined" orbits (specifically, 156929 numbered asteroids)."]http://ssd.jpl.nasa.gov/images/ast_histo.png[/img3]

<<A Kirkwood gap is a gap or dip in the distribution of main belt asteroids with semi-major axis (or equivalently their orbital period), as seen in the histogram below. They correspond to the location of orbital resonances with Jupiter.

For example, there are very few asteroids with semimajor axis near 2.50 AU, period 3.95 years, which would make three orbits for each orbit of Jupiter (hence, called the 3:1 orbital resonance). Other orbital resonances correspond to orbital periods whose lengths are simple fractions of Jupiter's. The weaker resonances lead only to a depletion of asteroids, while spikes in the histogram are often due to the presence of a prominent asteroid family.

The gaps were first noticed in 1857 by Daniel Kirkwood, who also correctly explained their origin in the orbital resonances with Jupiter while a professor at Jefferson College in Canonsburg, Pennsylvania.

More recently, a relatively small number of asteroids have been found to possess high eccentricity orbits which do lie within the Kirkwood gaps. Examples include the Alinda family and the Griqua family. These orbits slowly increase their eccentricity on a timescale of tens of millions of years, and will eventually break out of the resonance due to close encounters with a major planet.

The most prominent Kirkwood gaps (see diagram) are located at mean orbital radii of:

• * 2.06 AU (4:1 resonance)
  * 2.5 AU (3:1 resonance), home to the Alinda family of asteroids
  * 2.82 AU (5:2 resonance)
  * 2.95 AU (7:3 resonance)
  * 3.27 AU (2:1 resonance), home to the Griqua family of asteroids

Weaker and/or narrower gaps are also found at:

• * 1.9 AU (9:2 resonance)
  * 2.25 AU (7:2 resonance)
  * 2.33 AU (10:3 resonance)
  * 2.71 AU (8:3 resonance)
  * 3.03 AU (9:4 resonance)
  * 3.075 AU (11:5 resonance)
  * 3.47 AU (11:6 resonance)
  * 3.7 AU (5:3 resonance)

Note: Venus & Mercury orbital periods are in a ratio of ~23:9
[emulating Jupiter & the largest asteroid Ceres (2.7663 AU)]
or (46/45)ths past a narrow 5:2 'Kirkwood gap' resonance.
& (23/24)ths shy of a narrow weak 8:3 'Kirkwood gap' resonance.

[dougettinger]

Art, I understand the data but still do not comprehend the Kirkwood Gap Law or reason for such occurrences. Why do the resonances with simple fractions produce gaps or zones of less asteroid populations ?

Why do the resonances between the outer planets, which imply almost simple fractions, not show gaps, but indeed produce very large planets ? I only see inconsistency.

Daniel Kirkwood lived very close to where I live now. I am very surprised that enough asteroids were known in his time to produce any type of histogram that would reveal gaps.

Doug Ettinger
Pittsburgh, Pa

[neufer]

I understand the data but still do not comprehend the Kirkwood Gap Law or reason for such occurrences. Why do the resonances with simple fractions produce gaps or zones of less asteroid populations ?

Most resonances with simple fractions, e.g., 4:1, 3:1, 5:2, 7:3, 2:1 produce gaps because they induce cumulative perturbation effects.

Other resonances like the 3:2 Pluto/Neptune resonance are not cumulative and allow situations such as Pluto always avoiding Neptune where their orbits intersect.

Why do the resonances between the outer planets, which imply almost simple fractions, not show gaps, but indeed produce very large planets ? I only see inconsistency.

The Kirkwood Gap Law shows the effect of a large planet upon a small planet or asteroid.

It does not apply exactly to mutually interacting planets which appear to be attracted to Kirkwood Gaps but still probably must avoid an exact Kirkwood Gap resonance.

Note that the three inner Galilean moons revolve in a 4:2:1 resonance

Daniel Kirkwood lived very close to where I live now. I am very surprised that enough asteroids were known in his time to produce any type of histogram that would reveal gaps.

You live close to where Daniel Kirkwood lived but not exactly where he lived; otherwise your house would have been destroyed.

[dougettinger]

By "close" I mean about 15 miles. How do you know about Kirkwood's house ?

The period of a planet or asteroid is based on the orbital speed which is v = sqrt [(GM)/r] which is based on Kepler's third law and Newtonian mechanics. So how does the Law of Planetary Resonance as indicated by the outer planets and Jupiter's moons fit into this scheme of Newtonian mechanics ? What initially dictates a planet's parameters when it is forming ? Is it initial velocity ? Is it initial orbital radius from its parent body? And do the neighboring orbiting bodies have a say inisthe matter ?

I am tryiing to integrate this Kirkwood Gap concept into my overall thinking.

Doug Ettinger
Pittsburgh, PA

[jaksichj]

Although my expertise(?) is in chemistry--I have enrolled in a handful of astronomy courses--

I will try to not confuse the situation any further--if you are familiar with the condition of planetary resonances--I believe that "planetary illustration of Art's" points to the condition where a moon, planet, ateroid , etc will be locked tidally with its orbiting planet.

Unfortunaely ---I must cite the following reference --but it should work: * see references below---


1) The Kirkwood gap occurs in the asteroid belt
2) The asteroids in that gap are ejected because their orbital resonances are not integer multiples within the revolution of Jupiter


For instance--- a resonance scale is expressed as the number of revolutions of an asteroid at "some" distance from the Sun during one revolution of Jupiter.

Thus an asteroid can be kicked from a stable --orbit if it becomes non-resonant---


Herre are some of the references that I used:

Faure & Mensing---Introduction to Planetary Science--The Geological Perspective 2007

Chapman--cited in previous refer---
of which it is from:

Beatty, Petersen, & Chaikin The New Solar System 4th ed 1999

I hope this makes some sense

[neufer]

By "close" I mean about 15 miles. How do you know about Kirkwood's house ?

All I know is what you told me. (Is it made out of Kirkwood )

The period of a planet or asteroid is based on the orbital speed which is v = sqrt [(GM)/r] which is based on Kepler's third law and Newtonian mechanics. So how does the Law of Planetary Resonance as indicated by the outer planets and Jupiter's moons fit into this scheme of Newtonian mechanics ? What initially dictates a planet's parameters when it is forming ? Is it initial velocity ? Is it initial orbital radius from its parent body? And do the neighboring orbiting bodies have a say inisthe matter ?

I am tryiing to integrate this Kirkwood Gap concept into my overall thinking.

There is a large Kirkwood Gap in my own knowledge of these matters; but I'm learning slowly (I think).

<<In celestial mechanics, an orbital resonance occurs when two orbiting bodies exert a regular, periodic gravitational influence on each other, usually due to their orbital periods being related by a ratio of two small integers. Orbital resonances greatly enhance the mutual gravitational influence of the bodies (i.e., their ability to alter or constrain each others' orbits). In most cases, this results in an unstable interaction, in which the bodies exchange momentum and shift orbits until the resonance no longer exists. Under some circumstances, a resonant system can be stable and self correcting, so that the bodies remain in resonance. Examples are the 1:2:4 resonance of Jupiter's moons Ganymede, Europa and Io, and the 2:3 resonance between Pluto and Neptune. Unstable resonances with Saturn's inner moons give rise to gaps in the rings of Saturn. The special case of 1:1 resonance (between bodies with similar orbital radii) causes large Solar System bodies to clear the neighborhood around their orbits by ejecting nearly everything else around them; this effect is used in the current definition of a planet.

Except as noted in the Laplace resonance figure (below), a resonance ratio in this article should be interpreted as the ratio of number of orbits completed in the same time interval, rather than as the ratio of orbital periods (which would be the inverse ratio). The 2:3 ratio above means Pluto completes 2 orbits in the time it takes Neptune to complete 3.

History

Since the discovery of Newton's law of universal gravitation in the 17th century, the stability of the solar system has preoccupied many mathematicians, starting with Laplace. The stable orbits that arise in a two-body approximation ignore the influence of other bodies. The effect of these added interactions on the stability of the Solar System is very small, but at first it was not known whether they might add up over longer periods to significantly change the orbital parameters and lead to a completely different configuration, or whether some other stabilising effects might maintain the configuration of the orbits of the planets.

It was Laplace who found the first answers explaining the remarkable dance of the Galilean moons (see below). It is fair to say that this general field of study has remained very active since then, with plenty more yet to be understood (e.g. how interactions of moonlets with particles of the rings of giant planets result in maintaining the rings).

In general, an orbital resonance may

• * involve one or any combination of the orbit parameters (e.g. eccentricity versus semimajor axis, or eccentricity versus orbit inclination).
  
  * act on any time scale from short term, commensurable with the orbit periods, to secular, measured in 104 to 106 years.
  
  * lead to either long term stabilization of the orbits or be the cause of their destabilization.

A mean motion orbital resonance occurs when two bodies have periods of revolution that are a simple integer ratio of each other. Depending on the details, this can either stabilize or destabilize the orbit. Stabilization occurs when the two bodies move in such a synchronised fashion that they never closely approach. For instance:

* The orbits of Pluto and the plutinos are stable, despite crossing that of much larger Neptune, because they are in a 2:3 resonance with it. The resonance ensures that, when they approach perihelion and Neptune's orbit, Neptune is consistently distant (averaging a quarter of its orbit away). Other (much more numerous) Neptune-crossing bodies that were not in resonance were ejected from that region by strong perturbations due to Neptune. There are also smaller but significant groups of resonant trans-Neptunian objects occupying the 1:1 (Neptune trojans), 3:5, 4:7, 1:2 (twotinos) and 2:5 resonances with respect to Neptune.

* In the asteroid belt beyond 3.5 AU from the sun, the 3:2, 4:3 and 1:1 resonances with Jupiter are populated by clumps of asteroids (the Hilda family, 279 Thule, and the Trojan asteroids, respectively).

Orbital resonances can also destabilize one of the orbits. For small bodies, destabilization is actually far more likely. For instance:

* In the asteroid belt within 3.5 AU from the sun, the major mean-motion resonances with Jupiter are locations of gaps in the asteroid distribution, the Kirkwood gaps (most notably at the 3:1, 5:2, 7:3 and 2:1 resonances). Asteroids have been ejected from these almost empty lanes by repeated perturbations. However, there are still populations of asteroids temporarily present in or near these resonances. For example, asteroids of the Alinda family are in or close to the 3:1 resonance, with their orbital eccentricity steadily increased by interactions with Jupiter until they eventually have a close encounter with an inner planet that ejects them from the resonance.

* In the rings of Saturn, the Cassini Division is a gap between the inner B Ring and the outer A Ring that has been cleared by a 2:1 resonance with the moon Mimas. (More specifically, the site of the resonance is the Huygens Gap, which bounds the outer edge of the B Ring.)

* In the rings of Saturn, the A Ring's outer edge is maintained by a destabilizing 7:6 resonance with the moon Janus.

A Laplace resonance occurs when three or more orbiting bodies have a simple integer ratio between their orbital periods. For example, Jupiter's moons Ganymede, Europa and Io are in a 1:2:4 orbital resonance. The extrasolar planets Gliese 876e, Gliese 876b and Gliese 876c are also in a 1:2:4 orbital resonance (with periods of 124.3, 61.1 and 30.0 days).>>

[neufer]

<<* In the rings of Saturn, the Cassini Division is a gap between the inner B Ring and the outer A Ring that has been cleared by a 2:1 resonance with the moon Mimas. (More specifically, the site of the resonance is the Huygens Gap, which bounds the outer edge of the B Ring.)

Even MORE specifically, the Huygens Gap & Mimas are in a ratio of ~109/55
or (109/110)ths shy of a narrow 2:1 'Kirkwood gap' resonance.

The inside of the B Ring and Mimas are in a ratio of ~20/7
or (20/21)ths shy a narrow 3:1 'Kirkwood gap' resonance.

The C Ring's Maxwell Gap and Mimas are in a ratio of ~34/11
or (34/33)ths beyond a narrow 3:1 'Kirkwood gap' resonance.

The Maxwell Gap and the Maxwell Ringlet on its right side are above and right of center. The Maxwell Gap lies within the outer part of the C Ring. It also contains a dense non-circular ringlet, the Maxwell Ringlet. In many respects this ringlet is similar to the ε ring of Uranus. There are wave-like structures in the middle of both rings. While the wave in the ε ring is thought to be caused by Uranian moon Cordelia, no moon has been discovered in the Maxwell gap as of July 2008.

---

<<James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish theoretical physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and even optics into a consistent theory. Maxwell's equations demonstrated that electricity, magnetism and even light are all manifestations of the same phenomenon, namely the electromagnetic field. Subsequently, all other classic laws or equations of these disciplines were simplified cases of Maxwell's equations. Maxwell's achievements concerning electromagnetism have been called the "second great unification in physics", after the first one realised by Isaac Newton.

[At age 25 Maxwell's] mind was focused on a problem which had eluded scientists for two hundred years: the nature of Saturn's rings. It was unknown how they could remain stable without breaking up, drifting away or crashing into Saturn. The problem took on a particular resonance at this time as St John's College, Cambridge had chosen it as the topic for the 1857 Adams Prize. Maxwell devoted two years to studying the problem, proving that a regular solid ring could not be stable, and a fluid ring would be forced by wave action to break up into blobs. Neither met with observations, and Maxwell was able to conclude that the rings must comprise numerous small particles he called "brick-bats", each independently orbiting Saturn. Maxwell was awarded the £130 Adams Prize in 1859 for his essay On the stability of saturn's rings; he was the only entrant to have made enough headway to submit an entry. His work was so detailed and convincing that when George Biddell Airy read it he commented "It is one of the most remarkable applications of mathematics to physics that I have ever seen." It was considered the final word on the issue until direct observations by the Voyager flybys of the 1980s had confirmed Maxwell's prediction. Maxwell would also go on to mathematically disprove the nebular hypothesis (which stated that the solar system formed through the progressive condensation of a purely gaseous nebula), forcing the theory to account for additional portions of small solid particles.>>

[dougettinger]

from Wikipedia -

In general, an orbital resonance may

• * involve one or any combination of the orbit parameters (e.g. eccentricity versus semimajor axis, or eccentricity versus orbit inclination).
  
  * act on any time scale from short term, commensurable with the orbit periods, to secular, measured in 104 to 106 years.
  
  * lead to either long term stabilization of the orbits or be the cause of their destabilization.

Art, your references from Wikipedia are very enlightening. However, I am still in a quandary about Kirkwood Gaps and Orbital Resonances. The previous quote that states " _ _ either long term stabilization of the orbits or be the cause of their destabilization." leads me to believe a coin toss is needed. It is more satisfying to create some laws. Let me take a stab at some Kirkwood Gap/Orbital Resonance Laws without the aid of Maxwellian type mathematics.

#1. A forming planetoid in the early solar system, if it is inner to a much larger forming planetoid and its circumferential orbital location is close to a resonance with that larger outer planet, then it is very likely to be perturbed, ejected, and create a gap. The gap is the result of the small planetoid sweeping material and leaving behind other condensed materials in a belt that would have been eventually attracted to the ejected planetoid's gravity field.

This law helps to explain the Kirkwood Gaps in the Asteroid Belt.

#2. A very large forming planetoid in the early solar system will cause resonance with outer orbiting clumps of condensed materials in such a manner that the steady perturbations attract the clumps in their common orbit close to the point of resonance and aids in forming another planet.

This law explains the orbital resonance of the all the outer giant planets and of the primary moons of the outer planets.

#3. In similar fashion as Law #2, the overall array of planets that forms around a protostar are aided by orbital resonance. The first clumping to form a sizable planetoid begins the process of perturbing the clumping in other orbits and brings clumps in a common orbit together through repeated tugs of gravity to form another planetoid.

This law explains the orbital resonance of the solar system's major planets and exo-solar planets.

#4. The rings of the outer planets represent an on-going process of orbital resonance that human beings can actually observe. The dust and larger particles being gathered by these large masses or sources of gravity show gaps and boundaries. Clumping is created by orbital resonance that sweeps away material and creates gaps per Law #3. The forming planetoids are either observed in these gaps or they have been ejected by resonances of outer moons per Law #1.

This "law of the rings" actually provides proof for Laws #1 and #3.

I wish to further summarize and reduce some verbage.

#1 Smaller planetoids with inner orbits to a larger planetoid are normally ejected from their home orbit if they form in a location close to a resonance point of the outer body..

#2. Planetoids with outer orbits to a larger planetoid are more normally encouraged to increase their mass due to steady, periodical perturbations created by orbital resonance.

#3. The forming of one or more planetoids inside a protostar disk encourages the forming of more planetoids via orbital resonance.

#4. The observed status and evolution of the outer planetary rings is proof for these laws.

Art, take another stab. Try not to make it too bloody. Chris, you are invited, too. I feel like we are constructing a solar system.

Doug Ettinger
Pittsburgh, PA
12-19-2010

[neufer]

What I find most interesting (and useful to remember)
are the number of neighboring bodies near a 5:2 resonance:

• -----------------------------------------------------------------------------------
  Saturn & Jupiter orbital periods are in a ratio of ~77:31
  or just (154/155)ths shy of a narrow 5:2 'Kirkwood gap' resonance.
  
  Jupiter & Ceres orbital periods are in a ratio of ~47:19
  or just (94/95)ths shy of a narrow 5:2 'Kirkwood gap' resonance.
  
  Ceres & Mars orbital periods are in a ratio of ~22:9
  or just (44/45)ths shy a narrow 5:2 'Kirkwood gap' resonance.
  
  Venus & Mercury orbital periods are in a ratio of ~23:9
  or just (46/45)ths past a narrow 5:2 'Kirkwood gap' resonance.
  
  Mimas & B Ring orbital periods are in a ratio of ~5:2
  
  Enceladus & A Ring orbital periods are in a ratio of ~5:2
  -----------------------------------------------------------------------------------

Is the predominance of 5:2 resonances (corresponding to ~ 54.3% orbital ratios)
between neighboring bodies responsible, in part, for the Titius-Bode approximation?

[dougettinger]

Art, you posted an interesting data point, 54.3 % of orbital ratios are 5:2. Did you determine that percentage yourself ?

I believe that orbital ratios or resonances are independent of any process that determines orbital radii as the solar system formed. I also believe that these orbital ratios occur randomly, but of course via these small integers. The 5:2 ratio is probably a favorite due to the typical orbital spacings that were created. The orbital velocities and radii are determined more by the interactions over time of the growing central gravity source and the density gradients of materials spiralling inward toward the center.

The Titius-Bode theory deals with approximating the orbital radii. My hypothesis is that as a protostar grows in size, the central gravity force attracts the disk materials into circular bands or steps of increasing gradients of density. Vortices somehow peel off at the edges of these bands to initiate clumping or density gradients occurring circumferentially. Then the Kirkwood Gap/Orbital Resonance Laws take over to form a planetoid in this band of clumping. I am currently working on a better approximation for orbital radii than the Titius-Bode Rule that also works for the primary moons of Jupiter and Saturn.

Art, you helped significantly by emphasizing the importance of orbital resonance. Now the clumping that I envision in a band of orbiting dust can now come together to form a planetoid and become a resident planet or be ejected.

Doug Ettinger
Pittsburgh, PA
12/28/2010

[neufer]

Art, you posted an interesting data point, 54.3 % of orbital ratios are 5:2. Did you determine that percentage yourself ?

It is simply applying Kepler's 3rd law:

Pluto has an orbital period of 248 years so it lies out at (248)^2/3 ~ 39.5 AU (i.e., = it's semi major axis)].
If an object has an orbital period of 2/5 years it lies at (2/5)^2/3 ~ 0.543 AU.

(Learning to do a few math tricks is much more fun than speculating in the abstract, IMO.)

[Chris Peterson]

The Titius-Bode theory deals with approximating the orbital radii.

There is no such theory. Titius-Bode merely provides an observational fit to the positions of some (but not all) the major bodies in the Solar System. It has no theoretical basis at all, and there's no real reason to think it would apply to other systems. It is very likely a coincidence, or at best an inadvertent consequence of other factors, such as resonances.

My hypothesis is that as a protostar grows in size, the central gravity force attracts the disk materials into circular bands or steps of increasing gradients of density. Vortices somehow peel off at the edges of these bands to initiate clumping or density gradients occurring circumferentially.

Unless you can support this idea with some solid math, I'd say "hypothesis" is an inappropriate word.

Art, you helped significantly by emphasizing the importance of orbital resonance.

Don't confuse different resonance effects, however. Large bodies can find themselves in resonant orbits with other bodies, because such orbits can be stable. That means that the bodies gradually move into these resonances, and then stay there. But there are thousands of "virtual" resonances in the Solar System: orbital radii where there would be constructive or destructive resonances. These are most apparent in the asteroid belt, giving it its ring-like structure. But in considering the way the Solar System came together, you need to consider these virtual resonance zones (which are by-and-large empty). Indeed, that's a key component of the numerical simulations that are relied on today for explaining planetary system formation and dynamics.