#372 Pioneer Anomaly

Francois-Zinserling2026-06-08, edited 2026-06-22OpenGravitation Models/Derivations

Here’s an interesting summary of the Pioneer anomaly and how ‘heat’ was pinpointed as the culprit.

I wonder if heat was truly the issue, or if there was just another ‘loudest voice’ in the room.

Also interesting: “The flyby anomaly, in which probes like Galileo, NEAR, Cassini and Rosetta have picked up tiny unexpected velocity changes during Earth gravity assists, has not been fully explained.”

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Comments (3)

HanDeBruijn2026-06-08, edited 2026-06-27

Quoting from a paper on VM that is forthcoming.

The Pioneer anomaly is an extremely small acceleration towards the Sun, of $(8.74 \pm 1.33)\times 10^{-10}\mbox{ }m/s^2$. Previously proposed explanations include: The magnitude of the Pioneer effect is numerically quite close to the product [ .. ] of the speed of light [ .. ] and the Hubble constant [ .. ], ipse est $a_P \approx cH$. Wikipedia proceeds with: hinting at a cosmological connection, but this is now believed to be of no particular significance. Of no particular significance? Formula (8) in our second paper on Variable Mass is repeated here as a reminder.

$$ F = 2\pi G m \rho_c \left[ - \frac{1}{2\Gamma} + \frac{1}{2\Gamma} \right]= 0 \quad \mbox{with} \quad \Gamma = \frac{H}{c} $$

$G=$ Gravitational constant, $m=$ mass of test particle, $\rho_c=$ density (mass/volume) of cosmos here and now, $H=$ (intrinsic) Hubble parameter, $c=$ speed of light in empty space.

Some people think that a density $\rho_c$ of matter in the universe is not a meaningful concept. One reason is that exist huge irregular (fractal) structures out there. How to define an “infinitesimal” volume then in a sensible way? Let’s have a quote from The Case for a Hierarchical Cosmology by Gérard de Vaucouleurs. I have discussed on several occasions since 1953 the growing evidence for a Local Supercluster (23-25), encompassing the majority of the nearby galaxies and groups with a center approximately in or near the Virgo cluster. The influence of this supercluster on galaxy counts can be detected at least down to m = 16 in the northern galactic hemisphere (26). Our Galaxy is in an outlying location, in our Local Group, near the southern edge of the system.

Such an extreme case would possibly be covered by the following thought experiment. Suppose that all matter is at one side of the test particle and calculate an upper bound for the anomalous acceleration $a_P$, by modifying the above equation.

$$ F = 2\times 2\pi G m \rho_c \frac{1}{2\Gamma} \quad \mbox{with} \quad \Gamma = \frac{H}{c} $$

$$ \frac{F}{m} = 2\pi G \rho_c \frac{c}{H} = \frac{c}{H}\frac{6}{4}\left(\frac{4\pi}{3}G\rho_c\right) = \frac{3}{2}\frac{c}{H}H^2 = \frac{3}{2}c H \ \Longrightarrow \quad a_P \lt \frac{3}{2}c H $$

Use has been made of an equation for the Hubble parameter that is suggested in the book Origin of Inertia by Amitabha Ghosh in: 9.3 A Concept of Potential Energy in an Infinite Universe, footnote 11 on page 134:

$$ H=\sqrt{\frac{4}{3} G \rho_c} $$

Calculation details:

# Speed of light
c := 299792458;
# 1 megaParsec
Mpc := 3.08567758*10^22;
# Hubble parameter (2022-02-08)
H := 73.4*1000/Mpc;
# Pioneer Anomaly upper bound
3/2*H*c;

Resulting in a value $10.69688870\times 10^{-10}\mbox{ }m/s^2$. Indeed the observed acceleration of $a_P = (8.74\pm 1.33)\times 10^{-10}\mbox{ }m/s^2$ is somewhat below this upper bound.

Update

The second paper [2] on Variable Mass has been updated and the third one [3] has taken on a different content than anticipated. The formula from the second paper has become

$$ F = 2\pi G m \rho_c\mbox{ }\int_0^\infty e^{-\Gamma\mbox{ }u}\mbox{ }du \cdot \int_{-1}^{+1} v\mbox{ }dv = 0 \quad \mbox{with} \quad \Gamma = \frac{H}{c} \qquad (8) $$

And according to formula (10) in [3] we have

$$ \frac{H^2}{8\pi.G.\rho_c} = w \qquad (10) $$

Let us assume that only half of the universe is in play. Then there is a minor change in he integration bounds for $v$.

$$ F = 2\pi G m \rho_c\mbox{ }\int_0^\infty e^{-\Gamma\mbox{ }u}\mbox{ }du \cdot \int_{0}^{+1} v\mbox{ }dv \ne 0 \quad \Longrightarrow \quad a_P = 2 \pi G \rho_c.\frac{1}{\Gamma}.\frac{1}{2} = \pi G \rho_c . \frac{c}{H} $$

Substituting (10):

$$ a_P = \frac{8\pi.G.\rho_c.w}{8w}.\frac{c}{H} = \frac{H^2}{8w}.\frac{c}{H} = \frac{1}{8w}.Hc \quad \mbox{with} \quad w = 0.3883945571 $$

Giving an outcome $\approx 0,32\times Hc$ which is not even close to the desired one. But it’s interesting to notice that the acceleration $(Hc)$ pops up every time again.

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RedshiftDrift2026-06-09

I wonder if heat was truly the issue, or if there was just another ‘loudest voice’ in the room. - @Francois-Zinserling

It was the only explanation they could come up with, and they had to ‘tweak’ the heat leaked by the RTG to fit the measured acceleration.

The Pioneer anomaly is an extremely small acceleration towards the Sun, of $(8.74 \pm 1.33) \times 10^{-10}$ m/s$^2$. - @HanDeBruijn

There are some difficulties for Ghosh’s theory:

From [1] J. D. Anderson et al., “Study of the anomalous acceleration of Pioneer 10 and 11,” Physical Review D, vol. 65, no. 8, Apr. 2002, doi: 10.1103/PhysRevD.65.082004, the anomaly is not seen on the orbits of Earth and Mars (my highlight):

However, any universal gravitational explanation for the Pioneer effect comes up against a hard experimental wall. The anomalous acceleration is too large to have gone undetected in planetary orbits, particularly for Earth and Mars. NASA’s Viking mission provided radio-ranging measurements to an accuracy of about 12 m. If a planet experiences a small, anomalous, radial acceleration, $a_A$, its orbital radius $r$ is perturbed by $\Delta r = - (l^6 a_A)/(GM_\odot )^4$ → $-r a_A/a_N$ (Eq. (58)) where $l$ is the orbital angular momentum per unit mass and $a_N$ is the Newtonian acceleration at $r$. (The right value in Eq. (58) holds in the circular orbit limit.) For Earth and Mars, $\Delta r$ is about -21 km and -76 km. However, the Viking data determines the difference between the Mars and Earth orbital radii to about a 100 m accuracy, and their sum to an accuracy of about 150 m. The Pioneer effect is not seen.

The unexpected velocity changes picked up by probes like Galileo, NEAR, Cassini and Rosetta during Earth gravity assists only happened near Earth and depended on the inclination of the probe’s path.
The anomalous acceleration appears (for Pioneer 11) past the orbit of Saturn (although the accuracy is not that high for measurements at 9 AU but good at 6 AU.)

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ExpEarth2026-06-09, edited 2026-06-09

Here is my take on it. I can’t recall what Ghosh said, but the clue to this puzzle is that the acceleration shows up after the probe passes Saturn. After Saturn, it has picked up enough velocity to escape the Solar System. Before that point it was gravitationally speaking a part of the Solar System. It would not have shown an acceleration then. After Saturn the probe is operating in some larger gravitational frame of reference and only then can be seen to be accelerating. The acceleration is simply pointed against its direction of motion. The Earth and Mars would not be expected to show this effect, as they are fully bound to the Sun. You would only see it in them if an asteroid or something boosted them to escape velocity. For the same reason the MOND acceleration is only seen once stars are on the edge of achieving escape velocity from their galaxies. The acceleration continually acts against their motion and keeps them gravitationally bound.

At the same time, there is much evidence that the Moon’s orbital radius around the Earth, the Astronomical Unit, Titan’s orbit around Saturn and other orbital radii are all steadily increasing in proportion to the Hubble constant. This is related but not precisely the same thing as MOND or the Pioneer acceleration.

The heat explanation falls into the category of preferentially selecting explanations that don’t involve new physics.

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