Determination of Ice Crust Thickness from Flanking Cracks along Ridges on Europa

Introduction: The response of a planetary lithosphere to an applied load, such as a mountain or range of mountains, is regularly modeled as a plate deflecting under a mathematically defined load. The deflection caused by the load depends on the strength of the plate, which is defined by elastic parameters of the lithospheric material and by its thickness [1, 2]. A “forebulge”, where the crust warps upward at the surface, is characteristic of the deflection profile. The forebulge location has been used to characterize the deflection profile and to calculate crustal thickness at the time of load emplacement for locations on Earth [3–5], Mars [6, 7], the Moon [6], and Europa [8, 9]. However, the exact location of a forebulge crest can be difficult to identify in planetary surface images. Stresses caused by the downward deflection of the lithosphere can produce more readily identifiable extensional features such as cracks and graben between the forebulge and the load. The locations of maximum stresses can be correlated with the location of extensional features [10]. This correlation allows lithospheric elastic thickness at the time of load emplacement to be calculated. On Europa, the exact thickness of the ice crust is a contentious issue. Various methods have been employed to determine thickness, with results ranging from 0.2–30 km [11]. Lithospheric Flexure Due to Line Loading: Some of the many ridges of Europa are expected to load the ice crust enough to cause deflection [8, 12, 13]. Prior to the Galileo mission, Pappalardo and Coon [12] used a line load model developed by Turcotte and Schubert [2] to hypothesize that cracks might be seen flanking ridges caused by the load of the ridges, and that the flanking cracks would be located ~15 to 35 km from the ridge if the lithospheric thickness is ~2 to 6 km. More recent studies used the same method to determine the elastic thickness of Europa’s ice crust at two locations in Conamara Chaos: (1) 100 m to 500 m at a mound load [9]; and (2) 123 m to 353 m at a ridge load (“Ridge R”) [8]. In both cases, these estimates of ice crust thickness were determined using the distance from the load to the forebulge, which is difficult to locate precisely in Europa images. Deflection and Stress Due to Line Loading: For a line load on a broken plate (such as is produced by a ridge), the deflection profile is w=w0e αcos(x/α) [2], where w0 is the maximum deflection at the load, x is the distance from the ridge, and the flexural parameter α=[Eh/(3ρg(1-ν)]. The flexural parameter comprises Young’s modulus E, the thickness of the plate h, density of the plate ρ, the gravitational acceleration on the satellite g, and Poisson’s ratio ν. Maximizing the deflection equation gives the distance to the forebulge as xb=3!α/4. The stress profile is obtained from the deflection using the relationships: (1) strain ε=-y(dx/dy), where y is the horizontal distance from the center of the plate (downward is positive); and (2) stress σ=ε E/(1-ν). The stress profile can then be maximized to find the distance from the load to the maximum tensile stress. For a broken lithosphere, this maximum tensile stress occurs at the surface at xσ=!α/4. Tensile features are most likely to form at xσ and can be used to determine the plate thickness h by rearranging the above equations to obtain h=(3(4xσ/π) ρg(1-ν)/E). Stress and Flexure on Europa: Deflection due to line loading and the induced stress at the surface are shown for three hypothetical crustal thicknesses in Figure 1. In these calculations, appropriate values have been used for Europa: E is 6x10 Pa, ν is 0.3; g is 1.35 m/s and ρ is 1186 kg/m [9].