After visiting John Denker's web site, I decided to do a comparison of the
earth's surface gravity as predicted by the simple, rotating, uniform, rigid
sphere model I have described (hereafter termed the "sphere model") and the
Somigliana model. John had already compared the Somigliana model's
predictions with a large number of recent well-accepted gravity measurements
and found excellent agreement between Somigliana and the data. Admittedly,
this was easier for me than grappling with the actual gravity data, but the
comparison I did should be close to what would be achieved using the actual
data given the very close agreement between the data and the Somigliana
equation (as is evident from John's curves and data points). I found the
same magnitude of agreement between the sphere model and the Somigliana
model that I had found earlier between the sphere model and the few
measurements I got out of Young and Freedman (2012). Namely, +0.056% near
the equator and -0.12% near the poles. The agreement was perfect around
34.5 deg. I did note that John's "Barogenic + Centrifugal" model (which I
take to be the same model as the sphere model) didn't compare as well for
some reason I don't understand. For example, the Barogenic + Centrifugal
model differed from Somigliana by -0.17% at the equator and +0.32% at the
poles. So, not only are its differences with Somigliana considerably larger
than the sphere model but they are of opposite sign.
To be specific, by the sphere model, I mean the surface gravity is given by:
g(theta) = g_p*sqrt(1 - A*(2 - A)*cos^2(theta))
Where:
theta = latitude
g_p =gravity at pole = G*M/R^2
G = gravitational constant = 6.67428 x 10^-11 N*m^2/kg^2
M =mass of earth = 5.972 x 10^24 kg
R = mean earth radius = 6.371 x 10^6 m
A = R*w^2/g_p
w = earth's spin rate = 7.2921150 x 10^-5 rad/s
The above parameter choices were simply the best values for these parameters
I could find on the internet. No effort was made to arrive at them by
fitting the model to actual measurements.
For the Somigliana model, I used the equation and parameter values in David
Bowman's post of 8/8/15 at 5:14 pm:
g_e = gravity at equator = 9.7803268 m/s^2
k = 0.00193185
e = 0.081819191
John, could you explain to me why your Barogenic + Centrifugal model differs
from Somigliana more than the above sphere model (with the differences of
different sign). Aren't they the same model, or did you use different
parameter choices)?
Don
Dr. Donald G. Polvani
Adjunct Faculty, Physics, Retired
Anne Arundel Community College
Arnold, MD 21012