George E. Smith Tufts University

TURNING DATA INTO EVIDENCEThree Lectures on the Role of Theory in Science1. CLOSING THE LOOPTesting Newtonian Gravity, Then and Now2. GETTING STARTEDBuilding Theories from Working Hypotheses3. GAINING ACCESSUsing Seismology to Probe the Earth’s Insides George E. Smith Tufts University

THE USUAL VIEW • In science what turns a datum B into evidence for a claim A that reaches beyond it is a deduction from A of a sufficiently close counterpart of B. • In particular, historically what made celestial observations evidence for Newtonian gravity were the increasingly accurate predictions derived from the theory of these observations • The realization that Einsteinian gravity would all along have yielded no less accurate predictions tells us that scientists had all along over-valued the evidence for Newtonian gravity

Deduced predictions in celestial mechanics presuppose a proviso: no other forces (of consequence) are at work. The only evidence for this proviso is close agreement between the predictions and observation. But then a primary purpose of comparing deduced predictions and observation is to answer the question, Are other forces at work? How then is the theory of gravity tested in the process? SO, WHY NOT SIMPLY HYPOTHESIS TESTING BY MEANS OF DEDUCED PREDICTIONS?HEMPEL’S PROVISO PROBLEM

OUTLINE • Introduction: the issue • The logic, as dictated by Newton’s Principia • How this logic played out after the Principia A. “Then” – complications that obscure the logic B. “Now” – in light of the perihelion of Mercury • Concluding remarks

“GRAVITY RESEARCH” THEN AND NOW IN CELESTIAL MECHANICS: What are the true motions – orbital and rotational – of the planets, their satellites, and comets, and what forces govern these motions? IN PHYSICAL GEODESY: What is the shape of the Earth, how does the gravitational field surrounding it vary, and what distribution of density within the Earth produces this field?

CALCULATING PLANETARY ORBITS — 1680

NEWTON’S EVIDENCE PROBLEM IN THE PRINCIPIA “By reason of the deviation of the Sun from the center of gravity, the centripetal force does not always tend to that immobile center, and hence the planets neither move exactly in ellipses nor revolve twice in the same orbit. Each time a planet revolves it traces a fresh orbit, as in the motion of the Moon, and each orbit depends on the combined motions of all the planets, not to mention the action of all these on each other. But to consider simultaneously all these causes of motion and to define these motions by exact laws admitting of easy calculation exceeds, if I am not mistaken, the force of any human mind.” Isaac Newton, ca. December 1684 (First published by Rouse Ball in 1893)

INFERRING LAWS OF FORCE FROMPHENOMENA OF MOTION Phenomena: Descriptions of regularities of motion that hold at least quam proxime over a finite body of observations from a limited period of time The planets swept out equal areas in equal times quam proxime with respect to the Sun over the period from the 1580s to the 1680s. Propositions, deduced from the laws of motion, of the form: “If _ _ _ quam proxime, then …… quam proxime.” If a body sweeps out equal areas in equal times quam proxime with respect to some point, then the force governing its motion is directed quam proxime toward this point. • Conclusions: Specifications of forces (central accelerations) that hold at least quam proxime over the given finite body of observations Therefore, the force governing the orbital motion of the planets, at least from the 1580s to the 1680s, was directed quam proxime toward the Sun.

From Evidence that is Approximate to A Law that is Taken to be Exact Rule 3:Those qualities of bodies that cannot be intended and remitted and that belong to all bodies on which experiments can be made should be regardedas qualities of all bodies universally. Rule 4: In experimental philosophy, propositions gathered from phenomena by induction should be regarded as either exactly or very, very nearly true notwithstanding any con-trary hypotheses, until yet other phenomena make such propositions either more exact or liable to exceptions. This rule should be followed so that arguments based on induction may not be nullified by hypotheses.

PREREQUISITES FOR TAKING THE THEORY OF GRAVITY AS EXACT • The theory must identify specific conditions under which the phenomena from which it was inferred would hold exactly without restriction of time – e.g. • The area rule would hold exactly in the absence of forces from other orbiting bodies • The orbits would be perfectly stationary were it not for perturbing forces from other orbiting bodies • The theory must identify a specific configuration in which the macroscopic variation of gravity about a body would result from the microstructure of the body – e.g. • Gravity would vary exactly as the inverse-square around a body were it a sphere with a spherically symmetric distribution of density

TAKING THE THEORY TO BE EXACTTHE PRIMARY IMPLICATION Every systematic discrepancy between observation and any theoretically deduced result ought to stem from a physical source not taken into account in the theoretical deduction • a further density variation • a further celestial force

THE NEWTONIAN APPROACHCONTINUING EVIDENCE • Taking the law of gravity to hold exactly was a research strategy, adopted in response to the complexity of the true planetary motions. • Deductions of planetary motions etc. are “Newtonian” idealizations: approximations that, according to theory, would hold exactly in certain specifiable circumstances -- in particular, in the absence of further forces or density variations. • The upshot of comparing calculated and observed orbital motions is to shift the focus of ongoing research onto systematic discrepancies, asking in a sequence of successive approximations, whatfurther forces or density variations are at work? • Theory thus becomes, first and foremost, not an explanation (or even a representation) of known phenomena, but an instrument in ongoing research, revealing new “second-order” phenomena that can provide a basis for continuing testing of the theory.

THE LOGIC OF THEORY TESTING • The theory requires that every deviation from any “Newtonian” idealization be physically significant – i.e. every deviation must result from some further force or density variation. • Basic Testing: pin down sources of the discrepancies and confirm they are robust and physically significant (within the context of the theory) while achieving progressively smaller discrepancies between (idealized) calculation and observation. • Ramified Testing: keep incorporating previously identified physical sources of second-order phenomena into the (idealized) calculation, thereby progressively constraining the freedom to pursue physical sources for new second-order phenomena that then emerge. • The continuing evidence lies not merely in the aggregate of the individual comparisons with observation, but also in the history of the developmentof the sequence of successive approximations.

NEPTUNE AS AN EXAMPLE OF “PHYSICAL SIGNIFICANCE” seconds of arc

THE “GREAT INEQUALITY” AS A MORE TYPICAL EXAMPLE minutes of arc

OUTLINE • Introduction: the issue • The logic, as dictated by Newton’s Principia • How this logic played out after the Principia A. “Then” – complications obscuring the logic B. “Now” – in light of the perihelion of Mercury • Concluding remarks

Example Example Deviation of surface gravity from Newton’s ideal variation implies the value of (C-A)/Ma2 and hence a correction to the difference (C-A) in the Earth’s moments of inertia, and the lunar-solar precession implies the value of (C-A)/Cand hence a correction to the polar moment C; these two corrected values constrain the variation (r)of density inside the Earth, but they do not suffice to determine (r). Second-Order Phenomena Often UnderdetermineTheir Physical Source

RESPONDING TO UNDERDETERMINATION20TH CENTURY DETERMINATION OF (r) density core-mantle boundary density

Examples Mass of Moon inferred from lunar nutation supported by calculated tides and lunar-solar precession Mass of Venus inferred from a particular inequality in the motion of Mars supported by calculated perturbations of Mercury, Earth, and Mars The far reach of the gravity fields of Jupiter and Saturn supported by variations in period of Halley’s comet ROBUSTNESS OF PHYSICAL SOURCES

“The motion of the [lunar] perigee can be got [from observation] to within about 500,000th of the whole. None of the values hitherto computed from theory agrees as closely as this with the value derived from observation. The question then arises whether the discrepancy should be attributed to the fault of not having carried the approximation far enough, or is indicative of forces acting on the moon which have not yet been considered.” G. W. Hill, 1875 Newcomb’s Discordances, 1895 Mercury’s perihelion was 29 times probable error Venus’s nodes was 5 times probable error Mars’s perihelion was 3 times probable error Mercury’s eccentricity  was 2 times probable error PROBLEMS IN ISOLATING DISCORDANCES

ANOTHER EXAMPLE OF DIFFICULTY Many professional lives have been dedicated to the long series of meridian circle (transit) observations of the stars and planets throughout the past three centuries. These observations represent some of the most accurate scientific measurements in existence before the advent of electronics. The numerous successes arising from these instruments are certainly most impressive. However, as with all measurements, there is a limit to the accuracy beyond which one cannot expect to extract valid information. There are many cases where that limit has been exceeded; Planet X has surely been such a case.

In observations: Simple error – “bad data” Limits of precision Systematic bias in instruments Inadequate corrections for known sources of systematic error, incl. Imprecise fundamental constants Not yet identified sources of systematic error In theoretical calculations: Undetected calculation errors Imprecise orbital elements Imprecise planetary masses Insufficiently converged infinite-series calculations Need for higher-order terms Forces not taken into account Gravitation theory wrong THE MANY SOURCES OF DISCREPANCIES “Theultimate goal of celestial mechanics is to resolve the great question whether Newton’s law by itself accounts for all astronomical phenomena;the sole means of doing so is to make observations as precise as possible and then to compare them with the results of calculation. The calculation can only be approximate….”Henri Poincaré, 1892

18th Century: Acceleration in motion of Moon announced by Halley (1693) A physical source identified by Laplace (1787): 19th Century: Adams finds that Laplace has accounted for only half of the “secular” motion (1854) A further physical source: earth is slowing from tidal friction “SECULAR” MOTION OF THE MOON Owing to perturbations from gravity toward the planets, eccentricity of Earth’s orbit changing.

EXAMPLE OF SPECTACULAR SUCCESSSPENCER JONES (1939) • Residual discrepancies in the motions of Mercury, Venus, and Earth correlate with unaccounted-for discrepancy in lunar motion • Common cause => Earth’s rotation irregular (in more ways than one) • Expose a still further systematic observation error, requiring correction: • 1950: replace sidereal time with “ephemeris time”

This form of evidence can be very strong • It is evidence aimed at the question of the physical exactness of the theory, as well as the question of its projectibility • The sequence of successive approximations leads to new second-order phenomena of progressively smaller magnitude • New second-order phenomena presuppose not only the theory of gravity, but also previously identified physical sources of earlier second-order phenomena, thereby constraining the freedom to respond to these new phenomena • Theory becomes entrenched from its sustained success in exposing increasingly subtle details of the physical world without having to backtrack and reject earlier discoveries

Idealized calculated orbits presupposing theory and various physical details Comparison with astronomical observations Discrepancy with clear signature! Physical source of discrepancy: still further physical details that make a difference! New idealized calculation incorporating the new details and their further implications Ever smaller discrepancies Ever many more details that turn out to make a difference OVERALL HISTORICAL PATTERNA “FEEDBACK” LOOP

OUTLINE • Introduction: the issue • The logic, as dictated by Newton’s Principia • How this logic played out after the Principia A. “Then” – complications obscuring the logic B. “Now” – in light of the perihelion of Mercury • Concluding remarks

INEXACTNESS EXPOSED:THE PERIHELION OF MERCURY “The secular variations already given are derived from these same values of the masses, the centennial motion of the perihelion being increased by the quantity Dt = 43.″37 In order to represent the observed motion. This quantity is the product of the centennial mean motion by the factor 0.000 000 0806”

PERIHELION OF MERCURY: CURRENT

FROM NEWTONIAN TO EINSTEINIAN GRAVITY Discrepancy between Newtonian calculation and observation: 43´´.37 ± 2.1 ==> 43´´.11 ± 0.45 Increment from the Einsteinian calculation: 43´´. ==> 42´´.98 Newtonian gravity is the static, weak-field limit of Einsteinian! A limit-case idealization The orbital equation becomes, where μ = G(M+m), u = 1/r:

CONTINUITY OF EVIDENCE ACROSS THE CONCEPTUAL DIVIDE • 43´´ per century was a Newtonian second-order phenomenon • From limit-case reasoning, evidence for Newtonian gravity carried over, with minor qualifications, to Einsteinian • Earlier evidential reasoning for Newtonian gravity, even though requiring some qualifications, was not nullified • Previously identified physical sources of Newtonian second-order phenomena remained intact in Einsteinian

“… though the world does not change with a change of paradigm, the scientist afterward works in a different world…. I am convinced that we must learn to make sense of statements that at least resemble these.” Thomas S. Kuhn, SSR, p. 121 The continuity of evidence across the conceptual divide between Newtonian and Einsteinian gravity highlights an extremely important sense in which the scientist afterward works in the same world.

PRIMARY CONCLUSIONS • The most important evidence in classical gravitational research came from the complexities of the actual motions and of the gravitational fields surrounding bodies. • This evidence consisted of success in pinning down physical sources of deviations from “Newtonian” idealizations, in a sequence of increasingly precise successive approximations. • This evidence carried forward, continuously, across the tran-sition from Newtonian to Einsteinian gravity and remains an important source of continuing evidence today.

Idealized calculated orbits presupposing theory and various physical details Comparison with astronomical observations Discrepancy with clear signature! (Revised theory when deemed necessary) Physical source of discrepancy: still further physical details that make a difference! New idealized calculation incorporating the new details and their further implications Thrust of the Evidence: Not merely numerical agreement, a curve-fit Increasingly strong, still continuing evidence that certain physical details make specific differences CLOSING THE LOOP

THE KNOWLEDGE ACHIEVED IN GRAVITY SCIENCE • Interpenetration of theory and an ever growing multiplicity of details that make a difference • Details: evidence for theory and values for parameters • Theory: lawlike generalizations supporting counterfactual conditionals that license conclusions about differences a detail makes • Two requirements for generalizations to do this: • They must hold to high approximation over a restricted domain • They must be lawlike – i.e. they must be projectible over this domain • Just what Einstein showed about Newtonian gravity, and Newton took the trouble to show about Galilean gravity

George E. Smith Tufts University