Simple experimental evidence that Earth revolves around Sun

  • What are the simplest experiments or calculations that give evidence that the earth revolves around the sun? Can you please explain them and reference the history? Many simple explanations such as this cite observations such as that relative position of two stars are observed from earth vary every night - which would not be true if the stars orbited the earth. But isn't the observation also consistent with a model where the stars orbit the earth but do so at different speeds, while the earth still orbits the sun? Simple explanations would be helpful.


    Actually, as @MarkOlson notes, the geocentric view is actually quite correct for the Sun/Moon/stars, since we can view all motion as relative. The problem is with the planets: they clearly don't orbit the Earth in simple circles or even ellipses. You can compensate by using epicycles, but having the planets revolve around the Sun requires fewer artificial constructs. From there, it's a small leap to treating our solar system as heliocentric, instead of having the Sun and Moon orbit the Earth and the other planets orbit the Sun.

    It doesn't If the Earth tried to move that fast, the stack of turtles holding it up would fall apart.

    @barrycarter That's basically Occam's Razor, which is useful as a guiding principle, but not really a proof.

    Does "simple" include accepting the modern theory of gravity? Because if you start accepting the relative masses of the sun and planets, "Everything orbits the earth" can't work.

    @Barmar: Occam's razor works both ways. Indeed, eliminating observations which are unachievable by unaided eyesight alone, and applying Occam's razor, we would be forced to adopt a flat earth as the center of the known universe. After all, believing that all seven (classical) planets are of the same nature (with regards to their motion and composition) requires the smallest set of assumptions compared to believing that the sun is a star which does not revolve around the earth, that the moon is a satellite revolving around the latter, and the rest are (modern) planets, revolving around the sun.

    @Lucian Quite true. The ancient Greeks probably thought that assuming all rotational motion was based on circles was simplest, so epicycles seemed reasonable.

    Of course, the most accepted models don't exactly have the Earth orbiting the Sun so much as the two orbiting a common center which happens to exist very much near the a point we could describe as the Sun's center of mass

    The sun and the stars *do* orbit the earth--but the math is very complicated. The choice of reference frame (the earth is stationary, the sun is stationary, the mass-center of the solar system is stationary) is chosen for convenience, and "earth is stationary" makes the math really hard.

    Distant stars would not "orbit", they just bob back and forth every 6 months. And they do that with a different amplitude, depending on their distance, but all together, in perfect unison (save their proper motion). And proper motion of the stars can also be observed. But neither that not the parallax could be instrumentally observed in Galileo's time.

  • The answer is ironic: Without good instruments, there is no evidence. The people who thought that the Sun went around the Earth were perfectly correct as far as the actual evidence went until the early 1700s and mid-1800s when two lines of evidence opened up that showed that the Earth moved.


    Aberration of Starlight


    Wikipedia has a correct but over-complicated explanation. The easiest way to think about it is to imagine yourself at a stop sign in a car in the rain, and the rain is falling straight down. When you start moving, the rain's apparent direction of fall changes so that it appears to be falling from ahead of you and slanting down towards you. That's aberration.


    In the early 1700s, the stars were discovered to be shifting position, and in 1727, James Bradley correctly identified it as abberation of starlight due to the motion of the Earth around the Sun. (For any star in the ecliptic, the Earth is moving towards it at some time of the year and away from it six months later.)


    Parallax


    Wikipedia's article on parallax is better, and I refer you to it for details. Basically, if you hold your finger up before you and look at it with your left eye closed, and then with your right eye closed, it appears to jump with respect to the background -- the wall beyond or the trees outside or whatever. Switch back and forth between your eyes quickly to see it clearly.


    As the Earth circles around the Sun, nearby stars also appear to shift their position relative to the more distant stars. A key point here is that there were good scientific reasons to suppose that the stars were much smaller than the Sun. Seen through a telescope, stars showed disks and if they were like the Sun, their distance could be deduced from those disks. And they were close enough that if the Earth really went around the Sun, parallax should have been observed. But it wasn't and the lack of any noticeable parallax was a strong empirical argument against Heliocentric theories.


    In reality of course parallax exists, but the parallax of all stars is small, because they are much further away than was estimated from their disks. (The visible disks were actually diffraction disks and not true disks at all -- but it was not until nearly a century later that diffraction began to be understood.) Friedrich Bessel first measure the real parallax of a star in 1838.


    I'm not completely convinced -- doesn't the change in solar zenith over the seasons strongly suggest Earth is orbiting, since a geocentric Sun would have to have some pretty large perturbations in its orbit.

    The change in solar zenith was known from prehistoric times and convinced no one of a heliocentric world, so, no, it doesn't strongly suggest anything until you make other assumptions (e.g., that the Sun is massive w/respect to the Earth or that something like gravitation creates the motions of the heavenly bodies) that are incompatible with geocentrism. It's not direct evidence of heliocentrism. (It's worth remembering that the lack of a visible parallax was already in ancient times one of the arguments used *against* heliocentrism.)

    Part 9 of TheOFloinn's "The Great Ptlemaic Smackdown" details the historical accretion of the evidence you mention as well as Guglielmi's 1791 measurement of lateral Coriolis force showing *rotation*. The prior eight parts are also a fun read of the detailed replacement of geocentric with heliocentric models and probable evidence tampering against Galileo (by a justifiably angry large political institution).

    @Eric Towers: Wonderful and very useful to this question. (I'd lost track of Flynn, but in the past had attended many of his excellent talks at Boskone and Worldcon.) Many thanks!

    Good answer. We tend to think of early cosmologists as flat-earther,denying an obvious truth. In fact they had good technical arguments for believing in things like 'a fixed dome of stars'. Without a good understanding of optics, how point sources can appear much larger than they actually are, they thought distant stars would have to be vastly bigger than our Sun in order to show no parallax.

    Does your answer contradict David Hammen's answer?

    @Micha Wiedenmann: Not really. I believe that he's correct, but that he is not answering the question asked. The question asked for "simple experimental evidence" why the Earth goes around the Sun, while his answer gives an interesting theoretical discussion of why it's difficult to make absolute statements about pretty much anything. Nonetheless, in Galileo's apocryphal words, "Still, it goes around the Sun". (Well, the barycenter, anyway.)

    @MarkOlson I am not convinced since most certainly there is a frame of reference in which the earth is at rest and the sun moves around the earth.

    As you wish. It still doesn't answer the question.

    It is also worth noting the observation of phases of Venus (https://en.wikipedia.org/wiki/Phases_of_Venus) in 1610 which ruled out the possibility that planets orbit Earth, although it is consistent with both Earth orbiting the Sun and Sun orbiting Earth while other planets orbit the Sun.

    Are you sure none of this is predicted by general relativity when you use a reference frame with a stationary Earth? Because if it is (which seems likely given that relativity is based on the notion of all reference frames being equally valid), then this would be wrong.

    "No evidence" So how did Galileo know the heliocentric picture was correct? Surely he wasn't just taking a stab in the dark.

    @littleO: Not a stab in the dark, exactly, but it seems to have been a combination of him thinking the heliocentric hypotheses was more elegant -- which it was -- and his own cantankerous nature. (Even without the near sainthood later myth-makers gave him, he was a *very* good scientist for his age. But he was also one of the more unpleasant people around and enjoyed driving off his friends and benefactors. He probably liked it because it would annoy people.) Read Owen Gingerich's book on him -- or read the "The Great Ptlemaic Smackdown" recommended a dozen comments above.

    @littleO: please read the series of 9 blog posts by TheOFlonn mentioned above in comments. “He just knew” is the answer. He was so sure hist theory is correct that he defended it to the point of almost dishonesty or self-deception. This is not to belittle Galileo the scientist, no question he was a genius, just to understand who was Galileo the man. By the way, Einstein also famously noted that he is not interested in experimental confirmation of his GR, as an experiment contradicting it would certainly be invalid. This human side of science is fascinating!

    @jpmc26:there is no such thing as a "reference frame with stationary earth" in GR. There are no stationary objects in the 4D spacetime (except certain surfaces around singular solution points, but that's a technicality). The GR does away with the inertial/non-inertial frame distinction by design. The main Einstein impetus was to unite inertia and gravity. The fact that the $m$ in $F=m a$ (inertial mass) and in $F=G\frac{M m}{r^2}$ (gravitational mass) is so freaking same measured to umpteen decimals cried for an explanation. GR is based on the postulate that they are one and the same.

    @kkm Obviously, when I say stationary Earth, I'm talking about the Earth's overall position in the spatial dimensions, not the time dimension. Surely, it's not impossible to define a coordinate system where the center of the Earth stays at (0,0,0) and it's not rotating. Or if it is for some reason, then this answer needs to address that.

    @jpmc26: Stays for *which observer*? Remember, we are talking about a 4D space with a non-Euclidean metric, and you can project it on an arbitrary 3D hyperplane in an infinite number of ways (even with the simplified metric, e. g.. in a local SR approximation). You seem to be forgetting that GR is a purely geometric theory. Every object, both observers and observed, is described by a straight line, “there and forever”. Nothing *moves*, everything *is*. And this space is equipped with a complicated metric. You cannot separate time and space until you simplify it to the Newtonian dynamics.

    @jpmc26 “this answer needs to address that” -- I respectfully disagree. An answer does not *need* to address every imaginable misconception. The answer did not even mention the GR, your comment did. I am only trying to clarify things for you because of this comment, not commenting about the answer in general. Now, this is a meta-meta-comment. :)

    @kkm It's annoying to me that you're nitpicking at my comments but that you don't ask Mark to specify what observer is observing the aberration or parallax. It's not "addressing every imaginable misconception." The answer needs to explain why these phenomenon are impossible in a geocentric reference frame that accounts for GR. In particular, GR is commonly taught and understood to be largely based on the notion of that there is no preferred reference frame, and this answer blatantly contradicts that concept. That needs to be discussed if that conception is wrong.

    @jpmc26: I think I understand your point that a theory you call GR cannot account for the aberration. Unfortunately, the GR I had a privilege to be taught was the now-classic Ensteinian one, and apparently differs from that which you are talking about. I admit I never heard of it (or any other GR), despite its being commonly taught and easily accessible, as you have noted. I sincerely wish you good luck with your research on reconciling the “blatant contradictions” with observations that it apparently suffers. I am afraid I cannot help you with that, but hope you would meet someone who can.

    @MarkOlson: could you expand on the parallax argument? if other stars don't have noticeable parallax why would that be evidence of geocentric vs. heliocentric models?

    @user7496: I added some material. Does that work?

    @MarkOlson Im sorry I don't follow this "correctly identified it as abberation of starlight due to the motion of the Earth around the Sun. (For any star in the ecliptic, the Earth is moving towards it and some time of the year and away from it six months later.)" - how was aberration correctly identified with motion of the earth? On your second parallax argument, this makes it seem like there was no evidence until parallax was measured precisely by Bessel - but wasn't the heliocentric model thought be supported experimentally long before?

    @user7496 It's hard to explain in a comment. (You might try to wade through the Wikipedia artcile I cited, but I found it confusing.) Basically, think of standing still in a car in a rainstorm with drops are falling straight down. Start moving and the angle of the drops change so that they appear to come from in front of you, and more so the faster you go. Light behaves the same way, and it manifests to us as an annual change in the apparent position of the stars. The effect is large, but hard to observe, because all stars in a given portion of the sky are shifted by the same amount.

    @user7496 To your second point, no, the Heliocentric theory was *not* supported experimentally before parallax and aberration were observed. Venus' phases showed that Venus went around the Sun and not the Earth, but the Tychonic system also predicted that. Heliocentrism was more elegant for many, but -- especially in the Copernican version which insisted on circular orbits -- had nothing experimental to recommend it in preference to some of the other theories of the time. The Great Ptolemaic Smackdown website referred to around here somewhere has a great narrative of the whole thing.

  • You cannot prove that the Earth orbits the Sun rather than vice versa because this goes very much against the grain of all frames of reference being equally valid (but some make a lot more sense than others). For example, it makes much more sense to use an Earth-centered, Earth-fixed point of view rather than a non-rotating geocentric, heliocentric, barycentric, or galactocentric point of view when modeling the weather or the tides. One could, for example, use a heliocentric or even galactocentric point of view to model the Earth's weather, but doing so would be beyond stupid.



    On the other hand, when modeling the behavior solar system it makes much more sense to use a heliocentric, or even better, a solar system barycentric point of view. One could however use an Earth-centered, Earth-fixed point of view because all frames of reference are equally valid (in theory). Doing so would of course make the equations of motion quite ugly, and uglier yet on trying to make those equations of motion relativistically correct. A geocentric point of view nonetheless remains theoretically valid -- even for modeling the behavior of the Milky Way.



    The problem with a geocentric point of view isn't that it's invalid (which it isn't). The problem is that advocates of geocentricism argued (and sadly, continue to argue) that this is the one and only valid point of view. This argument is invalid, because once again, all frames of reference are equally valid.



    Note well: Just because inertial frames are special in some sense does not mean that non-inertial frames are invalid.


    As an aside, one of my favorite tests of the orbital dynamics framework I developed for NASA's Johnson Space Center was to place an object in orbit about the Earth's moon, but to model the time evolution of that object from the perspective a Neptune-centered inertial point of view. It worked, at least for a short period of time. While all frames of reference are equally valid in theory, some choices are rather dimwitted compared to others due to numerical accuracy concerns. My choice of Neptune-centered inertial was intentionally dimwitted.

    Nah, you just needed more numerical precision! :-)

    *all frames of reference are equally valid* Not true. Both Newtonian mechanics and general relativity distinguish between inertial and noninertial frames of reference. (In GR, an inertial frame is a free-falling frame.)

    @BenCrowell while equations of motion in inertial frames are generally nicer, this doesn't make non-inertial frames invalid – just introduces fictitious forces.

    Also, the basic postulates of general relativity apply in exactly the same way in all reference frames, inertial or otherwise. Newton's postulates do not.

    @BenCrowell, perhaps you meant the special relativity, not the GR. In the GR framework, everything is "free-falling" along their geodesics, pretty much by postulate from which the theory is developed.

    How would the Sun orbiting the Earth explain trigonometric parallax?

    The stars would also be going in circles, consistent with the coordinate system. It's simply a different coordinate system, and the key lesson of relativity is don't reify your coordinates, coordinates are just the language you have chosen.

    @KenG Why should all the stars be executing little circles? Orbiting their own planets I guess? But why all with a period of 1 year? This motion tells us that it is more correct to say that the Earth goes around the Sun and argues strongly against a geocentric point of view.

    You are making an essentially sociological point. But we are talking about coordinates-- and there is no problem applying the postulates of general relativity to coordinates in which all the stars are moving in either little circles to make the Earth stationary in its orbit, or even in very large circles to make the Earth not spin either. That's all just the mathematical language. Einstein made the physics work in all these languages, that was the crux of his great achievement. Philosophically, he used that this must be possible as a guiding principle of his theory.

  • If you start with the idea that the planets, the sun, the moon and the earth are all bodies that all move through space, exclude the apparently fixed stars, and then see what evidence there is as to how they move relative to each other, then in that context there is some evidence to be found in naked-eye astronomy aided by navigational instruments available even to the ancients.



    The patterns of observed movement of the planets is evidence of heliocentric orbit. The visible planets follow certain patterns. First, Mercury and Venus:




    • They are always seen in the vicinity of the sun.

    • The observed angular separations of both Mercury and Venus from the sun have a regular pattern.

    • Mercury has a much closer maximum separation than Venus, and its angular separation changes at a much faster rate.

    • Both planets stay close to the ecliptic, and never oscillate normal to it.

    • Both planets' orbits around the sun can be documented and predicted with relative ease. This can be done imprecisely even without a telescope, though it is much harder for Mercury, being so close to the sun.



    Beginning with the premise of bodies moving through the heavens, I believe the evidence is there for Mercury and Venus having a heliocentric orbit. Kepler described it precisely, but the ancient Greeks were able to model their motion very well without telescopes in the Antikythera Mechanism in geocentric terms.



    If an ancient Greek astronomer had wanted to precisely model the motion of the inner planets in heliocentric terms, he could have. The way to do it is to assume the fixed stars are rigidly fixed, and measure the angular distances between them all, and then plot the motions of the moving planets among them. Sextants and other devices were used by ancient mariners who were highly skilled even with primitive ones. So this could have been done to realize the "simple experiement or calculation" you are asking for. Whether it ever was done, with that question in mind, is a bit different issue.



    Now for the earth itself. Even in the ancient world the relationship between the sidereal day and the solar day has been well understood. The precession of the sun around the ecliptic plane is evidence of a heliocentric orbit. One just has to model it to make this clear. Ancient calculations relating to sidereal time and the Metonic cycle reveal that the earth's heliocentric motion could have been mathematically modeled, if conceived of and desired.



    As for the outer planets, to my mind this is the least intuitive, but there is evidence for a heliocentric orbit for them too, but only by building on the idea that earth and the inner planets orbit the sun. This comes from observing their retrograde motion. These planets will move retrograde against the "fixed background stars" at certain times, and those times can be correlated to their angular separation from the sun. Also the different planets move through the zodiac at different speeds, which also correlate with the amplitude of retrograde motion.



    If you simulate all this with a heliocentric orrery, it is very plainly evident that we on an inner, faster planet observe an outer, slower planet in its orbit. The ancient Greeks had enough skill to model the motions of Mars, Jupiter and Saturn in their Antikythera Mechanism in geocentric terms. So it follows that a precise, mathematical model of heliocentric motion for the outer planets was within their reach, if they ever reached for it.



    There is also some evidence that at least some ancient thinkers were able to decode all this into a heliocentric model. The ancient Greek Aristarchus of Samos had a heliocentric model. However, Plato and others seemed to disfavor it, and this reconstruction of the Antikythera Mechanism which is believed to come well after Aristarchus' day features a geocentric orrery which models planetary retrograde motion. And heliocentric thinking stayed within the minority in the west until the modern age. Perhaps the obvious geocentric orbit of the moon, or the question of the stars (whether they should be included in any correct model or not), or the lack of a universal theory of gravity, sufficiently obscured for them what to us is clear.


    I think you're disregarding the fact that the heliocentric model doesn't do a much better job of actually modeling the system until you give up on circles. The first attempts at heliocentric models (even at the time of Galileo) had the issue of having even more exceptions than the geocentric ones due to using circles which don't actually work well. http://tofspot.blogspot.com/2013/10/the-great-ptolemaic-smackdown-table-of.html does seem to do a great job of explaining this.

    @DRF You can probably tell I approached this from the point of view of, did the Greeks have enough *information* and *theory*, if not the insight, to prove heliocentricity at their level of mathematics, physics and technology? Following that same line, I don't know, but I wonder if you have to have good quality lenses in order to disprove circular orbits. Galileo had pretty good lenses, so maybe the Greeks were not capable of his level of precision. I'm not sure.

    The Antikythera Mechanism amazingly had an eccentric *gear* in its lunar module, accounting for the moon's elliptical orbit, which I imagine is close enough to us for a half-decent sextant to measure eccentricity. But for the others it looks like all circles in Antikythera, with the caveat that not all of the device was recovered. Nor have I seen any reference online to the Greeks discussing such issues with the visible planets.

    Although your blog author you linked to makes a pretty good case that the Greeks could have proved even elliptical orbits at their level, if they had followed all the thought processes of the later European astronomers, without lenses.

  • The best experimental evidence is probably retrograde motion. The data is not easily acquired: it takes a long time to collect, not to mention an astronomer would have to stay up every night keeping painstaking measurements of the positions of each object. But it can be done (ancient Greeks were aware of it) and in the modern world you can simply use a simulator like Stellarium.



    Download Stellarium, start it up, and navigate to your local position. Then set the simulation running and speed it up many times. You should see the sun and stars rotate around you. Then turn the ground off (so you can see through the Earth), turn the atmosphere off (so you can see stars during the day), switch to the equatorial mount (Ctrl + M; this is the mount where most of the sky is stationary), and zoom out until the Sun, the Moon, and all the planets appear to move in a circle.



    Now look carefully at the motions of all the planets. You should see that the Moon (and the Sun) goes in circles without ever slowing down. This is what you'd expect if they went around the Earth. However Mercury does not follow this motion - it visibly disappears around the Sun. Mars behaves differently as well: it goes round and round, then stops, goes backwards, and then goes round and round again. This last behavior is called retrograde motion and its explanation occupied a lot of ancient astronomy. Ancient Greeks came up with a complicated theory of epicycles to explain it, given that the planets orbited the Earth and moved in perfect circles (neither of these are true in modern knowledge).



    However retrograde motion can be easily explained if Mars didn't go around the Earth, but went around the Sun instead. This would simply mean that Mars goes retrograde when we overtake it on its orbit. In addition, this also explains how each time Mars goes retrograde, it is at its brightest, plus it is on the opposite side of the sky relative to the Sun. It also explains why Mercury does its loops around the Sun.



    This doesn't mean that the geocentric model is not able to account for the same observations, but it's drastically simpler. In the heliocentric model, every planet goes round the Sun on a simple path, an ellipse. In the geocentric model, every planet goes round the Earth, but on epicycle after epicycle. That's when we apply Occam's Razor and conclude that the simpler explanation is correct.


  • Well... the seasonal cycle is evidence enough that the Earth and Sun are orbiting each other. Whether A orbits B or B orbits A is an argument about relative mass. If you find that the movement of all the other planets are consistent with them orbiting the Sun but not the Earth, you can conclude that the Sun's mass is enormous and therefore barely affected by the pull of the Earth.


  • Detailed observations of any star in the sky reveal that the Earth moves in an elliptical orbit with a speed of approximately 30 km/s.



    When the line of sight velocities of stars are measured using the Doppler effect, they have to be corrected for the motion of the Earth. If they are not, then one would see an unexplained modulation of the velocities, with a period of 1 year and an amplitude of up to 30 km/s that would differ depending on the direction of the star with respect to the Earth-Sun orbital plane.



    Likewise, a geocentric model fails to explain why an observer on the Earth sees the positions of stars on the sky execute periodic ellipses on the sky with amplitudes (a.k.a. the trigonometric parallax) that appear to be inversely correlated with how far away they are, but all with a period of one year.



    Perhaps these are not the"simple" experiments that you were thinking of, but the universe cannot always be understood with what is visible to the naked eye and common sense.


  • This might oversimplify things but here's my go:




    • Create a flat surface (the larger the better as long as it stays flat), e.g. by placing a board on a still surface of water.

    • Put up a long pole (the longer the better) vertically on that surface at noon.

    • Measure its shadow (direction and length), which needs to be completely on the flat surface.

    • Have someone do the same (esp. same length of pole) at the same time far to the north of you (the further the better).

    • Have a third of the exact same measurements far to the south of you.



    Evaluating the measurements should establish:




    • Earth's surface is roughly spherical (actually earth is an oblate ellipsoid but you need more than 3 measurements to confirm that)

    • Earth diameter is within reported values (+/- expected deviation for measurement error and the fact that you only measured a very rough estimation)

    • Rough estimation of earth-sun distance by triangulating



    Using a pinhole camera you can now achieve a rough estimation of sun's actual diameter by its apparent diameter and the distance estimation from above. Even accumulating all the measurement errors, the difference in size between sun and earth should be some orders of magnitude.



    Attach two balls to the opposite ends of a rod (the lighter the rod compared to the balls the better). The balls need to be rough approximations of the above established measurements (e.g. you could guess the sun is pure hydrogen and the earth is pure iron to achieve an estimation of mass). Attach a string to the rod and find the point of balance. Most likely it will be way to the ball representing the sun (you need to accommodate for weight of the rod).



    You can now make the two balls circle each other while hanging from the string.



    Which one revolves around the other?


    Feel free to extend/correct this answer. I thought about how to have the described experiment/model as simple as possible. The only hope for this to achieve anything is that the difference in diameter and mass between earth and sun is so staggering large that the numbers work out although they are likely to be 50% (or more) off from the actual values.

  • With relatively simple equipment it is possible to observe the behaviour of the satellites of Jupiter. Assuming the hypothesis that Jupiter and all the planets rotate around the Earth, it should be expected that the occlusion of the satellites by Jupiter would happen on a highly regular basis. But what we see is the event happening at different times relative to Earth-bound clocks, even not very accurate ones, which proves that the orbit of Jupiter is not a simple epicycle around the Earth. Also the observation of any satellite not directly orbiting the Earth casts doubt on the Earth-centric view.


  • Very simply: because of relative motion, no proof exists. Any situation that you come up with can be explained by a tweaked geocentric module. Albert Einstein came to the same conclusion when he said "I have come to believe that the motion of the Earth cannot be detected by any optical experiment." and "...to the question whether or not the motion of the Earth in space can be made perceptible in terrestrial experiments. We have already remarked... that all attempts of this nature led to a negative result. Before the theory of relativity was put forward, it was difficult to become reconciled to this negative result."


    It really makes sense to elaborate on this particular quote. You are being downvoted, because this well-known quotation is often seen torn out of its contest to show as if E. supported the geocentric model. I am surprised, however, that no one except you has so far mentioned the GR in this context. This looks like an introduction to a very good and educational answer, if only abruptly ended.

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