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Earth, Space Science and Engineering
ESSE 3020
Anthony Anthony

1. b) The graph for 1b has been shown below. Average Height Above Earth's Surface 25 y = 0.2333x 20 15 10 Distance(km) 5 0 0 20 40 60 80 100 Colatitud, ᶱ 2. b) According to the article given, the French astronomer Jean Richer measured the period of a pendulum at Paris and at Cayenne. Cayenne is almost at the Equator, and Paris is closer to the poles than the Equator. The gravitational force in Cayenne was found to be less than that at Paris, and G.D. Cassini found that the Earth was a prolate ellipsoid. The current knowledge of the degree of flattening on Earth says that the poles have a slightly smaller radius than the equator. The equation for the gravitational force is: Given this equation, the gravitational force at the poles and the equator can be calculated. The reason that the gravitational force in Cayenne was found to be less than that at Paris is because the gravitational force is inversely proportional to the radius squared (Rowlinson, 2007). As the equator has a large radius, the gravitational force is less. In order to render the period of the pendulum exactly two seconds, it is known that the length of the pendulum depends on the acceleration due to gravity, g, and the length of the string. It is independent of mass. The formula for the period of a simple pendulum is √ 2 2π . Given this equation, at the equator, it is known that g=9.78 m/s 8 (Szeto, 2013). Therefore, π √ . To solve for L, this would be 9.691 m. The length of the pendulum was 9.69 m in order to render a period of exactly 2.00 seconds. Earth's Rotation 3. a) "The length of day(LOD) is only approximately constant", and this is due to the Earth's rotation. Earth does not have a uniform density, and the density of Earth is higher at the Earth's core than at the Earth's surface. There are variations in the Earth's rotation rate, which have been directly related to changes in atmospheric angular momentum. There is a LOD decrease of 2.4 milliseconds/century. The long-term changes in the LOD are caused by processes within the Earth's core. The motion in Earth's core generates the magnetic field of the Earth, and the changes in the fluid motion in the outer core that are inferred from the magnetic field match the longer period changes in the LOD. There is a precession cycle of 25,730 years (Szeto, 2013). LOD decreases long term because Earth's rotation slows down. Earth takes approximately 365.2422 days to orbit around the Sun, so every 400 years, 3 days are lost. Short-term changes in LOD periods are accounted for by atmospheric angular momentum, including the tilt of Earth's axis and the jet stream, but long-term changes are due to Earth's rotation. The Earth takes 365.2422 days to orbit around Earth, which is why there is a leap year every four years to synchronize the LOD. b) Ideally, if the solar year was 365.25 days in length, then inserting an additional day every 4 years would be perfect. However, the true solar year is not 365.25, it is actually 365.2422 days. Given this figure, there is a 0.0075 difference between this value and 365.25. In order to match up the number of days to have perfect synchrony, three days will need to be removed every 400 years. 3/400=0.0075, and this is the difference between 365.2422 and 365.25. The value of N would be 400/3=133.33 years. By this calculation, one day must be removed every 133.33 years to synchronize the number of days elapsed. c) A sidereal day is the amount of time that it takes the Moon to orbit around the Earth. The Moon goes around the Earth as the Earth goes around the Sun. The Moon issues a gravitational force on Earth. Different parts of Earth are different distances on the Moon due to the effect of differential forces. These differential forces form a bulge on Earth due to the deformation of gravity. A sidereal day lasts from where the Moon is at a point on the Earth's meridian until it is next at that point. The Earth has a 23.5° tilt away from or towards the Sun, depending on the hemisphere and time of year. Earth travels in a retrograde motion. The difference between the sidereal day and solar day occurs because over a year, Earth does one revolution of the Sun and there is one less solar day than sidereal day. The difference is 4 minutes a day. It takes Earth longer to go around 9 the Sun than the Moon. In one year, Earth takes 360° to travel around its orbital axis. The solar day is 0.986° greater. For one second of solar time, Earth rotates 0.015 arcs. A solar day is the period over a certain longitude, and it is the period that requires Earth to spin from 1/365.2422 days. It is exactly 24 hours, or 86,400 s. A sidereal day is 23:56 hours, or given that there are 3600 s in an hour, 86,160 s. The difference between a sidereal and a solar day is 240 s, or four minutes. To the second, taking the ratio of a solar day/sidereal day, the solar second is 1.0028 s longer than a sidereal second. A diagram has been shown below to illustrate the difference between a solar and a sidereal day (Haynes et al, 1999). As observable from the figure, the reason that the solar day is longer is due to the degree difference of 0.986°, and the Sun is further away from Earth by this additional angle. Figure 1 d) If I see the Moon rising at 11:15 PM one evening, there will be a different time that the Moon hits the Earth(this is called moonrise) the next evening. The Moon is 30 times slower than Earth, and it takes one hour for Earth to rotate exactly 15°(Szeto, 2013). As the Moon hits the Earth it introduces a lag angle, 2.
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