return to homepage
Part 1
Celestial mechanics has not made much progress since Kepler and Newton. Even General Relativity only recast the old concepts in new but basically equivalent terms. Einstein did not overthrow the fundamental mathematics of gravity and orbits. The old conceptualizations and equations still stand; they are still taught in schools everywhere. General Relativity only fine tunes them, by substituting a different but basically equivalent theory (curved space for action at a distance) and a nearly equivalent mathematics (tensor calculus for calculus). Einstein never implies that Kepler and Newton's theories were wrong—they are only incomplete. Import the finite speed of light and the tensor calculus into classical theory and you have current wisdom with regard to celestial mechanics.
There are many other similar mysteries about the stability of orbits, but I think I have made my point in regard to the circular orbit. Let us now graduate from the mysteries of the circular orbit to the mysteries of the elliptical orbit. As you know, Kepler told us that all orbits are ellipses, the nearly circular orbit being only a special case. Does an elliptical orbit solve any of the problems I have outlined above? Is it easier to explain the creation of orbits and the stability of orbits? No. Kepler does not address any of the things I have mentioned above. No one addresses these problems. Neither Kepler nor Newton nor Einstein nor anyone else has tried to build a necessary connection between the tangential velocity and the centripetal acceleration, not with elliptical orbits or any other orbits. Kepler's second law states that a planet sweeps out equal areas in equal times. This is achieved by varying the orbital velocity of the planet, obviously. In this case there is no tangential velocity, as least not as there was in the circular example, since only two tangents will be perpendicular to the line from the sun (perihelion and aphelion). So it is unclear where the initial velocity of the planet, before it was captured, has gone. Is it still a constant piece of the compositional velocity, or has it been lost? How can we explain the formation of the ellipse and its stability? Everyone interrupts here to scream at me that the orbiter will be going slower at aphelion. But everyone is so far missing my point completely. Yes, the orbital velocity will be slower at aphelion, but please concentrate for a moment on the curvature. If you are theoretically limited to two vectors like this, then if you slow the orbiter, you necessarily flatten out the curve. You can't draw an ellipse like this with two vectors. It's a magic trick. People have always accepted this diagram on faith, but it is a false diagram. You simply cannot draw the same amount of curvature at aphelion and perihelion and then claim that it is caused by a variable orbital velocity. An orbiter may sweep out equal areas in equal times, but not if its orbital velocity is determined by two vectors. It is a kinematic impossibility. It is impossible since the perpendicular vector has to stay the same length all the way around. The "innate motion" of the orbiter is a constant. It cannot vary. Most of those screaming at me scream that my two velocity vectors can't be equal, since the orbital velocity varies in an ellipse. But I repeat, those two velocities are not orbital velocities. They are drawn as and labelled as perpendicular or tangential velocities. They are a component of the orbital velocity but are not equivalent to it. The orbital velocity is a vector additon of the perpendicular velocity and the centripetal acceleration. The orbital velocity varies; the perpendicular velocity cannot, sinced the perpendicular velocity expresses Newton's "innate motion". This means the only primary vector you can vary is the acceleration vector. In any gravitational field, that is the only non-compound vector that can be varying, without cheating in some way. Look closely at the diagram above. If you vary only the length of the acceleration vector, in the vector addition, then you must vary the curvature. The orbiter is going more slowly at aphelion, and this slower orbital velocity is due to the smaller acceleration vector, and only to the smaller acceleration. But if this is true, then the orbiter can't be describing the curve that is drawn by the ellipse. An orbiter with a given "innate motion" and a larger acceleration cannot possibly be describing the same curve as that same orbiter with the same innate motion and a smaller acceleration. If we rigorously study the variable assignments of Kepler and Newton, what we find is this shape, not the ellipse: You see, the curvature cannot be the same on both sides if the innate motion or tangential velocity is a constant. This problem has been buried ever since Newton used his new calculus to find the orbital or curved velocity given the tangential velocity. In The Principia, he is given the tangential velocity or straight-line motion, and he derives the orbital or curved motion from it, using his ultimate interval (like a limit). The velocity variable in a=v^{2}/r must then be this new orbital velocity. So the old tangential velocity is lost. It has been buried from sight ever since. But in the ellipse, or any real orbit, we must continue to monitor the old tangential velocity, since we cannot allow it to vary without giving a mechanical explanation of that variation. If we see it varying in the ellipse, as I have shown, then we must ask how a planet can vary its innate motion to suit an orbit. How can either the planet itself, or the gravitational field, cause that velocity to vary? The planet cannot, because it is not self-propelled or self-correcting. The gravitational field cannot, because the gravitational field has no mechanism to influence that vector. Even Einstein admitted that the gravitational field had no influence at the tangent. This is not to say that ellipses don't exist. We can see that they do. But they must exist with the help of some field we have not included in our equations. Here's another problem. Say you want to recreate the ellipse with your ball and your string. You want to build a real mechanical ellipse with real forces, at a human scale. Well, you can't do it with string. But you can kind of do it with a rubber band. This will allow you to vary the centripetal force, to mimic the gravitational field of an ellipse. And yes, you can create the ellipse (sort of), and you can sweep out equal areas in equal times, and the orbital velocity is greater at perigee than at apogee. Everything looks great until you notice how your "gravitational field" is varying. The force on your rubber band is a lot greater at apogee than at perigee. What you have is a gravitational field inside out. What happened? What happened is that Kepler's ellipse is a myth. It can't be built in the real world, by unpropelled planets in a Newtonian orbit. The reason your ball on your rubber band exhibits an ellipse is because you are able to vary its orbital velocity from the focus. You do this by varying its perpendicular velocity at perigee and apogee. Try it and see. You will find that it is easiest to give the ball a boost between perigee and apogee, since at that point a sideways nudge is felt without throwing the ellipse out of round (mostly). But a planet cannot vary its perpendicular velocity. It has its initial velocity, and that is all. As further proof, go back to the paragraph where I am trying to build the ellipse. I start the planet at aphelion. The planet is then pulled into a tighter orbit, since its velocity is not great enough to achieve a circular orbit. I then say that the trajectory of the planet finally takes it below what its circular orbit would have been, giving it a sort of escape velocity. I imply that once it passes perihelion, its velocity allows it to begin increasing its orbital distance again. Well, this is not really true. It seems logical, and the textbooks always imply this, so it is easy to accept. But upon closer examination, it all begins to fall apart. What we imagine when we accept the ellipse as a logical-looking orbit is that it is simply a sort of squashed circular orbit. We think, well, maybe when a planet is captured, it first hits an orbital tangent at an angle, instead of at a perfect perpendicular. This throws its orbit a bit out of whack, but the orbit is somehow stable since the total area of the orbit is about the same. All very unscientific, but I would guess that many of us have assumed these things, without really questioning it very deeply. But, let's build that ellipse again, starting from aphelion. Let us draw the whole thing, just accepting that an ellipse must somehow be created, since we have evidence of them in the solar system. Finally, let us look for the "equivalent" circular orbit. Meaning that if we have the same planet with the same initial velocity and we want to put it into a circular orbit, where do we put it? Turns out that the circle is completely outside the ellipse, and that it has a lot greater area. Remember that the only way we can explain the planet in ellipse beginning to dive toward the sun as we move it past aphelion is that its velocity is not great enough to keep it in circular orbit. Therefore, to put it into a stable circular orbit, we must move it further away from the sun at aphelion. If we do that then aphelion becomes the radius of the circle, and we have our circular orbit. As you can see from the illustration, the path of the ellipse never crosses the path of the "equivalent" circle. If that is true, then the planet in ellipse can never reach a point where its perpendicular velocity overcomes the centripetal acceleration produced by the gravitational field. It never achieves a temporary escape velocity. No, it simply spirals into the sun. Its orbital velocity increases, yes. The "orbital velocity" continues to increase until the planet burns up in the sun's corona. As another argument, consider the standard example of the creation of an ellipse. Richard Feynman uses this example in his geometric “proof” of the elliptical orbit. Take two thumbtacks and put them some distance apart in a piece of paper. Tie one end of a piece of string to one thumbtack and the other end to the other, leaving the string with a lot of play. Now, take the point of a pencil and pull the string tight with it, making a triangle with the pencil tip and the two tacks. Pull the pencil to the left, drawing a line, at the same time keeping the string taut. Keep going as far as you can to the left and then go back as far as you can to the right. You will have half an ellipse. You can draw the other half by lifting the string over the tacks and continuing. The thumbtacks act as the two foci. Clearly, this works because the pencil tip is feeling forces from both foci. But in a planetary orbit, the planet can feel a force only from the sun, at one focus. This is why the ellipse cannot work for such an orbit, taking the forces and velocities as they are now understood. In the thumbtack example, the pencil is feeling the same forces at both perihelions (closest to each focus). A planet would be feeling different forces at those two points. An ellipse is simply not a potential orbit for the balancing of a tangential velocity and a single centripetal acceleration. You will say, but what about comets? We can see them. They have elliptical orbits. How do you explain that? I am not saying that elliptical orbits are impossible, I am saying that they are impossible to explain with current celestial mechanics. Elliptical orbits cannot be explained with current gravitational theory, not Kepler's, not Newton's, not Einstein's. In addition, as I have shown, stable circular orbits with moons are also impossible to explain. They should not work, since there is no reason for them to show the correctability they do show. Here is another argument against current theory. Consider Kepler's Third Law. It states that the ratio of the squares of the periods of all potential orbits are equal to the ratio of the cubes of their average distances. This law is still accepted. Einstein accepted it. It is in all current textbooks. Furthermore, it is confirmed by the most exacting modern measurements. To within a small fraction of error, the ratio r ^{3}/t^{2} for the nine planets is 3.34 x 10^{24}km^{3}/yr^{2}. What this means, of course, is that the orbit of the planet has nothing to do with the mass of the planet. According to Kepler's law, one must balance only the distance and the period. To see what I mean, take the Earth out to the distance of Jupiter and try to build an orbit. Could you do it? Of course. You just slow the Earth's orbital velocity down until it offsets the centripetal force from the sun. What you find is that the Earth will match the orbital velocity of Jupiter exactly. Somewhat surprising, isn't it? I assume that some readers will have thought that the Earth would be going slower, since it is smaller. It feels a smaller force from the sun, therefore it has less centripetal acceleration to offset with its velocity. But that is not how a gravitational field works. Yes, the force is different, but the acceleration is the same. F = ma. That is why all objects fall at the same rate in a vacuum, remember? Jupiter and the Earth fall toward the sun at the same rate—that is, the same acceleration—if they are at the same distance. You will say, "But the sun must pull harder on Jupiter, surely, to keep it in orbit, than on the Earth." Yes, surely. And that is my point. A gravitational field is a strange creature, and its characteristics have never been explained. They have been described, in several different ways, by Newton, Einstein, etc., but never explained. The gravitational field is not a force field, it is an acceleration field. When Newton or Einstein maps the varying numbers at varying distances in the field, he is mapping accelerations, not forces. Very mysterious, that. Notice that acceleration is a measurement of rate of change of motion. Acceleration is not directly a measurement of force. Movement, not force. That is very important.Nor has this problem been solved by General Relativity. More money is now being spent worldwide on finding the graviton that on any other scientific project. Billions, literally. It will not be found, but a good question to ask those who seek it is this: Would the sun need to send out bigger or more powerful gravitons to Jupiter than to the Earth, if they were both at the same orbital distance? If so, how does the sun know which to send? Perhaps we need to look for a messenger particle, one that precedes the graviton, and asks the orbiting object how much it weighs. I know that this all sounds like a joke, but the question must be addressed seriously by those who put "no action at a distance" on their t-shirts. The status quo in physics, made up of the biggest names in the field in the 20th century, still brags about this in the latest books. But their theories explain absolutely nothing. You may be asking yourself at this point, how has all this sloppiness stayed buried for so long? Stephen Hawking told us just twelve years ago that we were a decade away from knowing everything. The end of physics. Except for chasing the graviton, no one is even working on gravity anymore. It is a problem that is considered solved. The "great minds" are busy with superstring theory, and things like that. Tying gravity to quantum mechanics. But here I am saying that no theoretical progress has been made since Newton. How can that be? One word. Obstruction. The obstruction began with Newton himself. Newton derived Kepler's law from his own, to show that the two were consistent. He did it like this, roughly. Given that f = ma and that F = Gm _{1}m_{2}/r^{2} (Newton's famous equations, of course) Let f = F Gm _{1}m_{2}/r^{2} = m_{1}a Then let a = v ^{2}/r Gm _{1}m_{2}/r^{2} = m_{1}v^{2}/r Next, since all the orbits of the planets are nearly circular, let the distance travelled in each orbit equal the circumference of the orbit: v = 2πr/t Gm _{1}m_{2}/r^{2} = m_{1}[2πr/t] 2/r Gm _{2} /r^{3} = 4π^{2}/t^{2} t ^{2}/r^{3} = 4π^{2}/Gm_{2} Since the right side is a constant for all planets around the sun, the left side applies to all the planets, and all possible planets. This derivation is problematic not because we let the orbit be circular. That was only to simplify the math. The problem is in letting a = v ^{2}/r. As I showed above, this equation is applicable only when a is dependent upon v. If Newton or current textbooks want to use that equation, then they must explain how a is dependent upon v. Newton is implying that there is a necessary causal connection between the two, without providing us with a means of causation. For, I repeat, how can a gravitational field cause a velocity tangent to that field? Or, to make the analogy even tighter, how can the tangential velocity determine the field strength? That is what is happening with the ball on a string. Increased velocity causes a greater force on the string. Kepler's Third Law tells us unequivocally that a and v are dependent, but neither Newton nor Einstein nor anyone else can say how that dependence is arrived at. Newton ties his equations to Kepler's law by a kind of cheat. He slips an equation into his derivation that contains a gigantic theoretical leap, but then does nothing to support that leap. He hoped no one would notice, and apparently no one has for about 300 years. But the fact remains that there is no theoretical justification that has ever been offered for this leap. The theory of the gravitational field, either Newton's or Einstein's, cannot support Kepler's Third Law. People don't get famous and stay famous by putting up theories that have big obvious holes in them. So the smartest people learn to plaster up the holes and offer the theories as airtight. The very smartest people are just as good at plastering up holes, and painting over them, as they are at devising theories. Newton was one of the very smartest people. His greatest theories are full of chalk and mortar, and part of the greatness of the theories is how well the mortar has held over the centuries. But it is not just that. Subsequent scientists, unless they can devise a superior theory (which is obviously not so easy), prefer to let the mortar stand, even when it begins to show. They may even repaint over it themselves. They do this to maintain the prestige of the field. The history of physics is a history of geniuses. We all know that. And, since geniuses get paid better and make better copy, it is best to keep the field properly propped up. If Newton or Einstein is made to look foolish, we all look foolish, and our checks from the government vanish. Hobbyists stop reading about us with stars in their eyes, and Hollywood sticks to stories about old generals and ship captains and artists. To show how contemporary physicists have painted over Newton's crumbling mortar, one need look no further than the derivation of Schwarzchild's radius and the gravity wave. Both use the same unsubstantiated assumption Newton used, namely that a = v ^{2}/r. For a famous and typical example of the derivation of the gravity wave, I refer you to Appendix V of Peter Bergmann's The Riddle of Gravitation. [Forgive me for reading these things: I know we aren't supposed to check math in appendices, but I am just strange that way.] He says that Schwarzchild's radius is R = 2GM/c^{2}. If a = GM/r ^{2}, and a = v^{2}/r, then v ^{2}/r = GM/r^{2} v ^{2} = GM/r GM = Rc ^{2}/2 rv ^{2} = Rc^{2}/2 or 2v ^{2}/c^{2} = R/r This last equation is used to find the intensity of a gravity wave I ~ c ^{4}R^{5}/Gr^{7}The intermediate steps are not important, since they are all bombast anyway, but just note that the equation a = v ^{2}/r is still there big as an elephant and twice as invisible. It doesn't matter to anyone, even in the 20th century, that there is no reason why that acceleration should be dependent on that velocity. It was in everyone's high school textbook; why question it now. You may answer that a = v ^{2}/r is a necessary condition of a circular orbit. It is not a matter of "dependence," as I call it. It is a mathematical necessity. Any stable orbit, whether caused by the balancing of force and velocity, as in the earth/sun example, or by a combined velocity/force, as in the ball-on-a-string example, is defined by that equation. Maybe, but that is heuristics, not theory. Using a naked equation, without any theoretical underpinning, is dangerous. The fact ends up becoming the theory, and we have forgotten that we have anything to explain. We have forgotten that the orbit of the earth is problematical, since is does not work like the ball on the string. Another reason an equation unsupported by theory is dangerous is that it becomes dogma. It sits there, in all the same regalia as a supported equation, and we salute it in the same way throughout the centuries. It takes on the solidity of a fact, when it is not. And I am not just making airy accusations. I can show you that a = v ^{2}/r is not correct, even as it stands. It has sat on the pages of our books unquestioned since Newton. But it is false. [See my paper A Correction to a = v] ^{2}/rNext let us look at Nebular Theory. This is the theory of how the solar system was created. Kant and Laplace were the first to propose nebular theories to explain the orbits of the planets. This was about a half century after Newton. In short, according to this theory the solar system was created by the gravitational collapse of a cloud of gases. Those who took exception to my comments about the earth being “captured” by the sun above no doubt had the nebular theory in mind, since in it no capturing of planets is necessary. The planets and the sun form at the same time. First a solar disc is created, then this disc accretes into planets over millions of years. This nebular theory is still ascendant, with a few updates that I will mention in a moment. However, I would like to point out right now that Kant is known to history mainly as a philosopher, not as a scientist or mathematician. And yet, like Kepler, his theory still stands today as the basis for all modern nebular theory. This despite all we have been told by modern scientists about philosophers being inferior creatures, ones who should not dabble in science. It is perhaps just as well for scientists like Feynman that Kant predated the chance to take their advice. For about a hundred and fifty years the nebular theory was attacked mainly on the ground that it did not explain the lack of large angular momentum at the center of the collapse. The current solar system has larger angular momentum in the outer planets than it does at the center. That is, the sun has very little angular momentum itself. In a gravitational collapse of a nebula with some initial angular momentum, the center of the nebula would be expected to gain the most angular momentum. Many many competing theories were put forward in the 19th and early 20th centuries to explain the sun’s lack of angular momentum, but none were successful. As one recent textbook put it, they were “catastrophically” wrong. Not one was able to improve on the nebular theory. And so in recent times scientists have made their peace with the nebular theory. It has only one major flaw, whereas all the others had more than one. Modern science has created a theory of dissipation to explain the loss of angular momentum by the sun. Several mechanisms for dissipation have been presented, although none have been very successful at explaining it. To make up for this, current theory now assumes that all the theorized mechanisms are at work, and that the actual dissipation has been achieved by their combination over long periods. What exact combination is unknown at the present time. One of these mechanisms is the hypothesized separation of angular momentum by a two-element gas. Since nebular gases are known to be made up of hydrogen and helium (and traces of other gases), the differing forces upon these molecules during gravitational collapse is thought to have allowed angular momentum to dissipate outward. Even according to its proponents, this mechanism is not capable of explaining the actual dissipation that we see. But even more damaging to the theory is the fact that it would require that we now find a much greater concentration of hydrogen in the sun and a greater concentration of helium in the outer planets than is the case. Even if all the helium in the sun is assumed to have been created by fusion since the time of the formation of the solar system, this does not explain the ratio of hydrogen to helium in Jupiter or Saturn (the ratio is now assumed to be about that of the sun). Even if we assumed that all the helium in the pre-collapse nebula were initially positioned at the center of the cloud, and that it all was transported to the outer planets during system formation, even then that small amount of helium could not have carried all the angular momentum away from the sun. If, on the other hand, you assume that the helium would have been transported toward the center, displacing hydrogen to the outer planets, this also fails to account for the ratio of hydrogen to helium in Jupiter and the sun. Jupiter would be expected to have almost no helium and the sun would be expected to have a lot. The sun would have the helium from transport plus the helium from fusion. But the sun and Jupiter are thought to have about equal amounts of helium. The other mechanisms for dissipation—magnetic and non-magnetic turbulence—are equally tenuous. As is so-called long wavelength spiral structure dissipation. They all take on the form of ad hoc theories, and look very much like signs of desperation. Especially in light of my theory, which will show that dissipation is a complete myth. I will prove that the angular momentum of the sun and planets has nothing whatsoever to do with nebular collapse and that there is no nebular connection at all between the angular momentum of the sun and of Jupiter. As a lead into this, I want to first outline some of the other flaws in nebular theory. For a proto-star or solar disc or collapsing nebula to have angular momentum after the gas begins its gravitational collapse, it must have angular momentum before this collapse. In fact, this is already assumed, since we are told that interstellar nebula do in fact have angular momenta. This has been established empirically by Doppler shift measurements on different parts of the same nebula. But nebular theory never answers the first question—that being how could a pre-collapse nebula have angular momentum? It is like saying that a gas in a jar has angular momentum. There is absolutely no known force that could cause it. A gas is either in a state where thermal forces are ascendant or where gravitational forces are ascendant. It cannot be in a state where both are primary. If thermal forces are at work, then the gas would have no center. Gravitational forces would be trumped by thermal forces, and the nebula would show no signs of gravitational symmetry—that is, movement about a center. Only if the gas were thrown out of thermal equilibrium by some temporary concentration of molecules in a central region could a gravitational collapse begin. In that case, gravitational forces would become the primary structural forces, trumping thermal forces. Therefore, nebulae which show angular momenta must already be in the first stages of collapse. And the question then becomes, how does the cloud go from being a thermal cloud with no center to being a gravitational cloud with angular momentum? The answer is that it doesn’t. A nebula is never just a cloud of gas. A nebula requires the postulate of a center. This center is not created by the gravitational collapse, it causes the gravitational collapse. By that, I mean that the nebula does not create the sun and planets. Something must pre-exist as a center, that then creates the nebula. It has been theorized that a shock wave from a supernova or some other outside force may start the collapse of a nebula, but it is never explained how any outside force could impart angular momentum to a nebula. The shock wave from a supernova could only impart linear momentum to all or part of the nebula, but it could not start the nebula spinning unless the nebula was already arrayed about a center. A nebula therefore requires a seed. This argument soon becomes circular, since what I mean by seed is some massive body. But the nebular theory was proposed initially to explain the creation of massive bodies. So we require a massive body in order to create one. At first this may seem a small problem, since nebular theory explains star creation, and we need only a planet or planetoid to seed our gas cloud. But then we must explain the existence of our planetoid. Maybe it formed by simple accretion—not gravity but random collision. Accretion of what? Grain and fragments? And these grains were formed by accretion as well? If grains and planetoids formed by accretion from gas clouds, we would hardly need gravitational collapse to explain anything. But the fact is that helium and hydrogen gases do not accrete into planetoids. Gases don’t commonly just stick together for the convenience of theorists. To sum up: the problem with nebular theory is that it 1) utterly fails to explain how a gas sets up around a center, 2) fails to explain low angular momentum at the center, as in the case of the sun.
As a final example of the current state of the art in celestial mechanics, let me show you a specific example from If this paper was useful to you in any way, please consider donating a dollar (or more) to the SAVE THE ARTISTS FOUNDATION. This will allow me to continue writing these "unpublishable" things. Don't be confused by paying Melisa Smith--that is just one of my many |