## An interesting aside about gravity

In the past, when discussing the role of chaos in climate models, I’ve been known to argue that the complexity of multi-body dynamical systems means that we could probably not run a model of the formation of our Solar System that would actually produce a result entirely consistent with what we see today. That doesn’t mean, however, that we can’t use such models to understand how our Solar System formed and evolved. Similarly, that climate models are inherently chaotic does not mean that we cannot use them to understand how our climate might respond to changes in anthropogenic forcings. The response I would typically get is that gravity is verified/validated (or whatever other term the person chooses to use) but climate science is not (ignoring that much of the underlying physics is about as well understood as gravity).

Ignoring the complications of General Relativity, the gravitational force between two bodies of mass $M_1$ and $M_2$ a distance $R$ apart is

$F = \frac{G M_1 M_2}{R^2},$

where $G$ is the gravitational constant. If you look up the value of $G$ you’ll find that it is something like $G = 6.67384 \pm 0.0008 \times 10^{-11}$ m3 kg-1 s-2. However, John Mashey has pointed out in a comment that it may not be quite as well known as one might think (I’ve since found another article that discusses this). It turns out that the experiments that are trying to measure the gravitational constant disagree quite markedly. In fact, if you look at the figure below, they not even disagree with each other, some are actually quite discrepant with respect to the currently accepted value. Given that we know the value of some constants to many more significant figures than we do $G$, this is somewhat surprising.

So, we don’t know the value of $G$ nearly as precisely as we do other constants, and independent experiments don’t even agree particularly well. Is this some kind of controversy and does it really matter? Firstly, I’m not even sure how many people are actually aware of this issue. Also, the gravitational force is extremely small compared to the other forces, so it is maybe not surprising that it is extremely difficult to make a precise measurement. None of this, however, suggests that it isn’t a constant. Given that, not knowing the precise value is not really a particularly big deal.

One way we use gravity is to estimate the masses of astrophysical bodies (by observing the orbits of other bodies and then using Kepler’s laws). However, we don’t understand the internal structure of these bodies well enough to independently estimate their masses with more accuracy than we can get using gravity, even if $G$ isn’t precisely known. Therefore, a possible error of a fraction of a percent is not that significant. Also, most gravitational calculations/simulations are scale free; they assume $G = 1$. In this case, mass ratios might be important, but if all the masses have been estimated using a consistent value for $G$, then this will be fine. You can then convert from scale-free to real values by assuming a value for $G$. Therefore, even though we don’t have a precise value for $G$ doesn’t mean that we can’t trust simulations of the dynamical evolution of our Solar system, simulations studying the future paths of Earth-crossing asteroids, or even those that investigate the evolution of our universe. It doesn’t really matter.

So, since John pointed this out, I thought I might write a brief post. It’s certainly interesting in its own right, but also shows how people can recognise that even if we can’t measure something precisely, we can still do lots of interesting research as long as we understand the situation and as long as we’re consistent. Noone’s shouting that we can’t do any dynamical N-body calculations until we’ve estimated $G$ to 8 decimal places, because everyone recognises that it doesn’t really matter.

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### 37 Responses to An interesting aside about gravity

1. jsam says:

Think uncertainty is a bad thing? It’s actually a mark of sound science. – See more at: http://www.senseaboutscience.org/pages/uncertainty.html#sthash.sGoOKGQk.dpuf

2. jsam says:

Reblogged this on Gra Machree and commented:
Is the science settled?

3. jsam,
Indeed, that’s why I sometimes think confidence interval is better than uncertainty. Of course, what’s interesting here is that some of the recent experiments have confidence intervals that don’t overlap with the 2010 CODATA range. Of course, if you chose enough σs they would overlap, but that would just mean that it was highly unlikely that they were consistent. In this case, though, I suspect that this is because it is very difficult to remove all systematics from the experiments and so somewhere in the middle (as the figure seems to indicate) is probably a reasonable estimate.

4. Harry Twinotter says:

I seem to recall there is a hypothesis that G also breaks down on different scales ie you get a different value of the constant depending on how far apart the masses are. Somehow this can be used to test the idea that gravity ‘bleeds off’ to other dimensions, providing an explanation of why the gravity force appears so weak when compared to other forces.

5. anoilman says:

Were those measured at exactly the same location? Was is measuring it in XYZ?

6. BBP says:

I’m about twenty years out of date on this so take it with a grain of salt, but one area where it does matter, at least a little, is in understanding the evolution/structure of the sun. Uncertainty in the sun’s mass is proportional to uncertainty in G. This in turn effects the luminosity, so to get a model to match the sun’s luminosity at it’s current age the amount of helium (which is also uncertain) is tweaked. However, it didn’t stop solar physicists from being able to predict neutrino oscillations https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0CB4QFjAA&url=http%3A%2F%2Fen.wikipedia.org%2Fwiki%2FHomestake_experiment&ei=m9k2VPqPM5ejyATuhIKgDw&usg=AFQjCNGY7_IVjmSR-liilwl3H4W1M6qdmQ

7. anoilman says:

I have been enjoying this article on gravity’s effect on distribution of ocean level rise;
“The moving boundaries of sea level change: Understanding the origins of geographic variability. ”
http://www.tos.org/oceanography/archive/24-2_tamisiea.pdf

If a large ice sheet melts, it has little impact nearby since its gravitational tug has already drawn the water towards the ice. It will instead go further away from that ice sheet. This is a big reason for different ranges of sea level rise which is evident even with the most geologically stable locations.

In short, location matters.

8. AoM,
I think they weren’t at the same location.

BBP,
Interesting. Certainly the uncertainty in G does produce an uncertainty in the mass of the Sun, and I guess we know the age of the Sun quite precisely, so what you say makes sense. Although, I would have thought there would also have been uncertainty in its initial composition, which would also have influenced the evolution of the Sun’s luminosity.

9. Eli Rabett says:

Gravity is the weakest force (like about 10E-36 down on theelectromagnetic force), so why is this surprising? Any experiment is going to have to rely on the ability to measure a mass, which is not all that good if the mass has to be large, which it has to be to see any gravitational effect.

10. Eli,
I agree, I don’t think it is all that surprising. I must admit, though, that I didn’t know that experiments to measure $G$ were only consistent to about 2 decimal places. Of course, two decimal places is all that I ever use when working with $G$, so that may explain it 🙂

11. Harry,
I haven’t specifically heard what you mention. I know there are some studying modified Newtonian gravity in which the force behaves differently at large scales (galaxy clusters) compared to small scales (Solar system). This is an attempt to find an alternative to Dark Matter, which has never been observed. I don’t think it is particularly well regarded at the moment and most, I think, regard Dark Matter as being more plausible than modified Newtonian Gravity.

12. BBP says:

ATTP, there is uncertainty in the initial composition (I noted that parenthetically…) The initial He percentage is (or was 20 years ago) a parameter that gets adjusted until the model gets the right luminosity at the sun’s current age. There is also (or was) uncertainty in some of the nuclear reaction rates. The point I was making is it didn’t stop the models from getting the ‘right’ neutrino flux long before anyone knew about neutrino oscillation.

13. BBP.

14. jsam says:

I won’t believe any number until it is audited. Science demands.

15. Physics Today discusses these things extensively whenever the structure of the entire spectrum of SI units come up for review in CODATA. Another more extensive recalibration is already coming.

http://scitation.aip.org/content/aip/magazine/physicstoday/article/67/7/10.1063/PT.3.2448

16. C’mon, folks [1]. You don’t have to be interested in physics to ask for all the discussions between the NIST and the BIPM. Start FOIing them for all the emails that mentions the G word.

17. In the same issue they discuss the G problem extensively. Apologies if this was already nlinked.

http://scitation.aip.org/content/aip/magazine/physicstoday/article/67/7/10.1063/PT.3.2447

18. anoilman says:

Willard;
“C’mon, folks [1]. You don’t have to be interested in physics to ask for all the discussions between the NIST and the BIPM. Start FOIing them for all the emails that mentions the G word.”

The first person who mentions G spot will get called up before the senate. How about hiding the decline (in something or other…)

19. BBD says:

‘Gravity is the greatest scientific hoax of all time…’ Nope. It just sounds silly, doesn’t it?

20. Michael 2 says:

“Noone’s shouting that we can’t do any dynamical N-body calculations until we’ve estimated G to 8 decimal places, because everyone recognises that it doesn’t really matter. ”

It matters to the few people for whom it matters. For me, not so much since I am not trying to launch a probe to Mars hoping it will arrive at exactly the right place at exactly the right time. If I was, then it would matter.

If this nation was being asked to send 100 billion dollars a year to Africa based on the exact value of “G” then I’d be paying a lot more attention to it.

And yes, I know that several constants are reviewed and revised, numbers that exist pretty much just so that things “work out right”. Is anything absolute? Probably not. The speed of light is measured in meters and seconds; but what if either of those measures change? In the end it matters to those for whom it matters. The speed of light matters to me since my antenna calculations for my radio hobby depend on its value. The absolute value of a second doesn’t matter much so long as everyone I work with use the same value. Same with meter and kilogram. I’m not sure it matters so long as everyone uses the same value.

But “G” is somewhat special as are several other universal constants. It isn’t a thing that exists solely by agreement; it is a thing that exists because the universe exists and has decided what shall be those constants. Humans are limited to discovering those constants and then doing something with that knowledge.

21. izen says:

Nice to see good use made of uncertainty in science, planetary orbits are also a good example of how chaotic systems may display stable cyclic behaviour in which the chaotic variations occurs over much larger time-scales than the regular pattern.
Chaos does not mean inherently random, most often it means a quasi-cyclic behaviour with a power law distribution of frequency and magnitude.

May I repost this reply to Victor Venema from a month before John Mashey’s comment?
(GRIN)

izen says:
September 8, 2014 at 5:46 pm
@- Science has no formal ways for deciding, what should be considered as well established truth.

But there is a long established informal method. People who hold to ideas contradicted by known science are laughed at. Something like the Heliocentric solar system becomes an established truth(!) when there is no possible evidence that could refute the evidence already surpporting the theory.

@- When relativity or quantum mechanics showed classical mechanics wrong, bridges did not suddenly collapse. And when QM is shown wrong, our computers will keep on buzzing.

I would put my money on relativity being ‘wrong’. Note the problems with defining big G.
(Grin!)

22. It’s not a problem of defining G, it’s the limited precision in its measurement due to both its nature and the components of its definition. When we shuffle around CODATA terms nothing really changes, the various definitions are equivalent. What changes is just the structure, order and precision and accuracy of the measurements and definitions of the various CODATA terms and the various methods that the experimental portions of the terms are measured in practice.

This happens every decade or even on a yearly basis and is a standard part of science.

23. anoilman says:

BBD: ‘Gravity is the greatest scientific hoax of all time…’ Nope. It just sounds silly, doesn’t it?

Dang it! I wish you told me sooner! I just through the baby out the window cause it was safe!

You scientists should be locked up for this kind of garbage! 🙂

24. John Mashey says:

Note of course that reasoning about the different measurements requires some understanding of error bars. Whether “right”or not, some combinations are compatible, some are not, and a compellingly better measurement could rule some out, some not.

This of course relates to the reasoning or more often lack therof around the error band found in a certain temperature reconstruction. 🙂

25. Michael says:

All expensive space exploration should stop until this uncertainty about G is resolved.

In the meantime we could better spend the money helping the poor have accurate scales for weighing their rice.

26. …. always be careful with interpreting the confidence interval from a single study. http://blog.metasd.com/2012/07/capen-quiz/

It also highlights the value of examining the community-wide uncertainty. I am thinking specifically of the Bamber & Aspinall expert elicitation on future rates of ice loss. http://www.glaciology.net/Home/Miscellaneous-Debris/icesheetcontributionsfrombamberaspinall

Perhaps they should do something similar in the G community.

27. Perhaps they should do something similar in the G community.

I don’t think very many people are anticipating any great changes in G over the next century, You never know, though, something might come up. Best to be prepared.

28. The Very Reverend Jebediah Hypotenuse says:

Uncertainty in the fourth sig fig of G?
Pffft.

Before the mid-1990’s the Hubble constant (and therefore the age of the universe) was uncertain by a factor of 2!
(i.e H ≈ 50 – 100 (km/s)/Mpc).

Planck Mission estimate from 2013: 67.80±0.77 (km/s)/Mpc

Science for the win!

29. BBP says:

But we still have stars older than the universe 😉
Oldest Known Star

30. pendantry says:

Given that all our theorising and experimentation to date about gravity has been hampered by being at the bottom of a gravity well, I have often wondered what ‘reality’ is actually like between stars. The Voyager probes will ‘find out’ (by dint of exiting the heliosphere) whether gravitational conditions are different out there… but will this happenstance reveal anything useful to us? And if we were to, say, discover evidence suggesting that the distance of the probes from us was very different from that which we believe it to be, would we be able to accept that information for what it is — or would it be dismissed out of hand (‘instrumentation error’) as being ‘obviously wrong’?

31. pendantry,

or would it be dismissed out of hand (‘instrumentation error’) as being ‘obviously wrong’?

I doubt it would be dismissed out of hand. People took the Pioneer anomaly seriously for a while. Of course, people would be suspicious.

32. pendantry says:

I hadn’t heard of the Pioneer anomaly before. Thanks for expanding my mind! 🙂

Of particular interest to me is a) the 14 year gap before the anomaly was investigated, and b) that the end result was acceptance of a theory explaining the anomaly, one that could be interpreted cynically, by a layman such as myself (not that I would dream of doing such a thing!) as conveniently avoiding the need to revisit our view of reality. “The earth is flat after all, nothing to see here, move along, move along…” 🙂

33. pendantry,
I think one issue with the Pioneer anomaly being real was that they were inside the orbits of the planets. I think if the Pioneer anomaly was real, we would have expected to have seen something in the orbits of the outer planets. Voyager, on the other hand, is now well beyond the orbits of the planets and so if any anomaly was observed, there would be no immediate reason to dismiss it.

34. On astronomical scales, not only are small dynamical effects detectable, as in the Pioneer anomaly, but they are actually useful. Case in point :

http://www.sciencedirect.com/science/article/pii/S0094576512003876

35. Harry Twinotter says:

ATTP,

I found a link to the Scientific American article postulating gravity leakage to other dimensions. It’s speculative of course, but interesting.

http://web.stanford.edu/~savas/papers/Universe'sUnseenDimensions.pdf

36. Harry,
I wasn’t aware of that, thanks.