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6             Application to the Pioneers

In March 1972, the Pioneer 10 spacecraft was launched. Just over a year later – in April 1973 – Pioneer 11 followed. Both described orbits in the plane of the ecliptic, which took them past Jupiter and other planets, ultimately leaving our solar system in opposite directions.  

They generated a vast amount of research data about the solar system. One of the goals of the projects was to better understand the gravitational field in the solar system. The measurements of the spacecraft's accelerations were exceptionally precise, measuring them with an accuracy of 10^–10 m/s². However, this precision presented a new problem that remains unresolved.

 After passing Jupiter and completing final corrective manoeuvres, the Pioneers showed a faint but unmistakable, constant deceleration as they continued to increase their distance from the Sun, with a final calculated value – including the margin of error – of (8.74±1.33)x10^–10 m/s².
The reliability of the measurements is beyond dispute, but the result would imply that both spacecraft would eventually re-enter the solar system. This is completely contrary to common sense. However, science has not yet been able to offer an explanation for this.

 We believe the Obstruction Theory can provide this explanation. For simplicity, we will limit ourselves to the orbital motion of Pioneer 10. It will turn out to be an ill–considered way of measuring. From Earth's perspective, Pioneer should experience the anomalous acceleration from §4. But from Pioneer's perspective, Earth should also experience the same anomalous acceleration in the opposite direction. Both objects – Pioneer and Earth – are in the sun's gravitational field.

The measured anomalous acceleration of the Pioneer is found by adding the acceleration the Pioneer experiences due to the anomalous acceleration in the sun's gravitational field, measured from Earth, to the acceleration the Earth experiences in the opposite direction, as seen from the Pioneer. This therefore occurs between Earth and the spacecraft.

 

Together, this should yield 9.44x10^–10 m/s² near Jupiter. However, measurements near Jupiter could not yet be performed because the manoeuvres were still ongoing.

 How does the measurement work? A radio signal was regularly sent to the Pioneer and returned after processing. The Pioneer's distance, speed, and acceleration were determined from the signal's duration and frequency change.
These results naturally had to agree with the theory. The calculations were corrected for the time delay caused by the speed and gravitational field of the sun, as well as many other physical factors that could influence the speed. Relativistic mass and curved space were also taken into account. The only thing that was obviously not taken into account, because it was still unknown, was anomalous acceleration.

If we limit ourselves to Newtonian gravitational acceleration, we can derive Pioneer's acceleration from the calculations by subtracting the acceleration experienced by the Earth relative to the Sun from Pioneer's measured acceleration. This yields Pioneer's gravitational acceleration.  

However, the measurements measure not only the normal gravitational acceleration, but also the anomalous acceleration. Therefore, to find Pioneer's acceleration, we must also subtract the anomalous acceleration from the measurements. If we don't do this, the correction is insufficient we are left with the sum of the anomalous accelerations at Pioneer and Earth.  

That's the mistake NASA made. Therefore, at the end of the calculation, they should find an acceleration of Pioneer of at least 9.37x10^–10 m/s² towards Earth.
This result falls well within the margin of error cited in the literature for the measured anomalous acceleration of (8.74 ± 1.33)x10^–10 m/s².  

The reason the anomalous acceleration wasn't noticed until beyond Jupiter is likely related to the occasional accelerations Pioneer had to undergo before reaching Jupiter to steer it in the right direction. This prevented a usable, continuous series of measurements. Beyond Jupiter, this was possible, but initially, no anomaly was found.

This can be explained by the fact that the Pioneer was steered past Jupiter in such a way that it could make optimal use of the speed gain due to Jupiter's gravitational assistance.

This velocity gain is optimal when the Pioneer is flung away in a tangential direction relative to the distance r from the sun. The measured radial velocity relative to the Earth at j =90 degrees is zero at that moment. The Pioneer then no longer has any radial velocity relative to the Earth, so the anomalous velocity and anomalous acceleration can not occur. Pioneer's acceleration measurement then yields the traditional value of the sun's gravitational field at that location. The anomalous acceleration is then zero.  

Fig. 2 The orbits that the Pioneers described in the solar system

 After passing Jupiter, however, Pioneer will have reached a very high velocity, allowing it to escape the solar system. Its trajectory will increasingly become radial, with Pioneer moving in a straight line away from Earth.

We can then use (§4) the formula
     m/s² for Earth and Pioneer. The difference between the measured acceleration and the traditionally calculated acceleration is then found as:   m/s².

 We had to estimate the angle φ, the direction Pioneer is moving relative to the line with the Earth, using the drawing (see Fig 2) that NASA released of the orbits of Pioneer 10 and 11.

The sharp angles that the orbits exhibit upon encountering Jupiter are the result of the orbit being deflected by Jupiter's gravitational pull, which also results in a gain in speed.

 Using the data from the planets and an estimate of the angle j  and the last formula, we numerically calculated the anomalous acceleration Dg at different distances in the solar system (Table 1).

 As an approximation, we've positioned the Earth in the direction of the sun, but due to the Earth's orbital motion, the angle j should show an annual variation when calculated accurately. We've omitted this from these calculations.

The result of the calculations is also shown in a graph (Fig 3).

   

Table 1 The anomalous acceleration at the distances beyond Jupiter .

 

 

Figure 3 The anomalous acceleration of Pioneer 10 in the solar system according to the obstruction theory

 A comparison of the NASA data (Fig 4) with our results (Fig 3) reveals a remarkable agreement. However, there are also differences. For example, in our calculations, the level of constant acceleration is reached at 15 AU, while according to NASA's calculations, this only occurs at distances above 20 AU.
Furthermore, according to NASA measurements, the acceleration decreases in magnitude from that point onward, while the anomalous acceleration, according to the obstruction theory, becomes absolutely constant. This difference can be explained by assuming that NASA made incorrect estimates in its calculations of the spacecraft's acceleration corrections in an attempt to explain the anomalous acceleration. These incorrect corrections can have consequences even at great distances.

Figure 4 The anomalous acceleration as measured on Pioneers 10 and 11

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