Actual astronomic latitude (direction of gravity) does not match calculations taking into account centrifugal force of earth
ABBREVIATED QUESTION/PROBLEM:
The direction of earth's gravity vector does not point towards the center of the earth due to centrifugal force. Theoretically, it should point away by .1 degree. But it doesn't.
The difference between geodetic and geocentric latitude gives us the actual difference between the direction of gravity (geodetic) and the center of the earth (geocentric). Converting between geodetic to geocentric reveals a much greater difference than .1 degree.
Due to centrifugal force, earth's gravity should point away from the center of the earth by approximately a tenth of a degree.
The difference between the direction of gravity and the center of the earth is given by vertical deviation (astronomic latitude).
Astronomic latitude is given by $DEFLEC99$ for all points on earth. $DEFLEC99$ gives astronomic latitude as a difference in latitude from geodetic latitude. 371
However, the difference is mostly negligible ($1$ to $45$ arcseconds).
This means that geodetic and astronomic latitude are practically equal.
We can therefore calculate the actual difference between astronomic latitude (direction of gravity) and geocentric latitude (center of the earth) by subtracting geodetic latitude from geocentric latitude.
As an example, let's take the following:
Geodetic / Astronomic latitude - $49.26083$
Geocentric latitude - $49.07043926$
Difference in latitudes - $0.1903937385$
Now let's see what the difference should be according to the strength of centrifugal force:
Centrifugal force = $(2\pi/86164.091)^2 * (\cos<PHONE_NUMBER>]) * 6365904$ (radius at this latitude) $= 0.02217660948$
Gravity = $GM/R^2 - 3(J2)(GMa^2/R^4)(3/2)(\cos^2[49.<PHONE_NUMBER>) = 9.847454494$
To find the theoretical astronomical latitude, then, we would have to determine the $x$ and $y$ components of the vector sum of centrifugal force and gravity, which is the following:
x component = $\cos<PHONE_NUMBER>] * 9.847454494$
y component = $\sin<PHONE_NUMBER>] * <PHONE_NUMBER> - 0.02217660948$ (centrifugal force)
new angle = $\tan^{-1} (y component / x component) = 49.16806807$
difference between the latitude above and the geocentric latitude
$= <PHONE_NUMBER> - <PHONE_NUMBER> = 0.09762881339$
Compare this figure to the first figures:
Geodetic / Astronomic latitude - $49.260833$
Geocentric latitude - $49.07043926$
Difference in latitudes - $0.1903937385$
Calculated Astronomic latitude - $49.2187618253$
Geocentric latitude - $49.07043926$
Difference in latitudes - $0.09762881339$
Actual difference and calculated difference are off by a good $.1$ degree.
Keep in mind that this is not an isolated case particular to this latitude. The inequality persists at latitudes ranging from $20$ to $50$ degrees geodetic.
Is there a component of gravity I'm not solving for?
Explanation of different latitudes:
Geocentric: Imagine a line drawn from a point on the earth's surface to the center of the earth. The angle between this line and the line parallel to the equator is geocentric latitude.
Geodetic: Imagine a point on the earth's ellipsoid surface. The angle between the line perpendicular to the surface at this point and a line parallel to the equator is geodetic latitude.
Astronomic latitude: The angle between the direction of gravity (plumb line) and a line parallel to the equator. This angle is usually described as a deviation from geodetic latitude (a plus or minus value ranging from $1$ to $44$ arcseconds).
Because the difference between geodetic and astronomic latitude is so small, I tend to refer to them as equal for practical purposes.
SOURCES CITED:
https://physics.stackexchange. 371