The question asks about Kepler's First Law of Planetary Motion. This law describes the shape of planetary orbits around the Sun. We need to recall the specific geometric shape mentioned in this fundamental law of celestial mechanics.
C) An ellipse with the Sun at one focus is the correct statement of Kepler's First Law of Planetary Motion. An ellipse is a closed curve for which the sum of the distances from any point on the curve to two fixed points (called foci) is constant. In the case of planetary orbits, the Sun occupies one of these foci.
Kepler's Second Law, also known as the Law of Equal Areas, describes how the speed of a planet changes during its elliptical orbit around the Sun. We need to understand the implications of this law to determine where a planet moves fastest.
Understanding Kepler's Second Law: The law states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. This means that if you consider two equal time intervals, the area covered by the line from the Sun to the planet will be the same, regardless of where the planet is in its orbit.
Relating Area to Speed: For the area swept to be equal in equal time intervals, the planet's speed must vary. When the planet is closer to the Sun, the radius vector (line joining the planet and Sun) is shorter. To sweep out the same area as when it is farther away (where the radius vector is longer), the planet must travel a greater arc length in the same amount of time. A greater arc length in the same time implies a higher speed.
Identifying Perihelion and Aphelion: In an elliptical orbit, the point where the planet is closest to the Sun is called the perihelion. The point where it is farthest from the Sun is called the aphelion.
Applying the Law: According to Kepler's Second Law, for the area swept to be constant over time, the planet must move faster when it is closer to the Sun (at perihelion) and slower when it is farther from the Sun (at aphelion).
Conservation of Angular Momentum: This phenomenon can also be explained by the conservation of angular momentum. Angular momentum \(L\) is given by \(L = mvr\), where \(m\) is the mass of the planet, \(v\) is its orbital speed, and \(r\) is its distance from the Sun. Since there are no external torques acting on the planet-Sun system, angular momentum is conserved. Therefore, \(mvr = \text{constant}\). Since \(m\) is constant, as \(r\) decreases (when the planet is closer to the Sun), \(v\) must increase to keep \(vr\) constant. Conversely, as \(r\) increases, \(v\) must decrease.
B) Closest to the Sun (at perihelion) — When the planet is closest to the Sun (at perihelion), the radius vector is shortest. To sweep out an equal area in equal time, the planet must move faster along its orbit.
A) At a fixed constant speed throughout — This contradicts Kepler's Second Law. If the speed were constant, the area swept would not be equal in equal time intervals due to the varying distance from the Sun in an elliptical orbit.
C) Exactly midway in its orbit — The speed of the planet continuously changes throughout its elliptical orbit, being fastest at perihelion and slowest at aphelion. There isn't a single "midway" point where it's fastest; its speed depends on its distance from the Sun.
D) Farthest from the Sun (at aphelion) — At aphelion, the planet is farthest from the Sun. According to Kepler's Second Law (and conservation of angular momentum), it moves slowest at this point to ensure equal areas are swept in equal times.
The question asks about Kepler's Third Law, also known as the Law of Harmonies. This law describes the relationship between a planet's orbital period and a specific characteristic of its orbit. We need to recall the statement of this law to identify the correct quantity.
Correct Option: A) The semi-major axis of its orbit (distance from the Sun)
The question asks to identify the largest planet in our Solar System based on both mass and volume. This requires knowledge of the relative sizes and masses of the planets.
Correct Option: A) Jupiter is the largest planet in our Solar System, both in terms of mass and volume. Its mass is approximately \(1.898 \times 10^{27} \text{ kg}\), and its volume is about \(1.431 \times 10^{15} \text{ km}^3\).
The question asks to identify the smallest planet in our Solar System. This requires knowledge of the relative sizes of the planets.
Correct Option: B) Mercury
Mercury is the smallest planet in our Solar System. Its diameter is approximately \(4,879\) kilometers, making it only slightly larger than Earth's Moon.
The question asks to identify the hottest planet in our Solar System, specifically noting that it is not the closest to the Sun. This points to a planet with a significant greenhouse effect.
D) Venus is the hottest planet in the Solar System due to its thick carbon dioxide atmosphere causing a runaway greenhouse effect, trapping heat and resulting in extremely high surface temperatures, even though it is not the closest planet to the Sun.