Planetary Orbits
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AP PHYSICS
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Astronomy
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Feb 20, 2024
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planets with large orbits take a long time to complete an orbit
planets move faster when close to the sun Kepler’s 1st Law
Kepler’s 2nd Law
planets orbit the sun in elliptical paths
Kepler’s 3rd Law
only a force acting on an object can change its motion Newton’s 1st Law
Name: Planetary Orbit Simulator – Student Guide
Background Material
Answer the following questions after reviewing the “Kepler's Laws and Planetary
Motion” and “Newton and Planetary Motion” background pages.
Question 1: Draw a line connecting each law on the left with a description of it on the
right. Question 2: 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. b)
any object orbiting any star.
c)
any object orbiting any other object.
Question 3: The ellipse to the right has an eccentricity of about …
a)
0.25
b)
0.5
c)
0.75
d)
0.9
Question 4: 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.
NAAP – Planetary Orbit Simulator 1/8
Question 5: 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
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 6: For what eccentricity is the secondary focus (which is usually empty) located
at the sun? What is the shape of this orbit? The eccentricity is at 0.0, yes its located at the
sun. and the orbit shape is a circle. Question 7: 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
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.
Question 8: 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
10
30
Far Right
30
10
Question 9: 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. Question 10: 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 sun r1 and r2 is combined distance form foci
to the ellipse and it is double the length of the semi major axis. And yes it is true for all
ellipses.
Question 11: 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. Separation of the thumb tacks is equal to 20cm. the distance from the center of the ellipse
to a focus is equal to the eccentricity multiplied by the length of the semi- major axis. So
the length of the string equals to 40 cm; in fact, this is because the sum of the distances
NAAP – Planetary Orbit Simulator 3/8
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from the focii and any point along the ellipse is equal to the twice the value of the semi-
major axis.
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 12: 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 size does not change because the orbit is a uniform
circle
Question 13: 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?) The sweep segment is "skinniest" point farthest from the sun; in fact the
point closest to the sun is where the sweep segment is "fattest"; however, to astronomers,
the closest point is known as the perihelion and the farthest point is known as the
aphelion.
Question 14: What eccentricity in the simulator gives the greatest variation of sweep
segment shape? Eccentricity is 0.70 where we see the greatest variation of sweep segment
shape.
Question 15: 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)? NAAP – Planetary Orbit Simulator 4/8
For 6 months we can see how the comet travel through the perihelion, moving at
its fastest in the solar system. But when it moves out of the inner solar system, it gets
farther out of sight and slows down. And as an result, the comet spends the most of it's
orbital period very far from us, moving slowly and out of view.
NAAP – Planetary Orbit Simulator 5/8
↑ ↓
63
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 16: Use the simulator to complete the table below. Object
P (years)
a (AU)
e
P
2
a
3
Earth
1.00
1.00
0.017
1.00
1.00
Mars
1.89
1.52
0.093
3.54
3.54
Ceres
4.61
2.77
0.08
21.3
21.3
Chiron
50.7
13.6
0.38
2570
2570
Question 17: As the size of a planet’s orbit increases, what happens to its period? Question 18: Start with the Earth’s orbit and change the eccentricity to 0.6. Does
changing the eccentricity change the period of the planet? the size of the planet's orbit
increases, its period increase.
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 19: The acceleration vector is always pointing towards what object in the
simulator? No, changing the eccentricity does not change the period of the planet.
Question 20: 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. NAAP – Planetary Orbit Simulator 6/8
↑ ↓
θ = 61º 6161
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↑ ↓
82
↑ ↓
76
↑ ↓
94
↑ ↓
97
↑ ↓
115
↑ ↓
118
1 0= 61º 2 0= 63º
3 0= 82º
4 0= 97º
5 0= 115º
6 0=118º
7 0= 94º
8 0= 76º
Question 21: Where do the maximum and minimum values of velocity occur in the orbit?
Maximum velocity is at perihelion. Minimum velocity is at aphelion.
Question 22: 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 orbital velocity increases when the acceleration
increases from aphelion to perihelion. In fact the angle between the velocity and volume
acceleration vectors is 45° <θ < 90°. I think the orbital velocity decreases when
acceleration decreases from perihelion to aphelion, and the angle between the velocity
and acceleration vectors is 90° < θ < 135
NAAP – Planetary Orbit Simulator 7/8
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
planet’s do, Newton’s laws allow us to describe the nature of an orbit in fundamental
physical laws!
NAAP – Planetary Orbit Simulator 8/8