Kepler's Laws Summer 2021
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Apr 3, 2024
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1
Kepler’s Laws of Planetary Motion
Name: 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
.
Figure 3.
Figure 2.
Figure 1.
2
Kepler's second law basically says that the planets 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
Figure 5.
Figure 4.
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
° pertime 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?
Figure 6.
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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
360
Planet 2
1.57
229.51
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
2,880
1.92
0.5
0.35
1,018.23
1.42
0.75
0.65
554.26
1.15
1
1
360
1
1.25
1.40
257.60
0.89
1.5
1.84
195.96
0.81
1.75
2.32
155.51
0.75
2
2.83
127.28
0.70
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 decreases
(enter: “increases”, “decreases” or “stays the same”)
3.
Now start with Planet 2 at a
= 1 AU and e
= 0. Then adjust 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 mean
5
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
360
0.2
1
360
0.4
1
360
0.6
1
360
0.8
1
360
0.9
1
360
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?
An obit will be more elliptical
As you increase the eccentricity for Planet 2, does the Sun stay at the center of the orbit?
The distance to the sun becomes further away
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?
They are evenly spaced, but they are further apart
Look at the Distance to the Sun
graph. Is the distance of each planet to the Sun constant?
Yes
The distance from the Sun to Planet 1 is
1 AU
The distance from the Sun to Planet 2 is
1.5 AU
Do not forget to include units!
Look at the Orbital speed
graph. Are the orbital speeds of the planets constant?
Yes
The orbital speed of Planet 1 is
6.2 m/s
The orbital speed of Planet 2 is
5.1 m/s
Do not forget to include units!
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).
6
Is time step still 9 degrees (in columns B and O)? Yes
Is the true anomaly
(in columns I and V) the same as mean anomaly
(in columns C and P)? It’s 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, they are not evenly spaced out
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?
1.85
When is Planet 2 moving slowest?
0.9
Does each time step still represent 9 degrees?
Yes
Why or why not?
The degree never changes for the planets
Where does a planet spend more time in its orbital motion, near aphelion or near perihelion? Explain.
At the aphelion because that’s where they move the slowest
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)? Yes
Is the true anomaly
(in columns I and V) still the same as mean anomaly
(in columns C and P)? Yes
If not, what changed? Explain. Conclusion question:
Look at the data for major planets in worksheet from lab 2. 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.
Figure 7.
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7
Object
Eccentricity
Semi-major axis (AU)
Orbital period P
(years)
Venus
0.007
108.2
0.615
Pluto
0.25
5,915
248.6
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.
They appear to be constant
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, 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