# Is the speed of the Voyager probes accurately described by Newton's or Einstein's theories?

I have read a book claiming that the speed of the Voyager probes do not fit neither the Newton's laws of motion nor Einstein's relativity theory. Is it a recognized scientific fact? Where can I find the relevant information?

• I think you might be thinking of the Pioneer anomaly, not Voyager. That was accounted for by radiation pressure from waste heat. – DylanSp Feb 19 '16 at 12:52
• I agree with DylanSp that you are probably thinking of Pioneer rather than Voyager, but I have edited the title of your question to hopefully more accurately summarize the content of the question. You may want to edit further to at least indicate which book you have read that states this, and how it is stated in that book. – user Feb 19 '16 at 13:22
• Neither Newton nor Einstein's theories are sufficient on Earth, as they fail to predict friction. Now there's of course no planetary atmosphere in space, but it's not a perfect vacuum either. How much friction does that cause? How much trust is there from solar radiation? All those are inputs to the laws of motions, not predictions from the laws of motion. – MSalters Feb 22 '16 at 15:31
• @MSalters: you are confusing several things. It is correct that they don't predict friction, but this is the right thing to do. They give general formula of mechanics. Friction is one of force (or acceleration) that one should account when using Newton or Einstein equations. Then there is the predict part. Without measurements it is difficult to estimate the quantity of friction (chicken-egg problem when we have only few probes). – Giacomo Catenazzi Feb 28 '16 at 10:11
• Actually, they both account for friction just fine on microscopic level. It's just that friction being an effect of trillions of interactions per square millimeter of affected surface, the solution scales up badly, being still perfectly correct but completely impractical to solve, requiring far more calculations than anyone would be willing to undertake. – SF. Apr 30 '16 at 12:12

Wikipedia article "Pioneer anomaly" seems to address this. The accepted answer to this slowing down of the spacecraft slightly more than predicted is that the deceleration was caused by "an anisotropic radiation pressure caused by the spacecraft's heat loss."

So the anomaly was caused by radiation pressure caused by heat being emitted from the spacecraft.

The is also an article here at Nature Physics

• Excellent answer and link! I was happy to see that the actual PDF is available and not behind a paywall. A version of the actual Physical Review Letter article cited there can also be viewed via ArXiv – uhoh Apr 30 '16 at 12:24
• I've asked a related question. – uhoh Apr 30 '16 at 13:06

Is the speed of the Voyager probes accurately described by Newton's or Einstein's theories?

Fairly accurately but there are two important corrections that are necessary for long-term accuracy. One is General relativity and the other is higher order terms in the gravity fields of the planets that the Voyagers flew close to for assists. Far away they get really small, but for say a flyby of rapidly spinning and very oblate Jupiter, the trajectory will be very different if you ignore the oblateness and higher order terms and just treat it like a symmetric sphere or point.

As a vector, we can express the Newtonian acceleration of a spacecraft due to gravitation of other bodies as

$$\mathbf{a_{Newton}} = -GM \frac{\mathbf{r}}{|r|^3},$$

where $$r$$ is the vector from the body $$M$$ to the body who's acceleration is being calculated. Remember that in Newtonian mechanics the acceleration of each body depends only on the mass of the other body, even though the force depends on both masses, because the first mass cancels out by $$a=F/m$$.

The following approximation should be added to the Newtonian term:

$$\mathbf{a_{GR}} = GM \frac{1}{c^2 |r|^3}\left(4 GM \frac{\mathbf{r}}{|r|} - (\mathbf{v} \cdot \mathbf{v}) \mathbf{r} + 4 (\mathbf{r} \cdot \mathbf{v}) \mathbf{v} \right),$$

Oblateness ($$J_2$$ only):

The following should be added to the Newtonian term:

$$\mathbf{a_{J2}} = a_x \mathbf{\hat{x}} + a_y \mathbf{\hat{y}} + a_z \mathbf{\hat{z}}$$

where

$$a_x = J_2 \frac{x}{|r|^7} (6z^2 - 1.5(x^2+y^2))$$

$$a_y = J_2 \frac{y}{|r|^7} (6z^2 - 1.5(x^2+y^2))$$

$$a_z = J_2 \frac{z}{|r|^7} (3z^2 - 4.5(x^2+y^2))$$

Borrowed from this answer to How to calculate the planets and moons beyond Newtons's gravitational force?

• Higher-order terms in gravitational fields due to the bodies not being spherical aren't corrections to Newtonian gravitation or mechanics: they're corrections to the approximation that planets are homogenous spheres (so in particular the shell theorem does not apply). The GR corrections on the other hand are corrections to the theory itself. – tfb Apr 3 at 11:09