On this page, you will test
drive some of the ideas that we have developed so far. Your Mission is to dock
the Space Shuttle with the Mir 4 Space Station. Actually, you will guide the
Space Shuttle to within 250 Km of the station. At that point, if you
succeed, you will turn the controls over to your pilot for the final docking
maneuver.
The Mir 4 is moving at a
constant speed in a high circular orbit, at 3000 km above the surface
of the Earth. Since it is in a circular orbit at constant distance, we can deduce
its constant velocity.
Question 1: Show from the moving frames formula for orbital acceleration
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that if a satellite moves in a circular orbit under the influence of Newtonian
gravity, then it must move with constant speed. That is, 
is constant. Next, deduce this more easily from Energy considerations. If the
potential energy is constant, what follows?
The radius of the Earth is
roughly 
.
And according to Newton's law, there is a universal constant G
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with 
such that if 
represents the distance of the station from the center of the Earth, then the
acceleration due to gravity is:
measured in 
Thus, the acceleration is radial and must be due entirely to gravitation,
so
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Now for the space station's
circular orbit, 
are constant, so actually,
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The constant speed of the space station is
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so we can conclude that
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so the constant speed of the space station is approximately 6.521 km/sec
Question 2: Show that the conserved quantity 
for the space station is approximately 
.
We divide the Energy by the Mass because we defined earlier the Energy of a
unit mass (1 kg). The actual energy of the space station is the product of that
Energy with the mass of the space station. Compare this with the 
. What is the relation between
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for a satellite moving uniformly in a circilar orbit ?
You
will take control of the Shuttle at altitude 1000 Km above the surface of the
Earth. At that time, your speed will be 7.0479438 Km/sec. You have been injected
into the same plane of motion as that of the space station. Your thrusters are
set so that you will never leave that plane.
Question 3: Show that the quantity 
for the Space Shuttle will be approximately -29.2291 
at that time.
Of course, your 
will not be constant. It will change as you fire your thrusters to rendezvous
with the space station.
The values of various quantities will appear on the shuttle instrument panel to help you navigate.
The
first three entries (in dark blue) are the shuttle's Speed, Altitude (above
the Earth's surface) and .
These will be continually updated as you fly. The next two entries (in dark
red) are the constant values of Altitude and
of the space station. The last entry (bright red) is very important. It reports
your distance to the space station.
Of course, while instruments
are important, you will also have a view of the station itself (somewhat magnified)
on your monitor.
In this picture, the shuttle
is at altitude 2810.775 Km and is roughly 692.158 Km
away from the station, on final approach. You will attempt to bring the shuttle
to within 250 Km of the station. Your pilot will take it from there.
But there is an important consideration. You must not be moving too fast relative
to the station. In fact, your total
must be within 8 units of the
of the station. Also, if 
is your velocity and 
is the velocity of the station, then the length, 
km/sec.
You will learn right away
the first lesson of space flight. In Space, flying is coasting. Let momentum,
inertia, and gravity do most of the work. You should use your engines only when
you have to make course corrections. For example, we will show you shortly how
to coast. While you are coasting with your engines turned off, your altitude
and speed may change, but your total
will remain constant. In general, the greater the altitude, the greater the
potential Energy. Your present motion will inject you in an orbit (if it is
not on a line with the center of the Earth) only if 
.
Since the space station's 
is 
your motion must be orbital in the docking phase, and your orbit
must match that of the station (not going in the opposite direction, for example).
If you increase your shuttle's
speed, and then turn off the engines and coast, reciprocity will often slowly
decrease your speed as it increases your altitude. So to rise to 3000 km, just
give the shuttle a short burst of forward speed (tangent to the Earth) and coast
up to the higher altitude. Experienced shuttle captains take pride in the economy
of fuel that results from well-planned coasting. Of course, you will have to
lose some of the additional energy as you approach the shuttle, so keep an eye
on the instrument panel.
Exploration: Rocket Science 101
We will now explain how to
fly the ship. For that, you will use the control panel that gives you a different
view of the shuttle and space station. We have conveniently arranged for you
to begin in the same plane of motion as the space station, as we mentioned,
so a simultaneous 2-dimensional view of the process is appropriate. You will
see in the upper left corner our new state-of-the-art point-and-click navigation
control.
The navigation control shows
the position of the shuttle (blue dot) and the position of the space station
(red dot) at any time. The instrument panel just below gives more detailed information.
Now
the short blue and red vectors attached to the vehicle avatars indicate their
current velocities by the length and directions of those vectors. You also see
the thrust vector of the shuttle as a yellow vector that indicates the engines
are on.
Now to start flying your ship, press the Go button on the main panel
If you would like to hear
the rocket thrusters and shouts of acclamation on your success, or the Terminator's
chilly promise on failure, leave the Sound ? checkbox checked. Otherwise,
turn it off, and you will get visual messages from time to time. Before you
press Go, you should press Reset to go back to your starting position
after each try. We did not say that it would be easy!
Once you press Go,
you must continually move the cursor over the navigation control.
If you stop moving the cursor, the action is suspended until you start moving
it again. Move the cursor near the blue shuttle avatar to fire the rockets.
Move it in the direction you want the shuttle to go. The farther from the shuttle,
the greater the thrust. If the acceleration is perpendicular to the current
direction of motion, the effect is to turn the vehicle without changing its
speed. To slow down, move the cursor directly behind the shuttle.
Now if you want to coast,
that is, fly with the engines turned off, bring the cursor close to the center
of the Earth, and slowly move it around there.
You will see (and perhaps
hear) the engines light. The shuttle and space station velocities will slowly
change as you correct the course. But keep moving the cursor near the center
of the Earth to coast! That is very important. Finally, you must approach within
250 Kilometers of the Station to hand the job over to the pilot to dock. If
your velocity does not match closely enough to the Station's velocity, you will
hear a short message from The Terminator. All units are in Kilometers, Seconds,
and Kilograms (Metric units).
To stop the simulation, left-click
on the navigation panel. Be sure to press Reset before you Go
again.
Happy Contrails to You!
