Chapter 3 — Orbits and Gravity — Key Terms, Summary, Further Reading [Openstax]

The following are the Key Terms, Summary, For Further Exploration, and Problem Set for Chapter 3 —Orbits and Gravity of the Astronomy 2e textbook by OpenStax. (Citations are at the end of the page.)

1. Key Terms
2. Summary
3. For Further Exploration

Key Terms

1. angular momentum the measure of the motion of a rotating object in terms of its speed and how widely the object’s mass is distributed around its axis
2. aphelion the point in its orbit where a planet (or other orbiting object) is farthest from the Sun
3. apogee the point in its orbit where an Earth satellite is farthest from Earth
4. asteroid belt the region of the solar system between the orbits of Mars and Jupiter in which most asteroids are located; the main belt, where the orbits are generally the most stable, extends from 2.2 to 3.3 AU from the Sun
5. astronomical unit (AU) the unit of length defined as the average distance between Earth and the Sun; this distance is about 1.5 × 10⁸ kilometers
6. density the ratio of the mass of an object to its volume
7. eccentricity in an ellipse, the ratio of the distance between the foci to the major axis
8. ellipse a closed curve for which the sum of the distances from any point on the ellipse to two points inside (called the foci) is always the same
9. escape speed the speed a body must achieve to break away from the gravity of another body
10. focus (plural: foci) one of two fixed points inside an ellipse from which the sum of the distances to any point on the ellipse is constant
11. gravity the mutual attraction of material bodies or particles
12. Kepler’s first law each planet moves around the Sun in an orbit that is an ellipse, with the Sun at one focus of the ellipse
13. Kepler’s second law the straight line joining a planet and the Sun sweeps out equal areas in space in equal intervals of time
14. Kepler’s third law the square of a planet’s orbital period is directly proportional to the cube of the semimajor axis of its orbit
15. major axis the maximum diameter of an ellipse
16. mass a measure of the amount of material within an object
17. momentum the measure of the amount of motion of a body; the momentum of a body is the product of its mass and velocity; in the absence of an unbalanced force, momentum is conserved
18. Newton’s first law every object will continue to be in a state of rest or move at a constant speed in a straight line unless it is compelled to change by an outside force
19. Newton’s second law the change of motion of a body is proportional to and in the direction of the force acting on it
20. Newton’s third law for every action there is an equal and opposite reaction (or: the mutual actions of two bodies upon each other are always equal and act in opposite directions)
21. orbit the path of an object that is in revolution about another object or point
22. orbital period (P) the time it takes an object to travel once around the Sun
23. orbital speed the speed at which an object (usually a planet) orbits around the mass of another object; in the case of a planet, the speed at which each planet moves along its ellipse
24. perigee the point in its orbit where an Earth satellite is closest to Earth
25. perihelion the point in its orbit where a planet (or other orbiting object) is nearest to the Sun
26. perturbation a small disturbing effect on the motion or orbit of a body produced by a third body
27. satellite an object that revolves around a planet
28. semimajor axis half of the major axis of a conic section, such as an ellipse
29. velocity the speed and direction a body is moving—for example, 44 kilometers per second toward the north galactic pole

Summary

3.1 The Laws of Planetary Motion

Tycho Brahe’s accurate observations of planetary positions provided the data used by Johannes Kepler to derive his three fundamental laws of planetary motion. Kepler’s laws describe the behavior of planets in their orbits as follows: (1) planetary orbits are ellipses with the Sun at one focus; (2) in equal intervals, a planet’s orbit sweeps out equal areas; and (3) the relationship between the orbital period (P) and the semimajor axis (a) of an orbit is given by P² = a³ (when a is in units of AU and P is in units of Earth years).

3.2 Newton’s Great Synthesis

In his Principia, Isaac Newton established the three laws that govern the motion of objects: (1) objects continue to be at rest or move with a constant velocity unless acted upon by an outside force; (2) an outside force causes an acceleration (and changes the momentum) for an object; and (3) for every action there is an equal and opposite reaction. Momentum is a measure of the motion of an object and depends on both its mass and its velocity. Angular momentum is a measure of the motion of a spinning or revolving object and depends on its mass, velocity, and distance from the point around which it revolves. The density of an object is its mass divided by its volume.

3.3 Newton’s Universal Law of Gravitation

Gravity, the attractive force between all masses, is what keeps the planets in orbit. Newton’s universal law of gravitation relates the gravitational force to mass and distance:

\[ 𝐹_{gravity} = 𝐺⁡ \frac{𝑀_1⁢𝑀_2}{𝑅_2} \]

The force of gravity is what gives us our sense of weight. Unlike mass, which is constant, weight can vary depending on the force of gravity (or acceleration) you feel. When Kepler’s laws are reexamined in the light of Newton’s gravitational law, it becomes clear that the masses of both objects are important for the third law, which becomes a³ = (M₁ + M₂) × P². Mutual gravitational effects permit us to calculate the masses of astronomical objects, from comets to galaxies.

3.4 Orbits in the Solar System

The closest point in a satellite orbit around Earth is its perigee, and the farthest point is its apogee (corresponding to perihelion and aphelion for an orbit around the Sun). The planets follow orbits around the Sun that are nearly circular and in the same plane. Most asteroids are found between Mars and Jupiter in the asteroid belt, whereas comets generally follow orbits of high eccentricity.

3.5 Motions of Satellites and Spacecraft

The orbit of an artificial satellite depends on the circumstances of its launch. The circular satellite velocity needed to orbit Earth’s surface is 8 kilometers per second, and the escape speed from our planet is 11 kilometers per second. There are many possible interplanetary trajectories, including those that use gravity-assisted flybys of one object to redirect the spacecraft toward its next target.

3.6 Gravity with More Than Two Bodies

Calculating the gravitational interaction of more than two objects is complicated and requires large computers. If one object (like the Sun in our solar system) dominates gravitationally, it is possible to calculate the effects of a second object in terms of small perturbations. This approach was used by John Couch Adams and Urbain Le Verrier to predict the position of Neptune from its perturbations of the orbit of Uranus and thus discover a new planet mathematically.

For Further Exploration

Articles

Brahe and Kepler

Gingerich, O. “Johannes Kepler and the Rudolphine Tables.” Sky & Telescope (December 1971): 328. Brief article on Kepler’s work.

Rosenthal, J. “Kepler’s Dream, Today’s Reality.” Sky & Telescope (February 2020): 62. About Kepler’s book Somnium, about spaceflight and how it became reality.

Witkoski, F. “Governing the Planets.” Sky & Telescope (August 2019): 58. On the history of Kepler’s Third Law.

Newton

Christianson, G. “Newton’s Principia: A Retrospective.” Sky & Telescope (July 1987): 18. On the great book where he summarized his work.

Gingerich, O. “Newton, Halley, and the Comet.” Sky & Telescope (March 1986): 230.

Sullivant, R. “When the Apple Falls.” Astronomy (April 1998): 55. Brief overview of Newton and gravity.

The Discovery of Neptune

Bell, T. “Discovering Neptune: What Really Happened.” Sky & Telescope (September 2022): 32. New details about the prediction and discovery of the eighth planet.

Sheehan, W., “Finding Neptune.” Astronomy (February 2022): 26. History of the planet’s discovery.

Websites

Brahe and Kepler

Johannes Kepler: http://www.britannica.com/biography/Johannes-Kepler. Encyclopedia Britannica article.

Johannes Kepler: https://mathshistory.st-andrews.ac.uk/Biographies/Kepler/. MacTutor article with additional links.

Noble Dane: Images of Tycho Brahe: http://www.mhs.ox.ac.uk/tycho/index.htm. A virtual museum exhibit from Oxford.

Tycho Brahe: https://mathshistory.st-andrews.ac.uk/Biographies/Brahe/. MacTutor article with more links.

Newton

Isaac Newton’s Life: https://www.newton.ac.uk/about/isaac-newton/isaac-newtons-life/. From the Institute at Cambridge named for him.

Sir Isaac Newton: https://mathshistory.st-andrews.ac.uk/Biographies/Newton/. MacTutor article with additional links.

Sir Isaac Newton: http://www.luminarium.org/sevenlit/newton/newtonbio.htm. Newton Biography at the Luminarium.

The Discovery of Neptune

Adams, Airy, and the Discovery of Neptune: https://adsabs.harvard.edu/full/1988JHA….19..121C. A defense of Airy’s role by historian Alan Chapman.

Mathematical Discovery of Planets: https://mathshistory.st-andrews.ac.uk/HistTopics/Neptune_and_Pluto/. MacTutor article.

Neptune: The First Planet Discovered Mathematically: https://www.historyofinformation.com/detail.php?id=4004. Brief introduction with quotes.

Videos

Brahe and Kepler

Kepler’s Three Laws: The Mechanical Universe Episode 21: https://www.youtube.com/watch?v=uJvOGp1wzTI. Part of a public television series on physical science (28:46).

Renaissance Lives: Tycho Brahe and the Measure of Heavens: https://www.youtube.com/watch?v=JhXCC7ZKm1E. A talk by historian John Robert Christianson (1:07:54).

Solar System Dynamics: Orbits and Kepler’s Laws: https://www.youtube.com/watch?v=wjOOrr2uPuU. Brief discussion of Kepler’s career and how he came up with the laws of planetary motion, which are demonstrated; NASA video (6:54).

Tycho Brahe’s Life and Death: https://www.youtube.com/watch?v=Y6MIKP_mjDQ Talk by astronomer Ian Morrison (7:51).

Newton

Isaac Newton: Unhappy Scientific Genius: https://www.youtube.com/watch?v=OK1bCqkn6Vk. An episode of Biography that interviews many noted historians of science about Newton’s contributions. (44:30).

Newton’s Three Laws (with Bicycle): https://www.youtube.com/watch?v=JGO_zDWmkvk. A TED cartoon; teacher explains the three laws and how they apply to a bicyclist (3:32).

Sir Isaac Newton versus Bill Nye: Epic Rap Battles of History: https://www.youtube.com/watch?v=8yis7GzlXNM. A good-humored rap battle between Newton and the modern science popularizer (2:47).

The Discovery of Neptune

Conceptual Physics: The Discovery of Neptune: https://www.youtube.com/watch?v=ckOch3OAyFM Physics teacher Paul Hewitt explains how the eighth planet was found by its perturbing the orbit of Uranus. (2:36).

Richard Feynman: The Discovery of Neptune: https://www.youtube.com/watch?v=FgXQffVgZRs. A brief black-and-white lecture by the Nobel-prize winner and great explainer (4:33).

The Crazy Way We Found Neptune in the 1800s: https://www.youtube.com/watch?v=xO_zng-jeww. A good retelling of the history in the Solar System Snacks series (7:53).

Understanding Motion on Earth and in Space

Conservation of Angular Momentum: https://www.youtube.com/watch?v=1Bdyrv3cc0M. This student-produced video explores angular momentum with a number of good classroom demonstrations (5:20).

How You Wash Hair in Space: https://www.youtube.com/watch?v=uIjNfZbUYu8. Astronaut Karen Nyberg demonstrates how she washes her hair in free-fall aboard the International Space Station (2:54).

Orbital Motion Explained: https://www.youtube.com/watch?v=YSh_S2bHrHE. A short cartoon with discussion of what’s involved in a stable orbit, from NOAA (2:37).

[Note: This content is not the Copyright of Projeda. It is released by OpenStax under a CC BY-NC-SA 4.0 license allowing digital reproduction with proper attribution. Access for free at https://openstax.org/books/astronomy-2e/pages/1-introduction.]

[Andrew Fraknoi, David Morrison, Sidney Wolff. Astronomy 2e. OpenStax. Mar 9, 2022. Houston, Texas. Book URL: https://openstax.org/books/astronomy-2e/pages/1-introduction Section URL: https://openstax.org/books/astronomy-2e/pages/3-key-terms, https://openstax.org/books/astronomy-2e/pages/3-summary, https://openstax.org/books/astronomy-2e/pages/3-for-further-exploration]

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