Re: [Phys-L] down versus center-of-earth; geodetic latitude versus geocentric latitude

[John Denker <jsd@av8n.com>]

On 01/03/2013 07:46 AM, Philip Keller wrote:
> Somewhat related question:  later in the course, I may want to teach
> that there is a small difference between the "downward" we defined
> with a plumb line and "downward" toward the center of the earth
> because of the Earth's rotation.  Is there a way to measure this
> experimentally?

Wow, what a nifty question.

First let me describe a seemingly plausible measurement scheme 
and explain why it does *not* answer the question:
 -- Use a sextant to measure the height of the celestial pole 
  above the horizon, and compare that to your latitude.
 -- Equivalently, perhaps more accurately, use a sextant to 
  measure what stars are overhead, and compare their declination 
  to your colatitude.

As it turns out, that scheme doesn't work, because by tradition
the "latitude" is *defined* to be the geodetic latitude, i.e.
defined to be what you see in the sextant!  It is defined in
terms of the local notion of horizontal and vertical, which 
includes the centrifugal acceleration.  All that makes sense
if you think about how latitude has been used for navigation
over the centuries.

To say the same thing the other way, the unadorned term "latitude" 
does *not* correspond to the geocentric latitude, i.e. the angle 
as seen from the center of the earth.

As a consequence of the definition, parallels of latitude are 
not equally spaced.  This is easy to visualize if you extrapolate 
to a planet with extreme flattening, so that the geoid looks 
like a pancake.

===

So, the "experimental" scheme that does work requires nothing
more than a map, or (preferably) accurate survey data.  Measure
the north-south distance between two points, and compare it to 
the difference in latitude.  Repeat the operation for a pair of 
points near the equator and another pair far from the equator.

You can simulate the experiment using a good geodesy software 
package.  Beware that there are some not-so-good packages out
there.  I haven't done exhaustive checks, but so far I've been 
happy with GeographicLib by Charles Karney.
  http://geographiclib.sourceforge.net/

I calculate that one degree of latitude is
	111.694 km  near the pole
	110.574 km  near the equator.