Saturday, September 26, 2015

Kepler on Gravity: Newton who?

"If I have seen further, it is by standing on the shoulders of giants." - Sir Isaac Newton, modifying a quote attributed to 12th century philosopher Bernard of Chartres.

"Before Kepler, all men were blind, Kepler had one eye, and Newton had two eyes." -Voltaire who first referred to Newton's refinements of Kepler's contributions as "laws".

Kepler was aware of every aspect of universal gravity except explicitly saying 1/R^2, and that his most surprising gravity axiom shows he assumed either F=ma, or "Einstein's" equivalence principle between inertia and gravity.

Kepler's axioms of gravity contain the following statements.  This is my wording, but I'll spell out the caveats and give the exact wording further down. They are all correct.

1) objects at rest in space stay at rest, if not affected by gravity
2) gravity is proportional to mass
3) mass = density*volume
4) applies to all substances (corporeal). no substance is without mass
5) decreases with distance
6) extends forever if no other gravitational mass interferes
7) center of a mass has zero gravity
8) Earth and moon would collide if you let them go from resting positions. Two masses M1 and M2 released from rest at a distance R will collide after travelling D1=R*M2/(M1+M2) and D2=R*M1/(M2+M1). Closer to his wording and more simply written: D2/D1=M1/M2. It's correct. I believe it originates from Galileo's investigations, but Kepler was unsuccessful in convincing Galileo gravity extended from the moon to causes the tides. Can physicists prove 8 without using "Newton's" laws and "Newtonian" gravity?

The author of the newest translation of Kepler's "Astronomia Nova" (wherein Kepler states his gravity axioms and his 3 laws of planetary motion) is wrong in claiming that Kepler's gravity "[does not contain] the least notion that gravity extends to any other bodies than the Earth and the moon". It is so ingrained in modern thinking that Newton discovered gravity, that many excuses have to be made against Kepler's perfectly correct gravity (which was missing the 1/R^2 if you exclude the logical conclusion of his gravity and planetary laws). What was Newton's dwarf-on-shoulders of giants contribution? Kepler seemed to be simply unaware there was not an additional "motive" force other then Galileo's inertia that kept planets going around the Sun.

In Bethune's words from 1830 (London edition, not the 1932 Boston edition)
"[Kepler] also conjectured that the irregularities in the moon's motion were caused by the joint action of the sun and earth, and recognized the mutual action of the sun and planets, when he declared the mass and density of the sun to be so great that the united attraction of the other planets cannot remove it from its place."
Professor Forbes in 1909 in "History of Astronomy"
"it must be obvious that [Kepler] had at that time some inkling of the meaning of his laws--universal gravitation. From that moment the idea of universal gravitation was in the air, and hints and guesses were thrown out by many"  
In the words of Bethune again:
"Many who are but superficially acquainted with the History of Astronomy, are apt to suppose that Newton's great merit was in his being the first to suppose an attractive force existing in and between the different bodies composing the solar system. This idea is very erroneous .. the general notion of an attractive force between the sun, moon, and planets was very commonly entertained before Newton was born, and may be traced back to Kepler, who was probably the first modern philosopher who suggested it. "
Now compare this to Newton, known for denying credit to his peers, acting as if he discovered the attraction of gravity that was explained in the introduction to Kepler's book on planetary motion:
"Kepler’s laws, although not rigidly true, are sufficiently near to the truth to have led to the discovery of the law of attraction of the bodies of the solar system. The deviation from complete accuracy is due to the facts, that the planets are not of inappreciable mass, that, in consequence, they disturb each other's orbits about the Sun..."    
Here are more points in support of the view that Kepler was thinking in terms of a "universal" gravity:

1) Kepler thought magnetism must be in the Sun and planets because it was in the Earth, and he emphasized the similarities between gravity and magnetism (but did not claim they were the same but is accused of claiming magnetism was gravity).  He also correctly jumped to the conclusion that if the Earth spins, then so does the Sun, showing he viewed the Sun as a celestial body like the planets.

2) Kepler was justifiably accused of trying too hard to generalize basic ideas. Examples: "Harmonies" of the solar system being related to music, wondering if the Earth might breathe in some way analogous to fish (predating Gaia ideas), and imagining a story about travelling to the moon to meet potential inhabitants.

3) I have not seen anyhting to indicate he could be accused of thinking the Earth is different from any other planet. His comments that were inline with older thoughts of assigning personalities to planets seemed very much allegorical and an appeasement to the language and thought of the time.

4) He tried to apply his three planetary laws to the moon (see quote below), indicating he could think of the Earth acting as a "Sun" and the moon like a "planet", and since he stated gravity is always present, he had to be thinking it was an integral part of his planetary laws, and therefore applicable, if not equally applicable to all planets. As he states, you only need to make a correction for density if you know the volume in order to get the mass.

5) Kepler supported Bruno's opinion that stars were Suns with planets or moons in their orbits.

"Short History of Astronomy" by Arthur Berry, 1898
"The "Epitome of Copernican Astronomy" (1618)  contains the first clear statement that the two fundamental laws of planetary motion established for the case of Mars were true also for the other planets (no satisfactory proof being, however, given), and that they applied also to the motion of the moon round the earth, "
Universality of gravity aside, it is hard to underestimate the precision and completeness of Kepler's gravity axioms stated in the introduction to Astronomia Nova. It was presented as proof that the Earth can't be the center of the universe, and was in support of the idea that physics should be applied to astronomy. This was his most widely distributed text, and one of the few you can find for free on the internet. (This was his most famous book and it contains his 3 laws, but you can't find the rest of this text on the internet in any language except Latin. Only his 8 gravity axioms are easy to find.).  After reading his axioms, it is not possible to hold the opinion that Newton discovered gravity.  The only element of Newtonian gravity that is not blatant is 1/R^2, which is a geometrical effect of conserved "rays" of any type being emitted from a point source. It is also a derivable consequence of just the first two of Kepler's laws and Galileo's inertia. But he did not do this derivation (his math skills were easily up to the task which Einstein marveled at), possibly because he was thinking the planets were being subjected not only gravity, but some sort of friction and a sideways force from the Sun in the form of magnetism (or another unknown force) to overcome the friction.

Decades before Newton was thinking about falling objects, Kepler described with perfect precision in "meters" and "kg" how far the Earth would move "up" if we were able to stop the moon's orbit and drop it towards the Earth. He said this applied equally well to every mass. I can't derive this axiom unless I use F=ma and the knowledge that gravity causes an acceleration, i.e. Galileo's distance=1/2*a*t^2.  If you can derive this without using F=ma, please say so in the comments.

Other than not saying 1/R^2 explicitly, there are two potential problems with his gravity axioms.

One is that he used the phrase "cognate bodies" (mutuainter cognata corpora unitionem seu conjunctionem) which some have claimed means he was only talking about the Earth-moon system. There are many lines of reasoning that show this is a big assumption, as I've described above.

"If he knew gravity was universal, why didn't he talk more about it?"  He did place it in the introduction to his most famous work and call them axioms.  His friend Galileo could not believe the tides were caused by the moon and going further might have seemed premature. The Sun carrying such a force might have caused more problems with the church (other than trying to burn his mother at the stake). But more likely I think gravity is such an obvious everyday thing that Kepler could have considered it less important and trivial compared to determining the rules that govern the positions of the planets. Gravity was incredibly easier to see and understand compared to determining the rules that govern the wandering planets as viewed from Earth. We can easily say "ellipse" now, but determining it and proving it to others from the point of view of Earth's orbit in the midst of these other ellipses (that differed from circular by only 0.4% in the case of Mars) made Newton's mathematical restatement of Kepler's laws (and connecting it to Galileo's work) trivial (as Einstein explained). Kepler was no less shabby than Newton in optics and calculus-like work. What more needed to be said about gravity beyond Galileo?  Kepler had gravity correct far beyond Galileo's beliefs, but he could not see how it alone with inertia was enough to determine his laws without any other magnetic, animal, or friction forces. Gravity for him probably appeared to be obvious, everyday, simple, and solved, much less interesting than the solving the riddle of the planets.

Getting back to potential problems with his axioms: More advanced forms of the "cognate bodies" complaint are that he may have thought each planet ("cognate body") would have a different gravity such as a different "gravitational constant" or distance rule, especially if they had a magnet effect that was interfering with it and thereby modifying it.  Others think that maybe he thought masses were attracted only to their own "cognate" masses, which by some strange reasoning would include only their satellites but not other planets. Hence, they say, Newton was endowed with "universal" gravitation.

But if axiom 8 works on other planets with a different gravity constant, then (as I discuss below) the inertial force would have to change in lockstep with the change in the gravitational force (the "Einstein" principle of equivalency would remain intact because axiom 8 depends on it). This means the ratio of the gravity force to the inertial force would have to remain the same. But this is the same saying only the density has changed in other "cognate bodies" and he was fully aware of the effect of changes in density. His gravity adjusts for changes in density.

You might object that his knowledge of inertia was not advanced enough to make the above claim. Besides his comments on inertia and knowledge of Galileo (who discovered Newton's first law), his "Dream" book more than anything shows how well he understood inertia. He estimated that only with extreme precautions could aliens transport thin humans with strong bones (from riding goats since childhood) and opiates to the moon in only 4 hours by a "blast off" method accelerating into space, with great aliens standing on top of each other's shoulders to give the humans a boost "as if by gun powder", predating rocket ships.  He mentions easier travel after blast off, if it were not for the lack of air and the cold, and a similar deceleration process at the moon.  At 10 g, the most a person might be able to tolerate, I get that the acceleration would need to be for 300 seconds to get half way to the moon in 2 hours (4 hours for the full trip). That would need to be about a 3000 mile-high stack of aliens reaching out into space, all within the parameters of description. He had to understand inertia, falling bodies, and gravity well enough for this to come out so accurately in accordance with Newton's laws, with a good guess as to what a human body can withstand. I would have been hard-pressed to come this close without knowing 10 g, v=at, and s=1/2*a*t^2. He had some sort of access to the last two via Galileo. It is interesting that he made the travelling time "a little less than 4 hours" for another reason: at 1 g acceleration above Earth's gravity to half way and 1 g deceleration for the rest of the way, I get 3.88 hours. I could find no mention about gravity on the moon being less except his mentioning that the inhabitants grow to a very large size which is more likely when gravity is less.

Another potential problem is that in two different English translations he says the Earth is less "attracted" to a stone than the stone "seeks" the Earth. His latin is "trahat" and "petit", which can also be translated as "to pull forth" and "to strive for" or "travels to".  If he meant as some type of modern "force" then it is wrong, as the force is supposed to be equal on both. But this would conflict with his 8th axiom.  He may have meant it in a "distance" or "velocity" sense, in which case it is correct and in accordance with his 8th axiom.

Some might complain they did not have a distinct concept of "mass" so the translation could be in error. However, the latin was "moles" which is "mass", "weight", or "load", so I think it is sufficiently clear he meant it as you weigh it, and he distinctly indicated moles = density * volume which is mass.  He may not have thought about it as an absolute value of mass translatable to any planet as we know it, but he kept his comments on gravity perfectly correct by speaking in terms of ratios of mass.

Another complaint is that he thought magnetism was gravity but his two stones example did not require they be magnets. He entertained the idea that a magnetism or some other force from the Sun kept planets moving against a supposed friction. He also appears to have ascribed an "animal force" for this movement, inherent to bodies, but he may have specifically meant for this to mean inertia. He sometimes mentions gravity and mass have a parallel in the magnetic force in that magnets have an invisible type of action at a distance.

I can confirm axiom 8 only by using F=ma and Galileo's distance=1/2*a*t^2 for each mass, plugging in a=F/m, noting that they collide at the same time t, then thereby let F, 1/2, and t^2 cancel when I divide the two Galileo distance equations and get D1/D2=m2/m1 as Kepler states. This is an interesting simple result and it could be named "Kepler's law of mass attraction." It leaves open the possibility of gravity changing as any function of distance, but remains valid only if the well-known "Einstein" equivalency of the inertial force and gravity force on the mass are the same, allowing me to make the substation of "a" above.   How did he do it when he was not supposed to know F=ma?  Do his 1st two laws give this result when taken to the limit of an orbit of zero?  He is saying "if you take out my planetary laws by stopping the moon's orbit and remove all other forces on it, you will have a gravitational force that will cause the Earth and moon to come together at this particular point in space."  So he could see gravity clearly when he stopped thinking about "magnetic", "animal", and frictional forces in planets by making the orbits stop.

Finally I am getting to Kepler's own words, which are equal in weight to all the above. All the above was needed to defend against the various remarks made that try to diminish Kepler's contributions. Part of the problem is that there is no full English translation of Astronomia Nova, and no free version available on the internet in any language other than latin.  This is as astonishing as anything else. Everyone has had access to English Newton, but not German Kepler. And the dominance of British and US English speakers in world economics for the past 2+ centuries could not have helped Kepler's case.

In the quote below he indicates the earth/moon volume ratio is 54, but it is actually 49, showing he had a 3% error in the ratio of the diameters.

Kepler, on gravity in his introduction to "Astronomia Nova":
"It is therefore plain that the [common] theory of gravity is erroneous. The true theory of gravity is founded on the following axioms : Every corporeal substance, so far forth as it is corporeal, has a natural fitness for resting in every place where it may be situated by itself beyond the sphere of influence of a body cognate with it. Gravity is a mutual affection between cognate bodies towards union or conjunction (similar in kind to the magnetic virtue), so that the earth attracts a stone much rather than the stone seeks the earth. ...If two stones were placed in any part of the world near each other, and beyond the sphere of influence of a third cognate body, these stones, like two magnetic needles, would come together in the intermediate point, each approaching the other by a space proportional to the comparative mass of the other. If the moon and earth were not retained in their orbits by their animal force or some other equivalent, the earth would mount to the moon by a fifty-fourth part of their distance, and the moon fall towards the earth through the other fifty-three parts, and they would there meet, assuming, however, that the substance of both is of the same density. If the earth should cease to attract its waters to itself all the waters of the sea would he raised and would flow to the body of the moon. The sphere of the attractive virtue which is in the moon extends as far as the earth, and entices up the waters; but as the moon flies rapidly across the zenith, and the waters cannot follow so quickly, a flow of the ocean is occasioned in the torrid zone towards the westward. If the attractive virtue of the moon extends as far as the earth, it follows with greater reason that the attractive virtue of the earth extends as far as the moon and much farther; and, in short, nothing which consists of earthly substance anyhow constituted although thrown up to any height, can ever escape the powerful operation of this attractive virtue."
Part of the problem with not giving Kepler his due may be summed up by this Berry, in his "Short History of Astronomy,"
"as one reads chapter after chapter without a lucid, still less a correct
idea, it is impossible to refrain from regrets that the intelligence of
Kepler should have been so wasted, and it is difficult not to suspect at
times that some of the valuable results which lie embedded in this great
mass of tedious speculation were arrived at by a mere accident. On the
other hand it must not be forgotten that such accidents have a habit of
happening only to great men,"
But before believing any negative comments about Kepler, it's good to check Kepler's words against what others say he said.  Compare Sir David Brewster's comments to what Kepler actually said.

Brewster's libel:
"Although Kepler, in his Commentaries on Mars, had considered it
probable that the waters of our ocean are attracted by the moon, as iron
is by a loadstone, yet this opinion seems to have been a very transient
one, as he long afterwards, in his System of Harmonies, stated his firm
belief that the earth is an enormous living animal, and enumerates even
the analogies between its habits and those of known animated beings. He
considered the tides as waves produced by the spouting out of water
through its gills, and he explains their relation to the solar and lunar
motions by supposing that the terrene monster has, like other animals,
its daily and nightly alternations of sleeping and waking."
Here are Kepler's actual words:
"What so like breathing, especially of those fish who draw water into their
mouths and spout it out again through their gills, as that wonderful
tide! For although it is so regulated according to the course of the
moon, that, in the preface to my 'Commentaries on Mars,' I have
mentioned it as probable that the waters are attracted by the moon, as
iron by the loadstone, yet if anyone uphold that the earth regulates its
breathing according to the motion of the sun and moon, as animals have
daily and nightly alternations of sleep and waking, I shall not think
his philosophy unworthy of being listened to; especially if any flexible
parts should be discovered in the depths of the earth, to supply the
functions of lungs or gills."
Kepler seems sensible, fun, and open-minded to new ideas without abandoning his belief that the moon's gravity was the source of the tides..

Is it merely an English bias that has placed Newton so far above Germany's Kepler?  Einstein, another German, greatly admired Kepler's skill and grieved that Galileo did not give him more support.

The foundation of physics is the interaction between observation and mathematical condensation. Galileo successfully used observation to form ideas Kepler needed to form an idea of universal gravity. The physical "experimentalist" Brahe took careful measurements. Kepler religiously adhered to them, needing only a 0.4% error in Mars and a 2% error in Mercury (after a tremendously complex deduction of orbits) to abandon the perfect circles that he above others wanted to be true. He is very often accused of being soft, flaky, and religious in sentiment. But it was an incredibly strong faith in observation guiding theory without exception or imprecision, and with incredibly difficult mathematical work (according to Einstein), that enabled him to provide the physics Newton needed to do the mathematical combination of Kepler's gravity, Kepler's orbits, and Galileo's students' inertia.  In the English-speaking world Newton was the real great beginning or great leap forward in physics, but I think Kepler's work is closer to the truth.

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