Kepler to Newton -- Part One

It's spring break in the throes of a particularly harsh Wisconsin winter, and Virginia and Jimmy K. have joined me for some coffee and physics.

Jimmy K.: "You look agitated about something. What is it, CM?"

Celestial Mechanic: "In a recent thread on the 'death' of ATM, nutant_gene_71 said a few things about the need to use empirical science, instead of the theoretical. He left the impression that somehow our theories are just caprices, 'free inventions of the mind' as Einstein used to put it and Zanket (remember Zanket?) used to quote it all the time."

Virginia: "But of course our theories are tied to observations. It just isn't always that easy to trace every step in the argument."

CM: "But I think I've found one that serves quite well as an example. Most texts on celestial mechanics start from Newton's Law of Universal Gravitation and use it to show the validity of Kepler's Three Laws. But I just recently wondered if it could be run the other way, so that given Kepler's laws we can show that Newtonian gravity follows as a consequence."

V: "Certainly someone must have done this already."

CM: "In fact, Danby does something like this in his book, and of course Newton himself did do something like this in the Principia, Section III, Proposition XL, Problem VI. Alas, Newton's treatment is not very readable. I hope that mine will be. I will also have a few words about the measurement of masses in the Solar System, as an antidote to some of the nonsense being written here about masses and densities of the planets being overestimated here and underestimated there."

JK: "Well, I'm ready."

CM: "First let's state Kepler's laws:"

1. The orbit of each planet is an ellipse with the Sun at one focus.
2. The radius vector of each planet sweeps out equal areas in equal times.
3. The squares of the periods of any two planets are in the same ratio to one another as the cubes of their semimajor axes.

CM: "Now something that is implicit in all this is that the sizes, shapes, and orientations of the planets are not changing in time. We will see that this is not exactly true, but we will come to see that this motion is quite slow.

CM: "Let's put some equations to these laws:"

1. The radius vector r of an ellipse of semimajor axis a and eccentricity e is r=a*(1-e²)/(1+e*cos(f)), where the angle f, called the 'true anomaly', is the angle between the radius vector and a vector pointing from the focus of the ellipse (the Sun) to vertex of the ellipse (the perihelion point).
2. In polar coordinates the integral of the area between two values of f is (1/2)Integral^f2_f1(r²*df) which is equal to (1/2)Integral^t2_t1(h*dt) = (1/2)*h*(t2-t1). In other words, h=r²*df/dt is twice the constant rate of description of the area.
3. Lastly, P_i²/P_j²=a_i³/a_j³.

JK: "Are there any formulas for parabolas and hyperbolas like the one you gave for ellipses?"

CM: "Yes, in fact you can use the same formula for hyperbolas, provided you take a as negative. A much better formula is to substitute q=a*(1-e) in the formula for the ellipse, that is, r=q*(1+e)/(1+e*cos(f)). Here q is the minimum radius vector which is defined for all three types of conic section. This equation can be used whatever the value of e (provided it is positive, of course). While motion along parabolas and hyperbolas is possible, Kepler's laws were applied originally to ellipses since their motion is periodic and observable for a long time.

CM: "Now we are going to start connecting some of these formulas together in order to define some of the quantities that we will need for expressing the law of acceleration that will follow from the three laws above. First we start with the third law. For each planet define the rate of mean motion in radians per second, n, by n=2*pi/P. Then the third law may be written n_j²/n_i²=a_i³/a_j³. By cross-multiplying we find that:

n_i²*a_i³ = n_j²*a_j³ = ... = K.

CM: "I call this quantity K, which has SI units of m³/s², 'oomph'."

V: "Well there's a good scientific term for you!"

CM: "I've never seen a name for this particular term, and 'oomph' is as good as any. We will see that other bodies in the Solar System also have 'oomph', just not as much as the Sun.

CM: "Next we are going to connect the area description constant h with n, a, and e. During one period P the radius vector sweeps out an area of (1/2)*h*P which is equal to the area of the ellipse, pi*a*b, where b=a*sqrt(1-e²) is the semiminor axis of the ellipse. Using our definition of n, we find that h=n*a²sqrt(1-e²).

CM: "But as they say in those annoying infomercials, 'Wait! There's more!' If we square h we find:"

h²=n²*a⁴*(1-e²)=K*a*(1-e²).

CM: "We recognize the a*(1-e²) part as being the numerator from our expression for the radius vector, which we may rewrite as:"

r=(h²/K)/(1+e*cos(f)).

CM: "Now we are going to have to 'kick it up a notch', as Emeril would say. So far we have only considered quantities implied by Kepler's laws. But in order to get to Newton's law of gravitation we have to say something about directions, since Newton's law specifies the direction as well as the magnitude of the acceleration observed. Time for a refill!"

To be continued ...

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Kepler to Newton -- Part Two

It's spring break in the throes of a particularly harsh Wisconsin winter, and Virginia and Jimmy K. have joined me for some coffee and physics. Our object is to derive Newton's Law of Universal Gravitation from Kepler's Three Laws and maybe shed some light on the graviational constant, G, and on the masses of bodies in the Solar System.

Celestial Mechanic: "Now that our cups are refilled it's time to start putting in some vectors. To start with, we define:"

h = r x dr/dt.

Jimmy K.: "But don't we already have a scalar called h? Isn't this going to cause some confusion?"

CM: "No, because h the scalar is just the length of h the vector."

Virginia: "The vector dr/dt has two parts, one of magnitude dr/dt in the direction of r which does not contribute to the cross-product since it's parallel to r, and the other part perpendicular to r which has magnitude r*df/dt. Together their dot-product has magnitude r²*df/dt, which is precisely h."

CM: "Couldn't have said it better myself. Note that this vector h is perpendicular to both r and dr/dt, and also to rxh and dr/dtxh as well, which will become important later on."

CM: "Now we will turn our attention to the formula for the radius vector. Multiply both sides by the denominator to give:"

r + r*e*cos(f) = h²/K. (1)

CM: "The r*e*cos(f) part is reminiscent of a dot product of two vectors, r and e. r is just the radius vector of the planet, e is a vector of magnitude e pointing in the direction of the perihelion. It is called the 'eccentricity vector' or the 'Runge-Lenz vector' or even occasionally the 'Laplace-Runge-Lenz vector', which shows how long ago this was discovered. We may write:"

r+r*e*cos(f) = r+r*e = r*(r/r+e) = h²/K.

CM: "But wait! There's more! Consider this vector:"

dr/dtxh/K

JK: "Is K the 'oomph' you defined previously?"

CM: "Yes, it is. If we dot this vector with r we get a vector triple product and by using the cyclic property we find:"

r*dr/dtxh/K = h*rxdr/dt/K = h*h/K = h²/K.

CM: "This is precisely the same constant as in equation (1) above, so we may subtract to obtain:"

r*(e-dr/dtxh/K+r/r)=0.

V: "I've never seen the eccentricity vector derived like that! Usually it is defined as dr/dtxh/K-r/r and then it is proved to be conserved in the two-body problem!"

CM: "Well, we're not quite there yet, but close. You see, all we have shown is that the difference between e and its conventional definition is perpendicular to r. There may be additional parts perpendicular to r. We must show that they are zero. So write e as follows:"

e = dr/dtxh/K-r/r+B*(dr/dt-(r*dr/dt)r/r²)+C*h (2)

CM: "That weird thing multiplying B is just that part of dr/dt with its projection against r removed so that it is perpendicular to r. h is, of course, perpendicular to both of these. Now, since e is parallel to r when the planet is at perihelion, it must lie in the plane determined by r and dr/dt and thus be perpendicular to h. If we take the dot product of formula (2) above with h we have:"

0 = C*h².

CM: "Since for planetary motion h is non-zero, C must be zero. Now we turn our attention to B. Let us consider the planet to be at perihelion. Then r and e will be parallel to each other and dr/dt will be perpendicular to both of them."

JK: "Why?"

CM: "Consider r² as a function of time. It reaches its extrema when dr²/dt = 0, that is when r*dr/dt = 0. If we now multiply equation (2) by dr/dt at the time of perihelion passage we find:"

0 = B*dr/dt*dr/dt.

CM: "And so B equals zero as well. Therefore the eccentricity vector really is defined by ..."

e = dr/dtxh/K-r/r

CM: "... after all. This latter part is the toughest part of my proof. The rest of it is coasting downhill to the results."

V: "Most books I've seen define the eccentricity vector first, show that it is conserved and then go on to show how a conic section orbit results from it. It seems miraculous that anyone could find such a vector, because the definition just doesn't look very obvious."

CM: "Well, I have to admit that this approach has its own miracle. After showing that r dotted with (e+r/r) equals h²/K, who would imagine that r dotted with dr/dtxh/K equals the same thing? But the time has come for the majestic downhill coast. First, constancy of the vector h implies dh/dt = 0, which implies:"

0 = dh/dt = d/dt(rxdr/dt) = dr/dtxdr/dt + rxd²2r/dt².

CM: "The first term is automatically zero because it is a vector crossed with itself. The second term will vanish if d²r/dt² is parallel to r, that is if d²r/dt²=Ar, where A is to be determined. To determine it we turn to the eccentricity vector. Constancy of the eccentricity vector, de/dt=0, along with constancy of h implies:"

0 = de/dt = d/dt(dr/dtxh/K-r/r)

0 = Arxh/K-dr/dt/r+(r*dr/dt/r²)r

0 = -(Ar²/K+1/r)(dr/dt-(r*dr/dt/r²)r)

CM: "Filling in the omitted steps above is left as an exercise for the student."

JK: "Always a catch."

V: "But you don't expect to be given everything, do you?"

CM: "But we're so close! The last equation consists of a product of two terms, the first one a scalar and the second a vector. That vector is nothing more than the velocity vector with the radius vector projected out of it. It is zero if the velocity and the radius vector are parallel, but that occurs only for the uninteresting case of purely radial motion. For planetary motion the first term must be the one that is equal to zero, and this gives:"

Ar²/K+1/r=0,

CM: "... so A=-K/r³ and therefore d²r/dt²=-(K/r³)r, which is our acceleration law! Now, do you notice something missing in all this?"

JK: "I'm not sure."

V: "Well, there's no G in it."

CM: "Not only is there no G in it there's not a gram of mass in it anywhere. I deliberately avoided such words as 'force', 'mass', and 'momentum', but not for the purpose of showing off. Kepler's laws do not mention masses either. Kepler's laws are not only empirical, they are entirely geometrical. In that spirit I have shown how they lead to an acceleration law, rather than a force law. What other theory can you think of that is about trajectories rather than forces?"

V: "General relativity?"

CM: "Yes! There are no forces keeping objects in orbit, just objects following geodesics in spacetime. Unless they are charged and pushed or pulled off the geodesic by 'real' forces such as electromagnetism, that is."

JK: "I see that you have obtained the acceleration, but how do we get the force and Newton's law from this? Where does mass enter into it?"

CM: "After a refill."

To be continued ...

__________________
Microsoft is over if you want it.

The bar has been lowered for the promotion of ATM ideas; the bar for the acceptance of ATM ideas must remain high.

Kepler to Newton -- Part Three

It's spring break in the throes of a particularly harsh Wisconsin winter, and Virginia and Jimmy K. have joined me for some coffee and physics. Our object is to derive Newton's Law of Universal Gravitation from Kepler's Three Laws and maybe shed some light on the graviational constant, G, and on the masses of bodies in the Solar System.

Celestial Mechanic: "What I've done in the first two parts was to take Kepler's three laws of planetary motion and use them to infer a law of acceleration for it. Nowhere does Kepler mention mass, and nowhere did I, nor did I mention force or momentum. For all we know at this point only the Sun has the 'oomph' to send the planets orbiting around it, and this oomph could just as well be due to mass as to magnetism, ultramundane particles pushing everything around or even Decartes's vortices. The only fly in the ointment at this point is the observation that the values of K obtained from calculating n²*a³ for the planets then known were not all equal; the value of K calculated from Jupiter's orbit was larger by one-tenth of one percent than the smallest value, and all the other values lay in this narrow range."

Jimmy K.: "Our first anomaly!"

CM: "Indeed. And if Jerry and nutant_gene_71 had been around back then they probably would have demanded that Keplerian orbital mechanics be thrown out in favor of Descartes's vortices or jelly doughnuts or whatever the current fashionable 'alternative' was."

Virginia: "Never mind that Kepler's Rudolphine Tables were more accurate than both Copernicus's Prutenic Tables or the Alphonsine Tables based on Ptolemy's Syntaxis by at least two orders of magnitude."

CM: "If your mind is made up, why let facts stand in the way?"

JK: "Aren't you being just a bit unfair in your characterization of these two BAUT members?"

CM: "Have you seen Jerry's 'proof' that Newtonian gravity is off in its estimate of Mars's mass?"

JK: "No."

CM: "Well, then have a look."

At this point Celestial Mechanic fired up his trusty computer Gurnemanz, logged on to BAUTForum, and showed JK the relevant posts.

JK: "This is horrible! If I gave an argument like that on a term paper I would get a big, fat 'F' on it!"

CM: "And that 'F' wouldn't be for fine, either. nutant_gene_71 still alludes to a theory in which the gravitational constant varies linearly with distance from the Sun. This would imply a seasonal variation of G as measured on Earth in Cavendish-type experiments of over three percent in range, something that is just not observed. He still hasn't reconciled this.

CM: "But enough of the bad ideas of others; let's look for a solution to our 'first anomaly'. At about the time Kepler's works were published or shortly thereafter satellites were discovered around Jupiter and then Saturn. And it was found that these satellite systems also obeyed the Keperian laws, just with a different amount of 'oomph' for Jupiter and Saturn's systems. Now one problem is that initially we did not know the distances to the Sun and planets in ordinary, earthly units like mile, leagues, or kilometers. But we did know the ratios of the distances with respect to our mean distance from the Sun, so we could take that as a unit and use it as a place-holder until we could measure the distances in SI units."

V: "But what about Earth? We know that objects dropped at the surface accelerate at 9.81 m/s², and that the equatorial radius is 6,378 km, so ignoring the complications due to the shape of the Earth and the centrifugal effects, we know that K_E/R_E²=g and so K_E=g*R_E². From that formula you gave in that one paper they adopted a value of 398,600.4 km³/s² for what we are calling the 'oomph' of the Earth."

CM: "Right you are! We do know the 'oomph' of the Earth in SI units. And for reasons I will give shortly we also know the 'oomph' of the Moon as well in SI units. But the calculation of the Moon's 'oomph' rests on the resolution of Jimmy K.'s 'first anomaly'.

CM: "One of the questions that I most frequently ask is, 'am I forgetting something?' The fact that Jupiter and Saturn were shown to have satellite systems of their own suggests that bodies other than the Sun must have 'oomph' as well, just not nearly so much of it. And if Jupiter has 'oomph' then the Sun, indeed everything else in the Solar System, must also accelerate towards Jupiter. So the magnitude of the relative acceleration of the Sun and Jupiter must be:"

(K_Sun+K_Jupiter)/r²

CM: "... and so for Jupiter we must write:"

K_Sun+K_Jupiter = n²*a³.

CM: "Not only does this illustrate the importance of 'am I forgetting something', it is also an illustration of one of my other guiding principles: 'amend and extend'. By extending this principle to the Earth and Moon, we see that:"

K_Earth+K_Moon = n²*a³,

CM: "... where n and a are the sidereal orbital period and the mean distance."

JK: "But how do we know the mean distance to the Moon?"

CM: "We know the size of the Earth from surveying. In the case of the Moon, we know its distance because we can measure a parallax for it. This parallax is about one degree, which puts the Moon about 60 Earth radii away on average. It is then a simple matter of using the amended Kepler's third law to compute the total 'oomph' of the Earth and Moon, subtract the 'oomph' that Virginia found for it earlier and discover that the Moon has only one-eightieth the 'oomph' of the Earth. Actually, if you try to do this computation using the values in, say, Danby's book, you won't get quite the correct results. That's due to a different definition of a in the lunar theory, which I can't go into here."

V: "So why don't we measure the parallax of the Sun?"

CM: "Well, we can, but it is only about 8.8 seconds of arc, which puts that Sun about 400 times farther away than the Moon. That was one of the reason that observations of transits, particularly those of Venus were sought after, in order to nail down the parallax of the Sun and hence the value of the astronomical unit. I will also add that close approaches of the minor planet 433 Eros were also useful for determining this distance."

JK: "But ultimately we can determine the 'oomph' for the Sun, the planets, and even our own Moon. How do we get from 'oomph' to mass?"

CM: "As you see, I have given a treatment of celestial mechanics that never mentions mass, that only deals with 'oomph'. But ultimately we have to account for where this 'oomph' comes from. We start by taking the acceleration of object i towards object j:"

d²r_i/dt² = K_j*(r_j-r_i)/r_ij³,

CM: "... where r_ij is the distance between objects i and j. We know the acceleration, so all we have to do is multiply both sides by m_i, the inertial mass of object i, to obtain the force of j acting on i:"

F_ij = m_i*d²r_i/dt² = K_j*m_i*(r_j-r_i)/r_ij³.

CM: "We also have for the force exerted by i upon object j:"

F_ji = m_j*d²r_j/dt² = K_i*m_j*(r_i-r_j)/r_ij³.

CM: "Now we invoke Newton's third law which in this case means that, in order for the total momentum to be conserved, F_ij+F_ji=0 for all pairs of objects. This will be true if and only if for all pairs of objects:"

K_i*m_j = K_j*m_i.

CM: "We can rearrange this to read:"

K_i/K_j = m_i/m_j.

CM: "... which means that the ratio of any pair of 'oomphs' is equal to the ratio of the masses. And not just any old masses, but the inertial masses of the objects.

CM: "But it gets even better. We can also rearrange the proportion to read:"

K_i/m_i = K_j/m_j = ... = G.

CM: "... thus defining G and showing that it must be universal in order to conserve momentum. Finally, we may write for the 'oomph':"

K_i = G*m_i.

CM: "All that remains is to determine the value of G using a Cavendish-type experiment and we get the masses of Solar System bodies in SI units if we want to use them. Be forewarned, though, that Cavendish experiments are difficult to perform and so G is the physical constant known with the least precision."

JK: "Wow! That's going to take some time to sink in!"

V: "You've pretty much laid variable G in the coffin and thrown on the first handful of dirt!"

CM: "Not so fast! Don't do that victory lap just yet. There is General Relativity beyond Newtonian gravity, and possibly (almost inevitably) new physics waiting to be discovered beyond that. The laws we have found above may well be subject to quite a bit of amendment. Consider this: suppose that the 'oomph' of a mass m is defined this way:"

K = G*M*sinh(m/M)

CM: "... where M is a new and possibly universal constant. For M very large compared to any m we have in the Solar System, our new formula will approach our old one in the limit. Any violation of conservation of momentum and Newton's third law would be suppressed by a factor of (m/M)² and difficult to detect."

V: "Do you really think that 'oomph' behaves this way?"

CM: "Not really, I just give this as an example of an extension. One thing I would like to point out here is that adding an extension or amendment often results in adding a new constant somewhere along the way. For example, nutant_gene_71's idea for linear growth of G adds at least one constant, the linear growth factor, which has units of inverse energy, by the way. The problem with his theory is that it involves the addition of even more constants, because linear growth cannot continue forever, so somehow G has to level off at some distance from the Sun as I've pointed out. And something I did not point out before is that if G literally increases at 1 'Earth-G' per AU, then G would be zero at the Sun and the Sun would fly apart! Only by having some non-zero value for G at the Sun and using a lower growth coefficient and having some kind of asymptotic limit can that theory be saved. That's three constants in place of one Newton's constant."

JK: "Occam shaves clean again."

CM: "More coffee?"

To be continued ...

__________________
Microsoft is over if you want it.

The bar has been lowered for the promotion of ATM ideas; the bar for the acceptance of ATM ideas must remain high.

Kepler to Newton -- Part Four

It's spring break in the throes of a particularly harsh Wisconsin winter, and Virginia and Jimmy K. have joined me for some coffee and physics. Our object is to derive Newton's Law of Universal Gravitation from Kepler's Three Laws and maybe shed some light on the graviational constant, G, and on the masses of bodies in the Solar System.

Celestial Mechanic: "Now it is time to sum up and to say a few words about the masses of bodies in the Solar System.

CM: "First, celestial mechanics can be developed using only accelerations and 'oomphs'. Kepler's laws are entirely geometric, and allow a law of acceleration to be inferred from them. The consistent and diligent use of this acceleration law allows the motions of nodes and perihelia to be calculated, for the precession and nutation of the Earth's rotation, the lunar theory, and much else besides.

CM: "Second, the introduction of mass and Newtonian mechanics allows us to show that 'oomph' is really proportional to mass, and this constant of proportionality is G, the Newtonian gravitational constant."

CM: "Third, our findings are only valid to the extent that Newtonian mechanics is valid. All velocities in the Solar System are of the order of 10^-4 or less, so relativistic effects will be small. Our key result rests on the conservation of momentum, as expressed by Newton's third law. If that can be relaxed, then the way is open for G to be variable, but even then it is doubtful that it will vary by that much over the size of the Solar System.

CM: "Lastly, given the practically constant value for G in the Solar System, the values of the masses for the planets follow from this and the modified third Kepler law. There is no room for Titan to be ten times or two times or even 1.05 times the mass known for it. Those who would seek an explanation of anomalous descents of space probes should look to atmospheric physics for their answers."

Jimmy K.: "One last question, please? How are the masses of planets without satellites determined?"

CM: "Through their perturbations on other planets. This is not as easy or as accurate as the case with a satellite, and it is necessary to use it for Venus and Mercury, and it had to be used with Mars as well until its satellites were discovered in 1877."

Virginia: "Why is a defined differently for the Moon?"

CM: "The Moon represents a kind of a special case. The perturbations due to the Sun are so large that it is actually better to ignore eccentricities and inclinations initially and concentrate on what is called the 'variational' orbit first."

V: "Where is the equivalence principle in all this?"

CM: "Kepler's laws, by virtue of being geometrical, implicitly assume the equivalence principle in its weakest form, namely that planets will follow elliptical orbits regardless of their internal structure and makeup. It should then be no surprise that Newtonian gravity, derived from it, should incorporate it, again implicitly."

This being spring break, conversation turned to UW's chances in the then upcoming March Madness. Thus ended another discussion of physics over coffee.

__________________
Microsoft is over if you want it.

The bar has been lowered for the promotion of ATM ideas; the bar for the acceptance of ATM ideas must remain high.

All would be well and good...if it were not for the stubborn new anomalies emerging. You can wring the same first-order effects you have delineated above out of VHF electromagnetic tensors; the differences are small, but important: Eddy currents, resonances, harmonics and a catch-all bin of nonlinear effects: resistance, stray impedance and phantom couplings. An electromagnetic system is never completely linear: No one would ever blame uncalculated ineffiencies of a windmill upon dark matter.

Nowhere in your equations do you justify the Newtonian assumption of equivalence between gravitational and inertial systems: The Twentieth century physics you expound ignored Mach's pale of water. and it will all come tumbling down.

How can I know this? In the Newtonian assumption of equivalence, when a probe passes over a planetary topological feature, such as a mountain or a deep chasm, if you carefully integrate the small change in gravametric forces (or space-time curvature, if you prefer) you can calculate the masses of the features. There is no rational reason the deep rifts we have found on Mars are broadly under-dense, but that is what we observe when the Newtonian Equivalence Principle is assumed in mass/density calculations. At one time, this was supposed to be because there are deep layers of ice in the valleys of Mars. But subsequent radar sounding has excluded this possibility (not to mention there is no more logic in assuming deep ravines near the equator of Mars are filled with ice than there is to assume there is permafrost in the Grand Canyon.)

We have volumes of Newtonian data telling us the material in the volcanic mountains of Mars are over-dense relative to the surrounding terrain, while on Venus exactly the opposite occurs: under-dense mountain (that also appear to be volcanic) and over-dense chasma. Simple mechanical orbits, no matter how exquisite the math, render only close approximations. Newton knew that. If Newton would have known these facts, he would have immediately realized the ramifications, and volumously reconsidered his untested assumption of equivalence.

Newton also made a rather peculiar assumption that the infinite size of the universe would stabilize stellar systems - this is not true - Newtonian orbits remain elliptical unless electromagnetic effects dampen them. This is why it is universally assumed that the planets of the solar system condensed from a dusty ring; and why comets were assumed to be made of ice and primal dirt. Our close encounters with comets have greatly complicated this simple model, leaving the 'condensation' theory cold when we find clays and other heat-treated compounds in comet nuclei.

The anomalous acceleration of Cassini, Messenger, Galileo, Near and possibly other probes during gravitational assists is consistent with a flawed equivalence principle, a position NG and I were both arguing before the publication of this data. I also stated that Messenger would experience anamolous acceleration during the gravitanal assists from Mercury (confirmed) and Venus (as yet unconfirmed).

Newton and Keplars' assumptions about the perfect order of the spheres were based upon their belief there was, at one time, a perfect order in all of heaven and Earth. Today, we do not have that luxury: We know the heavens are as chaotic as our own DNA. We get to marvel at the order we see, but we cannot conclude it is perfect in any way, nor assume that our understanding of even the simplist things is complete.

__________________
jwj

It's ok not to know. We should try harder to find out.

Are you implying that gravitation is some "first-order effect" that can be derived from "VHF electromagnetic tensors"? I'd like to see that; I'd like to see some real math from you, something better than "assume all low-albedo terrestrial planets have the same density".
Uh, dude, the pail of water is from Newton's Principia. Mach did comment on it, though.

One problem Jerry: the equivalence principle has been shown to be true to within 1 part in 10¹³ in numerous experiments. Newton himself verified it to about 1 part in a thousand.
As I indicated above, the equivalence principle was not untested by Newton. He had tested it to about one part in a thousand. And all our "close approximations" in celestial mechanics are quite close, far, far, closer than all of your so-called results of all your hand-waving.
Sorry, the equivalence principle is not at fault here. The order of the flyby anomaly seems to be 10^-6, which is larger than any possible violation of the equivalence principle. It is also larger than any frame-dragging effect, which is what makes this anomaly so interesting.
Well, the perfect order of the spheres was just so much philosophical moonshine, long ago refuted. I'm not after perfection, but understanding. Our understanding of the simplest things is quite advanced, thank you. Celestial mechanics regularly predicts the positions of planets, satellites (natural and man-made), eclipses, occultations, transits with far more accuracy than you or nutant_gene_71 have ever been able to muster. I will say it again: it is only because of our current theories and their incredible accuracy that it is possible to detect these anomalies. Without these theories that you are so eager to "efface and replace", you would have no measurements of anomalies and nothing to speculate about.

I think that the mainstream understands the simplest things quite well, even if not as completely as some would like, certainly better than those with no understanding on which to base judgments of completeness.

__________________
Microsoft is over if you want it.

The bar has been lowered for the promotion of ATM ideas; the bar for the acceptance of ATM ideas must remain high.

Dude, you know that math is just the mainstream physics version of the Illuminati!
No ATM proponent is going to burn his fingers at something like that!

BTW, thanks for a nice discourse, again!!!

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Optimism does not change the laws of physics. (T'Pol)
A good scientist has freed himself of concepts and keeps his mind open to what is. (Dao De Jing 27)
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Martin ( http://www.geocities.com/DrMartinV )

Yes, very nicely done. This is a great piece. Thanks!

__________________
Forming opinions as we speak

Great post CM!!

Not to butt in on your very fine development CM, really impressive. But since you brought this up, I suppose it’s okay to reply, though there is no easy answer to this Cavendish-Eotovos gravity experiment conundrum. I did reply on a similar problem elsewhere, with this (I’m copy and paste here):

“I think the answer to this conundrum, if there was a variable G scenario at work within the Earth’s orbital eccentricity, lies with the Equivalence Principle. This would mean that whether the Earth is at perihelion or aphelion, assuming Newton’s G changes with distance from the Sun (which I think is about 1G per 1 AU - as per my posts), then the equivalence of mass as expressed in kilograms would of necessity adjust for any difference in G. So measuring G using the usual Cavendish or Eotovos experiments, since we are using kilograms that self-adjust per Equivalence, will not show a meaningful difference. This is an unresolvable problem, unless one uses some other method of independently verifying inertial mass, such as spin up or spin down measurements of gyroscopes in orbit (what I believe ESA has proposed, by using a detached mass) to gauge if any changes in inertial mass occur.”

So the real reason why at a 3% eccentricity no such variation in G is evident, if this is true, is because our Earth based ‘kilograms’ already adjust for it. Same mass, same G adjusted kilos, regardless of where measured, because they all self-adjusted already. This means of necessity that measuring for G using kilograms that self-adjust for G becomes difficult to notice any meaningful variations.

On your other point, a good one too, there is a non-zero value of G for the Sun’s mass, obviously, though there may be a near-zero (or very low) value at the Sun’s hottest portion of the corona where gravity fails (their inertial mass is drastically reduced), which means molecules get spit out at high velocities, which may also be the origin of the ‘solar wind’ catapulted into space. So in yours:

There are no new constants involved, as far as I can tell, except that G has a linear growth (possibly) with distance from the hot energy of the Sun, and where it levels out at about the MOND level some distance out, at about 100,000 AU, for reasons too complicated to go into here (and ATM to boot!). The measured ‘mass’ of all distant bodies are what they are, given we used a non-variable G (Newton’s universal constant G) in our calculations, and if G is different for them, then the mass would change in ‘local units’ only, per inertia-gravity Equivalence (but we wouldn’t see it if kg adjusted), but not change the mass as calculated. (It’s still the same body, same mass.) However, where it might show is in any variations to atmospheric density and size, since atmospheric gas is most easily adjusted (as opposed to solids), it would show up in higher G regions where they would be thicker and taller (perhaps denser) if G were higher there. It may also show up in how space probes (though orders of magnitude smaller than host body’s mass) respond in flybys and landings, or their gyros spin up and down. (The Pioneer Anomaly is merely another clue.) Does that violate Occam’s Razor? I would think it gives a closer and smoother shave, rather than complicate things, with a simplified theory. However, it might make a messy reconciliation with standard cosmology such as it is today, not to mentioned what a higher G would do to space dust and gas (what we call Dark Matter) for line of sight redshift at cosmological distances (where light is coming out of a very deep long well of space dust and gas gravity well). Which is right? Empirical evidence would act as a fair umpire, I would think, so we must find a way to measure for G ‘out there’. The real trick is how do you measure for a variable G if the kilos move with it? Yikes!

Otherwise, very enjoyable dialogue and read, my name is not mentioned in vain. Carry on. I really think it’s cool how you independently developed force and mass from Keplerian orbital math! Well done.

Jerry, we no doubt will see more of these anomalies, now that we know to look for them. Astronomy has never been this exciting since the days of Newton! We should know within a few decades whether Einsteinian space-time General Relativity has the same 'oomph' as a variable G adjusted Newtonian-Keplerian new physics. But I wouldn't throw dirt on that coffin quite yet!

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Credibility is simply incredible... sometimes even to me.
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[quote=nutant gene 71;1217956]

SNIPPET
. This is an unresolvable problem, unless one uses some other method of independently verifying inertial mass, such as spin up or spin down measurements of gyroscopes in orbit (what I believe ESA has proposed, by using a detached mass) to gauge if any changes in inertial mass occur.”[/i][/indent]
nutant. I think I'll disagree with a SNIPPET here. It's not spin up or down of a gyroscope that is suspect. A gyroscope contains a collection of atoms with their collective spins randomly oriented....a spin glass...as it were. It will be with polarized targets with a relatively high degree of polarization that should show the anomaly.
Two near.~identical masses with spin populations opposed by 180 degrees should show a distinct difference. The geologically quiescent Moon with scant atmosphere, should show Galileo's Tower of Pisa experiment with one consistently outstripping the other. I bet a hot fudge sundae, with double cherries on top....and walnuts, lots of crushed walnuts. pete

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A third rate theory forbids
A second rate theory explains after the fact
A first rate theory predicts...A. Lomonosov

Thanks trinitree88, I was not familiar with the 'polarized targets' part of a gyroscopic anomaly. Perhaps I should have worded it differently, that a separate 'test mass' would be used, not necessarily gyroscopes, if this polarization factor is to be considered. Perhaps the suggested ESA test had that in mind, at least in theory?

Thanks for the 'fudge sundae' offer, but I'm a diet of whole grains and low sugar. But the nuts are okay!

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Credibility is simply incredible... sometimes even to me.
disclaimer