Kepler's Laws of Planetary Motion, Newton's Law of Gravitation | Physics

Learning objectives

Teacher Support

The learning objectives in this section will help your students master the following standards:

(4) Science concepts. The student knows and applies the laws governing motion in a variety of situations. The student is expected to:
- (C) analyze and describe accelerated motion in two dimensions using equations, including projectile and circular examples.

In this section students will apply Kepler’s laws of planetary motion to objects in the solar system.

[BL][OL] Discuss the historical setting in which Kepler worked. Most people still thought Earth was the center of the universe, and yet Kepler not only knew that the planets circled the sun, he found patterns in the paths they followed. What would it be like to be that far ahead of almost everyone? A fascinating description of this is given in the program Cosmos with Carl Sagan (Episode 3, Harmony of the Worlds).

[AL] Explain that Kepler’s laws were laws and not theories. Laws describe patterns in nature that always repeat themselves under the same set of conditions. Theories provide an explanation for the patterns. Kepler provided no explanation.

Section Key Terms

aphelion	Copernican model	eccentricity
Kepler’s laws of planetary motion	perihelion	Ptolemaic model

Concepts Related to Kepler’s Laws of Planetary Motion

Examples of orbits abound. Hundreds of artificial satellites orbit Earth together with thousands of pieces of debris. The moon’s orbit around Earth has intrigued humans from time immemorial. The orbits of planets, asteroids, meteors, and comets around the sun are no less interesting. If we look farther, we see almost unimaginable numbers of stars, galaxies, and other celestial objects orbiting one another and interacting through gravity.

All these motions are governed by gravitational force. The orbital motions of objects in our own solar system are simple enough to describe with a few fairly simple laws. The orbits of planets and moons satisfy the following two conditions:

The mass of the orbiting object, m, is small compared to the mass of the object it orbits, M.
The system is isolated from other massive objects.

Based on the motion of the planets about the sun, Kepler devised a set of three classical laws, called Kepler’s laws of planetary motion, that describe the orbits of all bodies satisfying these two conditions:

The orbit of each planet around the sun is an ellipse with the sun at one focus.
Each planet moves so that an imaginary line drawn from the sun to the planet sweeps out equal areas in equal times.
The ratio of the squares of the periods of any two planets about the sun is equal to the ratio of the cubes of their average distances from the sun.

These descriptive laws are named for the German astronomer Johannes Kepler (1571–1630). He devised them after careful study (over some 20 years) of a large amount of meticulously recorded observations of planetary motion done by Tycho Brahe (1546–1601). Such careful collection and detailed recording of methods and data are hallmarks of good science. Data constitute the evidence from which new interpretations and meanings can be constructed. Let’s look closer at each of these laws.

Kepler’s First Law

The orbit of each planet about the sun is an ellipse with the sun at one focus, as shown in Figure_07_01_circle. The planet’s closest approach to the sun is called perihelion and its farthest distance from the sun is called aphelion.

View original figure (external; image not reproduced in this text edition)

Original figure description: Three ellipses are shown. In image (a) the ellipse is divided into quadrants. In image (b) the sun and a planet are shown. In image (c), the sun and planet are shown, and the aphelion and perihelion.

(a) An ellipse is a closed curve such that the sum of the distances from a point on the curve to the two foci (f₁ and f₂) is constant. (b) For any closed orbit, m follows an elliptical path with M at one focus. (c) The aphelion (ra) is the furthest distance between the planet and the sun, while the perihelion (rp) is the closest distance from the sun.

If you know the aphelion (r_a) and perihelion (r_p) distances, then you can calculate the semi-major axis (a) and semi-minor axis (b).

\begin{matrix} a = \frac{(r_{a} + r_{p})}{2} \\ b = \sqrt{r_{a} r_{p}} \end{matrix}

Teacher Support

[AL] If any students are interested and proficient in algebra and geometry, ask them to derive a formula that relates the length of the string and the distance between pins to the major and minor axes of an ellipse. Explain that this is a real world problem for workers who design elliptical tabletops and mirrors.

[BL][OL] Impress upon the students that Kepler had to crunch an enormous amount of data and that all his calculations had to be done by hand. Ask students to think of similar projects where scientists found order in a daunting amount of data (the periodic table, DNA structure, climate models, etc.).

View original figure (external; image not reproduced in this text edition)

Original figure description: A picture shows a hand holding a pen, drawing a blue ellipse, with 2 black pins inside the ellipse. A red string is attached to the pins. The pen is held inside the red string, to draw the ellipse.

You can draw an ellipse as shown by putting a pin at each focus, and then placing a loop of string around a pen and the pins and tracing a line on the paper.

Kepler’s Second Law

Each planet moves so that an imaginary line drawn from the sun to the planet sweeps out equal areas in equal times, as shown in Figure_07_01_planet.

View original figure (external; image not reproduced in this text edition)

Original figure description: A diagram shows a circle, divided into quadrants. The sun (labeled M) is inside the circle. The circle represents a planet’s orbit around the sun. The planet is labeled m. Shaded regions are labeled A1, A2, and A3

The shaded regions have equal areas. The time for m to go from A to B is the same as the time to go from C to D and from E to F. The mass m moves fastest when it is closest to M. Kepler’s second law was originally devised for planets orbiting the sun, but it has broader validity.

Kepler’s Third Law

The ratio of the periods squared of any two planets around the sun is equal to the ratio of their average distances from the sun cubed. In equation form, this is

\frac{T_{1}^{2}}{T_{2}^{2}} = \frac{r_{1}^{3}}{r_{2}^{3}},

where T is the period (time for one orbit) and r is the average distance (also called orbital radius). This equation is valid only for comparing two small masses orbiting a single large mass. Most importantly, this is only a descriptive equation; it gives no information about the cause of the equality.

History: Ptolemy vs. Copernicus

Before the discoveries of Kepler, Copernicus, Galileo, Newton, and others, the solar system was thought to revolve around Earth as shown in Figure_07_01_solar (a). This is called the Ptolemaic model, named for the Greek philosopher Ptolemy who lived in the second century AD. The Ptolemaic model is characterized by a list of facts for the motions of planets, with no explanation of cause and effect. There tended to be a different rule for each heavenly body and a general lack of simplicity.

Figure_07_01_solar (b) represents the modern or Copernican model. In this model, a small set of rules and a single underlying force explain not only all planetary motion in the solar system, but also all other situations involving gravity. The breadth and simplicity of the laws of physics are compelling.

View original figure (external; image not reproduced in this text edition)

Original figure description: Image a shows the Ptolemaic view of the solar system. This view shows Earth at the center of the solar system with all other planets orbiting clockwise, the moon circling the Earth, Mercury and Venus which go in wide circles as it orbits, which are called Epicycles. The Sun orbits earth between Venus and Mars. Mars circles opposite of the others, its own path shown with a looping arrow. Jupiter and Saturn come next. On the outside of the solar system are stars. Image b shows the Copernican view of the solar system which is the more traditional view of the solar system with the Sun in the center and the other planets orbiting the sun.

(a) The Ptolemaic model of the universe has Earth at the center with the moon, the planets, the sun, and the stars revolving about it in complex circular paths. This geocentric (Earth-centered) model, which can be made progressively more accurate by adding more circles, is purely descriptive, containing no hints about the causes of these motions. (b) The Copernican heliocentric (sun-centered) model is a simpler and more accurate model.

Nicolaus Copernicus (1473–1543) first had the idea that the planets circle the sun, in about 1514. It took him almost 20 years to work out the mathematical details for his model. He waited another 10 years or so to publish his work. It is thought he hesitated because he was afraid people would make fun of his theory. Actually, the reaction of many people was more one of fear and anger. Many people felt the Copernican model threatened their basic belief system. About 100 years later, the astronomer Galileo was put under house arrest for providing evidence that planets, including Earth, orbited the sun. In all, it took almost 300 years for everyone to admit that Copernicus had been right all along.

Explain why Earth does actually appear to be the center of the solar system.

Earth appears to be the center of the solar system because Earth is at the center of the universe, and everything revolves around it in a circular orbit.
Earth appears to be the center of the solar system because, in the reference frame of Earth, the sun, moon, and planets all appear to move across the sky as if they were circling Earth.
Earth appears to be at the center of the solar system because Earth is at the center of the solar system and all the heavenly bodies revolve around it.
Earth appears to be at the center of the solar system because Earth is located at one of the foci of the elliptical orbit of the sun, moon, and other planets.

Calculations Related to Kepler’s Laws of Planetary Motion

Kepler’s First Law

Refer back to Figure_07_01_circle (a). Notice which distances are constant. The foci are fixed, so distance $\bar{f_{1} f_{2}}$ is a constant. The definition of an ellipse states that the sum of the distances $\bar{f_{1} m} + \bar{m f_{2}}$ is also constant. These two facts taken together mean that the perimeter of triangle $Δ f_{1} m f_{2}$ must also be constant. Knowledge of these constants will help you determine positions and distances of objects in a system that includes one object orbiting another.

Kepler’s Second Law

Refer back to Figure_07_01_planet. The second law says that the segments have equal area and that it takes equal time to sweep through each segment. That is, the time it takes to travel from A to B equals the time it takes to travel from C to D, and so forth. Velocity v equals distance d divided by time t: $v = d / t$ . Then, $t = d / v$ , so distance divided by velocity is also a constant. For example, if we know the average velocity of Earth on June 21 and December 21, we can compare the distance Earth travels on those days.

The degree of elongation of an elliptical orbit is called its eccentricity (e). Eccentricity is calculated by dividing the distance f from the center of an ellipse to one of the foci by half the long axis a.

(e) = f / a

When $e = 0$ , the ellipse is a circle.

The area of an ellipse is given by $A = π a b$ , where b is half the short axis. If you know the axes of Earth’s orbit and the area Earth sweeps out in a given period of time, you can calculate the fraction of the year that has elapsed.

Kepler’s First Law

At its closest approach, a moon comes within 200,000 km of the planet it orbits. At that point, the moon is 300,000 km from the other focus of its orbit, f₂. The planet is focus f₁ of the moon’s elliptical orbit. How far is the moon from the planet when it is 260,000 km from f₂?

Solution

Perimeter of $f_{1} m f_{2} = 200, 000 km + 100,000 km + 300,000 km = 600,000 km.$

$m f_{1} = 600,000 km - (100,000 km + 260,000 km) = 240,000 km.$

Discussion

The perimeter of triangle f₁mf₂ must be constant because the distance between the foci does not change and Kepler’s first law says the orbit is an ellipse. For any ellipse, the sum of the two sides of the triangle, which are f₁m and mf₂, is constant.

Kepler’s Second Law

Figure_07_01_ellipse shows the major and minor axes of an ellipse. The semi-major and semi-minor axes are half of these, respectively.

View original figure (external; image not reproduced in this text edition)

Original figure description: A drawing shows as ellipse, divided into quadrants. The horizontal line is labeled major axis; the vertical line is labeled minor axis. A focus point is marked at each end of the major axis.

The major axis is the length of the ellipse, and the minor axis is the width of the ellipse. The semi-major axis is half the major axis, and the semi-minor axis is half the minor axis.

Earth’s orbit is very slightly elliptical, with a semi-major axis of 1.49598 × 10⁸ km and a semi-minor axis of 1.49577 × 10⁸ km. If Earth’s period is 365.26 days, what area does an Earth-to-sun line sweep past in one day?

Solution

\begin{matrix} area per day & = & \frac{total area}{total number of days} \\ = & \frac{π a b}{365 d} \\ = & \frac{π (1.496 \times 10^{8} km) (1.496 \times 10^{3} km)}{365 d} \\ = & 1.93 \times 10^{14} {km}^{2} / d \end{matrix}

The area swept out in one day is thus $1.93 \times 10^{14} {km}^{2}$ .

Discussion

The answer is based on Kepler’s law, which states that a line from a planet to the sun sweeps out equal areas in equal times.

Kepler’s Third Law

Kepler’s third law states that the ratio of the squares of the periods of any two planets (T₁, T₂) is equal to the ratio of the cubes of their average orbital distance from the sun (r₁, r₂). Mathematically, this is represented by

\frac{T_{1}^{2}}{T_{2}^{2}} = \frac{r_{1}^{3}}{r_{2}^{3}} .

From this equation, it follows that the ratio r³/T² is the same for all planets in the solar system. Later we will see how the work of Newton leads to a value for this constant.

Kepler’s Third Law

Given that the moon orbits Earth each 27.3 days and that it is an average distance of $3.84 \times 10^{8} m$ from the center of Earth, calculate the period of an artificial satellite orbiting at an average altitude of 1,500 km above Earth’s surface.

Strategy

The period, or time for one orbit, is related to the radius of the orbit by Kepler’s third law, given in mathematical form by $\frac{T_{1}^{2}}{T_{2}^{2}} = \frac{r_{1}^{3}}{r_{2}^{3}}$ . Let us use the subscript 1 for the moon and the subscript 2 for the satellite. We are asked to find T₂. The given information tells us that the orbital radius of the moon is $r_{1} = 3.84 \times 10^{8} m$ , and that the period of the moon is $T_{1} = 27.3 days$ . The height of the artificial satellite above Earth’s surface is given, so to get the distance r₂ from the center of Earth we must add the height to the radius of Earth (6380 km). This gives $r_{2} = 1500 km + 6380 km = 7880 km$ . Now all quantities are known, so T₂ can be found.

Solution

To solve for T₂, we cross-multiply and take the square root, yielding

\begin{matrix} T_{2}^{2} = T_{1}^{2} {(\frac{r_{2}}{r_{1}})}^{3}; T_{2} = T_{1} {(\frac{r_{2}}{r_{1}})}^{\frac{3}{2}} \\ \begin{matrix} T_{2} = (27.3 d) (\frac{24.0 h}{d}) {(\frac{7880 km}{3.84 \times 10^{5} km})}^{\frac{3}{2}} = 1.93 h . \end{matrix} \end{matrix}

Discussion

This is a reasonable period for a satellite in a fairly low orbit. It is interesting that any satellite at this altitude will complete one orbit in the same amount of time.

Practice Problems

A planet with no axial tilt is located in another solar system. It circles its sun in a very elliptical orbit so that the temperature varies greatly throughout the year. If the year there has 612 days and the inhabitants celebrate the coldest day on day 1 of their calendar, when is the warmest day?

Day 1
Day 153
Day 306
Day 459

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Check Your Understanding

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True or false—According to Keplers laws of planetary motion, a satellite increases its speed as it approaches its parent body and decreases its speed as it moves away from the parent body.

True
False

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Section Summary

All satellites follow elliptical orbits.
The line from the satellite to the parent body sweeps out equal areas in equal time.
The radius cubed divided by the period squared is a constant for all satellites orbiting the same parent body.

Key Equations

Kepler’s third law	$\frac{T_{1}^{2}}{T_{2}^{2}} = \frac{r_{1}^{3}}{r_{2}^{3}}$
eccentricity	$e = \frac{f}{a}$
area of an ellipse	$A = π a b$
semi-major axis of an ellipse	$a = (r_{a} + r_{p}) / 2$
semi-minor axis of an ellipse	$b = \sqrt{r_{a} r_{p}}$

Concept Items

A circle is a special case of an ellipse. Explain how a circle is different from other ellipses.

The foci of a circle are at the same point and are located at the center of the circle.
The foci of a circle are at the same point and are located at the circumference of the circle.
The foci of a circle are at the same point and are located outside of the circle.
The foci of a circle are at the same point and are located anywhere on the diameter, except on its midpoint.

Comets have very elongated elliptical orbits with the sun at one focus. Using Kepler's Law, explain why a comet travels much faster near the sun than it does at the other end of the orbit.

Because the satellite sweeps out equal areas in equal times
Because the satellite sweeps out unequal areas in equal times
Because the satellite is at the other focus of the ellipse
Because the square of the period of the satellite is proportional to the cube of its average distance from the sun

True or False—A planet-satellite system must be isolated from other massive objects to follow Keplers laws of planetary motion.

True
False

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Critical Thinking

In the figure, the time it takes for the planet to go from A to B, C to D, and E to F is the same.

View original figure (external; image not reproduced in this text edition)

Compare the areas A₁, A₂, and A₃ in terms of size.

A₁ ≠ A₂ ≠ A₃
A₁ = A₂ = A₃
A₁ = A₂ > A₃
A₁ > A₂ = A₃

A moon orbits a planet in an elliptical orbit. The foci of the ellipse are 50, 000 km apart. The closest approach of the moon to the planet is 400, 000 km. What is the length of the major axis of the orbit?

400, 000 km
450, 000, km
800, 000 km
850, 000 km

In this figure, if f₁ represents the parent body, which set of statements holds true?

View original figure (external; image not reproduced in this text edition)

Original figure description: The image shows an oval with small irregular pie shapes shaded in. The left smaller pie shape is labeled X and the right skinny pie shape is labeled Y.

Area X < Area Y; the speed is greater for area X.
Area X > Area Y; the speed is greater for area Y.
Area X = Area Y; the speed is greater for area X.
Area X = Area Y; the speed is greater for area Y.

Problems

The closest Earth comes to the sun is 1.47×10⁸ km, and Earths farthest distance from the sun is 1.52×10⁸ km. What is the area inside Earths orbit?

2.23×10¹⁶ km²
6.79×10¹⁶ km²
7.02×10¹⁶ km²
7.26×10¹⁶ km²

Earth is 1.496×10⁸ km from the sun, and Neptune is 4.490×10⁹ km from the sun. What best represents the number of Earth years it takes for Neptune to complete one orbit around the sun?

10 years
30 years
160 years
900 years

Test Prep Multiple Choice

A planet of mass m circles a sun of mass M. Which distance changes throughout the planet’s orbit?

$\bar{f_{1} f_{2}}$
$\bar{m M}$
$\bar{M f_{2}}$
$\bar{M f_{1}}$

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An artificial satellite orbits the Earth at a distance of 1.45×10⁴ km from Earths center. The moon orbits the Earth at a distance of 3.84×10⁵ km once every 27.3 days. How long does it take the satellite to orbit the Earth?

0.200 days
3.07 days
243 days
3721 days

Earth is 1.496×10⁸ km from the sun, and Venus is 1.08×10⁸ km from the sun. One day on Venus is 243 Earth days long. What best represents the number of Venusian days in a Venusian year?

0.78 days
0.92 days
1.08 days
1.21 days

Test Prep Short Answer

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Test Prep Extended Response

The orbit of Halley's Comet has an eccentricity of 0.967 and stretches to the edge of the solar system.
Part A. Describe the shape of the comets orbit.
Part B. Compare the distance traveled per day when it is near the sun to the distance traveled per day when it is at the edge of the solar system.
Part C. Describe variations in the comet's speed as it completes an orbit. Explain the variations in terms of Kepler's second law of planetary motion.

Part A. The orbit is circular, with the sun at the center. Part B. The comet travels much farther when it is near the sun than when it is at the edge of the solar system. Part C. The comet decelerates as it approaches the sun and accelerates as it leaves the sun.
Part A. The orbit is circular, with the sun at the center. Part B. The comet travels much farther when it is near the sun than when it is at the edge of the solar system. Part C. The comet accelerates as it approaches the sun and decelerates as it leaves the sun.
Part A. The orbit is very elongated, with the sun near one end. Part B. The comet travels much farther when it is near the sun than when it is at the edge of the solar system. Part C. The comet decelerates as it approaches the sun and accelerates as it moves away from the sun.

For convenience, astronomers often use astronomical units (AU) to measure distances within the solar system. One AU equals the average distance from Earth to the sun. Halleys Comet returns once every 75.3 years. What is the average radius of the orbit of Halleys Comet in AU?

0.002 AU
0.056 AU
17.8 AU
653 AU

Glossary

aphelion: the point farthest from the sun in the path of an orbiting planet (called apoapsis for other celestial bodies)
Copernican model: the model of the solar system where the sun is at the center of the solar system and all the planets orbit around it; this is also called the heliocentric model
eccentricity: a measure of the separation of the foci of an ellipse
Kepler’s laws of planetary motion: three laws derived by Johannes Kepler that describe the properties of all orbiting satellites
perihelion: closest distance between a planet and the sun (called periapsis for other celestial bodies)
Ptolemaic model: the model of the solar system where Earth is at the center of the solar system and the sun and all the planets orbit around it; this is also called the geocentric model

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