sign in sign up
<<
Home , Libration, Tidal locking

１

Interaction Mechanism and Response of Tidal Effect on the Shallow Geology of 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 and 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 locking, orbital resonance

1. INTRODUCTION According to existing observations by Galileo radio-tracking/ 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 ’s three system--, Europa and , these are in the 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 is separated in two parts: an equilibrium 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 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 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 (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 . When tidal interaction occurs, Europa’s body is approximately assumed to be an , 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 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. It can be ∆휃 A B found that all the point pairs’ positions where the gravitational torque increases and prevents the

Fig. 2 Schematic diagram of tidal locking rotation are (휑푖, 휑푗) ∈ 휋 According to figure 2, investigate the two states (0, ) while 푴푮 increases and reverses 풆흎 2 of the single satellite model. Europa reaches position 휋 ( , 휋) while 푴푮 decreases and aligns with 풆흎 2 after rotating from position 1 for a small period of 2 휋 time ∆푡 . In the process from position 1 to 2, the For (0, ), combined with the above equation, we can 2 micro precession angle of Europa is 푑휃, and 푑훼 is get the angle of rotation. It’s obvious that 2 푑훼 − 푑휃 = (휔 − 휔 )푑푡 = ∆휔푑푡 퐴퐵 ∆휔푑푡 푆,푟푒 푆,푟표 푑푴푮(휃, 푡) = − 푑퐹 cos 휃 풆흎 Tidal energy Q can be generated due to the 2푟 ４

Therefore, in a certain angular range ， the was captured by Jupiter. Since multi-body problem gravitational torque will reduce |∆흎| and occurs an can’t be analyzed and the interaction between Jupiter irreversible process of tidal heat generation, and and its satellites is much larger than that between finally tends to be stable where|∆흎| = 0, proves the satellites, the interaction between the three satellites inevitability of tidal locking under certain conditions. was ignored in process 1, and mechanical energy is Besides, we can see that tidal effect is a periodic considered to be composed only of the gravitational interaction and is related with fluid’s oscillation in potential energy and its own kinetic energy brought chapter 3, where tidal heating’s diffusion and decay by Jupiter. play a decisive role. For Europa, the blocking of gravitational torque on rotational kinetic energy is much smaller than that 2.3. Orbital Resonance’s Impact on Tidal Locking of gravitational force on translational energy, so the Since Europa is not only tidally locked by variation of rotational energy is ignored in this Jupiter but also has a Laplace resonance with IO and section. Ganymede, which also happen to be tidally locked For any satellite’s process in beginning to form by Jupiter, due to this coincidence, we suspect that in orbit, being captured by Jupiter's gravity and orbital resonance plays a role in the tidal locking achieving stable , apparently these gravitations process of the celestial body. The cause and from other satellites can be neglected, and Jupiter’s deformation of tidal phenomena can be analyzed and mechanical energy in satellites’ gravitational field explained from the perspective of force, but it would can be considered as constant, therefore, satellite in be extremely difficult to analyze and solve the force process 1 can be considered to satisfy the problem of multi-body resonance. Therefore, we conservation of the sum of mechanical energy and discuss the role of orbital resonance by analyzing the thermal energy. process of orbital evolution and the variation and Might as well set 푟 up for infinitely long time transformation of quantities such as tidal heating. ∞ after the ideal stable orbits, and set the initial capture Any satellite shall go through the following position 푟 as the zero-potential point. In later three processes from the initial state to orbital 0 discussion, the selection of 푟 must satisfy certain resonance (as shown in Fig. 3): 0 conditions. So we have 1. Get captured from free motion and enter the orbit. 푟∞ 1 2. Make self-coordination to achieve tidal locking. 퐸 = [퐸 − ∫ 퐹(푟)푑푟] + 푚푣2 ∞ 푝0 2 ∞ 3. Achieve orbital resonance by long-term gravity by 푟0 2 퐺푀푚 푣∞ 퐺푀푚 other satellites which cannot be ignored. with 퐹(푟) = − 2 and 푚 2 = 2 . So we get 푟 푟∞ 푟∞

1 1 퐸∞ = ( − )퐺푀푚 푟0 2푟∞ Due to the conservation of satellite’s energy which is Center 1 1 assumed above, 퐸 = 퐸 = 퐸 + 푚푣2 = 푚푣2, celestial ∞ 0 푝0 2 0 2 0 body we can get the steady-state orbit radius after single satellite‘s capture which is 1 푟 = ∞ 2 푣2 Fig. 3 − 0 The Capture of Any Satellite 푟0 퐺푀 We can see that 푟 is only related to the mass of ∞ Jupiter, 푟 and initial capture velocity rather than The orbit radius and of a satellite 0 푣2 are determined by its initial kinetic energy when it mass of itself. And notice that 0 > 0, there must be 퐺푀 ５

푟0 the satellite), which are respectively the angular 1 < < 2 푟∞ momentum and kinetic energy of the system. As the It’s no doubt that the initial capture position satellite rotation slows down, 푟 increase and

푟0′푠 selection is important. Based on previous system's mechanical energy will decrease. In other discussion, it should be defined as the maximum words, during the satellite deceleration, tidal heat is boundary radius where gravitational effect from generated due to the friction between surface and Jupiter cannot be ignored. internal matter, which is This can explain IO, Europa and Ganymede’s 훿푄 = 퐸(푟) − 퐸(푟 + 푑푟) relative angular positions are very close to 0: 휋: 0 Boundary conditions for equations above are

(as shown in Fig. 4). The angular velocity of steady- 휔푆,푟표(푟푙) = 휔푆,푟푒(푟푙) state satellite is mainly related to its orbital radius, 푄푙 = 퐸(푟푙) − 퐸(푟0) followed by the position of the surrounding satellite. While subscript l refers to the locking state in

The difference in angular velocity between corresponding quantities. 푟0 indicates the initial perihelion and aphelion is very small, and the spacing where satellite is just captured by the Jupiter. perturbation of the three satellites can be ignored. Then we investigate the orbital evolution Therefore, its angular velocity of revolution only process of multiple satellites. Since the interaction depends on the orbital radius. among satellites is much smaller than that between satellite and Jupiter, it can be considered that the joint action of IO, Europa and Ganymede on Jupiter is approximately three act separately on the linear superposition, which avoid the impossible analysis about multi-body interaction. Same as Jupiter above, we have 3

퐿푆 = ∑[퐽푆푖,푟표휔푆푖,푟표(푟) + 퐽푆푖,푟푒휔푆푖,푟푒(푟)] 푖=1

퐿푆 + 퐿푃 = 퐿0 3 1 퐸 = ∑[퐽 휔 2(푟) + 퐽 휔 2(푟)] Fig. 4 Orbital Resonance Positions of IO, 푘푆 2 푆푖,푟표 푆푖,푟표 푆푖,푟푒 푆푖,푟푒 푖=1 Europa and Ganymede 3 퐺푀푚푖 When the moon has not entered the tidal locking 퐸푘푃 + 퐸푘푆 − ∑ = 퐸(푟) 푟푖 state, it will slow down and rotate under the action of 푖=1 the moment of resistance according to 2.2. The 1 while 퐿 = 퐽 휔 (푟) 퐸 = 퐽 휔 2(푟). evolution of the single-moon and Jupiter system 푃 푃,푟표 푃,푟표 푘푃 2 푃,푟표 푃,푟표 follows Notice that the periodic interaction of orbital resonance is much longer than the locking process. 퐿 = ∑ 퐿 0 푖 The satellite has been tidally locked until reaching

퐺푀푚 Laplace resonance, for 푖 = 1,2,3 and 푟 = 푟푙, 퐸(푟) = ∑ 퐸푘푖 − 푟 휔푆푖,푟표(푟) = 휔푆푖,푟푒 (푟푙) while ∑ 퐿푖 = 퐽푃,푟표휔푃,푟표(푟) + 퐽푆,푟표휔푆,푟표(푟) + When it reaches Laplace resonance, we have 1 휔 : 휔 : 휔 = 퐴 : 퐴 : 퐴 = 4: 2: 1 퐽 휔 (푟) and ∑ 퐸 = [퐽 휔2 (푟) + 푆1,푟푒 푆2,푟푒 푆3,푟푒 1 2 3 푆,푟푒 푆,푟푒 푘푖 2 푃,푟표 푃,푟표 The above is the conservation equations followed by 2 2 퐽푆,푟표휔푆,푟표(푟) + 퐽푆,푟푒휔푆,푟푒 (푟)] (푟 is orbit radius of the state variables in evolution process, and the ６ initial value condition is just the orbital parameters ipsilateral collinear, and qualitatively, the state of the in discussion of process 1. ipsilateral collinear will result in the interaction Let the initial orientation of satellites be between satellite is the strongest, before and after

휃0푖 ( 푖 = 1,2,3) , according to existing observations, reach the state of the ipsilateral collinear, its changes their angular positions are respectively such as torque, the relative potential energy, also is

휃1 = 휃01 + 휔1푡 = 2휋푡⁄푇1) the strongest, the overall is not conducive to long- 휃2 = 휃02 + 휔2푡 = 휋(1 + 푡⁄푇1) term stability. 휃3 = 휃03 + 휔3푡 = 휋푡⁄2푇1 When any satellite has yet to reach locking state, Assuming three satellites can reach the same during the process, go through the rest of other two side of Jupiter and collinear, there must be 푘1, 푘2 ∈ satellites and multiple disturbance of Jupiter, mainly 푍 make 휃1 − 휃2 = 2휋푘1 , 휃2 − 휃3 = 2휋푘2 . Get reflected in the torque’s value and direction and rapid rid of 푡 and we can get changes in potential energy. Each with a satellite

2푘1 − 4푘2 + 3 = 0 formation tidal locking, its gravitational torque effect no integer solutions for 푘1, 푘2, contradiction, so the becomes 0, at the same time reduces the frequency assumption is wrong. of the disturbance of the rest of the satellite. In a long 1:2:4 for IO, Europa and Ganymede system period of a large number of periodic interactions, the therefore, is unlikely to be the location of the formation of such a stable resonance state. 3. SHALLOW GEOLIGICAL RESPONSE TO TIDAL ACTION Based on Galileo and its SSI observations and earlier voyager thermal simulations, Europa has four major geological features: 1. Highly differentiated geological crust, including metal/metal sulfide core, rocky mantle, outer (near surface) water layer or 80-170km deep ocean, surface ice. 2. Ductile deformation exists in the inner part, the lithosphere transversal migration rate is in the tens of kilometers, and local melting began in modern times. 3. Geologically active: surface craters retain an age of 10-100 million years. The surface craters are estimated to have remained for 30 million years (10 kilometers in diameter) and are found to be consistent with the 10 kilometers of ocean ice. 4. Has vast glacial rift valleys on surface which stretches for hundreds of kilometers and local heat flows may also exist. ice crust, liquid ocean in the middle, surface 3.1. Geological Processes from Tidal Effect fragile lithosphere beneath. The discovery of a large amount of liquid water under Europa and other explorations are still in an surface imperfect stage and lack of explanation of the creation of the shallow geological structure. Currently, there are four main international conjectures (as shown in Fig. 5): Ⅰ. A thin, brittle, electrically conductive crust of Ⅰ Ⅱ Ⅲ Ⅳ ice covering a deep global ocean. Ⅱ. The crust layer of almost completely solid water, consisting of the thin, brittle ice lithosphere in the upper part and the warm, convective ice ퟐ asthenosphere in the lower part. ~ퟏퟎ 풌풎 Ⅲ. between Ⅰ and Ⅱ: A thin layer of ocean, global or scattered, beneath thick, convective ice and Fig. 5 Four Main Conjectures brittle lithosphere of Europa’s Shallow Geology

Ⅳ. between Ⅰ and Ⅱ: Outer layer thick convection According to analysis in chapter 2, the tidal ７ strain of the surface of the celestial body affected distribution of tidal heat quickly change to by tide is uniform distribution after the internal first reaches 4푅퐺푀푚푒푞 1 the melting point at a certain position, and thus 휀 = 3 퐸푆퐴푒푞 푟 forms a large area of liquid ocean. If the crystal in the geological layer is a Ⅱ. Occurs in the region with the highest fracture perfect crystal of the same material with high brittleness. The purity, crystallinity and grain state symmetry and no defects, the deformation of tidal of the ice layer are different in different positions. effect on geological layer is exactly the same at the Under the action of local stress caused by tidal same thickness from the center of the celestial forces, cracks occur in high-risk places and heat body. Therefore, for tidal thermal effect, the heat up rapidly with relative slip and friction. Unlike Ⅰ, generation power of deep geological layer is lower the local mechanical stress may be so large that it than that of shallow geological layer. But because precedes the thermal concentration, causing the

Europa’s mean radius 푅̅̅2̅ = 0.245Ea. = 1561km, fluid to erupt violently. The weight of for shallow geology 100km about from the surface, transgranular and intergranular fractures on the the difference of tidal strain between different surface of the earth is different due to the thickness is less than 10%. Therefore, depth is not difference of the size of the crystal, the orientation the main cause of tidal effects in shallow geology. of the crystal plane and the energy at the grain The observation results suggest that there are boundary. local violent processes such as eruption and What the main conjectures have in common explosion of frozen liquid or mud, as well as a are a hard, brittle surface, electrically conductive large area of glacial rift valley on the surface. We ice, and a softer, convective interior, such as a soft suggest two possible formation mechanism: ice sheet or a liquid ocean.Varying degrees of Ⅰ. Occurs in areas where geologic layer friction internal softening may be due to tidal heat.For or plate action is most intense. In the process of example, during the formation of liquid ocean, tidal locking, due to different angular positions tidal heat power tends to be tidally locked with and depths of the geological element by the Europa, and the total tidal heat production tends to magnitude and direction of gravity is different, an upper limit with time evolution, which that is, tidal action, combined with its own determines the proportion of liquid ocean in the geological heterogeneity, will lead to the local first total shallow volume.When it is stable, it forms a crack and slip layer, friction and collision, and state of overall orderly stratification and local rapid increase and expansion. Due to deep solid chaos and turbulence, where surface glaciers and matter is relatively shallow liquidity is poorer, tide shallow layers have liquid ocean or soft ice, where heat by convection and heat conduction, gradually dynamic equilibrium convection and rapid and accumulation will lead to local warming rapidly, massive renewal exist at the junction of the two. so that the solid ice crystal melting or amorphous 3.2 Geological Action After Reaching Resonance impurities into the viscous flow, increase liquidity, In locking and Laplace resonance state, heat conduction more quickly to the surrounding Jupiter has no gravitational torque on the three water ice, not melt local overheating will cause moons, but the weak gravitational torque between explosion. On the other hand, as the modulus E the three moons still exists. Due to the action of decreases due to the phase transition, the strain Jupiter is much stronger than that of other satellites, variable of tidal effect increases, further increasing it is very difficult to change the kinetic energy the relative displacement and friction heat caused by the gravitational coupling moment generation between liquids. This positive between other satellites. Most of the potential feedback effect makes the concentrated energy is converted into tidal heat instead of ８ rotational kinetic energy.Therefore, after reaching maintain the resonance state and change their orbit the orbital resonance state, the tidal thermal power radius synchronously, and the effects such as tidal will decrease rapidly, but will not stop. Due to the heat will gradually approach 0, forming a stable constant period ratio, the three satellites will geological structure and activity form. 4. SUMMARY AND CONCLUTIONS Above all, we work mainly lies in: Europa and Jupiter interaction system through reasonable assumptions and approximation, toy model analyses the dynamic mechanism of the , tidal locking of the forming process, and get the result that the gravitational torque is a tidal locking the direct cause of formation, modeling of Jupiter and its moons system are analyzed theoretically and analyzes the evolution process of Laplace resonance characteristics and contact, etc. Will affect the state of tidal locking orbital resonance, and tending to tidal locking this process, it is such as tidal fever, the causes of surface deformation and a series of tidal effect. The explanation of the surface glacier and the shallow layer activity is put forward. On the other hand, the shortcomings in our exploration process mainly include: the actual situation of the interaction of multiple celestial bodies is too complex to be accurately modeled and the analytic solution can be obtained. The author hopes to improve the calculation and other technical details at the same time, the corresponding conjecture can be more general and reasonable. In addition, the weak effects of orbit perturbation and of Europa are ignored in the analysis process, which need to be further improved. At the same time, the discussion of geological activities, such as internal periodic vibration and the role of molten fluid, still has a lot of room for innovation.

９

REFERENCES [1] Xing Wei. Surface Tide on a rapidly rotating body, Institute of Natural Sciences: Shanghai 200240, 2019 [2] Rosemary A. Mardling. Long-term tidal evolution of short-period with companions, Victoria: Australia 2007 [3] G. I. Ogilvie, D. N. C. Lin. Tidal dissipation in rotating giant planets, arXiv:astro-ph/03012118v1 8 Oct 2003 [4] J. S. Kargel, J. Z. Kaye, James W. Head, Giles M. Marion, Roger Sassen, James K. Crowley, O. P. Ballesteros, S. A. Grant, D. L. Hogenboom. Europa’s Crust and Ocean: Origin, Composition, and the Prospects for , Icarus 148, 226 –265 (2000)