# Clemence 1947 — The Relativity Effect in Planetary Motions **By G. M. Clemence** U. S. Naval Observatory, Washington, D. C. Published in *Reviews of Modern Physics*, Volume 19, Number 4, October 1947, pp. 361–364. Four pages from the man whose job was computing planetary orbits for the U.S. government. This is not a theoretical paper. It is an observational error budget. Clemence catalogues every source of uncertainty in the perihelion measurement and lays bare how fragile the "confirmation" of GR actually is. --- ## The Subtraction Chain The 43" is not an observation. It is what remains after subtracting two layers of large calculated numbers from the raw observation. Nobody has ever directly observed a 43"/century anomaly. What is observed is Mercury's total apparent perihelion motion, as seen from Earth. The "anomaly" lives entirely in the gap between calculated numbers. $\boxed{43'' = 5600'' - 5025'' - 532''}$ $\text{Residual} = \text{Raw observation} - \text{Earth's wobble} - \text{Planetary tugs}$ ### What each term is | Term | Value ("/century) | What it is | How it's determined | Uncertainty | |---|---|---|---|---| | **Raw observation** | ~5600 | Mercury's total apparent perihelion motion as seen from Earth | Direct observation of Mercury's position against the equinox over centuries | ±0.41 (Clemence) | | **Precession of the equinoxes** | ~5025 | Earth's own axial wobble. The reference point (equinox) drifts because Earth's axis traces a cone over ~26,000 years. This has nothing to do with Mercury — it is an artifact of observing from a tilting platform. | Calculated from star position measurements over decades. Different authorities get different values (Oort vs de Sitter disagree by 50% of the uncertainty). | ±0.50 (Clemence); de Sitter says ±0.75 | | **Planetary perturbations** | ~532 | Gravitational tugs from Venus (278"), Jupiter (154"), Earth (90"), and other planets pulling Mercury's orbit around | Calculated from Newtonian mechanics using planetary masses. "Significantly different results have been obtained by different computers." | ±0.85 (combined) | | **Residual ("anomaly")** | ~43 | What's left after both subtractions | Not observed — calculated as a difference of differences | ±0.94 | ### The subtraction in DMS | Term | Value ("/century) | DMS | Uncertainty | | ----------------------- | ----------------- | ---------------- | ----------- | | Raw observation | 5599.74" | 1° 33' 19.74" | ±0.41" | | Precession of equinoxes | 5025.645" | 1° 23' 45.645" | ±0.50" | | Venus perturbation | 277.856" | 0° 4' 37.856" | ±0.68" | | Jupiter perturbation | 153.584" | 0° 2' 33.584" | ±0.01" | | Earth perturbation | 90.038" | 0° 1' 30.038" | ±0.08" | | All other + oblateness | ~10.07" | 0° 0' 10.07" | ~±0.03" | | Theoretical sum | 5557.18" | 1° 32' 37.18" | ±0.85" | | **Residual** | **42.56**" | **0° 0' 42.56"** | **±0.94"** | | **GR prediction** | **43.03**" | **0° 0' 43.03"** | **±0.03"** | | **Gerber prediction** | **43.03**" | **0° 0' 43.03"** | **—** | The residual is zero degrees, zero arcminutes, 42.56 arcseconds per century. You are subtracting 1° 23' 45" (Earth's wobble) from 1° 33' 19" (raw observation) to isolate a signal of 0° 0' 42". The subtraction is 120× larger than the signal. ### The problem in plain terms The raw observation (~5600") is dominated by Earth's own wobble (~5025"). That wobble is **90% of the total** and must be subtracted before you can see anything about Mercury. Then the planetary perturbations (~532") are subtracted. What's left over — the 43" — is less than **1%** of the raw observation. To isolate the 43", you need to know the 5025" to better than 1% accuracy. The uncertainty Clemence assigns to it (±0.50) is already more than half the residual. De Sitter estimated the uncertainty at 50% larger (±0.75). A shift of just 0.5" in the precession constant (0.01% of its value) moves the residual by more than its own error bar. > [!danger] The 43" is the difference of differences > You never observe the anomaly. You observe Mercury from a wobbling, orbiting platform (Earth), subtract your own calculated wobble, subtract the calculated gravitational tugs from every other planet in the solar system, and call whatever survives "the anomaly." The signal is 1% of the raw data. The dominant subtraction (Earth's wobble) is "one of the most difficult problems of positional astronomy" (Clemence). Both subtractions carry uncertainties that are comparable to the signal itself. --- ## Page 1 (p.361) ![[1947_Clemence_Relativity_Effect_Planetary_Motions.png]] ### Introduction It is well known that, according to the general theory of relativity, the elliptical orbit of a planet referred to a Newtonian frame of reference rotates in its own plane in the same direction as the planet moves, with a speed that is given by $\frac{\delta\bar{\omega}}{\varphi} = \frac{12\pi^2 a^2}{c^2 T^2(1-e^2)}$ In this formula $\delta\bar{\omega}/\varphi$ is the amount of rotation (commonly called the motion of the perihelion) per revolution of the planet about the sun, $a$ is half the major axis of the ellipse, $c$ is the velocity of light, $T$ is the time required for one revolution of the planet, and $e$ is the eccentricity of the ellipse; if $a$, $c$, and $T$ are measured in centimeters and seconds. The fraction of a revolution through which the perihelion advances during one revolution of the planet is represented by $\delta\bar{\omega}/\varphi$, a dimensionless number. > [!info] The formula > This is the same perihelion formula from Einstein (Eq. 14) and Gerber, written slightly differently. Note Clemence writes it as $12\pi^2 a^2/(c^2 T^2(1-e^2))$ — which equals $24\pi^3 a^2/(T^2 c^2(1-e^2))$ when you account for the $2\pi$ in his $\varphi$. Same equation. Same kinematic content: $a$, $T$, $c$, $e$, and geometric constants. No mass. [[perihelion_calc]] ### Table I: Theoretical Values | Planet | $\bar{\omega}'$ | $e\bar{\omega}'$ | |---|---|---| | Mercury | 43".03 | 8".847 | | Venus | 8.63 | 0.059 | | Earth | 3.84 | 0.064 | | Mars | 1.35 | 0.126 | | Jupiter | 0.06 | 0.003 | The last column gives the motion of the perihelion multiplied by the eccentricity. The size of this quantity is a measure of the angular displacement of the planet when it is at perihelion, and hence this is the quantity that fixes the accuracy with which the effect can be determined by analysis of observations. > [!info] Why $e\bar{\omega}'$ matters > The precession itself ($\bar{\omega}'$) is not what you observe directly. What you can measure is the displacement of the planet from its predicted position when it's at perihelion. That displacement is proportional to $e \times \bar{\omega}'$. For Mercury ($e = 0.206$), $e\bar{\omega}' = 8".8$ — detectable. For Venus ($e = 0.007$), $e\bar{\omega}' = 0".06$ — buried in noise. The effect is only measurable for Mercury because it has the highest eccentricity and the largest precession. Einstein himself noted this in 1915. ### The Three Complications Clemence identifies **three reasons** the observations are not simple: > **(1)** Observations of Mercury are among the most difficult in positional astronomy. They have to be made in the daytime, near noon, under unfavorable conditions of the atmosphere; and they are subject to large systematic and accidental errors arising both from this cause and from the shape of the visible disk of the planet. > **(2)** The planet's path in Newtonian space is not an ellipse but an exceedingly complicated space curve due to the disturbing effects of all of the other planets. The calculation of this curve is a difficult and laborious task, and significantly different results have been obtained by different computers. > **(3)** The observations cannot be made in the Newtonian frame of reference. They are referred to the moving equinox, that is, they are affected by the precession of the equinoxes, and the determination of the precessional motion is one of the most difficult problems of positional astronomy, if not the most difficult. > [!danger] The weight of these admissions > This is not a critic of GR speaking. This is the U.S. Naval Observatory's chief orbit computer, in *Reviews of Modern Physics*, listing the reasons the 43" confirmation should be viewed with caution: > > 1. Mercury is the hardest planet to observe (daytime, near the Sun, atmospheric distortion) > 2. The Newtonian perturbation calculation is so complex that different experts get different answers > 3. The observations are geocentric, and the conversion to an inertial frame requires knowing the precession of the equinoxes — itself one of the hardest measurements in astronomy > > The 43" residual is less than 1% of the total precession (~5600"). It is extracted by subtracting effects that are each larger than the signal, using methods that "significantly different results have been obtained by different computers." ### The Belief Statement > *"I am not aware that relativity is at present regarded by physicists as a theory that may be believed or not, at will."* > [!info] What Clemence is saying > Read carefully: he is NOT saying relativity is proven. He is saying it is treated as mandatory belief — not as a testable hypothesis. In 1947, 32 years after Einstein's paper, the man who knows the observational uncertainties better than anyone is noting that the theory has been placed beyond question rather than confirmed beyond doubt. He then proceeds to present the evidence and let the reader judge. --- ## Page 2 (p.362) ![[1947_Clemence_Relativity_Effect_Planetary_Motions-1.png]] ### Previous Work Clemence references his earlier comprehensive study: observations of Mercury from 1765 to 1937, which was intended to exhaust the useful observational evidence available. In that discussion the relativity effect in the motion of Mercury is confirmed, and some slight evidence of the effect is found in the motion of the Earth as well. He also references H. R. Morgan's analysis of observations of the Sun which concludes the effect is present in Earth's motion. ### Updated Calculations To make the subject more comprehensible, results are presented in a different form from those previously published. Three changes: 1. Doolittle's calculation of the Newtonian motions is used, with certain corrections, instead of Newcomb's 2. Some new values of the planetary masses are introduced 3. Oort's most recent value of the precession is adopted The observational results remain unchanged. The effect of the alterations has been to make the agreement between observations and theory slightly worse instead of better, but not significantly so. > [!info] Making it worse, not better > This is remarkable honesty. Clemence updated the calculations with newer, presumably better data and methods. The result: the agreement got **slightly worse**. He publishes this without spin. In modern science, a result that gets worse with better data would raise serious questions. Clemence simply reports it. ### The Observed Motions of the Perihelia of Mercury and the Earth > *"Unfortunately, the observational material is so extensive and the methods of analysis so complex that it is not practicable here to present any evidence that will enable the reader to form an independent judgment of the errors involved."* > [!important] You cannot check this yourself > Clemence is telling the reader of *Reviews of Modern Physics* that the observational analysis is so complex that he cannot present enough information for independent verification. You have to trust the specialists. This is the opposite of reproducible science. The 43" that "proves" GR rests on calculations that cannot be independently audited from the published literature. ### Error Terminology Clemence defines "probable error" as the quartile error (multiplied by 0.6745 to get the standard deviation). He notes that quartile error measures only accidental discordances of a set of data, with **no allowance being made for systematic errors**, which in an analysis of a very extended series of observations are likely to be much more important than the accidental discordances. > [!info] Systematic errors not included > The error bars on the 43" do NOT include systematic errors. They only measure the scatter in the data. Systematic biases (wrong precession constant, wrong planetary masses, atmospheric distortion patterns) would shift the entire result up or down without showing up in the error bars. Clemence knows this and flags it explicitly. ### Two Types of Mercury Observations 1. **Meridian observations**: observations of Mercury's spherical coordinates on the celestial sphere when it is on the meridian. Extend from 1765 to 1937, numbering about 10,000 in each coordinate. 2. **Transit observations**: observations of the time at which Mercury's disk is tangent to the disk of the Sun when Mercury crosses the Sun's face. 17 transits have been used, extending from 1799 to 1940. --- ## Page 3 (p.363) ![[1947_Clemence_Relativity_Effect_Planetary_Motions-2.png]] ### Table II: Contributions to the Motion of the Perihelia of Mercury and the Earth This is the key table. Every contribution to Mercury's perihelion motion, with error estimates: | Cause | $m^{-1}$ | | Motion of Perihelion (Mercury) | Motion of Perihelion (Earth) | |---|---|---|---|---| | Mercury | 6,000,000 | ±1,000,000 | 0".005±0".00 | −137".75±2".8 | | Venus | 408,000 | ±1,000 | 277.856±0.68 | 345.43±0.8 | | Earth | 329,300 | ±300 | 90.038±0.08 | — | | Mars | 3,098,000 | ±3,000 | 2.536±0.00 | 97.09±0.1 | | Jupiter | 1,047.39 | ±0.03 | 153.584±0.01 | 699.55±0.0 | | Saturn | 3,499 | ±4 | 7.302±0.01 | 18.74±0.0 | | Uranus | 22,800 | ±200 | 0.141±0.00 | 0.57±0.0 | | Neptune | 19,000 | ±300 | 0.042±0.00 | 0.13±0.0 | | Solar oblateness | | | 0.010±0.02 | 0.03±0.0 | | Moon | | | — | 7.08±0.0 | | General precession (Julian century, 1850) | | | 5025.645±0.50 | 5025.55±0.5 | | **Sun** | | | **5557.18 ±0.85** | **6170.1 ±2.5** | | **Observed motion** | | | **5599.74 ±0.41** | **6183.7 ±1.1** | | **Difference** | | | **42.56 ±0.94** | **4.6 ±2.7** | | **Relativity effect** | | | **43.03 ±0.03** | **3.8 ±0.0** | > [!danger] What this table reveals > The "observed" precession is 5599.74"/century. The "theoretical" (Newtonian + precession of equinoxes) is 5557.18"/century. The difference is 42.56 ± 0.94. > > The relativity prediction is 43.03 ± 0.03. > > The match looks good (42.56 vs 43.03). But look at the error structure: > > - The **general precession** (Earth's axial wobble) contributes **5025.645 ± 0.50** — this single term is 90% of the total, and its uncertainty alone (±0.50) is more than half the residual being measured > - **Venus** contributes **277.856 ± 0.68** — its uncertainty is 1.6% of the residual > - The **observed motion** has uncertainty **±0.41** > - The **theoretical sum** has uncertainty **±0.85** > - The **difference** therefore has uncertainty **±0.94** — which is 2.2% of the 43" signal > > The measurement is barely 2σ. A 2σ result in any other field would not be considered definitive. > > For Earth: the difference is 4.6 ± 2.7 vs a prediction of 3.8. That's less than 1σ. Consistent with GR, but also consistent with zero. ### The Planetary Mass Problem The contributions of the planets are directly proportional to their several masses, which are not all known with the desired accuracy. The quantities denoted by $m^{-1}$ are the reciprocals of the adopted masses, the Sun's mass being taken as unity. The uncertainties in the masses of Mercury and Venus contribute most to the uncertainty of the final results. > [!info] Mass uncertainty feeds directly into the residual > The masses used in the subtraction come from orbital mechanics (the $a^3/T^2$ ratios of their satellites). If Mercury's mass is wrong by its stated uncertainty (±1,000,000 in $m^{-1}$), that shifts the "theoretical" total, which shifts the residual. The 43" is not a clean measurement — it inherits the uncertainties of every mass in the solar system. Clemence notes that until the masses of Mercury and Venus are better determined, "the motions of the perihelia of Mercury and the earth can be observed more accurately than they can be calculated." --- ## Page 4 (p.364) ![[1947_Clemence_Relativity_Effect_Planetary_Motions-3.png]] ### Solar Oblateness The effect of the rotational oblateness of the Sun produces a small additional contribution to the perihelion motions. If the Sun were a homogeneous gas sphere, the contribution would be 1".2. For the actual Sun this value must be multiplied by $4K/3$, $K$ being a dimensionless constant depending on the interior constitution. The value of $K$ is very small for a highly concentrated gas sphere (which the Sun is believed to be). The latest theoretical determination is by Motz, who finds 0.006. Clemence adopts twice its amount as the probable error, giving 0".010 ± 0".02 for Mercury. > [!info] Solar oblateness is uncertain > The Sun is not a perfect sphere — it bulges at the equator due to rotation. This bulge produces a small extra perihelion precession. But the size of the effect depends on the Sun's internal mass distribution ($K$), which is modeled, not measured. Different models give different $K$ values. This is another source of systematic uncertainty that feeds into the residual. ### The Precession of the Equinoxes > *"The precession is that resulting from Oort's latest discussion; the attached probable error is my estimate."* Clemence notes that de Sitter in 1938 estimated the probable error to be fifty percent larger than the value given here. If de Sitter's estimate is correct, the penultimate line of Table II would read 4".6 ± 3".7 instead of 4".6 ± 2".7 for Earth. > [!info] The precession constant is contested > The general precession (5025.645"/century) is the single largest number in the entire subtraction chain. Different authorities (Oort vs de Sitter) disagree on its uncertainty by 50%. Since this number is 120× larger than the residual being measured, even small fractional errors in it could swamp the 43" signal. ### Conclusion > *"The theoretical relativity effect in the motion of Mercury's perihelion is 43".03 ± 0".03; the value obtained by subtracting all other known effects from the total observed motion is 42".56 ± 0".94. For the earth's perihelion the corresponding figures are 3".8 ± 0".0 and 4".6 ± 2".7. The confirmation by observation of the relativity effect is regarded as satisfactory for both Mercury and the earth."* > [!important] "Satisfactory" > Clemence calls the confirmation "satisfactory" — not "precise," not "definitive," not "beyond doubt." For Mercury: 42.56 ± 0.94 vs 43.03 ± 0.03 — a match within the error bars, but the error bars are large (±0.94 on a 43" signal). For Earth: 4.6 ± 2.7 vs 3.8 — the uncertainty is 70% of the signal. "Satisfactory" is doing a lot of work in that sentence. > > Clemence also notes that once the gravitational theory of Mars is completed, the relativity effect in Mars's motion "should be easily detected with higher precision than has been found for the earth." As of 1947, this had not been done. --- ## Summary: What Clemence Establishes | Point | Detail | |---|---| | Observations are geocentric | Cannot be made in a Newtonian frame; affected by precession of equinoxes | | Precession of equinoxes is the largest subtraction | 5025"/century — 120× the signal, with contested uncertainty | | Planetary perturbation calculations differ between experts | "Significantly different results have been obtained by different computers" | | Mercury is the hardest planet to observe | Daytime, near the Sun, atmospheric distortion, shape of visible disk | | Systematic errors not included in error bars | Quartile errors only; systematic biases could shift the entire result | | Mass uncertainties feed into the residual | Mercury and Venus masses contribute most uncertainty | | Solar oblateness depends on interior model | $K$ factor is modeled, not measured | | The match is ~2σ for Mercury | 42.56 ± 0.94 vs 43.03 ± 0.03 | | The match is <1σ for Earth | 4.6 ± 2.7 vs 3.8 ± 0.0 (consistent with zero) | | Updated calculations made agreement slightly worse | Newer data and methods did not improve the fit | | The analysis cannot be independently verified | "Not practicable here to present any evidence that will enable the reader to form an independent judgment" | | Clemence calls the confirmation "satisfactory" | Not definitive, not precise — satisfactory | --- ## Citation Clemence, G. M. "The Relativity Effect in Planetary Motions." *Reviews of Modern Physics* 19, no. 4 (October 1947): 361–364. --- ## See Also - [[2020_Vankov_Perihelion_Critique]] — Vankov uses the Clemence quote as his epigraph - [[1915_Perihelion_Motion_of_Mercury]] — Einstein's derivation of the 43" - [[1898_Spatial_and_Temporal_Propagation_of_Gravity]] — Gerber's derivation of the same formula, 17 years earlier - [[perihelion_calc]] — Numerical verification that the formula is pure kinematics - [[00_Gravitational_Waves_Index#Step 0 The Problem — Mercurys Perihelion and Newtons Crisis]] — Presentation context