Chapter 7 Section 3 : Motion in space

Objectives

• Describe Kepler’s laws of planetary motion.

• Relate Newton’s mathematical analysis of gravitational force to the elliptical planetary orbits proposed by Kepler.

• Solve problems involving orbital speed and period.

Planetary movement

• People have studied the motions of the planets since ancient times. Until the middle of the 16th century, most people believed that earth was at the center of the universe. Originally, it was believe that the sun and other planets orbited earth in perfect circles. However these model did not account for all of the observations of planetary motion.

• In 1543, the polish astronomer Nicolaus Copernicus (1473-1543) published the book On the revolutions of the heavenly spheres , in which he proposed that earth and other planets orbit the sun in perfect circles.

Kepler’s laws

• The astronomer Tycho Brahe(1546-1601) made many precise observations of the planets and stars. However some of Brahe’s data did not agree with the copernical model.

• In the early 1600s, Johannes Kepler proposed three laws of planetary motion. Kepler was able to summarize the carefully collected data of his mentor - Tycho Brahe - with three statements that described the motion of planets in a sun-centered solar system. Kepler's efforts to explain the underlying reasons for such motions are no longer accepted; nonetheless, the actual laws themselves are still considered an accurate description of the motion of any planet and any satellite.

Kepler’s three laws of planetary motion

Kepler’s laws describe the motion of the planets.

• First Law: Each planet travels in an elliptical orbit around the sun, and the sun is at one of the focal points.

• Second Law: An imaginary line drawn from the sun to any planet sweeps out equal areas in equal time intervals.

• Third Law: The square of a planet’s orbital period (T2) is proportional to the cube of the average distance (r3) between the planet and the sun.

Kepler’s first law

• Kepler's first law - sometimes referred to as the law of ellipses - explains that planets are orbiting the sun in a path described as an ellipse.

Kepler’s second law

• Kepler's second law - sometimes referred to as the law of equal areas - describes the speed at which any given planet will move while orbiting the sun. The speed at which any planet moves through space is constantly changing. A planet moves fastest when it is closest to the sun and slowest when it is furthest from the sun. Yet, if an imaginary line were drawn from the center of the planet to the center of the sun, that line would sweep out the same area in equal periods of time. For instance, if an imaginary line were drawn from the earth to the sun, then the area swept out by the line in every 31-day month would be the same. This is depicted in the diagram below. As can be observed in the diagram, the areas formed when the earth is closest to the sun can be approximated as a wide but short triangle; whereas the areas formed when the earth is farthest from the sun can be approximated as a narrow but long triangle. These areas are the same size. Since the base of these triangles are shortest when the earth is farthest from the sun, the earth would have to be moving more slowly in order for this imaginary area to be the same size as when the earth is closest to the sun.

Kepler’s second law

• According to Kepler’s second law, if the time a planet takes to travel the arc on the left (∆t1) is equal to the time the planet takes to cover the arc on the right (∆t2), then the area A1 is equal to the area A2.

Kepler’s third law

• Kepler's third law - sometimes referred to as the law of harmonies - compares the orbital period and radius of orbit of a planet to those of other planets. Unlike Kepler's first and second laws that describe the motion characteristics of a single planet, the third law makes a comparison between the motion characteristics of different planets. The comparison being made is that the ratio of the squares of the periods to the cubes of their average distances from the sun is the same for every one of the planets.

Kepler’s third law

As an illustration, consider the orbital period and average distance from sun (orbital radius) for Earth and mars as given in the table below.

Planet

Period

(s)

AverageDist. (m)

T2/R3

(s2/m3)

Earth 3.156 x 107

s

1.4957 x 1011

2.977 x 10-19

Mars 5.93 x 107 s

2.278 x 1011

2.975 x 10-19

 

Kepler’s third law

Kepler’s third law

• Kepler’s third law leads to an equation for the period of an object in a circular orbit. The speed of an object in a circular orbit depends on the same factors:

T 2r3

Gm vt G

m

r

•Note that m is the mass of the central object that is being orbited. The mass of the planet or satellite that is in orbit does not affect its speed or period. •The mean radius (r) is the distance between the centers of the two bodies.

Example 1

Galileo is often credited with the early discovery of four of Jupiter's many moons. The moons orbiting Jupiter follow the same laws of motion as the planets orbiting the sun. One of the moons is called Io - its distance from Jupiter's center is 4.2 units and it orbits Jupiter in 1.8 Earth-days. Another moon is called Ganymede; it is 10.7 units from Jupiter's center. Make a prediction of the period of Ganymede using Kepler's law of harmonies.   

solution

• Given:

• Io: Rio = 4.2 and Tio = 1.8

• Ganymede: Rg = 10.7 Tg=???

• Use Kepler's 3rd law to solve.

• (Tio)^2/(Rio)3 = 0.04373;

• so (Tg)^2 / (Rg)3 = 0.04373

• Proper algebra would yield (Tg)^2 = 0.04373 • (Rg)3

• (Tg)2 = 53.57 so Tg = SQRT(53.57) = 7.32 days

• Answer: T = 7.32 days

EXAMPLE 2

Period and Speed of an Orbiting Object

Magellan was the first planetary spacecraft to be launched from a space shuttle. During the spacecraft’s fifth orbit around Venus, Magellan traveled at a mean altitude of 361km. If the orbit had been circular, what would Magellan’s period and speed have been?

Example 3

Suppose a small planet is discovered that is 14 times as far from the sun as the Earth's distance is from the sun (1.5 x 10 11 m). Use Kepler's law of harmonies to predict the orbital period of such a planet. GIVEN: T2/R3 = 2.97 x 10-19 s2/m3.

 

solution

Use Kepler's third law:(Te)^2/(Re)^3 = (Tp)^2/(Rp)

3

Rearranging to solve for Tp:

(Tp)^2=[ (Te)2 / (Re)

3] • (Rp)3

or (Tp)2 = (Te)

2 • [(Rp) / (Re)]3 where (Rp) / (Re) = 14

so (Tp)2 = (Te)

2 • [14]3 where Te=1 yr

(Tp)2 =(1 yr)2 *[14]^3 = 2744 yr2

Tp = SQRT(2744 yr2)

Answer: Tplanet = 52.4 yr 

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Student guided practice

• The average orbital distance of Mars is 1.52 times the average orbital distance of the Earth. Knowing that the Earth orbits the sun in approximately 365 days, use Kepler's law of harmonies to predict the time for Mars to orbit the sun.

•  

solution

Given: Rmars = 1.52 • Rearth and Tearth = 365 days

Use Kepler's third law to relate the ratio of the period squared to the ratio of radius cubed(Tmars)

2 / (Tearth)2 • (Rmars)

3 / (Rearth)3

(Tmars)2 = (Tearth)

2 • (Rmars)3 / (Rearth)

3

(Tmars)2 = (365 days)2 * (1.52)3

(Note the Rmars / Rearth ratio is 1.52)

Tmars = 684 days

Weight and weightlessness

• Weightlessness in Orbit

• Astronauts who are orbiting the Earth often experience sensations of weightlessness. These sensations experienced by orbiting astronauts are the same sensations experienced by anyone who has been temporarily suspended above the seat on an amusement park ride. Not only are the sensations the same (for astronauts and roller coaster riders), but the causes of those sensations of weightlessness are also the same. Unfortunately however, many people have difficulty understanding the causes of weightlessness.

Testing understanding

Test your preconceived notions about weightlessness:

Astronauts on the orbiting space station are weightless because...a. there is no gravity in space and they do not weigh anything. b. space is a vacuum and there is no gravity in a vacuum.c. space is a vacuum and there is no air resistance in a vacuum.d. the astronauts are far from Earth's surface at a location where gravitation has a minimal affect.

Answer

• None of these are appropriate reasons for the weightless sensations experienced by orbiting astronauts. Continue reading this part of Lesson to find out the real reason.

Weightlessness

Weightlessness is simply a sensation experienced by an individual when there are no external objects touching one's body and exerting a push or pull upon it. Weightless sensations exist when all contact forces are removed. These sensations are common to any situation in which you are momentarily (or perpetually) in a state of free fall. When in free fall, the only force acting upon your body is the force of gravity - a non-contact force. Since the force of gravity cannot be felt without any other opposing forces, you would have no sensation of it. You would feel weightless when in a state of free fall.

weightlessness

• These feelings of weightlessness are common at amusement parks for riders of roller coasters and other rides in which riders are momentarily airborne and lifted out of their seats.Suppose that you were lifted in your chair to the top of a very high tower and then your chair was suddenly dropped. As you and your chair fall towards the ground, you both accelerate at the same rate - g. Since the chair is unstable, falling at the same rate as you, it is unable to push upon you.

Weight and weightlessness

• Weightlessness is only a sensation; it is not a reality corresponding to an individual who has lost weight. As you are free falling on a roller coaster ride (or other amusement park ride), you have not momentarily lost your weight. Weightlessness has very little to do with weight and mostly to do with the presence or absence of contact forces. If by "weight" we are referring to the force of gravitational attraction to the Earth, a free-falling person has not "lost their weight;" they are still experiencing the Earth's gravitational attraction. Unfortunately, the confusion of a person's actual weight with one's feeling of weight is the source of many misconceptions.

Weight and weightlessness

Technically speaking, a scale does not measure one's weight. While we use a scale to measure one's weight, the scale reading is actually a measure of the upward force applied by the scale to balance the downward force of gravity acting upon an object. When an object is in a state of equilibrium (either at rest or in motion at constant speed), these two forces are balanced. The upward force of the scale upon the person equals the downward pull of gravity (also known as weight)

Homework

• Do worksheet problems 1through 5 in the kepler’s laws worksheet.

Have a great day!!