Lecture_03 oblateness Earth

GG612 Lecture 3 2/1/11 1 Clint Conrad 3-1 University of Hawaii LECTURE 3: EARTH’S FIGURE, GRAVITY, AND GEOID Earth’s shape, tides, sea level, internal structure, and internal dynamics, are all controlled by gravitational forces. To understand gravitation and how it affects Earth, we start with Newton’s laws: Gravitational Potential For a point mass: Newton’s law of gravitation: ! r F =mr a ="GmMr2 Then the acceleration due to gravity is: ! g="GMr2r The work by a force F on an object moving a distance dr in the direction of the force is: dW = Fdr The change in potential energy is: dEp = -dW = -Fdr The gravitational potential UG is the potential energy per unit mass in a gravitational field. Thus: ! mdUG="Fdr="mgdr Then the gravitational acceleration is: ! r g ="r # U="$$x,$$y,$$z% & ' ( ) * U The gravitational potential is given by: ! UG="GMr For a distribution of mass: If a mass is distributed within a body of volume V, then we can integrate to find the total potential: ! UG="G#r r ()rV$dV For the special case of a spherical shell of thickness t, applying this integral yields: ! UG="GMr as if the sphere were concentrated at the center. Thus, everywhere outside a sphere of mass M: ! UG="GMr GG612 Lecture 3 2/1/11 2 Clint Conrad 3-2 University of Hawaii ω ac θ r x Centrifugal Potential For a rotating body such as Earth, a portion of gravitational self-attraction drives a centripetal acceleration toward the center of the Earth. When viewed in the frame of the rotating body, the body experiences a centrifugal acceleration away from the Earth’s axis of rotation. Angular velocity: ! "=d#dt=vx where ! x=rsin" Centrifugal acceleration: ! ac="2x=v2x But ! r a c="r # Uc, so we can calculate the centrifugal potential by integrating: ! Uc="12#2x2="12#2r2sin2$ Figure of the Earth Earth’s actual surface is an equipotential surface (sea level), a surface for which ! UG+Uc=constant. The figure of the Earth a smooth surface that approximates this shape and upon which more complicated topography can be represented. The earth approximates an oblate spheroid, which means it is elliptically-shaped with a longer equatorial radius than a polar radius. The flattening (or oblateness) is the ratio of the difference in radii to the equatorial radius: For earth, f=0.00335287, or 1/298.252, and the difference in the polar and equatorial radii is about 21 km. ! f=a"ba GG612 Lecture 3 2/1/11 3 Clint Conrad 3-3 University of Hawaii z x y P O θ The International Reference Ellipsoid is an ellipsoid with dimensions: Equatorial Radius: a = 6378.136 km Polar Radius c = 6356.751 km Radius of Equivalent Sphere: R = 6371.000 km Flattening f = 1/298.252 Acceleration Ratio ! m=aCaG="2a3GME=1/288.901 Moment of Inertia Ratio ! H=C"AC=1/305.457 Hydrostatic equilibrium predicts that the flattening should be: f=1/299.7 This is smaller than the observed flattening by about 113 m [see Chambat et al., Flattening of the Earth: further from hydrostaticity than previously estimated, Geophys. J. Int., 183, 727-732, 2010]. The gravitational potential of an ellipsoid is given by: ! UG="GMEr"GC"A()r33cos2#"1()2="GMEr"GC"A()r3P2cos#() where A and C are the moments of inertia about the equatorial and polar axes. More generally: ! UG="GMEr1"Rr# $ % & ' ( n=2)*2JnPncos+()# $ % % & ' ( ( Where Pn are the Legendre polynomials and the coefficients Jn are measured for Earth. The most important is the dynamical form factor: ! J2=C"AMER2=1082.6#10"6 The next term, J3, describes pear-shaped variations: a ~17 m bulge at North pole and ~7 m bulges at mid-southern latitudes (~1000 times smaller than J2) GG612 Lecture 3 2/1/11 4 Clint Conrad 3-4 University of Hawaii λ λg g a c r The gravitational potential of the Earth (the geopotential) is given by: ! Ug=UG"12#2r2sin2$="GMr+Gr3(C"A)3cos2$"12% & ' ( ) * "12#2r2sin2$ The geopotential is a constant (U0) everywhere on the reference ellipsoid. Then: At the equator: ! U0="GMa+G2a3(C"A)"12#2a2 At the pole: ! U0="GMc+Gc3(C"A) Then: ! f=a"cc=(C"A)MEa2a2c2+2ca# $ % & ' ( +12a2c)2GME*32J2+12m Where we have approximated a~c on the right hand side. Gravity on the Reference Ellipsoid To first order: ! r=a1"fsin2#() Geocentric latitude = λ (measured from center of mass) Geographic latitude = λg (in common use) To first order: ! sin2"#sin2"g$fsin22"g The acceleration of gravity on the reference ellipsoid is given by: ! r g ="r # Ug Performing this differentiation gives: ! g=GMr2"3GMEa2J2r23sin2#"12"$2rcos2# Rewriting and simplifying gives: ! g=ge1+2m"32J2# $ % & ' ( sin2)* + , - . / Writing in terms of λg gives: ! g=ge1+52m"f"1714mf# $ % & ' ( sin2)g+f28"58mf# $ % & ' ( sin22)g* + , , - . / / ! g=9.7803271+0.0053024sin2"g+0.0000059sin22"g[] Equatorial gravity is: ! ge=GMa21"32J2"m# $ % & ' ( =9.780327 m/s2 This allows us to compute the polar gravity: ! gp=9.832186 m/s2 GG612 Lecture 3 2/1/11 5 Clint Conrad 3-5 University of Hawaii The poleward increase in gravity is 5186 mgal, and thus only about 0.5% of the absolute value (gravity is typically measured in units of mgal = 10-5 m/s2). Gravity decreases toward to pole because the pole: (1) is closer to the center of Earth than the equator (6600 mgal) (2) does not experience centrifugal acceleration (3375 mgal) But the equator has more mass (because of the bulge), which increases the equatorial gravity. Together these three affects yield the 5186 mgal difference. Earth’s Geoid The geoid is the equipotential surface that defines sea level, and is expressed relative to the reference ellipsoid. Temporal variations in the geoid are caused by lateral variations in the internal densities of the Earth, and by the distribution of masses (primarily hydrological) upon the surface of the Earth. Mass excess (either subsurface excess density or positive topography) deflects the geoid upwards. GG612 Lecture 3 2/1/11 6 Clint Conrad 3-6 University of Hawaii GG612 Lecture 3 2/1/11 7 Clint Conrad 3-7 University of Hawaii Spherical Harmonics The geoid (and any function on a sphere) can be expressed in terms of spherical harmonics of degree n and order m: ! Ynm=anmcosm"+bnmsinm"()Pnmcos#() Top view Side view Top view Side view The power spectrum of the geoid is given by: ! Pn=anm2+bnm2()m=0n" The dominance of the low-harmonic degrees in the geoid power spectrum indicate that the dominant shape of the geoid is controlled by structures deep within the mantle.