Section 22: Orbits, tides, and precession 1

Transcription

1 Section 22: Orbits, tides, and precession 1 Tides Are there tides in Iowa? One effect of the Moon's gravitational attraction on Earth is that Earth's shape becomes less spherical and more like an ellipsoid: There are two "tidal bulges" due to the moon's gravity, one directed towards the moon and one exactly opposite. As the Earth rotates, most locations pass through both bulges, so most locations experience two high tides per day, approximately. Because the oceans are liquid, they are easily distorted into the bulging shape of the tides. The solid mantle of the Earth resists being reshaped by the tides, so the "rock tides" are much smaller than the ocean tides. Thus we notice tides usually only when we are near the coast, where the water level changes with respect to the coastline. We do, however, have tides that raise and lower Iowa by something like 10 cm. Because the moon orbits the Earth once every 27.3 days, we will have the moon on our Meridian once every 24 + (1/27.3)x24 = hours rather than once every 24 hours. So in fact there are two high tides every hours, or about every 25 hours, most places. Moon's position at this high tide Moon's position at the last high tide Direction of Earth's rotation Direction of Moon's orbital motion

2 Section 22: Orbits, tides, and precession 2 Not all high tides are equal. The very highest lunar tide comes when the moon passes through the zenith or the nadir: line of latitude N line of latitude S Equator The observers at the northern and southern latitude lines in the above sketch get one very high tide and one barely noticeable high tide each day. On a given day, will an observer at the Equator have two high tides that are about equal, or can they be very different? The height of the high tide has to do with how far the moon is above or below the horizon: closer to Zenith or Nadir gives higher tides, closer to the horizon gives lower ones. Thus we can use a Meridian diagram to figure out whether two tides will be nearly equal or quite different. When the moon's declination is zero, then it passes through both zenith and nadir for people living at on the Earth. When the moon's declination is not zero, a little playing around with meridian diagrams should convince you that the two tides of a given day are likely to be unequal for everyone except someone living at the Earth's equator. To the summer Sun, the Summer New Moon, or the Winter Full moon at noon NCP To the summer Sun, the Summer New Moon, or the Winter Full moon at midnight To the Sun, the New Moon, or the Full moon at To the Sun, the New Moon, or the Full moon at

3 Section 22: Orbits, tides, and precession 3 EXERCISE 22.1In the diagram below, the dashed lines are the directions to the Moon at two times of a single day between March and September. With that as a clue, figure out (and label) where the NCP, the CE are. Hint: start by labeling the Zenith and the Horizon. The sun also raises tides on Earth -- but smaller ones because the sun is so much farther away. When the solar tides and the lunar tides are acting together, when the Sun, Earth and Moon are at syzygy, then the tides are largest -- this occurs at what phases of the moon? Look at your Moon Phase Diagram to check the relationship between the Moon, the Sun, and the Earth at different phases of the Moon. Syzygy of the Sun, Earth and Moon occurs at and moon. "Spring tides" are the higher ones, occurring at full or new moon. "Neap tides" are lower, and they occur near first and third quarter phases of the moon. Why are "neap tides" lower?

4 Section 22: Orbits, tides, and precession 4 EXERCISE Not all tides are equal. Consider this sketch of the "tidal bulge" at full moon: On this sketch indicate the direction to the sun. What season is shown? All else being equal, where on Earth would you expect the largest tides when the moon is full in June? (Circle all that apply and/or fill in the blank): Arctic, temperate latitudes, Tropic of Cancer, Equator, Tropic of Capricorn, "Spring tides" occur at moon, "neap" tides at moon. When would you expect the highest spring tides at the latitude of Ames -- Summer, winter, spring or fall? Explain! There are two tides every 25 hours, but these tides need not be equally high. At our latitude (42 o ) at what season would you expect two equal tides at full or new moon? Make a sketch similar to the one above to illustrate your explanation!

5 Section 22: Orbits, tides, and precession 5 Causes and Effects of Tides Tides result from an imbalance between gravity and the force needed to maintain an objects orbit (the "centripetal" force). Objects orbiting a mass M at a distance r with v = GM/r move in a circle. If v > GM/r the object will move in an orbit that is elliptical with a>r (or will escape if v> 2GM/r, the escape velocity). Objects with v < GM/r will orbit in an ellipse with a < r. Consider the moon, moving about the Earth and keeping one side always facing the Earth. The center of mass of the moon is moving at v = GM/r (assuming, for simplicity, the moon is moving in a circle -- it is a pretty good approximation). The part of the moon closest to Earth is going around the Earth once in the same amount of time as the center of the moon, but it makes a smaller circle, so it moves more slowly. This means it moves at v < GM/r, and so, from the discussion above, it will try to move in a smaller orbit and this will show up as an apparent force towards the Earth. Similarly, the back side of the moon also goes around in the same amount of time, but it must move more quickly to do so. By this argument, there will be an apparent force, the tidal force, on the near side of the moon directed towards the Earth, and on the far side of the moon directed away from the Earth. This argument was made for a special case, but the result is in fact quite generally true. For example, for Earth, one bulge is directed towards the moon and one is directed away.

6 Section 22: Orbits, tides, and precession 6 Tides cause water to move across the land; there is friction; this takes some of Earth's rotational energy and converts it to heat. This causes the Earth to slow down its rotation. Conservation of angular momentum in the Earth-Moon system then requires that the moon's orbit get bigger. One can visualize how this could happen: Moon: Once around in 27.3 days Earth: once around in 24 hours Friction in the Earth's ocean floors and rocky mantle prevent the Earth from adjusting instantly to keep the bulges lined up exactly with the moon. So the bulge gets ahead of the moon, and thus pulls the moon forward (into a bigger orbit) while the moon pulls the bulge back, and slows down the Earth's rotation. Based on how effective this process is now, and some detailed modeling calculations for the Earth-Moon system in the past, the best prediction is that this process will end when the moon's orbital period (the month) is equal to the rotation period of the Earth (the day) and both are about 44 of our present days long. Going the other direction, it is possible to calculate how much faster the Earth was rotating (and how much shorter the month was) in the past. This type of calculation can actually be checked against historical records of eclipses. In general, the agreement is good, except that some of the eclipse records appear to have been somewhat distorted by the all too human tendency to associate them with important historical events -- an eclipse that happened two days before a big battle and 20 miles away may get moved, in later accounts, to the day and place of the battle. Precession Another effect of the Moon's gravitational pull on the Earth is the "precession of the equinoxes", usually referred to simply as "precession". This is analogous to what happens when you spin a top: Now Later The Earth's axis changes direction in such a way that the North Celestial Pole makes a great circle in the sky. This happens slowly -- it takes about 26,000 years to make one full circle. As a result

7 Section 22: Orbits, tides, and precession 7 of this motion, the Vernal Equinox also shifts; it moves all the way around the Celestial Equator once in every 26,000 years. Since right ascension and declination are found relative to the Celestial Equator, North Celestial Pole, and Vernal Equinox, precession causes small, steady changes in the coordinates of stars. Thus if you look in a star catalog or a star chart, it will probably say somewhere "Epoch 1900", "Epoch 1950", or "Epoch 2000", meaning the coordinates are given precessed to 1900, 1950, or Deneb Precession of the North Celestial Pole follows this circle Vega Precession causes the position of the Vernal Equinox shift by 30 o in about 2200 years years ago, when astrology was invented, the Vernal Equinox was in the constellation Pisces. Now it is in Aries, moving into Aquarius. So if you were born in late February through early March, popular horoscopes will label you as a Pisces, because the sun was in Pisces at that time of year 3000 years ago, but now, the sun is in Pisces in late January and early February and is in Aries in late February and early March. to the NCP now to the NCP about 13,000 years from now

8 Section 22: Orbits, tides, and precession 8 These sketches from Voyager show the part of the sky around the Vernal Equinos at 2000 AD (roughly, now), and at 0 AD: 2000 AD 0 AD An enlargement of the area around the Vernal Equinox at 2000AD and 0AD: 2000 AD 0 AD Polaris is the Pole star now, but 2000 years ago there was no good pole star. Right now, Polaris is at declination 89 o 13'. A hundred years from now, Polaris will be at declination 89 o 33'; so it will be an even better pole star. Then the pole will start moving away from Polaris AD 0 AD When Columbus sailed (500 years ago) the North Celestial Pole was about 3 degrees from Polaris. Columbus' sailors worried because they saw the Pole star making a circle in the sky; Columbus knew that that was normal, but some of the sailors thought it was a sign they shouldn't be sailing out into uncharted waters. EXERCISE 22.3 It takes about 26,000 years for the precession of the NCP to make a full circle, and therefore, it takes 26,000 years for the Vernal Equinox to precess once around the Celestial Equator. About how big a change does this make it the right ascension of the stars in Orion's belt (which lie along the Celestial Equator) each year? Precession in RA for stars with declination = 0 is about h m s. Compare your answer with the 2000 year precession shown in the illustrations at the top of this page to check that you have (at least approximately) the right answer.