Lecture 2.1: Gravity Field of the Earth
Gravitational Potential vs Gravity Potential
Gravitational potential V is the potential generated only by the Earth’s masses (attraction of all particles of matter) while Gravity Potential W is the total potential we actually observe on Earth’s surface, which is the sum of gravitational potential and centrifugal potential.
If Earth did not rotate, then gravity potential = gravitational potential. Because Earth rotates, we must add centrifugal potential to get the true vertical direction (plumb line) and the real shape of the geoid.
How is centrifugal potential analytic?
Centrifugal potential is analytic since it is just a polynomial in coordinates which is infinitely differentiable, no singularities, smooth everywhere in space. In mathematics, a function is analytic if it can be expressed as a convergent power series in a neighborhood of every point. Since Φ is analytic, all “complicated” or irregular behavior in Earth’s total potential W = V + Φ comes from V (the mass distribution)
Level Surfaces
A level surface is an equipotential surface: a surface on which the total gravity potential W has the same constant value everywhere.
They are perpendicular to the gravity vector at every point. They are “horizontal” in the physical sense
Plumb Lines
A plumb line is the curve that is everywhere tangent to the gravity vector
Plumb lines always intersect level surfaces at right angles, since gravity is perpendicular to equipotentials.
Contrast: analytical vs. non-analytical surfaces
Analytical surface: smooth, infinitely differentiable, can be represented locally by a power series (e.g., ellipsoid, sphere, paraboloid).
Non-analytical surface: has singularities, discontinuities, or irregularities (e.g., geoid is not strictly analytical because it reflects Earth’s irregular mass distribution — mountains, ocean trenches).
The centrifugal potential (and its level surfaces) are analytic → easy to model, predictable, no irregularities.
The gravitational potential depends on mass distribution → its level surfaces (the real geoid) are not perfectly analytic (they have small undulations).
That’s why in geodesy we often approximate Earth with an ellipsoid of revolution (an analytic surface) instead of the geoid for calculations.
Height (Orthometric Heights)
Orthometric height is the real “height above sea level,” measured along the true vertical (plumb line). As you go upward by dH, the potential decreases by gdH. Gravity is thus the negative gradient of the potential, which shows why potential is central to defining the geoid and heights.
Requires knowledge of the gravity field because it follows curved vertical lines
Ellipsoidal height
Geometric height above the reference ellipsoid (a smooth mathematical surface approximating Earth)
Measured along the ellipsoid normal
Does not consider the Earth’s gravity field — purely geometric
Curvature of Level Surfaces
tells how “bumpy” equipotential (sea level) surfaces are in different directions
Curvature of Plumb Lines
tells how much the true vertical bends relative to a straight line
Equation Relating the Two
Vertical part: arises from the rate of change of gravity magnitude along the plumb line (height direction).
Horizontal part: comes from the curvature of the plumb line itself, which causes the gravity vector to turn sideways; its size is gκg and it points along the principal normal.
Bruns Equation
formula in geodesy that relates the height anomaly (the separation between the geoid and the reference ellipsoid) to the disturbing potential (the difference between the Earth's actual gravity potential and the normal gravity field) and the normal gravity at a given point