The Origin and Evolution of the Solar System

28 The structure of the Solar SystemFigure 1.14. The geometry of the Earth–Moon system.´Å¨Ö · ѵ, if the bodies can be regarded as spherically symmetric. Thetidal effect is due to the Moon, mass Ñ, but part of its potential gives an accelerationof magnitude ÑÊ ¾ to the Earth. This arises from a force field withcorresponding potential ÑÜÊ ¾ where Ü is the coordinate relative to the centreof the Earth in the Earth–Moon direction. The effective tide-generating potentialis thus ½ ÜÎ Ñ Ê ¾ (1.6)The components of the tidal acceleration at P are given by the partial derivatives δĐÜ ĐÝ ĐÞµ ÑÜ ÎÝ ÎÞ Ê Ü ½ ¿ Ê ¾ ÑÝ ¿Assuming that Ê Ö the accelerations at points A, B and D are ¾ÑÖ´ĐÜ ĐÝ ĐÞµ Ê ¿ ¼ ¼ ¾ÑÖ´ĐÜ ĐÝ ĐÞµ ´ĐÜ ĐÝ ĐÞµ ¼Ê ¿ ¼ ¼ÑÖÊ ¿ ¼ ÑÞ ¿ (1.7)(1.8a)(1.8b)(1.8c)The tidal forces over a polar section of the Earth are shown in figure 1.15.The observed physical effect of this tidal force is the semi-diurnal rise andfall of sea level in the oceans. There are two high tides per day because the tidalforces cause bulges on opposite sides of the Earth rather than just on the sidefacing the Moon. The difference in sea levels in mid-ocean is around 1 m but thiscan be amplified considerably by coastal effects. The highest tides, spring tides,are experienced fortnightly when the Sun and the Moon are in line with the Earthbut when the Earth–Sun and Earth–Moon directions are perpendicular the effectsare subtracted giving the so-called neap tides.