Emission Theories and Their Refutation

The Michelson–Morley null result does not prove special relativity. That sentence is the reason this article exists. In 1887 the fringes did not move and the ether died — but a second theory predicts exactly the same null, for a completely different reason, and it walked away from Cleveland untouched. The theory is Walther Ritz's, published in 1908: light is a projectile. It leaves its source at $c$ relative to the source and thereafter coasts, exactly as a bullet leaves a moving gun with the gun's velocity added on. Write the observed speed as $c_{\text{obs}} = c + kv$ , where $v$ is the source's velocity toward you — $k = 1$ for pure ballistics, $k = 0$ for relativity. Now look at Michelson's apparatus. The lamp is bolted to the bench, so source and mirrors are mutually at rest, and on an emission theory the light travels at $c$ relative to the apparatus, isotropically, whatever the bench is doing. Both arms take $2L/c$ exactly, and the fringe shift is zero — not small, not second order, but identically zero, for every $k$ . Ritz passes the most famous experiment in physics without breaking stride. So 1887 kills the ether and leaves two survivors. To separate them you need what an interferometer structurally cannot give you: a source moving fast, relative to the observer, over a very long baseline. That is not a laboratory; that is the sky. In 1913 Willem de Sitter pointed out that a star in a spectroscopic binary swings toward us and away from us every orbit, so on an emission theory its light arrives out of order — and the further the star, the worse the scrambling. The orbits we see are clean.

Key takeaways

An emission (ballistic) theory says light leaves its source at $c$ relative to the source: the observed speed is $c_{\text{obs}} = c + kv$ , with $v$ the source's velocity along the line of sight. Pure Ritz is $k = 1$ ; relativity is $k = 0$ . Such theories keep the principle of relativity and throw away the constancy of $c$ — the opposite trade from Lorentz's ether.
The owned result: Michelson–Morley is degenerate. Its source is bolted to its bench, so on an emission theory light moves at $c$ relative to the apparatus in every direction and $t_\parallel = t_\perp = 2L/c$ exactly, giving $\Delta N = 0$ . The ether predicts $0.3735$ ; emission and relativity both predict $0$ ; the observation was $\lesssim 0.01$ . The 1887 null kills the ether and leaves two survivors side by side.
De Sitter's kill (1913): a binary star's light, emitted at $c \pm kv$ half an orbit apart, arrives split by $\Delta T \approx 2kvD/c^2$ . For a binary at $300$ light-years, pure ballistics smears one $4$ -day orbit across 73 days. Observed orbits are Keplerian, so $k$ is tiny.
That bound was then voided by Fox (1962): interstellar gas absorbs and re-radiates starlight, resetting its speed within about a parsec. The kill shot had to be extinction-free — Alväger et al. (1964, CERN) timed $\gamma$ -rays from $\pi^0$ decays whose source moved at $\beta = 0.99975$ , and found $k = (-3 \pm 13)\times10^{-5}$ .
A null result only refutes the theories that predicted a signal. Michelson–Morley predicted one on the ether theory alone, so it could only kill the ether; the bullet theory had to be shot down separately, by binary stars and by a beam of pions.

The core insight

Light as a bullet is the natural hypothesis: it saves Galilean kinematics, needs no ether, no contraction, no strange time — and it passes Michelson–Morley trivially, because that experiment carries its own lamp. What it cannot survive is a source that moves relative to you. On an emission theory, light emitted while a binary star approaches would overtake light emitted half an orbit later while it recedes, arriving out of sequence after a journey of centuries and scrambling the star's apparent motion beyond recognition. The sky shows clean Keplerian orbits. Light is not a bullet.

Where this lives today

Emission theory is dead, but the test it inspired is very much alive, because "does the speed of a signal depend on its source?" is a question you can ask of every messenger the universe sends. Modern searches for Lorentz-invariance violation reuse de Sitter's logic wholesale: take a source at cosmological distance and let the baseline amplify a speed anomaly into a timing anomaly. The 2017 neutron-star merger GW170817 ran exactly that experiment across messengers: a gravitational wave and a gamma-ray burst from the same event $130$ million light-years away arrived $1.7$ seconds apart, pinning the speed of gravitational waves to that of light to about one part in $10^{15}$ . De Sitter's lever is now standard equipment. He just used it first.

The physical picture

A machine gun on a moving truck fires bullets whose ground speed is the muzzle speed plus the truck's. Emission theory says light is like that: the "muzzle speed" is $c$ relative to the lamp, and a moving lamp adds its own velocity on top. Maxwell says the opposite — light is a wave whose speed is fixed by the properties of empty space, indifferent to whoever launched it. The two pictures are indistinguishable whenever the source sits still in your frame, which is exactly the situation in every bench-top optics experiment ever performed, Michelson's included. To tell them apart you need a lamp that is moving, fast, and far away. Nature runs that experiment continuously: it is called a binary star.

Figure

Emission theory applied to a binary star. Light emitted while the star approaches leaves at c + v; light emitted half an orbit later, while it recedes, leaves at c − v. Over a baseline of D = 300 light-years the fast beam overtakes the slow one, and the star's apparent motion is scrambled.

Now the whole logical landscape in one figure — because the mistake this article exists to correct is a mistake about which experiment refuted what.

Figure

PUNCHLINE — the decision tree, with the numbers on it. Michelson–Morley 1887 kills the ether and NOTHING else: emission theory predicts the same null. The ladder below is the real execution, and every rung is a separate experiment.

Read the bottom ladder carefully, because its shape is the history. Ritz sits at $k = 1$ ; de Sitter pushes to $2\times10^{-3}$ ; Fox pulls the rug out, because that bound was measured through interstellar gas; Alväger re-establishes it where no extinction argument can touch it; Brecher reaches $2\times10^{-9}$ . Emission theory was not refuted once — it was refuted, un-refuted, and refuted again.

The core equations

An emission theory is defined by a single parameter, and the cleanest way to state it is to let that parameter interpolate between the two rival worlds. Let a source move with velocity component $v$ along the line of sight toward an observer, and let the light it emits be observed to travel at

c_{\text{obs}} = c + k\,v \qquad (1)

where: $c$ — the emission speed relative to the source; $v$ — the source's velocity toward the observer, in the observer's frame; $k$ — the emission coefficient. Physical reading: $k = 1$ is pure ballistics — light is a bullet, and Galilean velocity addition holds for it exactly (Ritz). $k = 0$ is postulate 2 of Einstein's Two Postulates — light forgets its source. The experimental history of this subject is the history of $k$ being squeezed toward zero. Now feed a Michelson interferometer to equation (1), and watch it refuse to discriminate. The source is bolted to the bench, so the source's velocity relative to the observer — which is what enters (1) — is zero, whatever the bench is doing relative to anything else. Hence $c_{\text{obs}} = c$ in every direction of the apparatus, and both arms are trivial.

t_{\parallel} = t_{\perp} = \frac{2L}{c} \quad\Longrightarrow\quad \Delta N_{\text{emission}} = 0 \qquad\text{versus}\qquad \Delta N_{\text{ether}} = \frac{2L\beta^{2}}{\lambda} \qquad (2)

where: $L$ — arm length; $\lambda$ — wavelength; $\beta = v_{\oplus}/c$ — the apparatus's speed through the hypothetical ether. Physical reading: the emission-theory null is exact, not a leading-order accident — there is no $\beta^2$ residue to hunt for, and it holds for every $k$ . The ether prediction, which The Michelson–Morley Experiment owns, is $0.3735$ fringes. The experiment separates ether from not-ether; it does not separate Ritz from Einstein. To break the degeneracy the source must move relative to the observer, and the effect must be given room to grow. A spectroscopic binary supplies both: the star's radial velocity swings by $\pm v$ each orbit, and its light then crosses a distance $D$ of hundreds of light-years. Light emitted on the approach travels at $c + kv$ and arrives early; light emitted half an orbit later travels at $c - kv$ and arrives late.

\Delta T = \frac{D}{c - kv} - \frac{D}{c + kv} = \frac{2kvD}{c^{2} - k^{2}v^{2}} \;\approx\; \frac{2kvD}{c^{2}} \qquad (3)

where: $D$ — distance to the binary; $v$ — the star's orbital speed along the line of sight; $\Delta T$ — the spread in arrival times between the two extreme orbital phases. Physical reading: the effect is linear in $k$ and linear in $D$ — and $D$ is astronomical, which is the entire trick. A coefficient far too small to see on a bench becomes overwhelming after centuries of flight. When $\Delta T$ grows to the orbital period $P$ , wavefronts from different phases arrive intermingled and the apparent orbit ceases to make sense.

Where the refutation comes from

Derivation AMichelson–Morley cannot tell Ritz from Einstein

This derivation kills a piece of folklore that appears in many otherwise careful books: that the Michelson–Morley null establishes the constancy of the speed of light. It does not. A null result refutes exactly those theories that predicted a signal — and emission theory never predicted one. What follows proves the emission-theory null is exact: not small, not second order, but identically zero at every speed and for every $k$ . The Michelson–Morley Experiment owns the ether calculation; it is recalled for contrast, not re-derived.

Step 1. State the emission postulate precisely, because the whole argument turns on a frame. Light leaves its source at $c$ relative to the source; in any other frame its velocity is the source's velocity plus a vector of magnitude $c$ . Why: " $c + kv$ " is meaningless until you say $v$ relative to what. It is the source's velocity relative to the observer — and in an interferometer that number is zero.

Step 2. Take the strictest case, $k = 1$ , in the "space" frame where the bench drifts at $v$ . Outbound along the drift the light moves at $c + v$ while the far mirror runs away at $v$ : closing speed $(c+v) - v = c$ . On the return it moves at $-(c-v)$ while the splitter advances at $+v$ : closing speed $(c-v) + v = c$ . Each leg takes $L/c$ , so $t_\parallel = 2L/c$ . Why: the drift cancels twice and <strong>exactly</strong> — once against the mirror's recession, once against the splitter's approach. No residue at any order.

Step 3. Do the transverse arm. In the source's frame the beam is aimed straight across at $c$ ; in the space frame its velocity is $(v, c)$ — it inherits the bench's sideways motion. But the transverse mirror inherits it too, so relative to the apparatus the light's velocity is $(0, c)$ : a clean perpendicular crossing, and $t_\perp = 2L/c$ . Why: the light is dragged along by its source exactly as the apparatus is dragged by the Earth — so the ether's $\sqrt{1-\beta^2}$ aiming penalty never arises.

Step 4. Subtract. $t_\parallel - t_\perp = 0$ , identically in $v$ — no expansion, no small parameter. So $\Delta N_{\text{emission}} = 0$ , and by Step 1 this holds for every $k$ . Set it beside the ether's $\Delta N = 2L\beta^2/\lambda = 0.3735$ and the observed $\lesssim 0.01$ . To discriminate, the source must move relative to the observer — and no instrument carrying its own lamp can arrange that. Why: this is the payload. Two of the three theories on the table predicted zero, and zero is what was seen. A null result must always be quoted together with the list of theories it was capable of killing. *Verified symbolically in verify_aq-sr-emission-theories-refutation.py: sympy confirms $t_\parallel = L/((c+v)-v) + L/((c-v)+v) = 2L/c$ and $t_\perp = 2L/c$ , identically in $v$ ; the ether comparison gives $\Delta N = 2 \times 11 \times 10^{-8} / (589\times10^{-9}) = 0.3735$ .*

Go deeper · Derivation B · 2 minDe Sitter's binary stars and the bound on k

Derivation BDe Sitter's binary stars and the bound on k

Now the experiment that can discriminate. The goal is to turn "the orbits look normal" into a number — an upper bound on $k$ — and then to be honest about how weak that number is, because the crude version of this argument is routinely quoted as though it delivered the tight modern limits. It does not. The literature bounds are sourced, not derived here; what this derivation owns is the mechanism and the scaling.

Step 1. Set up the split. A star in a binary at distance $D$ has line-of-sight velocity $+v$ at one orbital phase and $-v$ half an orbit later. The two beams leave at $c \pm kv$ , so their arrival times are spread by equation (3): $\Delta T = 2kvD/(c^2 - k^2v^2) \approx 2kvD/c^2$ . Why: the effect accumulates over the whole flight, so distance is not a nuisance but the amplifier — the one lever an astronomer has that a laboratory physicist does not.

Step 2. Put numbers to it. Take a representative spectroscopic binary: $D = 300$ light-years $= 2.84\times10^{18}\ \mathrm{m}$ , period $P = 4$ days, line-of-sight orbital speed $v = 100\ \mathrm{km/s}$ . For pure ballistics ( $k = 1$ ), $\Delta T = 2vD/c^2 = 6.32\times10^{6}\ \mathrm{s} = \mathbf{73.1\ days}$ . Why: the orbit lasts four days and its light arrives smeared over seventy-three — a factor of <strong>18</strong>. There is no sense in which such a star has an apparent orbit at all.

Step 3. Convert the absence of scrambling into a bound. Distortion becomes gross once $\Delta T$ reaches $P$ ; demanding that it does not, $2kvD/c^2 \lesssim P$ , gives $k \lesssim Pc^2/(2Dv) = \mathbf{0.055}$ . Why: pure ballistics is excluded by a factor of eighteen — decisively, but only just. And the bound goes as $1/D$ , so a binary ten times more distant tightens it tenfold, to $k \lesssim 0.0055$ at $3000$ light-years. Distance is the whole argument.

Step 4. Be honest about the gap, and record the loophole. The bound above rests on "we would have noticed the star in two places at once", and lands near $10^{-2}$ — not $10^{-9}$ . De Sitter's own $k < 2\times10^{-3}$ used a sharper criterion: the absence of spurious orbital eccentricity in the radial-velocity curves. And then Fox (1962) voided all of it — starlight crossing the interstellar medium is absorbed and re-radiated within an extinction length of order one parsec, after which it travels at $c$ relative to the gas, not the star. Against a $300$ -light-year path that is under two percent. Why: a serious objection, not a quibble — it reopened the whole question for two decades.

Step 5. Close the loophole. The answer needed a fast source, radiation that does not interact, and a short vacuum path — and it came from an accelerator. Alväger, Farley, Kjellman and Wallin (1964, CERN) timed $\gamma$ -rays from $\pi^0 \to \gamma\gamma$ decays whose parent pions moved at $\beta = 0.99975$ . Pure ballistics predicts photons at $1.99975\,c$ — nearly double $c$ , an enormous target. They measured $c$ : $k = (-3 \pm 13)\times10^{-5}$ . Brecher (1977) later reached $k < 2\times10^{-9}$ from X-ray binary pulse timing. Why: the CERN result is extinction-free by construction — $\gamma$ -rays barely interact, and the flight path is a vacuum pipe. It is the experiment de Sitter would have run if he could have built it. *Verified in verify_aq-sr-emission-theories-refutation.py: sympy derives $\Delta T = 2kvD/(c^2-k^2v^2)$ and its series $2kvD/c^2$ ; the representative binary gives $\Delta T = 73.1$ days $= 18.3P$ at $k=1$ and $k \lesssim 0.0547$ ; the literature bounds $2\times10^{-3}$ , $(-3\pm13)\times10^{-5}$ and $2\times10^{-9}$ are asserted as sourced, non-derived values.*

Limiting cases

Case	Result	Meaning
$k = 0$	$\Delta T \equiv 0$	special relativity: light forgets its source, and no arrival-time split exists at any distance
$k = 1$	$\Delta T = 2vD/c^2$	pure Ritz ballistics — 73 days of smear on a 4-day orbit at 300 ly
$0 < k < 1$	$\Delta T \propto k$	partial re-emission models; halving $k$ halves the smear
Galilean limit $c \to \infty$	$\Delta T \to 0$ , bound vacuous	the argument lives on the finiteness of $c$ : infinitely fast light arrives all at once
$D \to 0$ (a lab baseline)	$k \lesssim Pc^2/(2Dv) \to \infty$	no bound at all — astronomy supplies the lever arm $D$ , the accelerator the lever arm $v$
Source at rest ( $v = 0$ )	$c_{\text{obs}} = c$ for every $k$	the Michelson–Morley degeneracy — the experiment cannot see $k$ , because its lamp does not move

Worked examples

1 · The 1887 fringe shift under all three theories

The Michelson–Morley apparatus had effective arm length $L = 11\ \mathrm{m}$ and used sodium light, $\lambda = 589\ \mathrm{nm}$ ; the Earth's orbital speed gives $\beta \approx 10^{-4}$ . What does each theory predict, and what was seen?

Step 1. The stationary ether. The Michelson–Morley Experiment owns this calculation: rotating the table by $90^\circ$ swaps the arms and gives $\Delta N = 2L\beta^2/\lambda = 2 \times 11 \times 10^{-8} / (5.89\times10^{-7}) = \mathbf{0.3735\ fringes}$ — a signal the instrument could see forty times over.

Step 2. The emission theory. The lamp is at rest in the apparatus, so the source velocity entering $c_{\text{obs}} = c + kv$ is $v = 0$ , and the light travels at exactly $c$ in every direction of the bench. Both arms take $2L/c = 7.34\times10^{-8}\ \mathrm{s}$ , the difference is zero, and $\Delta N_{\text{emission}} = \mathbf{0}$ — exactly, and for every $k$ . Special relativity says the same: postulate 2 makes $c$ isotropic in every inertial frame, so $\Delta N_{\text{relativity}} = \mathbf{0}$ too.

Step 3. The observation: $\Delta N \lesssim \mathbf{0.01}$ . Compare the four numbers — $0.3735$ , $0$ , $0$ , $\lesssim 0.01$ — and the verdict writes itself. The ether is excluded by a factor of nearly forty. Emission theory and relativity are both perfectly consistent with the data, and the experiment cannot choose between them. That is what a degenerate experiment looks like.

2 · What the binary stars actually bound

Take a representative spectroscopic binary: distance $D = 300$ light-years, orbital period $P = 4$ days, line-of-sight orbital speed $v = 100\ \mathrm{km/s}$ . How badly would pure ballistics scramble it, and what bound on $k$ follows?

Step 1. Convert. $D = 300 \times 9.4607\times10^{15} = 2.838\times10^{18}\ \mathrm{m}$ ; $P = 4 \times 86400 = 3.456\times10^{5}\ \mathrm{s}$ ; $v = 1.0\times10^{5}\ \mathrm{m/s}$ .

Step 2. The arrival-time split at $k = 1$ , from equation (3): $\Delta T = 2vD/c^2 = 2(1.0\times10^{5})(2.838\times10^{18})/(2.998\times10^{8})^2 = 6.32\times10^{6}\ \mathrm{s} = \mathbf{73.1\ days}$ . Against a $4$ -day orbit that is a smear of $\mathbf{18.3\ orbital\ periods}$ — light from eighteen different orbits arriving together. The star would not look eccentric; it would look like several stars.

Step 3. Invert to a bound. Requiring $\Delta T \lesssim P$ gives $k \lesssim Pc^2/(2Dv) = \mathbf{0.055}$ . Pure ballistics ( $k=1$ ) is excluded by a factor of eighteen.

Step 4. Be honest about what this is. The crude bound is $\sim 10^{-2}$ , not $10^{-9}$ . De Sitter's own $2\times10^{-3}$ needs the sharper criterion of absent spurious eccentricity; Brecher's $2\times10^{-9}$ needs X-ray pulse-timing residuals, not merely "the orbit looks Keplerian". And every optical bound is exposed to Fox's extinction objection. The clean, extinction-free number is Alväger's: $k = (-3 \pm 13)\times10^{-5}$ .

Physics of the portrait

De Sitter is remembered for cosmology — the empty expanding universe that bears his name. But the 1913 paper that concerns us is barely two pages, contains no field equations, and settles a question by looking at stars that had already been catalogued. It is one of the cheapest decisive experiments ever performed: the data were free, and the theory it killed was killed with arithmetic.

Origins · 2 minWillem de Sitter and how this was discovered

Willem de Sitter

1872–1934 · Sneek · Groningen · Cape Town · Leiden

Born in Sneek in the Netherlands, de Sitter read mathematics at Groningen and was pulled into astronomy by J. C. Kapteyn, who sent him to the Cape Observatory in South Africa from 1897 to 1899. He became professor of astronomy at Leiden in 1908 and director of its observatory in 1919. His lasting fame is in relativistic cosmology: in 1917 he found a solution to Einstein's field equations describing an expanding universe empty of matter — "de Sitter space" — and his correspondence with Eddington from neutral Holland during the First World War carried general relativity across the lines to Britain. But in 1913, before any of that, he published a short note pointing out that the spectroscopic binaries already in the catalogues ruled out Ritz's ballistic light. It required no new observation, no new instrument and no new mathematics — only the realisation that the sky had been running the experiment all along.

History

1908

— Walther Ritz (1878–1909), the same Ritz as Rayleigh–Ritz, publishes his emission theory in Annales de Chimie et de Physique: light is a projectile, emitted at c relative to its source, and the ether is unnecessary.

1909

— Ritz and Einstein publish a joint note in Physikalische Zeitschrift recording their disagreement over the radiation problem. Ritz dies later that year, aged 31.

1910

— Daniel Frost Comstock, and independently Richard Tolman, raise the double-star argument against emission theories; Tolman publishes "The Second Postulate of Relativity" in the Physical Review.

1913

— Willem de Sitter, in Physikalische Zeitschrift 14, 429, argues that spectroscopic binaries would show grossly distorted orbits if light inherited its source's velocity, and bounds the emission coefficient at k < 0.002. Zurhellen sharpens it to k < 10⁻⁶ in 1914.

1962

— J. G. Fox points out in the American Journal of Physics that starlight is absorbed and re-radiated by the interstellar medium within an extinction length, erasing any source-velocity signature and voiding all the optical binary bounds. He reviews the whole field again in 1965.

1964

— Alväger, Farley, Kjellman and Wallin, at CERN, measure the speed of γ-rays from π⁰ decays whose source moves at β = 0.99975, and find k = (−3 ± 13)×10⁻⁵: the extinction-free kill.

1977

— Kenneth Brecher, using the pulse timing of X-ray binaries, reaches k < 2×10⁻⁹, a millionfold improvement on de Sitter.

Going further

Texts. Rindler, Introduction to Special Relativity, 2nd ed., §1 · Ritz, Ann. Chim. Phys. 13 (1908) 145 · de Sitter, Physik. Zeitschr. 14 (1913) 429 · Fox, Am. J. Phys. 30 (1962) 297 and 33 (1965) 1 · Alväger, Farley, Kjellman & Wallin, Phys. Lett. 12 (1964) 260 · Brecher, Phys. Rev. Lett. 39 (1977) 1051.

Provenance. Everything numeric is recomputed in verify_aq-sr-emission-theories-refutation.py: the exact emission-theory null $t_\parallel = t_\perp = 2L/c$ (symbolic, identically in $v$ ); the ether's $\Delta N = 0.3735$ at $L = 11\ \mathrm{m}$ , $\lambda = 589\ \mathrm{nm}$ , $\beta = 10^{-4}$ ; the de Sitter split and its leading order $2kvD/c^2$ ; and the representative-binary numbers ( $73.1$ days, $18.3P$ , $k \lesssim 0.055$ ). The literature bounds — de Sitter $2\times10^{-3}$ , Alväger $(-3\pm13)\times10^{-5}$ , Brecher $2\times10^{-9}$ — are sourced values, asserted but not derived here. Three base errors, corrected. (i) The base's $c_{2\text{-way}} = c - k^2v^2/c$ feeds the apparatus a light speed of $c \pm kv$ when the source is co-moving — a wrong premise, contradicting the base's own derivation card, which correctly gets $t_\parallel = t_\perp = 2L/c$ . (ii) Its CERN number $\beta_\gamma = 0.9987 \pm 0.0021$ is not reproducible from the paper's own $k = (-3\pm13)\times10^{-5}$ , which implies $\beta_\gamma = 1.00000 \pm 0.00013$ ; it is not printed. (iii) It dates Fox's extinction objection to 1965; the objection is Fox 1962.

Builds on. Galilean Relativity · The Michelson–Morley Experiment · Electromagnetism and the Failure of Galilean Invariance. Leads to. The Constancy of the Speed of Light · The Principle of Relativity.

If Michelson–Morley did not prove relativity, what did it prove?

It excluded the stationary luminiferous ether, and decisively — $\Delta N \lesssim 0.01$ against a prediction of $0.3735$ , a factor of nearly forty. That removed the preferred frame nineteenth-century physics had built itself around. But excluding the ether leaves two theories standing, not one. Relativity says light forgets its source; Ritz says light remembers it. Both predict a null fringe shift in an apparatus whose lamp is bolted to the bench, so choosing between them needed different evidence entirely.

Why doesn't the emission theory produce at least a second-order residue, like the ether?

Because there is nothing for it to be second-order in. The ether theory has a real parameter — the bench's speed $v$ through the ether — on which the arm times depend, so expanding in $\beta = v/c$ makes sense and yields the $\beta^2$ term. On an emission theory the light's speed is set relative to the source, and the source is at rest in the apparatus, so the bench's motion through anything else never enters the arm times at all. There is no $\beta$ in the problem. The null is exact, not asymptotic — a much stronger statement than "too small to see".

Emission theory keeps Galilean relativity and needs no ether. Wasn't it the more conservative option?

It was, and that is exactly why it was taken seriously — by Ritz, and briefly by Tolman. Emission theory keeps the principle of relativity, absolute time and Galilean velocity addition, and explains Michelson–Morley for free. Its cost is Maxwell: the vacuum wave equation fixes the propagation speed from $\varepsilon_0$ and $\mu_0$ alone, with no slot in which to hide the emitter's motion, so $c_{\text{obs}} = c + kv$ cannot come out of electrodynamics without rewriting it. Ritz was willing to rewrite it. The stars said no.

How can Fox's extinction objection undo an argument about light in interstellar space? Isn't space empty?

Not empty enough. The interstellar medium is a very thin gas, but the light path is enormous, and what matters is the extinction length — the distance over which the incident wave is absorbed and re-radiated by the atoms it passes, after which it propagates at $c$ relative to them rather than to whatever emitted it. For the interstellar medium that length is of order a parsec, about $3.3$ light-years: against the $300$ -light-year path of our example binary, under two percent of the journey, so any source-velocity signature is scrubbed out in the first fiftieth of the flight. It is why the decisive test had to be run indoors.

Why do X-ray binaries escape the extinction objection when optical binaries do not?

Because the extinction length depends strongly on wavelength and on the medium's refractive index, and hard X-rays interact with the thin interstellar plasma far more weakly than visible light — their extinction length is much longer than the source distance, so the radiation arrives still carrying the memory of its source. That is what lets Brecher (1977) use the pulse timing of X-ray binaries — a far sharper observable than "the orbit looks Keplerian" — to reach $k < 2\times10^{-9}$ .

Is emission theory just wrong, or is it ruled out only within some error bar?

Strictly the latter, as with every experimental claim. The tightest published bound is Brecher's $k < 2\times10^{-9}$ ; the cleanest extinction-free measurement is Alväger's $k = (-3 \pm 13)\times10^{-5}$ , consistent with zero. What makes the case overwhelming is not any single number but that the bounds come from radically different regimes — pion decays in an accelerator, X-ray pulse timing across the galaxy — and that emission theory has no independent successes to weigh against them, while flatly contradicting Maxwell's equations. It is a corpse with a well-measured temperature.