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Astronomy
Date
Dec 6, 2023
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Figure 1.
Figure 2.
Figure 3.
1
Kepler’s Laws of Planetary Motion
Name:
Dasol
Lee
Introduction:
In this lab you investigate Kepler’s Laws of Planetary motion and their consequences.
Learning Goals:
Students will become familiar with Kepler’s laws of planetary motion and what they tell us about planets’
trajectories.
Learning tools:
Excel spreadsheet with planetary motion simulator (provided).
Background
:
Johannes Kepler published three laws of planetary motion, the first
two in 1609 and the third in 1619. The laws were made possible by
planetary data of unprecedented accuracy collected by Tycho Brahe.
The laws were both a radical departure from the astronomical
prejudices of the time and profound tools for predicting planetary
motion with great accuracy. Kepler, however, was not able to describe in a significant way why the laws
worked.
1
st
Law: Law of Ellipses
The orbit of a planet is an ellipse where one
focus of the ellipse is the Sun.
How “elliptical” an orbit is described by
its
eccentricity
(see lab 2 for description of
eccentricity).
2nd Law: Law of Equal Areas
A line joining a planet and the Sun sweeps out
equal areas in equal time intervals.
With elliptical orbits a planet is sometimes closer
to the sun than it is at other times. The point at
which it is closest is called
perihelion
. The point at
which a planet is farthest is called
aphelion
.
Kepler's second law basically says that the planets
Figure 4.
Figure 5.
2
speed is not constant – moving slowest at aphelion and fastest at perihelion. The law allows an astronomer to
calculate the orbital speed of a planet at any point.
3
rd
Law: Law of Harmonies
The square of planet’s orbital period
P
(expressed in years) equals to the cube of its average orbital
distance
a
(or semi-major axis) expressed in astronomical units (AUs).
P
2
=
a
3
This equation is only good for objects that orbit our Sun.
Later on Isaac Newton was able to derive a more
general form of the equation using his Law of Gravitation.
This law tells us that not only planets that are further from the Sun take longer to orbit it, something that could
be anticipated since the planet in a more distant orbit will have a longer trajectory to travel through as its orbit
has a greater circumference, but that planets that are further away actually also move more slowly (see orbital
speed in the worksheet for lab 2).
Isaac Newton was able to explain that: the further a planet is from the Sun,
the weaker the force of gravity that the Sun exerts on it, which results in slower orbital motion of a more
distant object.
Kepler developed his three laws to describe the motion of the planets in our solar system.
When the third law
is derived using Newton’s Laws of Motion and his Universal Law of Gravitation (see appendix A), which allows
to take the mass of the orbited body into account, Kepler’s three laws apply to other objects e.g. moons that
orbit a planet, planets that orbit other stars and etc.
Planetary motion simulator
(in your Excel worksheet for this lab)
Open the Excel worksheet simulator.
The simulator contains a spreadsheet called
Input
and graphs—
Orbit
,
Distance to the Sun
,
and
Orbital Speed
.
Look at the
Input
spreadsheet
contains cells in a table titled “input
quantities” (see figure 5) where you
input the variables that you’ll adjust:
the semi-major axis,
a
, and
eccentricity,
e
, for two planets.
Note that the minimum and
maximum values for semi-major axis
a
and eccentricity
e
are given in parentheses. Do not exceed them,
the simulator is not designed to handle values beyond this range. Two other variables can also be adjusted
(angle between the orbits and inclination), but in this experiment we’ll leave them at their default values What
you will vary are semi-major axes and eccentricities of the two planets.
For Planet 1, the default value of
a
is 1 AU, for Planet 2, the default value of
a
is 1.5 AU.
By default,
eccentricities of both planets are set to zero, thus both planets are in perfectly circular orbits. Click on tab
Orbits and look at the graph. It shows the orbits of the two planets.
Are they in fact circular?
Yes
both orbits
are in fact circular. The orbit of Planet 1 is just a small circle than the orbit of Planet 2.
Figure 6.
3
Return to the Input spreadsheet. Below input quantities, there is a table called “calculated quantities” (see
figure 6).
Based on input
quantities and Kepler Laws, orbital
periods for the two planets are
calculated and displayed there.
In
addition, this table contains
calculated data for mean annual
motion denoted by letter
N
.
Mean annual motion
, measured in degrees, is a quantity that tells you what angle a line joining the planet and
the Sun has swept during one Earth year (365.25 days). For the default setting of 1 AU for Planet 1, the orbital
period
P
is equal to 1 year (that’s pretty much the Earth setting, since Earth is 1 AU from the Sun).
During 1
year a planet in orbit with orbital radius 1 AU, like the Earth, completes the full orbit i.e. the mean annual
motion is 360 degrees.
For Planet 2, which by default has semi-major axis of 1.5 AU, i.e. is 50% further from
the Sun (fairly close to Mars’ semi-major axis), orbital period is 1.84 years.
This planet takes longer to
complete one trip around its orbit, so it does not complete a full circle in a single year, in fact its mean annual
motion is 195.96 degrees which means that it completes a little bit more than half of its circular orbit (half
would be 180 degrees).
In rows 23-63 of the simulator, the data for Planet 1 (columns B-N) and for Planet 2 (columns O-AC) is
computed in 40 steps (see column A). What happens is that the program divides the orbital period of a planet
period into 40 equal time intervals called
time steps
.
Since the period is different for the Planet 1 and Planet
2, the amount of time represented by each time step is also different.
For planet 1 which default setting is
semi-major axis of 1 AU and, thus, orbital period of 1 year, time step is
time step
default for Planet
1
=
1
year
40
steps
=
0.025
years
For Planet 2, for which the default setting is 1.5 AU, and thus orbital period is 1.84 years, time step is
time step
default for Planet
2
=
1.84
year
40
steps
=
0.046
years
Thus the time step depends on Planet’s period and is always equal to 1/40
th
of that orbital period.
The
calculation of a planetary orbit starts with a planet at the perihelion and then advances in 40 time steps of
equal duration, each equal to that planet’s time step (which in turn depends on its orbital period). Look at the
Orbit
graph and notice that, the forty data points are equally spaced around circular orbits.
The spacing
between consecutive data points represents equal intervals of time. That means that planets are moving at a
constant speed in these circular orbits covering the same distance in each equal time interval.
Look at the
graph
Orbital speed
and note that it too shows that each planet moves at a constant speed (also note that
Planet 2, which is more distant from the Sun, moves at a lower speed than Planet 1)
Because completing one full orbit around the Sun takes 360 degrees and because it is done in 40 time steps,
for each planet the mean angle swept out by an imaginary line between the planet and the Sun (called the
mean anomaly
) is 9
°
for each time step.
360
°
40
time steps
=
9
° per time step
This time step between two consecutive data points will not change if the eccentricity changes.
For
e
= 0 the
calculated time steps amount to 9-degree intervals between the data points. You will investigate what happens
when the eccentricity changes: does the time step change? Does the angle change?
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4
Procedure:
1.
Adjust the input quantities settings to the following: for Planet 1:
a
= 1 AU and
e
= 0 and for Planet 2:
a
= 1.35 AU and
e
= 0.
What are the values of the period
P
and the mean annual motion
N
?
Record the results in table 1 below.
Table 1.
Object
Orbital period
P
(years)
Mean annual motion
N
(degrees)
Planet 1
1 year
360 degrees
Planet 2
1.57 years
229.51 degrees
2.
Adjust the values of the semi-major axis for Planet 2 to values listed in table 2 below.
For each value of
semi-major axis, record the period
P
and the mean annual motion
N
. From
Orbital speed
graph, read
the orbital speed of Planet 2 for each value of semi-major axis.
Record it in table 2 below.
Table 2.
Semi-major
axis (AU)
Orbital period
P
(years)
Mean annual motion
N
(degrees)
Orbital speed (AU/year)
0.25
0.13 years
2880.00 degrees
0.13 AU
0.5
0.35 years
1018.23 degrees
0.35 AU
0.75
0.65 years
554.26 degrees
0.65 AU
1
1 year
360 degrees
1 AU
1.25
1.4 years
257.60 degrees
1.4 AU
1.5
1.84 years
195.96 degrees
1.84 AU
1.75
2.32 years
155.51 degrees
2.20 AU
2
2.83 years
127.28 degrees
2.69 AU
Based on this data, complete the following statements:
As the value of the semi-major axis increases, the period
increases
(enter: “increases”, “decreases” or “stays
the same”)
As the value of the semi-major axis increases, the
mean annual motion
decreases
(enter: “increases”,
“decreases” or “stays the same”)
As the value of the semi-major axis increases, the orbital speed
increases
(enter: “increases”, “decreases” or
“stays the same”)
3.
Now start with Planet 2 at
a
= 1 AU and
e
= 0.
Then a
djust the value of eccentricity
e
for Planet 2
to the values listed in table 3.
For each value of eccentricity, record the period and the period
P
and the
5
mean annual motion
N
. Also, for each value of eccentricity, look at the graph showing the orbits. Record
the data in table 3 below.
Table 3.
eccentricity
Orbital period
P
(years)
Mean annual motion
N
(degrees)
0
1 year
360 degrees
0.2
1 year
360 degrees
0.4
1 year
360 degrees
0.6
1 year
360 degrees
0.8
1 year
360 degrees
0.9
1 year
360 degrees
Based on this data, complete the following statements:
As the value of the eccentricity increases, the period
stays the same
(enter: “increases”, “decreases” or “stays
the same”)
As the value of the eccentricity increases, the
mean annual motion
stays the same
(enter: “increases”,
“decreases” or “stays the same”)
Describe how the shape of the orbit changes as eccentricity increases?
As the eccentricity increases the
shape of the orbit changes to more of a oval than a perfectly circular orbit.
As you increase the eccentricity for Planet 2, does the Sun stay at the center of the orbit?
Not at the center
but more so as a perihelion. The sun is on the right most hand side of Planet 2’s orbit path.
4.
Adjust the input quantities settings to the following: for Planet 1:
a
= 1 AU and
e
= 0 and for Planet 2:
a
= 1.5 AU and
e
= 0.
Look at the
Orbits
graph.
Are the fourty data points still evenly spaced around the orbit?
Yes, all 40 points
are evenly spaced around the orbit
Look at the
Distance to the Sun
graph
.
Is the distance of each planet to the Sun constant?
Yes the distance of
each planet to the sun is constant.
The distance from the Sun to Planet 1 is
1.8 years
The distance from the Sun to Planet 2 is
1 year
Do not forget to include units!
Look at the
Orbital speed
graph
.
Are the orbital speeds of the planets constant?
The orbital speeds of the
planets are constant. All the steps are all constatnt to one another.
The orbital speed of Planet 1 is
1 AU
The orbital speed of Planet 2 is
1.84 AU
Do not forget to include units!
Figure 7.
6
In the
Input
spreadsheet, look at the data table for Planet 1 (columns B, C and I; rows 23-63) and Planet 2
(columns O, P and V; rows 23-63).
Is time step still 9 degrees (in columns B and O)?
No
Is the true anomaly (in columns I and V) the same
as mean anomaly (in columns C and P)?
Yes it is
the same
True anomaly
is an angle that the line between the
planet and the Sun at any given time makes with
the line drawn between the planet’s perihelion and
the Sun (see figure 7). True anomaly tells us where a
planet is in its orbit.
5.
Change the eccentricity of Planet 2 to
e
= 0.8.
Look at the
Orbits
graph
.
Are the fourty data points for Planet 2 still evenly spaced around its orbit?
No, as
Planet 2 gets closer to the sun the points are more spread apart as to when the Planet is furthest from the
sun the points are very close to one another.
Look at the
Distance to the Sun
graph.
Is the distance of Planet 2 to the Sun still constant?
No
Look at the
Orbital speed
graph
.
Is the orbital speeds of Planet 2 constant?
No
When is Planet 2 moving fastest?
When the Planet is closer to the Sun
When is Planet 2 moving slowest?
When the Planet is further from the Sun
Does each time step still represent 9 degrees?
No
Why or why not?
Because the orbit path is not constant,
there will be certain times when the Planet is turning more than other times. So it will not be a constant 9
degrees for each step.
Where does a planet spend more time in its orbital motion, near aphelion or near perihelion?
Explain
.
Aphelion, because when the Planet is closer to the Sun due to its gravitational pull pulling on the Planet it
causes the orbit to increase in speed. With that being said when a Planet doesn’t have the Sun’s
gravitational pull pulling on it, then it will spend more time in its orbital motion.
In the
Input
spreadsheet, look at the data table for Planet 1 (columns B, C and I; rows 23-63) and Planet 2
(columns O, P and V; rows 23-63).
Is time step still 9 degrees (in columns B and O)?
No
Is the true anomaly (in columns I and V) still the same as mean anomaly (in columns C and P)?
No
If not,
what changed? Explain
.
For Planet 2, column P and V do not match, P still shows 9 degree steps but column
v shows the it starting from 88 degrees and going up to 115.
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7
Conclusion question:
Look at the data for major planets using NASA’s
Planetary Fact Sheet
.
Which major planet has the most
eccentric orbit? Which major planet has the most eccentric orbit? Record their data in table 4 below:
Table 4.
Object
Eccentricity
Semi-major axis (AU)
Orbital period
P
(years)
Least eccentric
major planet
0.007
0.7 AU
0.615 years
Most eccentric
major planet
0.244
49.3 AU
248.11 years
Set Planet 1 parameters in the simulator to semi-major axis and eccentricity of least eccentric of the real major
planets and those for Planet 2 to those for the most eccentric.
Look at the
Orbits
graph.
Describe how the
most and least eccentric of the real major planets’ orbits appear on the simulation.
The least eccentric
Planet’s orbit looks pretty regular, nothing extordinary just a smaller orbit of Earth’s orbit. The most
eccentric Planet’s data can not be displayed due to the AU being above the maximum cap.
Copy the Orbits graph (made for two major planets listed in table 4) and paste it here:
Submission details:
Submit into this lab’s drobox on Blackboard:
MS Word report (this document with your entries) only,
8
Appendix A
Newton’s three Laws of motion:
1.
An object at rest will remain at rest and an object in motion will continue to move at a constant speed
along a straight line, unless it experiences a net external force
2.
Object’s acceleration equals to the net external force divided by its mass.
3.
When a body A exerts a force on body B, body B will exert an equal amount of force on body A in opposite
direction.
Newton’s Law of Gravitation:
The force of gravity between two masses (
M
1
and
M
2
) is directly proportional to the product of the two
masses and inversely proportional to the square of the distance
d
between them:
F
=
G
M
1
×M
2
d
2
Here
G
is the universal gravitational constant
G
= 6.67
×
10
-11
N
×
m
2
/kg