THE ORIGIN OF THE TIDES Summary The ocean tides occur because the gravitational forces of the Moon and the Sun acting on the Earth vary from place to place over the surface of the Earth. The result of lunar gravity is small horizontal forces (that is, parallel to the Earth's surface) that accelerate parts of the oceans toward the sub-lunar point and other parts toward the anti-lunar position. This produces a build up of water at the places closest to and furthest from the Moon, a double tidal bulge in the oceans. Solar gravity also produces horizontal forces that accelerate the oceans toward the sub-solar and anti-solar points, and yield a double tidal bulge about half as high as the lunar bulge. The solid Earth rotates under this complex ocean pattern, and the result is a semi-diurnal pattern of approximately two high tides and two low tides each day at most locations on the Earth. Origin of the Lunar Tides At first we will ignore the rotation of the Earth about its axis, and the motion of the Earth about the Sun, and concentrate on the motion of the Earth and Moon under their mutual gravitation. The Earth and Moon orbit their common center of mass ("COM," labeled "Z" below), each taking a lunar month to make a full cycle. The COM lies ~1/82 of the way from the center of the Earth toward the center of the Moon, a position that is about 1600 km below the surface of the Earth. The first key to the tides is understanding the motion of the Earth as it revolves about the COM of the Earth-Moon system. Figures 1-4 below show the location of four places labeled "A"-"D" on the surface of the Earth, as the Earth "E" and the Moon "M" orbit their mutual COM. In each case we are looking down on the north pole of the Earth, and the Moon and Earth each orbit the COM in a counter-clockwise direction. Since we ignore the rotation of the Earth, the relative orientations of points A through D do not change as the Earth revolves about the COM. .................................... (1) A D E ZB M C ................................... Figure 1: The Earth-Moon system viewed from above the Earth's north pole. The points "E" and "M" are the centers of the Earth and Moon, respectively. Points A-D lie on the equator of the Earth, separated by 90 degrees in longitude. The center of mass of the Earth-Moon system is the point Z, which lies about 1600 km under the Earth's surface. The relative size of the Earth (radius R) and the Earth-Moon distance d are not drawn to scale; in reality d = 60 R. ................................... (2) M A Z D E B C .................................... Figure 2: Same as Figure 1, but 1/4 lunar month later. Here we are ignoring the rotation of the Earth about its axis, so the points A-D are located in the same places as in Figure 1. .................................... (3) A M DZ E B C .................................... Figure 3: The Earth-Moon system 1/4 lunar month after Figure 2. .................................... (4) A D E B Z C M ................................... Figure 4: The Earth-Moon system 1/4 lunar month after Figure 3. In this motion the COM remains fixed in space. A few moments reflection should convince you that when the Earth moves in this way, each and every part of the Earth moves in a circle whose radius is the distance from the center of the Earth to the COM of the Earth-Moon system. (The location of the COM inside the Earth is NOT a fixed point in the Earth, but rather moves around in a circle as seen in the frame of the Earth.) The motion of the Earth around in such a circle is produced by a centripetal acceleration of each part of the Earth. Since the whole Earth moves together, these centripetal accelerations are the same for each point on and in the Earth (see Figure 5). These accelerations are parallel to the line joining the centers of the Earth and Moon; they are horizontal and to the right in Figure 1, vertical and upward in Figure 2, horizontal and to the left in Figure 3, and vertical and downward in Figure 4. In Figure 5 we show the Earth-Moon system as seen from a point in the plane of their orbits. With the Moon situated to the right of the Earth, the centripetal acceleration of each part of the Earth is horizontal and to the right. .................................... (5) + + N--> + + D--> E Z B--> M + + S--> + + .................................... Figure 5: The Earth-Moon system viewed from a point in the plane of their orbit at a moment when the Earth is approaching the viewer and the Moon is receding. In this picture the north pole of the Earth is "N" and the south pole "S" (this view is different from that of Figures 1-4!). The axis about which the Earth-Moon system rotates is indicated by the vertical line "+++". The centripetal accelerations "-->" experienced at points N, B, S, and D are all the same since each point in the Earth moves in the same circle about the COM of the Earth-Moon system. These centripetal accelerations are, of course, caused by the gravitational force of the Moon. According to Newton's second law, these accelerations are equal to the average gravitational force/mass of the Moon acting on the Earth. The average force/mass exerted by the Moon on the Earth is that required to accelerate the center of the Earth in this way, i.e., Fm(avg) = G M(moon)/d^2, where d is the distance between the centers of the Earth and Moon. This force is parallel to the line between the center of the Earth and the center of the Moon. However, the local forces/mass Fm exerted by the Moon on different parts of the Earth are in general different from Fm(avg) because in general (1) each place is at a different distance from the center of the Moon, and (2) the lines between each place and the center of the Moon point in slightly different directions. THE DIFFERENCES BETWEEN THE LOCAL FORCES ACTING AT VARIOUS PLACES ON THE EARTH AND THE AVERAGE FORCE DRIVE THE TIDES. These difference forces are called "tractive forces." The local forces/mass Fm try to produce accelerations that are slightly different from the mean acceleration of the Earth, but as the Earth is solid it moves as a whole with a single average acceleration. The additional forces needed to satisfy Newton's second law locally at each place in the Earth (and to produce the average acceleration) are provided by solid body (elastic) forces. On the other hand, THE OCEANS ARE FREE TO RESPOND TO THE LOCAL FORCE/MASS EXERTED BY THE MOON, AND ARE THEREBY ACCELERATED DIFFERENTLY THAN THE SOLID EARTH AROUND AND UNDER THEM. The Tractive Forces Due to the Moon The vector nature of the tractive forces may be discerned by considering separately the side of the Earth closest to the Moon (the "near side of the Earth"), and the side of the Earth farthest from the Moon (the "far side of the Earth"): (1) Everywhere on the Earth, the local force/mass of the Moon acting on the Earth is a vector directed toward the center of the Moon. (2) On the near side of the Earth the force/mass of the Moon is a vector coming upwards out of the surface of the Earth. On the far side of the Earth, the force/mass of the Moon is a vector pointing downwards into the interior of the Earth. (3) On (almost all of) the near side of the Earth, the local force/mass Fm is greater than the average force/mass Fm(avg), and points in a different direction than Fm(avg). (4) On the far side of the Earth, the local force/mass Fm is smaller than the average force/mass Fm(avg), and points in a different direction than Fm(avg). On each side of the Earth, the difference between Fm and Fm(avg) is defined to be the local value of the tractive force/mass Ft due to the Moon. At any location on the surface of the Earth, the tractive force/mass may be resolved into a component directed towards the center of the Earth (the vertical component), and a component perpendicular to the vertical component, one that points along the surface of the Earth (the horizontal component): (5) Everywhere on the Earth, the vertical component of the tractive force/mass due to the Moon is tiny compared to the gravitational force/mass exerted by the Earth (which is the mean acceleration of gravity g), and may be neglected. (6) The horizontal component of the tractive force/mass is zero along the boundary between near and far sides, and zero at the sub-lunar and anti-lunar points. At all other points on the surface of the Earth the horizontal component of the tractive force/mass due to the Moon has an amplitude that is about 1 part in ten million of the force/mass due to the Earth itself (i.e., it has a value of about one-millionth of a meter per second-squared). (7) Despite their relative size compared to g, the horizontal components of the tractive force/mass due to the Moon have a significant effect on the oceans, and may not be neglected. This is true for two reasons. First, Ft is the only horizontal force since the Earth's gravity is vertical, and second, Ft is a "body force," that is, a force that acts on every part of the ocean, rather than simply acting at one of its boundaries. The direction of the horizontal component of Ft is different on the near and far sides of the Earth from the Moon: (8) On the near side of the Earth, the horizontal component of Ft at a point P is directed along a great circle that flows from P towards the sub-lunar point. (9) On the far side of the Earth, the horizontal component of Ft at a point P is directed along a great circle that flows from P towards the anti-lunar point. (10) Thus, on the near side of the Earth to the Moon, compared to the solid Earth, the oceans are accelerated along the surface of the Earth toward the sub-lunar point, and on the far side of the Earth from the Moon, the oceans are accelerated along the surface of the Earth towards the anti-lunar point. (11) The result of these tidal tractive forces is that the oceans reach a quasi-equilibrium in which there is a build-up of water around the sub-lunar and anti-lunar positions, and a relative deficit of water around the boundary between the near and far sides of the Earth. These are the "lunar tidal bulges." The Solar Tides Exactly the same considerations apply to the gravitational force exerted by the Sun at various places on the Earth. In other words, everywhere above where we have written "Moon" and "lunar," you may substitute "Sun" and "solar." Even though the Sun is far more massive than the Moon, because it is also much more distant Earth, the resulting solar tidal bulges are only about 50% as high as the lunar bulges. The Net Tides The net tidal bulges are the sum of the bulges produced separately by the Moon and Sun. Each of these bulges produces an approximately semi-diurnal pattern of high and low tides. The rotation of the Earth causes the solid portion of the Earth to move with respect to the oceans, carrying a given location on Earth from low to high tide and back about twice each day. The sum of the lunar and solar tides results in a complex pattern since they have similar but slightly different periods. Then the lunar and tidal bulges overlap the tides are especially large, and when they are at right angles to each other the tides are minimal. A complete theory of the tides must include the effects of local geographical features on the freedom of the ocean to be accelerated by the tractive forces. It also includes the tilt of the Earth's axis, the tilt of the orbit of the Moon, and the varying distance of Moon and Sun from the Earth. Solid Earth Tides The same tractive forces that act on the oceans also act on the solid Earth. The Earth does not deform as easily under tractive forces as do the oceans. However, the deformation is measurable (even though it is only a few centimeters in height) by lunar laser ranging or very high precision radio astronomy. Conclusions The lunar and solar tides on the Earth are the result of the different gravitational force/mass that is exerted on different parts of the Earth and its oceans by the Moon and by the Sun. The differences between the actual forces/mass and the mean force/mass accelerate the mobile oceans with respect to the immobile solid Earth. The direction of these tractive forces is such that they drive the oceans on the half of the Earth near the Moon toward the sub-lunar point, and the oceans on the far side of the Earth toward the anti-lunar point, giving rise to a double tidal bulge in the oceans. A similar but smaller double tidal bulge is created by the gravitation of the Sun. The rotation of the Earth carries the land under these tidal bulges, and produces approximately two high tides and two low tides per day at typical mid-latitude locations. Acknowledgement The first satisfactory explanation of the tides was (not surprisingly) due to Isaac Newton. This treatment is based on S. Pond and G. L. Pickard,"Introductory Dynamical Oceanography," 2nd ed., Pergamon (1983), which in turn credits G. H. Darwin, "The Tides and Kindred Phenomena in the Solar System," Houghton Mifflin (1911), reprinted by Freeman (1962).