Looking for the origin of the idea that the Sun’s gravity diminishes as distance increases, I found this on Wikipedia:
Ismaël Boulliau known as Bullialdus was a friend of Pierre Gassendi, Christiaan Huygens, Marin Mersenne, and Blaise Pascal, and an active supporter of Galileo Galilei and Nicolaus Copernicus. It is for his astronomical and mathematical works that he is best known. Chief among them is his Astronomia philolaica, (published 1645). In this work he strongly supported Kepler‘s hypothesis that the planets travel in elliptical orbits around the Sun, but argued against the physical theory the latter had proposed to explain them.^{[1]} In particular, he objected to Kepler’s proposal that the strength of the force exerted on the planets by the Sun decreases in inverse proportion to their distance from it. He argued that if such a force existed it would instead have to follow an inversesquare law:^{[2]}

As for the power by which the Sun seizes or holds the planets, and which, being corporeal, functions in the manner of hands, it is emitted in straight lines throughout the whole extent of the world, and like the species of the Sun, it turns with the body of the Sun; now, seeing that it is corporeal, it becomes weaker and attenuated at a greater distance or interval, and the ratio of its decrease in strength is the same as in the case of light, namely, the duplicate proportion, but inversely, of the distances that is, 1/d².^{[3]}
Brilliant deduction, but then he dropped the ball. Wiki continues:
However, Bullialdus did not believe that any such force did in fact exist.^{[2] }After writing the abovequoted passage, he then went on to write:

… I say that the Sun is moved by its own form around its axis, by which form it was ignited and made light, indeed I say that no kind of motion presses upon the remaining planets … indeed [I say] that the individual planets are driven round by individual forms with which they were provided …^{[3]}
In his Principia Mathematica of 1687, Isaac Newton acknowledged that Bullialdus’s determination of the sizes of the planets’ orbits ranked with Kepler’s as the most accurate then available.^{[4]}
This seems like a sop to soothe Newton’s own conscience (if he had one).
Bullialdus was one of the earliest members of the Royal Society, London, having been elected on April 4, 1667, seven years after its founding. The Moon‘s Bullialdus crater is named in his honor.
1645 for publication and 1667 for acceptance to the Royal Society are interesting dates in relation to others who claimed primacy for their conception of the Inverse Square Law. From Wikipedia again:
Robert Hooke and Giovanni Alfonso Borelli both expounded gravitation in 1666 as an attractive force^{[1]} (Hooke’s lecture “On gravity” at the Royal Society, London, on 21 March; Borelli’s “Theory of the Planets”, published later in 1666). Hooke’s 1670 Gresham lecture explained that gravitation applied to “all celestiall bodys” and added the principles that the gravitating power decreases with distance and that in the absence of any such power bodies move in straight lines. By 1679, Hooke thought gravitation had inverse square dependence and communicated this in a letter to Isaac Newton. Hooke remained bitter about Newton claiming the invention of this principle, even though Newton’s “Principia” (Published in 1687) acknowledged that Hooke, along with Wren and Halley, had separately appreciated the inverse square law in the solar system,^{[2]} as well as giving some credit to Bullialdus. From Kepler’s third law, which states that the period of a planet’s orbit is proportional to the cube of it’s semimajor axis, Newton derived his law of gravitation:
So where in his publications did Kepler allude to the inverse proportionality as Bullialdus attests?
In an annotation to the second edition of the Mysterium Cosmographicum published in 1621 Kepler reified his earlier views on the origin of motive force:
If you substitute for the word “soul” the word ”force,” you have the very principle on which the celestial physics of the treatise on Mars etc. is based…Formerly I believed that the cause of the planetary motion is a soul, fascinated as I was by the teachings of J.C. Scaliger on the motory intelligences. But when I realized that these motive causes attenuate with distance from the sun, I came to the conclusion that this force is something corporeal, if not so properly, at least in a certain sense.
William Gilbert, after experimenting with magnets decided that the center of the Earth was a huge magnet. An unreferenced claim on Wikipedia is that his theory had led Kepler to think that a magnetic force from the Sun drove planets in their own orbits. However, according to historian of science Alexander Koyre the traditional assumption of scholastic physics was that the power of gravitational attraction remained constant with distance whenever it applied between two bodies, and such was assumed by Kepler and also by Galileo in his mistaken universal law that gravitational fall is uniformly accelerated, and also by Galileo’s student Borrelli in his 1666 celestial mechanics.
But is Koyre correct? If Bullialdus objected to Kepler’s proposal that the strength of the force exerted on the planets by the Sun decreases in inverse proportion to their distance from it then it would appear not. And if Gilbert’s magnetism of the Earth led Kepler “to think that a magnetic force from the Sun drove planets in their own orbits”, then this is more evidence he understood the concept of a diminishing force. The dropoff of the field around a magnet is far more easily observed than the change in gravitational acceleration of a free falling body at different altitudes. But perhaps there’s something subtler going on here. Could it be that Kepler wasn’t thinking of a magnetic force of gravity (how would nonmagnetic items such as sheep or apples be pulled to Earth?), but a magnetic force which “drove planets in their own orbits” not only towards the Sun, but around it too? Some clues can be gleaned from this paper by H.H. Ricker.
Fully answering that question will require some more source material and research, which will have to wait for a day or two, but it’s worth considering that Newtons alternative – the ‘innate force’ wouldn’t have been appealing to an earlier natural philosopher surrounded by an ‘Aether’: Too much drag, not enough momentum. Something would have to be ‘topping up’ the planets angular velocity.
The problem of Newton’s ‘innate force’ is still with us today. Orbital energy is lost to friction, tidal losses etc. Why hasn’t the system slowed down more than it has, if the only thing keeping the planets going round avoiding a Sundive, is the momentum they acquired at the formation of the solar system 4.5 billion years ago? Is a continual transition to lower orbits maintaining their speed? Not that we’ve been able to measure.
Tallbloke,
You have a fantastic feeling for digging up interesting topics. This one need a lot of discussion. Just one comment now due to time limitation.
The gravitational force proportional to the inverse distance squared is only working at 2 moments in one orbital circuit. No one can figure out the geometry of the solar system or the motion of the planets and satellites based upon the Newtonian formula. Kepler provided observational evidence for the constancy of mv no theoretical understanding.
Obviously something very important is lacking.
Why would the planets slow down in their orbits?
An orbit is the trajectory of least resistance already, to slow an orbiting body down one has to accelerate it in some way.
I suppose I should note that I’ve spent a good portion of the last 25 or 26 years learning about the interaction of matter with spacetime. So one could say I’m a bit familiar with the subject at this point.
Treating gravity as a force doesn’t work very well, luckily it does make sense as a geometrical property of spacetime.
If you took two hypothetical solar systems frozen at a moment in time, one with the sun, and one without the sun, you could draw a circle at various orbital distances, right?
If you measured the radius and circumference of that circle, you could then calculate the value of pi.
For the solar system without the sun present, the radius at the average distance of the Earth would be 1 AU, the circumference should then match the calculated value.
When you compare that with the sun present, you may be surprised when you learn that the circumference is less than 2pi*r, which Feynman referred to as “excess radius”, and which I like to describe as things being deeper towards the center of a gravity well.
When you’re standing on the ground, if you drew a circle around the planet at say, the height of your shoulders, you could measure it and calculate what the radius should be.
If you then measured the actual distance to the center of the planet, you would find it is larger than your calculated value, so it is literally deeper in that direction.
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Alternatively, you could think of the circle you drew around the planet being smaller than you would expect.
Similarly, the solar system is deeper in the direction of the sun (and to some extent jupiter) than any other direction.
At a large enough distance the effect this has is to change what we would naively expect an undisturbed trajectory to be for an object moving past the sun.
Naively you would expect the body to follow a straight line and continue off without stopping.
Actually you find that the body curves around towards the sun, following the straightest trajectory allowed for a body with that mass and velocity.
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To make a body follow what we would consider a “straight” line, you have to push it away from the gravitationally defined “straight” line.
Similarly, to slow a body following said path, one must act upon it. There is no friction from spacetime itself, only the solar wind, magnetic fields, and the gravity wells of the other planets can affect it.
[Reply] Can I have the reference to Feynman’s ‘excess radius’ please.
Max,
If I understand what you have said correctly, then if I use a laser to measure from a point in space to another point in space and I get a distance of say 1Au, if at one of the points I then introduce a gravitational singularity with a gravitational force similar to or larger, even much larger than the Sun, I re measure and then get a distance greater than the 1 Au I measured previously.
Even though to an external observer or frame of reference the distance has not changed.
Does this tie up with Miles Mathis saying that at solar orbital sizes pi is 4 not 3.142…
Something is still missing though, what keeps it all stable. I am reminded of a youtube (?) video showing an object mysteriously held a distance above a superconducting magnet, and the object could be pushed or pulled nearer or further away from the magnet and it returned to it’s previous position, and it did this even when held upside down.
Thus magnetism can simultaneously push and pull. Perhaps when it comes to planets gravity can also push and pull, or the two in concert.
[Reply] Mathis has some interesting new papers on the Pi=4 argument involving the ‘Manhatten Taxi’ problem. A reader of his has also found a term in the Lagrangian dynamic equations that pushes instead of pulling. Which is an odd thing for gravity to be doing. I’ll link the paper when I find it again.
Or, have I interpreted that incorrectly, is it the other way around ?
The radial distance hasn’t changed, but the circumferential distance has.
Six Not So Easy Pieces, Page 119 is where he explains it in terms of drawing a circle on a sphere and measuring the radius across the surface of the sphere.
Six Not So Easy Pieces, Page 124 is where he extends the description to 3 dimensions.
Six Not So Easy Pieces, Page 126 is where he explains Einstein’s derivation of spacetime curvature in this manner.
Great book in general, I highly recommend it and Six Easy Pieces, both by Feynman of course.
I disagree with Wikipedia completely. They are assuming, incorrectly, that he is speaking of the radial force in that paragraph. The way I read it, Bullialdus is speaking tangentially. Let me annotate his paragraph the way I read it:
I do not see this even speaking of the inverse square gravitational attraction between the bodies. Many at that time thought the spinning of the sun about its axis was somehow pushing the planets around in their orbits and this seems to be what he is addressing.
Oh, didn’t mean to skip over your post, J Martin.
It’s not that the radius or circumference has changed per se, it’s that we aren’t in a flat world.
The curvature of spacetime can be understood as, noted in the post above, excess radius when you compare direct measurements to calculations.
Your post about a singularity is uh, well, singularities are a problem for everyone in theoretical physics still, but yes, if you were to introduce a massive compact object into the center of a formerly flat region of spacetime with a circle that had a measured radius upon measuring again you would find the actual center to be around half a kilometer further from the edge of the circle than you expected it to be.
No clue who Mathis is, so I can’t say what he is talking about, I’m just going off of Feynman and Einstein here.
Standing on the shoulders of giants.
Whilst I can’t say I understand much of this stuff I do wonder where there speed of gravity comes into it.
”Too much drag, not enough momentum. Something would have to be ‘topping up’ the planets angular velocity.’
A clue might be found in the flux transfer events NASA discovered a few years ago.
‘During the time it takes you to read this article, something will happen high overhead that until recently many scientists didn’t believe in. A magnetic portal will open, linking Earth to the sun 93 million miles away. Tons of highenergy particles may flow through the opening before it closes again, around the time you reach the end of the page.’
http://science.nasa.gov/sciencenews/scienceatnasa/2008/30oct_ftes/
Tim,
Tom van Flandern’s speed of gravity paper was really interesting. Apparently, some Japanese scientists recently claimed to have proven gravity does move at light speed.
https://tallbloke.wordpress.com/2012/04/13/tomvanflandernthespeedofgravitywhattheexperimentssay/
Oldbrew: I’m going to be researching what Kepler thought about magnetism and the Sun. That guy was way ahead of his time. Perhaps that 1604 supernova really did inspire him.
Something relevant to this thread is that Kepler was aware that light followed an inverse square law, and why it does. So it seems strange he used the area law for gravity. More investigation needed.
>.>
Gravity doesn’t move.
Curvature changes propagate at c, so if you were to magically disappear the sun instantly, the planets would continue orbiting until the last photons to leave the sun reach them.
For 8 minutes after the sun vanished, we would continue orbiting it as usual, before the sky went dark.
Not sure what sort of gravitational ripples you would get from ripping a solar mass out of a region of spacetime, I would be surprised if there were none though.
Remember though, gravity isn’t a result of force carrier exchanges. Orbiting bodies aren’t made to follow their trajectories due to absorption/emission of particles or force field attraction/repulsion, an orbit is the path of least resistance geometrically, that’s all there is to it.
Max says: ‘an orbit is the path of least resistance geometrically, that’s all there is to it’
Max: do you have any theory about the elliptical shape of orbits?
Does anyone see what I’m seeing here? Clue: ratio of the radii.
Max says: ‘an orbit is the path of least resistance geometrically, that’s all there is to it’
Once some unspecified ‘action at a distance’ has curved the spacetime around massive bodies.
oldbrew posted a link from http://en.wikipedia.org/wiki/Ellipse#Ellipse_as_hypotrochoid
Can you see the little moon whose gravity vector (!) helps to create the barycenter into which its planet “falls” 😉
@ Chaeremon 2:29 pm
The caption says: ‘An ellipse (in red) is a special case of the hypotrochoid with R = 2r’
Could there be another reason why it’s a ‘special case’? If the point of intersection on the radius line isn’t 6, but 6.18 then: if R = 10, r = 6.18, and R/r = 1.618, which is the golden ratio.
‘The special case’ is confirmed here.
http://www.maths.surrey.ac.uk/hostedsites/R.Knott/Fibonacci/phi2DGeomTrig.html
What ratio of circle sizes (radii) makes the ellipse equal in area to the ring between the two circles?
The answer is again when the inner radius is 0.618 of the larger one, the golden ratio.
The calcs that prove it are included.
@ oldbrew 4:25 pm, 5:37 pm; Yes the math looks good, good find.
What use you make of this, what directions?
@ Chaeremon
Allow me to speculate a bit. If Phi can be related to a type of ellipse, and planetary orbits are always elliptical, the two facts could be linked.
Quote:
‘Newton’s Law of Universal Gravitation is F = G M m / R 2
Kepler’s Third Law is R 3/ T 2 = constant ( the value of this constant depends on the mass of the object that is being orbited)
Newton’s law can also be written as F = m 4 (pi) 2 R / T 2
Combining this form of Newton’s law with Kepler’s law gives
F = m 4 (pi) 2 R / T 2 = G M m / R 2
so R 3 / T 2 = 4 (pi) 2 G M so the constant in Kepler’s law depends only on the mass of the object since 4, pi and G are constant’
http://wiki.answers.com/Q/How_does_newton's_laws_relate_to_kepplers_law
Key phrase: ‘Kepler’s law depends only on the mass of the object’ (the rest are constants).
One of the constants is ‘pi’. If it is replaced by another one that has the same value, the same result must be obtained.
Part of the equation is 4pi = 12.56637 (*all figs rounded off)
pi / 1.2 = 2.618 = Phi^2
4.8 x Phi^2 = 12.56656 (*)
So if Phi replaces pi, with a tiny discrepancy factor is about 0.000015 (*), the Newton/Kepler gravitation law still works.
@oldbrew, the Kepler+Newton equations you posted.
😦 I cannot study, understand or assess this (c&p’ed) concoction of mathematical symbols where so many are missing 😦 can the equations be unobfuscated (or have better ref?)
But I agree, generally, with your suggestion at this level: circle, ellipse and eggish oval [1] can be made equivalent by equal area, without violating Kepler’s or Newton’s laws, etc. Nobody can object this, nobody can demand any nonmathematical role for the ellipse’s 2nd focus (without blowing up cosmology, I mean), anyways.
If phi can play a role, so be it; there be unthoughtof physical phenomena around the corner 😉
[1] observable view of orbital path, faster at the tip (as demanded by Newton’s F), slower at the other side (F again):
http://mathworld.wolfram.com/Oval.html
Oldbew: two bodies produce two curved regions of spacetime, and both have their trajectory altered by the presence of the other body.
It is not quite an ellipse though, and precession complicates that further, though truthfully it isn’t appropriate to ignore the motion of the sun/solar system as well.
“Once some unspecified ‘action at a distance’ has curved the spacetime around massive bodies.” ~tallbloke
Uh, the stress energy tensor curves spacetime where the body is located the greatest, that curvature decreases with distance, the action is local, and drops off as you move away from the source of the curvature towards infinity.
Indeed, the fact that the distance between two massive bodies is not the same as it would be if they were massless means it would be more accurate to say ‘action on the definition of distance’, rather than ‘action at a distance’, if anything.
Now, I’m fairly certain that there is no way to explain observations of gravitational lensing by invoking some sort of force carrier exchange ‘acting at a distance’, so we can safely toss any such model into the dustbin, observations trump theory, after all.
Thanks Max. Correct me if you see it another way, but first I’ll repeat an earlier quote:
What ratio of circle sizes (radii) makes the ellipse equal in area to the ring between the two circles? The answer is again when the inner radius is 0.618 of the larger one, the golden ratio.’
In that case a series of imaginary concentric circles around the sun could be replaced by ellipses, without altering the average distance of each of them from it (much)? Obviously that’s not a 3D representation but most planets are in similar alignment to the ecliptic plane.
http://en.wikipedia.org/wiki/Invariable_plane
I like the idea that gravity is due to a state of repulsion between matter and space, rather than by matter to matter attraction. How it would manifest locally would depend on both absolute and relative scale, as well as distance. And the vector nature of the engagement between matter and space, would be the means for everything to keep moving, right down to atomic levels.
The interaction isn’t a vector, it’s a tensor, Ulric.
As for the way we orbit the sun: as I said, it’s not quite as simple as you think, the main component of our velocity is towards Vega as the solar system spirals around the galaxy at about 200 km/s, we orbit the sun at around 48 km/s, and the rotation of the planet is between 100 and 300 m/s roughly to the east.
Some more on it: http://calgary.rasc.ca/howfast.htm
Max, the fact remains that the Earth follows a Keplerian orbit around the EarthSun barycentre to with 0.00013% or so. I’m not saying the improved accuracy of the latest model isn’t great. It’s just that its much easier to visualise the relative motion using accessible concepts. To be sure the steel balls on the rubber sheet looks great. I don’t find it a great help with the dynamics of multiple bodies though. Each to their own I guess.
Well, if you look at it from a heliocentric frame, it’s actually not following a circular/elliptical orbit at all, it’s tracing a spiral, which itself is dragged sideways by the motion of the milky way and virgo supercluster, respectively.
General Relativity and geometrical explanations of gravity can both predict and explain this, but a force exchange model is unable to do so.
If spacetime were flat, and if the presence of the other planets could be ignored, and if the solar system weren’t moving, it would resemble a keplerian orbit.
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Oh, small correction, the smaller debris all orbits the centerofmass of the solar system, which is mostly determined by the Sun and Jupiter: http://zidbits.com/wpcontent/uploads/2011/09/solarsystembarycenter.jpg
The influence of the Earth changes the EarthSun barycenter by a hundred or so miles, last I checked.
The motion of the solar system means that every second the point which WAS the location of the barycenter is displaced by about 150~ miles in the direction opposite Vega relative to the actual location.
The time it takes for the curvature changes to propagate means the Earth is orbiting a point which trails about 72,000 miles behind the centerofmass of the Solar System.
http://spaceplace.nasa.gov/barycenter/
EarthSun
Ohh, this looks fun to play with: http://www.orbitsimulator.com/gravity/articles/ssbarycenter.html
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That is all before we even take into consideration what effects GR has on things, being unable to define something like “these two events happened at the same time” for objects at different locations in a gravity well has a tendency to complicate the calculations of something like a barycentric coordinate system.
@Max
I do mean vector, and the movement towards Vega is irrelevant.
Max: The time it takes for the curvature changes to propagate means the Earth is orbiting a point which trails about 72,000 miles behind the centerofmass of the Solar System.
If that was the case we’d be seeing extra swings in TSI of around 30W/m^2 over a ~20 year cycle consonant with the motion of the Sun WRT the centerofmass of the Solar System. Which we don’t. We see a smooth swing of ~120W/m^2 at TOA over the annual orbital cycle. The Earth orbits the EarthSun barycentre, to within 0.00013%, not the centerofmass of the Solar System. QED. It may possibly lag behind that orbit due to the propagation of curvature, but that doesn’t change the fact it follows the Sun, not the centre of mass of the whole system.
Well, tallbloke, I myself do not like the rubber sheet model either, as it gives the awful impression that the curvature is “down” relative to some bisecting a spherical body.
If a rubber sheet analogy is to be used, it should be noted that the only time it remotely resembles the actual curvature is for a plane which is tangent to the surface of the sphere.
If you take all of the planes tangent to every point on a sphere, you can get a tangent bundle representing the curvature tensor at that distance, if you then do the same for every spherical surface inside and outside of the sphere you will have a pretty good representation of the curvature induced by a single body.
Here’s an animation along those lines.
If you add another body, the situation becomes dramatically more complicated, and three or more bodies… well, there is a reason we only have approximate solutions for even the VASTLY more simple Newtonian case.
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The Earth orbits the Sun, the Sun orbits the center of mass, see where I’m going with that?
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Ulric, a vector depends on the coordinate system chosen, a vector is a tensor, but a tensor can be used to give a mathematical representation regardless of the coordinate system, and most importantly it can be used to transform between coordinate systems.
The movement towards Vega means the orbital trajectories are spirals, not ellipses, we never return to the same position relative to our position at any point in the past, only the arrangement relative to the other bodies in the solar system is repeated, and even then it isn’t exact.
Watch the video I linked above for the three or more bodies example, notice when it zooms over to the system of bodies orbiting each other as they leave the central cluster, that’s what I mean when I say the solar system is a collection of bodies following spiral trajectories which only look like elliptical orbits from a coordinate system centered on and moving with the Sun.
“It may possibly lag behind that orbit due to the propagation of curvature, but that doesn’t change the fact it follows the Sun, not the centre of mass of the whole system.”
Yes, tallbloke, the speed of gravity, a fine and mind boggling problem! 😉
I read the paper you put up above. And I’ll read it again later as my head stops hurting! Now just how is curvature instant (or very, very close)? That makes you rearrange some of your longheld visualizations of just how gravity does operates, curved spacetime and all. That was one addition, light time delay, I was going to add to “improve” my ephemeris integrator but looks like that might be exactly the wrong tack to take.
Ok, see, the reference to Flandern explains why my point about the motion of the Sun was deemed unimportant.
If you don’t account for the motion of the bodies you might think the propagation was instantaneous.
Max: The Earth orbits the Sun, the Sun orbits the center of mass, see where I’m going with that?
Agreed, that’s what I pointed out on the ‘expert opinion’ thread recently to Lawrence Wilson. The difference between this situation and what you said earlier that I disagreed with is important for Earth’s energy balance and climatic variation. I’m not sure yet ‘where you’re going with that’.
the solar system is a collection of bodies following spiral trajectories which only look like elliptical orbits from a coordinate system centered on and moving with the Sun.
They still look very much like elliptical orbits from a coordinate system centred on the centre of mass of the system (solar system barycentre SSB) and moving with it. Just that those elliptical orbits have their individual planetarySun barycentre as the focus of the individual orbit, not the Sun’s centre.
I agree that from a galactic frame of reference, everything in the solar system is moving in spirals. The choice of reference frame isn’t entirely arbitrary though. In the context of understanding the change in the forces the bodies are subject to, the difference in acceleration towards the galactic centre experienced by planets as they orbit within the solar system is truly tiny. So from that perspective, I think you need to justify why we need to complicate our visualisation of the system with the spiral motion.
only the arrangement relative to the other bodies in the solar system is repeated, and even then it isn’t exact.
Agreed. However, it’s similar enough at certain periodicities for similar effects to be recorded in paleo proxies. This is how we come to identify certain ‘cycles’ such as Hale, Gleissberg, De Vries, Halstatt etc. Again, how we characterise these things depends on the relevance of the ‘near enough’ repeating events to the object of study. In our case, the solar variation object and the Earth climate object.
The orientation relative to the Sun as far as things like insolation go isn’t when you need to consider the various ways in which the system is moving.
In the case of gravitational effects, the mass of a body as well as the state of motion must be considered.
A body can only be influenced by events within the past light cone for that body, i.e. only effects which propagate at c or less, otherwise causality goes *splorch* and everything explodes into frogs or something.
A body in motion has a tilted light cone, this tilt is responsible for the appearance that the Earth is orbiting the position of the Sun at this very moment, rather than the position of the Sun 8 minutes ago.