kepler's law of planetary motion assignment

13.5 Kepler's Laws of Planetary Motion

Learning objectives.

By the end of this section, you will be able to:

• Describe the conic sections and how they relate to orbital motion
• Describe how orbital velocity is related to conservation of angular momentum
• Determine the period of an elliptical orbit from its major axis

Using the precise data collected by Tycho Brahe, Johannes Kepler carefully analyzed the positions in the sky of all the known planets and the Moon, plotting their positions at regular intervals of time. From this analysis, he formulated three laws, which we address in this section.

Kepler’s First Law

The prevailing view during the time of Kepler was that all planetary orbits were circular. The data for Mars presented the greatest challenge to this view and that eventually encouraged Kepler to give up the popular idea. Kepler’s first law states that every planet moves along an ellipse, with the Sun located at a focus of the ellipse. An ellipse is defined as the set of all points such that the sum of the distance from each point to two foci is a constant. Figure 13.16 shows an ellipse and describes a simple way to create it.

For elliptical orbits, the point of closest approach of a planet to the Sun is called the perihelion . It is labeled point A in Figure 13.16 . The farthest point is the aphelion and is labeled point B in the figure. For the Moon’s orbit about Earth, those points are called the perigee and apogee, respectively.

An ellipse has several mathematical forms, but all are a specific case of the more general equation for conic sections. There are four different conic sections, all given by the equation

The variables r and θ θ are shown in Figure 13.17 in the case of an ellipse. The constants α α and e are determined by the total energy and angular momentum of the satellite at a given point. The constant e is called the eccentricity. The values of α α and e determine which of the four conic sections represents the path of the satellite.

One of the real triumphs of Newton’s law of universal gravitation, with the force proportional to the inverse of the distance squared, is that when it is combined with his second law, the solution for the path of any satellite is a conic section. Every path taken by m is one of the four conic sections: a circle or an ellipse for bound or closed orbits, or a parabola or hyperbola for unbounded or open orbits. These conic sections are shown in Figure 13.18 .

If the total energy is negative, then 0 ≤ e < 1 0 ≤ e < 1 , and Equation 13.10 represents a bound or closed orbit of either an ellipse or a circle, where e = 0 e = 0 . [You can see from Equation 13.10 that for e = 0 e = 0 , r = α r = α , and hence the radius is constant.] For ellipses, the eccentricity is related to how oblong the ellipse appears. A circle has zero eccentricity, whereas a very long, drawn-out ellipse has an eccentricity near one.

If the total energy is exactly zero, then e = 1 e = 1 and the path is a parabola. Recall that a satellite with zero total energy has exactly the escape velocity. (The parabola is formed only by slicing the cone parallel to the tangent line along the surface.) Finally, if the total energy is positive, then e > 1 e > 1 and the path is a hyperbola. These last two paths represent unbounded orbits, where m passes by M once and only once. This situation has been observed for several comets that approach the Sun and then travel away, never to return.

We have confined ourselves to the case in which the smaller mass (planet) orbits a much larger, and hence stationary, mass (Sun), but Equation 13.10 also applies to any two gravitationally interacting masses. Each mass traces out the exact same-shaped conic section as the other. That shape is determined by the total energy and angular momentum of the system, with the center of mass of the system located at the focus. The ratio of the dimensions of the two paths is the inverse of the ratio of their masses.

Interactive

You can see an animation of two interacting objects at the My Solar System page at Phet . Choose the Sun and Planet preset option. You can also view the more complicated multiple body problems as well. You may find the actual path of the Moon quite surprising, yet is obeying Newton’s simple laws of motion.

Orbital Transfers

People have imagined traveling to the other planets of our solar system since they were discovered. But how can we best do this? The most efficient method was discovered in 1925 by Walter Hohmann, inspired by a popular science fiction novel of that time. The method is now called a Hohmann transfer . For the case of traveling between two circular orbits, the transfer is along a “transfer” ellipse that perfectly intercepts those orbits at the aphelion and perihelion of the ellipse. Figure 13.19 shows the case for a trip from Earth’s orbit to that of Mars. As before, the Sun is at the focus of the ellipse.

For any ellipse, the semi-major axis is defined as one-half the sum of the perihelion and the aphelion. In Figure 13.17 , the semi-major axis is the distance from the origin to either side of the ellipse along the x -axis, or just one-half the longest axis (called the major axis). Hence, to travel from one circular orbit of radius r 1 r 1 to another circular orbit of radius r 2 r 2 , the aphelion of the transfer ellipse will be equal to the value of the larger orbit, while the perihelion will be the smaller orbit. The semi-major axis, denoted a , is therefore given by a = 1 2 ( r 1 + r 2 ) a = 1 2 ( r 1 + r 2 ) .

Let’s take the case of traveling from Earth to Mars. For the moment, we ignore the planets and assume we are alone in Earth’s orbit and wish to move to Mars’ orbit. From Equation 13.9 , the expression for total energy, we can see that the total energy for a spacecraft in the larger orbit (Mars) is greater (less negative) than that for the smaller orbit (Earth). To move onto the transfer ellipse from Earth’s orbit, we will need to increase our kinetic energy, that is, we need a velocity boost. The most efficient method is a very quick acceleration along the circular orbital path, which is also along the path of the ellipse at that point. (In fact, the acceleration should be instantaneous, such that the circular and elliptical orbits are congruent during the acceleration. In practice, the finite acceleration is short enough that the difference is not a significant consideration.) Once you have arrived at Mars orbit, you will need another velocity boost to move into that orbit, or you will stay on the elliptical orbit and simply fall back to perihelion where you started. For the return trip, you simply reverse the process with a retro-boost at each transfer point.

To make the move onto the transfer ellipse and then off again, we need to know each circular orbit velocity and the transfer orbit velocities at perihelion and aphelion. The velocity boost required is simply the difference between the circular orbit velocity and the elliptical orbit velocity at each point. We can find the circular orbital velocities from Equation 13.7 . To determine the velocities for the ellipse, we state without proof (as it is beyond the scope of this course) that total energy for an elliptical orbit is

where M S M S is the mass of the Sun and a is the semi-major axis. Remarkably, this is the same as Equation 13.9 for circular orbits, but with the value of the semi-major axis replacing the orbital radius. Since we know the potential energy from Equation 13.4 , we can find the kinetic energy and hence the velocity needed for each point on the ellipse. We leave it as a challenge problem to find those transfer velocities for an Earth-to-Mars trip.

We end this discussion by pointing out a few important details. First, we have not accounted for the gravitational potential energy due to Earth and Mars, or the mechanics of landing on Mars. In practice, that must be part of the calculations. Second, timing is everything. You do not want to arrive at the orbit of Mars to find out it isn’t there. We must leave Earth at precisely the correct time such that Mars will be at the aphelion of our transfer ellipse just as we arrive. That opportunity comes about every 2 years. And returning requires correct timing as well. The total trip would take just under 3 years! There are other options that provide for a faster transit, including a gravity assist flyby of Venus. But these other options come with an additional cost in energy and danger to the astronauts.

Visit this site for more details about planning a trip to Mars.

Kepler’s Second Law

Kepler’s second law states that a planet sweeps out equal areas in equal times, that is, the area divided by time, called the areal velocity, is constant. Consider Figure 13.20 . The time it takes a planet to move from position A to B , sweeping out area A 1 A 1 , is exactly the time taken to move from position C to D , sweeping area A 2 A 2 , and to move from E to F , sweeping out area A 3 A 3 . These areas are the same: A 1 = A 2 = A 3 A 1 = A 2 = A 3 .

Comparing the areas in the figure and the distance traveled along the ellipse in each case, we can see that in order for the areas to be equal, the planet must speed up as it gets closer to the Sun and slow down as it moves away. This behavior is completely consistent with our conservation equation, Equation 13.5 . But we will show that Kepler’s second law is actually a consequence of the conservation of angular momentum, which holds for any system with only radial forces.

Recall the definition of angular momentum from Angular Momentum , L → = r → × p → L → = r → × p → . For the case of orbiting motion, L → L → is the angular momentum of the planet about the Sun, r → r → is the position vector of the planet measured from the Sun, and p → = m v → p → = m v → is the instantaneous linear momentum at any point in the orbit. Since the planet moves along the ellipse, p → p → is always tangent to the ellipse.

We can resolve the linear momentum into two components: a radial component p → rad p → rad along the line to the Sun, and a component p → perp p → perp perpendicular to r → r → . The cross product for angular momentum can then be written as

L → = r → × p → = r → × ( p → rad + p → perp ) = r → × p → rad + r → × p → perp L → = r → × p → = r → × ( p → rad + p → perp ) = r → × p → rad + r → × p → perp .

The first term on the right is zero because r → r → is parallel to p → rad p → rad , and in the second term r → r → is perpendicular to p → perp p → perp , so the magnitude of the cross product reduces to L = r p perp = r m v perp L = r p perp = r m v perp . Note that the angular momentum does not depend upon p rad p rad . Since the gravitational force is only in the radial direction, it can change only p rad p rad and not p perp p perp ; hence, the angular momentum must remain constant.

Now consider Figure 13.21 . A small triangular area Δ A Δ A is swept out in time Δ t Δ t . The velocity is along the path and it makes an angle θ θ with the radial direction. Hence, the perpendicular velocity is given by v perp = v sin θ v perp = v sin θ . The planet moves a distance Δ s = v Δ t sin θ Δ s = v Δ t sin θ projected along the direction perpendicular to r . Since the area of a triangle is one-half the base ( r ) times the height ( Δ s ) ( Δ s ) , for a small displacement, the area is given by Δ A = 1 2 r Δ s Δ A = 1 2 r Δ s . Substituting for Δ s Δ s , multiplying by m in the numerator and denominator, and rearranging, we obtain

The areal velocity is simply the rate of change of area with time, so we have

Since the angular momentum is constant, the areal velocity must also be constant. This is exactly Kepler’s second law. As with Kepler’s first law, Newton showed it was a natural consequence of his law of gravitation.

You can view an animated version of Figure 13.20 , and many other interesting animations as well, at the School of Physics (University of New South Wales) site.

Kepler’s Third Law

Kepler’s third law states that the square of the period is proportional to the cube of the semi-major axis of the orbit. In Satellite Orbits and Energy , we derived Kepler’s third law for the special case of a circular orbit. Equation 13.8 gives us the period of a circular orbit of radius r about Earth:

For an ellipse, recall that the semi-major axis is one-half the sum of the perihelion and the aphelion. For a circular orbit, the semi-major axis ( a ) is the same as the radius for the orbit. In fact, Equation 13.8 gives us Kepler’s third law if we simply replace r with a and square both sides.

We have changed the mass of Earth to the more general M , since this equation applies to satellites orbiting any large mass.

Example 13.13

Orbit of halley’s comet.

This yields a value of 2.67 × 10 12 m 2.67 × 10 12 m or 17.8 AU for the semi-major axis.

The semi-major axis is one-half the sum of the aphelion and perihelion, so we have

Substituting for the values, we found for the semi-major axis and the value given for the perihelion, we find the value of the aphelion to be 35.0 AU.

Significance

Check your understanding 13.9.

The nearly circular orbit of Saturn has an average radius of about 9.5 AU and has a period of 30 years, whereas Uranus averages about 19 AU and has a period of 84 years. Is this consistent with our results for Halley’s comet?

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Access for free at https://openstax.org/books/university-physics-volume-1/pages/1-introduction

• Authors: William Moebs, Samuel J. Ling, Jeff Sanny
• Publisher/website: OpenStax
• Book title: University Physics Volume 1
• Publication date: Sep 19, 2016
• Location: Houston, Texas
• Book URL: https://openstax.org/books/university-physics-volume-1/pages/1-introduction
• Section URL: https://openstax.org/books/university-physics-volume-1/pages/13-5-keplers-laws-of-planetary-motion

© Jul 23, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

• Sign in / Register
• Administration
• Edit profile

The PhET website does not support your browser. We recommend using the latest version of Chrome, Firefox, Safari, or Edge.

Kepler's Three Laws of Motion ( OCR A Level Physics )

Revision note.

Kepler's Three Laws of Motion

Kepler's first law.

• Kepler's First Law describes the shape of planetary orbits
• It states: 

The orbit of a planet is an ellipse, with the Sun at one of the two foci

The orbit of all planets are elliptical, and with the Sun at one focus

• Some planets, like Pluto, have highly elliptical orbits around the Sun
• Other planets, like Earth, have near circular orbits around the Sun

Kepler's Second Law

• Kepler's Second Law describes the motion of all planets around the Sun

A line segment joining the Sun to a planet sweeps out equal areas in equal time intervals

• The consequence of Kepler's Second Law is that planets move faster nearer the Sun and slower further away from it

Kepler's Third Law

• Kepler's Third Law describes the relationship between the  time of an orbit and its  radius

The square of the orbital time period T is directly proportional to the cube of the orbital radius  r

• Kepler's Third Law can be written mathematically as:
• Which becomes:
• T = orbital time period (s)
• r = mean orbital radius (m)
• k = constant (s 2 m –3 )
• In the case of our solar system,  k is constant for all planets orbiting the Sun 

You are expected to be able to describe Kepler's Laws of Motion, so make sure you are familiar with how they are worded. 

Applications of Kepler's Third Law

• Kepler's Third Law, the fact that  T 2 ∝  r 3 applies to  any body in orbit about some larger body
• The moons orbiting other planets, like the four moons of Jupiter (Io, Europa, Callisto and Ganymede)
• Exoplanets in orbit about foreign stars
• Therefore,  useful  and  interesting data about the  mass of orbital systems can be deduced from experimental data

You've read 0 of your 0 free revision notes

Get unlimited access.

to absolutely everything:

• Downloadable PDFs
• Unlimited Revision Notes
• Topic Questions
• Past Papers
• Model Answers
• Videos (Maths and Science)

Join the 100,000 + Students that ❤️ Save My Exams

the (exam) results speak for themselves:

Did this page help you?

Author: Katie M

Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.

• History & Society
• Science & Tech
• Biographies
• Animals & Nature
• Geography & Travel
• Arts & Culture
• Games & Quizzes
• On This Day
• One Good Fact
• New Articles
• Lifestyles & Social Issues
• Philosophy & Religion
• Politics, Law & Government
• World History
• Health & Medicine
• Browse Biographies
• Birds, Reptiles & Other Vertebrates
• Bugs, Mollusks & Other Invertebrates
• Environment
• Fossils & Geologic Time
• Entertainment & Pop Culture
• Sports & Recreation
• Visual Arts
• Demystified
• Image Galleries
• Infographics
• Top Questions
• Britannica Kids
• Saving Earth
• Space Next 50
• Student Center

What does Kepler’s first law mean?

What is eccentricity and how is it determined, what is the meaning of kepler’s third law, why is a planet’s orbit slower the farther it is from the sun, where is earth when it is traveling the fastest.

Kepler’s laws of planetary motion

Our editors will review what you’ve submitted and determine whether to revise the article.

• NASA Science - Orbits and Kepler’s Laws
• University of Rochester - Department of Physics and Astronomy - Johannes Kepler: The Laws of Planetary Motion
• University of Nebraska, Lincoln - Astronomy Education: Kepler's Laws of Planetary Motion
• Physics LibreTexts - Kepler's Laws of Planetary Motion
• BCcampus Open Publishing - The Laws of Planetary Motion – Brahe and Kepler
• Kansas State University - Mathematics Department - Kepler's Laws of Planetary Motion and Newton's Law of Universal Gravitation

Kepler’s first law means that planets move around the Sun in elliptical orbits . An ellipse is a shape that resembles a flattened circle. How much the circle is flattened is expressed by its eccentricity. The eccentricity is a number between 0 and 1. It is zero for a perfect circle .

The eccentricity of an ellipse measures how flattened a circle it is. It is equal to the square root of [1 - b*b/(a*a)]. The letter a stands for the semimajor axis, ½ the distance across the long axis of the ellipse. The letter b stands for the semiminor axis, ½ the distance across the short axis of the ellipse. For a perfect circle, a and b are the same such that the eccentricity is zero. Earth ’s orbit has an eccentricity of 0.0167, so it is very nearly a perfect circle.

How long a planet takes to go around the Sun (its period, P) is related to the planet’s mean distance from the Sun (d). That is, the square of the period, P*P, divided by the cube of the mean distance, d*d*d, is equal to a constant. For every planet, no matter its period or distance, P*P/(d*d*d) is the same number.

A planet moves slower when it is farther from the Sun because its angular momentum does not change. For a circular orbit , the angular momentum is equal to the mass of the planet (m) times the distance of the planet from the Sun (d) times the velocity of the planet (v). Since m*v*d does not change, when a planet is close to the Sun, d becomes smaller as v becomes larger. When a planet is far from the Sun, d becomes larger as v becomes smaller.

It follows from Kepler’s second law that Earth moves the fastest when it is closest to the Sun . This happens in early January, when Earth is about 147 million km (91 million miles) from the Sun. When Earth is closest to the Sun, it is traveling at a speed of 30.3 kilometers (18.8 miles) per second.

Kepler’s laws of planetary motion , in astronomy and classical physics , laws describing the motions of the planets in the solar system . They were derived by the German astronomer Johannes Kepler , whose analysis of the observations of the 16th-century Danish astronomer Tycho Brahe enabled him to announce his first two laws in the year 1609 and a third law nearly a decade later, in 1618. Kepler himself never numbered these laws or specially distinguished them from his other discoveries.

Kepler’s three laws of planetary motion can be stated as follows: ( 1 ) All planets move about the Sun in elliptical orbits , having the Sun as one of the foci. ( 2 ) A radius vector joining any planet to the Sun sweeps out equal areas in equal lengths of time. ( 3 ) The squares of the sidereal periods (of revolution) of the planets are directly proportional to the cubes of their mean distances from the Sun. Knowledge of these laws, especially the second (the law of areas), proved crucial to Sir Isaac Newton in 1684–85, when he formulated his famous law of gravitation between Earth and the Moon and between the Sun and the planets, postulated by him to have validity for all objects anywhere in the universe . Newton showed that the motion of bodies subject to central gravitational force need not always follow the elliptical orbits specified by the first law of Kepler but can take paths defined by other, open conic curves; the motion can be in parabolic or hyperbolic orbits, depending on the total energy of the body. Thus, an object of sufficient energy—e.g., a comet —can enter the solar system and leave again without returning. From Kepler’s second law, it may be observed further that the angular momentum of any planet about an axis through the Sun and perpendicular to the orbital plane is also unchanging.

The usefulness of Kepler’s laws extends to the motions of natural and artificial satellites , as well as to stellar systems and extrasolar planets . As formulated by Kepler, the laws do not, of course, take into account the gravitational interactions (as perturbing effects) of the various planets on each other. The general problem of accurately predicting the motions of more than two bodies under their mutual attractions is quite complicated; analytical solutions of the three-body problem are unobtainable except for some special cases. It may be noted that Kepler’s laws apply not only to gravitational but also to all other inverse-square-law forces and, if due allowance is made for relativistic and quantum effects, to the electromagnetic forces within the atom .

Orbits and Kepler’s Laws

Planetary Physics: Kepler's Laws of Planetary Motion

Kepler's three laws describe how planetary bodies orbit the Sun. They describe how (1) planets move in elliptical orbits with the Sun as a focus, (2) a planet covers the same area of space in the same amount of time no matter where it is in its orbit, and (3) a planet’s orbital period is proportional to the size of its orbit (its semi-major axis).

Explore the process that Johannes Kepler undertook when he formulated his three laws of planetary motion.

The planets orbit the Sun in a counterclockwise direction as viewed from above the Sun's north pole, and the planets' orbits all are aligned to what astronomers call the ecliptic plane.

The story of our greater understanding of planetary motion could not be told if it were not for the work of a German mathematician named Johannes Kepler. Kepler lived in Graz, Austria during the tumultuous early 17th century. Due to religious and political difficulties common during that era, Kepler was banished from Graz on August 2nd, 1600.

Fortunately, an opportunity to work as an assistant for the famous astronomer Tycho Brahe presented itself and the young Kepler moved his family from Graz 300 miles across the Danube River to Brahe's home in Prague. Tycho Brahe is credited with the most accurate astronomical observations of his time and was impressed with the studies of Kepler during an earlier meeting. However, Brahe mistrusted Kepler, fearing that his bright young intern might eclipse him as the premier astronomer of his day. He, therefore, led Kepler to see only part of his voluminous planetary data.

He set Kepler, the task of understanding the orbit of the planet Mars, the movement of which fit problematically into the universe as described by Aristotle and Ptolemy. It is believed that part of the motivation for giving the Mars problem to Kepler was Brahe's hope that its difficulty would occupy Kepler while Brahe worked to perfect his own theory of the solar system, which was based on a geocentric model, where the earth is the center of the solar system. Based on this model, the planets Mercury, Venus, Mars, Jupiter, and Saturn all orbit the Sun, which in turn orbits the earth. As it turned out, Kepler, unlike Brahe, believed firmly in the Copernican model of the solar system known as heliocentric, which correctly placed the Sun at its center. But the reason Mars' orbit was problematic was because the Copernican system incorrectly assumed the orbits of the planets to be circular.

After much struggling, Kepler was forced to an eventual realization that the orbits of the planets are not circles, but were instead the elongated or flattened circles that geometers call ellipses, and the particular difficulties Brahe hand with the movement of Mars were due to the fact that its orbit was the most elliptical of the planets for which Brahe had extensive data. Thus, in a twist of irony, Brahe unwittingly gave Kepler the very part of his data that would enable Kepler to formulate the correct theory of the solar system, banishing Brahe's own theory.

Since the orbits of the planets are ellipses, let us review three basic properties of ellipses. The first property of an ellipse: an ellipse is defined by two points, each called a focus, and together called foci. The sum of the distances to the foci from any point on the ellipse is always a constant. The second property of an ellipse: the amount of flattening of the ellipse is called the eccentricity. The flatter the ellipse, the more eccentric it is. Each ellipse has an eccentricity with a value between zero, a circle, and one, essentially a flat line, technically called a parabola.

The third property of an ellipse: the longest axis of the ellipse is called the major axis, while the shortest axis is called the minor axis. Half of the major axis is termed a semi-major axis. Knowing then that the orbits of the planets are elliptical, johannes Kepler formulated three laws of planetary motion, which accurately described the motion of comets as well.

Kepler's First Law: each planet's orbit about the Sun is an ellipse. The Sun's center is always located at one focus of the orbital ellipse. The Sun is at one focus. The planet follows the ellipse in its orbit, meaning that the planet to Sun distance is constantly changing as the planet goes around its orbit.

Kepler's Second Law: the imaginary line joining a planet and the Sun sweeps equal areas of space during equal time intervals as the planet orbits. Basically, that planets do not move with constant speed along their orbits. Rather, their speed varies so that the line joining the centers of the Sun and the planet sweeps out equal parts of an area in equal times. The point of nearest approach of the planet to the Sun is termed perihelion. The point of greatest separation is aphelion, hence by Kepler's Second Law, a planet is moving fastest when it is at perihelion and slowest at aphelion.

Kepler's Third Law: the squares of the orbital periods of the planets are directly proportional to the cubes of the semi-major axes of their orbits. Kepler's Third Law implies that the period for a planet to orbit the Sun increases rapidly with the radius of its orbit. Thus we find that Mercury, the innermost planet, takes only 88 days to orbit the Sun. The earth takes 365 days, while Saturn requires 10,759 days to do the same. Though Kepler hadn't known about gravitation when he came up with his three laws, they were instrumental in Isaac Newton deriving his theory of universal gravitation, which explains the unknown force behind Kepler's Third Law. Kepler and his theories were crucial in the better understanding of our solar system dynamics and as a springboard to newer theories that more accurately approximate our planetary orbits.

(mov) (78.67 MB)

Alternate #1

(m4v) (27.32 MB)

Alternate #2

(webm) (30.67 MB)

(txt?emrc=66b8a3edca7bc) (5.49 KB)

EN - English

DE - Deutsch

ES - Español

IT - Italiano

FR - Français

PT - Português

Table of content

Full table of contents

In the early 17th century, German astronomer and mathematician Johannes Kepler postulated three laws for the motion of planets in the solar system. His first law states that all planets orbit the Sun in an elliptical orbit, with the Sun at one of the ellipse's foci. Therefore, the distance of a planet from the Sun varies throughout its revolution around the Sun.

While in an elliptical orbit, the total energy of the planet is conserved. Therefore, the planet slows down when it is at apogee and speeds up when it is at perigee. These conclusions led Kepler to state his second law, that the radius vector of the planet sweeps out in equal areas in equal time. This means that if a planet takes the same time to travel from A to A 1 , and then from B to B 1 , then the areas AF 1 A 1 and BF 1 B 1 are equal, as shown in Figure 1.

Since the area is constant in a given time interval, the planet's sector velocity remains constant. Since the sector velocity is proportional to the planet's angular momentum, Kepler’s second law implies that the angular momentum of a planet in an elliptical orbit is conserved.

This text is adapted from Openstax, University Physics Volume 1, Section 13.5 Kepler’s Laws of Motion.

Terms of Use

Recommend to library

JoVE NEWSLETTERS

JoVE Journal

Methods Collections

JoVE Encyclopedia of Experiments

JoVE Business

JoVE Science Education

JoVE Lab Manual

Faculty Resource Center

Publishing Process

Editorial Board

Scope and Policies

Peer Review

Testimonials

Subscriptions

Library Advisory Board

Copyright © 2024 MyJoVE Corporation. All rights reserved