Lecture 3

• the interesting variations of gravity (those that tell us about the Earth's internal structure) are tiny compared with the background attraction of the whole Earth
• we construct a model Earth and calculate its gravity. The model predicts things we already know about the Earth that cause its gravity to vary from place to place. The difference between the observed and predicted gravity is called a gravity anomaly and is written as Δg. The gravity anomaly then just depends on the things we did not know, like its subsurface density structure
• gravity varies because of the Earth's

  • shape
  • rotation
  • internal density distribution
• 'shape' involves both: 'height above sea-level' - orography and the shape of the 'mean sea-level surface' - the geoid
• horizontal is bumpy
• background gravity of the bulk Earth + effect of local high density mass = the resultant net gravity is deflected
• 'vertical' is the direction of net gravity
• 'horizontal' is a surface at right angles to vertical
• on the 'horizontal' a marble does not roll away because there is no component of gravity parallel to the surface - no where else on it is 'down hill'. the surface of a fluid at rest will be horizontal
• the shape of the geoid (mean sea-level) is approximately by an ellipsoid
• GPS measures position with respect of the center of the Earth, or, equivalently height above the ellipsoid
• the height of the geoid above a best fitting ellipsoid (range less than 200 m) shows up lateral variations in sub-surface density - not correlated with surface geology
• the Earth has an ellipsoid shape - it has an equatorial bulge
• the equatorial radius: a = 6378 km is about 21 km larger than the polar radius b
• normal Earth model: total mass M, spin rate Ω, size and shape
• the Earth's flattening f is f = $f=$$\frac{a-b}{a}$ = 1/298.254
• the Coriolis force

  only acts when you move with respect to the Earth - deflects your motion to the right in the NH and to the left in the SH - central to understanding motion of the atmosphere and ocean currents but not important for gravimetry
• the centrifugal force

  pushes objects away from the spin axis
• fictitious 'centrifugal' vs real 'centripetal' force

  in the internal (non-rotating) reference, there is an acceleration/force towards the center of rotation - this is the centripetal acceleration/force

  however, it is often easier to formulate equations in a rotating (non-inertial) reference frame. Viewed from this reference frame, we have to introduce accelerations/forces - this is the centrifugal acceleration/force

  centrifugal accelerations/forces are equal in magnitude, but opposite in direction to centripetal accelerations/forces
• centrifugal acceleration (force on unit mass) = distance from spin axis x (spin-rate)^2
• spin rate Ω = 2π/T where T is period of rotation
• vertical component of centrifugal acceleration R(cosΦ)^2Ω^2 where R is the distance from spin axis
• the centrifugal force weakens gravity
• effect of rotation is largest on the equator and zero at the poles
• Why has the Earth got an equatorial bulge?

  the centrifugal force pushes material away from the spin axis. If the Earth were naturally spherical and has some strength to resist the centrifugal force, it would be less flattened than if it had no strength (like fluid). we can calculate what the flattening would be if the Earth had no strength and behave like a liquid -> Earth has very little long-term strength
• gravity weaker at the pole (less nearby mass) gravity stronger on the equator (more nearby mass)
• opposing effect - the inverse square law and the shape surface of the equatorial bulge

  at the poles, the surface of the Earth is close to the attracting center of the bulk Earth than on the equator -> surface gravity larger at the poles
• the gravitational attraction of the normal Earth model is called normal gravity, γ
• normal gravity includes effects making gravity stronger at the poles

  • rotation (m)
  • shape of the surface (f)

  as well as another effect making it weaker - extra mass in the equatorial bulge
• gravity anomalies at sea - that is on an Earth with no topography

  normal gravity takes into account the Earth's rotation and the roughly ellipsoidal shape of mean sea level but will not include the effect of the Earth's topography