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Liftoff to Learning:
The Mathematics of Space-Rendezvous

Video Title: The Mathematics of Space -
Rendezvous

Video Length: 17:00
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Mathematics Standards:
Mathematics as Problem Solving
Mathematics as Communication
Mathematics as Reasoning
Computation and Estimation
Algebra
Geometry
 

Measurement

Subjects: The mathematics of spacecraft orbital rendezvous.

Description: This program addresses the basic mathematical operations of spacecraft rendezvous in Earth orbit. Middle school students in a mathematics class work to solve some problems that permit the Space Shuttle to rendezvous and dock with the Russian Space Station Mir. The video has stopping points to permit viewers to work the problems as well.


Table of Contents

Background
In a few years, the International Space Station (ISS) will be ready for full-time occupancy by crews of astronauts. From their vantage point in space, astronauts will study Earth's environment and conduct a variety of scientific and technological experiments that will ultimately help to improve life on Earth.

One of the critical tasks to the construction and use of a space station is the ability to rendezvous in space. Consequently, the first phase of the international Space Station program is a series of Space Shuttle mission rendezvous and docking activities with the Russian Space Station Mir. By bringing crew and equipment to Mir, Shuttle astronauts have gained valuable practice in the maneuvers they will need to work with the ISS. These maneuvers are complex and docking is a delicate operation. Each maneuver employs extensive use of mathematics to achieve the objective.

The objective of this video is to invite students to work out some of the fundamental equations Shuttle crews use for rendezvous. The equations presented in the video are listed in a following section of this guide.

Some students may become confused with some of the mathematical operations shown. The video is designed with stopping places for you to work the mathematics with your students.

One rendezvous concept presented in the video may need some additional explanation. By firing its rocket engines to increase velocity, the Space Shuttle actually slows down. Conversely, firing engines to slow down causes the vehicle to speed up. These paradoxical statements are easier to understand when you remember that movement in space is a three-dimensional problem. Changes in velocity leads to changes in altitude.

On Earth, rendezvous between two automobiles is a relatively simple operation. Both drive at specific speeds to arrive at a specific location. When the automobiles arrive at the right place, they stop. In space, the rendezvous location is over a specific place on Earth, but it is also at a specific altitude. When the two spacecraft arrive, they cannot stop. Doing so will cause them to fall back to Earth. Instead, their rendezvous is at a specific location and altitude (three dimensions), and at a specific time. Time is important when you consider that the spacecraft will be traveling at 5 or more kilometers per second. A mere 5-second error will cause the spacecraft to miss each other by 25 kilometers.

The nature of space rendezvous is also complicated by some basic physical laws. Altitude and velocity of a spacecraft are related. Spacecraft in low orbits travel very fast because the gravitational pull is strong. In higher orbits, spacecraft travel slower because the force of gravity is less. The force of gravity between two objects (Earth and the Shuttle) is determined by the following mathematical relationship that was first formulated by Isaac Newton and later modified by Henry Cavendish.

formula f=G times m1m2 over r squared

G in the equation is the gravitational constant. The r in the equation is the distance between the center of Earth and the center of the Shuttle (not the altitude of the Shuttle over Earth's surface). As you can see, r has an inverse square relationship in the equation. That means that the closer the centers of the two bodies are to each other, the greater the force of attraction. It also means that increasing the distance between the centers decreases the attraction by an inverse square.

The difference in gravitational attraction with change in distance (orbital altitude) is where the speed-up/slow-down paradox comes in. You must travel faster in a lower orbit than a higher one to stay in orbit. If you want to go to a higher orbit, you must fire your rocket engines to accelerate. The acceleration causes your spacecraft to climb higher above Earth. As you climb higher, your velocity diminishes until you are traveling at the right velocity for the higher orbit. It is a slower velocity than you were traveling before the firing of the engines. In other words, you sped up so that you could slow down in a higher orbit.

The reverse is true if you want to go to a lower orbit. To descend, you fire rocket engines in the opposite direction you are traveling. This causes your spacecraft to slow. Earth's gravity pulls your spacecraft downward, and, as you fall, your speed increases until you are at the right speed for the new altitude. Your speed is greater in the lower orbit. Thus, you slowed down to speed up.

Although somewhat complicated, this paradox helps to accomplish rendezvous. For example, when the two spacecraft are on opposite sides of Earth from each other, having one spacecraft in a lower orbit will enable it to close the distance. In the lower orbit, the spacecraft will not only travel faster than the higher spacecraft, but the orbit has a smaller circumference as well. After closing the distance, the lower spacecraft can begin maneuvers to adjust to the right altitude for the rendezvous. How long to fire rocket engines, when to do it, and in what directions is determined with mathematics.

Equations Used In the Program  contents

Degrees longitude orbital ground track shifts eastward with each orbit.
equation

Number of orbits so that Mir flies over Moscow.
Mir - Moscow = Longitude Distance
 105 degrees minus 37 = 68 degrees  formula


Time for Mir's orbit to cross Moscow.

formula representing above info


Mir's orbital speed.

distance over time = speed


Distance (circumference) Mir travels during one orbit. (The altitude is the distance from Earth's center to Mir.)

formula for information above

Mir's orbital speed.

formula for MIR's orbital speed

Shuttle speed change needed to raise orbit 7 kilometers. (It is stated in the video that a change in velocity of 0.4 meters per second raises the Shuttle 1 kilometer.)

formula for shuttle speed

Classroom Activities  contents

How High?

Materials:

Earth globe
Metric ruler

Objective: To learn why it is necessary to exaggerate altitudes when orbits are shown in model form.

Procedure:
Measure the diameter of the globe you are using for the model. Determine its scale. To do this, you will need to know the actual diameter of the Earth (12,756 km). Using the same scale, determine how high above the globe's surface the Space Shuttle and Mir would be (400 km).

Discussion:
Diagrams of planets and spacecraft orbiting them are difficult to portray accurately. The diagrams of Earth, the Space Shuttle, and Mir Space Station used in the video greatly exaggerate the distance the orbiting spacecraft are above Earth. Without doing this, the orbits would lie so closely to the surface of Earth that the lines would be indistinguishable.

Extension: Using the same scale, determine how far the Moon would be from Earth.

Sine Curve Orbits?

Materials:

Earth globe
Paper
Tape
Marker
Scissors

Objective: To show why a sine curve orbital plot is created when an orbit is portrayed on a flat map.

Procedure:
When shown on a flat map, Space Shuttle orbits resemble sine curves. To show why this happens, wrap and tape a cylinder of paper around an Earth globe. Use a marker pen to draw an orbit around the cylinder. Start with an orbit inclined 28 degrees. Draw the line around the cylinder so that it falls on a plane inclined to the globe's equator by 28 degrees. Remove the cylinder and cut the paper along the line you drew. If you drew the line carefully, the edge of the cut will fall on a plane. Unwrap the cylinder and look at the shape of the orbit.
 illustration of set up

Discussion:
Orbital maps displayed in Mission Control at the NASA Johnson Space Center show three Space Shuttle orbits at a time. A small Space Shuttle orbiter is displayed on one of the orbits over the geographic position the actual orbiter is flying. The curve of the orbits resembles a sine curve. The steepness of the curve is determined by the angle in which the Space Shuttle was launched in respect to Earth's equator.
orbital map

Many Shuttle orbits are inclined at 28.5 degrees. This is the geographic latitude of the Kennedy Space Center. When a Shuttle is launched due east, its orbit is inclined 28.5 degrees. This happens because an orbit must be concentric with the center of Earth. In geographic terms, the orbit must be a great circle.

Extension:
Create other cylinders for different orbits such as 35 degrees and 51.6 degrees (orbit of the International Space Station). Compare the steepness of the curves when the cylinders are flattened.

globe with orbits drawn

STS-84 Crew Biographies  contents

Commander: Charles J. Precourt (Col., USAF)
Pilot: Eileen M, Collins (Lt. Col., USAF)
Payload Commander, Mission Specialist: Jean-Francois Clervoy (Ingenieur en Chef de l Armement, ESA Astronaut)
Mission Specialist: Edward T. Lu (Ph.D)
Mission Specialist: Carlos I. Noriega (Major, USMC)
Mission Specialist, NASA-Mir 4: Jerry M. Linenger (Capt., USN)
Mission Specialist: Elena V. Kondakova
Mission Specialist, NASA-Mir 5: C. Michael Foale (Ph.D.)

To obtain biographic information, click on highlighted names

 
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