Starship Asterisk*

It is quite apparent that collisions have taken place not only with asteroids and comets but amongst the major celestial bodies in our solar system. The main affects are orbital changes, rotational changes, and spin axes tilts. I am primarily interested, for the moment, in how spin axes tilts are affected by a massive collision, as for instance a Mars size body hiting Earth or an appropriately sized body rolling Uranus about 90 degrees on its axis.

Assuming that the masses of the primary body and its impactor, their translational velocity vectors, the rotational velocity of the primary, and the amount of offset from the equator of the primary body"s impact zone are inputted, what equation is needed to determine the amount of tilt ? Assume that the bodies have uniform densities, the velocity vector of the impactor is in the longitunal center plane, and the angle of impact is normal to the surface of the primary. I imagine the gyroscopic stability of the spinning primary has to be considered (?)

The more simple answer is what relative size bodies can cause what amount of tilt in a collision of two celestial bodies.

Doug Ettinger
Pittsburgh, PA

G'day Dougettinger

You may find this paper interesting.

http://arxiv.org/abs/0912.0181
A collisionless scenario for Uranus tilting

Authors: Gwenaël Boué, Jacques Laskar
(Submitted on 1 Dec 2009 (v1), last revised 9 Feb 2010 (this version, v2))

Abstract: The origin of the high inclination of Uranus' spin-axis (Uranus' obliquity) is one of the great unanswered questions about the Solar system. Giant planets are believed to form with nearly zero obliquity, and it has been shown that the present behaviour of Uranus' spin is essentially stable. Several attempts were made in order to solve this problem. Here we report numerical simulations showing that Uranus' axis can be tilted during the planetary migration, without the need of a giant impact, provided that the planet had an additional satellite and a temporary large inclination. This might have happened during the giant planet instability phase described in the Nice model. In our scenario, the satellite is ejected after the tilt by a close encounter at the end of the migration. This model can both explain Uranus' large obliquity and bring new constraints on the planet orbital evolution.

harry wrote:You may find this paper interesting.

http://arxiv.org/abs/0912.0181
A collisionless scenario for Uranus tilting

It isn't clear that collisions have been an important factor in modifying any planet's orbit or rotation. The only good case for an actual collision is the formation of the Moon (and even here there is no need for the orbit or rotation to have been significantly modified). The discussion in this paper emphasizes that collisions are not necessary. More to the point, collisions are statistically far less likely than near misses- that is, you would expect quite a few near misses for every actual collision.

harry wrote:G'day Dougettinger

You may find this paper interesting.

http://arxiv.org/abs/0912.0181
A collisionless scenario for Uranus tilting

Authors: Gwenaël Boué, Jacques Laskar
(Submitted on 1 Dec 2009 (v1), last revised 9 Feb 2010 (this version, v2))

Abstract: The origin of the high inclination of Uranus' spin-axis (Uranus' obliquity) is one of the great unanswered questions about the Solar system. Giant planets are believed to form with nearly zero obliquity, and it has been shown that the present behaviour of Uranus' spin is essentially stable. Several attempts were made in order to solve this problem. Here we report numerical simulations showing that Uranus' axis can be tilted during the planetary migration, without the need of a giant impact, provided that the planet had an additional satellite and a temporary large inclination. This might have happened during the giant planet instability phase described in the Nice model. In our scenario, the satellite is ejected after the tilt by a close encounter at the end of the migration. This model can both explain Uranus' large obliquity and bring new constraints on the planet orbital evolution.

G,day, Harry. Thanks for the reference and abstract. This reason might very well be the best for Uranus' tilt after I recently learned about the sensitivity of spin axis stability.

Doug Ettinger
Pittsburgh, PA

Chris Peterson wrote:

harry wrote:You may find this paper interesting.

http://arxiv.org/abs/0912.0181
A collisionless scenario for Uranus tilting

It isn't clear that collisions have been an important factor in modifying any planet's orbit or rotation. The only good case for an actual collision is the formation of the Moon (and even here there is no need for the orbit or rotation to have been significantly modified). The discussion in this paper emphasizes that collisions are not necessary. More to the point, collisions are statistically far less likely than near misses- that is, you would expect quite a few near misses for every actual collision.

I am very happy to learn that Uranus does not need a collision scenario for creating its spin axis anomaly. I know you have a great deal of confidence in the Nice theory which is also required. And, yes, collisions are far less likely than near misses (which also may be dangerous to life on Earth) and perturbations. That is why I am writing in this forum today. Nevertheless, one collision of two major bodies early in the history of the solar system is presently unavoidable in trying to explain the Earth-Moon system. I have a different hypothesis for explaining their existence. That is why I yearn for an equation or an explanation that gives the correlation between the mass of an impactor and the resulting amount of axis tilt of the body that was struck. I need to know the relative amounts of large energy losses in my collision scenario.

Doug Ettinger
Pittsburgh, PA

dougettinger wrote:Nevertheless, one collision of two major bodies early in the history of the solar system is presently unavoidable in trying to explain the Earth-Moon system.

I disagree. The collision theory of formation is certainly the best supported by evidence, which is why it has become so widely accepted. But it isn't the only possibility.

G'day

The link that I posted is for information. Not trying to prove a point.

I thought it maybe of interest.

Chris Peterson wrote:

dougettinger wrote:Nevertheless, one collision of two major bodies early in the history of the solar system is presently unavoidable in trying to explain the Earth-Moon system.

I disagree. The collision theory of formation is certainly the best supported by evidence, which is why it has become so widely accepted. But it isn't the only possibility.

I am curious to know the most current next best theory for the formation of the Earth-Moon system outside of collision theories (?) Perhaps the Nice theory is able to address the tilts of the outer planets, but that leaves the tilt axis of Earth and Mars. There are other data that support possible collision scenarios for Earth and Mars with their high inclinations of 23 degrees. And my suspicion is that their spin axis tilts were possibly related to the collision. Can a correlation be made between the an impactor's mass, velocity, and its location of impact with the amount of tilt of the primary spinning body ? Perhaps this is mathematically indeterminate (?)

Doug Ettinger
Pittsburgh, PA

dougettinger wrote:I am curious to know the most current next best theory for the formation of the Earth-Moon system outside of collision theories (?)

I don't know that other theories can be ordered. Certainly, theories involving fission during planetary formation and theories involving a capture remain viable. Such theories are not popular, because there are numerous observations that they don't accommodate very well. But they have not been excluded so completely that the impact theory remains the only possible answer. It is just the best theory.

Perhaps the Nice theory is able to address the tilts of the outer planets, but that leaves the tilt axis of Earth and Mars.

The Nice theory is not required to explain planetary tilts. It is just one model of solar system evolution. Multibody planetary systems are inherently unstable. When you have a combination of axial tilt (even at a tiny angle) and orbital eccentricity (again, even very small), you have a system where angular momentum can be transferred between bodies, and where variable precession and nutation can be converted to different axial tilts. And of course, you still have the likelihood of collisions and near-misses early in the development of the Solar System. I don't think that the deviation of many planets' spin axes from the invariant plane is of much concern to those who study how our system formed (which isn't to say it isn't of interest).

Chris, I am supremely impressed with your answer. It gives me even more crazy ideas. Some geologists postulated and wrote a book that a near miss for Earth occurred about 11,500 years ago that caused the Great Flood and other related catastrophes. The closeness of it gravity field shifted the Earth's crust. You are making their ideas more sensible.

As you have probably guessed, I am still digging around for that spin axis tilt equation. I have other reasons for believing that someone shook the pinball machine. You may not be old enough to know that pinball machines indicated "tilt" and stopped working if you tried to tilt them in order to guide the pinball.

Doug Ettinger
Pittsburgh, PA

dougettinger wrote:Some geologists postulated and wrote a book that a near miss for Earth occurred about 11,500 years ago that caused the Great Flood and other related catastrophes. The closeness of it gravity field shifted the Earth's crust. You are making their ideas more sensible.

Don't say that! I'm a Research Associate in Geology at the Denver Museum of Nature and Science, and if anybody thought I took such a theory seriously, I'd probably be booted out. And I'd be the first to agree I should be.

There is absolutely no merit to that idea. There is no evidence of any worldwide catastrophe occurring at that time. There is a controversial theory that a (probably cometary) impact occurred around then, creating the Younger Dryas, a period of cool climate in the northern hemisphere. Evidence for such an event remains ambiguous (and the topic is a matter of hot discussion in the geology and meteoritics communities). Even if it occurred, however, it was an "ordinary" impact of a small body (or a cluster of bodies) such as occurs every few tens of thousands of years. Such bodies can produce short term global climate effects, but will have no impact at all on our orbit or axial tilt.

FWIW, Earth's orbital eccentricity, day length, and axial tilt can be reliably determined for at least the last 600 million years using geologic data.

As you have probably guessed, I am still digging around for that spin axis tilt equation. I have other reasons for believing that someone shook the pinball machine. You may not be old enough to know that pinball machines indicated "tilt" and stopped working if you tried to tilt them in order to guide the pinball.

I built a Pong machine from scratch while in high school, and in college worked on one of the first consumer video game systems (what eventually became the Mattel Intellivision). At the time, people still associated arcades with pinball machines, although the first video arcade games were just showing up. So I do understand the "tilt" reference <g>.

Chris Perterson wrote:
"Don't say that! I'm a Research Associate in Geology at the Denver Museum of Nature and Science, and if anybody thought I took such a theory seriously, I'd probably be booted out. And I'd be the first to agree I should be.

There is absolutely no merit to that idea. There is no evidence of any worldwide catastrophe occurring at that time. There is a controversial theory that a (probably cometary) impact occurred around then, creating the Younger Dryas, a period of cool climate in the northern hemisphere. Evidence for such an event remains ambiguous (and the topic is a matter of hot discussion in the geology and meteoritics communities). Even if it occurred, however, it was an "ordinary" impact of a small body (or a cluster of bodies) such as occurs every few tens of thousands of years. Such bodies can produce short term global climate effects, but will have no impact at all on our orbit or axial tilt.

Wow! You like geology. Forget everything I said about a catastrophic event 11,500 years ago. My tilt scenario has nothing to do with that event; it goes back 3.9 bya.

Anyway, give me your idea for the difference between the the Earth's magnetic poles and spin axis poles (?) And why are the magnetic poles drifting slowly toward the spin axis ?

Doug Ettinger
Pittsburgh, PA

dougettinger wrote:Anyway, give me your idea for the difference between the the Earth's magnetic poles and spin axis poles (?) And why are the magnetic poles drifting slowly toward the spin axis ?

There is no reason to expect any planet with a magnetic field to have it precisely aligned with the spin axis. Planetary magnetic fields are created by the internal flow of conductive fluids (iron in the case of the Earth). This flow is heat driven convection, and is somewhat chaotic (as is often the case with moving fluids). It is the characteristics of this flow pattern that determine the axes of the approximately dipole magnetic field, not the spin axis of the planet. As the flow varies, so does the position of the magnetic poles. There are also quadrupole and higher moment fields, which distort the primary dipole field. As a result, the magnetic poles aren't even perfectly opposed on the surface of the Earth.

dougettinger wrote: ...I have a different hypothesis for explaining their existence. That is why I yearn for an equation or an explanation that gives the correlation between the mass of an impactor and the resulting amount of axis tilt of the body that was struck. I need to know the relative amounts of large energy losses in my collision scenario.

...And my suspicion is that their spin axis tilts were possibly related to the collision. Can a correlation be made between the an impactor's mass, velocity, and its location of impact with the amount of tilt of the primary spinning body ? Perhaps this is mathematically indeterminate (?)

I've put some time into quoting or providing a simplified collision model and subsequent equation you yearn for. Which should not be a suprise to you, there has been much work, mostly numerical analysis, to explore collision behavior - the fundamental theory has been in place. In particular, interesting modeling behind making video affects and gaming more realistic is progressing deeper into rigid-body collision mechanics. Also, the concept of rotation axis change after a collision is not new, and to reiterate, applying collision theory to explain planetary orbital and rotational parameters is not new.

The bottom line in calculating a change in rotation axis is that it's complicated; what you want is not readily derivable, although I can't say that an rough-order-of-magnitude, simplified solution is indeterminate either. Put simply, a change in rotation axis requires a moment (torque) and as a result, angular momentum does not have to be conserved. And in this case, we're dealing with impulse momentum which is difficult to quantify. Of course, assumptions can be made that permits an analytical solution but I don't have the knowledge to justify them. For example, one could assume angular momentum components are conserved and the change in AM vectors lead to an equivilent / predictable change in rotation vectors. Unfortunately, I do know this is not true and that is all I know. Certainly this is an interesting problem, the challenge being to express a simplified but applicable model. Calculating results within 10x of reality using an "applicable model" I would consider very good given the simplistic assumptions that have to be made. Even 10x might be optimistic.

...The more simple answer is what relative size bodies can cause what amount of tilt in a collision of two celestial bodies.

I can't answer this, but one interesting bookend might be to consider what mass is needed to stop the earth's rotation. A mass = 5% of the earth (~4.5x moon's mass) impacting the earth's equator at grazing incidence (30km/sec) has enough AM to stop the earth's rotation. But what about the energy? Only 2% of the energy is used to stop the rotation, which leaves 98% of the energy left to do other things! A "simple" calculation suggests that this impact will obliterate the earth with energy to spare! The point is that energy cannot be ignored, the reality of heat and ejecta are needed to calculate the effecs from large mass collisions. If you don't, there is no reason for anyone to take your results serously when applied to real objects like moons and planets.

In the last post I wrote: ... A mass = 5% of the earth (~4.5x moon's mass) impacting the earth's equator at grazing incidence (30km/sec) has enough AM to stop the earth's rotation.

This statement is wrong, there is a decimal point error. It should read: "A mass = 0.5% of the earth (~1/2 moon's mass) impacting the earth's equator at grazing incidence (30km/sec) has enough AM to stop the earth's rotation"

The gist of the rest of the paragraph is correct, it's only the value of the impactor mass that needs correction.

Alter-ego, thanks for paying serious attention to my collision scenario and the resulting axis tilt.

Where has all the energy gone? In my collision scenario small portions of the energy go into heat, ejecta, increasing or decreasing the Earth's spin, and very possibly changing the axis tilt. Assuming most of the collision is elastic; and, an almost direct hit absorbed the largest portion of the impactor body the majority of the energy, 80% or more, (my personal idea) goes into changing the orbital speed and direction. An almost perfect elastic collision is assumed since the bodies should be very mushy, but cooling at a high rate, in the early solar system.

Then what torque or impactor distance from the equator would cause a certain amount of tilt if the initial masses, vector velocities and spin of the primary were guessed and inputted? I am only looking for a factor of 10 error at the very best for this determination.

Doug Ettinger
Pittsburgh, PA

dougettinger wrote: Where has all the energy gone? In my collision scenario small portions of the energy go into heat, ejecta, increasing or decreasing the Earth's spin, and very possibly changing the axis tilt. Assuming most of the collision is elastic; and, an almost direct hit absorbed the largest portion of the impactor body the majority of the energy, 80% or more, (my personal idea) goes into changing the orbital speed and direction. An almost perfect elastic collision is assumed since the bodies should be very mushy, but cooling at a high rate, in the early solar system.

Where has all the energy gone? (sounds like a song) First, for my following comments I assume you meant "inelastic" not "elastic". For inelastic collisions, your estimates for kinetic energy and ejecta / heat energy are orders of magnitude off. Here is a basic principle for inelastic collisions: Energy left over after Momentum Conservation is expressed equivalently as Heat.

Let's look at the example in my last post, but instead of grazing incidence, let the 0.5% Earth's mass impact the Earth on center and assume no angular momentum transfer. Linear momentum conservation determines the velocity (and therefore kinetic energy) after the collsion. V_final = V_initial x m_impactor/M_earth (since m << M, ignore m in the total mass). Calculating the after-to-before kinetic energy ratio you get: E_ratio = m/M! That means that for a direct hit, 99.5% of the energy turns to heat, where only 0.5% remains as kinetic energy. Although a 100% elastic model is not reality, I think it's much better than an ideal elastic one. In this collision, and any large object collision, billiard balls they are not. That's why large-mass collisions are depicted (and simulated) as being catastrophic, at least for present day "cold" bodies. But hot, mushy protoplanets will likely behave even more like putty and therefore inelastic, so the above low-percentage, post-collision kinetic energy may be a reasonable order-of-magnitude approximation.

I certainly enjoy your passion and creative thinking in the collision process, but if you want more than just a good feeling, you need frame your hypotheses within the natural laws (physics in our discussion). If you want to pursue the truth, your ideas need a solid foundation in the mechanics. Excuse my frankness, but you can only go so far winging it. Maybe take a class or two in introductory mechanics to warm up to the math side?

Lastly, I must confess I floundered in this collision exercise also! I made a fundamental error in the expression I derived for you in an earlier thread. I mistakenly used energy conservation, but as I've correctly stated here, I should have used momentum. Momentum is a vector quantity, and energy is a scalar. Energy only adds, vectors can subtract. I will post a correction to that derivation, even if only to satisfy my need to fix my own mistakes.

alter-ego wrote:

dougettinger wrote: Where has all the energy gone? In my collision scenario small portions of the energy go into heat, ejecta, increasing or decreasing the Earth's spin, and very possibly changing the axis tilt. Assuming most of the collision is elastic; and, an almost direct hit absorbed the largest portion of the impactor body the majority of the energy, 80% or more, (my personal idea) goes into changing the orbital speed and direction. An almost perfect elastic collision is assumed since the bodies should be very mushy, but cooling at a high rate, in the early solar system.

Where has all the energy gone? (sounds like a song) First, for my following comments I assume you meant "inelastic" not "elastic". For inelastic collisions, your estimates for kinetic energy and ejecta / heat energy are orders of magnitude off. Here is a basic principle for inelastic collisions: Energy left over after Momentum Conservation is expressed equivalently as Heat.

Let's look at the example in my last post, but instead of grazing incidence, let the 0.5% Earth's mass impact the Earth on center and assume no angular momentum transfer. Linear momentum conservation determines the velocity (and therefore kinetic energy) after the collsion. V_final = V_initial x m_impactor/M_earth (since m << M, ignore m in the total mass). Calculating the after-to-before kinetic energy ratio you get: E_ratio = m/M! That means that for a direct hit, 99.5% of the energy turns to heat, where only 0.5% remains as kinetic energy. Although a 100% elastic model is not reality, I think it's much better than an ideal elastic one. In this collision, and any large object collision, billiard balls they are not. That's why large-mass collisions are depicted (and simulated) as being catastrophic, at least for present day "cold" bodies. But hot, mushy protoplanets will likely behave even more like putty and therefore inelastic, so the above low-percentage, post-collision kinetic energy may be a reasonable order-of-magnitude approximation.

I certainly enjoy your passion and creative thinking in the collision process, but if you want more than just a good feeling, you need frame your hypotheses within the natural laws (physics in our discussion). If you want to pursue the truth, your ideas need a solid foundation in the mechanics. Excuse my frankness, but you can only go so far winging it. Maybe take a class or two in introductory mechanics to warm up to the math side?

Lastly, I must confess I floundered in this collision exercise also! I made a fundamental error in the expression I derived for you in an earlier thread. I mistakenly used energy conservation, but as I've correctly stated here, I should have used momentum. Momentum is a vector quantity, and energy is a scalar. Energy only adds, vectors can subtract. I will post a correction to that derivation, even if only to satisfy my need to fix my own mistakes.

Yes, you are right in correcting me. I did mean an inelastic collision and not an elastic collision. The collision would be similar to an aluminum nut or chunk of ice be thrown at the centeline of a slightly hard poached egg; the impactor cracks and penetrates the egg shell and lodges itself inside the slightly hardened yolk.

I have taken statics and dynamics courses years ago. I have hunted through my textbooks and some modern testbooks, too. My main weakness is applying calculus when necessary. I have not found a solved problem that resembles my problem. The basic problem is how an outside, off-centered force on a celestical body affects the spin axis tilt of this body that is orbiting a star well inside the star's gravity field. Assume, initially, that the force only affects the tilt and no other dynamical properties of the primary body.

Is there a simple answer to this supposely simple case ?

I understand about scalar and vector quantities. Where are you posting your correction to your previous subject derivation ? Thank you very much for this correction.

Doug Ettinger
Pittsburgh, P{A

dougettinger wrote: Where are you posting your correction to your previous subject derivation ? Thank you very much for this correction.

I have posted the correction in original thread. "Off-Center Collision of Two Celestial Bodies"
If I make any headway with a "simple" model to estimate a rotation axis change, I'll let you know.

I will be anxiously awaiting your model. I am assuming in my collision scenario that some energy transfer goes into tiling the axis. Any collision scenario should not be centered either longitudinally or latitudinally.

Doug Ettinger
Pittburgh, PA

Hello All,

WRT the Earth/Moon senario I can suppose that this is out?

http://asterisk.apod.com/vie ... on#p112121

astrolabe wrote:WRT the Earth/Moon senario I can suppose that this is out?
http://asterisk.apod.com/vie ... on#p112121

Pretty much. It doesn't seem the Earth ever spun fast enough for that to happen, and there are issues with respect to the mineralogical differences and system angular momentum that this theory can't handle.

Hello Chris,

Thank you. One of the chief benefits of being a member of this fine Forum is getting reasonable answers to things that some one (like ME) have had on their minds for years. Joining was one of the good things I have done for myself. And besides, back in Nov. '09 when I composed that post, it turned out to be the last of 13 posts in the thread so I never got a response. It happens sometimes- at first I thought no one could be bothered with such a ridiculous idea so i do appreciate your responding this time around.

BTW IYO would speed of rotation have to be that great for this to be conceivable?