# Rømer's Method with Modern Ephemerides **A Speed-of-Light Null Test** *Release from Aether Cosmology Research Group* ## Abstract Rømer's 1676 method derives the speed of light from the timing of Jupiter satellite eclipses observed from different positions of Earth in its orbit. The method has been part of standard astronomy for three centuries and was used during that period to determine longitude on land via published predictive tables of Galilean satellite eclipses. The method's validity, given correct distances and correct timings, is not in dispute. The question this report puts is the inverse: with modern eclipse timings (resolved to seconds) and modern heliocentric ephemerides (Earth-Jupiter distances given to kilometres), does the procedure return the textbook value of c? It does not. Across the four Galilean moons of Jupiter and the major moons of Saturn, the slope of detrended eclipse-timing residual against detrended Earth-satellite range gives speeds-of-light spanning +116,000 to +460,000 km/s and including a negative value. None of the moons recover c=CODATA. No single global value of c reconciles them. After the slope fit, the detrended residual of Io and Europa is then 91-94% explained by a sum of two sinusoids at Earth's tropical year (365.256 d) and the Sun-Earth-Jupiter synodic period (398.884 d), with the two moons sharing the same amplitude ratio and phase offset even though the fits were independent. The Newtonian gravitational calculation rules out Earth's gravity as the source: the Earth-on-Io tidal timing shift is on the order of nanoseconds, the observed residual is on the order of hundreds of seconds, and the matching reciprocal effect that Jupiter's gravity should produce on the Earth-Moon system at the millisecond level is empirically absent in lunar laser ranging. These observations comprise an internal critique of the heliocentric model. The geometric component of the model (the synodic-period repetition of Sun-Jupiter-Io alignment as seen from Earth) is intact and frame-invariant. The dynamical component of the model (the prediction that Jovicentric eclipse timing depends only on Sun-Jupiter-Io physics and is independent of Earth) is contradicted by the data. --- ## 1. Historical Background ### 1.1 Rømer's 1676 announcement In 1676 Ole Rømer, working at the Paris Observatory under Cassini, presented to the Académie Royale des Sciences a hypothesis that successive Io eclipses, observed from Earth, do not appear at the regular interval that Io's orbital period would predict, but vary with the Earth-Jupiter distance. From the timing variation Rømer concluded that light has a finite velocity and that the variation is the time required for light to traverse the changing Earth-Jupiter range. The argument was published as *Démonstration touchant le mouvement de la lumière trouvé* in the *Journal des Sçavans*, 7 December 1676 [^cohen44]. The geometric construction Rømer presented in the *Demonstration* defines the method. Letting A be the Sun, B Jupiter, C the first satellite (Io) entering Jupiter's shadow, D the point where it exits, and EFGHKL Earth at six successive positions in its orbit, the construction establishes that the Earth-Jupiter distance varies as Earth orbits the Sun, and that successive Io emersions observed from different Earth positions will arrive offset in time by the change in Earth-Jupiter range divided by the speed of light. ![[1944_Cohen_Roemer_First_Determination_p58_demonstration_geometry.png]] Rømer in the *Demonstration* argues that the per-revolution effect is small (~42.5 hours per Io revolution; 210 Earth-diameters of range change in that interval) but that 40 successive revolutions accumulate a measurable delay across half the synodic period: > "After M. Romer had examin'd the thing more nearly, he found, that what was not sensible in two revolutions, became very considerable in many being taken together, and that, for example, forty revolutions observed on the side F, might be sensibly shorter, than forty others observed in any place of the Zodiack where Jupiter may be met with; and that in proportion of twenty two for the whole interval of HE, which is the double of the interval that is from hence to the Sun." [^cohen44] The 22-minute figure across the diameter of Earth's orbit (2 AU) corresponds to a speed of light of approximately 220,000 km/s, ~26% below the modern value. The shortfall is attributed in modern scholarship to the timekeeping precision of 1670s pendulum clocks, not to any flaw in the method. Shea (1998) reconstructs Rømer's reasoning [^shea98] and confirms that the method itself is unquestionably valid; only the timekeeping accuracy of 1676 was at the edge of what was needed to demonstrate the predicted effect. > "Although the method Rømer conceived is unquestionably valid, his original and only paper on the subject left out much of the detail necessary to determine whether his measurements were adequate to the task of demonstrating the effect he claimed to have observed." [^shea98] ![[1998_Shea_Romer_Io_Doppler_p01_abstract.png]] [^cohen44]: Cohen, I. B., 1944. *Roemer and the First Determination of the Velocity of Light.* The Burndy Library. See [[1944_Cohen_Roemer_First_Determination]]. [^shea98]: Shea, J. H., 1998. *Ole Rømer, the speed of light, the apparent period of Io, the Doppler effect, and the dynamics of Earth and Jupiter.* American Journal of Physics 66 (7), 561. See [[1998_Shea_Romer_Io_Doppler]]. ### 1.2 The Galilean satellites as a celestial timekeeper Rømer's method did not arrive in isolation. From Galileo's 1610 telescopic discovery of the four Galilean satellites onward, Jupiter's moons were treated as a celestial clock. Their eclipses recur with predictable periodicity, and that predictability was used to determine longitude on land for two centuries. Higgitt (2021) summarises the role of the Galilean satellites in longitude determination [^higgitt21]: > "Galileo's observations showed that these satellites would appear and disappear as they circled the planet with a regularity that could be predicted and used as a celestial timekeeper." > "This method of establishing longitude became, with the publication of more accurate predictive tables by Giovanni Cassini at the Paris Observatory in 1668, an important tool for precision land survey but it remained a very difficult observation to make on board ship." ![[2021_Higgitt_Longitude_p06_galileo_cassini_tables.png]] Cassini's 1668 tables, published at the Paris Observatory eight years before Rømer's announcement, are the same Galilean-satellite eclipse predictions that Rømer studied at Cassini's institution. Cassini's own introduction to the tables, reproduced in Cohen 1944 [^cohen44], states the operational purpose plainly: > "Avant mon départ de Bologne au mois de Mars 1668. je m'étois pressé de publier mes premières Tables du mouvement des Satellites de Jupiter... afin que les Astronomes qui n'en avoient pas d'autres qui pussent servir à cette usage... eussent la commodité d'observer de concert les configurations & les Eclipses de ces Satellites, pour les faire servir à l'invention des longitudes : ce qui a eu l'effet que j'en avois esperé, ces Tables & ces Ephémerides n'ayant pas plûtôt paru que les Astronomes de diverses Nations s'en servirent pour observer de concert ces Satellites, & pour tirer du rapport de ces Observations la difference des longitudes des lieux éloignez où elles ont été faites." In English: I hastened to publish my first Tables of the motion of the Satellites of Jupiter so that astronomers who had no others suitable for this use would have the convenience of observing in coordination the configurations and eclipses of these satellites, to make them serve the determination of longitudes; which had the effect I had hoped for, these tables and ephemerides no sooner appeared than astronomers of various nations used them to observe these satellites in coordination, and to derive the difference in longitudes of the distant places where the observations were made. ![[1944_Cohen_Roemer_First_Determination_p21_cassini_quote.png]] The tables were authoritative enough to be the basis of land survey by national observatories from 1668 well into the 19th century. The angular and temporal relationships between Earth, Jupiter, and Jupiter's moons that the tables predicted were treated as physically valid throughout that period of operational use. The same Galilean-satellite eclipse data Cassini collected for longitude survey is the data Rømer used to argue for finite light velocity. [^higgitt21]: Higgitt, R., 2021. *Longitude.* Encyclopedia of the History of Science, Carnegie Mellon University Libraries. See [[2021_Higgitt_Longitude]]. ### 1.3 Newton's testimony and the method's accepted validity When the British Parliament in 1714 took up the question of which methods could establish longitude, Newton listed the Galilean-satellite eclipse method among the established theoretical solutions, calling it "true in the Theory" and noting only that operational difficulty at sea (telescope length, ship motion) limited its maritime application [^higgitt21]: > "Problems remained with these methods in 1714. As Newton said, they were 'true in the Theory, but difficult to execute'. Eclipses of Jupiter's satellites could not yet be observed at sea because of 'the Length of Telescopes requisite' and 'the Motion of a Ship'." ![[2021_Higgitt_Longitude_p07_newton_1714_testimony.png]] The two-century career of the method as the standard tool for longitude survey is direct evidence that the timing relationships of Galilean eclipses, predicted from Cassini's tables and observed from Earth, were treated as physically valid. Eclipse timing differences correspond, in the heliocentric model, to differences in the Earth-Jupiter range divided by the speed of light. If the timing difference and the speed of light are taken as known, the range follows. If the range and the speed of light are taken as known, the timing difference follows. If two of the three quantities are known, the third is constrained. Longitude survey took c and the predictive tables as known, and used eclipse timings to derive position. Rømer took eclipse timings and the heliocentric distance scale as known, and derived c. ### 1.4 Rømer's worked computation and the per-cycle residual Cohen 1944 reproduces a numerical reconstruction of Rømer's actual reasoning using a specific pair of Io eclipse observations from Rømer's holograph data [^cohen44]: - 24 October 1671, immersion at 297d 17h 59m 15s mean time - 12 January 1672, immersion at 12d 9h 8m 45s mean time (next year) - Interval: 79 days 15 hours 9 minutes 30 seconds - Mean Io period (from Cohen's tabulation of Rømer's data): 1d 18h 28m 30s - 45 revolutions × 1d 18h 28m 30s = 79d 15h 22m 30s predicted The observed January 12 immersion arrived **13 minutes earlier** than the constant-period extrapolation predicted. Cohen attributes the difference to the change in Earth-Jupiter range across the 79-day interval (Earth approaching Jupiter shortens the light-travel time, and the immersion arrives sooner). ![[1944_Cohen_Roemer_First_Determination_p34_worked_computation.png]] This is the operational form of Rømer's argument: a per-eclipse residual (observed minus constant-period extrapolation) of order 10-15 minutes for an interval of half a synodic cycle. The slope-fit form used in the modern JPL ephemeris pipeline (§4 below) extracts the same information from the full multi-year sequence of residuals. Cohen also reproduces Dr. Meyer's tabulation of the mean Io period across nine observational periods in Rømer's holograph manuscript, separating means computed from emersions and from immersions: | Period | Mean (1d 18h 28m...) | Type | |---|---|---| | I | 47s | Emersions | | II | 18s | Immersions | | III | 35s | Emersions | | IV | 27s | Immersions | | V | 46s | Emersions | | VI | 48s | Emersions | | VII | 20s | Immersions | | VIII | 47s | Emersions | | IX | 30s | Immersions | ![[1944_Cohen_Roemer_First_Determination_p33_mean_period_table.png]] Means computed from emersions cluster around 1d 18h 28m 46s; means from immersions cluster around 1d 18h 28m 28s. The systematic ~18-second difference is the signature of light-time delay across the Earth-Jupiter geometry: emersions and immersions sample different phases of the synodic cycle and therefore different signs of the range derivative. This is the same systematic pattern modern reproductions reproduce. ### 1.5 Cassini's prior hypothesis and his subsequent objection Cassini publicly proposed the finite-light-velocity hypothesis in August 1675, more than a year before Rømer's December 1676 paper [^cohen44]: > "Cette seconde inégalité paroit venir de ce que la lumière emploie quelque temps à venir du satellite jusqu'à nous, et qu'elle met environ dix à onze minutes à parcourir un espace égal au demi-diamètre de l'orbite terrestre." In English: This second inequality appears to come from the fact that light takes some time to come from the satellite to us, and that it takes about 10 to 11 minutes to traverse a space equal to half the diameter of the Earth's orbit. ![[1944_Cohen_Roemer_First_Determination_p27_cassini_1675_announcement.png]] Cassini withdrew the hypothesis when the second inequality could not be reconciled across the four Galilean satellites: > "Cassini perceived that the successive propagation of light explained the irregularities in the eclipses of the first satellite when the Earth was in different positions of her orbit; but finding that it did not account in an equally satisfactory manner for the irregularities of the other satellites, he rejected it altogether, and instead of it he used in the tables of the first satellite an empiric equation depending on the relative positions of the Earth and Jupiter." ![[1944_Cohen_Roemer_First_Determination_p29_cassini_objection.png]] Cassini's grounds for withdrawing the hypothesis are notable in light of the modern cross-moon finding (§5.1 below): a single value of $c$ does not collapse the residual across moons in 1670s data or in 2026 JPL ephemeris data. The disagreement Cassini identified at the per-satellite level in the 17th century persists in the modern dataset. ### 1.6 Halley's 1694 endorsement and Newton's 1704 adoption Edmond Halley in 1694 published Cassini's tables of the first Galilean satellite (reduced to Julian style and the meridian of London) and explicitly endorsed Rømer's hypothesis of the progressive motion of light [^cohen44]: > "Monsieur ROEMER did most ingeniously explain [this second inequality] by the Hypothesis of the progressive Motion of Light; to which yet Cassini by his manner of calculus seems not to assent, though it be hard to imagine how the Earth's Position in respect of Jupiter should any way affect the motion of the Satellites..." ![[1944_Cohen_Roemer_First_Determination_p37_halley_1694_endorsement.png]] Halley's specific puzzlement, "though it be hard to imagine how the Earth's Position in respect of Jupiter should any way affect the motion of the Satellites," anticipates the same puzzle the modern data raises in §6: an Earth-anchored periodicity in jovicentric eclipse timing, which in the heliocentric model should not exist. Halley accepted that Rømer's finite-light-velocity hypothesis explained the effect; the alternative — that Earth's position genuinely affects Io's motion — he found "hard to imagine." Newton in his 1704 *Opticks* cited the Galilean-satellite eclipse argument as evidence that light is propagated in time: > "But by an Argument taken from the Aequations of the times of the Eclipses of Jupiter's Satellites, it seems that Light is propagated in time, spending in its passage from the Sun to us about seven Minutes of time : And therefore I have chosen to define Rays and Refractions in such general terms as may agree to Light in both cases." ![[1944_Cohen_Roemer_First_Determination_p38_newton_1704_opticks.png]] Newton's seven-minute Sun-to-Earth light-time figure (revised from the 10-minute figure of his first Principia edition to 7-8 minutes in later editions) corresponds to a speed of light close to the modern value, suggesting the central tendency of well-averaged 17th-century data was accurate even where individual measurements scattered. Newton's adoption of the method in the *Opticks* established its authority for the next two centuries. ### 1.7 Modern reproductions The 2014 *Rømer Revisited* report from the Orwell Astronomical Society Ipswich and the Hampshire Astronomical Group describes a coordinated amateur reproduction of Rømer's method using telescope timings of Galilean eclipses observed during 2012–14 [^oasi14]. The authors note that despite the method's textbook simplicity, very few modern attempts to reproduce the measurement have been published. > "It is a comparatively simple matter to repeat his approach and, using modern information about the scale of the solar system, to take a step further and estimate the speed of light (denoted c) but, surprisingly, few astronomers have reported tackling the challenge." The OASI authors also note that the natural quantity returned by the slope-fit form of the method is 1/c, expressed as light-time per AU. [^oasi14]: Orwell Astronomical Society Ipswich and Hampshire Astronomical Group, 2014. *Rømer Revisited: A Modern Estimation of the Speed of Light from Observations of Jupiter's Galilean Satellites.* See [[2014_OASI_Romer_Revisited]]. --- ## 2. The Question The Galilean satellites of Jupiter have been a celestial clock for over four centuries. The angular relationships between the Sun, Jupiter, and its moons, as expressed in Cassini's 1668 tables and now in JPL's DE441 ephemeris, are the operational basis on which Rømer derived c, on which 18th-century surveyors derived longitude, and on which modern spacecraft trajectories are computed. Modern eclipse timing resolves to seconds. Modern ephemerides report Earth-Jupiter geometry to kilometres. The signal-to-noise should be enormous compared to Rømer's pendulum-clock observations. If the heliocentric model is consistent and its asserted distances are physically meaningful, then any modern dataset combining accurate Galilean eclipse timings with accurate ephemeris distances must reproduce the textbook value of c. The question this report puts: **Do the physically asserted distances in the heliocentric model hold true under the same procedure that defined the method?** If yes, modern ephemeris distances divided by modern eclipse timing residuals must return c=299,792 km/s. If they do not, then either the timings are wrong, or the distances are wrong, or the dynamics are wrong. The angular relationships and the timings have been validated for two centuries of land surveying and for modern spacecraft navigation. The dynamics inherits the assumption of the heliocentric model. The asserted physical distances inherit the assumption that the heliocentric distance scale, computed from Earth-based observations and Newtonian dynamics, is correct. --- ## 3. Null Hypothesis **H₁:** The heliocentric model's distance scale and dynamical predictions are jointly consistent. Rømer's method, applied to modern Galilean eclipse timings using JPL ephemeris distances, returns c=CODATA. **H₀:** The heliocentric model's distance scale and dynamical predictions are not jointly consistent. Rømer's method, applied to modern Galilean eclipse timings using JPL ephemeris distances, does not return c=CODATA. **Result:** Failed to falsify H₀ across multiple independent extraction methods, six moons of Jupiter and Saturn, and a 3-year baseline. --- ## 4. Method ### 4.1 Eclipse detection For each Galilean satellite of Jupiter and the major moons of Saturn, hourly heliocentric ecliptic-J2000 Cartesian state vectors are pulled from JPL Horizons over the window 2023-01-01 to 2026-01-01. The eclipse detection geometry is built entirely in the **Sun-Parent-Satellite frame** with no Earth coordinates entering the calculation: - Sun-parent unit vector $\hat{u}_{PS}$ - Along-axis depth $b$ of the satellite behind the parent - Perpendicular distance $\rho$ from the sun-parent axis - Umbra cone radius $r_{\text{umbra}}(b) = R_{\text{parent}} - b \cdot (R_\odot - R_{\text{parent}}) / \lvert\vec{r}_{P}\rvert$ Ingress is the first hourly sample where $b > 0$ and $\rho < r_{\text{umbra}}$. The exact ingress time is refined with `scipy.optimize.brentq` on the signed-shadow function. The refined time `t_emit` is the jovicentric (or saturnicentric) eclipse time in JD on the TDB scale. **The eclipse event time we extract is jovicentric or saturnicentric by construction: Earth's position never enters the umbra geometry.** This will become important when the residual carries an Earth-anchored periodicity that, by construction, should not be present. For each refined eclipse time the Earth-Sat range $R$ is computed by interpolating Earth's heliocentric position at that JD and forming $R = \lvert\vec{r}_S + \vec{r}_P - \vec{r}_E\rvert$. This is geometric, no light-time iteration applied. ### 4.2 Constant-period fit and detrending For each moon, a linear fit in event index gives the orbital period $P_s$ and reference epoch $T_0$: $t_s(n) = T_0 + P_s \cdot n + \text{resid}_{\text{raw}}(n).$ The raw residual is then detrended with an order-2 polynomial in time-from-start to remove slow drift. The same order-2 detrend is applied to $R(t)$. The detrended quantities are `R_detr` and `resid_detr`. ### 4.3 Slope fit for c_implied Linear fit of detrended residual against detrended range: $\text{resid}_{\text{detr}} = a + b \cdot R_{\text{detr}}.$ If the residual is a pure light-time signal at some constant $c$, then $c_{\text{implied}} = \frac{1}{1/c_{\text{true}} + b}.$ A synthetic-data check (build $t = T_0 + P n + R/c_{\text{true}}$ exactly, run the same pipeline) recovers $c_{\text{true}}$ to within 1 km/s. The fitter is correct. --- ## 5. The Empirical Result ### 5.1 Per-moon c_implied | System | N events | $\sigma_{\text{detr}}$ (s) | corr | $c_{\text{implied}}$ (km/s) | |---|---:|---:|---:|---:| | Jupiter / Io | 619 | 133.4 | −0.888 | +460,005 | | Jupiter / Europa | 308 | 633.5 | +0.848 | +116,235 | | Jupiter / Ganymede | 153 | 120.0 | −0.650 | +389,182 | | Jupiter / Callisto | 15 | 100.3 | −0.761 | −420,787 | | Saturn / Rhea | 144 | 208.6 | +0.594 | +217,763 | | Saturn / Titan | 23 | 234.0 | +0.417 | +174,288 | CODATA $c = 299{,}792.458$ km/s. **No moon recovers CODATA. The values span both signs and differ across moons by more than the CODATA value itself.** This is the central observation. A single global error in $c$ would give the same biased value across every moon. The values are not the same. A single global error in the heliocentric distance scale would give the same fractional bias across every moon. The fractional biases are not the same. The "wrong c" hypothesis and the "wrong AU" hypothesis are both falsified by the cross-moon comparison. ### 5.2 The 56% framing tightened A live presentation summarised the result as "c is off by ~56% from Io alone." The data is sharper than that summary suggests. No single global $\Delta c$ reconciles the moons. The residual is therefore not a $c$-effect at all. The headline reading of the data is: > No single value of c collapses the residual against range across the moons. The residual is not a light-time signal at any single c. ### 5.3 The headline plot — Io detrended residual vs Earth-Io range After detrending, plotting Io's residual (red) against the Earth-Io range (blue) on a dual-axis 3-year time series shows the two curves oscillating in phase with each other at the Sun-Earth-Jupiter synodic period (~399 days). The Pearson correlation between detrended residual and detrended range is $-0.876$ for Io. If the residual were a pure Rømer light-time signal at $c=$ CODATA, the slope would be $-1/c \approx -3.34 \times 10^{-6}$ s/km. The slope the data exhibits is $-1.16 \times 10^{-6}$ s/km, corresponding to $c_{\text{implied}} = 460{,}005$ km/s. The data delivers about 35% of the slope CODATA $c$ would produce. --- ## 6. The Earth-Anchored Periodicity ### 6.1 Two-component fit at Earth's year and the Earth-Jupiter synodic The detrended residual is fit by least squares to a basis of constant plus four sinusoids at fixed Earth-related frequencies: $\text{resid}(t) = C_0 + A_y \sin(\omega_y t) + B_y \cos(\omega_y t) + A_s \sin(\omega_s t) + B_s \cos(\omega_s t)$ with $\omega_y = 2\pi/365.256\,\text{d}$ (Earth's tropical year) and $\omega_s = 2\pi/398.884\,\text{d}$ (Sun-Earth-Jupiter synodic period). Recovered amplitudes $R_y = \sqrt{A_y^2 + B_y^2}$, $R_s = \sqrt{A_s^2 + B_s^2}$ and phases $\varphi_y, \varphi_s$. The phase difference $\Delta\varphi = (\varphi_y - \varphi_s + \pi) \bmod 2\pi - \pi$ is wrapped to $[-\pi, \pi)$. | Moon | $\sigma_{\text{before}}$ (s) | $\sigma_{\text{after}}$ (s) | Variance reduction | $R_y / R_s$ | $\Delta\varphi$ (rad) | |---|---:|---:|---:|---:|---:| | Io | 133.4 | 40.7 | **91 %** | 0.46 | −2.39 | | Europa | 633.5 | ~150 | **94 %** | 0.49 | −2.46 | Io and Europa, fit independently with no cross-moon coupling, land at essentially the same $(R_y/R_s, \Delta\varphi)$ coordinate. The shared coordinate is not a fit constraint; it is a fit result. ### 6.2 The 436-day puzzle resolved as an unresolved sum A Lomb-Scargle periodogram of the detrended Io residual on the 3-year baseline shows a dominant peak at 436 d, not at 365 d or 399 d directly. On a 3-year baseline the Rayleigh frequency resolution is $\frac{1}{T} = \frac{1}{1095\,\text{d}} = 9.1 \times 10^{-4} \,\text{cycles/day}$ while the actual frequency separation between Earth's year and the Earth-Jupiter synodic is $\left|\frac{1}{P_y} - \frac{1}{P_s}\right| = \left|\frac{1}{365.256} - \frac{1}{398.884}\right| = 2.3 \times 10^{-4} \,\text{cycles/day}.$ The actual separation is 4× smaller than the Rayleigh resolution. The two real periods cannot be resolved separately on a 3-year baseline; their unresolved sum produces a single peak at a beat-shifted location. A direct synthesis test (sample $\sin(2\pi t/399) + A\sin(2\pi t/365 + \varphi)$ on the actual eclipse-time grid, vary $(A, \varphi)$) confirms the 436 d peak emerges precisely at the $(A, \varphi)$ coordinates the constrained fit independently recovers. After subtracting the two-component fit, the 436 d peak vanishes. **The 436-day peak is not a new period.** It is the unresolved alias of two real Earth-related periods on a baseline too short to separate them. ### 6.3 What the residual carries that should not be there The eclipse-time residual contains 91-94% of its variance at exactly two periods: Earth's tropical year (365.256 d) and the Sun-Earth-Jupiter synodic (398.884 d). Both are Earth-anchored quantities. Neither involves Jupiter's intrinsic orbital frequency directly. In the heliocentric model, the eclipse detection geometry is jovicentric: a hypothetical observer at the Sun-Jupiter line predicts Io's umbra ingress from Sun-Jupiter-Io geometry alone. Earth never enters that prediction. Yet the timing residual after constant-period subtraction is dominated by Earth-related periodicity at amplitudes (∼200 seconds peak) that are 9 orders of magnitude larger than anything Earth's gravity could plausibly produce (see §7 below). A jovicentric eclipse calculation that returns Earth-anchored periodicity in the residual is, on the heliocentric model, internally inconsistent. --- ## 7. The Newtonian Dynamic Refutation If the residual is a real perturbation of Io's orbital motion, the only mechanism the heliocentric model permits is gravitational. The Newtonian calculation can be done in closed form. ### 7.1 Earth's tidal pull on Io For a perturber of mass $M$ at distance $r_{\text{dist}}$ from a parent body, with a satellite at orbital radius $r_{\text{orb}}$ from the parent, the tidal acceleration on the satellite relative to the parent is $\Delta a_{\text{tidal}} = \frac{2 G M r_{\text{orb}}}{r_{\text{dist}}^3}.$ The quasi-static radial displacement at the satellite's orbital frequency $\omega = 2\pi / P_{\text{orb}}$ is $\Delta r = \Delta a_{\text{tidal}} / \omega^2$, and the along-track timing shift is $\Delta t = \Delta r / v_{\text{orb}}$. For Earth perturbing Io across an Earth-Jupiter distance of 5.2 AU: - $\Delta a_{\text{tidal}} \sim 10^{-12}$ m/s² - Quasi-static Io displacement $\sim 0.5$ mm - Along-track timing shift $\Delta t \sim 30$ ns **The observed residual is 200 seconds. The gravitational prediction is 30 nanoseconds. The ratio is 7 × 10⁹.** ### 7.2 The reciprocal: Jupiter's tidal pull on the Moon Same physics, swap masses ($M_\oplus \leftrightarrow M_J$) and inner-orbit radii ($r_{IJ} \leftrightarrow r_{EM}$). Same Earth-Jupiter distance. Because Jupiter is 318 times more massive than Earth, Jupiter's tidal acceleration on the Moon is 7.7 × 10⁴ times stronger than Earth's tidal acceleration on Io across the same distance. The predicted Moon along-track timing shift is on the order of milliseconds. **Lunar Laser Ranging resolves Moon position to a few millimeters, equivalent to about 10 microseconds of along-track timing.** That is 100× finer than the predicted Jupiter effect on the Moon. The Jupiter effect does not appear in the LLR residuals. If Earth-on-Io is a real gravitational perturbation, then Jupiter-on-Moon must exist at a much larger amplitude, and it must be visible to LLR. It is not. ### 7.3 Sun-on-Moon as the calibration anchor The Sun's tidal pull on the Earth-Moon system is large, well-measured, and produces the parallactic inequality, evection, and variation terms in lunar theory. The same formula applied to the Sun gives a Moon along-track timing shift on the order of minutes, matching the well-measured terms in lunar theory. The formula is calibrated against a known correct case. **The triplet — Earth-on-Io 9 orders too small, reciprocal Jupiter-on-Moon empirically absent, Sun-on-Moon validated against measurement — is hard to wave off.** Heliocentric Newtonian gravity does not allow a 200-second perturbation of Io by Earth. --- ## 8. The c-Circularity in the JPL Pipeline A common objection at this point is "JPL would have caught a c-error if there were one." This rests on a misunderstanding of how the JPL ephemeris is constructed. ### 8.1 c is wired into the ephemeris at three layers The JPL planetary and lunar ephemerides DE430/DE431 (Folkner et al 2014 [^folkner14]) and DE440/DE441 (Park et al 2021 [^park21]) are produced by numerically integrating the equations of motion of solar-system bodies, fit by least-squares against observational data sets. The construction commits to $c = c_{\text{CODATA}}$ at three distinct layers: 1. **Time scale.** The TDB-TT transformation contains $c$ in the gravitational potential terms (Folkner 2014 Eq. 5; Park 2021 Eq. 3). The time argument of every state vector returned by Horizons inherits this conversion. 2. **Equations of motion.** The parameterised post-Newtonian (PPN) equation of motion (Park 2021 Eq. 27) contains $c$ at multiple orders, in the velocity-dependent and gravitational-potential terms. ![[2021_Park_DE440_DE441_p05_PPN_eom.png]] 3. **Residual format.** For range observations and lunar laser ranging, the residual fed into the least-squares fit is $\text{residual} = (t_{\text{meas}} - t_{\text{comp}}) \cdot c / 2$, converting one-way light-time to distance using $c$ as the conversion factor (Folkner 2014 page 18; Park 2021 page 10). ![[2014_Folkner_DE430_DE431_p18_residual_format.png]] ### 8.2 Implication Any timing anomaly that disagrees with $c=$ CODATA cannot manifest as a $\Delta c$ in the JPL output. It must propagate into a different distance estimate during the fit. The JPL distances are not measurements independent of $c$; they are outputs of a fit that holds $c$ fixed. When we apply Rømer's method to JPL distances and JPL eclipse timings and ask whether the slope returns $c=$ CODATA, we are asking whether the joint fit closes on itself. The answer is no. The fit does not close. The residual we recover is the part of the JPL fit that has not been absorbed into either the distance estimate or the dynamical model. [^folkner14]: Folkner, W. M., Williams, J. G., Boggs, D. H., Park, R. S., Kuchynka, P., 2014. *The Planetary and Lunar Ephemerides DE430 and DE431.* JPL IPN Progress Report 42-196. See [[2014_Folkner_DE430_DE431]]. [^park21]: Park, R. S., Folkner, W. M., Williams, J. G., Boggs, D. H., 2021. *The JPL Planetary and Lunar Ephemerides DE440 and DE441.* The Astronomical Journal 161:105. See [[2021_Park_DE440_DE441]]. --- ## 9. Frame-Independence of the Result The synodic period is a kinematic quantity. It is invariant under a change of inertial reference frame, including the change between heliocentric and strict stationary-Earth coordinate descriptions of Sun-Jupiter-Io geometry. The eclipse detection is geometric in the Sun-Parent-Satellite triangle. A coordinate transformation does not change the angles or the timing. The dynamical claims of the heliocentric model are not invariant under such a transformation. A model that asserts Earth's gravitational influence on Io is negligible (which Newtonian gravity confirms at the 10⁻⁹ s level) but produces residuals dominated by Earth-anchored periodicity at the 200 s level is internally inconsistent in its dynamical claims, regardless of the coordinate system used to write down the eclipse detection. The geometric component of the result therefore holds regardless of frame choice. The dynamical component is what is contradicted. --- ## 10. What We Claim and What We Do Not Claim ### What we claim 1. Modern Galilean-satellite eclipse timings, combined with modern heliocentric ephemeris distances, do not return $c=$ CODATA via Rømer's method. 2. The cross-moon dispersion of $c_{\text{implied}}$ rules out a single global $c$ error or distance-scale error as the source. 3. The detrended residual for Io and Europa is 91-94% explained by sinusoids at Earth's year and the Sun-Earth-Jupiter synodic period. 4. Io and Europa land at the same $(R_y/R_s, \Delta\varphi)$ coordinate under independent fits. 5. The 436-day Lomb-Scargle peak is the unresolved alias of $P_y$ and $P_s$ on a 3-year baseline, not a new period. 6. The Newtonian gravitational calculation places Earth's influence on Io 9 orders of magnitude below the observed residual, and the matching reciprocal effect on the Moon is empirically absent in LLR. 7. Within the heliocentric Newtonian framework, the empirical signal is present but cannot be a consequence of the dynamics the model specifies. ### What we do not claim 1. We do not claim that Earth physically affects Io. 2. We do not claim that any particular alternative physical mechanism is operating. 3. We do not claim that the heliocentric model is wrong about geometry, or that any specific geocentric model is correct. 4. We do not claim a numerical value for the speed of light other than what the data returns. The result is an internal critique of the heliocentric model's dynamical claims. The geometric and timing relationships of the Galilean satellites have served two centuries of land surveying and continue to serve modern spacecraft navigation. The geometric relationships are intact. The dynamical claim that jovicentric eclipse timing depends only on Sun-Jupiter-Io physics is contradicted. --- ## 11. Sources - [[1944_Cohen_Roemer_First_Determination]] — Cohen, I. B., 1944. *Roemer and the First Determination of the Velocity of Light.* Burndy Library. - [[1998_Shea_Romer_Io_Doppler]] — Shea, J. H., 1998. *Ole Rømer, the speed of light, the apparent period of Io, the Doppler effect, and the dynamics of Earth and Jupiter.* American Journal of Physics 66 (7), 561. - [[2014_OASI_Romer_Revisited]] — Orwell Astronomical Society Ipswich and Hampshire Astronomical Group, 2014. *Rømer Revisited.* - [[2021_Higgitt_Longitude]] — Higgitt, R., 2021. *Longitude.* Encyclopedia of the History of Science. - [[2014_Folkner_DE430_DE431]] — Folkner et al, 2014. *The Planetary and Lunar Ephemerides DE430 and DE431.* JPL IPN Progress Report 42-196. - [[2021_Park_DE440_DE441]] — Park et al, 2021. *The JPL Planetary and Lunar Ephemerides DE440 and DE441.* The Astronomical Journal 161:105. - Rømer, O., 1677. *Démonstration touchant le mouvement de la lumière trouvé.* Journal des Sçavans, 7 December 1676. - JPL Horizons On-Line Ephemeris System: https://ssd.jpl.nasa.gov/horizons/ - CODATA recommended values of the fundamental physical constants 2018: https://physics.nist.gov/cuu/Constants/ **Repository (analysis code, data, notebooks):** https://github.com/AlanSpaceAudits/romer_c --- ## See Also - [[Speed_of_Light_Null]] — Einstein-Maxwell c-independence of the value of c - [[../Earth_Anchored_Signals/Synodic_Period_Eclipse_Residual]] — Detailed treatment of the Earth-anchored residual - [[../Earth_Anchored_Signals/Dynamic_Heliocentric_Null]] — Formal null on the dynamic refutation - [[../Earth_Anchored_Signals/00_Index]] — Earth-Anchored Signals folder index - [[00_Null_Hypothesis_Index]] — Master null hypothesis index