12.201/ Essentials of Geophysics Geodesy and Earth Rotation Prof. Thomas Herring

10/13/ / Today’s Topics More complete analysis of Earth rotation variations and what they tell us about the Earth’s interior and exterior –Rigid body rotation review –Incorporation of fluid effects namely the fluid inner core and external fluids such as oceans and atmosphere Measurements of Earth rotation –Very long baseline interferometry: Measurement relative to an extra-galactic reference frame –Global Positioning System (GPS): Measurement relative to Earth orbiting spacecraft –Satellite Laser Ranging: Measurement relative to orbiting body but one with better dynamical behavior than GPS.

10/13/ / Rigid Body Rotation Fundamental equation of a rotating body: The rate of change of the angular momentum in a non-rotating frame equals the applied torque. Defining the angular momentum H with See: scienceworld.wolfram.com/physics/AngularMomentum.html

10/13/ / Rigid body rotation With the angular momentum defined, and torque defined as L=rxF where F is the force applied, we have You will see both forms used. Non-rotating form used for externally applied torques such as those from Sun and Moon: Yields motion of rotation axis in space (precession nutation) Rotating form for terrestrial problems such as atmospheric excitation of motion of rotation axis with respect to the crust (polar motion)

10/13/ / Moments of Inertia Moments of inertia are related to the Earth’s gravity through the Stokes coefficients Cnm and Snm ie., when gravity potential expanded as: Substituting P 20, P 21 and P 22 forms in the above equations yields:

10/13/ / Moments of Inertia Relationship between gravity field coefficients and moments of inertia: Notice that there are only 5-Stokes coefficients but 6 moments of inertia. (Also need R radius of the Earth and M its mass). To obtain all the moments of inertia: We use the precession constant but care is needed

10/13/ / Precession constant The rate of precession of the rotation due to constant part of the luni-solar torques (the lunar orbit is near the plane of the Earth’s orbit about the sun, and the Earth’s equator is inclined by 23.5 degrees to the orbit plane (  obliquity of the ecliptic).  is average rotation rate and M and R are mass and average distance to body

10/13/ / Cautions with Precession constant Equations are strictly solved for a biaxial ellipsoidal body Precession is the secular motion of the rotation axis in space. Orbit perturbations introduce periodic terms called nutation. Precession constant is measured over a finite duration of time (a mere 20-years for current estimate using very-long-baseline interferometry). Historically, optical telescope data used for about 1 century. There are long period nutation contributions from planetary orbits There is a general relativistic contribution to the secular motion called geodetic precession In modern use; all the nutation terms and precession are used to obtain (C-A)/C called the “dynamic ellipticity”

10/13/ / Rotation: Rigid body Using the equations of angular momentum and moments of inertia in a rotating frame (Euler’s equations) we obtain:

10/13/ / Chandler Wobble Solution to the previous equations when the torque is zero generates  3 =  constant and a resonance frequency of  1 and  2 of [(C-A)/A]  cw Notice that once the bi-axial assumption is made, the Euler equations decouple into rotation rate terms and rotation axis direction. Rotation rate variations are measured as changes in the length of day (LOD) relative to atomic time standards (post 1950’s). Prior to 1950, ephemeris time was used (time that made Newton’s equations of motion work). The direction of the rotation relative to the crust of the Earth is called polar motion.

10/13/ / Nature of temporal variations In the figures shown: –Short period variations (periods less than a year) are driven mainly by atmospheric angular momentum variations. –For polar motion: Signal is dominated by “beat” of 433-day period Chandler Wobble with annual signal (A beat generates signals with sum and difference of frequencies) –Secular drift of polar motion thought to be due to post-glacial rebound changing the position of the maximum moment of inertia. –Oceans have sizable contribution to polar motion. –Long period LOD changes thought to be due to exchange of angular momentum between fluid core and mantle. Couple mechanism still not clear. –For atmosphere and oceans infer angular momentum from measurements and assimilation models –For fluid core: Angular momentum comes from magnetic field variations.

10/13/ / Chandler wobble variations

10/13/ / Recent polar motion

10/13/ / Length of Day

10/13/ / Recent length of day

10/13/ / Including deformation and fluid core For a deformable body, the equations of motion are written as a complex equation for polar motion: The last equation is the “excitation” function and involves changes in moments of inertia (mass term) and angular momentum of components of the system (such as the atmosphere). Care is needed here since it is possible to compute the torque from the atmosphere (L) or include its angular momentum

10/13/ / Solution to equation If the excitation function is known (through atmospheric and ocean models), the previous equation can be integrated to yield: In the periodic form, the amplitude of the polar motion depends on 1/(  cw -  ) where  wc is the Chandler wobble frequency

10/13/ / Addition of fluid core When the fluid core is added to the equations, two sets of coupled equations are obtained Where n is rotation vector of fluid core relative to mantle fixed axes, f c is dynamic ellipticity of the core and subscript m means mantle. In this derivation, the fluid core acts like a rigid body and the momentum associated with the fluid flow needed to rotate inside the elliptical core-mantle boundary is second order (and neglected). The flattening of the fluid core is a proxy for the coupling between the mantle and the core. In newer versions f c is replaced with terms that include magnetic coupling

10/13/ / Fluid core effects The solution to the coupled equations still has a Chandler wobble resonance but now the frequency is  cw =(A/A m )(1+f c )  r where  r is the rigid body Chandler wobble frequency A new resonance appears in the system and this term has a period very close to one-day. Its frequency is approximately:  fcn = -  [1+(A/A m )f c ] This term is called the “free core nutation” and is very well observed with VLBI. The equivalent to polar motion in nutations are nutations in obliquity and longitudexsin(obliquity). Next figure shows the differences between measured VLBI nutation angles and predictions from hydrostatic earth model. Three different analysis centers are shown.

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10/13/ / Recent papers of topic Modern theories of the nutation include also the effects of the solid inner core with both mechanical and electromagnetic coupling. Ocean and atmospheres are not coupled in current solutions (rather there angular momentum is computed and applied). Mathews, P. M., T. A. Herring, and B. A. Buffett, Modeling of Nutation- Precession: New nutation series for nonrigid Earth, and insights into the Earth's interior, in press J. Geophs. Res., ETG 3-26, 2001JB000390, Buffett, B. A., P. M. Mathews, and Herring, T. A., Modeling of Nutation- Precession: Effects of the Magnetic Coupling, J. Geophs. Res., ETG 5- 14, 2001JB000056, Herring, T. A., P. M. Mathews, and B. A. Buffett, Modeling of Nutation- Precession: Very long baseline interferometry results, J. Geophs. Res., ETG 4-12, 2001JB000165, 2002.

10/13/ / Rotation variation angles Changes in rotation rate are expressed as length-of-day (LOD) changes. The integrated effected is called Universal Time 1 (UT1). The atomic clock derived time is called UT coordinated (UTC). VLBI measures UT1-UTC and LOD is obtained by differentiation. Polar motion: Position of rotation axis with respect to the crust. “Long period” (> 1 day) variations in this frame Nutation: Position of rotation axis in space. “Long period” in inertial space. Separation between polar motion and nutation not-unique. Long period variations in space are nearly diurnal in a crust fixed frame (due to rotation of earth). Only three angles needed to rotate between crust fixed frame and inertial frame, separation is based on frequency content; nearly diurnal variations are nutations

10/13/ / Measurement of Earth Rotation Main techniques: –Very long baseline interferometry (VLBI): Measures difference in arrival times of radio signal (2 and 8 GHz usually) from extragalatic radio sources. Time difference measured to about 10 mm accuracy. Requires large (usually fixed) radio telescopes (10-30 m diameters). Limited number of systems around the world Reference system is stable, inertial frame radio sources (quasars normally). Extremely good for UT1 and nutation angles. –Global Positioning System (GPS): Measures propagation delay and phase of signals from ~28 orbiting high-altitude satellites (20,000 km). Measurement accuracy a few millimeters for phase. System inexpensive (5- 10K$). Hundreds of systems around the world Reference system is satellite orbits and so orbit perturbations need to be modeled (and estimated). High altitude helps but satellites are active and this complicates orbit determination. Very good a polar motion (many stations) and LOD. Even nutation angle rates of change are useful but limited.

10/13/ / Measurements of Earth rotation Satellite laser ranging –Measure two-way travel time from ground based telescope to orbiting satellite with corner cube reflectors (including the Moon). Range accuracy of order 10 mm for good systems but only one satellite at a time Reference frame is satellite orbits (such as LAGEOS). Orbital dynamics is better than GPS but only a few good geodynamics satellites. Polar motion not as good at GPS (few stations), LOD probably not as good either. International Earth Rotation Service (IERS) –Combines earth rotation measurements from many different groups and produces official earth rotation values – is their web page with links to components.