No, latitude doesn't follow gravity (as @mkennedy notes, it follows the
normal to the ellipsoid).
And, no, gravity doesn't follow your hyperbolic curve (nor a straight
The simplest model for gravity of the earth which accounts for its
ellipsoidal shape and its rotation is "normal gravity". (And the
formulas for normal gravity are conveniently expressed in terms of
ellipsoidal coordinates.) Unfortunately, the Wikipedia articles on this
gravity and normal
are deficient in that the height variation is treated only
approximately. (I haven't yet had the energy to fix this!) However, I
have written up some detailed notes on normal gravity
Here's the figure from those notes showing the field lines (green) and
level surfaces (blue) for an exaggerated model of the earth:
The red curve is the surface of the ellipsoid. Normal gravity
is only uniquely defined outside the ellipsoid because the gravity inside the
ellipsoid depends on the mass distribution (which is not specified in
the derivation of normal gravity). In this figure, normal gravity has
been extended inside the ellipsoid assuming that the mass is all
concentrated on a disc on the equatorial plane.
By the way, falling bodies don't follow field lines. Because this is
a rotating system, Coriolis forces come into play. In addition the
bodies interia will cause the body to deviate from a curved field line.
The field lines follow hyperbolas if the ellipsoid is not rotating.
Two possible mass distributions which then result
in a constant gravitational potential on the reference ellipsoid (i.e.,
which satisfy the conditions for normal gravity) are:
All the mass is sandwiched uniformly between the ellipsoid and a
slightly smaller similar ellipsoid. In this case the potential is
constant inside the ellipsoid. Such an ellipsoidal shell is called a
A massive circular disk of radius E, where E2 =
a2 - b2, with mass distribution proportional
to 1/sqrt(E2 - R2), for radius R < E.
This is the limiting case of the homoeoid.
If a < b (the ellipsoid is prolate), the disk is replaced by a
massive rod with uniform mass distribution.
Details are given in my
A uniform mass distribution is a possible solution to the problem of
normal gravity. This is the so-called
In this case the flattening is given by the rotation (instead of being
independently specified). In this case, the level surfaces inside the
ellipsoid are concentric similar ellipsoids and the field lines all
terminate at the center of the ellipsoid. (The field outside the
ellipsoid is normal gravity, of course.) Here are the level surfaces
(blue) and field lines (green) inside the ellipsoid for f = 1/5: