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ESS 9 Fall 2009
Week 1 Discussion
Oct. 1-2, 2009
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Orbits and Gravity
Due Date:
Oct 7-8
YOU WILL NEED A RULER
AND CALCULATOR
Name____________________________ ID#___________ Section: ___________
Summary. Bodies orbit their common center of mass because they gravitationally attract each
other and they have a component of angular velocity. Gravitational attraction is described by
Newton's law of gravity and the orbits obey Kepler's laws, meaning:
1. The orbits are ellipses;
2. The orbits sweep out equal areas in equal times;
3. There is a relationship between the period and size of the orbit (P2 = a D3).
where P = period of orbit, D = semi-major axis (distance from sun), and
“a” is a constant.
The “Orbital Demonstrator Lab” displays Venus and Earth in orbit around the sun and allows
you to vary the mass of all three objects as well as the relative velocities and distances. Note that
all the units are normalized with respect to Earth, so that Earth’s mass, distance and velocity are
all set to “1”, and the sun’s mass, for example, is set to 300,000. Venus’ mass is initially set to
0.10, although in reality Venus is more massive than this. These values were chosen by the
programmer rather arbitrarily and are for demonstration purposes. The main purpose of the lab
is to illustrate how altering these three parameters results in different and interesting orbits.
Exercise 1: Learning to fly
Goal: Learn how to use the simulator
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Press 'start' to set things in motion
Click on 'vectors' and see what happens (vectors show gravity accelerations)
Click on 'trails' and see what happens
On the timer, try the 'lap' and ‘zero’ buttons
Press 'stop'
Click on one of the planets. The boxes in the top left will show the color of the planet.
Change one or more parameters (velocity and distance have most effect)
Press 'start' and see how things behave differently
Experiment with using the buttons and changing parameters until you are familiar with
the module
 'Reset' (at the bottom) sets the parameters to their default values
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ESS 9 Fall 2009
Week 1 Discussion
Oct. 1-2, 2009
2
Exercise 2: Going into orbit
Goal: To see how initial velocity affects the planet's trajectory
 Press 'reset' in the CONTROLS section to reset the parameters
 Turn off the 'Venus' toggle. We will now focus on one planet only.
 Click on the blue planet and set Distance=1.2, Velocity=0.00 (Note: These are
arbitrary units)
 Turn on 'Vectors' (These vectors represent the size and direction of the gravitational force
between the two bodies).
 Click 'Start'
Which direction does the planet move in? ___________________________________________.
What happens to the force (acceleration) of gravity as the planet gets closer to the Sun?
________________________________.
What happens to the planet's velocity as it gets closer to the Sun? ________________________.
 Now set the velocity to some small value, like 0.5, and run it again.
 Watch it with 'vectors', and watch it with 'trails'
Describe the path of the planet:___________________________________________________.
What happens to the force of gravity as the planet gets closer to the Sun?
_______________________________________________________________________.
What happens to the planet's velocity as it gets closer to the Sun (you can see this more clearly if
you switch off 'vectors')? ________________________________________________________.
 Try a few more velocity values, up to 1.0 or so.
What happens to the planet's trajectory as you increase the velocity?_____________________
_______________________________________________________________________.
What happens to the planet's closest distance to the Sun as you increase velocity?
___________________________________________________________________________.
Exercise 3: Ellipses and Kepler's laws
Goal: Familiarity with the eccentricity and semi-major axes of an ellipse, and 'testing' Kepler's
third law.
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ESS 9 Fall 2009
Week 1 Discussion
Oct. 1-2, 2009
3
Kepler's first law states that planets travel in elliptical orbits with the Sun at one focus of the
ellipse. Here we will make some measurements of orbital ellipses, and the corresponding orbital
periods.
The experiment:
 Press 'reset' in the CONTROLS section.
 Focus on Earth: 'Venus' can be on or off, it doesn't affect the exercise
 Change Earth's velocity between 0.6 and 1.1, and fill in the table below
How to make measurements:
 Use a ruler to measure the lengths of the major axis (2D). This is the largest distance
across the ellipse. Then calculate (D), the semi-major axis.
 Measure the perihelion rmin, which is the closest distance between the planet and the Sun.
 Calculate eccentricity, e = 1 – (rmin/D)
 Measure orbital periods P using the timer to time one orbit.
rmin
Equations needed:
1.
e = 1- (rmin / D)
2.
P2 / D3 = constant
(D)
Major axis (2D)
major
velocity
axis = 2D
D
rmin
eccentricity
=1–
(rmin/D)
P
P2
D3
P2/D3
0.6
0.8
0.9
1.1
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ESS 9 Fall 2009
Week 1 Discussion
Oct. 1-2, 2009
4
From your experience with the simulations, circle the correct answer to the following questions
regarding Kepler’s laws:
Kepler’s Second Law: Planets travel faster / slower when they are closer to the Sun, and faster
/ slower when they are further away from the Sun, so that an imaginary line connecting the
planet to the Sun sweeps out equal areas in equal times.
As far as we can tell from the measurements above, P2/D3 for an orbit changes a lot/ seems
constant.
Extra credit (5 Points): On a separate sheet of paper, plot P2 vs. D3.
the plot should give you a (roughly) straight line.
If P2 / D3 is a constant,
Exercise 4: Heavy planets, and Moons
Goal: Recognizing the orbit of the Sun, and finding out what parameters are needed to get a
moon in orbit around a planet.
In the previous experiments, the planets were extremely light compared to the mass of the Sun,
so they had negligible effect on the Sun, and also on each other. Now we will change that!
 Reset the parameters
 Click on the Sun and note its mass below: _________________________________
 Set the mass of the Earth to 1/3 of the Sun's mass, click 'Start' and watch for a while
Does the Sun move? yes/no. Describe the Sun's trajectory:
___________________________________________________________________
With these parameters, it is clear that the Earth is no longer “orbiting the Sun”, but the Earth and
Sun are both orbiting around their common center of mass, which is about 1/3 of the way
between the Sun and Earth. This motion of the star is what astronomers use to detect planets
around distant stars.
Now (leaving the Earth’s mass the same) we are going to put Venus in orbit around Earth, rather
than the Sun:
 Keep the same parameters as in exercise 3, but set Earth's velocity to 0.9 so its orbit is
roughly circular
 Set the 'Venus' distance to 0.85 (close to Earth)
 Keep running the simulator and adjusting Venus' velocity until it has a stable orbit around
the Earth
 Hints: (i) you need to increase Venus' velocity. (ii) there is a range of velocities for
which it will orbit Earth: fiddle around until it has a reasonably circular orbit around
Earth.
I find that Venus will orbit Earth for an initial velocity of ________________.
 Turn 'trails' on and watch the pattern! Describe in the orbits in words/picture. (use back
of sheet if necessary)
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