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When YOU release a toy gyroscope on its pedestal, it has NO precessional motion around that pedestal. An instant later, it is merrily precessing around the pedestal! Both Kinetic Energy and Angular Momentum now exist where they did not 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. But the Angular Momentum of the Precession which just appeared does not come from anywhere!
The Earth is an even more impressive example of this same effect. In school, you were taught that the Earth "wobbles" (precesses) in about 26,000 years. Even your Teacher did not know that on every March 21 and September 21, that Precession STOPS COMPLETELY. And it becomes about twice the average rate around June 21 and December 21. The Earth is coming up with massive amounts of Angular Momentum, ALL THE TIME, where there is NO SOURCE FOR IT TO HAVE COME FROM!
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. Given those two conditions, it is fairly easy to 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.
In general, that reasoning and that conclusion is true. However, this presentation shows that there IS one process which violates the Conservation of Angular Momentum, which then PERMITS planets to (very slowly) alter each other's orbital radii.
They and others must certainly have seen toy gyroscopes that APPEAR to have very constant Precession, and made that incorrect assumption.
They had overlooked an obvious fact that when a toy gyroscope is FIRST released, its Precessional motion MUST ACCELERATE from the intial 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 the Violation can and does occur.
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). 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 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, and the amounts get returned to the Sun in 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 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 AVERAGE Precession rate of the Earth's wobble is as we all learned in school, but it ACTUALLY is constantly 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. It 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 that are 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.
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!
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.
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. |
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 * (dw3/dt) + (I2
- I1) * w2 *
w1
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 (dw3/dt) which is what we need to determine. The second term includes three terms that cannot change and one which could (w2). 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, devined 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!)
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.
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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. 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!
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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 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). 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. 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 would come 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 arise in that Plane. THIS is the actual 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!
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 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.
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!
A Campus where Builders could learn to Build 14 different Houses using these methods.
C Johnson, Physicist, Physics Degree from Univ of Chicago