1. Imagine that the Earth¡¦s spin were to be reversed, so that the planet rotated about its axis from East to West, at the same rate at which it now rotates from West to East. If Earth¡¦s orbital motion about the Sun were unchanged, which of these describe the changes we would observe?
A. The duration of the sidereal day would not change
B. The sidereal day would be 8 minutes longer than it is now
C. The duration of the Solar day would not change
D. The Sun would appear to move along the celestial sphere from East to West once a year
E. The Solar day would be 8 minutes longer than it is now
F. The Sun would appear to move along the celestial sphere from West to East once a year
G. The Solar day would be 8 minutes shorter than it is now
2. Imagine that Earth¡¦s orbital motion about the Sun were in a plane perpendicular to its axis (no tilt). If Earth¡¦s spin were unchanged, which of these describes the changes we would observe?
A. The apparent rotation of the celestial sphere would be precisely unchanged
B. The Sun would appear to move about the celestial equator to the West once a year
C. The seasonal changes in climate and the variations in the amount of daylight would disappear
D. The Sun would appear to move about the celestial equator to the East once a year
E. The apparent rotation of the celestial sphere would be reversed
F. The seasonal changes in the visibility of stars would be largely unchanged
G. The seasonal changes in climate and variations in the amount of daylight would become more dramatic
3. The satellite orbits, as we shall see next week, in a circle in the plane of the Earth¡¦s equator, centered on the Earth¡¦s center, with a radius of approximately 5.6 times the Earth¡¦s radius. This means we cannot assume with high precision that it appears in the same apparent position in the sky from all points on Earth. For example, while the celestial equator appears to coincide with the horizon as viewed from either pole, the satellite will appear below the horizon. Use the small angle approximation to estimate the latitude at which the satellite will appear to lie on the horizon.
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1. To determine the changes we would observe if the Earth's spin were reversed, we need to consider the effects on the duration of the sidereal day and the solar day, as well as the apparent motion of the Sun along the celestial sphere.
- The sidereal day is the time it takes for the Earth to complete one rotation relative to a distant star. Since the Earth's orbital motion around the Sun is unchanged, reversing the Earth's spin would not affect the duration of the sidereal day. So, option A is correct.
- The solar day is the time it takes for the Sun to return to the same position in the sky. If the Earth's spin were reversed, the solar day would be affected. When the Earth rotates from West to East, the Sun appears to move from East to West. So, if the Earth spun from East to West, the apparent motion of the Sun would be reversed. Therefore, the solar day would be 8 minutes longer than it is now. Thus, option E is correct.
- With the reversed spin, the Sun would appear to move along the celestial sphere from East to West once a year, opposite to its current apparent motion. Therefore, option F is correct.
So, the correct answers for the changes we would observe are options A, E, and F.
2. To determine the changes we would observe if Earth's orbital motion around the Sun were in a plane perpendicular to its axis (no tilt), while Earth's spin remains unchanged, we need to consider the effects on the apparent rotation of the celestial sphere, the motion of the Sun, the seasonal changes in climate and daylight, and the visibility of stars.
- The apparent rotation of the celestial sphere is due to the Earth's rotation on its axis. If the Earth's tilt were removed, its spin would remain the same, resulting in an unchanged apparent rotation of the celestial sphere. Therefore, option A is correct.
- The motion of the Sun along the celestial equator is caused by the Earth's tilt. If the tilt were removed, the Sun would no longer appear to move along the celestial equator. Therefore, option B is incorrect.
- The seasonal changes in climate and the variations in the amount of daylight are mainly caused by the Earth's tilt. If the tilt were removed, these seasonal changes and variations would disappear. Therefore, option C is correct.
- With the removal of the tilt, the Sun would no longer appear to move about the celestial equator. Therefore, option D is incorrect.
- The apparent rotation of the celestial sphere is not reversed by the removal of Earth's tilt. Therefore, option E is incorrect.
- The seasonal changes in the visibility of stars would still occur even without the Earth's tilt being present. Therefore, option F is incorrect.
- As mentioned earlier, the removal of the Earth's tilt would result in the disappearance of the seasonal changes in climate and variations in the amount of daylight, making the changes more dramatic. Therefore, option G is correct.
So, the correct answers for the changes we would observe are options A and C, and option G for the third scenario.
3. To estimate the latitude at which the satellite would appear to lie on the horizon, we can use the small angle approximation. The small angle approximation states that for small angles, the sine of the angle is approximately equal to the angle in radians.
Given that the satellite orbits in a circle in the plane of the Earth's equator, centered on the Earth's center, with a radius of approximately 5.6 times the Earth's radius, we can consider the right triangle formed between the observer at the satellite's position, the Earth's center, and the satellite.
The angle between the horizon and the line connecting the observer and the satellite is the same as the angle between the observer, the Earth's center, and the point on the Earth's equator directly below the satellite. This angle can be considered as the latitude at which the satellite would appear to lie on the horizon.
Using the small angle approximation, we can set up the following equation:
sin(latitude) ≈ (Earth's radius) / (satellite's distance from center)
Since the satellite's distance from the center is approximately 5.6 times the Earth's radius, we can substitute this value into the equation:
sin(latitude) ≈ (Earth's radius) / (5.6 * Earth's radius)
Simplifying the equation:
sin(latitude) ≈ 1 / 5.6
Taking the inverse sine (sin^(-1)) of both sides to solve for latitude:
latitude ≈ sin^(-1)(1 / 5.6)
Using a calculator, we find that the approximate latitude at which the satellite would appear to lie on the horizon is around 11.53 degrees.
Therefore, the estimated latitude at which the satellite would appear to lie on the horizon is approximately 11.53 degrees.