On 11/3/14, 1:50 PM, Tom Van Baak wrote:
I have a question about that. If I understand correctly, recent IAU
resolutions have decoupled the definition of the SI second from the
terrestrial geoid, which is too fuzzy to be used for a definition. Instead
the geoid potential is held fixed by (or defined by) a constant. Potential
with respect to what exactly? "At infinity" is all very well, but there
are local gravity sources (solar, even galactic) that would seem to
complicate any operational realization of this definition.
Sorry if this is a bit off-topic. I'd like a simple, clear explanation for
the layman that drills down on exactly how the current definitional scheme
can be realized to arbitrary precision. For example, assume that we must
go off-earth at some point to get a better timescale. How fuzzy is the
solar potential ("soloid")?
Cheers,
Peter
Hi Peter,
Based on mass and radius, a clock here on Earth ticks about 6.969e-10 slower
than it would at infinity. The correction drops roughly as 1/R below sea level
and 1/R² above sea level. For practical and historical reasons we define the SI
second at sea level.
The non-local gravity perturbations you speak of are 2nd or 3rd order and so you probably
don't need to worry about them. Then again, if you want to get picky, it's easy to
compute how much the earth recoils when you stand up vs. sit down. So it's best to avoid
the notion of "arbitrary" precision; that's for mathematicians. For normal
people, including scientists, we know that precision and accuracy have practical limits.
The most obvious gravitational perturbation is that of the Moon. You can
predict, and even measure, that g changes in the 7th decimal place as the moon
orbits the earth. This is so minor it cannot as yet be measured by the best
atomic clocks, but it has been measured by the best pendulum clocks (because
pendulum clock make better gravimeters than atomic clocks). For details, see:
http://leapsecond.com/hsn2006/
Sun and Moon are of about the same gravity magnitude, and, of course,
you get approx one cycle/day for both.
Wikipedia says about 2E-6 m/sec^2 (e.g. 7th digit, as Tom said)
Wikipedia also provides some math models for variation with latitude, etc.
Interestingly, they say that the variation among different cities
amounts to about 0.5% (Anchorage high, Kandy low)
for height..
g(h) = g(0) * (Re/(Re+h))^2
Change of 0.08% for 0 to 9000 meters
Since the period of a pendulum goes as Sqrt(1/g), the sun/moon effect is
about 1E-7.. Set up a 10 meter long pendulum, which will have a period a
bit longer than 6 seconds. Set it swinging, and time it for 200
swings (about 20 minutes) (I think it will run that long if you've got a
nice heavy bob, etc.) Accurately(!) time that 1200 second interval with
100 microsecond precision and you might *just* be able to see the effect.
I started down this measurement path in the 70s in high school, but
encountered several logistics problems.
-> big pendulums are subject to environmental effects. You might do
better with a shorter pendulum in a vacuum, which would eliminate air
drag and reduce temperature effects.
-> this kind of timing implies that you've got a counter stable to 1E-8
over the measurement period (notionally 12 hrs)
And at this precision, there's all kinds of other effects one should
take into account (for instance, the period is only approximately =
2*pi*sqrt(L/g).. that depends on the sin(theta)=theta small angle
approximation.
However, i've always wanted to set up a rig where there's one of those
big Foucault pendulums and see if you can do it. I suspect the drive
system on the big ones would perturb the system, but maybe you could do
an off hours experiment and let it just swing down to zero.
->
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