When you release a toy gyroscope on its pedestal, it initially has no precessional rotation around that pedestal. A moment later, it is merrily precessing around the pedestal! Both Kinetic Energy and Angular Momentum now exist where they did not exist a moment earlier. We will discuss below that the Kinetic Energy is easy to explain, as appearing due to the body of the gyroscope dropping a tiny fraction of a millimeter in the Earth's gravitational field, giving up exactly the correct amount of Potential Energy. So Conservation of Energy definitely does apply as we have always known. But the Angular Momentum of the Precession which just appeared does not come from anywhere! This is therefore a Violation of the Conservation of Angular Momentum, which still appears to be universally true for all other situations than gyroscopic-type precessional motions.
The Earth is an even more impressive example of this same effect. In school, you were taught that the Earth SLOWLY "wobbles" ( precesses), somewhat resembling a child's wobbling top, in about 26,000 years. Even your Teacher did not know that on every March 21 and September 21, that Precession absolutely STOPS COMPLETELY. (This statement is technically only true for the SOLAR portion of the Earth's Precession, and a similar statement can be made about twice every month regarding the Moon's portion of it . We are simplifying a little here in just considering the Sun's for now! ) And then it increases to become about twice the average rate around June 21 and December 21. During this complex dance, the Earth's Precessional motion CREATES massive amounts of Angular Momentum, between Mar 21 and Jun 21, and then gives it back up, between Jun 21 and Sep 21.
There is NO SOURCE FOR THAT ENORMOUS AMOUNT OF ANGULAR MOMENTUM TO HAVE COME FROM, OR TO HAVE GONE BACK TO! (It is related to the gravitational field of the Sun or Earth, but in a peculiar way. ) I have a different web-page that discusses what acts as a VIOLATION OF THE CONSERVATION OF ANGULAR MOMENTUM, in that precession related angular momentum appearing and disappearing on a regular basis.
Energy is what is called a scalar quantity, a number without any direction, and energy can be converted back and forth between being Potential Energy and Kinetic Energy or other forms. Angular Momentum is very different, as it is a Vector quantity. It has a direction! In both the toy gyroscope and the Earth, the axis if the Precession is what can be referred to as vertical . It cannot be converted or changed into anything else, even any vector into any other direction, and it could not have come from anything else or even any angular momentum vector along any other axis. A true problem!
Down below, we will show the calculations of these effects. As to Kinetic Energy of the Earth's Solar Precession, that amount of energy is zero on March 21 and it increases to the Earth having 10,000 trillion joules of Precessional Kinetic Energy just three months later . This is a significant quantity of energy being transferred into the Precessional motion, which then has to all be returned to its source during the next three months to again become zero. The RATE of energy transfer is rather impressive! The fact that it is spread out over a three-month period means that the average rate of energy transfer (power) into and out of Precession is about 1,250,000 kiloWatts, but which constantly varies itself.
For the record, if we could ever figure out a way of capturing that energy and making it into electricity, that's more than a nuclear power plant could provide!
In fact, the similar Precession caused by the Moon on the Earth is actually both larger and faster changing. Roughly 40,000 trillion joules is transferred into and out of the Earth's Lunar Precession over approximately a week interval. This results in a RATE of energy transfer being far greater than that due to the Sun, being around 63,000,000 kiloWatts as an average. This SEEMS potentially interesting to us humans as that AVERAGE amount of power from just Lunar Precession is actually comparable to the ENTIRE electricity production of ALL the 104 nuclear power plants in the United States ! Unfortunately, mankind does not seem to know of any way of CAPTURING any of that power! (All these calculations are included below).
Actually, the two examples of the toy gyroscope's precession and the Earth's Precession are OPPOSITE each other in one way. The effect of the Earth's gravity on a toy gyroscope acts to try to INCREASE THE TILT of the spin axis of the gyroscope, which then causes the Precessional motion (per Euler's Differential Equations) to tend to fall over. In contrast, the effect of the Sun's gravity on the tilted Earth acts to try to STAND UP the Earth's spin axis, which then causes the opposite sort of Precessional motion.
We provide another web-page which is specifically on Precession which provides the math to support both of these situations.
The work of Euler in developing the three-dimensional equations of Newton's one- and two-dimensional Mechanics (which are generally called the set of three Euler Differential Equations) has rarely been fully comprehended or used.
LaPlace, LaGrange, Leverrier and other brilliant early astronomers all made an ASSUMPTION that now appears to have been slightly wrong! They correctly recognized that Conservation of Energy must apply to planets orbiting the Sun. They also accepted that Angular Momentum is always Conserved which turns out not to always be true! Given those two conditions, it is rather easy to mathematically show that two planets might perturb each other in a variety of ways, but NEVER by ever being able to change the radius of each other's orbits. And so they all agreed on an absolute statement to that effect, that planets could never alter the semi-major radius of each other's orbits. THAT statement is now seen to be clearly incorrect, but ONLY when gyroscopic-types of effects apply.
Their reasoning and their resulting conclusion is generally true. However, this presentation shows that there IS one process which violates the Conservation of Angular Momentum, which then PERMITS planets to (very, very slowly) alter each other's orbital radii. It is the gyroscopic precessional processes by which planets can affect each other.
Those brilliant scientists and mathematicians and others must certainly have seen toy gyroscopes that APPEAR to have very constant Precession, and made that incorrect assumption. They even developed the calculations to show that such a toy gyroscope gradually precesses FASTER as friction gradually slows the rotor spinning down, but that turns out to be a rather different effect.
They had overlooked an obvious fact that when a toy gyroscope is FIRST released, its Precessional motion MUST ACCELERATE from the initial non- precessing situation. Angular Momentum around the precession axis simply BEGINS out of nothing! ZERO Angular Momentum around the precession axis rather quickly becomes some non-zero number, which then remains essentially constant thereafter. The so-called Law of Conservation of Angular Momentum says that cannot happen! This is the simplest and most obvious example of a clear Violation of that Law of Conservation of Angular Momentum.
As long as gyroscopic precession effects do not occur, then it IS true that Angular Momentum IS Conserved, and the mathematics and the conclusions of those brilliant scientists are true. It is ONLY when precession occurs that this Violation can and does occur.
Among the solar system, IF a perturbing object (planet, moon or sun) happens to be on the equator of a spinning other object, that second object is NOT perturbed in this gyroscopic manner. However, most of the time, the perturbing object is either ABOVE or BELOW that equatorial plane, and a precessional effect DOES occur. Kinetic Energy is then transferred from one to the other, an effect which has long been known, and which results in effects such as Regression of the Nodes of the orbit of the target object. It has always been ASSUMED that Angular Momentum was also Conserved, but that is NOT precisely true, due to this Violation of the Conservation of Angular Momentum.
However, in practical terms, the effect of this Violation is quite small, and it has apparently always been neglected. The fact that this is such a small effect for objects like the planets and moons of the solar system, makes the effects VERY small and therefore slow to cause changes great enough for detection.
It also turns out that the common belief that planetary Precession is constant is not remotely true! In fact, for the Earth's Precession, which school children learn takes around 26,000 years to complete one wobble, the actual motion is extremely complex. EVERY March 21 (or more precisely, at the instant of the Spring Equinox), Precession due to the Sun's effect (Solar Precession) briefly vanishes! At that instant, there is ZERO Solar Precession! The effect then rapidly increases, up to a maximum Precession rate around June 21 (Summer Solstice), when it has become roughly TWICE what the average annual Precession rate is measured to be. Then it decreases again to again become zero at the Autumnal Equinox (around Sept 21) and again increases to a (positive) maximum at the Winter Solstice (around Dec 21).
The math behind these statements are fairly simple and they are provided in a linked page about Precession.
The huge amounts of Kinetic Energy involved in these rather rapid and enormous changes of speed of the Earth's Precession, actually do NOT violate Conservation of Energy! The energy "which appears" actually came from an identical amount of Kinetic and Potential energy of the Sun-Earth system, and those amounts get returned to the Sun-Earth system during the following three months! But as to Angular Momentum (of the Earth-Sun system), it is NOT Conserved, as neither the Sun nor Earth loses any Z-axis Angular Momentum in those three months while the Earth's Precessional motion is accelerating (Mar 21-Jun 21 or Sep 21-Dec 21).
This all turns out to be immensely important ! The usual Euler Equations are used, but they must be Integrated, in order to analyze the VARIATIONS in the rates of Precession and the consequences of that IN A DIFFERENT AXIS (direction). THIS relatively simple mathematical procedure can then explain WHY AND HOW a toy gyroscope has a precessional motion "appear" where it had not previously existed. At least, it explains where the Energy came from!
This then results in a VIOLATION of an universal concept in science, the Conservation of Angular Momentum! (The results of this effect are rather small and for planets are only of major effect after many thousands of orbits, thousands of years for planets and moons.)
And THAT then allows all sorts of incredible consequences! LaPlace and the others ASSUMED that when planets perturbed each other, that they COULD NOT alter the overall orbital radius (because that would be impossible if both Energy and Angular Momentum are conserved) . But that is now seen to be wrong, which then provides explanations for many astronomical phenomena which have never been properly explained before. Much of Astrophysics probably needs to be re-examined and re-written to be more correct. The differences are very slow in effect. For the four Galilean moons of Jupiter to have gotten themselves into a fascinating mutual pattern, certainly took many millions of their orbits to create . And IF these effects would ever cause an EXACT commensurability of two moons or planets, the same effects would destabilize the situation. So the four Galilean moons are not QUITE in perfect commensurability, but tantalizingly close! Even the amount of this meta-stability can be calculated, with results that are very close to observed differences of those moons, the relationship between Jupiter and Saturn, aspects of Saturn's rings and moons, the Kirkwood Gaps in the asteroid belt, and much more. Such patterns are not just coincidences or accidental as some scientists claim! They are not only logically sound, but the math confirms that fact!
The AVERAGE Precession rate of the Earth's wobble is as we all learned in school, but it ACTUALLY is constantly rather wildly CHANGING, constantly accelerating and decelerating.
All those CHANGES in the rates of precession have effects similar to those seen when releasing a gyroscope. There are EFFECTS which cause the planets to constantly be increasing and decreasing in their precessional rates. They are called Perturbations, and they are forever causing slight changes in the shape and orientation of each planet's orbit, and in the rotation and orientation of each planet's spinning . The REALITY in the solar system is as though we would see a toy gyroscope's precession keep starting and stopping in a herky-jerky motion!
Precession between planets can act on planets in two very different ways. There is the spinning precession like that we are all familiar with, where the effects of changes in the rate of precession only has the effect of tiny wobbles in the precise orientation of the spin axis. The Earth shows such wobbling of a small fraction of a mile, where the North Pole axis is actually never quite where maps say the North Pole is! It is actually about 900 feet away from that map location, gradually circling around the map location of the Pole, in a very complex dance.
But what is of more interest here is that the second way that precessional effects can occur is by ORBITAL changes such as what is called Regression of the Nodes.
IF the previous assumption had been true, then the cumulative effect of many years of Regression of the Nodes would always have resulted in no net change in the orbital radius. But since Angular Momentum is NOT Conserved in this specific situation, it turns out that, very slowly and very gradually, planets CAN cause modifications to the orbital radius of each other. They DO still comply with the Law of Conservation of Energy!
This all means that in SHORT-TERM viewing, there does not SEEM to be any observed effects of semi-major radii being altered due to precession effects, but that over LONG-TERM periods, such changes can and do occur.
After many thousands or millions of orbits, the two planets (or moons) therefore wind up having orbits which have changed and that are now CLOSE to having some simple fractional proportion regarding orbital period. They CANNOT have stable orbits which are EXACTLY commensurable, as there are severe effects of instability then. But the result is a meta-stable pair of orbits which are quite close to seeming commensurable, and which can maintain themselves in that relationship for very long periods of time.
I recommend that mathematical Physicists set up such a problem in a computer, of any two planets orbiting a much more massive primary, and given RANDOM orbital periods to start. This process also works regarding differences in orbital inclination and regarding the eccentricity of both orbits, but those processes seem to take longer to occur. A computer experiment where two co-planar planets of minimal orbital eccentricitys, shows these changes in the semi-major axis dimension of both, where during a million orbits, significant changes will have occurred.
This then provides an explanation for WHY the four large Galilean Moons of Jupiter have orbits which are very close to being in the ratio of 1 : 2 : 4 : 8 regarding their orbital periods, BUT they are not EXACT in those proportions but are necessarily slightly different for the meta-stability to be possible.
It also explains the Long Inequality of Jupiter and Saturn, the Kirkwood Gaps in the Asteroids due to Jupiter, very distinct gaps in the rings of Saturn, and many other orbital relationships in the Solar System, possibly even including Titius-Bode's Law regarding the orbital periods of all the planets!
There are astounding implications for Nuclear Physics as well. In the 1930s, Physicists saw that electrons only appear to revolve around nuclei in very specific orbits. THAT was the basis for the development of Quantum Dynamics, to provide an explanation for that apparent "graininess" of many small phenomena. But this new concept indicates that if an electron is thrown into an atom with ANY initial orbit, within a few million orbits (maybe a trillionth of a second, too short for us to ever detect or be aware of), the electrons could MUTUALLY PERTURB each other into those orbitals that Nuclear Physicists always see! (I refer to this as that we have "slow eyes" to not be able to see the gradual perturbations of the orbits).
The point here is that, possibly, there is a far better explanation for countless nuclear processes, of actually using Classical Mechanics (but with far smaller time graininess than we could ever detect) instead of using the assumptions on which Quantum Dynamics is based. There seems a valid chance that much of Nuclear Physics may need to be re-examined to become more correct.
Newton, Joule and others established several Laws of Nature, including Newton's Laws of Motion, Conservation of Energy, Conservation of Momentum and Conservation of Angular Momentum. These laws have been used extensively to develop many later aspects of science, particularly in Astronomy and Astrophysics.
Specifically, regarding Perturbations of planets or other objects by other planets, it has always been assumed that the orbital radius (called the semi-major axis) cannot be affected by perturbations of other planets. The reasoning always seemed sound. If both Conservation of Energy and Conservation of Angular Momentum apply, then the semi-major axes could not change. If the TOTAL (kinetic) energy of the two objects remained the same (one becoming greater and the other less, in the exact same amount), then the TOTAL Angular Momentum of the two could not remain the same, if their orbital radii had been altered. The reason is that the kinetic energy is proportional to the SQUARE of the velocities in orbit, while the angular momentum is proportional to the velocities themselves. Kepler's work showed us that the velocity has to change with the distance from the central body, which always seemed to mean that Perturbations might affect other Orbital Parameters but could never affect the actual average distance from the central body. This is a universally accepted conclusion among Astrophysicists today.
It is incorrect!
But only in a very specific and very peculiar way and the effects could only arise very slowly, over very long time periods, longer than is ever considered by any existing perturbation theories. In practical situations, this effect is never seen, as Conservation of Angular Momentum is seen as being valid to within measurable amounts.
Those statements ARE true, if only one plane of motion is considered . However, all of those brilliant people ASSUMED a situation which neglected a central result of the process of gyroscopic precession.
The most obvious way of first presenting this is with a high-quality child's gyroscope. Consider one where all the support bearings are perfect, that is, there is no friction whatever, and it is operated in a total vacuum, where there is no air friction, such that the gyroscope rotor will spin forever and never slow down. Placed on the usual pedestal in a axle-horizontal position, we all know that the gyroscope will do two unusual things: it "hangs there", apparently defying gravity, and it also precesses (slowly revolves) around the pedestal. But we note here the important fact that the gyroscope does not START OUT precessing! The current question is now related to the issue that, " when the gyroscope is released, it necessarily ACCELERATES up to the final precessional rate. So what is the source of that energy that is used up in that acceleration?"The gyroscope starts out with NO angular momentum around the precessional (Z) axis, but quickly develops a NEW angular momentum due to the precession.
According to the conventional description, this is a clear violation of the Conservation of Angular Momentum! The rotor did not slow down, so that was not the source of any angular momentum.
These are the Euler Equations, the expression of Newton's Laws of Motion as Differential Equations for motion in three dimensions. As usually interpreted for a child's gyroscope, the first Equation considers the motion about the gyro spin axis (which is horizontal in the simplest case), in other words, bearing friction and air resistance and any motors that might affect the rate of spin of the gyro rotor. In this case, there are no changes and this equation is 0 = 0.
The second Equation considers the motion about the "2" axis (which is also horizontal but normal to the 1 axis), in this case the effect of the gyro falling due to gravity, and therefore attempting for the gyroscope body to rotate around the support point at the top of the pedestal. The third Equation considers the motion about the "3" (vertical) axis, that is the precessing motion of the whole gyroscope body about the vertical axis (also around the support point at the top of the pedestal).
We need to now look carefully at the second and third Equations, which will be
seen to be inter-related. The third Euler Equation,
for this horizontal gyroscope, is:
M3 = I3 * (d ω3/dt) + (I2 - I1) * ω2 * ω1
We can first look at the situation AFTER the precessional motion has fully developed. This is the equation that describes the dynamics of the motion around axis 3, the precession. There is no external Moment applied (around the 3 axis), so M3 = 0 . The other two terms must therefore always add to zero . In other words, once the precession is at its correct rate, this equation is 0 = 0.
Now we can look at the situation as the gyroscope is first released, where there is initially zero precessional velocity. A precessional angular acceleration is therefore required. The M3 term on the left is the EXTERNALLY APPLIED Moment (torque) which is zero for this situation, which is still always zero. The first term on the right involves the angular acceleration of the precession (d ω3/dt) which is what we need to determine. The second term includes three terms that cannot change and one which could ( ω2) . Both of these potentially variable terms therefore become non-zero for a brief period, immediately after the gyroscope is released . As the precession accelerates (around the "3" axis), the gyroscope slightly lowers (around the "2" axis) . In the case of a toy gyroscope, this all usually occurs in a fraction of a second, and the distance the body of the gyroscope drops is extremely small.
The SOLUTION to the long-standing error of assumption is seen if we use the set of Euler Equations but Integrate them. The directions of the (acceleration) vectors are similar, defined by the standard Vector Calculus procedures. We can then see that a (downward, gravitational) ACCELERATION of the axis-2 "dropping" of the gyroscope body (an acceleration vector along the 2-axis) causes an ACCELERATION in the axis-3 precessional motion. Once it has given the appropriate precessional velocity, the effect then works in the opposite direction to STOP accelerating the precessional speed and also stops the downward acceleration of the body of the gyroscope.
The usual Right-Hand-Rule applies, which establishes which way the precessional motion will accelerate, and therefore which direction the gyroscope will precess.
This is NOT instantaneous, but both these accelerations proceed in a sine-wave curve . This insight now allows calculating HOW LONG it takes to have the precession accelerate up to its final speed, and also how much of a downward angle the body of the gyroscope "falls" during that time interval. In a related presentation on Precession, linked below, those calculations are done for a representative toy gyroscope . We show there that less than one one-millionth of a joule of energy is transferred from the one axis to the other, and that amount of energy is provided as the body of the gyroscope falls around one four-thousandth of a millimeter vertically. We also calculate there that the entire process for a toy gyroscope occurs in around one ten-thousandth of a second.
The entire process of the precessional speed RISING from zero to its expected rate, in a smooth process, as well as the lowering of the body of the gyroscope (also in a smooth process), is therefore calculable, where the entire process is unambiguously described by the mathematical differential and, more specifically, second-differential equations.
This effect has apparently been overlooked because all practical-sized gyroscopes seem to achieve their proper precessional speed extremely rapidly, and no one seems to have realized the incredible importance of this effect ! (A toy gyroscope gets up to its proper precessional speed in around 0.0001 second and it drops less than 0.001 millimeter, which makes it seem to be essentially instantaneous and of no noticeable effect other than the new precession!)
We can Integrate both variable terms in either the second or third Euler Equation over the entire time interval of the precession acceleration, and we wind up with terms which represent ω3 (the actual final precessional rate) and M2 (a change of angle of the tilt of the gyro axis).
The Precession page, linked below, provides the calculations for an actual toy gyroscope, and the results indicate that the gyroscope physically drops down a tiny fraction of a degree while the precession accelerates up to speed . (This commonly represents a lowering of the body of the gyroscope by around 0.00026 millimeter, a distance that would be hard to notice and is also even hard to detect! I have confirmed this experimentally.) The precessional kinetic energy which appears in our toy gyroscope is just under one one-millionth of a joule (or newton-meter), which is EXACTLY the same as the amount of potential energy that was released as the gyroscope dropped that tiny fraction of a millimeter, which properly shows the Conservation of Energy.
The result is that there is an angular acceleration of the precessional motion (around the 3-axis), which is due to (vertical, dropping) motion in a different plane (around the 2-axis)! The support angle of the gyro body is very slightly lowered, which gives up some gravitational potential energy, which is then converted into the kinetic energy of the precessional motion. Conservation of Energy is actually exactly maintained. It would not appear to be Conserved if just the precessional motion was examined in just the horizontal plane (or along the "3" axis). There was initially zero kinetic energy of the precessional motion and some kinetic energy would seem to just "appear"!
The significant fact is that this demonstrates a transfer of (potential) Energy from one plane ("2") to another (as kinetic energy) which seems to give the appearance of NOT conserving Energy in the process! It actually DOES Conserve Energy, but it cannot and does not also Conserve Angular Momentum in the process! Before being released, only the rotor is moving, spinning (in the 1-axis), so there is no Angular Momentum along axes "2" or "3". Once released, the Angular Momentum of the rotor is not changed, and after the precession has gotten up to proper speed, there is again no Angular Momentum along the "2" axis, but now there IS Angular Momentum along the "3" axis, in the form of the Angular Momentum associated with the precessional motion.Even though the precessional motion appears to begin without any source of energy, it actually has a source in the potential energy in the vertical axis (in the gravitational field). Conservation of Energy therefore still applies. Energy is a SCALAR quantity, which has no direction or orientation associated with it.
However, Conservation of Angular Momentum is violated, where it is always otherwise true . As the precessional motion begins, angular momentum "appears" (along the "3" axis) where it had not existed before. This is in disagreement with the universal acceptance of Conservation of Angular Momentum in the field of Physics!
Technically, this is NOT really a Violation because there is an external force acting on the gyroscope, which is the Earth's gravitation . In the Solar System, there is also an external force acting on the Perturbation of planets with each other, the Sun's gravitation
This indicates that the long-held assumption that Angular Momentum is always conserved is not really necessarily true when more than one plane of motion is considered, and gyroscopic precession certainly shows that flaw of that reasoning.
It has been assumed by all astronomers and Physicists that planets can perturb several parameters of the orbits of each other, BUT that they could never alter the semi-major radii of each others' orbits . That conclusion WOULD be true IF all the objects in the Solar System orbited in exactly the same Plane. But they certainly do not.
The Solar System objects move in various planes. This fact results in effects that are similar to the non-Conservation of Angular Momentum of the toy gyroscope. Examples are the Earth's Precession, the Regression of the Nodes of the Moon's Orbit (and all other orbits), and any other perturbations where the Z-axis is involved . Planets ARE causing precessional effects in each other . Now that the precessions are all established, no significant violations of Conservation of Angular Momentum SEEM to occur, but whenever each of those precessions CHANGES, that is, they ACCELERATE, they certainly represented clear violations!
For example, the earth has an equatorial bulge that is rotating in a plane where each of the Sun and Moon nearly always are acting to gravitationally try to tilt that plane (trying to stand the Earth more upright), which causes the Precession that the Earth experiences. There seems to be a common misconception that this Precession of the Earth is constant, and we all learned of the 26,000 year time period of the (wobble) Precessional motion in Elementary School. However, that is not even close to being true! TWICE each year, the precessional effect of the Sun on the Earth entirely vanishes, at the instant when the Earth's orbital motion causes the Sun to appear to exactly cross the celestial Equator (around March 21 and September 21 each year). The Precession of the Earth due to the Sun ENTIRELY STOPS on those two days each year! After that, the precessional speed ACCELERATES during the following three months, up to a point where the precessional SPEED is greatest around June 21 and December 21 each year. After that, there is a DECELERATION of the precessional speed during the next three months, to get back down to the zero precessional speed.
Also, consider a "new earth" exactly like ours but not precessing at all. It would (somehow) START to precess, in other words, the Precessional motion of the earth would ACCELERATE up to the rate it is now at. A motion which takes 26,000 years to occur might seem to not involve very much energy, and so it seems that it has always been neglected by Researchers regarding energy considerations. But the Earth is quite massive and since that Precessional motion starts and stops twice every year, there is actually an enormous amount of energy involved.
This represents a good deal of kinetic energy of the Precessional motion, and the Conservation of Energy insists that a source for that energy have provided it . The energy that would supply that motion comes from slight variations in the tilt of the Earth's rotation axis, so Kinetic Energy would be conserved, even with the "precessional acceleration up to the new precession rate". However, Angular Momentum in the Plane of the Ecliptic would NOT be conserved! New Angular Momentum would constantly arise and disappear in that Plane . The AVERAGE of this is the actual observed velocity of the Precessional motion.
In fact, a "new Earth" would be no different than our current Earth, since our precessional motion (due to just the Sun) entirely STOPS twice every year! That significant amount of Kinetic Energy involved in the Earth's precessional motion is CREATED and then CANCELLED OUT twice every year! The processed being discussed therefore involve significant energy transfers! These energy transfers occur because of the gravitational field of the Sun (and Moon).
In fact, since the precessional effects of various solar system bodies on each other are constantly CAUSING ACCELERATIONS AND DECELERATIONS in the precessional speeds, this necessarily indicates that the Earth and other planets are also doing a very slight tilt-axis dance that has always simply been considered a part of Solar Nutation! It is quite a small effect, but experimentally measurable!
The same effect occurs as planets perturb the orbits of other planets and satellites, sometimes also referred to as precession but more commonly called Regression of the Nodes. These effects are commonly presented as though they are constant effects, but they are NOT constant at all! As a perturbing planet varies from being above or below the orbital plane of a perturbed planet, it crosses that orbital plane twice in each synodic period . This causes the precessive effect to constantly be oscillating, from zero effect to a maximum, during the synodic period of the two bodies. This effect occurs for BOTH of the orbital motion and the rotational motion, and affects both bodies involved. This indicates that there must certainly CONSTANTLY be MANY small violations of Conservation of Angular Momentum occurring.
The effect described here is fairly small, and the cumulative effects are very slow . In all practical situations, Conservation of Angular Momentum will be seen to appear true. It is only where Euler's equations transfer energy from one plane to another that any variances with that Conservation can occur. Conservation of Energy appears to still always be true.
If we consider that on March 21 of any year, the Earth has NO Precessive effect due to the Sun, we can easily calculate some things here. We first calculate how much kinetic energy there is in our AVERAGE precession. It is 1/2 * I * ω2 . We know that the rotational inertia (I) of the earth is 8.07 * 1037 kg-meters2. We know that ω is one precessional revolution in 25,800 years or one radian in 1.296 * 1011 seconds . Therefore, the AVERAGE kinetic energy the Earth has in precessing is around 2.4 * 1015 joules . In planetary dynamics, that is not very much, but it still is kinetic energy that did not used to exist!
The AVERAGE kinetic energy of the precessional motion is that amount . However, we know that around Mar 21 and Sep 21 each year (considering only the Sun's contribution) that amount is briefly zero, and arould Jun 21 and Dec 21, it is much greater than that average amount. Actually, since the Precessional MOTION is twice as fast at those instants, there was FOUR TIMES AS MUCH kinetic energy transferred. Note that this means that around 10 quadrillion joules of energy is ADDED to the Earth's (Solar driven) Precessional motion in a three-month interval, and then the same amount is REMOVED from the Earth's precessional motion in the following three months! This is a significant transfer of energy into and out of that motion, on a very regular basis! The Moment (torque) of this constantly fluctuating amount of Precessional Kinetic Energy is related to a slight axis tilt change of the Earth's spin axis . The double Integration of the Euler Equations shows that the energy involved is always Conserved, but that it is simply transferred back and forth between a slight fluctuating tilt of the Earth's rotation axis and the varying Precessional speeds.
Ten quadrillion joules might sound like a lot, but since it is spread out over a three-month interval, that is about an average of 1,250,000 kiloWatts (because a watt is a joule/second) . That might not be worth the bother regarding trying to build any equipment to try to capture it! But it gets WAY better!
In addition, similar calculations show that around 40 quadrillion joules of energy is ADDED TO the Earth's (Lunar driven) precession in about a WEEK, and then the same amount is removed during the following week! Many people have noticed and measured the very small-scale wobbling that the Earth does (collectively called Nutation) but I have never seen that anyone has realized that it was actually (primarily) due to a side effect of the constantly varying Precessional effect.
It might be noted that the energy transfer due to the Moon's effect here is relatively significant from a human perspective! There is about 40 quadrillion joules transferred in a period of about one week (or 637,000 seconds) which means that an average power transfer of that quotient is occurring, or about 63 billion joules/second or 63 billion watts or 63 million KiloWatts! That 63,000 Megawatts is comparable to the entire output of electric power from ALL US nuclear generating plants! But I do not see how it could ever be captured by anything that we humans could ever do! Maybe some human far smarter than me can see some way to capture that energy, and we would then have an enormous supply of power, essentially forever!
The puny little Moon causes this effect around 50 times greater than the enormously massive Sun does! Interesting!
More significantly, each time when the Precession Effect is non-zero, there is Angular Momentum that did not used to exist ! This fact means that one of the two pre-conditions that Laplace, LaGrange and everyone else have always applied is often (slightly) invalid. In short-term motions or perturbations, these effects are not seen, as they are comparatively small amounts of energy and angular momentum involved, and Angular Momentum appears to be conserved. However, over very long periods of time, these effects of continuously modifying small amounts of "new" angular momentum ALLOWS planets to mutually alter their semi-major axes ! (Which is currently assumed impossible.) This then allows some very slow perturbation effects that are so small that they have not been yet detected (or understood). However, they certainly occur, because there is extensive evidence of near-commensurability in orbits of planets, satellites, asteroids, ring particles, and more. These are therefore not mere coincidences, but the very long-term effects of this new category of mutual perturbations where the semi-major axes are altered . Again, the Hamiltonian remains true, and Energy is Conserved, but slight changes in Angular Momentum certainly occur.
We have known for thousands of years that the Moon causes and creates spectacular amounts of energy that exist in the Oceans' Tides (every day). We know that this is due to some gravitational effects of the Moon on the Earth and on the waters of the oceans . Here, we are discussing a DIFFERENT gravitational effect of the Moon on the Earth, which is actually due to the Earth having an Equatorial Bulge so that a Precessional motion can be caused on the Earth by the Moon. In other words, IF we could just figure out HOW to collect some of the precessional effects of the Moon on the Earth, we might easily collect enough energy to power much of modern civilization ! This presentation is a discussion of one (of many) possible approaches we might try. If done effectively and efficiently, we probably could harvest spectacular amounts of energy which exists due to the Moon's existence orbiting our planet.
This presentation was first placed on the Internet in Sept 2006.
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C Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago