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**Modificado de: Earth’s Tides Simulation**

Craig Brubaker Union College 2/28/04

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Agenda Introduction Tidal Force Tidal Bulges Orbits

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**Tidal Force Recall Force of Gravity:**

Proportional to the product of the masses over the square of the distance. Tidal Force: Proportional to the product of the masses over the CUBE of the distance.

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**Laws of gravity and tides**

The gravitational attraction between two bodies is determined by Newton’s law of gravitation: A = gravitational attractive force. m1 and m2 = masses of the bodies attracting each other. r = distance between the centers of the two bodies. G = the gravitational constant. The tide generating force is derived from Newton’s law. This force is related to the difference between the gravitational attraction at Earth’s center and surface. Basically, all of the tidal celestial tidal influences can be derived from Newton’s law of gravitation. F = the effective tidal force. µ = proportional to.

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**Major tidal influences on Earth**

Though much more massive than the moon, the sun is so much farther away that its tidal influence is less than half. Because the centrifugal acceleration effect on tides is tied to the gravitational effect, the two effects together yield tidal effects of identical magnitude on both sides of the Earth from the sun or moon. Sun Moon Mass 2.0*1033 g 7.3*1025 g Distance 150,000,000 km 385,000 km Tidal effect 0.46 1.00

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**Tidal Bulge Tidal Bulge Interaction of lunar and solar tides**

Tidal Force produces TWO high tide points. Point on Earth closest to orbital body. Point on Earth farthest from orbital body. Tidal Force produces a low tide line equidistant to both high tide points. The greater the high tide points, the lesser the low tide line. Interaction of lunar and solar tides At any point it is a simple sum.

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**Gravitational tidal bulge**

If we pretend that the moon does not orbit the Earth, gravitational attraction would raise a tidal bulge directed toward the moon. Ignore the rotation of the Earth about its polar axis. Assuming that the moon does not orbit the Earth but instead is suspended in space, there would be a tidal bulge directed toward the moon. Naturally there would be low water on the opposite side of the Earth. © Kurt Hollocher, 2002 The gravitational forces are directed exactly toward the moon’s center.

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**Earth-moon system: resultant forces on Earth’s surface**

The centrifugal and gravitational tidal vectors are shown resolved over the Earth’s surface. The vectors show that water is drawn into the tidal bulges, reaching maximum bulge elevation at points nearest and farthest from the moon. The lower diagram shows how, in a simple-minded model, two low and two high tides are expected daily. Both bulges are the result of vector sums of gravitational and centrifugal accelerations. Indeed, centrifugal component is identical everywhere. It is the decrease in the moon’s gravitational attraction with distance from the moon that permits the bulge opposite the moon to rise high. The tidal effects are therefore the result of relative force strengths at each point on the surface. Notice that according to this simple minded model, the poles should experience no tides.

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**Earth-sun-moon system: tide systematics**

Recall that the tidal bulges are identical on the sides of the Earth facing or opposite the moon. The same is true of the tides raised by the sun, except the solar tides are only ~46% as large as those of the lunar tides. Because the tides are waves, solar and lunar tides ignore one another. However, they do interfere to produce variable tidal ranges. When the moon and the sun are aligned (full and new moon phases), the tidal waves are aligned and constructively interfere (spring tides). When the moon and sun directions are at right angles (commonly called half moon phases), the tidal waves destructively interfere (neap tides). This period is called the lunar day. Solar tides follow the 24 hour day, and lunar tides follow the hour day, so solar and lunar tides gradually move in and out of phase resulting in the spring and neap tide ~2 week cycle. Spring tides are therefore tides with an especially high tidal range (high high tide and low low tide) that occur when the sun and moon are aligned. Spring tides therefore occur approximately every 2 weeks, not just in the spring. This meaning of the word “spring” is to jump or leap. Neap tides are therefore tides with an especially low tidal range (low high tide and high low tide) that occur approximately a week after and a week before spring tides, part of the same ~2 week cycle.

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**Orbits Kepler’s Laws of Planetary Motion**

I. The orbits of the planets are ellipses, with the Sun at one focus of the ellipse.

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**Orbits Kepler’s Laws of Planetary Motion**

II. The line joining the planet to the Sun sweeps out equal areas in equal times as the planet travels around the ellipse.

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**Orbits Kepler’s Laws of Planetary Motion**

III. The ratio of the squares of the revolutionary periods for two planets is equal to the ratio of the cubes of their semimajor axes:

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**Ellipticity of Earth and moon orbits**

The orbits of the moon around Earth and of the Earth around the sun are ellipses. The Earth-moon orbital distance varies from 375,200 to 405,800 km, so the lunar tidal effect varies by 11.8% over a lunar month (inverse cube law, remember). The Earth-sun distance varies from 148,500,000 to 152,200,000 km, so the solar tidal effect varies by 3.7% over a year. These cycles can go in and out of phase, so the total variation in tide range due to ellipticity is 8.1 to 15.5%. Where the average tide range is 2 m, for example, this can change tidal range by up to 0.31 m. The orbital ellipticities vary on a long cycle as well. The ellipticity cycle for Earth’s orbit is roughly 100,000 years.

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