Uncertainty of gravitational constant
Bill Jefferys
> There are two gravitational constants:
> 1 - Gravitational constant measured in laboratories on the Earth;
> 2 - Gravitational constant calculated by methods of a celestial
>mechanics from the theories for Solar system.
> These gravitational constants should have identical values,
>but they differ from each other.
>
> Who can explain this paradox?
>
>[Moderator's note: do they really differ from each other? - jb]
Traditionally, the gravitational constant as used in celestial mechanics
has been a _defined constant_, not a measured quantity. The reason for
this is that masses and distances were poorly known in "real" units. For
example, until about 40 years ago, the metric equivalent of the length of
the astronomical unit (the unit of distance adopted by celestial
mechanics) was known only to about 4 significant digits; and the mass of
the Sun (the unit of mass used by celestial mechanics) was known to no
better precision. However, the quantities usually measured to determine
positions of planets were known to much higher precision...these were
angles, and typically were known with precisions of 1 second of arc or
better. To have adopted metric units and the metric value of the
gravitational constant would have been to have thrown away the intrinsic
precision of the measurements.
Hence, the units in celestial mechanics are: Mass of Sun=1, Unit of Time=1
day (originally 1 mean solar day, now 1 day of 86,400 seconds of atomic
time), k=Gaussian constant of gravity=0.01720209895. However, the Gaussian
constant of gravity has to be squared to have length, mass and time units
equivalent to the SI version of G. This is for convenience: it avoids
taking a square root when computing angular velocities. Remember, this was
all worked out by Gauss before calculators. Gauss had to take square roots
by dividing the log by 2 and looking up the antilog. This convention saved
some calculational labor. The derived unit of length is the Astronomical
Unit (AU).
Note that if you multiply k by 365.256, the number of days in a sidereal
year, you get 2*pi almost exactly. This reflects the fact that the
semimajor axis of the Earth's orbit is very nearly 1 AU. It isn't quite 1
AU because (1) the numbers Gauss used when calculating k were a little
off, and (2) the semimajor axis of the Earth's orbit isn't constant in any
case, because of perturbations by the other planets. It would be more
inconvenient to keep the semimajor axis of the Earth's orbit exactly equal
to 1 and have a variable unit of length than to keep k constant and allow
the semimajor axis of the Earth's orbit to vary; so we have kept Gauss'
definition.
So, do they really differ from each other, as John Baez asks? Clearly, the
interpretation is different. But k^2 and G play the same roles, and once
you straighten out the interpretations it's clear that they can't be
fundamentally different. Certainly one can for example calculate the
Earth's semimajor axis in meters (possible because the AU in meters is now
known to very high precision owing to radar ranging of the planets and
artificial space vehicles), plug in the sidereal period of the Earth in
seconds (also known to high precision) and get a very precise estimate of
the mass of the Sun (+ Earth-moon system), by using the ordinary SI
version of Kepler's third law on these metric quantities.
Bill
--
Bill Jefferys/Department of Astronomy/University of Texas/Austin, TX 78712
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Matthew J McIrvin
> There are two gravitational constants:
> 1 - Gravitational constant measured in laboratories on the Earth;
> 2 - Gravitational constant calculated by methods of a celestial
>mechanics from the theories for Solar system.
> These gravitational constants should have identical values,
>but they differ from each other.
There are not two gravitational constants.
The only way that we know the masses of the planets and Sun is by
assuming that G is the same for them as it is for the earth. The only
way that G is ever measured is by using known masses which applies only
to things with measurable density such as laboratory objects and
mountains. That is why G is so difficult to measure.
-- Ray Tomes -- http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm --
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Ray Tomes
Esa A E Peuha
> There are two gravitational constants:
> 1 - Gravitational constant measured in laboratories on the Earth;
> 2 - Gravitational constant calculated by methods of a celestial
> mechanics from the theories for Solar system.
> These gravitational constants should have identical values,
> but they differ from each other.
That depends on what you mean by "differing". The obvious difference is
that constant 1 has kilogram as its unit of mass, while constant 2 has the
mass of Sun. However, the *only* currently available way to get the mass
of Sun in kilograms is to equate these two constants, so they simply can't
have different values.
--
Esa Peuha
student of mathematics at the University of Helsinki
http://www.helsinki.fi/~peuha/
Robert Hill
> a_n_ti...@hotmail.com wrote:
>
> > There are two gravitational constants:
> > 1 - Gravitational constant measured in laboratories on the Earth;
> > 2 - Gravitational constant calculated by methods of a celestial
> >mechanics from the theories for Solar system.
> > These gravitational constants should have identical values,
> >but they differ from each other.
>
>
> The only way that we know the masses of the planets and Sun is by
> assuming that G is the same for them as it is for the earth. The only
> way that G is ever measured is by using known masses which applies only
> to things with measurable density such as laboratory objects and
> mountains. That is why G is so difficult to measure.
In other words, if M is the mass of a Solar System body (including
the Earth), we can measure GM (and can do so with great accuracy if
the body has a natural or artificial satellite[1]). If M, M' are the
masses of two such bodies, we can know M/M' with great accuracy.
But our absolute knowledge of M is only as good as our
laboratory-derived knowledge of G, which is quite poor.
---
[1] What we actually measure is G(M+m) where m is the mass of the
satellite, byt usually m << M, so this is essentially the same thing
as measuring GM.
--
Robert Hill
University Computing Service, Leeds University, England
"Though all my wares be trash, the heart is true."
- John Dowland, Fine Knacks for Ladies (1600)
john baez
Bill Jefferys <bi...@warthog.as.utexas.edu> wrote:
>Traditionally, the gravitational constant as used in celestial mechanics
>has been a _defined constant_, not a measured quantity. The reason for
>this is that masses and distances were poorly known in "real" units. For
>example, until about 40 years ago, the metric equivalent of the length of
>the astronomical unit (the unit of distance adopted by celestial
>mechanics) was known only to about 4 significant digits; and the mass of
>the Sun (the unit of mass used by celestial mechanics) was known to no
>better precision. However, the quantities usually measured to determine
>positions of planets were known to much higher precision...these were
>angles, and typically were known with precisions of 1 second of arc or
>better.
This is related to an interesting symmetry of the equations governing
gravity: if you have a solution and you multiply all distances, times,
and masses by the same constant, you get a new solution.
This is true both for Newtonian gravity and general relativity. The
reason is that the constants appearing in these theories, namely the
gravitational constant G and the speed of light c, have the following
units:
G = L^3/MT^2
c = L/T
They remain unchanged when we multiply L, M, and T by the same constant!
So if you think about it for a while, you'll see that Newtonian gravity
and general relativity have this sort of "scaling" symmetry. In
particular, they have no intrinsic length scale.
(Of course, *quantum* gravity has an intrinsic length scale, since
it involves a third constant, Planck's constant:
hbar = ML^2/T
which *does* change when you multiply L, M, and T by the same constant.)
Scaling symmetry changes distances and masses but not angles. Angles
are easy to measure in astronomy, but distances and masses are not.
This is why it's hard to tell the overall scale of astronomical systems.
But wait a minute: *times* are easy to measure! Why can't we use
*times* to measure the scale of an astronomical system?
Well, we *can* in contexts where c is relevant. E.g. we can see how
long it takes light to bounce off the moon and use that to determine
the distance to the moon quite accurately. But in old-fashioned
Newtonian astronomy, only G was relevant. The constant G is also
unchanged by another symmetry, where we leave time alone, but
multiply lengths by some constant and multiply masses by that constant
cubed. So if make all the planets heavier and suitably farther apart,
they trace out paths that look just the same in the sky, moving along
at the same rate through the "celestial sphere".
(Folks familiar with conformal geometry will recognize this "celestial
sphere" as an old friend, the "sphere at infinity". No surprise:
conformal geometry is all about transformations that preserves angles
but not necessarily distances.)
Of course, G is also unchanged by various combinations of the two
symmetries described already. For example, it's unchanged when we
leave masses alone, multiply lengths by some constant, and multiply
times by the that constant to the 3/2 power. This is the symmetry
built into Kepler's famous third law: " the square of the period is
equal to the cube of the major axis".