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Home , Equatorial bulge, Spheroid

Review

• What is the international reference ? • What is the (polar) ? What’s the ’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 ?

1 Normal

• How to define “” on a ? – How do we define the “vertical” and the “horizontal” direction at a point on the ? – 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 + 1 sin + 2 sin 2)

GE 3 27 g = 1+f m + f 2 fm where e a2 2 14 5 ✓ 15 17 ◆ = m f + m2 fm 1 2 4 14 1 5 = 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. – that vanishes at the poles. – extra mass under the due to the . – Think about the sign and magnitude of each contribution. 5 : 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 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 m + S sin m) P (cos ✓) r r nm nm nm n=0 m=0 X ✓ ◆ X • Coefficients, Cnm and Snm are calculated based on 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)