１ Interaction Mechanism and Response of Tidal Effect on the Shallow Geology of Europa Yifei Wang 1 Beijing University of Aeronautics and Astronautics Department of Materials Science and Engineering Undergraduate ABSTRACT Europa has been confirmed to have a multilayered structure and complex geological condition in last two decades whose detail and cause of formation remains unclear. In this work, we start from analyzing the mechanism of tidal effect on satellite’s surface and discuss if interaction like tidal locking and orbital resonance play an important role in tidal effect including heating, underground convection and eruption. During discussion we propose the main factors affecting Europa’s tidal heat and the formation mechanism of typical shallow geological features, and provide theoretical support for further exploration. Keywords: Europa, tidal heating, tidal locking, orbital resonance 1. INTRODUCTION According to existing observations by Galileo radio-tracking/gravity data and Galileo SSI images, Europa has a complex structure and distribution of different components including solid ice layer, warm liquid ocean, local convection and eruption between solid and liquid region, and hard brittle lithosphere. And for Jupiter’s three moon system--IO, Europa and Ganymede, these are in the rotation and revolution of synchronous pattern which called tidal locking. Besides, it is confirmed that IO, Jupiter and Ganymede are in the situation of orbital resonance, whose orbital period ratio is about 1:2:4 and the ratio remains constant with time. Their effect on the surface is periodic during revolution. From Ogilvie and Lin’s work2, a response to tidal force is separated in two parts: an equilibrium tide and a dynamic tide which is irrelevant with time and the angular position to Jupiter and other satellites because of periodic interaction to the subsurface structure. However, there is no recognized and unified explanation for the geological structure, composition and formation for Europa so far. And For shallow landscape especially the large area of glacial rift valley, a convincing explanation is missing. In this paper, we propose that tidal effects which mainly includes deformation in surface and inner structures and tidal heating, are mainly caused by the process where Europa tends to be tidal locked, and the gravitational torque is the motivation, and orbital resonance, or Laplace among 3 satellites, minimizes tidal thermal effects. We also propose a reasonable forming process which may provide theoretical support and predictions for further detection. 2. DYNAMIC ANALYSIS AND MODELING 2.1. Tidal Deformation on Europa’s Surface To simplify the analysis, we consider system For the angle between Jupiter’s equatorial plane which only includes a central celestial body and a and Europa’s orbital plane is 0.470°.And it’s easy to single satellite. Since the size of Jupiter is much prove that the vector diameter 풓 of Europa satisfies larger than Europa and the deformation of tidal effect 퐴풓̈ + 퐵풓̇ + 퐶풓 = ퟎ on Jupiter is much larger than that on Europa, the A, B and C are constants which are irrelevant with satellite was considered as SNREI body which is the position vector, 풓̈ 풓̇ and 풓 are in the same spherically symmetric, nonrotational and elastic certain plane, accordingly, below we will treat it as isotropic, as shown in figure 1. the approximation of coplanar. 1 Contact Information: [email protected] Address: 37 Xueyuan Road, Haidian District, Beijing, China 2 See in REFERENCES [3] ２ while 푀푒푞 is the equivalent mass acted on by Europa, and 푟푒푞1 and 푟푒푞2 are the equivalent 풗 푆,푟표 distance acted on the front and rear surfaces respectively. These equivalent quantities have the Jupiter same dimension with corresponding quantities, and lurevolu in positive correlation with them, but can’t be precisely solved due to practical geometrical shape 흎푆,푟표 and situation. For two bodies with constant mass and volume, 푀푒푞 is a constant value. And for Europa, 푟 is approximately equal to the distance from Fig. 1 푒푞 Europa's surface to the center of Jupiter. Schematic diagram of tidal phenomena Investigate the surface strain due to tidal force. Suppose Europa’s mass is 푚 and the central Suppose the magnitude of unilateral deformation is celestial body’s mass is 푀, Europa’s radius 푅, and ∆푟 when it reaches stability, and the equivalent the gravitational constant is 퐺. The radius from any tensile area of Jupiter is 퐴푒푞, where 퐴푒푞 is the sum point on Europa to the center of mass of Jupiter is 풓 effect of external tension and vertical internal with unit vector 풆풓. Take the center of Europa as the pressure on both sides. This rationality is due to the reference frame, the inertial forces at all points 풓 fact that the vertical internal pressure effect is reverse from the vector diameter, which is linearly proportional to the tensile effect on both 퐺푚푑푀 sides. According to dimensional relation we have 푑푭ퟎ = − 2 풆풓 2 푟 퐴푒푞 ∝ 푅 . Suppose the young's modulus at any Therefore, microbody 푑푀 is taken from the nearest given position on the surface of Europa to be 퐸푆 , and farthest points of Europa from the center of there is Jupiter, and the force received by the two places, 푅、퐸 、퐴 = 푐표푛푠푡. namely the tide-generating force, was respectively 푆 푒푞 1 1 So we can get the stress-strain relationship which is 푑푭 = 퐺푚푑푀( − )풆 ퟏ (푟 − 푅)2 푟2 풓 ∆푟 ∆퐹 = 퐸 퐴 ∝ ∆푟 1 1 푆 푅 푒푞 푑푭ퟐ = 퐺푚푑푀( − )풆풓 (푟 + 푅)2 푟2 It can be seen that the ∆퐹 - ∆푟 presents a linear Therefore, different distances from Europa relationship, and the size of ∆퐹 directly determines cause the different gravitational attraction to the degree of Europa’s tidal effect. Substitute this microbody. And the parts facing and departing formula into the above formula to define the tidal Europa have opposite directions of tide-inducing strain ε whose size represents the deformation force, which makes the surface be stretched. effect, so we get Qualitatively, the integral result must be 푭1 − 푭2 > ∆푟 4푅퐺푚푀푒푞 1 1 ε = = 3 ∝ 3 0, which causes deformation and reset of the surface 푅 퐸푆퐴푒푞 푟 푟 shape and elevation. So ∆푟-푟3 is inversely proportional. In conclusion, If Europa is only attracted by Jupiter's gravity, the 1. larger mass and radius 2. the smaller distance then due to the fact that 푟 > 400푅, we have 푟2 ≫ 3. the larger young's modulus of the tidal body: the 푅2. The sum of two tide-generating forces more significant the deformation effect tidal 퐺푚푀푒푞 퐺푚푀푒푞 4퐺푚푀푒푞 interaction can cause. Due to the complexity of ∆퐹 = − ≈ 푟 2 푟 2 푟 푅4 Europa's geological composition and structure 푒푞1 푒푞2 (푟2 − 2푅2 + ) 푅 푟2 Different areas of equivalent young's modulus and 4퐺푚푀푒푞푅 density will be the first motivation of different = 푟3 ３ deformation conditions, causing a lot of collisions interaction between the surface of the earth (such as and a series of physical changes in shallow geology. squeezing, tribological heating, etc.). After a limited period of time, there are 푑훼 − 푑휃 = 0(푡 = 푡 ) and 2.2. The Evolution of Tidal Locking 푙 푑훼 푑휃 The tidal effect on Jupiter works in exactly the = = 0(푡 ≥ 푡 ) , which means tidal locking 푑푡 푑푡 푙 same way on the Earth caused by the difference of state is approached and 휔 = 휔 (푡 ≥ 푡 ) . Moon’s gravity. Tidal effect will make the system 푆,푟표 푆,푟푒 푙 Since Europa’s radius is much smaller than Jupiter’s, gradually close to the tidal locking state. When a it can be considered that the projection of the force satellite has a tidal effect, its surface will deform to a exerted by mass microbodies at A and B on the certain extent, and the celestial body can no longer connection direction of the center of the two bodies be regarded as a sphere. From this analysis below we is equal, and the difference mainly lies in the will see the reason why the tidal phenomenon leads projection size perpendicular to the connection the system to approach tidal locking, and this process direction. Due to geometrical relation, the small is the second motivation of geological interaction angle between 퐴퐵̅̅̅̅ and the line satisfy and generation of tidal heating. 퐴퐵 We first define the system’s initial state as: 푑휃퐴퐵 = ∆휔푑푡⁄푟 Europa is captured by Jupiter and has just formed a 2 So the vertical components of force at A and B are stable orbit. When tidal interaction occurs, Europa’s body is approximately assumed to be an ellipsoid, 푑퐹퐴푦 = 푑퐹퐴 ∙ sin 푑휃퐴퐵 = 푑퐹 ∙ 퐴퐵∆휔푑푡⁄2푟 and the direction of its long axis is the line between the center of Jupiter and Europa. Due to the tidal 푑퐹퐵푦 = 푑퐹퐵 ∙ sin 푑휃퐴퐵 = 푑퐹 ∙ 퐴퐵∆휔푑푡⁄2푟 phenomenon, the surface of the earth is deformed. They have equal value and opposite direction, which is Correspondingly, the surface which is perpendicular equivalent to a gravitational torque. The differential form to both sides of the long axis will fall or concave. is For most celestial bodies, the direction of 2 rotation is the same as the direction of revolution, 푑푴푮(휃 = 0) = −푑퐹 ∙ 퐴퐵 ∆휔 ∙ 푑푡⁄2푟 ∙ 풆흎 which is only discussed later. When Europa was while 풆 is a unit vector in the same direction as initially captured, the angular velocity was satisfied 흎 the net angular velocity, and only for a pair of special 휔 ≠ 휔 . 푆,푟푒 푆,푟표 points (A, B). B Investigate the state of mass point pairs around ∆훼 satellite’s surface. The specified point pair (A, B)’s A angular position (휑퐴, 휑퐵) = 0 and the angle O increases with respect to counterclockwise.