Gravitational Capture Potential for Planets:
I have been interested in the origin of planet-satellite systems for several decades. After coming to Denison University in 1972, I got associated with Ronald Winters and Michael Mickelson (Physics and Astronomy Dept.) and we did some joint projects in the planetary sciences. From 1985 to 1988 a very talented physics student, David Mehringer (DU,’88), got associated with the project and implemented an energy-dissipation subroutine for a three-body numerical simulation program (fourth-order Runge-Kutta integration procedure) which made gravitational capture simulations possible. Our first set of capture simulations (in the Spring of 1987) was for capture of a lunar-mass planetoid from an earth-like heliocentric orbit into a geocentric orbit. The results were exciting in that no one had ever done this before!
In the era of 1990-1993, two physics/computer science students (Wentao Chen, DU,’92 and Albert Liau, DU,’93) made minor corrections to the numerical code and developed programs for plotting tidal amplitudes as well as tidal ranges directly from the geocentric orbital data. In general, these three very talented undergraduate students (Mehringer, Chen, and Liau) were super important in the development of this planetoid capture project. It is interesting to note here that of all the articles in the planetary science literature on gravitational capture (or articles mentioning the process before 1987) no investigator had done a successful numerical simulation of capture. I think the main reason that it had not been done is because of some fundamental paradoxes that are associated with gravitational capture processes in general.
(NOTE: A good summary of lunar origin models can be found in a book by Peter Cadogan published in 1981 titled “The Moon: Our Sister Planet”. This book was published before the Giant Impact Model became dominant.)
The first paradox is that the planetoid (the smaller body) must absorb most of the energy for its own capture. Successful capture is a matter of h’s and Q’s. The h (displacement Love number) is a measure of the deformability of the planet or planetoid and the Q (specific dissipation factor) is a measure of the quantity of the energy stored by tidal distortion during a close gravitational encounter that is subsequently dissipated during a single encounter (a tidal oscillation). The fraction of stored energy that is dissipated during one tidal oscillation is 1/Q. A reasonably warm lunar-mass planetoid could have an h in the range of 0.1 to 0.3. For capture the Q of this planetoid must be very low: 1, 2, or 3. In contrast to the planetoid, the planet is a passive bystander furnishing a strong gravitational field for tidal deformation of the planetoid. The h of the Earth at present is about 0.54 (a very deformable body). The Q of the Earth is very high (about 200). The h and Q for the primitive Earth (about 4.0 billion years ago) would not have been greatly different (perhaps h=0.7 and Q=200). The bottom line is that the earth, at no time in its history, would be an effective energy sink for tidal capture.
A second paradox is that larger planetoids are more capturable than smaller ones. The reason for this is that as the planetoid radius decreases, the h for storing the energy for capture increases. Thus, if all candidate planetoids of lunar density have a low Q value, only those larger than about 0.2 lunar mass can temporarily store, by tidal distortion, the energy for their own capture.
A third paradox is that “cool” planetoids are more capturable than “hot” planetoids. Martin Ross and Gerald Schubert did a set of calculations at UCLA in the middle 1980’s on “Q vs. Viscosity” of a planetoid. Their conclusions were that a low Q is associated with an intermediate value of planetary viscosity. In other words, if a body is too cold, it will be too rigid (high viscosity) to absorb much energy. If a body is too hot (low viscosity), the body can be easily deformed but it will be too “mushy” to dissipate more than a small fraction of the tidal energy of any one tidal oscillation. The low Q values, then, are associated with an intermediate viscosity value. A reasonable assumption is that for a lunar-like planetoid the intermediate viscosity condition would be when the lowest melting component of the lunar magma ocean (probably at a depth of about 200 km) would be just below the melting point. Thus, with an h in the range of 0.20 to 0.25 and a Q near 1, sufficient energy could be dissipated for capture during one tidal oscillation. These are the plantoid body conditions necessary for tidal capture.
I have now calculated and plotted literally thousands of successful capture scenarios and about as many encounter sequences that result in collision on some subsequent encounter as well as a number of escape sequences. The results of this large quantity of capture, collision and escape scenarios have led to the concept of a stable capture Stable Capture Zone (SCZ) is a restricted region of parameter space (planet anomaly vs. planetoid orbital eccentricity) in which capture can occur IF sufficient energy is dissipated in a combination of interacting bodies to insert the body of the planetoid into a geocentric orbit. The philosophical value of the geometry of these SCZs is that one can estimate (calculate) directly the probability of capture when the h and Q values are within the permissible range for gravitational capture.
I have found that there are four stable capture zones for each planet-planetoid combination: two prograde SCZs and two retrograde SCZs. That is, one can capture a planetoid from an orbit that is slightly larger or slightly smaller than the orbit of the planet and the encounter can be in a counterclockwise direction (prograde) or in a clockwise direction (retrograde) as viewed from the north pole of the Solar System.
Other Problems to be Addressed:
Two other problems that need to be addressed by a Gravitational Capture Model for the Origin of the Earth-Moon System are (1) the place of origin of the pre-capture planetoid (Luna) and (2) the similarity of oxygen isotope ratios of the Earth and Moon. Without going into technical details and to shorten the story considerably, a favorite place of origin at present is inside the orbit of planet Mercury at about 0.1-0.2 AU (astronomical unit). Such a place or origin is compatible with the anhydrous and refractory nature of lunar rocks (no water of hydration). And according to Evans and Tabachnik in a 1999 article in the journal Nature, planetoid orbits can be stable in this region of space for hundreds of millions of years. I think, then that this region is a good source for the formation of refractory planetoids.
The Similarities of Oxygen Isotope Ratios between Earth and Moon:
The problem of the similarity of oxygen isotopes ratios between Earth and Moon is a bit more challenging. It is known that Earth, Moon, enstatite chondrites, and the HED asteroids all have oxygen isotope ratio similarities. Mars and a few other asteroids that we think we have samples from via meteorites, are enriched in the heavier oxygen isotopes. However, there is no known information from Venus or Mercury that would relate to whole body oxygen isotope ratios, and until there is, we will not know what constitutes a trend in the inner solar system. My prediction is that the oxygen isotope ratios for planets Mercury and Venus will be very similar to those of planet Earth and Luna (the Moon).
(NOTE: A good summary and discussion of the oxygen isotope ratios of various solar system bodies as well as information on many other lunar and planetary science topics is found in a book by Ross Taylor published in 2001.)
The X-Wind Model and the Place of Origin of Luna:
Frank Shu [formerly at UC-Berkeley and now at Academia Sinica (Taipei, Taiwan)] has been working on models for the early history of sun-like stars (for example, Lada and Shu, 1990; Shu et al., 1991, Shu et al, 2000, and Shu et al., 2001). The name of the model is derived form the geometry of magnetic flux lines that intersect in the form of the letter “X”. They refer to their model as the X-Wind Model. This X-Wind Model appears to very successfully explain the origin of Calcium-Aluminum Inclusions (CAI’s) in meteorites but has been less successful in explaining the origin of chrondrules. This X-Wind would operate during the early history of the Sun before it settles into the main sequence of burning. There are some high-temperature pulses involved which result from reconnection of solar magnetic field lines and Shu thinks that there is enough energy associated with this flaring action to evaporate Oxygen-16 rich dust near the inner edge of the solar accretion disk and thus enrich this region in Oxygen-16 (light oxygen). I think that this excess O-16 could enrich the preplanetary nebular cloud and subsequent planetoid-building materials, all the way out to the vicinity of planet Earth. A prediction based on the X-Wind Model is that Venus and Mercury would fall along the same oxygen-isotope trend as the Earth and Moon. Thus, there is a possibility that the Gravitational Capture Model may be compatible with the similarity of oxygen-isotope ratio patterns of Earth and Moon.
Planet Venus has about the same radius as Earth and it is just a bit closer to the Sun but it has radically different features. The rotation rate is very slow in the retrograde (counterclockwise) direction, there is virtually no obliquity (tilt angle), and it has a very dense atmosphere composed mainly of carbon dioxide. It also has no old rocks on its surface, nothing older than 1.0 billion years and maybe only one half that age. An obvious question is: why the radical differences between sister planets?
A retrograde capture scenario for planet Venus may help explain the differences. The scenario goes something like this. Planet Venus captures one of these refractory planetoids formed in this belt between Mercury and the Sun into a retrograde orbit. The highly elliptical orbit then circularizes in a few tens of millions of years after capture (mainly due to energy dissipation in the planetoid) and continues to become smaller after circularization (mainly because of rock and ocean tidal energy dissipation mainly in the planet). The time scale of interaction is long (billions of years), but the trend is for the circular orbit to get smaller and smaller as the prograde spin rate of the planet decreases and the tides (both ocean and rock tides) become higher in amplitude and frequency. Eventually, after about 3.0 to 3.5 billion years of orbital evolution, the satellite is in a small circular orbit and stirs the mantle so much, in a unidirectional manner, that the original crust is subducted into the mantle and the planet gets a global resurfacing of basalt. The satellite eventually comes so close to the planet that it is at the Roche limit for a solid body. The satellite breaks up in orbit (within about 1.6 venus radii) and the pieces fall to the surface of Venus via atmospheric drag. The dense atmosphere would consist mainly of carbon dioxide, the main volcanic gas generated by the super (planet-wide) volcanic action. A satellite with about one-half the mass of Earth’s Moon would be sufficient to despin Venus from a primordial rotation rate of about 15 hours/day to zero and then cause it to rotate very slowly in the opposite (retrograde) direction.
If this story holds up for planet Venus, then Earth and Venus have something in common: they both have gravitationally captured satellites. The difference is in the details. Earth captured a planetoid in the PROGRADE direction and Venus captured a planetoid in the retrograde direction. The results for the evolution of the “sister planets” are radically different!
(NOTE: A retrograde capture scenario was first proposed by Fred Singer in a 3-page article in the journal Science in 1970. But I am the only one to pursue the geological consequences of such a retrograde capture scenario.)
Planet Neptune is the most distant gassy planet in the Solar System. It has two satellites that have been known for many years and many smaller one that have been discovered recently via the recent Ulysses mission to the outer planets. Nereid is a small satellite in a large, somewhat eccentric prograde orbit and does not pose much of a problem for explanation. Triton, in contrast, is a large satellite in a small, circular retrograde orbit that is inclined about 20 degrees to the plane of the planets. Triton is a major problem to be explained!
Most investigators agree that Triton was somehow captured into a retrograde orbit. The currently most popular explanation is that published by Peter Goldreich and others in 1989 in the journal Science suggesting that Triton is a product of collisional capture. In this model, planetoid Triton, would just happen to crash into a small existing satellite as it approaches planet Neptune. The orientation of the existing satellite and that of planetoid Triton was just right to take enough energy from planetoid Triton’s orbit to change it from a heliocentric orbit to an elliptical retrograde neptocentric orbit. Then over geologic time the orbit would circularize via tidal dissipation within the body of Triton. The authors’ suggested place of origin of the planetoid is the Kuiper Belt of icy planetoids, which is located just beyond the orbit of Pluto.
My model for the origin of the Neptune-Triton system has much in common with the above model. The difference is in the capture mechanism. It so happens that icy, triton-mass planetoids can have a low Q value at intermediate viscosity conditions just like rocky ones. Numerical simulations of gravitational capture show that it is possible to capture planetoid Triton into a large, but stable, retrograde orbit by way of energy dissipation within the body of Triton during one close encounter with planet Neptune using very reasonable h and Q values for Triton. The SCZs for retrograde capture are fairly large for encounters from either an orbit slightly larger or slightly smaller than Neptune’s orbit. The SCZs for prograde capture, however, are very limited and much more energy dissipation is necessary for prograde capture. Thus, retrograde gravitational capture is highly favored over prograde gravitational capture for a Triton-mass planetoid. After successful retrograde capture, in this model, the highly elliptical orbit then circularizes over a long period of time (billions of years) with nearly 100% of the tidal energy for orbit circularization dissipated in the body of Triton. Rocky planets are poor absorbers of tidal energy because of a high Q value, but gassy planets are even poorer absorbers because the Q values are an order of magnitude higher than those of their rocky relatives. Thus, the orbit of Triton is essentially “frozen” at its present location and the body of Triton will probably not coalesce with Neptune in the history of the Solar System.
Back to Planet Earth:
It appears difficult for planetary scientists (including geologists) geophysicists, astronomers, etc) to think of planet Earth as part of the Earth-Moon system. But the Moon seems to have had a subtle, but emphatic, effect on the history of our planet. For example, the rotation rate on a moonless Earth would be in the range of 12 to 14 hours per day. The tidal mechanism, mainly the ocean tides, has slowed the rotation to the present, very comfortable, 24 hours/day. A number of earth and planetary scientists have also suggested that unidirectional mantle rock convection may cause some unidirectional trends in the movement of tectonic plates. They suggest that these plate movements may be powered by the unidirectional operation of earth (rock) tides (which really have been operating ever since the Moon got associated with planet Earth). An article by Nelson and Temple in the 1972 volume of the Bulletin of the American Association of Petroleum Geologists is interesting reading. A recent (2000) book by Robert Bostrom (University of Washington) titled Tectonic Consequences of Earth's Rotation is also very interesting reading. The most recent summary of the possible influence of Earth tides on plate tectonics was published by B. Scoppola and others in 2006 in the Geological Society of America Bulletin. [Some of the Biological Consequences caused by the evolution of the Earth-Moon system are covered in a book by Peter Ward (geologist) and Donald Brownlee (physicist) from the University of Washington titled Rare Earth: Why Complex Life is Uncommon in the Universe which was also published in 2000.] In general, I think that the study of the subtle rock and ocean tidal effects of the Moon on the Earth (that is, the Evolution of the Earth-Moon System) is a potentially fruitful endeavor!
Planet Orbit - Lunar Orbit Resonances:
Still another interest of mine along the “Rare Earth” theme is on planet orbit – lunar orbit resonances and their possible geological effects on our planet. In recent years I have been working on the long-term tidal evolution of the Earth-Moon system from the time of capture (about 3.9 billion years ago) to the present. I am searching for eras in the history of the system in which the lunar orbit may enter into resonances with the orbital motion of planets, mainly Jupiter and Venus. The main concept here is that, for a resonant condition, the period of the perigean (or apsidal) cycle of the satellite (the prograde progression of the position of the moon at perigee) is equal to, or nearly equal to, the period of the heliocentric orbit of the perturbing planet. A four-body (sun, earth, moon, jupiter) numerical program is used for calculation of a Jupiter Orbit—Lunar Orbit resonance. Such an orbital resonance can cause a forced eccentricity of the lunar orbit consequently causing an era of higher than normal ocean and rock tides on both the planet and satellite. For the case of Jupiter this orbital resonance would be most effective when the Moon is at an orbital distance of about 53.4 earth radii (probably between 1.0 and 0.6 billion years before present).
This is a time of significant events in the rock record of Earth: (1) There is good evidence of at least two major “global” glaciations. (2) There is the development of several major continental rift zones. (3) There is a major change in the pathway of organic evolution from an algal-bacterial regime to a world with metazoans.
(NOTE: The first mention of a Jupiter Orbit – Lunar Orbit resonance was mentioned in an article by Peale and Cassen in the journal Icarus in 1978. They did mention, and I am paraphrasing, that if the resonance is stable, there could be significant thermal implications for both Earth and Moon. I am the only one to my knowledge to pursue these geological and lunar ramifications.)
I am also exploring the possible effects of Venus Orbit – Lunar Orbit resonances. These planet orbit—lunar orbit resonances would be somewhat weaker gravitational influences than the Jupiter-powered resonance. If any of these Venus-induced orbit resonances are stable enough over a significant period of time, they could cause a significant increase in eccentricity of the lunar orbit and thus could cause increased rock and ocean tidal activity in the equatorial zone of the planet. The enhanced tidal action, in turn, could cause some favorable environmental conditions for evolution of life forms on the planet during certain eras of our planet’s history. During these eras of increased orbital eccentricity, the global tectonic regime would also be effected by an increase in “tidal vorticity induction”, a process described in Robert Bostrom’s book mentioned above.
The Cool Early Earth Model:
Finally, there is another model in which I have a special interest – The Cool Early Earth Model. John Valley (Univ. of Wisconsin) and coworkers suggest that the near constant range of oxygen-isotope ratios in zircon crystals implies somewhat temperate condition on Earth from about 4.4-2.6 billion years ago). This time-frame includes most of the Hadean Eon as well as all of the Archean Eon. In the article (Valley et al., 2002) the authors question whether the Giant-Impact Model for the origin of the Earth’s moon is compatible with this new information because the putative earth-shattering impact which results in the formation of the Moon must happen and the Earth must cool to a condition where oceans can form (a temperature of about 200 degrees C) before 4.45 billion years ago. Valley et al. (2002) suggest that maybe a planetoid capture model should be considered for the origin of the Earth-Moon system.
(NOTE: In addition to the Valley et al. (2002) article, there is a very well illustrated article by John Valley in the Oct. 2005 issue of Scientific American and there is a good summary of the zircon crystal information in the journal Elements in 2006).
My gravitational capture model which features capture at about 3.9 billion years ago is very compatible with the Cool Early Earth Model. In this capture model the Earth would be moonless from the time of its origin until the time of capture. Thus the planet could accrete, the metallic core could settle to the center of the planet, the planet could cool, and ocean water could gradually accumulate on the surface as the atmospheric temperature decreases. Another factor that could aid in this cooling is that solar radiant energy would be about 20% less during this era than at present. The capture episode would be registered as a high temperature event on the captured planetoid because most of the energy for capture must be dissipated, over a short period of time, in that body. In great contrast, the Earth is a passive bystander in this model. Indeed, the rock and ocean tides on Earth are high and irregular for a geologically brief era, and much of the primitive crust would be recycled into the mantle in the equatorial zone of the planet. But the temperature of the earth’s interior is increased very little by the capture episode and the crust in the polar regions is sheltered from the tidal deformation (which occurs mainly in the equatorial zone of the planet). In other words, surface conditions on Earth are significantly effected, but the thermal regime of the surviving crust (and the associated zircon crystals) as well as the ocean-atmosphere system is not effected significantly.
Five publications that relate to gravitational capture issues:
- Malcuit, R. J., Winters, R. R., and Mickelson, M. E.,1977, Is the Moon a captured body?: Abstracts Volume, Eighth Lunar Science Conference, p. 608-610.
- Winters, R. R., and Malcuit, R. J., 1977, The Lunar Capture Hypothesis Revisited: The Moon, v. 17, p. 353-358.
- Malcuit, R. J., Mehringer, D. M., and Winters, R. R., 1989, Numerical simulation of gravitational capture of a lunar-like body by Earth: Proceedings, 19th Lunar and Planetary Science Conference, Cambridge Univ. Press and Lunar and Planetary Institute (Houston), p. 581-591.
- Malcuit, R. J., Mehringer, D. M., and Winters, R. R., 1992, A gravitational capture origin for the Earth-Moon system: Implications for the early history of Earth and Moon: in Glover, J. E., and Ho, S. E., eds, The Archaean: Terrains, Processes and Metallogeny: Gelogy Dept. (Key Center) and Univ. Extension, The Univ. of Western Australia, Publication No. 22, p. 223-235.
- Malcuit, R. J., and Winters, R. R., 1996, Geometry of stable capture zones for planet Earth and implications for estimating the probability of stable gravitational capture of planetoids from heliocentric orbit: Abstracts Volume, 27th Lunar and Planetary Science Conference, Lunar and Planetary Institute (Houston), p. 799-800.
One publication on the retrograde capture of a satellite for Venus:
- Malcuit, R. J., and Winters, R. R., 1995, Numerical simulation of retrograde gravitational capture of a satellite by Venus: Implications for the thermal history of the planet: Abstracts Volume, 26th Lunar and planetary Science Conference, Lunar and planetary Science Institute (Houston), p. 829-830.
Two publications on a capture origin for the Neptune-Triton system:
- Malcuit, R. J., Mehringer, D. M., and Winters, R. R., 1991, Numerical simulation of retrograde tidal capture of a triton-like planetoid by a neptune-like planet: Abstracts Volume, 22nd Lunar and Planetary Science Conference, Lunar and Planetary Institute (Houston), p. 845-846.
- Malcuit, R. J., Mehringer, D. M., and Winters, R. R., 1992, Numerical simulation of retrograde tidal capture of a triton-like planetoid by a neptune-like planet: Two-dimensional limits of a stable capture zone: Abstracts Volume, 23rd Lunar and Planetary Science Conference, Lunar and Planetary Institute (Houston), p. 827-828.
One publication on a Jupiter Orbit – Lunar Orbit resonance model:
- Malcuit, R. J., and Winters, R. R., 2001, A jupiter orbit – lunar orbit resonance model which may relate to Noproterozoic events on Earth and Moon: Geological Society of America, Abstracts with programs (National Annual Meeting), v. 33, no. 6, p. A-143.
One publication relating to the “Cool Early Earth Model”:
- Malcuit, R. J., and Winters, R. R., 2002, The cool early earth model and the lunar capture model: Are they compatible?: Geological Society of America, Abstracts with Programs (National Annual meeting), v. 34, no. 6, p. 273.
Two publications on the capture model and the primitive Earth:
- Malcuit, R. J., and Winters, R. R., 2006, The Cool Early Earth and the Tidal Capture Model: thermal and tectonic effects on Earth and Moon: Geological Society of America, Abstracts with Programs (National Annual Meeting), v. 38, no. 7, p. 386.
- Malcuit, R. J., and Winters, R. R., 2007, Early Archean Ophiolites and the Cool Early Earth: Can they be explained in the context of a Tidal Capture Model for the origin of the Moon?: Geological Society of America, Abstracts with Programs (National Annual Meeting), v. 39, no. 6, p. 333-334.
Other articals/authors referred to in this “soliloquy” on Planetary Science:
- Bostrom, Robert C., 2000, Tectonic Consequences of Earth’s Rotation: Oxford University Press, 266 p.
- Cadogan, Peter H., 1981, The Moon – Our Sister Planet: Cambridge University Press, 391 p.
- Evans, N. Wyn, and Tabachnik, Serge, 1999, Possible long-lived asteroid belts in the inner Solar System: Nature, v. 399, p. 41-43.
- Goldreich, Peter, Murray, N., Longaretti, P. Y., and Banfield, D., 1989, Neptune’s Story: Science, v. 245, p. 500-504.
- Lada, Charles J., and Shu, Frank H., 1990, The Formation of Sun-like Stars: Science, v. 248, p. 564-572.
- Nelson, T. H, and Temple, P. G., 1972, Mainstream mantle convection: A geological analysis of plate motion: American Association of Petroleum Geologists Bulletin, v. 56, p. 226-246.
- Ross, Martin, and Schubert, Gerald, 1986, Tidal dissipation in a viscoelastic planet, in Proceedings of the 16th Lunar and Planetary Science Conference, part 2: Journal of Geophysical Research, v. B91, p. D447-D452.
- Scoppola B., Boccaletti, D., Bevis, M., Carminati, E., and Doglioni C., 2006, The westward drift of the lithosphere: A rotational drag?: Geological Society of America Bulletin, v. 118, p. 199-209.
- Singer, S. F., 1970, How did Venus lose its angular momentum?: Science, v. 170, p. 1196-1198.
- Shu, F. H., Ruden, S. P., Lada, C. J., and Lizano, S., 1991, Star formation and the nature of bipolar outflows: Astrophysical Research Letters, v. 370, p. L31-L34.
- Shu, Frank H., Najita, Joan R., Shang, Hsein, and Li, Zhi-Yun, 2000, X-Winds: Theory and Observations, in Manning, V., Boss, A. P., and Russell, S. S., eds. Protostars and Planets IV: University of Arizona Press, p. 789-813.
- Shu, F. H., Shang, H., Gounelle, M., Glassgold A. E., and Lee, T., 2001, The origin of chondrules and refractory inclusions in chondritic meteorites: The Astrophysical Journal, v. 548, p. 1029-1050.
- Taylor, S. Ross, 2001, Solar System Evolution: A New Perspective, 2nd ed.: Cambridge University Press, 460 p.
- Valley, J. W., Peck, W. H., King, E. M., and Wilde, S. A., 2002, A Cool Early Earth: Geology, v. 30, p. 351-354.
- Valley, John W., 2005, A Cool Early Earth?: Scientific American, v. 293, p. 58-65.
- Valley, John W., 2006, Early Earth: Elements, v. 2, p. 201-204.
- Ward, Peter D., and Brownlee, Donald, 2000, Rare Earth: Why complex life is uncommon in the Universe: Copernicus (Springer-Verlag, New York), 333 p.