The
Geometry
in Space
Project
Sponsored by the Indiana
Space Grants Consortium and Ball State University
Orbital
Mechanics: From Earth to Mars
The Solar
System
The solar system consists of
an average star we call the sun, the planets Mercury, Venus, Earth, Mars,
Jupiter, Saturn, Uranus, Neptune, and Pluto. It includes the satellites of the
planets, numerous comets, asteroids, meteoroids, and the interplanetary medium,
which permeates interplanetary space.
Using the information resources found at
Basics of Space Flight: The Solar System and Mars Exploration, answer the
following questions:
1)
What percent
of the total mass of the solar system is found in the sun? In the planets? Comets, asteroids (minor
planets), and meteors?
2)
What is an
astronomical unit (AU)? How many
kilometers are there in 1 AU?
3)
What is a
light year? How many AU are there
in a light year? How many
kilometers?
Orbits
Every planet, asteroid, and comet in the
solar system circles the sun, following a path called an orbit. The first person to mathematically
describe planetary orbits was Johannes Kepler. Kepler discovered that planets and
comets circle the sun in elliptical orbits, with the sun at one focus of the
ellipse. The oval shape seen in
Figure 1 is an example of such a curve.
In this curve, as in all ellipses, the sum of the distances from each
focus (Distance A + Distance B) to each point of the ellipse is a constant. If this constant is changed, a different
ellipse is obtained. If the
separation between the foci is changed, a different ellipse is obtained (See Basics of Space
Flight: Gravitation and Mechanics).
4)
What is the major axis of an
ellipse? The minor
axis?
5)
If the two foci coincide,
what sort of curve is obtained?
6)
What is the eccentricity of
an ellipse? How is it
computed?
Figure 1 Definition of an
Ellipse
Click on Figure 2 to start a
java applet that will allow you to explore elliptical orbits directly. When the applet starts, reposition the
blue dot (corresponding to a planet) by clicking and dragging it to a new
position relative to the sun (red dot).
The applet also computes the eccentricity of the ellipse. Using the applet and your imagination,
answer the following questions:
7)
Does the planet move at a
steady rate around the sun, like the hands of a clock around its face? How would you describe the motion of the
planet?
8)
Describe the relationship
between the eccentricity of an ellipse and its shape.
Figure 2 Ellipses &
Eccentricity
If you have access to the
Geometers Sketchpad, you may continue your exploration of ellipses using the
file Ellipse.gsp
(See Figure 3). To see the
geometric construction behind the animation, select Show All Hidden from the
Display menu.
9)
Study the construction and
write an explanation for why it draws an ellipse.
Figure 3 Geometers Sketchpad
Ellipse
Click on Figure 4 to experiment with a
java applet showing the relative positions of the planets and their orbits on
any day. Note: While the planets
are positioned correctly in their orbits, the distances between the planets are
not shown to scale.
10) Plot the orbits of the planets for
today. For the day, month, and year
of your birth. For your birthday
this year. On which of these three
days were Mars and Earth closest?
11) When are Mars and Earth closest in
2001?
12) What is the length of the Martian year in
Earth days?
Figure 4 Planetary Orbits
Using the tool Orbit
Xplorer, you may animate the movement of the planets. Click on Figure 5 and launch Orbit
Xplorer. When the program has
loaded, experiment with the following files:
13) Inner Solar System Sim. Be sure that your parameter settings
include the following values:
Figure 5 Inner Solar System Sim. Parameter Settings
Figure 6 Inner Solar System Sim. Simulation
a)
Which orbit seems most
circular?
b)
Which orbit appears to be
most eccentric?
14) Rocket Launch Sim. Experiment with the parameters to try
and achieve a successful launch (See Figure 8). Be sure that the parameter settings
include the selections seen in Figure 7.
Figure 7 Rocket Launch Sim. Parameter Settings
Figure 8 Rocket Launch Sim. Simulation
a)
What happens when you use
the default velocity, vy = 9000 m/s? Why?
b)
Set vy = 0 and
let vx be at least 7000 m/s.
i)
What is the smallest
vx that results in an orbit?
ii)
Convert this velocity to
miles per hour.
iii) Convert it to miles per
second.
iv) Is the orbit circular or
elliptical in appearance?
v)
When the simulation stops,
click on the Graphs tab. Describe
the shape of the Distance graph.
(1) Why does the graph have this
shape?
(2) Approximately how many
seconds are required for one orbit of the Earth? This is known as the period of
the orbit.
(3) Convert the period to hours,
minutes, and seconds.
c)
Experiment with other
successful launch parameters, trying to achieve a more eccentric
orbit.
i)
On each orbit, how close
does the satellite get to the Earth?
How far away?
ii)
Describe the shape of the
Distance graph (See Figure 9).
iii) What happens to the Distance
graph as the orbit become more eccentric?
iv) What happens to the period
of the orbit as it becomes more eccentric?
Figure 9 Distance Graph
15) Satellite Around the Earth
Sim. Experiment with the parameters
to try and achieve a circular orbit.
What parameters produce a circular orbit?
Figure 10 Inner Solar System
Simulation
The
following resources provide additional information on both Kepler and the Laws
of Planetary Motion.
·
Johannes Kepler, His Life,
His Laws and Times
http://www.kepler.arc.nasa.gov/johannes.html
·
Johannes Kepler: The Laws of
Planetary Motion
http://csep10.phys.utk.edu/astr161/lect/history/kepler.html
·
Kepler's Three Laws of
Planetary Motion
http://observe.ivv.nasa.gov/nasa/education/reference/orbits/orbit_sim.html
·
Kepler’s
Laws
http://www.cvc.org/science/kepler.htm
Transfer
Orbits
A
spacecraft traveling between the planets is said to follow a transfer orbit,
which is itself an ellipse with the sun at one focus. Figure 11 shows such an orbit. The orbital transfer used by the Mars
Odyssey spacecraft is shown in Figure 12.
Clearly, the transfer orbit of the spacecraft must intersect the orbits
of both Earth and Mars.
16) What other requirement(s)
must be met for the transfer to be a success?
17) Where is the
Mars Odyssey spacecraft right now?
Detailed discussions of
transfer orbits are available at the following resources.
|
Figure 11 Transfer
Orbit |
Figure 12 Odyssey Transfer
Orbit |
Arrival at
Mars
As
the spacecraft nears Mars its trajectory is changed to place it in orbit around
the planet. This process is called
orbital insertion. Orbital
insertion requires precise positioning, timing, and controlled deceleration. As
the spacecraft's trajectory is bent by the planet's gravity, the command
sequence aboard the spacecraft places the spacecraft in the correct attitude,
and fires its engine(s) at the proper moment and for the proper duration. Once
the retro-burn has completed, the spacecraft has been captured into orbit by its
target planet. Having been captured
by the planet's gravity, the spacecraft is maneuvered into the particular orbit
planned for its scientific observations.
Additional details on the goals and methods of orbital insertion are
available at the Basics of Space
Flight: Encounter Phase
Figure 13 Orbital Insertion
Send Your Name to Mars
If you have enjoyed this simulated trip to Mars, you may wish to send
your name there aboard the Mars
Exploration Rover-2003 mission.
Your name will be written into a Compact Disk that will be enclosed in
one of the 2003 Mars Rover missions!