Limits on the Strength of Europa’s Icy Shell from Topographic Spectra

Introduction: Radio echo sounding on Europa may provide estimates of the depth to the brittle-ductile transition. This information should be consistent with the topography spectrum. Here we derive an upper bound on the amplitude and shape of the topog-raphic spectrum that can be supported by the strength of the lithosphere. Topographic features that exceed this upper bound must be supported by local compensation. Lithospheric Strength: Byerlee [1] showed that the frictional resistance on a fault t depends on the fault normal stress sn and the coefficient of friction and is largely independent of material properties. Suppose that the brittle layer of ice has closed fractures oriented in all directions and a differential stress † Ds = s 1-s 3 () is applied. Frictional sliding will occur on optimally-oriented fractures when the resolved shear stress exceeds the Byerlee criteria. The magnitude of this differential stress is called the yield strength [2] and it is primarily a function of the over-burden pressure as follows: † Ds =-3rgz (horizontal compression) Ds = 0.7rgz (horizontal extension) For lack of additional information about the ductile layer, we'll assume it has a linear decrease in strength with depth and zero strength is achieved at twice the depth of the brittle ductile transition (Figure 1). One can now calculate the maximum bending moment that can be maintained by the lithosphere before it fails Ms. The saturation bending moment is the integral of the yield strength over depth times the distance to the brit-tle/ductile transition † M s = Ds(z)(z-z b)dz 0 2z b Ú = 0.6rgz b 3 where r is the density (1000 kg m-3) and g is the acceleration of gravity (1.3 m s-2). The saturation bending moment increases as the cube of the depth to the brit-tle/ductile transition. Topographic Moment: This estimate of saturation bending moment provides an upper bound on the amplitude of the topography that is supported by stress in the shell. The vertical load of positive topography must compensated by a nearby negative topographic load. This high to low dipole must be maintained by the strength of the lithosphere. Consider sinusoidal topography of wavelength l and amplitude w 0. The moment that must be applied at the origin to maintain the topography is given by the following formula. † M = gr w o sin 2px l Ê Ë Á ˆ ¯ ˜-l / 4 l / 4 Ú xdx = …