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Tidal Deformation and the Interior Structure of Europa
Hussmann H., F. Sohl, and J. Oberst
DLR Berlin Institut fЭr Planetenforschung

International Workshop, Moscow, Feb. 2009: Europa Lander: Science Goals and Experiments


Introduction

Europa's complex surface is the consequence of physical processes in the ice shell and in the deep interior. Tidal heating and tidal deformation is an important factor for these processes. Tidal deformation induces pressure changes and therefore may drive currents in the ocean.


Ellipsoidal Shape (static tides and rotation)
z c - north pole ­ north pole
height above mean radius

+

-
a - to Jupiter x ­ to Saturn b - along orbit

-y ­ along orbit

related to moment of inertia if hydrostaticity is assumed determined by J2 and C22 coefficients


Tidal Potential

Dynamical Tides

pericenter

+

apocenter

-

Time-variable radial & librational tide. Periodic deformation on the time-scale of the orbital period (3.55 days)


Solid Body Tide Love Numbers
Measures for tidally-induced global distortion of a solid planet or satellite hn ... radial displacement kn ... potential variation ln ... tangential displacement complex numbers (amplitude, phase) depending on the interior structure (density profile), tidally-effective rheology, and period of tidal forcing.

n ... degree of spherical harmonic representation of tidal potential


Detecting Oceans from Orbit
maximum amplitude Europa Ganymede Callisto ~ 20 ­ 30 m (~60 cm) ~ 3 ­ 4 m (~20 cm) ~ 2 ­ 3 m (~10 cm) h
2

k

2

~1.16 ­ 1.26 (~0.03) ~1.0 ­ 1.5 (~0.2) ~0.9 ­ 1.5 (~0.2)

~0.1 ­ 0.3 (~0.01) ~0.5 (~0.08) ~0.3 (~0.08)

in red: without ocean

reference: Moore and Schubert 2000, 2003

Determined by Doppler Tracking and Laser Altimetry. Additional constraints come from radar sounding and the magnetic response of the ocean. This allows for the determination of the ice shell thickness.

cold brittle ice

thin ice layer

completely frozen

ocean


Europa's Response to Tidal Forcing
h
2

Elastic response of the ice shell. Radial displacement h2, and tidal potential k2.
k

2

The signal will clearly show if there is an ocean present in Europa. There is a dependence of h2 on ice thickness and on rheological properties of the ice shell. Additional constraints from radar and the induced magnetic field will additionally constrain the ice thickness.


Tidal Distortion Measured at the Surface
Combinations of body tide Love numbers hn, kn, ln are measured. Astronomical latitude changes (star-field observations with a lander camera) latitude factor = 1 + kn ­ ln Gravity changes due to the displacement of mass (gravimeter) gravimeter factor = 1 + 2/n hn ­ (n + 1)/n kn Tilt changes (tiltmeter) diminishing factor = 1 + kn - h

n

Linear-, area-, volume strain changes (strainmeter) strain factor = 2 hn ­ n (n + 1) ln

n ... degree of spherical harmonic representation of tidal potential


Measurements at the Surface
Gravimetric factor Latitudinal factor

Additional equations to determine the tidal response, independent from the determination from orbit, are provided by measuring linear combinations of h,k and l directly at the surface. The rigidity or other rheological properties and the ice thickness can be better constrained.


Measurements at the Surface
Deminishing factor Strain factor

Additional equations to determine the tidal response, independent from the determination from orbit, are provided by measuring linear combinations of h,k and l directly at the surface. The rigidity or other rheological properties and the ice thickness can be better constrained.


Measurements at the Surface
Lander Camera changes of latitude by monitoring star-fields. Gravimeter/Long-period Seismometer changes of gravitational acceleration using test masses suspended on metal or quartz springs. Tiltmeter changes of surface tilting using vertical and horizontal pendulum. Strainmeter/Horizontal Extensiometer changes of relative distances (N-S, E-W-components), length standards provided by invar wires, quartz rods and laser beams. These are standard methods to measure solid-body tides on Earth.


Summary
Measuring the tidal distortion at Europa's surface ... linear combinations of the Love numbers h, k, and l can be obtained. ... is complementary to the determination of h2 and k2 from orbit. ... can further constrain the ice thickness and/or the rheological properties of the ice shell. ... have the advantage of a fixed location ( => no integration times) during the tidal cycle. Recommended instruments: Gravimeter (long-period seismometer); camera observations of star-field; tiltmeter, strain-meter.


Summary
Advantages: Single station sufficient, tidal excitation-/source functions well known, time series analysis standard methods. At Europa: large tidal amplitudes, no surface loads (atmosphere, surface oceans) Disadvantages: Instrument drift (3.55-day tidal cycle), temperature variations, coupling to surface, tectonics; harsh radiation environment (short life-time); at least one tidal cycle (3.55 days) should be covered The possible improvement of constrainig the thickness and other properties of Europa's ice shell has to be assessed in further studies.