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I read that when the stellar aberration of light is measured with the help of a telescope filled with water, the value of the aberration is equal as in the case of a telescope filled with air.

Just to draw your attention: note that it were the problems concerning aberration that led to SRT.

Report of G. Airy: Proc. Roy. Soc. Lond. 1871 Vol. 20 page 35ff.

E.g it is stated in this reference that

In 1810 François Arago performed a similar experiment and found that the aberration was unaffected by the medium in the telescope, providing solid evidence against Young's theory. This experiment was subsequently verified by many others in the following decades, most accurately by Airy in 1871, with the same result.

I didn't know that and am surprised. Articles on the internet that try to explain this are confusing in so far as they connect this with historical discussions about aether-drag theories and/or wave-particle theories.

I am looking for a possible explanation in terms of ordinary space-time coordinates in a variation of the 17th century particle model of light.
In the end only SRT can explain aberration correctly, but lets asume for one moment that we are in the 17th century, we use a particle model for light and we use $c_{\text{water}} < c$ which I know is historically incorrect.

Assuming a star perpendicular to earths surface, a simple argument would be as follows:
$$ \alpha_{\text{air}} = \frac{v}{c}$$ with $\alpha$ the stellar aberration angle as measured by the telescope filled with air ($\tan{\alpha}\approx \alpha$) and $v$ earth's velocity in the orbit around the sun and $c$ the velocity of light in vacuum.

Because $$c_{\text{water}} = \frac{c}{n}$$ the angle of aberration should be multiplied with $n$: $$ \alpha_{\text{water}} = n \;\alpha_{\text{air}}. $$

But there is refraction at the opening of the telescope filled with water. The angle of incidence is $\alpha_{\text{air}}$ and the angle of refraction is: $$\alpha_{\text{water}} = \frac{\alpha_{\text{air}}}n$$ using Snell's law and $\sin{\alpha}\approx \alpha$.

My question is: In the context of this model, do these two effects really compensate each other and is would that be a reason for stellar aberation to be independant of the medium?

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    $\begingroup$ Is your eye immersed in water when it looks through the telescope? If not, then the starlight also has to exit the water, at which point it regains its original aberration. $\endgroup$ – probably_someone Mar 20 '18 at 21:49
  • $\begingroup$ Yes, but I suppose the measurement is done in the inside of the water medium, cf. fen.bilkent.edu.tr/~gurses/Project_2.pdf, part V, page 4. $\endgroup$ – Gerard Mar 20 '18 at 21:58
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I have been thinking whether a classical description of aberration in a water-filled telescope could be as follows (I'm interested in hearing opinions from others!)

According to the classical explanation for normal aberration without water: $$ \cot{\theta'} = \frac{\cos{\theta} + \frac{v}{c}}{\sin{\theta}} $$ with $\theta' - \theta$ the angle of aberration.

For small angles this may be written as $$\theta' - \theta = \frac{v}{c} \sin{\theta}. \quad (\star)$$

$\theta = 90$ is a star in the zenith. Let's assume such a star in the zenith and the aberration angle measured is $\frac{v}{c}.$

Now the telescop is filled with water and because the telescope is slightly tilted but the star is still in the zenith, refraction occurs according to the starting point to use only classical arguments.

The refraction angle will be $\frac{v}{nc}$ with refractive index $n>1$. The "light ray" or particle that previously (without the water) moved vertically through the tube of the telescope, now appears to come from the direction $\theta = 90 -(\frac{v}{c} - \frac{v}{nc})$.

According to $(\star)$ the expected aberration angle for which one has to set up the telescope, becomes $ \theta' - \theta = \frac{v}{c} \cos{(\frac{v}{c} - \frac{v}{nc})}$.

Because for such small angles $\cos{\alpha} = 1 - \frac{1}{2}\alpha^2$ this is a second order effect and the expected aberration angle in first order does not change because of refraction.

So because the velocity of light in water is smaller $c_{\text{water}} = \frac{c}{n}$ the aberration angle should be greater. Refaction cannot change that fact (i.e. when the above reasoning makes sense).

The experiment of Airy showed that the aberration angle does not change, which shows that all these classical arguments are wrong. And only special relativity is able to explain stellar aberration.

The idea that a light ray is going vertically through the slightly tilted tube of the telescope is wrong, Airy's experiment refutes that idea completely.
More explicitly: the aberration effect has nothing to do with the telescope, the medium, not even the relative velocity of star and observer.
It is a relativity effect between two different observers (it may be the same observer at a different time). I find the invalidity of all classical arguments somewhat shocking.

Side note: Pauli comments about the Airy experiment, that it only shows the trivial fact that in both cases (water or no water) there is normal incidence.

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My question is: In the context of this model do these two effects really compensate each other, and is: would that be a reason for stellar aberation to be independant of the medium?

The shortest explanation, using 17th century science with a tiny sprinkle of current science, is:

The velocity of starlight does not depend on the state of motion of the medium of transmission (vis. ether or air). Why? Because if it did, starlight that entered the Earth’s atmosphere would be swept along by the moving air (or the ether with the embedded light in it would be dragged along by the Earth moving through the ether) and the aberration effect would not exist or would vary in certain directions. (Bergmann, pp. 21 – 22).


Source: "Early Attempts to Understand the Velocity of Light - Chapter 7":

Introduction: "It can be a daunting task to attempt to sort out and explain, let alone understand, the labyrinth of false assumptions, invalid theories, irrelevant equations, false conjectures, paradoxes, and misinterpreted experiments that (during the last two centuries) have confused and distorted physics in general, and Maxwell’s concept of the constant transmission velocity of light at c in particular. Most prominently included within this labyrinth are the arbitrary concepts of stationary ether as an absolute reference frame; Newton’s absolute space and absolute time; the theories of ‘ether drag’ and ‘partial ether drag;’ the Michelson & Morley null results; Fitzgerald’s, Lorentz’s and Einstein’s contraction of matter theories; the misapplication of Galileo’s Relativity to light; the Lorentz transformation equations; and above all Einstein’s theories of relativity.

Since a basic understanding of each of the above is necessary to an appreciation of the current untenable situation and to its ultimate solutions, we will do our best, in this and later chapters, to state and explain such confusion and distortions in as straightforward, simple and understandable terms as possible. We will thereafter set forth and explain the real facts and the real solutions for the false assumptions, paradoxes and other problems that have been created and still exist.".

...

" Does the constant transmission velocity of a light ray at velocity c relative to the medium of empty space vary, depending upon the linear motions of the bodies toward which such light ray propagates? The first experiment that deals with this question was conducted around 1728, the year after Newton’s death. British astronomer James Bradley (1693 – 1762) devised an optical experiment designed to measure the magnitude of observed stellar parallaxes. (see Figure 7.6A) But in the process Bradley discovered that he had to tilt the telescope slightly in the direction of the Earth’s motion around the Sun in order to keep the viewed star in the center of the telescope’s field of view. (Figure 7.6B)

This tilting requirement, which Bradley had discovered by accident, was later called the “aberration of starlight.” (Goldberg, pp. 429-432) The angle that the telescope must tilt in order to keep the viewed star in the center of the field of view is called the “angle of aberration.” (Id., p. 431) Since Bradley already knew the Earth’s approximate solar orbital distance, he also knew the approximate orbital speed v of the Earth (30 km/s). He computed the distance which he had to tilt the upper end of the telescope (in order to compensate for the orbital speed v of the Earth) compared to the distance light had to travel from the upper end of the telescope to his eye (at the velocity of light). This ratio v/c was approximately 1:10,000. (Bergmann, pp. 21 – 23) From this ratio Bradley also computed the approximate finite transmission velocity of light to be 303,000 km/s. (Hoffmann, 1983, p. 49)

Among other things, the aberration of starlight also demonstrated that the velocity of starlight does not depend on the state of motion of the medium of transmission (vis. ether or air). Why? Because if it did, starlight that entered the Earth’s atmosphere would be swept along by the moving air (or the ether with the embedded light in it would be dragged along by the Earth moving through the ether) and the aberration effect would not exist or would vary in certain directions. (Bergmann, pp. 21 – 22)

The aberration of starlight also implied that light had a constant transmission velocity relative to the medium of empty space, regardless of the relative linear speed of its source body (the star). Why? Because the angle of aberration (the ratio 1:10,000) was always the same for every star, regardless of the star’s speed or direction of motion relative to the Earth. (Id., pp. 21 - 23)

In addition, and very importantly, the aberration of light implied that the relative speed or direction of motion of the body (i.e. the Earth) toward which such starlight propagates did not alter the transmission velocity of the starlight, again because the angle of aberration (the ratio 1:10,000) was always the same for every star, regardless of the relative linear speed of the Earth in two opposite directions during its solar orbital motion.".

[Bolding done by me, to permit skipping through the text rapidly.]

SRT:

"The theory is "special" in that it only applies in the special case where the curvature of spacetime due to gravity is negligible. In order to include gravity, Einstein formulated general relativity in 1915. Special relativity, contrary to some outdated descriptions, is capable of handling accelerations as well as accelerated frames of reference.

As Galilean relativity is now considered an approximation of special relativity that is valid for low speeds, special relativity is considered an approximation of general relativity that is valid for weak gravitational fields, i.e. at a sufficiently small scale and in conditions of free fall. Whereas general relativity incorporates noneuclidean geometry in order to represent gravitational effects as the geometric curvature of spacetime, special relativity is restricted to the flat spacetime known as Minkowski space. A locally Lorentz-invariant frame that abides by special relativity can be defined at sufficiently small scales, even in curved spacetime.".

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    $\begingroup$ I appreciate the effort, but you totally missed the question at hand. $\endgroup$ – Gerard Mar 29 '18 at 8:58
  • $\begingroup$ The measurement is not done within the medium, the telescope wasn't designed that way; if that was your point/question. $\endgroup$ – Rob Mar 29 '18 at 10:48
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Parallax: Substantially correct, but simply stated, the matter is this: if indeed the earth moves in its orbit around the sun, then the place from which we observe the stars will be continually changing. Therefore, we shall see a nearby star as moving in a small circle against a background of more distant stars. What is more: looking at such a nearby star A from point M in March, and point S in September, and knowing the distance SM to be about 3x108 km, we can by means of triangulation obtain the distance to A.

Now Bradley found that Gamma Draconis indeed does describe a small circle with a radius of 20.5 seconds of arc (20”.5). The problem facing him was how to explain this phenomenon. Did it indeed result from the earth's revolution about the sun, and hence relative to the array of fixed stars? That is, did it show the parallax he had hoped to find or was the motion caused by the sun and stars circling with respect to an earth “at rest?” Bradley was forced to opt for the first alternative, but then had to reject it; for Gamma Draconis did not circle against the backdrop of stars, but all the stars joined in the motion which would imply that they were all at the same distance from earth. In other words, to accept the phenomenon as a parallax would mean re-introducing the discarded medieval concept of a Stellatum, a gigantic shell of stars centered on the sun which revolves about us. Since this was considered to be impossible, another interpretation of the observational facts had to be found. The circlets were decidedly not offering any Parallax.

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The point is this: Bradley's 20”.5 angle of aberration depends on the ratio between the speed of light and the orbital velocity of the earth. The latter, Boscovich reasoned, we cannot change; but the former we are able to reduce by means of observing the stars through a telescope filled with water. This will slow down the light, and consequently increase the angle of aberration. A water-filled telescope will thus have to be tilted more than an air-filled one.

Enter Airy

In 1871 G. B. Airy (1802-1892) implemented the verification of Bradley's aberration hypothesis proposed by Boscovich. As already noted, if the experiment indeed would show a larger aberration then this hypothesis would have been logically and irrefutably verified. Its modus tollende tollens logic by denying the consequent would also definitely disprove the geocentric theory of an earth at rest. Of course, Airy's water-filled instrument did not deliver the desired proof of the Copernican paradigm. Agreeing with somewhat similar tests already performed by Hoek and Klinkerfusz, the experiment demonstrated exactly the opposite outcome of that which had to be confidently expected. ”Actually the most careful measurements gave the same angle of aberration for a telescope with water as for one filled with air.

To say that in fine there was the devil to pay is not an overstatement, even when taken literally. Since the earth, as every right-thinking person is supposed to know, goes around the sun, it had to be possible somehow to explain away Airy's failure and to affirm Bradley's truth. One approach to do this stood in readiness. By means of Fresnel's so-called ”dragging coefficient,” the appearances could be saved. “It is, however, possible generally to prove Fresnel's theory entails that no observation whatsoever enables us to decide whether the direction in which one sees a star has been altered by aberration. By means of aberration one therefore cannot decide whether the earth is moving or the stars; only that one of the two moves with regard to the other can be established.

This note very helpful, ambiguous escape-hatch from an ”unthinkable” truth did, however, not have a very long life. In 1887 Michelson and Morley's once-and-for-all experiment to confirm the heliocentric creed turned out, as is well known, to be a dismal failure. Not only did it demonstrate an earth virtually at rest in omnipresent aether-space; the outcome also served “entirely to refute Fresnel's explanation of aberration.”

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