Re: Accuracy / precision / resolution / etc.

[John Denker <jsd@MONMOUTH.COM>]

At 02:15 PM 9/1/99 -0700, Leigh Palmer wrote:
> >  a) the mass of the sun is known to about one percent.
      ^^^^^^^^^^^^^ bogus statement ^^^^^^^^^^^^
> >  b) Newton's gravitational constant is known to a little better than a
> > tenth of a percent.
> >  c) The product of these two quantities is known to about one part in ten
> > to the eighth.  There's no way you could have known that from the numbers
> > themselves.  You have to understand the physics.
>
> I guess I don't understand the physics here. It seems to me that,
> by my physics, the mass of the Sun and the gravitational constant
> must be known to the *same* relative uncertainty.

A better explanation might be that Leigh *does* understand the physics
here, and was paying attention.  It is easy to show (even without using
physics, just mathematics) that I wasn't paying attention on this one.
Specifically:

1) Rather than talk about relative errors in G or M, let's talk about
absolute errors in log(G) and log(M) -- which comes to the same thing.

2) In general, to an appropriate approximation, the uncertainty in "log(G)
versus log(M)" space can be represented by an ellipse.  If we had
independent measurements of G and M, the ellipse would be oriented parallel
to the axes, and would be fully characterized by its width in the G and M
directions.  But clearly in this case the ellipse is quite tilted:


 log(M)|
       |
       |    \
       |     \
       |      \
       |
       |________________
                        log(G)


The alleged uncertainty on G (.1%) implies a minimum length to the ellipse;
 assuming maximum eccentricity the ellipse must be oriented within a
fraction of a milliradian to ensure that the projection on the GM direction
does not exceed the alleged uncertainty.  Given this length and this
orientation, Leigh's point is proved, to an excellent approximation.

==============

Of course, knowing a little physics doesn't hurt either.  It's a good bet
that the product GM is known precisely because of the way it appears in
Kepler's 1-2-3 law:  (1 over 4 pi^2) GM Period^2 = Radius^3.  Then a
laboratory measurement of G lets us infer M.

==============

The point I was *trying* to make remains: It's possible for the
error-ellipse to be quite tilted.

An interesting example of this is that if you measure the mass of the earth
in units of solar masses, the uncertainty is much less than if you measure
it in kilograms.

My apologies for posting a bogus number for the uncertainty in the solar
mass.  Better numbers can be found at
  http://ssd.jpl.nasa.gov/astro_constants.html
  http://aibn91.astro.uni-bonn.de/~jbraun/w3_apd.html

Cheers --- jsd