Chapter 4: Problem 20
Two comets are leaving the vicinity of the Sun, one traveling in an elliptical orbit and the other in a unbound orbit. What can you say about the future of these two comets? Would you expect either of them to return eventually?
Short Answer
Expert verified
The comet in the elliptical orbit will return periodically. The comet in the unbound orbit will not return.
Step by step solution
01
- Understand the Orbits
Identify the types of orbits: One comet is in an elliptical orbit and the other in an unbound orbit. An elliptical orbit is a closed path around the Sun, while an unbound orbit is an open path, meaning the comet will escape the gravitational pull of the Sun.
02
- Analyze the Elliptical Orbit
For the comet in the elliptical orbit, it follows a Keplerian orbit around the Sun. This means that it will periodically return to the Sun after completing one orbit. The key characteristic of an elliptical orbit is that it is bound and closed, ensuring the comet returns after each period.
03
- Analyze the Unbound Orbit
The comet traveling in an unbound orbit, such as a parabolic or hyperbolic trajectory, is not gravitationally bound to the Sun. As a result, this comet will continue moving away from the Sun indefinitely and will not return.
04
- Summary of Analysis
Summarize the findings: One comet in the elliptical orbit will return to the Sun periodically, while the other comet in the unbound orbit will travel away from the Sun and not return.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
elliptical orbits
Comets traveling in elliptical orbits move along closed paths around the Sun. These paths vary from nearly circular to highly elongated ellipses. The shape and size of the ellipse determine the orbital period—the time it takes for the comet to complete one full trip around the Sun. Because elliptical orbits are closed, a comet following this trajectory will always come back to the Sun after completing its orbit. Kepler's First Law of Planetary Motion tells us that the Sun sits at one of the two foci of the ellipse.
This means the comet's distance from the Sun changes constantly over its journey. When close to the Sun, the comet travels fast, and when farther away, it moves slower. This periodic motion ensures that as long as the comet remains in its elliptical orbit, it will return repeatedly.
This means the comet's distance from the Sun changes constantly over its journey. When close to the Sun, the comet travels fast, and when farther away, it moves slower. This periodic motion ensures that as long as the comet remains in its elliptical orbit, it will return repeatedly.
unbound orbits
Unbound orbits are paths that allow a comet to escape the gravitational pull of the Sun. These orbits can be hyperbolic or parabolic. In a parabolic orbit, a comet just reaches the velocity needed to break free from the Sun, while in a hyperbolic orbit, the comet exceeds this escape velocity. Because these orbits are open, they do not loop back around to complete a circle or an ellipse.
A comet in an unbound orbit will continue moving away from the Sun forever once it has enough energy to overcome the Sun's gravity. This means such comets will not return to the inner solar system.
A comet in an unbound orbit will continue moving away from the Sun forever once it has enough energy to overcome the Sun's gravity. This means such comets will not return to the inner solar system.
Keplerian orbits
Keplerian orbits are governed by Kepler's laws of planetary motion. These laws describe how objects orbiting the Sun, like comets, move:
- First Law: The orbit of a planet (or comet) is an ellipse with the Sun at one of its two foci.
- Second Law: A line connecting a planet to the Sun sweeps out equal areas in equal times, meaning comets move faster when closer to the Sun and slower when farther away.
- Third Law: The square of the orbital period of a planet (or comet) is proportional to the cube of the semi-major axis of its orbit.
gravitational interactions
Gravitational interactions shape the orbits of comets and other celestial bodies. The Sun's gravity is the dominant force acting on comets, pulling them into various orbital shapes. For a comet in an elliptical orbit, this gravitational pull is strong enough to keep it bound, ensuring it periodically returns after each orbit.
For comets in unbound orbits, the gravitational interaction is not strong enough to capture them permanently. Instead, these comets have enough energy to escape the Sun's gravitational influence. Over time, gravitational perturbations from other planets, particularly Jupiter, can alter a comet's path significantly, potentially changing an elliptical orbit to an unbound orbit or vice versa.
For comets in unbound orbits, the gravitational interaction is not strong enough to capture them permanently. Instead, these comets have enough energy to escape the Sun's gravitational influence. Over time, gravitational perturbations from other planets, particularly Jupiter, can alter a comet's path significantly, potentially changing an elliptical orbit to an unbound orbit or vice versa.
closed and open paths
Orbital paths can either be closed or open. Closed paths, like elliptical or circular orbits, keep comets bound to the Sun. These comets will recur, returning to the same region of space over and over. Elliptical orbits fall into this category, ensuring the comet remains within the solar system.
Open paths, on the other hand, include hyperbolic and parabolic trajectories. These orbits allow comets to escape the Sun's gravitational hold, traveling indefinitely into space. Once a comet enters an open path, it will not return to the vicinity of the Sun. Understanding whether a comet's trajectory is closed or open helps predict its long-term behavior and movement in space.
Open paths, on the other hand, include hyperbolic and parabolic trajectories. These orbits allow comets to escape the Sun's gravitational hold, traveling indefinitely into space. Once a comet enters an open path, it will not return to the vicinity of the Sun. Understanding whether a comet's trajectory is closed or open helps predict its long-term behavior and movement in space.