Source: http://www.physicsclassroom.com/class/circles/Lesson-4/Kepler-s-Three-Laws
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The Physics Classroom » Physics Tutorial » Circular Motion and Satellite Motion » Kepler's Three Laws
Circular Motion and Satellite Motion - Lesson 4 - Planetary and Satellite Motion
Kepler's Three Laws
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Kepler's Three Laws
Circular Motion Principles for Satellites
Mathematics of Satellite Motion
Weightlessness in Orbit
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Energy Relationships for Satellites
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 can be described as follows:
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The path of the planets about the sun is elliptical in shape, with the center of the sun being
located at one focus. (The Law of Ellipses)
An imaginary line drawn from the center of the sun to the center of the planet will sweep
out equal areas in equal intervals of time. (The Law of Equal Areas)
The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes
of their average distances from the sun. (The Law of Harmonies)
The Law of Ellipses
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. An ellipse can easily
be constructed using a pencil, two tacks, a string, a sheet of
paper and a piece of cardboard. Tack the sheet of paper to
the cardboard using the two tacks. Then tie the string into a
loop and wrap the loop around the two tacks. Take your
pencil and pull the string until the pencil and two tacks make
a triangle (see diagram at the right). Then begin to trace out
a path with the pencil, keeping the string wrapped tightly
around the tacks. The resulting shape will be an ellipse. An
ellipse is a special curve in which the sum of the distances
from every point on the curve to two other points is a constant. The two other points
(represented here by the tack locations) are known as the foci of the ellipse. The closer
together that these points are, the more closely that the ellipse resembles the shape of
a circle. In fact, a circle is the special case of an ellipse in which the two foci are at the
same location. Kepler's first law is rather simple - all planets orbit the sun in a path that
resembles an ellipse, with the sun being located at one of the foci of that ellipse.
The Law of Equal Areas
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 31day 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.
The Law of Harmonies
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. 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)
Average
Distance (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
Observe that the T2/R3 ratio is the same for Earth as it is for mars. In fact, if the
same T2/R3 ratio is computed for the other planets, it can be found that this ratio is
nearly the same value for all the planets (see table below). Amazingly, every planet has
the same T2/R3 ratio.
Planet
Period
Average
T2/R3
(yr)
Distance (au)
(yr2/au3)
Mercury
0.241
0.39
0.98
Venus
.615
0.72
1.01
Earth
1.00
1.00
1.00
Mars
1.88
1.52
1.01
Jupiter
11.8
5.20
0.99
Saturn
29.5
9.54
1.00
Uranus
84.0
19.18
1.00
Neptune
165
30.06
1.00
Pluto
248
39.44
1.00
(NOTE: The average distance value is given in astronomical units where 1 a.u. is equal to the
distance from the earth to the sun - 1.4957 x 1011 m. The orbital period is given in units of
earth-years where 1 earth year is the time required for the earth to orbit the sun - 3.156 x
107 seconds. )
Kepler's third law provides an accurate description of the period and distance for a
planet's orbits about the sun. Additionally, the same law that describes the T2/R3 ratio
for the planets' orbits about the sun also accurately describes the T2/R3 ratio for any
satellite (whether a moon or a man-made satellite) about any planet. There is
something much deeper to be found in this T2/R3 ratio - something that must relate to
basic fundamental principles of motion. In the next part of Lesson 4, these principles
will be investigated as we draw a connection between the circular motion principles
discussed in Lesson 1 and the motion of a satellite.
How did Newton Extend His Notion of Gravity to Explain
Planetary Motion?
Newton's comparison of the acceleration of the moon to the acceleration of objects on
earth allowed him to establish that the moon is held in a circular orbit by the force of
gravity - a force that is inversely dependent upon the distance between the two objects'
centers. Establishing gravity as the cause of the moon's orbit does not necessarily
establish that gravity is the cause of the planet's orbits. How then did Newton provide
credible evidence that the force of gravity is meets the centripetal force requirement for
the elliptical motion of planets?
Recall from earlier in Lesson 3 that Johannes Kepler proposed three laws of planetary
motion. His Law of Harmonies suggested that the ratio of the period of orbit squared
(T2) to the mean radius of orbit cubed (R3) is the same value k for all the planets that
orbit the sun. Known data for the orbiting planets suggested the following average
ratio:
k = 2.97 x 10-19 s2/m3 = (T2)/(R3)
Newton was able to combine the law of universal gravitation with circular motion
principles to show that if the force of gravity provides the centripetal force for the
planets' nearly circular orbits, then a value of 2.97 x 10-19 s2/m3 could be predicted for
the T2/R3 ratio. Here is the reasoning employed by Newton:
Consider a planet with mass Mplanet to orbit in nearly circular motion about the sun of
mass MSun. The net centripetal force acting upon this orbiting planet is given by the
relationship
Fnet = (Mplanet * v2) / R
This net centripetal force is the result of the gravitational force that attracts the planet
towards the sun, and can be represented as
Fgrav = (G* Mplanet * MSun ) / R2
Since Fgrav = Fnet, the above expressions for centripetal force and gravitational force are
equal. Thus,
(Mplanet * v2) / R = (G* Mplanet * MSun ) / R2
Since the velocity of an object in nearly circular orbit can be approximated as v =
(2*pi*R) / T,
v2 = (4 * pi2 * R2) / T2
Substitution of the expression for v2 into the equation above yields,
(Mplanet * 4 * pi2 * R2) / (R • T2) = (G* Mplanet * MSun ) / R2
By cross-multiplication and simplification, the equation can be transformed into
T2 / R3 = (Mplanet * 4 * pi2) / (G* Mplanet * MSun )
The mass of the planet can then be canceled from the numerator and the denominator
of the equation's right-side, yielding
T2 / R3 = (4 * pi2) / (G * MSun )
The right side of the above equation will be the same value for every planet regardless
of the planet's mass. Subsequently, it is reasonable that the T2/R3 ratio would be the
same value for all planets if the force that holds the planets in their orbits is the force of
gravity. Newton's universal law of gravitation predicts results that were consistent with
known planetary data and provided a theoretical explanation for Kepler's Law of
Harmonies.
Investigate!
Scientists know much more about the planets than they did in Kepler's days. Use The
Planets widget bleow to explore what is known of the various planets.
The Planets
Select a planet from the pull-down menu to retrieve information about it.
Retrieve Information
Check Your Understanding
1. Our understanding of the elliptical motion of planets about the Sun spanned several
years and included contributions from many scientists.
a. Which scientist is credited with the collection of the data necessary to support the
planet's elliptical motion?
b. Which scientist is credited with the long and difficult task of analyzing the data?
c. Which scientist is credited with the accurate explanation of the data?
See Answer
2. 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.
See Answer
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 1011 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
See Answer
4. 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.
See Answer
Orbital radius and orbital period data for the four biggest moons of Jupiter are listed in
the table below. The mass of the planet Jupiter is 1.9 x 1027 kg. Base your answers to
the next five questions on this information.
Jupiter's Moon
Period (s)
Radius (m)
T2/R3
Io
1.53 x 105
4.2 x 108
a.
Europa
3.07 x 105
6.7 x 108
b.
Ganymede
6.18 x 105
1.1 x 109
c.
Callisto
1.44 x 106
1.9 x 109
d.
5. Determine the T2/R3 ratio (last column) for Jupiter's moons.
See Answer
6. What pattern do you observe in the last column of data? Which law of Kepler's does
this seem to support?
See Answer
7. Use the graphing capabilities of your TI calculator to plot T2 vs. R3 (T2 should be
plotted along the vertical axis) and to determine the equation of the line. Write the
equation in slope-intercept form below.
See Answer
See graph below.
8. How does the T2/R3 ratio for Jupiter (as shown in the last column of the data table)
compare to the T2/R3 ratio found in #7 (i.e., the slope of the line)?
See Answer
2
3
9. How does the T /R ratio for Jupiter (as shown in the last column of the data table)
compare to the T2/R3 ratio found using the following equation? (G=6.67x1011
N*m2/kg2 and MJupiter = 1.9 x 1027 kg)
T2 / R3 = (4 * pi2) / (G * MJupiter )
See Answer
Graph for Question #6
Return to Question #6
Next Section:
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Circular Motion Principles for Satellites
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