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Normal Gravity Formula (Derivative of 2 2 the Geopotential): Gn = Ge(1 + 1 Sin + 2 Sin 2)

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Normal Gravity Formula (Derivative of 2 2 the Geopotential): Gn = Ge(1 + �1 Sin � + �2 Sin 2�) Review • What is the international reference ellipsoid? • What is the (polar) flattening? What’s the Earth’s flattening and its implication for the internal structure? • What’s the McCullagh’s formula? – What’s the meaning of the two terms? – Why do we usually ignore the third or higher terms? • What is the geopotential? 1 Normal Gravity • How to define “latitude” on a spheroid? – How do we define the “vertical” and the “horizontal” direction at a point on the reference ellipsoid? – Gravity, by definition, perpendicular to an equipotential surface. – “Vertical” is not equal to “radial.” – Geocentric latitude vs. geographic latitude. 2 Normal Gravity 3 Normal Gravity • As a part of an Earth model like WGS84, the theoretical value of gravity on the rotating ellipsoid is calculated according to the normal gravity formula (derivative of 2 2 the geopotential): gn = ge(1 + β1 sin λ + β2 sin 2λ) GE 3 27 g = 1+f m + f 2 fm where e a2 − 2 − 14 5 ✓ 15 17 ◆ β = m f + m2 fm 1 2 − 4 − 14 1 5 β = f 2 fm 2 8 − 8 -2 -3 and ge= 9.780318 m s , β1=5.3024x10 , and −6 β2=−5.87x10 . • Note that f & m ≈ 1/300. So, the quadratic terms, f2, m2 and fm, are all about 300 times smaller than f and m. 4 Normal Gravity • Dropping all the second order terms, we get, for λ=90°, g g 5 p − e = m f ge 2 − – Called Clairaut’s theorem. -2 -2 – gp = 9.832177 m s (cf. ge = 9.780318 m s ) – corresponding to increase in gravity by 5.186x10-2 m s-2. • What causes this increase in gravity? – distance to the center of mass of the Earth. – centrifugal force that vanishes at the poles. – extra mass under the equator due to the equatorial bulge. – Think about the sign and magnitude of each contribution. 5 Geoid: The“Real” shape of the Earth • Reference ellipsoid and the normal gravity formula are idealizations for convenience. • There can be numerous propositions about what is the “real shape” of the Earth and how to represent it. • Geoid: the equipotential surface* that coincides with the mean sea level over the ocean. *doesn’t have to be an ellipsoid! 6 Geoid: “Real” shape of the Earth • Geoid is represented by the spherical harmonics. n= n m=n E 1 R U = G (C cos mφ + S sin mφ) P (cos ✓) − r r nm nm nm n=0 m=0 X ✓ ◆ X • Coefficients, Cnm and Snm are calculated based on satellite geodesy and published. Anyone can then construct U at desired locations (r, ϑ, ϕ). 7 Geoid: “Real” shape of the Earth • Geoid undulation or geoid anomaly height – defined as (geoid radius – ellipsoid radius). – can be related to anomaly in gravitational potential: • gn ΔN = ΔU = (U – Ug), where ΔN (unit: m) is the geoid undulation, gn is the normal gravity, and Ug the geopotential. 8 https://en.wikipedia.org/wiki/EGM96 Geoid: “Real” shape of the Earth 5000 km < ¸ < 15,000 km 9 (King, JGR, 2002) Geoid: “Real” shape of the Earth ¸ < 5,000 km 10 (King, JGR, 2002) .
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