On the Annual and Diurnal Variations of the Anomalous Acceleration of Pioneer 10
| Scientific Paper | |
|---|---|
| Title | On the Annual and Diurnal Variations of the Anomalous Acceleration of Pioneer 10 |
| Read in full | Link to paper |
| Author(s) | Amitabha Ghosh |
| Keywords | anomalous acceleration, redshift |
| Published | 2007 |
| Journal | Apeiron |
| Volume | 14 |
| Number | 3 |
| No. of pages | 12 |
| Pages | 288-299 |
Read the full paper here
Abstract
An apparent anomalous acceleration of about 8 × 10−8 cm/s2 (directed towards the Sun) has been detected in the Doppler residuals of Pioneer 10 and 11. A considerable amount of effort has been made in searching for a conclusive origin of this apparent acceleration, however, without success till date. Detailed study of the data has revealed that an annual and a daily variation of the data exist and these can be interpreted as the fluctuating components of the apparent acceleration superimposed on the steady anomalous acceleration. Since these components are definitely related to the Earths motion an explanation has been found for these annual and diurnal fluctuations. The doppler effects due to the motions of the Earth are already incorporated in the model; there should thus be no residual redshift present in the results. It has been shown that the excess redshift of the signal between the Earth and Pioneer 10 due to inertial induction can manifest itself as the apparent acceleration of the spacecraft. It has been shown that the annual and the diurnal components can be accounted for by the excess redshifts due to inertial induction. Both the magnitude and the temporal phase match with the observation.
Overview
Ghosh, writing from the Indian Institute of Technology Kanpur, does something unusual in the Pioneer anomaly literature: he goes after the periodic parts of the residual rather than the headline number. The steady sunward acceleration of about 8 × 10−8 cm/s2, roughly independent of distance beyond 20 AU, had by 2007 resisted every proposed explanation. But Anderson and Turyshev's own analyses had also shown two small oscillating terms riding on it — an annual component of amplitude about 1.6 × 10−8 cm/s2 and a daily one of about 0.03 × 10−8 cm/s2 or a little less. Since the Doppler contributions from the Earth's orbital and spin motions are already in the tracking model, Ghosh argues, any residual at those two periods must come from an additional frequency shift that the model does not contain.
His candidate is velocity-dependent inertial induction, the extension of Mach's principle he had been developing since 1984 and had applied to a range of astronomical problems in Pramana, Astrophysics and Space Science and his 2000 Apeiron book Origin of Inertia. In that theory a body moving with respect to another experiences a drag; applied to a photon it produces a distance-dependent redshift over and above the Doppler shift. The paper's claim is narrow and is stated honestly: the mechanism accounts for the annual and diurnal fluctuations in both magnitude and phase, and does not account for the steady term, which "may have a separate origin."
The argument
The inertial-induction drag on a photon
The drag force between two bodies in relative motion is taken from Ghosh's earlier work as
F = GmAmBv2/(c2r2)
— the ordinary Newtonian attraction multiplied by (v/c)2 — and it always opposes the relative motion, whether B recedes from or approaches A. Applying it to a photon by setting v = c and mB = hν/c2 gives
F = GmAhν/(c2r2)
Over a displacement dr the photon loses energy dE = −F dr, and since dE = h dν, the frequency obeys dν/ν = −(GmA/c2) dr/r2. Integrating between the observing station at rE and the source at r:
ln(ν0/ν) = (GmA/c2)(1/rE − 1/r)
This is the whole physical content; everything after it is geometry and arithmetic.
Converting the excess redshift into an apparent acceleration
For the Sun–Earth–Pioneer system, mA becomes the solar mass, rE = 1 AU, and r > 40 AU over the period analysed. Pioneer 10's heliocentric latitude is only about 3° past 50 AU, so the problem is treated in the ecliptic plane, and because rE << r the Sun and Earth are taken as coincident. The exponent is small, so the round-trip fractional shift is
Δν/ν0 ≈ 2(GMS/c2)(1/rE − 1/r)
with the factor 2 for the outward and return legs. Differentiating with respect to time — only the 1/r term varies — and equating to the standard relation between a drifting Doppler residual and an apparent acceleration, aap = (c/2ν0) d(Δν)/dt, yields
aap ≈ (GMS/cr2) × dr/dt
with
dr/dt ≈ V − v sin θ − vd sin φ
where V is Pioneer 10's speed, v the Earth's orbital speed and vd the equatorial surface speed from the Earth's daily rotation. The three terms are exactly the steady, annual and diurnal components of the residual, and they are separated because the three speeds differ by orders of magnitude.
The numbers
Ghosh evaluates the three terms at three epochs using V = 1.22 × 104 m/s, v = 3 × 104 m/s and vd = 4.65 × 102 m/s:
| Component | 1990 (48 AU) | 1994 (59 AU) | 1997 (69 AU) | Observed |
|---|---|---|---|---|
| Annual, theory | 2.6 × 10−8 | 1.7 × 10−8 | 1.25 × 10−8 | 1.6 × 10−8 |
| Diurnal, theory | 0.04 × 10−8 | 0.026 × 10−8 | 0.02 × 10−8 | 0.03 × 10−8 |
| Steady, theory | 1.05 × 10−8 | 0.6 × 10−8 | 0.5 × 10−8 | 8 × 10−8 |
(all in cm/s2). These reproduce on recomputation: at r = 48 AU, GMS/cr2 = 8.6 × 10−15 s−1, which multiplied by v, vd and V gives 2.57, 0.040 and 1.05 × 10−8 cm/s2 respectively, and the 1/r2 scaling then reproduces the 1994 and 1997 columns.
Ghosh states the negative result plainly: "it is quite clear that the inertial induction effect cannot explain the steady part of the observed apparent acceleration of Pioneer 10", being an order of magnitude short.
The phase test
The more distinctive part of the paper is a check on when the annual maximum should occur. The extremes of dr/dt fall where the Earth–Sun line is perpendicular to the Sun–Pioneer line. Pioneer 10's ecliptic longitude drifts slowly from 71° at 40 AU to 75.6° at 69 AU, so an average of 73° is used; the Earth's ecliptic longitude on 1 January is 99.8°. The maximum therefore falls 63.2° of orbital phase after 1 January — "about 1/6th of a year, i.e. about 2 months" — and Ghosh reports that the peaks in the published residuals do occur about two months after 1 January each year. He adds that the model predicts a slow decline in amplitude with increasing r, and that there is "a faint suggestion of a gradual decrease of the amplitude over the years, though a reasonable quantitative analysis is difficult."
Assessment
Two things about this paper are genuinely good. The first is its restraint: Ghosh had a theory of inertia he had been promoting for over twenty years and an unexplained anomaly in front of him, and he reports that his mechanism misses the headline number by a factor of eight and says so in the body text, in the table and in the concluding remarks. That is not the usual pattern in this literature. The second is that he attacks the part of the data that carries the most information. A steady offset can be produced by almost anything; a periodic term carries an amplitude, a period and a phase, and a model that gets all three has said something. Predicting the annual maximum at roughly two months after 1 January from nothing but two ecliptic longitudes is a real test, and the arithmetic behind it is correct.
The arithmetic elsewhere is correct too — every entry in Table 1 reproduces from the stated inputs and the 1/r2 scaling — which makes it possible to see what the table actually shows. The annual prediction is not a fixed 1.6 × 10−8 cm/s2; it falls by a factor of 2.1 across the seven years tabulated, from 2.6 to 1.25, straddling the single observed value. So the model matches the observation at one epoch and is 60 per cent high or 20 per cent low at the others, and the decisive test is precisely the decline the author concedes he cannot quantify. A firm measurement of that decline would have settled the question in either direction; the paper leaves it open, resting instead on a "faint suggestion" that the author declines to quantify.
The steady term is worse than the paper admits. Ghosh notes only that his prediction is an order of magnitude too small. But his formula also gives the steady component a 1/r2 dependence, falling from 1.05 to 0.5 × 10−8 cm/s2 between 48 and 69 AU, whereas the defining observational property of the anomaly — stated in the paper's own introduction — is that it is "reasonably independent of the distance from the Sun beyond 20 AU". The mismatch is in the functional form as well as the magnitude, which is a stronger objection than the one raised.
A further difficulty is not addressed at all. The redshift derived here, (GM/c2)(1/rE − 1/r), has exactly the magnitude and the same 1/r form as the ordinary gravitational redshift. For light leaving the solar photosphere, GMS/c2RS corresponds to 636 m/s — which is the measured solar gravitational redshift, determined from Fe I lines at 638 ± 6 m/s and agreeing with the standard prediction to about one per cent. If inertial-induction drag is an additional effect it doubles a quantity that is measured and already accounted for; if it is meant to replace the gravitational redshift, the paper needs to say so, and the round-trip factor of 2 in equation (11) implies the shifts are being added rather than substituted. Nothing in the text resolves which is intended, and the point is not peripheral: it is the same constant GM/c2 doing the work in both cases.
Finally, the history has moved. The steady anomaly that Ghosh could not explain was resolved shortly after this paper by Turyshev and collaborators, who recovered the early Pioneer telemetry, built a thermal model of the spacecraft, and showed that anisotropic emission of heat from the radioisotope generators and the electronics compartment produces a recoil of the right size and — the decisive point — one that decays with the 87.7-year half-life of the plutonium fuel, a decline subsequently found in the extended Doppler data. The annual and diurnal terms that this paper set out to explain were attributed by the same analyses to errors in the modelling of the Earth's orientation and orbit and to station-dependent troposphere and clock effects, which is what one would expect of residuals whose periods are exactly the Earth's two rotation periods. Ghosh's own framing — that these terms "must be linked to the orbital and spin motions of the Earth" — is right, but that link is at least as naturally an artefact of how those motions are modelled as it is a new force. His paper is best read as a careful, self-critical demonstration of what a photon-drag term of this size can and cannot do, published at the point when the question was still genuinely open.