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. 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 peculiar way and the effects could only arise very slowly, over very long time periods, longer than is ever considered by any perturbation theories. In practical situations, this effect is never seen as Conservation of Angular Momentum is valid to within measurable amounts.
Those statements ARE true, if only one plane of motion is considered. However, Euler expanded Newton's equations of motion for three dimensions, and a new possibility then arises.
The most obvious way of first presenting this is with a high-quality child's gyroscope. Consider one where the support bearings are perfect, that is, there is no friction whatever, and it is operated in a total vacuum, 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 around the pedestal. But we note here the important fact that the gyro does not START OUT precessing! The 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 axis, but quickly develops the 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, 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, in this case the effect of the gyro falling due to gravity. The third Equation considers the motion about the "3" axis, that is the precessing motion of the whole gyro about the vertical axis.
We need to now look 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 are again using
the letter d for the differential symbol.)
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, 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. 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 terms therefore become non-zero for a brief period, immediately after the gyroscope is released. As the precession accelerates (in the "3" axis), the gyroscope slightly lowers (in the "2" axis). In the case of a toy gyroscope, this all usually occurs in a fraction of a second.
If we Integrate both terms over the time interval of the precession acceleration, we wind up with terms which include w3 (the actual final precessional rate) and q2 (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 drops down a tiny fraction of a degree while the precession accelerates up to speed. (This represents around 0.03 millimeter, a distance that would be hard to notice!) The precessional kinetic energy which appears in our toy gyroscope is about one one-millionth of a newton, [roughly] the same as the amount of potential energy that was released as the gyroscope dropped that tiny fraction of an inch, which Conserves Energy.
The result is that there is an angular acceleration of the precessional motion, which is due to (vertical, dropping) motion in a different plane! 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 maintained, but it would not appear to be Conserved when 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) gives the appearance of NOT conserving Energy in the process! It actually DOES Conserve Energy, but it cannot also Conserve Angular Momentum in the process! Before being released, only the rotor is moving, spinning, so there is no Angular Momentum along axes "2" or "3". Once released, the Angular Momentum of the rotor is not changed, and 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 becomes violated, where it is always true otherwise. 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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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 when each of those precessions first began, they certainly represented clear violations!
For example, the earth has an equatorial bulge that is rotating in a plane where the Sun and Moon nearly always act to gravitationally try to tilt that plane (trying to stand the Earth more upright), which causes the Precession that the Earth experiences. Consider a "new earth" exactly like ours but not precessing. It would START to precess, in other words, the Precessional motion of the earth would ACCELERATE up to the rate it is now at. The energy that would supply that motion would come from a slight Z-axis (Solar-System-vertical) relative movement of the Sun/Earth and Moon/Earth (or other perturbing body), 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.
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, also are not precisely constant, and so there must certainly CONSTANTLY be very small violations of Conservation of Angular Momentum.
The effect described here is very small, and 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.
For that "new Earth" that is initially not precessing, we can easily calculate how much kinetic energy there is in our precession. It is 1/2 * I * w2. We know that the rotational inertia (I) of the earth is 8.07 * 1037 kg-meters2. We know that w is one precessional revolution in 25,800 years or one radian in 1.296 * 1011 seconds. Therefore, the 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!
More significantly, there is now Angular Momentum that did not used to exist. This fact means that one of the two pre-conditions that Laplace, LaGrange and everyone else has always applied can sometimes be invalid. In short-term motions or perturbations, these effects are not seen, and Angular Momentum appears to be conserved. However, over very long periods of time, these effects of continuously adding small amounts of "new" angular momentum ALLOWS planets to mutually alter their semi-major axes! (Currently considered impossible.) This then allows some very slow perturbation effects that are so small that they have not been yet detected. However, they certainly occur, because there is extensive evidence of near-commensurability in orbits of planets, satellites, asteroids, ring particles, and more. These are 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.
( http://mb-soft.com/public/index.html )
C Johnson, Physicist, Physics Degree from Univ of Chicago