Has anyone looked into the possibility that the effect is caused by the increased mass of the Sun within Pioneer’s reference frame? As objects move faster their momentum functions like additional mass. From our reference frame here on Earth, the Pioneer probes are moving very fast and appear to be slightly heavier. But the situation is symmetrical. From the perspective of the probes, the Sun is moving very fast and also appears to be slightly heavier. That small increase in mass should lead to a small increase in the amount of gravitational attraction the sun exerts on the probes, slowing them down.
Now maybe scientists have already factored this into their calculations of the expected trajectory of the probes. But none of the articles that I’ve read about the anomaly mention it. It seems like the effect should be large enough to notice – if GPS satellites are moving fast enough for relativistic time dilation to affect their accuracy, it seems like the Pioneer probes should be moving fast enough for relativistic mass to affect their trajectories.
First of all, while objects moving at great speed have more momentum than Newton would have calculated, this momentum does not function like extra mass. Second, the theories of relativity (special or general, the latter being the relevant one in gravitational problems) cannot explain the Pioneer anomaly.
Third of all, though, there are a great many other things which can explain it, and most of them are pretty boring. For instance, it could be that the interplanetary medium is slightly more dense than we’re assuming, or many other such mundane possibilities. We would want to rule out all of those mundane explanations very thoroughly before we should start thinking about any more esoteric explanations involving changes to the laws of physics as we know them.
Do we have any other probes headed into deep space that will give us more data along these lines? This is one of those things that probably has a mundane explanation, but we still need to do the grunt work of figuring it out.
That Chronos certainly isn’t me. I’m a bit surprised; I would have thought I’d have noticed someone with the same name who was that prolific in physics discussions online.
And the business about whether kinetic energy is or is not mass is a bit more complicated than it might at first appear. Mass is a property of a system of objects, not just of individual objects. Basically, for any system of objects, there is some reference frame where the total energy of the system is a minimum. The energy in that reference frame is the mass.
So, for the example of the Sun-Pioneer system: The reference frame in which the energy is minimized is the one where the center of mass is at rest. Since the Sun is much, much more massive than the Pioneer, this is almost exactly the same as the frame where the Sun is at rest. So almost none of the energy in this frame is due to the Sun’s kinetic energy, so almost none of the Sun’s kinetic energy contributes to the mass.
On the other hand, if you have two stars (or subatomic particles) moving rapidly on a collision course with each other, then the center of mass is right between them, and both have a significant speed relative to the center of mass. So in this case, in the COM frame, there is still significant kinetic energy from both, which does contribute to the mass of the system.
As for the synchrocyclotron, the compensation there is not due to the mass changing, but due to the fact that momentum is no longer mass times velocity. Basically, it’s only if you use P = m*v as a definition for mass that you get this notion of “relativistic mass”.
I’m pretty sure that the effects of relativistic speeds on gravity are well enough understood that the scientists would know what its effect would be on such probes and take it into account. The fact that it isn’t mentioned in the articles you read is because all the scientists know it has nothing to do with the situation, and thus have no reason to bring it up when discussing it in an interview with a news source.
I would find it highly surprising if they took any special effort to calculate orbital modifications due to relativistic changes for a Mars orbit, as the difference is somewhere near the order of 1 m, and thus well within the error bounds for measurement. It may be that the standard codes they are using factor in relativistic changes (which is important for calculating precise positions for multi-year journeys going to the outer system) but the effects are insignificant for an Earth-Mars transit.
The fact that mars is big enough, and close enough to make the order of magnitude irrelevant to actually getting there, the instrumentation used relies on time measurement of an extremely high accuracy on both the vehicle, and on ground control. Keeping those in sync requires very close attention to relativistic variations.