Keplers 3 laws-1
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Astronomy
Date
Apr 3, 2024
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NAME______________________
CLASS_________
Instructions:
Go to web site http://astro.unl.edu
. Click on the Nebraska astronomy applet project and then go to NAAP Modules(at top of screen) and pick Planetary Orbital Simulator. Read the materials and complete the guide below and complete the exercises and complete the document below—the background materials will help you answer the questions—the flash demonstration will help you complete the rest.
ON LINE LAB 04
Nebraska Astronomy Applet Project
Student Guide to the
Planetary Orbit Simulator
_____________________________
Background Material
Answer the following questions after reviewing the “Kepler's Laws and Planetary
Motion” and “Newton and Planetary Motion” background pages.
Draw a line connecting each law on the left with a description of it on the right. Question 1: When written as P
2
= a
3
Kepler's 3rd Law (with P in years and a in AU) is
applicable to …
a)
any object orbiting our sun. Lab 4 NAAP – Planetary Orbit Simulator 1/8
Kepler’s 1
st
Law
Kepler’s 2
nd
Law
Kepler’s 3
rd
Law
Newton’s 1
st
Law
planets orbit the sun in elliptical paths
planets with large orbits take a long time to complete an orbit
planets move faster when close to the sun only a force acting on an object can change its motion
b)
any object orbiting any star.
c)
any object orbiting any other object.
Question 2: The ellipse to the right has an eccentricity of about …
a)
0.25
b)
0.5
c)
0.75
d)
0.9
Question 3: For a planet in an elliptical orbit to “sweep out equal areas in equal amounts
of time” it must …
a)
move slowest when near the sun.
b)
move fastest when near the sun.
c)
move at the same speed at all times.
d)
have a perfectly circular orbit.
Question 4: If a planet is twice as far from the sun at aphelion than at perihelion, then the
strength of the gravitational force at aphelion will be ____________ as it is at perihelion.
a)
four times as much b)
twice as much c)
the same
d)
one half as much
e)
one quarter as much Kepler’s 1st Law
If you have not already done so, launch the NAAP Planetary Orbit Simulator
.
Open the Kepler’s 1
st
Law tab if it is not already (it’s
open by default).
Enable all 5 check boxes.
The white dot is the “simulated planet”. One can click on
it and drag it around.
Change the size of the orbit with the semimajor axis slider. Note how the
background grid indicates change in scale while the displayed orbit size remains
the same.
Change the eccentricity and note how it affects the shape of the orbit. Be aware that the ranges of several parameters are limited by practical issues that
occur when creating a simulator rather than any true physical limitations. We have
Lab 4 NAAP – Planetary Orbit Simulator 2/8
Tip:
You can change
the value of a slider
by clicking on the
slider bar or by
entering a number in
the value box.
limited the semi-major axis to 50 AU since that covers most of the objects in which we
are interested in our solar system and have limited eccentricity to 0.7 since the ellipses
would be hard to fit on the screen for larger values. Note that the semi-major axis is
aligned horizontally for all elliptical orbits created in this simulator, where they are
randomly aligned in our solar system.
Animate the simulated planet. You may need to increase the animation rate for
very large orbits or decrease it for small ones.
The planetary presets set the simulated planet’s parameters to those like our solar
system’s planets. Explore these options.
Question 5: For what eccentricity is the secondary focus (which is usually empty) located
at the sun? What is the shape of this orbit? When the eccentricity is at zero the shape is a
circle. Question 6: Create an orbit with a = 20 AU and e = 0. Drag the planet first to the far left
of the ellipse and then to the far right. What are the values of r
1
and r
2
at these locations?
r
1 (AU)
r
2 (AU)
Far Left
20
20
Far Right
20
20
Question 7: Create an orbit with a = 20 AU and e = 0.5. Drag the planet first to the far
left of the ellipse and then to the far right. What are the values of r
1
and r
2
at these
locations? r
1 (AU)
r
2 (AU)
Far Left
19
21
Far Right
21
19
Question 8: For the ellipse with a = 20 AU and e = 0.5, can you find a point in the orbit
where r
1
and r
2
are equal? Sketch the ellipse, the location of this point, and r
1
and r
2
in the
space below. Lab 4 NAAP – Planetary Orbit Simulator 3/8
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Question 9: What is the value of the sum of r
1
and r
2
and how does it relate to the ellipse
properties? Is this true for all ellipses? The suns value is 40 AU, and it is twice the size
of the ellipse’s axis. It is true for all ellipses. Question 10: It is easy to create an ellipse using a
loop of string and two thumbtacks. The string is
first stretched over the thumbtacks which act as
foci. The string is then pulled tight using the pencil
which can then trace out the ellipse.
Assume that you wish to draw an ellipse
with a semi-major axis of a = 20 cm and e = 0.5. Using what you have learned earlier in
this lab, what would be the appropriate distances for a) the separation of the thumbtacks
and b) the length of the string? Please fully explain how you determine these values. The thumbtacks should be 20 centimeters apart because c= 10 and then you multiply it by
2 to get the distance between them. The string should be 60 cm for the same reason. The
foci = 30 and multiply by 2. Kepler’s 2nd Law
Use the “clear optional features” button to remove the 1st Law features.
Open the Kepler's 2nd Law tab.
Press the “start sweeping” button. Adjust the semimajor axis and animation
rate so that the planet moves at a reasonable speed.
Adjust the size of the sweep using the “adjust size” slider.
Click and drag the sweep segment around. Note how the shape of the sweep
segment changes, but the area does not.
Add more sweeps. Erase all sweeps with the “erase sweeps” button.
The “sweep continuously” check box will cause sweeps to be created
continuously when sweeping. Test this option.
Question 11: Erase all sweeps and create an ellipse with a = 1 AU and e = 0. Set the
fractional sweep size to one-twelfth of the period. Drag the sweep segment around.
Does its size or shape change? The shape and size stay the same.
Lab 4 NAAP – Planetary Orbit Simulator 4/8
Question 12: Leave the semi-major axis at a = 1 AU and change the eccentricity to e =
0.5. Drag the sweep segment around and note that its size and shape change. Where is
the sweep segment the “skinniest”? Where is it the “fattest”? Where is the planet when it
is sweeping out each of these segments? (What names do astronomers use for these
positions?) It is skinniest to the right and fattest to the left. Question 13: What eccentricity in the simulator gives the greatest variation of sweep
segment shape? When the eccentricity is 0.7.
Question 14: Halley’s comet has a semimajor axis of about 18.5 AU, a period of 76
years, and an eccentricity of about 0.97 (so Halley’s orbit cannot be shown in this
simulator.) The orbit of Halley’s Comet, the Earth’s Orbit, and the Sun are shown in the
diagram below (not exactly to scale). Based upon what you know about Kepler’s 2
nd
Law, explain why we can only see the comet for about 6 months every orbit (76 years)? It is traveling very fast when the comet gets close to our sun and orbit, explaining why we
only see it for 6 months of its own orbit.
Lab 4 NAAP – Planetary Orbit Simulator 5/8
Kepler’s 3
rd
Law
Use the “clear optional features” button to remove the 2nd Law features.
Open the Kepler's 3rd Law tab.
Question 15: Use the simulator to complete the table below. Object
P (years)
a (AU)
e
P
2
a
3
Earth
1
1.00
0.17
1
1
Mars
1.88
1.52
0.93
3.54
3.54
Ceres
4.56
2.77
0.08
20.8
20.8
Chiron
50.7
13.8
0.38
2630
2630
Question 16: As the size of a planet’s orbit increases, what happens to its period? The
period increases. Question 17: Start with the Earth’s orbit and change the eccentricity to 0.6. Does
changing the eccentricity change the period of the planet? It has no effect on the period of
the planet. Newtonian Features
Important: Use the “clear optional features” button to remove other features.
Open the Newtonian features tab.
Click both show vector boxes to show both the velocity and the acceleration of
the planet. Observe the direction and length of the arrows. The length is
proportional to the values of the vector in the plot. Question 18: The acceleration vector is always pointing towards what object in the
simulator? It is always pointing toward the sun. Question 19: Create an ellipse with a = 5 AU and e = 0.5. For each marked location on
the plot below indicate a) whether the velocity is increasing or decreasing at the point in
the orbit (by circling the appropriate arrow) and b) the angle θ between the velocity and
acceleration vectors. Note that one is completed for you. Lab 4 NAAP – Planetary Orbit Simulator 6/8
↑ ↓
θ = 61º 6161
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Question 20: Where do the maximum and minimum values of velocity occur in the orbit?
The maximum and minimum values occur at the furthest point left and right in the orbit.
Question 21: Can you describe a general rule which identifies where in the orbit velocity
is increasing and where it is decreasing? What is the angle between the velocity and
acceleration vectors at these times? The velocity increases as it approaches the perihelion
and decreases when it approaches the aphelion. As it increases it will be between 0-90
degrees and decreasing between 90-100 degrees. Astronomers refer to planets in their orbits as “forever falling into the sun”. There
is an attractive gravitational force between the sun and a planet. By Newton’s 3
rd
law it is
equal in magnitude for both objects. However, because the planet is so much less massive
than the sun, the resulting acceleration (from Newton’s 2
nd
law) is much larger. Acceleration is defined as the change in velocity – both of which are vector
quantities. Thus, acceleration continually changes the magnitude and direction of
velocity. As long as the angle between acceleration and velocity is less than 90°, the
magnitude of velocity will increase. While Kepler’s laws are largely descriptive of what
Lab 4 NAAP – Planetary Orbit Simulator 7/8
↑ ↓
θ = ↑ ↓
θ = ↑ ↓
θ = ↑ ↓
θ = ↑ ↓
θ = ↑ ↓
θ = ↑ ↓
θ =
planet’s do, Newton’s laws allow us to describe the nature of an orbit in fundamental
physical laws!
Lab 4 NAAP – Planetary Orbit Simulator 8/8