Keplers 3 laws_
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Lab 4 NAAP –
Planetary Orbit Simulator 1/8 NAME__Jocelyn Townsend_ CLASS 1354 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. 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
Lab 4 NAAP –
Planetary Orbit Simulator 2/8 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 1
st 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. Tip:
You can change the value of a slider by clicking on the slider bar or by entering a number in the value box.
Lab 4 NAAP –
Planetary Orbit Simulator 3/8 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 5:
For what eccentricity is the secondary focus (which is usually empty) located at the sun? What is the shape of this orbit? At 0.0 eccentricity, the secondary focus is located at the sun. The shape of the orbit is circular. 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 AU 20 AU Far Right 20 AU 20 AU 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 10 AU 30 AU Far Right 30 AU 10 AU 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.
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Lab 4 NAAP –
Planetary Orbit Simulator 4/8 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 value of the sum of r1 and r2 is the combined distances from the foci to any point of the ellipse and it is equivalent to double the length of the semi-major axis. Yes, this is true for all ellipses. Question 9:
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 separation of the thumbtacks is 20cm because the distance from the center of the ellipse to a focus is equal to the eccentricity times by the length of the semi-major axis. The length of the string is equal to 40cm. and that’s because the sum of the distance from the foci and any point along the ellipse is equal to 2x the value of the semi-major axis. Kepler’s 2
nd 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.
Lab 4 NAAP –
Planetary Orbit Simulator 5/8 Question 10:
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 doesn’t change because the orbit is a uniform circle. Question 11:
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 “
fatt
est”?
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 the “skinniest” from the point farthest from the sun. The point that’s closest to the sun is where the “fattest” point lies. The names astronmers use for these points are perihelion (close/fattest) and aphelion (farthest/skinny) Question 12:
What eccentricity in the simulator gives the greatest variation of sweep segment shape? At .70 eccentricity is where you see the greatest variation of the sweep segment shape Question 13:
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)? For about 6 months we can see the comet travel through the perihelion, moving at its fastest in the inner part of the solar system, when it moves out it gets farther out of sight and slows down. The result ends with the comet spending the majority of it’s orbital period very far from us, moving slowly and out of our view.
Lab 4 NAAP –
Planetary Orbit Simulator 6/8 Kepler’s 3
rd
Law •
Use the “
clear optional features
” b
utton to remove the 2nd Law features. •
Open the Kepler's 3rd Law tab. Question 14:
Use the simulator to complete the table below. Question 15:
As the size of a planet’s orbit increases, what happens to its period? As the size of the planet’s orbit increases, the period increases. Question 16:
Start with the Earth’s orbit and c
hange the eccentricity to 0.6. Does changing the eccentricity change the period of the planet? No, changing the eccentrinity does not change the perood 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 17:
The acceleration vector is always pointing towards what object in the simulator? The acceleration vector is always pointing towards the sun. Question 18:
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 Object P (years) a (AU) e P
2
a
3
Earth 1.00 1.00 0.017 1.00 1.00 Mars 1.88 1.52 0.093 3.54 3.54 Ceres 4.61 2.77 0.08 21.3 21.3 Chiron 50.7 13.7 0.38 2570 2570
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Lab 4 NAAP –
Planetary Orbit Simulator 7/8 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. Question 19:
Where do the maximum and minimum values of velocity occur in the orbit? Maximum velocity is at perihelion and the minimum velocity is at aphelion. Question 20:
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 acceleration increases, from aphelion to perihelion, and the angle between the velocity and acceleration vectors is 45 degrees <0 < 90 degrees. The orbital velocity decreases when acceleration decreases from perihelion to aphelion and the angle between the velocity and acceleration vectors is 90 degrees < 0 < 135 degrees. ↑ ↓
θ = ↑ ↓
θ = ↑ ↓
θ = ↑ ↓
θ = ↑ ↓
θ = ↑ ↓
θ = ↑ ↓
θ = ↑ ↓
θ
= 61º 6161
Lab 4 NAAP –
Planetary Orbit Simulator 8/8 A
stronomers 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 fun
damental physical laws!