土衛八伊阿珀托斯赤道脊之彈性折彎模型模擬

Abstract

土衛八伊阿珀托斯可能是太陽系中最具特異性之一顆衛星。在平均半徑約為735公里的星球上，有一列平均高度約10公里的山脊精準地落在其赤道上，這高聳的山脊甚至讓伊阿珀托斯的外型看起來像核桃般特異。此列山脊大概佔據了伊阿珀托斯超過75%的赤道總長，吾人即據此事實名之「赤道脊」。赤道脊在2005被卡西尼號探測器發現，但它的形成原理卻因缺少進一步的觀測證據，至今還在爭辯中；目前已經有數種不同的形成假說，並可以粗略地被分為內生性成因(板塊運動)或是外生性成因(環遺骸堆積物)。 另外，伊阿珀托斯的外型呈現扁橢球形，並可對應至一個自轉週期為16小時的液體球平衡外型。但伊阿珀托斯目前已受土星之潮汐鎖定，導致其自轉週期為79天。由於此衛星具有古老且隕石坑眾多的表面，潮汐鎖定事件和赤道脊形成的事件應也在衛星形成的早期(> 4000 Ma)即完成。如此一來，在赤道脊形成之時，伊阿珀托斯之表面應具有相當高之熱流值，使得地表易受到負重或板塊作用力的影響而彎曲。因此，只要利用此點，就可算出在赤道脊形成時，伊阿珀托斯表面之材料特性，再利用這些限制求出較可能之赤道脊起源。 本研究主要目的，即是利用彈性岩石圈理論，運用解析法及數值法，建構伊阿珀托斯赤道脊之折彎模型(Flexure Model)。赤道脊在此方法中被視為在伊阿珀托斯的硬外殼(彈性板)上完美之線狀負載，伊阿珀托斯的數值高程模型(DTM)不僅被作為建構赤道脊負載的量化依據，同時也顯露出赤道脊山腳有數公里深之凹陷證據。如此深的凹陷意味著彈性板厚度可能極小，如此一來伊阿珀托斯的硬外殼可以被視為平坦的單一薄板。另外從地質特徵觀之，伊阿珀托斯並無板塊作用(側向應力的存在)的證據。在解析式模型中，赤道脊被設定為一負載點，以上條件可令折彎與負載的關係簡化成一維線性常微分方程，輸入彈性板的厚度後，就可求得被折彎之地表的情形。而在一維數值模型中，負載之函數則依赤道脊的外貌剖面建立，並使用有限差分法模擬折彎之地表。另外，因為隕石撞擊事件可能影響彈性板厚度，本研究也嘗試使用彈性板厚度隨位置變化之數值模型做為參考。 模擬結果顯示了與前人研究一致之訊息：超過100公里厚之彈性岩石圈將不會造成任何顯著的地表折彎。然而DTM高程提供的訊息(尤其是隆起處與赤道脊中心之距離)卻顯示其較有可能為5-10公里厚之彈性岩石圈作用下的結果。數值模擬結果也顯示了赤道脊區域的地型主要由隕石撞擊事件，以及薄彈性層折彎這兩因子所塑造。此一新結果雖與前人研究相異，但除了一部份疑似受隕石撞擊事件施以彈性層水平應力之區域外，對於DTM地形卻高度相符。如此薄的岩石圈，也說明了赤道脊形成時，伊阿破托斯內部具有高熱流通量(~18 mWm-2)。如此一來16小時週期之橢球外貌的形成，時間上應晚於赤道脊之形成。折彎模型中也顯示了赤道脊的原始高度可能較現今高出一倍，如此一來原始坡度就與環殘骸堆疊(外生性成因)後自然形成的堆積坡休止角較為相近而不矛盾。綜觀以上結果，由於並無明顯的證據支持內生性成因，因此較有可能之赤道脊起源仍屬外生性之環殘骸假說。 簡而言之，從地表地型資料與衛星熱史推論，伊阿珀托斯的彈性折彎模型顯示了赤道脊可能負載於一較薄(5-10公里)之彈性板上，並且提供了更多關於伊阿珀托斯，這顆太陽系中有趣的衛星，其起源的更多線索。

Iapetus may be the most peculiar satellite in the Solar System. This Saturnian moon has a mean radius of 735 km, but an averagely 10-kilometer-high mountain ridge lies precisely on its 75% equatorial circumference. The ridge is so high that Iapetus appears walnut shaped, and it is named “equatorial ridge” after this amazing truth. The ridge was discovered by the Cassini spacecraft in 2005, but the formation theory is still under debate because of the lack of observational data. Several hypotheses, which are roughly divided into endogenic (tectonic buckling) and exogenic (ring remnant) processes, are attributed to explain its origin. Previous studies also noted that the shape of Iapetus is an oblate spheroid related to a hydrostatic spin period of 16 h, but Iapetus now is tidally synchronized with a 79-day period. Because the surface of Iapetus is old and heavily cratered, the formation of the ridge and the oblate spheroid had finished in the early stage of Iapetus (> 4000 Ma). Thus, it’s plausible to assume that Iapetus had a high thermal flux when the equatorial ridge formed. The assumption leads to a result that the surface would bend when the applying load like the ridge exerted. Therefore, upon calculating the deflection of the surface, we could obtain some constraints for the thermal history of Iapetus, and the proper origin model of the equatorial ridge. According to this idea, we attempted to construct analytical and numerical flexural models of the equatorial ridge by utilizing elastic lithosphere theory. The equatorial ridge is treated as a perfectly linear load on Iapetus’ hard shell (i.e. elastic layer of Iapetus). The Digital Terrain Model (DTM) data are inputted and transformed to a vertical load function, and also reveals that large deflection exists in some foothills area. This few-kilometre deflection implies a very thin elastic layer enough to regard it as a flat plate. Moreover, there are no tectonic signals on Iapetus, so the flat-Earth and one-plate condition could adapt to the flexure model. To obtain an analytical solution, the equatorial ridge is simplified to a central loading point. This can be rearranged into an explicit deflecting function in the 1-D coordinate system, so the deflection can be computed if the elastic thickness is given. In the numerical model, the point vertical force is replaced by a loading map. The finite difference method is used to solve the ODE flexural function. Consider the elastic thickness may vary with different areas; we also set a variable-thickness program for the numerical modelling. The modelling results illustrate that an over 100-km elastic layer would not cause any significant deflection; it coincides with the previous suggested. However, a deflecting curve with a range of 5-10 km elastic thickness well fits the terrain data, especially for the distance between a bulge and the ridge. Numerical solution also shows that there are 2 factors contributing the geomorphological changes: cratering and the flexure. Cratering created a deep hole and a thinner elastic layer. These new results seem controversial to the previous studies, but the modelled surface profile is highly consistent with numerical ridge DTM profile except the plateau regions which are suspected to be caused by cratering end load pressure. Such a thin shell implies that the ridge formed when the heat flux stayed high (~18 mWm-2). Therefore, the formation of the ridge probably happened before the despin (oblate shaping) event. The thin-layer flexure model also solves the problem of the angle of response because the ridge sank in the deflected surface, lowered the slope from the angle of response to the observed slope of the ridge. Since there is no evidence relating to endogenic processes, the exogenic origin is in favour. In conclusion, the flexural model of Iapetus’ equatorial ridge reveals the possibility of thinner (5-10 km) hard shell, fits the surface profile and thermal history, and supplies more clues to the origin of Iapetus, the interesting satellite in the Solar System.