Download Unit 3, Prelab Unit 3

UNIT
3
UNIT 3
THE PLANETARY ORBITS
Goals
After mastery of this unit, you should:
a. be able to use the small angle equation.
b. have an understanding of the gravitational force, including the inverse square law.
c. have an understanding of the general structure of the solar system.
d. know the general history of ideas of the structure of the solar system.
e. know Kepler’s laws, Newton’s laws, and Newton’s version of Kepler’s laws.
➤ Text References
1.2; 2.2; 3.1 through 3.4; 5.1 through 5.5; A.1 through A.2; A.4; A.6
See the online Text Guide to DOSC to find the links to the listing by objectives.
➤ Mastery
will be evaluated in two Parts:
Part I will be a 15-question multiple-choice evaluation with 13 correct responses
required.
Part II will be performing the laboratory “Orbits and Positions of the Planets.”
Part I will be based upon the following:
➤ Objectives
You should be able to recognize:
+1. the difference between linear measure and angular measure;
2. that a circle can be divided into 360 degrees, each degree into 60 minutes of arc,
and each minute into 60 seconds of arc, and that the symbols (°, ', '') are used for
degrees, minutes, and seconds, respectively;
3. the result of multiplying and dividing numbers written in powers-of-ten notation;
4. and use the small angle equation (Optional Basic Equation I);
+5. the following about significant figures:
(a) the meaning of the term significant figures
(b) the correct significant figures in a simple calculation
+6. from a list of the planets (or any subset of planets) the correct order of increasing or
decreasing distance from the Sun;
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+7. the symbols commonly used to denote the Sun and the planets;
+8. from a list a planet’s approximate:
(a) distance from the Sun in AU
(b) radius relative to the Earth’s radius
9. a description of:
(a) Ptolemaic model and problems with that model
(b) Occam’s razor
10. the names of the five key figures in the Copernican Revolution and the major
contributions of each;
+11. the definition of an ellipse and a description of an ellipse’s eccentricity. Given a
drawing of an ellipse, you should be able to label the semi-major and semi-minor
axes;
+12. by choosing an ellipse with the largest or smallest eccentricity, when given two or
more ellipses with equal semi-major axes and unequal semi-minor axes;
13. from either words or a sketch, among an ellipse, a parabola, and a hyperbola and
under what conditions of launch from the Earth, which of these three would be the
resulting orbit;
+14. that an ellipse, a parabola, and a hyperbola are conic sections, and why the term
conic section is used;
+15. the meaning of the following dealing with planetary position:
(a) opposition and conjunction
(b) synodic and sidereal period
(c) perihelion and aphelion
(d) inferior and superior planets
16. a description of Kepler’s three laws and to associate each with its number. Also
recognize the name often given to the third law and its equation form;
17. a description and the approximate value of the Astronomical Unit and the proper
abbreviation for the Astronomical Unit;
18. the following for Galileo Galilei:
(a) made basic investigations into physical motions
(b) was among the first to use a telescope for astronomical observations
19. a statement, and associate a number with, Newton’s three laws of motion, examples
of simple applications, and a statement of Newton’s law of gravitation;
20. a comparison of Kepler’s empirical approach to Newton’s deductive approach;
+21. the following about Newton’s modification of Kepler’s Laws:
(a) a description of the concept of center of mass for a two-body system
(b) that, using his three laws of motion and the gravitational force law, Newton
could deduce modified versions of Kepler’s three laws
(c) that Newton modified Kepler’s first law, putting the center of mass of the Sunplanet system at the focus of the ellipse and allowed parabolic and hyperbolic
orbits
(d) that the second law of Kepler was unchanged by Newton
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THE PLANETARY ORBITS
UNIT
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(e) that the third law of Kepler was expanded by Newton to include the sum of the
masses of the two gravitating bodies and that a proper choice of units eliminates the constants and the appropriate choice of units for the solar system and
for the Earth-Moon system
(f) the equation form of the third law as modified by Newton, and be able to use it
with some simple numbers
22. The evidence that the heliocentric theory is correct, i.e., the Earth orbits the Sun.
➤ Streaming Videos — Unit 3, Prelab Unit 3
— Meeks videos “The Motions of Attracting Bodies” and
“Planetary Motion and Kepler’s Laws”
NOTES–If there are no notes, all the information is in the relevant sections of the text.
Obj. 1. Linear measure is the straight-line distance between two points; it is commonly
measured in meters or kilometers. Another example of a linear measure is the length of
an object. Angular measure is a measure of the difference in direction between two points
as seen from a specified third point. Angular measure is commonly specified in degrees,
minutes, and seconds of arc.
Obj. 2. 60 seconds of arc (60'') = 1 minute of arc (1'); 60 minutes of arc (60') = one
degree (1°)
60' = 21,600', 360°. ______
3600" = 1,296,000''
1 circle = 360° = 360° × ___
1°
1°
Obj. 3.
Multiplying and dividing powers of ten is relatively simple if we follow five rules.
Rule
1.
Process
Pose problem
Example
(6 × 105) × (9 × 10− 2)
__________________
Collect all numbers together and all powers of
ten terms together.
6×9×
_____
4
4 × 102
(105 × 10− 2)
__________
102
−2
(10 × 10 )
__________
5
(13.5) ×
2.
Multiply and divide the numbers.
3.
If powers of ten are multiplied, add exponents.
10
10
(13.5) × _____
13.5 × ___
102
102
4.
If powers of ten are divided, subtract exponents.
(numerator exponent minus denominator
exponent).
10 = 13.5 × 103 – 2 = 13.5 × 10
13.5 × ___
2
10
5.
Use the powers of ten rules to put the first
number into a form between 1 and 10.
5−2
2
10
3
3
(13.5) × 10 = (1.35 × 10) × 10 = 1.35 × 102
Obj. 4. The small angle equation is discussed in Section 3.3 of DOSC (the online text).
For the form we will use, the angular size must be expressed in seconds of arc, and the
diameter and distance both must be in the same units. Whenever d is much less than D (we
write this as d << D) the results are the arc S which is nearly d in Fig. SG 3-1. For larger
angles a measures S which is larger than d. This difference can be corrected.
Figure SG 3-1 S is the arc length while d is the straight line.
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For this objective, you will also need the skills of rearranging equations discussed in Unit 1.
In other words, with powers of ten notation, the equations can be written in three forms:
d
α " = 2.06 × 105 × __
Equation (1)
D
α" × D
d = __________
Equation (2)
2.06 × 105
d
D = 2.06 × 105 × ___
Equation (3)
α"
are all the same, each being solved for a different quantity. To use the small angle equation,
it is necessary to know, or be given, two of the three quantities (α ", d, and D) that appear in
the equation.
If we know d and D and want α ", Equation (1) is the useful form.
If we know α " and D and want d, Equation (2) is the most useful form.
If we know α " and d and want D, Equation (3) is the most useful form.
The sample questions for this objective give further examples. In each case, draw a figure
similar to SGI Fig. 3-1 to help visualize what is desired.
Obj. 5. Physical measurements can never be exact. Usually the last significant figure is
uncertain by 1 or more; this is often expressed by giving the uncertainty. Thus if a wooden
block has a length of 3.52 ± 0.02 cm (read as 3.52 plus or minus 0.02 cm), it probably is at
least 3.52 − 0.02 = 3.50 cm long and probably no longer than 3.52 + 0.02 = 3.54 cm. The
± 0.02 is termed the uncertainty. If no uncertainty is given, all physical quantities in this
material are assumed to be good to ±0.5 in the last significant figure. That is, if I say the
angular size of the Moon is 0°.50, I mean 0°.50° ± 0°.05. The use of significant figures will
be of primary importance in multiplication and division.
The number 3.1 has two significant figures. The number 0.0034 has two significant figures.
The number 345 has three significant figures. The number 2.5 × 107 cm, has two significant
figures; while 2.50 has three significant figures.
Now, some examples using area = length × width:
(a) (3.06 × 103 cm) × (1.2 × 102 cm) = 3.7 × 105 cm2 (2 figures)
(b) (3.05 × 103 cm) × (4.2 × 102 cm) = 1.3 × 106 cm2 (2 figures)
(c) (0.0012 m) × (0.0122 m) = 1.5 × 10–5 m2
–2
(d) (0.0421 m) × (1.034 m) = 4.35 × 10 m
2
(2 figures)
2
(3 figures)
2
The cm is read as square centimeters and the m as square meters. Both denote an area
measurement.
In cases (a)–(c), since only two figures are known for one of the lengths, keep only two
significant figures in the final answer. In case (d), three and four figures are known for the
lengths, so the answer is known to only three figures.
Taking example (b), we can justify this rule as follows: 3.05 × 103 is probably between (3.045
3
2
2
and 3.055) × 10 cm, and 4.2 × 10 cm is probably between (4.15 and 4.25) × 10 cm. The
extremes occur when we multiply 3.045 by 4.15 or 3.055 by 4.25). The former gives 1.26 ×
106 cm; the latter gives 1.29 × 106 cm. Thus the 1.3 × 106 cm2 [which means (1.30 ± 0.05) ×
106 cm2] includes the probable answer.
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THE PLANETARY ORBITS
UNIT
3
Division is treated in the same fashion.
For example:
2
7.9 cm = 2.1,
________
225 cm = 1.9 × 102 cm,
________
3.71 cm
1.2 cm
and
8.52 cm = 2.07
________
4.12 cm
However, if you are asked to find 1/2 of 3.71, this is 1.86 not 2, as the 1/2 is assumed to be
known perfectly. For all evaluation questions dealing with significant figures, the number
of significant figures left is the same as the least number given for any physical quantities
(that is, excluding numerical factors such as 2, which are assumed to be known perfectly).
Note from the above examples that the ratios of numbers measured in the same units have
no units.
The correct significant figures on a calculator are obtained by rounding to the number of
figures the calculator produces using the given rules. For example:
If you have three significant figures and the number calculated is 1.273 give the answer
1.27 but if you have three significant figures and the number calculated is 1.278, give the
answer 1.28. Round off — do not just drop extra numbers.
Obj. 6-8. Questions will be based on the following table:
Planet
Symbol
Distance from Sun (AU)
Radius of Planet in Earth Radii
Mercury
C
2/5
2/5
Venus
D
3/4
1
Earth
O
1
1
Mars
E
3/2
1/2
Jupiter
F
5
11
Saturn
G
10
9
Uranus
or N
20
4
Neptune
I
30
4
In 2006 the IAU (International Astronomical Union) defined the word “planet.”
1. The orbit must be circular enough not to cross the orbit of another planet.
2. A planet must also have enough mass to become nearly spherical.
3. And a planet must be able to sweep other small astronomical objects out of its
orbital path.
Using this definition, Pluto is not a planet. In 2008 objects satisfying 2 & 3 and usually
outside Neptune’s orbit were named Plutoids. Don’t forget that
planet radius _______________
planet diameter
_____________
=
Earth radius
Earth diameter
Obj. 10. Copernicus’ proposal of a heliocentric theory provided the first explanation
of retrograde motion by relative motion of the Earth and planet. Kepler used Brahe’s
observational data to get his empirical laws.
Obj. 11 and 12. There are a number of ways of defining ellipses. The simplest is to take
two points (each point is called a focus) and define an ellipse as the set of all points such
that the sum of the distance from each focus is a constant. This is shown in Fig. SG 3-2.
Two numbers are sufficient to describe an ellipse. Often we use the semi-major axis and the
semi-minor axis. These are shown in Fig. SG 3-3. They are also the quantities labeled “a”
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Figure SG 3-3 Showing semi-minor and semimajor axis of an ellipse.
Figure SG 3-2 Definition of an ellipse.
F + G = F’ + G’ = F” + G” = constant.
The foci are indicated by the *’s.
and “b” in Fig. SG 3-2. That is, “2a” is the major axis and “2b” is the minor axis. Another
way is to use the semi-major axis and the eccentricity, ε. For planetary motion, the semimajor axis is the time average distance of a planet from the Sun.
The eccentricity is a number between 0 and 1 that describes the flatness of the ellipse.
A circle has an eccentricity of 0. As the eccentricity increases, the ellipse becomes flatter.
As the eccentricity approaches one, the ellipse becomes more and more like a parabola.
Obj. 14. Each of the four possible orbits (circle, ellipse, parabola, and hyperbola) may be
generated by passing a plane through a right circular cone (imagine an inverted ice cream
cone). If the plane is exactly perpendicular to the axis, we produce a circle. If the plane is
nearly perpendicular, we produce an ellipse. If the plane is parallel to one edge of the cone,
we produce a parabola. A more steeply inclined plane produces a hyperbola (see Fig. SG 3-5,
on the next page, for two of the four cases).
Obj. 15a. Once we admit that the Earth circles the Sun like the other planets some
new terminology and different interpretation of old terminology become necessary. For
example, in Fig. SG 3-6, if a planet is in position A relative to the Sun, it is said to be in
opposition. If it was in position B the planet is said to be in conjunction.
Obj. 15b. Since the Earth is moving about the Sun we must be careful when we refer
to a period of a planet. An obvious period would be from opposition to opposition. (The
synodic period of the Moon, for example, is the interval between full Moons.) This period,
with respect to the Sun, is called a synodic period. The other period is the sidereal period
that is the interval with respect to the fixed stars. These two periods differ because the
Earth is moving in orbit about the Sun.
Obj. 15c. If the planetary orbits are ellipses with the Sun at one focus, then the distance
between the Sun and a planet varies as the planet moves about the Sun. When the planet
is closest to the Sun, it is said to be at perihelion. When the planet is most distant from the
Sun it is said to be in aphelion.
Obj. 15d. Planets closer to the Sun than the Earth are termed inferior planets, while
planets further from the Sun than the Earth are superior planets.
Obj. 16 and 17.
Summary of Kepler’s three laws:
1. Orbits are ellipses with Sun at one focus.
2. Line joining planet and Sun sweeps out equal areas in equal times.
3. Squares of sidereal periods are proportional to cubes of semi-major axis:
P2 = K a3
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K = universal constant, P the sidereal period, and a the semi-major axis.
THE PLANETARY ORBITS
UNIT
3
Exercise:
Take a length of thread and
pierce it with two pins, one near
each end. (If you don’t need to
know the length of the thread
between the pins, you may knot
the thread about the pins. Poke
two pins vertically into a soft
board (or cardboard) with a pin
separation less than the length of
the thread between the pins. Draw
an ellipse with a pencil by using
the pencil to make the thread taut
against the pins. Move the pencil
on the board, keeping the thread
taut. Remove pencil and start at a
new point to keep the thread from
wrapping around the pins. See Fig.
SG 3-4. Each pin is a focus of the
ellipse and the length of thread
between the pins is the quantity
“A + B” of Fig. SG 3-1. Try again with
the pins closer, then with the pins
farther apart.
Figure SG 3-4 Drawing an ellipse.
Figure SG 3-5 Conic sections.
Figure SG 3-6 Showing opposition (at A) and conjunction (at B).
Define the Astronomical Unit as follows:
1 Astronomical Unit (AU) is the average distance of the Earth from the Sun:
1 AU = 1.5 × 1011 meters
Now make the following choice of units:
time
— Earth years;
distance
— Earth-Sun distances (AU)
then Kepler’s third law for the Earth becomes:
2
3
(1 yr.) = K × (1 AU)
so
(1 year)2
K = ________
(1 AU)3
In these units, K has a numerical value of unity.
Thus we may write
P =a
2
3
if P is sidereal period in years and a is distance in AUs.
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Consider two examples of using Kepler’s third law in these units:
Suppose an object is orbiting the Sun with a 4.0 AU semi-major axis, then
P 2 = a3 = (4)3 = 64 = (8)2,
so
P = 8 years.
Suppose an object is orbiting the Sun with a period of 5.0 years, then
P 2 = a3 = (5)2 = 25 = (2.9)3,
so
a = 2.9 AU.
If the square root or cube root is not an integer, the answers will be spaced so you should
be able to estimate and then choose the correct answer. For the second example, choices
might be 1.0, 2.9, 5.8, 29, and 36.
The three laws describe the following:
1. The shape of a planet’s orbit.
2. How a planet moves in its orbit.
3. The relation of the average speed to the size of the orbit.
If possible, go to a planetarium or use an orrery to help you visualize these laws.
Obj. 20. Kepler’s approach to the problem of planetary motion was empirical. That is; he
had the data on the observed motions of the planets. By a process of systematic trial and
error on one set of data (Mars), he arrived at laws that fit that data. He then checked them
against the data for the other planets.
Newton’s approach, on the other hand, was deductive. He first postulated certain basic
laws of the motion of matter, and from these deduced the laws of planetary behavior.
Newton began with basic ideas of space (length) and time. These were absolute concepts
that were to exist even if matter did not exist.
For our purposes, you can take the intuitive idea of space and time that you probably now
have, also adding the idea of force. (Weight is a good example of force.) This way Newton
arrived at his three laws of motion and the gravitational law. From these he deduced the
modified forms of Kepler’s law.
Obj. 21. For the application of Newton’s laws to planetary motion, we need the idea of
center of mass. For two bodies this is a point located between the two such that the mass of
one body times its distance from the center of mass is equal to the mass of the other body
times its distance from the center of mass. In terms of the diagram in Fig. SG 3-7, the center
of mass is such that if the separation is r = r1 + r2, then
r1 m1 = r2 m2
and
v1 r2 = v2 r1
where v1 and v2 denote the motions of m1 and m2 relative to some observer.
Figure SG 3-7 Showing how center of mass is related to the mass of two bodies and their separation.
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THE PLANETARY ORBITS
UNIT
3
From his laws Newton deduced that the center of mass of an isolated system is not
accelerated (Newton’s first law), so he could best describe the motion of the planets in
terms of motion about center of mass. This leads to Kepler’s laws as modified by Newton.
Newton’s essential changes from Kepler’s laws are as follows:
First Law: Planets and the Sun move in ellipses about a common center of mass.
This center of mass is at one focus. Orbits can be ellipses, parabolas, or
hyperbolas. (See Chapter 4, Objective 14 for a discussion of these conic
sections.) In the solar system, the Sun is so massive that the center of mass
always lies inside the surface of the Sun.
Second Law: Unchanged
Third Law: If two bodies revolve mutually about each other, the sum of their masses
times the square of their sidereal period of mutual revolution is in
proportion to the cube of the semi-major axis of the relative orbit of one
about the other. As an equation:
(M1 + M2)P 2 = A r 3
where A is a universal constant. If we measure mass in units of Solar
masses, P is in years and r is in AUs, we can eliminate A as was done
with Kepler’s version of this law and get:
(M1 + M2)P 2 = r3
Suppose there was a planet of 0.1 Min orbit 1.1 AU from the Sun, then
(1.1)3
r3
P 2 = ________
= ________
so P 2 = (1.1)2, and P = 1.y1
M1 + M2 (1 + 0.1)
Within our solar system only Jupiter with an MJupiter = 0.001 M0 produces an easily
measured motion of the Sun. But when we look outside the solar system, there are many
star systems with more than one star. In those cases the modification of Newton is very
important. We will return to this in Unit 27.
EV tip: Think of Kepler’s three laws as a concept cluster. Do the same for Newton’s three
laws and Newton’s version of Kepler’s three laws. You may ask for scratch paper
to write these as you take your EV.
➤ Concepts from Unit 3 Objectives, Notes, and Text
linear measure
semi-major axis of an ellipse
inferior planets
angular measure
eccentricity of an ellipse
Sun as a star
small angle equation
parabola
16a. solar system
parallax
hyperbola
third law of motion
significant figures
astronomical unit
center of mass
Kepler
planet in opposition
transits of planets
Newton
mass
inverse square law
Galileo
weight
law of gravitation
Copernicus
retrograde motion
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Astronomers, like all scientists, are storytellers in the classical sense. We
weave our observations into an explanatory picture of the universe.
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THE PLANETARY ORBITS
UNIT
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UNIT 3
THE PLANETARY ORBITS
➤Word Practice
1. __________ used an empirical method to discover three laws of planetary motion.
2. Newton’s __________ __________ __________ __________ states that for every action
there is an equal and opposite reaction.
3. The measurement in the difference in direction between two points as seen from a
third point is the __________ measure.
4. __________ used a deductive method to discover three laws of planetary motion.
5. __________ was a Polish astronomer who proposed the heliocentric theory.
6. __________ is a measure of the amount of matter in a body.
7. The Sun is a __________.
8. A satellite launched from Earth with a velocity greater than escape has an orbit that
is a __________.
9. The shift in apparent position of a star, due to the motion of the Earth around the
Sun, is called the __________ of the star.
10. __________ is a force.
11. __________ figures are the number of digits known for certain in quantity.
12. The __________ __________ equation relates the apparent angular size of an object
to the object’s true size and its distance from the observer (if the distance is large).
13. The __________ __________ __________ of the Earth-Sun lies along a line joining
them. It lies very close to the center of the massive Sun.
14. The __________ motion of the planets can most easily be understood by having the
Earth revolve about the Sun.
15. The __________ __________ is comprised mostly of the Sun and its planets.
16. __________ was the first known astronomer to use the telescope.
17. The __________ __________ of an ellipse is half its long dimension.
18. __________ of inferior planets occur when they pass between the Earth and the Sun.
19. The __________ measure is the distance between 2 points.
20. The __________ __________ is the average distance of the Earth from the Sun.
21. Newton proposed the __________ __________ Law for the strength of the
gravitational interaction.
22. Newton’s __________ __________ __________ together with his three laws of motion,
enabled him to predict planetary motion.
23. A planet in __________ rises at sunset.
24. The __________ is a measure of the “flatness” of an ellipse.
25. A spacecraft launched at exactly escape velocity moves along a __________.
26. Planets closer to the Sun than the Earth are called __________ planets.
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