Both theories are mathematical. And the same problem is going to be met when the moon is sideways to the earth. Gravitational forces would be trumped by thermal forces, and the nebula would show no signs of gravitational symmetry—that is, movement about a center. The tangential motion and the centripetal motion must be perfectly balanced or the orbit will deteriorate immediately in one direction or another (inward or outward). How can you postulate the applying of torques with a gravitational field—a field that is absolutely incapable of creating mechanical torques? They should not work, since there is no reason for them to show the correctability they do show. Chaos theory is often said to be deterministic, but this meaning is imprecise at best. Well, the moon is going to pull the earth into a fractionally lower orbit. Finally, let us look for the "equivalent" circular orbit. The same is true of every other sub-field. So the earth must have an initial tangential velocity, and it retains this velocity after it is captured by the sun. Jupiter and the Earth fall toward the sun at the same rate—that is, the same acceleration—if they are at the same distance. For Newton, the mystery was in understanding how the sun influenced the earth, for instance. 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. The history of celestial mechanics is a history of mathematical analysis that is very short on theory. By giving us the ball-on-a-string illustration, the book leaves the impression that the analogy is complete; that is, that the tangential velocity and the acceleration are conceptually connected in both instances. It cannot study the world or even data directly; it must look through the heavy lens of a dense mathematics. This kind of math was not created to be determinate, it was created to be useful and “fruitful”, which it is. But here I am saying that no theoretical progress has been made since Newton. They are going to be missing chunks. This problem is one of the great problems in celestial mechanics. Most of those screaming at me scream that my two velocity vectors can't be equal, since the orbital velocity varies in an ellipse. Given two theories that have the same content—the same power of prediction—always choose the one that has the fewest moving parts, the fewest postulates. Saturn cannot go higher due to a perturbation from Jupiter, unless that perturbation is repulsive at some point in the long cycle. I myself am a new researcher in the field, but I find simple mistakes and basic holes on a daily basis—mistakes that are almost remedial and holes that almost audibly yawn. Most books don’t have a single page on the theory of orbits. How is an orbit like this created? “Celestial Mechanics and Astrodynamics: Theory and Practice” also presents the main challenges and future prospects for the two fields in an elaborate, comprehensive and rigorous manner. Kepler and Newton believed that a gravitational field was produced by a massive object, that space (if not the field) was rectilinear, and that the massive object acted directly—though in an unknown way—upon any matter within the field. This book is composed of 17 chapters, and begins with the concept of elliptic motion and its expansion. So, for the sake of argument, let us say that a planet has arrived at aphelion due to some fortuitous collision. For instance, Kepler's theory of ellipses still pertains to this day. Besides, current math theory is quite simply wrong when it claims that calculus is deterministic. Either the forces balance or they do not. The next two weeks of corrections cannot offset this. There is one perfect distance that creates a stable orbit for a given velocity. Obstruction. He slips an equation into his derivation that contains a gigantic theoretical leap, but then does nothing to support that leap. To see what I mean, take the Earth out to the distance of Jupiter and try to build an orbit. So this motion must be independent of the gravitational field. These cannot be done simultaneously. A moon creates a perturbation that cannot correct itself. Mathematically I am pointing at differentials. My other papers might be called a compilation of these holes. Therefore, nebulae which show angular momenta must already be in the first stages of collapse. Most books don’t have a single page on the theory of orbits. So far so good. If you speak of simple logic you are immediately dismissed as that most dangerous of all scientific demons—a philosopher. How was this possibly achieved, given the current list of forces and causes of forces? Almost no one thinks that the moon’s orbit is caused by the rotation of the earth about its own axis. The problem is in letting a = v2/r. There is no margin of error. I will discuss a selected number of these problems. There is no force at a distance nor any attractive force (including negative charge). The latter is understood to be the problem of calculating the motion of three point-like masses which move under the sole influence of … So you must accelerate the earth out of the orbit. Ultimately I must take exception to Kepler's theory of ellipses. There is no way to eject an object from the center of its future orbit with a velocity tangential to that orbit. It is not a matter of "dependence," as I call it. If we had to read an article that told us the truth—that we know next to nothing about the way things work in the universe, we would not be able to sleep at night. If the differentials are showing a variation, this variation must be explained by an external force. Physicists like Richard Feynman abused both the philosophy departments and the mathematics departments, and physicists cheered him. Or, to be more precise, they have a lot dust around them. 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. At each point on the ellipse, the orbital velocity of the planet is the vector addition of the perpendicular velocity (which is no longer tangential) and the “instantaneous” centripetal velocity. Kepler's laws still hold, Newton's laws still hold. The current solar system has larger angular momentum in the outer planets than it does at the center. To within a small fraction of error, the ratio r3/t2 for the nine planets is 3.34 x 1024km3/yr2. Newton ties his equations to Kepler's law by a kind of cheat. The boy whirls the ball around him, and a circular orbit is created. So we require a massive body in order to create one. It sits there, in all the same regalia as a supported equation, and we salute it in the same way throughout the centuries. Richard Fitzpatrick University of Texas at Austin. This new theory of gravity and orbits makes use of the current E/M field, Relativity, and classical equations. It is stated explicitly that the earth had this velocity before it entered the orbit. Lagrangian mechanics, Eulerian rigid body rotation theory) are derived in the text. The usual answer to this is to show a summing of potential and kinetic energies in a closed loop and prove mathematically that all energy is conserved. The moon has pulled the earth closer to the sun: in order for it to now pull it back two weeks later, it would have to be bigger. Any math that is based on calculus must be indeterminate, by current axioms, and if you add averaging to that math you have left determinacy far behind. Even General Relativity only recast the old concepts in new but basically equivalent terms. Every book you will find in the section on celestial mechanics at even the largest university libraries concerns creating equations to explain orbits based on observations. We can’t possibly see it. Kepler's second law states that a planet sweeps out equal areas in equal times. And this applies to the ellipse just as it does the circle. Its failures are the same failures as classical theory. In neither case can you mechanically explain a torque. The problem is that planets arriving in orbits immediately after collisions are going to be damaged planets. This allows its tangential velocity to eventually counteract gravity, pulling it back into ever-increasing distances from the sun. And this is where it gets silly. If Newton or Einstein is made to look foolish, we all look foolish, and our checks from the government vanish. For him they were equivalent abstractions or ideas. As I said above, this analysis began with Newton when he described circular motion in Proposition I of The Principia. The solutions of the equations of CM show great sensitivity to initial conditions: very close initial conditions may lead to totall… It is one of the great unseen gaps in kinematics. Besides, these lists of unsolved mysteries are always long on big theoretical problems—the sort of things that might be expected to remain even though we are brilliant masters of the universe. Why? This would certainly be a reductio ad absurdum. A nebula requires the postulate of a center. I show that orbital differentials betray a variance in the "innate motion" (tangential velocity) of the orbiter, a variance that cannot be explained mechanically or mathematically by the current theory. They all take on the form of ad hoc theories, and look very much like signs of desperation. This is the theory of how the solar system was created. He went even further, though, for he believed that "the field" and "space" were two words for the same thing. General Relativity fails on this basis alone; it is cut by Occam’s Razor. Gravity is a centripetal force, not a tangential force, so that resonances are beyond explanation. So you must accelerate the earth out of the orbit. We know the moon goes there occasionally because we see total eclipses. They have elliptical orbits. CONTENTSIntroduction § 1. This was a classical problem in celestial mechanics at the turn of this century and any Say you want to recreate the ellipse with your ball and your string. Any satellite engineer knows this. For Einstein, no carrier of influence was necessary in a normal gravitational field, since matter in the field "felt no force." To answer this, we must go back to the circular illustration. There is no proposed mechanism. Matter in a gravitational field moves as it does because the field is curved, not because it is being influenced by sub-particles. But this is absurd. There is no mechanism to link the orbit of the solar disc to the spin of the sun either, and yet it is accepted at face value. And if the sun is creating tangential perturbations in the gravitational field, the theory must mechanically explain how they are produced. This is achieved by varying the orbital velocity of the planet, obviously. Is it still a constant piece of the compositional velocity, or has it been lost? Chaos theory is often said to be deterministic, but this meaning is imprecise at best. It's a magic trick. Does an elliptical orbit solve any of the problems I have outlined above? But it does not show why the empirical data is what it is. Chaos math and theory closely parallels that of quantum mechanics. For instance, if you think that the moon simply pulls the earth back out of danger two weeks later when it is farthest from the sun, you are not thinking right. The reason that physicists have moved to such maths may be shown by this quote from the Wikipedia article on chaos: "Everyday predictable non-chaotic deterministic systems (like good billiard tables) might seem boring because, in most cases, scientists discovered exactly how they work centuries ago, and nobody who knows how they work will ever be very surprised by them." Neither Triton nor Neptune is an ideal body. There is much talk and work currently on gravitons, but none have been found. Any other orbit requires the satellite to speed up or slow down—to make corrections. Since then the science magazines have followed suit, encouraging a smug self-satisfaction while at the same time promoting increasing levels of obstruction. The mystery of influence remains unsolved, and the mechanism now has two steps rather than one. They do this to maintain the prestige of the field. Laplace's equations contain that repulsion (in hiding) but cannot explain it. Newton and Leibniz and the invention of calculus, unsolved problems in celestial mechanics became the experimental laboratory for the discovery of new mathematics. If the field around a massive object was curved, then space was. What happened? If the earth had only a tangential velocity, its trajectory would not curve. Einstein did not overthrow the fundamental mathematics of gravity and orbits. But in the illustration of the orbiting earth, the sun does not swing the earth—there is no implication of that. 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. It is true that the orbit of Triton is decaying, so that the orbit is not in fact completely stable. 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 accept c, Relativity, QED, Newton's math and most of his theory, and I give no qualities or quantities to space, including expansion. I have listed many of the holes in orbital theory above, but every modern theory is full of invisible holes. Only mathematics exists. He agreed that the field was produced by the massive object, but he theorized that the object acted on the field rather than on matter in the field. Exponents cause even more problems, since they may magnify any initial imprecision. Both theories are mathematical. Another reason an equation unsupported by theory is dangerous is that it becomes dogma. If you look at individual motions in any orbit that has three or more bodies, you will find that the differentials show a variation in the tangential velocity of the orbiting body. Given that f = ma and that F = Gm1m2/r2 (Newton's famous equations, of course) You must have angular momentums working at a distance. 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). If not, then Kepler has a problem. We have a heuristic theory that allows us to put our own objects into orbit, what else do we need? Scientists use this complex field to explain the motions of Neptune’s other moons. When we are shown the illustration of circular motion in our physics textbooks, we are always shown the accompanying illustration, which is that of a ball on a string. We must differentiate between the tangential velocity and the orbital velocity and it is easiest to do this in the simpler illustration. If you are a Paypal user, there is no fee; so it might be worth your while to become one. A balancing of velocities like this cannot be self-correcting. Saturn cannot go higher due to a perturbation from Jupiter, unless that perturbation is repulsive at some point in the long cycle. For Einstein, the mystery becomes in understanding how the sun influences the space around it, which then influences the earth. Preface. The errors our equations are lending us are infinitesimal compared to the errors we buy freely by failing to look closely at the world in front of us. In saying that a massive object curves space, Einstein was in many ways begging the question. If there are huge holes in the gravitational theories of Newton and Einstein, what does it matter? These stars are of roughly solar mass and they have what appear to be discs. Modern science has created a theory of dissipation to explain the loss of angular momentum by the sun. The orbiting body is assumed by Newton to have a velocity due to "its innate force." rv2 = Rc2/2 or novaforjava is a general purpose, double precision, Celestial Mechanics, Astrometry and Astrodynamics library. This is a problem since imperfect planets create perturbations in orbits. We are taught to bow to the implications of chaos theory and to devote endless hours of modeling time to discovering which mathematical operations cause the most error. Newton derived Kepler's law from his own, to show that the two were consistent. No one thinks this because there is no mechanism to link the rotation of the earth to the orbit of the moon. Clearly, this works because the pencil tip is feeling forces from both foci. Mathematical Preambles Chapter 1. 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). It is perpendicular to that field, whether the field is rectilinear or curved. His popular book was a sign of the times, and a sign of things to come. The first is that if the disc is not in our plane of sight, then it can’t be the cause of any obscuring of shifts, red or blue. Besides, philosophy became obsolescent in the 20th century. And it must regain a degree of rigor and self-criticism. What exact combination is unknown at the present time. Celestial Mechanics, An Introduction to Modern Astrophysics 2nd - Bradley W. Carroll, Dale A. Ostlie | All the textbook answers and step-by-step explanations Numerical Methods Many other fatal problems are also addressed. Instead we see a long slow decay. Notice that if the earth had no velocity tangential to the sun's gravitational field as it was captured by that field, it would simply crash directly into the sun. History of celestial mechanics . And the same problem is going to be met when the moon is sideways to the earth. The planet cannot, because it is not self-propelled or self-correcting. But upon closer examination, it all begins to fall apart. But so what? It can't be built in the real world, by unpropelled planets in a Newtonian orbit. The velocity variable in a=v2/r must then be this new orbital velocity. We can only imagine that it must be because our planet is not moving fast enough to achieve a circular orbit. Any math that is based on calculus must be indeterminate, by current axioms, and if you add averaging to that math you have left determinacy far behind. Newton calls it the body's "innate motion." In the time of Newton, for instance, scientists attacked eachother publicly and directly. It remains more mysterious than the E/M field even at the quantum level (one might say especially at the quantum level—where gravity becomes a ghost, a vanishing "force" that has no known influence). They cannot make any corrections. But this is not the question. This would be a bit more convincing if there weren't so many fundamental problems still embedded in classical theory and linear maths. Specifically, the lack of angular momentum in the Sun is explained in Part IV. It never achieves a temporary escape velocity. The problems I am enumerating here are mostly not known to exist anymore, for the very reason that all study of orbits and gravity is now strictly mathematical. This nebular theory is still ascendant, with a few updates that I will mention in a moment. Where do the tangential components of the tensors come from? Yes, the orbital velocity will be slower at aphelion, but please concentrate for a moment on the curvature. But this answers nothing, for current theory fails to explain how this primordial disc of pre-planets or planetoids achieved its tangential motion in the first place (see below). We can’t possibly see it. A sub-field of physics that thinks that it is perfect or near-perfect will not make any effort to improve. The phase field usurps the physical field, and reality evaporates. As the planet continues on, the centripetal acceleration begins to overpower the tangential velocity, and it gets closer and closer to the sun. There appear to be constraints on decay and escape far beyond what would be logically expected. But so what? If it doesn’t involve computer modeling or advanced mathematics, they can’t be bothered to look at it. Its mathematics becomes its reality. But what has happened is that first perturbation theory and then chaos theory have engulfed and defined all serious analysis in orbital theory, so that no one even remembers what else was going on before Poincaré. 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. Gravity is a centripetal force, not a tangential force, so that resonances are beyond explanation. This has left us with orbital math that is a very precise heuristics. It isn't. That is, the tangential velocity is not caused by the gravitational field. Next, since all the orbits of the planets are nearly circular, let the distance travelled in each orbit equal the circumference of the orbit: The planets affect eachother by applying small torques to one another, we are told. No, it simply spirals into the sun. Sound familiar? But how can an object entering a gravitational field decelerate? And this is not even taking into account the sun's effect on the moon's orbit, which causes further uncorrectable perturbations. No one doubts they exist, but how can gravity explain them? Neither Triton nor Neptune is an ideal body. Let us now graduate from the mysteries of the circular orbit to the mysteries of the elliptical orbit. Most consider it beneath them. Neither Newton nor Kepler nor Einstein have anything to say on the subject. But to do this they must give the earth slightly eccentric little accelerations and decelerations, which they never explain. When Earth, the Moon, and the Sun are considered to be point masses, this particular three-body problem is called “the main problem of the lunar theory,” which has been studied extensively with a variety of methods beginning with Newton. His assumption has never been seriously questioned. 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). At this lower orbit the acceleration toward Neptune is even faster. I encourage you to try it. If it is coming from outer space into the field of the sun, it must somehow decelerate in order to fall into its current position. Most physicists can’t afford to spend the necessary time learning basic physics, much less basic logical analysis. On the chaotic motions and the integrability of the planar 3-centre problem of Celestial Mechanics. As a lead into this, I want to first outline some of the other flaws in nebular theory. A similar problem is caused by any three-body analysis. All the other perturbations of the solar system are likewise mysterious. Velocity requires two separate measurements: it requires a measurement of distance, then it requires a measurement of how much distance per time. Einstein never implies that Kepler and Newton's theories were wrong—they are only incomplete. How could it be? At the beginning of the 20th century, G.D. Birkhoff called the N body problem of celestial mechanics the most celebrated problem in mathematics. Laplace's equations contain that repulsion (in hiding) but cannot explain it. You cannot physically measure at a point or an instant, and this must affect all your final numbers. We are left with a fait accompli: since the two motions are tied to one another with the ball on a string, the two motions must be tied in the earth/sun example, and there is nothing to explain. I am not complaining here about a sum or an integral. Small denominators § 4. Perturbation and chaos theory is not completely without merit or use. But as I have shown in my paper on the foundation of the calculus, this is a misassignment. But you can kind of do it with a rubber band. Lagrange and Laplace did much work on perturbation theory, which is basically a differential or series analysis of the restricted three-body problem. v2/r = GM/r2 This new theory of gravity and orbits makes use of the current E/M field, Relativity, and classical equations. Look at the vector additions at aphelion and perihelion, using this diagram. How was this possibly achieved, given the current list of forces and causes of forces? We have a partial answer for why the stars don’t fly out into space: gravity. But I for one think it was preferable when this warfare was in the open. They have been described, in several different ways, by Newton, Einstein, etc., but never explained. The successes of physics in the 20th century allowed it to appear to transcend any need for cooperation with “lesser” men and women. But the sun and Jupiter are thought to have about equal amounts of helium. The question is, can we connect up the ellipse? Einstein never implies that Kepler and Newton's theories were wrong—they are only incomplete. 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. Let's work backwards and see if we can imagine how the earth might get to that optimum distance, with just the optimum tangential velocity. And, I must say that this is only to be expected, since science has become the new religion. Most can’t follow a straightforward algebraic analysis with variables clearly assigned to physical objects and fields, since they are used to dealing with extremely complex and abstract variables and extremely complex mathematical manipulations. It takes a greater force to nudge a planet into a higher orbit than it does to nudge it into a lower orbit. The answer is that it doesn’t. We are now taught, in courses influenced by the thinking of Einstein, that the elegance of a scientific theory resides, in part, in its simplicity. Be constraints on decay and escape far beyond the scope of this constant it is true then. Mass and they have a heuristic theory that explores the results of nescience initial... 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