Kepler's laws Transitional Reasoning Download PDF

Questions

1. 1.

   Earth travels around the Sun in an elliptical orbit with the Sun located at one focus. When the orbit is divided into two halves by the minor axis passing through the geometric center, Earth spends approximately 179 days in the half containing perihelion and approximately 186 days in the half containing aphelion. Based on Kepler's laws of planetary motion, which of the following correctly explains this difference in time?

   • A) The gravitational force from the Sun produces a net torque that increases the areal sweep rate throughout the perihelion half.
   • B) The elliptical path length of the perihelion half is significantly shorter than that of the aphelion half due to the Sun's focal displacement.
   • C) The position vector from the Sun sweeps out a smaller area in the perihelion half, requiring less time because the areal sweep rate is constant.
   • D) The orbital speed of Earth reaches its minimum near perihelion, allowing the position vector to sweep out equal areas in less time.
2. 2.

   An interplanetary research probe travels in an eccentric elliptical orbit around the Sun with an orbital eccentricity of $e = 0.60$. Mission controllers measure the probe's orbital speed at perihelion (its point of closest approach to the Sun) to be $v_p$. Based on Kepler's laws of planetary motion, what is the probe's orbital speed at aphelion ($v_a$) in terms of $v_p$?

   • A) $v_a = 0.40\,v_p$
   • B) $v_a = 0.25\,v_p$
   • C) $v_a = 0.50\,v_p$
   • D) $v_a = 0.16\,v_p$
3. 3.

   An astronomical survey tracks a newly observed comet travelling in an eccentric elliptical orbit around the Sun. Ground-based instruments determine that the comet's perihelion distance is 2.0 AU, while its aphelion distance is 16.0 AU. Based on Kepler's first and third laws of planetary motion, what is the orbital period of this comet?

   • A) 3.0 Earth years
   • B) 9.0 Earth years
   • C) 64 Earth years
   • D) 27 Earth years
4. 4.

   Two communication satellites, Satellite A and Satellite B, are placed into distinct circular orbits around Earth. Telemetry data reveals that the orbital speed of Satellite B is half that of Satellite A. Based on the principles of circular gravitation and Kepler's third law, how does the orbital period of Satellite B ($T_B$) compare to the orbital period of Satellite A ($T_A$)?

   • A) $T_B = 2 T_A$
   • B) $T_B = 4 T_A$
   • C) $T_B = 8 T_A$
   • D) $T_B = 16 T_A$
5. 5.

   An uncrewed research probe travels in an eccentric elliptical orbit around the Sun. Flight telemetry confirms that as the probe travels from aphelion toward perihelion, its orbital speed continuously increases until reaching its maximum at perihelion. According to Kepler's laws and the principles of Newtonian gravitation, which of the following correctly explains this increase in speed?

   • A) The gravitational force remains perpendicular to the velocity vector at every point on the ellipse, but the increasing orbital curvature forces an acceleration.
   • B) The gravitational force exerts a net torque about the Sun on the probe during this segment, which continuously increases its orbital angular momentum.
   • C) The gravitational force has a component directed along the probe's velocity vector, performing positive work that increases its kinetic energy.
   • D) The rate at which the probe's position vector sweeps out area increases as it approaches the Sun, necessitating a proportional increase in speed.
6. 6.

   An orbital monitoring station tracks a satellite moving in an eccentric elliptical orbit around a planet. The position vector from the planet's center to the satellite sweeps out an area of $2.0 \times 10^7\text{ km}^2$ in 4.0 hours during one phase of its orbit. Later along the trajectory, the satellite sweeps out an area of $5.0 \times 10^7\text{ km}^2$. According to Kepler's second law and the principles of gravitation, how much time does the satellite take to sweep out this second area, and which conserved quantity explains this constant areal sweep rate?

   • A) 10.0 hours; orbital angular momentum
   • B) 10.0 hours; orbital kinetic energy
   • C) 2.5 hours; orbital linear momentum
   • D) 25.0 hours; orbital mechanical energy
7. 7.

   An astronomical survey identifies two exoplanets, Planet 1 and Planet 2, each moving in a circular orbit of radius $R$ around its respective host star. The host star of Planet 1 has a mass of $M$, whereas the host star of Planet 2 has a mass of $4M$. Assuming each planet interacts gravitationally only with its host star, how does the orbital period of Planet 2 ($T_2$) compare to that of Planet 1 ($T_1$)?

   • A) $T_2 = T_1$, because the ratio $T^2 / R^3$ is a universal constant for all planetary systems regardless of the star's mass.
   • B) $T_2 = \frac{1}{2} T_1$, because the orbital period is inversely proportional to the square root of the central star's mass.
   • C) $T_2 = 2 T_1$, because the greater gravitational force exerted by the more massive star increases the orbital period proportionally.
   • D) $T_2 = \frac{1}{4} T_1$, because the orbital period is inversely proportional to the mass of the central star.
8. 8.

   An aerospace agency deploys two scientific probes around Mars: Probe X follows a circular orbit of radius $R$, while Probe Y travels in an eccentric elliptical orbit whose semi-major axis is also equal to $R$. Mission controllers need to coordinate periodic data relay windows based on the orbital periods of both spacecraft. Based on Kepler's laws of planetary motion, which of the following correctly compares the orbital periods of Probe X ($T_X$) and Probe Y ($T_Y$)?

   • A) $T_X < T_Y$, because the elongated trajectory of Probe Y has a greater perimeter than the circle, requiring more time to complete one revolution.
   • B) $T_X > T_Y$, because Probe Y attains a much higher orbital speed near periapsis, which significantly reduces its overall travel time.
   • C) $T_X = T_Y$, because Kepler's third law dictates that the orbital period around a given central body depends solely on the semi-major axis.
   • D) $T_X < T_Y$, because gravitational force performs net negative work over each cycle of an eccentric orbit, causing Probe Y to take longer.
9. 9.

   Two exoplanets, Aethel and Boreas, travel in circular orbits around the same host star. Aethel orbits at an average distance of 4 astronomical units (AU) with an orbital period of 8 Earth years, whereas Boreas orbits at an average distance of 9 AU. According to Kepler's third law, what is the orbital period of Boreas?

   • A) 12 Earth years
   • B) 18 Earth years
   • C) 36 Earth years
   • D) 27 Earth years
10. 10.

    An interplanetary probe moves in an eccentric elliptical orbit around the Sun. At perihelion, the probe is at a distance of 1.0 AU from the Sun and travels at an orbital speed of 30 km/s. At aphelion, its distance increases to 3.0 AU from the Sun. Considering Kepler's second law and the conservation of angular momentum about the Sun, what is the speed of the probe at aphelion, and how does the rate at which the position vector sweeps out area compare between perihelion and aphelion?

    • A) The speed is 10 km/s, and the areal velocity is three times greater at perihelion than at aphelion.
    • B) The speed is 10 km/s, and the areal velocity is identical at both perihelion and aphelion.
    • C) The speed is 90 km/s, and the areal velocity is three times greater at aphelion than at perihelion.
    • D) The speed is 90 km/s, and the areal velocity is identical at both perihelion and aphelion.