Lesson 4: Planetary and
Satellite Motion
Kepler's Three Laws
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 which 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:
- The path of the planets about the sun are 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)
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 which 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 which resembles an
ellipse, with the sun being located at one of the foci of
that ellipse.
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 will be constantly
changing. A planet moves fastest when it is closest to the
sun and slowest when it is furthest from the sun. Yet, if a
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 that line were drawn from
the earth to the sun, then the area swept out by the line in
every month would be the same. This is depicted in the
diagram below. As can be noted 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
(0.5*base*height) are the same size. Since the base
of these triangles are longer when the earth is furthest
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.
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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 the first and second laws, which 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
Dist.
(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
will be found that this
ratio shows approximate agreement with the same
value for both earth and mars (see table below). Amazingly,
every planet has the same
T2/R3
ratio.
Planet
|
Period
(yr)
|
Ave.
Dist.
(au)
|
T2/R3
(yr2/au3)
|
Mercury
|
.241
|
.39
|
0.98
|
Venus
|
.615
|
.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 which 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 ratio - something which 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 object's on earth
allowed him to establish that the moon
is held in a circular orbit by the force of
gravity - a force which 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 which 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 which 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 * R) / T2 = (G*
Mplanet * MSun ) /
R2
By cross-multiplication, 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 which holds the planets in their orbits
is the force of gravity. Newton's universal law of
gravitation predicts results which were consistent
with known planetary data and provided a
theoretical explanation for Kepler's Law of
Harmonies.
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Check
Your Understanding
1. Today it is widely believed that
the planets travel in elliptical paths around the sun.
- Who gathered the data
to support this fact?
- Who analyzed the data to
support this fact?
- Who provided the accurate
explanation for why this occurs?
2. Galileo discovered four moons of
Jupiter. One moon - Io - which he measured to be 4.2 units
from the center of Jupiter and had an orbital period of 1.8
days. Galileo measured the radius of Ganymede to be 10.7
units from the center of Jupiter. Use Kepler's third law to
predict the orbital period of Ganymede.
3. If a small planet were located
eight times as far from the sun as the Earth's distance from
the sun (1.5 x 1011 m), how many years would it
take the planet to orbit the sun. GIVEN:
T2/R3 = 2.97 x 10-19
s2/m3
4. On average, the planet Mars is 1.52 times further from
the sun as is Earth. Given that the Earth orbits the sun in
approximately 365 earth days, predict the time required for
Mars to orbit the sun.
The information below represents the R-T data for the
four biggest moons of Jupiter. The mass of the planet
Jupiter is 1.9 x 1027 kg. Use this information to
answer the next five questions.
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. Fill in the last column of the data table.
6. What do you notice about the values in the last
column? What law is this?
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 graph below.
8. How does the
T2/R3
ratios 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)?
9. 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 using the following equation?
(G=6.67x10-11 N*m2/kg2 and
MJupiter = 1.9 x 1027 kg)
T2 / R3
= (4 * pi2) / (G * MJupiter
)
Graph for question #6
![graph]()
Return to Question #6
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