F=G

m m

r

1

2

2 F=G

m m

r

1

2

2

Educational Product

National Aeronautics and Teachers Grades 5-12

Space Administration

Educational Brief

Subject: Mathematics

Topic: Microgravity

THE MATHEMATICS OF

MICROGRAVITY
The Mathematics of

Microgravity

This publication is in the Public Domain and is not protected by copyright.

Permission is not required for duplication.

EB-1997-02-119-HQ

National Aeronautics and Space Administration

Office of Life and Microgravity Sciences and Applications

Microgravity Science and Applications Division

Office of Human Resources and Education

Education Division
Acknowledgments

This publication was developed for the

National Aeronautics and Space

Administration.

Writers:

Melissa J. B. Rogers, MS

TAL-CUT Company

NASA Lewis Research Center

Cleveland, OH

Gregory L. Vogt, Ed.D.

Teaching From Space Program

NASA Johnson Space Center

Houston, TX

Michael J. Wargo, Sc.D.

Microgravity Science and Applications

Division

NASA Headquarters

Washington, DC

How To Use This Educational Brief

The main text of the educational brief uses

large print and is located in the wide

column. Subjects relating to mathematics

and physics principles are highlighted in

bold. Mathematics standards, questions for

discussion, examples, and derivations are

provided in smaller print in the narrow

column of each page.

Editor:

Carla B. Rosenberg

Teaching From Space Program

NASA Headquarters

Washington, DC

Each area highlighted in the text has a

corresponding section in the narrow

column. This corresponding section first

lists the applicable Mathematics Standards,

indicated by grade level. We have

attempted to position the appropriate

mathematics discussion as close as

possible to the relevant highlighted text.
1

The Mathematics of Microgravity

Amplitude

λ

Distance

Mathematics Standards

Grades 5-8 (∆); Grades 9-12 (❑)

❑ Mathematics as Communication

❑ Mathematical Connections

∆ ❑ Number and Number Relationships

∆ Number Systems and Number Theory

The electromagnetic spectrum is generally

separated into different radiation categories

defined by frequency (units of Hertz) or

wavelength (units of meters). Wavelength is

commonly represented by the symbol λ .

Examples:

 Approximate

Name Wavelength (m)

X-rays = 10-15 to 10-9

Ultraviolet = 10-8 to 10-7

Visible Light = 10-7 to 10-6

Infrared = 10-6 to 10-3

Microwave = 10-3 to 10-1

Television = 10-1 to 1

AM Radio = 102 to 103

Mathematics Standards

Grades 5-8 (∆); Grades 9-12 (❑)

∆ ❑ Algebra

∆ ❑ Mathematics as Communication

∆ Measurement

 1 g ≅ 9.8 m/s2

≅ 32 ft/s2

Units conversion is very important in all areas of

life, for example in currency exchange, airplane

navigation, and scientific research.

Introduction

Space flight is important for many reasons. Space

flight carries scientific instruments and sometimes

humans high above the ground, permitting us to see

Earth as a planet and to study the complex

interactions of atmosphere, oceans, land, energy,

and living things. Space flight lifts scientific instruments above the filtering effects of the atmosphere,

making the entire electromagnetic spectrum

available and allowing us to see more clearly the

distant planets, stars, and galaxies. Space flight

permits us to travel directly to other worlds to see

them up close and sample their compositions.

Finally, space flight allows scientists to investigate

the fundamental states of matter—solids, liquids,

and gases—and the forces that affect them in a

microgravity environment.

The study of the states of matter and their interactions in microgravity is an exciting opportunity to

expand the frontiers of science. Investigations

include biotechnology, combustion science, fluid

physics, fundamental physics, and materials

science.

Microgravity is the subject of this brief. This publication identifies the underlying mathematics and physics principles that apply to microgravity and acts as a

supplement to Microgravity: Activity Guide for

Science, Mathematics, and Technology Education

(EG-1997-02-110-HQ).

What Is Microgravity?

The presence of Earth creates a gravitational field

that acts to attract objects with a force inversely

proportional to the square of the distance between

the center of the object and the center of Earth.

When we measure the acceleration of an object

acted upon only by Earth's gravity at Earth's surface,

we commonly refer to it as one g or one Earth

gravity. This acceleration is approximately

9.8 meters per second squared (m/s 2).
The Mathematics of Microgravity

2

Mathematics Standards

Grades 5-8 (∆); Grades 9-12 (❑)

∆ ❑ Mathematical Connections

∆ ❑ Mathematics as Communication

1 micro-g or 1 μ g = 1 x 10-6 g

Questions for Discussion

• What other common prefixes or abbreviations

for powers of ten do you know or can you

find?

• In what everyday places do you see these

used? Grocery stores, farms, laboratories,

sporting facilities, pharmacies, machine

shops.

Common prefixes for powers of ten:

10-9 nano- n

10-3 milli- m

10-2 centi- c

103 kilo- k

106 mega- M

109 giga- G

Mathematics Standards

Grades 5-8 (∆); Grades 9-12 (❑)

∆ ❑ Algebra

∆ Computation and Estimation

❑ Conceptual Underpinnings of Calculus

❑ Discrete Mathematics

∆ ❑ Mathematical Connections

∆ ❑ Mathematics as Problem Solving

∆ ❑ Mathematics as Reasoning

∆ Number and Number Relationships

Calculate the times in these examples. Teachers

can use these examples at several different

scholastic levels.

Provide the equation as:

t = 2d

a or 2d

a



 

1 2

Provide the equation as d=(1/2) at2, and have the

students re-order the equation.

Making measurements and calculating results

involve the concepts of accuracy and precision,

significant figures, and orders of magnitude.

With these concepts in mind, are the drop times

given in the text "correct"?

We can interpret the term microgravity (μg) in a

number of ways, depending upon the context. The

prefix micro- (μ) derives from the original Greek

mikros, meaning "small." By this definition, a microgravity environment is one that imparts to an object a

net acceleration that is small compared with that

produced by Earth at its surface. We can achieve

such an environment by using various methods including Earth-based drop towers, parabolic aircraft

flights, and Earth-orbiting laboratories. In practice,

such accelerations will range from about one percent

of Earth's gravitational acceleration (aboard aircraft in

parabolic flight) to better than one part in a million (for

example, aboard Earth-orbiting free flyers). Earthbased drop towers create microgravity environments

with intermediate values of residual acceleration.

Quantitative systems of measurement, such as the

metric system, commonly use micro- to mean one

part in a million. By this second definition, the acceleration imparted to an object in microgravity will be

one-millionth (10-6) of that measured at Earth's

surface.

The use of the term microgravity in this guide

corresponds to the first definition: small gravity levels

or low gravity. As we describe how low-acceleration

environments can be produced, you will find that the

fidelity (quality) of the microgravity environment will

depend on the mechanism used to create it. For

illustrative purposes only, we will provide a few simple

quantitative examples using the second definition.

The examples attempt to provide insight into what

might be expected if the local acceleration environment were reduced by six orders of magnitude from

1 g to 10-6 g.

If you stepped off a roof that was five meters high,

you would reach the ground in just one second. In a

microgravity environment equal to one percent of

Earth's gravitational pull, the same drop would take 10

seconds. In a microgravity environment equal to

one-millionth of Earth's gravitational pull, the same

drop would take 1,000 seconds, or about

17 minutes!
3

The Mathematics of Microgravity

Figure 1. Acceleration and weight

The person in the stationary elevator car experiences normal

weight. In the car immediately to the right, weight increases slightly

because of the upward acceleration. Weight decreases slightly in

the next car because of the downward acceleration. A scale does

not measure weight in the last car on the right because of free fall.

Lighter than

normal

Normal

weight

Mathematics Standards

Grades 5-8 (∆); Grades 9-12 (❑)

∆ ❑ Algebra

∆ Computation and Estimation

∆ ❑ Mathematical Connections

∆ ❑ Mathematics as Problem Solving

∆ ❑ Mathematics as Reasoning

∆ Measurement

Questions for Discussion

• How far away is the Moon?

• How far away is the center of Earth from the

center of the Moon?

• Why did we ask the previous question?

• How far away is the surface of Earth from the

surface of the Moon?

• What are the elevations of different features of

Earth and the Moon?

• How are elevations measured?

Scientists can create microgravity in two ways. Because gravitational pull diminishes with distance, one

way to create a microgravity environment is to travel

away from Earth. To reach a point where Earth's

gravitational pull is reduced to one-millionth of that at

the surface, you would have to travel into space a

distance of 6.37 million kilometers from Earth

(almost 17 times farther away than the Moon).

This approach is impractical, except for automated

spacecraft, since humans have yet to travel farther

away from Earth than the distance to the Moon.

However, the act of free fall can create a more practical microgravity environment.

We will use a simple example to illustrate how free fall

can create microgravity (Figure 1). Imagine riding in

an elevator to the top floor of a very tall building. At

the top, the cables supporting the car break, causing

the car and you to fall to the ground. (In this example,

we discount the effects of air friction and elevator

safety mechanisms on the falling car.) Because you

No measured

weight

Heavier than

normal l
The Mathematics of Microgravity

4

and the elevator car are falling together, acted on only

by gravity, you will float inside the car if you lift your

feet off the elevator floor. In other words, you and the

elevator car accelerate downward at the same rate,

which is due to gravity alone. If a scale were

present, your weight would not register because

the scale would be falling too.

Gravity

Gravitational attraction is a fundamental property of

matter that exists throughout the known universe.

Physicists identify gravity as one of the four types of

forces in the universe. The others are the strong and

weak nuclear forces and the electromagnetic force.

More than 300 years ago the great English scientist Sir

Isaac Newton published the important generalization

that mathematically describes this universal force of

gravity. Newton was the first to realize that gravity

extends well beyond the domain of Earth. The basis of

this realization stems from the first of three laws he

formulated to describe the motion of objects. Part of

Newton's first law, the law of inertia, states that objects

in motion travel in a straight line at a constant velocity

unless acted upon by a net force. According to this law,

the planets in space should travel in straight lines.

However, as early as the time of Aristotle, scholars knew

that the planets travelled on curved paths. Newton

reasoned that the circular motions of the planets are the

result of a net force acting upon each of them. That

force, he concluded, is the same force that causes an

apple to fall to the ground—gravity.

Newton's discovery of the universal nature of the

force of gravity was remarkable. To take the familiar

force that makes an apple fall to Earth and be able to

recognize it as the same force that keeps the planets

on their quiet and predictable paths represents one of

the major achievements of human intellectual endeavor. This ability to see beyond the obvious and

familiar is the mark of a true visionary. Sir Isaac

Newton's pioneering work epitomizes this quality.

Mathematics Standards

Grades 5-8 (∆); Grades 9-12 (❑)

∆ ❑ Mathematical Connections

∆ ❑ Mathematics as Reasoning

Questions for Discussion

• How does a scale work?

• What does a scale measure?

• How many different kinds of scales can you

list?

• Do they need gravity for them to work?

• Would you get different results on the Moon

or Mars?

• How can you measure the mass of an object in

microgravity? Hint: See Inertial Balance

Activity in the Microgravity Activities Guide,

EG-1997-02-110-HQ.

Mathematics Standards

Grades 5-8 (∆); Grades 9-12 (❑)

∆ ❑ Algebra

∆ Computation and Estimation

❑ Functions

∆ ❑ Mathematics as Communication

∆ Number and Number Relationships

∆ Patterns and Functions

Introduction of concepts of proportionality

versus equality and mathematical expressions

versus equations.

indicates proportionality

indicates equality

is an expression

is an equation

G

m1 m2

r

2

F ∝ m1 m2

r

2

F = G

m1 m2

r

2

F = G

m1 m2

r

2
5

The Mathematics of Microgravity

Newton’s experimental research into the force of

gravity resulted in his elegant mathematical statement that is known today as the Law of Universal

Gravitation. According to Newton, every mass in the

universe attracts every other mass. The attractive

force between any two objects is directly proportional to the product of the two masses being

measured and inversely proportional to the

square of the distance separating them. If we let

F represent this force, r represent the distance

between the centers of the masses, and m1 and m2

represent the magnitude of the masses, the relationship stated can be written symbolically as:

(We define ∝ mathematically to mean "is proportional to.") From this relationship, we can see that the

greater the masses of the attracting objects, the

greater the force of attraction between them. We

can also see that the farther apart the objects are

from each other, the less the attraction. Note that the

inverse square relationship with respect to distance

is important. In other words, if the distance between

the objects doubles, the attraction between them

diminishes by a factor of four, and if the distance

triples, the attraction is only one-ninth as much.

The eighteenth-century English physicist Henry

Cavendish later quantified Newton’s Law of Universal Gravitation. He actually measured the gravitational force between two one kilogram masses

separated by a distance of one meter. This attraction was an extremely weak force, but its determination permitted the proportional relationship of

Newton’s law to be converted into an equation. This

measurement yielded the universal gravitational

constant or G. Cavendish determined that the value

of G is 0.0000000000667 newton m2/kg2 or 6.67 x

10-11Nm2/kg2. With G added to the equation, the

Universal Law of Gravitation becomes:

Mathematics Standards

Grades 5-8 (∆); Grades 9-12 (❑)

∆ ❑ Algebra

∆ ❑ Mathematical Connections

∆ ❑ Mathematics as Communication

∆ Measurement

The internationally recognized Systeme

International (SI) is a system of measurement

units. The SI units for length (meter) and mass

(kg) are taken from the metric system. Many

dictionaries and mathematics and science

textbooks provide conversion tables between the

metric system and other systems of measurement.

Units Conversion Examples

Questions for Discussion

• What common objects have a mass of about

 1 kg?

• What are the dimensions of this sheet of paper

in cm and inches?

• How many liters are there in a gallon?

Mathematics Standards

Grades 5-8 (∆); Grades 9-12 (❑)

∆ Computation and Estimation

∆ ❑ Mathematics as Communication

∆ Number and Number Relationships

∆ Number Systems and Number Theory

Scientific notation makes it easier to read, write,

and manipulate numbers with many digits. This

is especially useful for making quick estimates

and for indicating the number of significant

figures.

Examples:

0.001 = 10-3

 10 = 101

 1000 = 103

Which is bigger, 6 x 10-3 or 8 x 10-4? 6 x 103 or

8 x 104? How much bigger?

1 kg 2.2 lb

1 liter 1 qt

≅

≅

1 in 2.54 cm

1 yd 0.9 m

≅

≅

F ∝

m 1 m 2

r 2

F = G

m 1 m 2

r 2
The Mathematics of Microgravity

6

Creating Microgravity

Drop Towers and Tubes

In a practical sense, we can achieve microgravity

with a number of technologies, each depending

upon the act of free fall. Drop towers and drop

tubes are high-tech versions of the elevator analogy presented in a previous section. (The large

version of these facilities is essentially a hole in the

ground.)

NASA's Lewis Research Center in Cleveland, Ohio

has a 145 meter drop tower facility that begins on

the surface and descends into Earth like a mine

shaft. The test section of the facility is 6.1 meters

in diameter and 132 meters deep. Beneath the

test section is a catch basin filled with polystyrene

beads. The 132 meter drop creates a microgravity

environment for a period of 5.2 seconds.

To begin a drop experiment, one places the experiment apparatus in a cylindrical test vehicle that can

carry experiment loads of up to 450 kilograms.

The vehicle hangs suspended from a cap that

encloses the upper end of the facility. Air is

pumped out of the facility until a vacuum of

Deep In Space

You can use the inverse square relationship, with respect to distance, of the Law of

Gravitation to determine how far to move a

microgravity laboratory from Earth to achieve

a 10-6 g environment. Distance (r) is measured between the centers of mass of the

laboratory and of Earth. While the laboratory

is still on Earth, the distance between their

centers is 6,370 km (equal to the approximate

radius of Earth, re). To achieve 10-6 g, the

laboratory has to be moved to a distance of

1,000 Earth radii. In the equation, r then

becomes 1,000 re or r= 6.37 x 106 km.

Mathematics Standards

Grades 5-8 (∆); Grades 9-12 (❑)

∆ ❑ Algebra

∆ Computation and Estimation

❑ Conceptual Underpinnings of Calculus

❑ Discrete Mathematics

∆ ❑ Mathematical Connections

∆ ❑ Mathematics as Problem Solving

∆ ❑ Mathematics as Reasoning

∆ Number and Number Relationships

The inverse square relationship can also be used

to determine the acceleration due to gravity at

any distance from the center of Earth, r.

Mathematics Standards

Grades 5-8 (∆); Grades 9-12 (❑)

∆ ❑ Mathematical Connections

∆ ❑ Mathematics as Reasoning

The mass of an object describes how much the

object accelerates under a given force. In British

units (commonly used in the United States),

force is given in units of pounds. The force due

to gravity is called weight. It is incorrect,

however, to refer to the mass of a substance in

terms of pounds. The British unit of mass

corresponding to 1 pound force is the slug.

Question for Discussion

• What can you think of that has a mass of

about 450 kilograms?

• Would it fit in a 6.1 meter diameter tube?

r

2

 Gmem

r

e

2 F =

 Gme

re

2 g =

r2 a = = Gme

a =

 re

2

 = mg

= re

2

r2

 a

g

= Force of gravity on mass,

m, at Earth’s surface

Acceleration due to

gravity at distance, r

From Newton’s second law

g
7

The Mathematics of Microgravity

10-2 torr is achieved. (Atmospheric pressure is

760 torr.) Doing so reduces the acceleration effects

caused by aerodynamic drag on the vehicle to less

than 10-5 g. During the drop, cameras within the

vehicle record the action and experiment data are

telemetered to recorders.

The NASA Marshall Space Flight Center in Huntsville,

Alabama houses a smaller facility for microgravity

research. It is a 100 meter high, 25.4 centimeter

diameter evacuated drop tube that can achieve microgravity for periods of as long as 4.5 seconds. A

stainless steel bell jar fits in the upper end of the tube.

For solidification experiments, scientists mount an

electron bombardment or an electromagnetic levitator

furnace inside the jar to melt the test samples. After

the sample melts, drops form and fall through the tube

to a detachable catch fixture at the bottom of the tube

(Figure 2).

Figure 2. 100 Meter Drop Tube at the NASA

Marshall Space Flight Center.

Mathematics Standards

Grades 5-8 (∆); Grades 9-12 (❑)

∆ ❑ Algebra

∆ Computation and Estimation

❑ Discrete Mathematics

❑ Functions

∆ ❑ Mathematical Connections

∆ ❑ Mathematics as Problem Solving

∆ Number and Number Relationships

∆ Patterns and Functions

Questions for Discussion

• What percentage of atmospheric pressure is

10-2 torr?

• By what percentage is the pressure in the

facility reduced?

• Atmospheric pressure differs at different

locations on Earth. Investigate how it varies.

On what types of things does this variation

depend?

Mathematics Standards

Grades 5-8 (∆); Grades 9-12 (❑)

∆ ❑ Mathematical Connections

∆ Measurement

∆ ❑ Statistics

Telemetry is the process of transmitting data

from a remote measurement location to a

receiving station for recording and analysis.

Questions for Discussion

• What are different ways you transmit and

record data?

• What errors do these methods involve?

Optical Pyrometer

Mass Spectrometer

Mechanical 

Roughing 

Pump

Time Calibration 

Detector

Indium-Antimony Detector

Inert Gas Supply

Silicon Detector

Turbo-Molecular

Vacuum Pumps

Detachable 

Catch

System

100-Meter Drop Tube

Mechanical 

Roughing 

Pump

Indium Antimonide

Detector
The Mathematics of Microgravity

8

Mathematics Standards

Grades 5-8 (∆); Grades 9-12 (❑)

∆ ❑ Algebra

∆ Computation and Estimation

❑ Conceptual Underpinnings of Calculus

❑ Discrete Mathematics

❑ Functions

∆ ❑ Mathematical Connections

∆ ❑ Mathematics as Problem Solving

∆ ❑ Mathematics as Reasoning

∆ Patterns and Functions

∆ ❑ Statistics

Questions for Discussion

• What is the functional relationship between

acceleration, distance, and time?

Use the three sets of drop tower data

points given in the text and the additional

data set: (24 meters, 2.2 seconds)

This data set is for the drop tower in

Figure 3.

How can you get another data set?

(0 meters, 0 seconds).

Suggested solution methods: Use different

types of graph paper. Use a computer

curve fitting program.

• Knowing that g = 9.8 m/s2

, what equation

can you write to incorporate acceleration,

distance, and time?

• Assume it costs $5000 per meter of height to

build a drop tower.

How much does it cost to build a drop

tower to allow drops of 1 second,

2 seconds, 4 seconds, 10 seconds?

Why does it cost so much more for the

longer times?

What would be an inexpensive way to

double low-gravity time in a drop tower?

Shoot the experiment package up from the

bottom.

NASA Field Centers and other countries have additional drop facilities of varying sizes to serve different

purposes. A 490 meter deep vertical mine shaft in

Japan has been converted to a drop facility that can

achieve a 10-5g environment for up to 10 seconds.

Figure 3. 2.2 Second Drop Tower at the NASA Lewis Research

Center.
9

The Mathematics of Microgravity

Figure 4. Parabolic Flight Characteristics.

Mathematics Standards

Grades 5-8 (∆); Grades 9-12 (❑)

❑ Conceptual Underpinnings of Calculus

❑ Functions

∆ ❑ Mathematical Connections

∆ Patterns and Functions

Microgravity carriers and other spacecraft follow

paths best described by conic sections. The

aircraft and sub-orbital rockets trace out

parabolas. Orbiting spacecraft are free falling on

elliptical paths. When a meteoroid is on a path

that is influenced by Earth or any other planetary

body but does not get captured by the

gravitational field of the body, its motion, as it

approaches then moves away from the body,

traces out a hyperbolic path.

Aircraft

Airplanes can achieve low-gravity for periods of about 25

seconds or longer. The NASA Johnson Space Center in

Houston, Texas operates a KC-135 aircraft for astronaut

training and for conducting experiments. The plane is a

commercial-sized transport jet (Boeing 707) with most of

its passenger seats removed. Padded walls protect the

people inside.

Although airplanes cannot achieve microgravity conditions of as high quality as those produced in drop towers

and drop tubes (since they are never completely in free fall

and their drag forces are quite high), they do offer an

important advantage over drop facilities—experimenters

can ride along with their experiments.

A typical flight lasts 2 to 3 hours and carries experiments

and crew members to a beginning altitude about 7 km

above sea level. The plane climbs rapidly at a 45 degree

angle (pull up), traces a parabola (pushover), and then

descends at a 45-degree angle (pull out) (Figure 4).

During the pull up and pull out segments, crew and

experiments experience between 2 g to 2.5 g. During the

parabola, at altitudes ranging from 7.3 to 10.4 kilometers,

net acceleration drops as low as 1.5 x 10-2 g for more than

15 seconds. On a typical flight the KC-135 flies 40 parabolic trajectories. The gut-wrenching sensations produced on the flight have earned the plane the nickname

of the "vomit comet." Altitude (kilometers)

P b li Fli ht Ch t i ti

6.0

7.5

9.0

10.5

1x10 g (5-15 sec)

1x10 g

(15-25 sec)

45 deg

2-2.5 g

Max.

2-2.5 g

Max.

-3

-2

Plane

Right Cone

Ellipse

Plane

Right Cone

Circle

Plane

Right Cone

Parabola

Plane

Right Cone

Hyperbola
The Mathematics of Microgravity

10

NASA also operates a DC-9 for low-gravity research

out of the NASA Lewis Research Center. Flying on a

trajectory similar to the one followed by the KC-135, the

DC-9 provides a low-acceleration environment of

1.5 x 10-2 g to 2 x 10-2 g for more than 15 seconds.

Rockets

Small rockets provide a third technology for creating

microgravity. A sounding rocket follows a suborbital

trajectory and can produce several minutes of free

fall. The period of free fall exists during its coast,

after burn out, and before entering the atmosphere.

Acceleration levels are usually at or below 10-5 g.

NASA has employed many different sounding rockets

for microgravity experiments. The most comprehensive series of launches, from 1975 to 1981, used

SPAR (Space Processing Application Rocket) rockets

for fluid physics, capillarity, liquid diffusion, containerless processing, and electrolysis experiments. The

SPAR could lift 300 kg payloads into free fall

parabolic trajectories lasting four to six minutes

(Figures 5, 6).

Figure 5. Rocket Parabolic Flight Profile

Launch

High-g acceleration

Payload separation

Microgravity

Parabolic Trajectory

Deceleration

Recovery

Telemetry

Figure 6. Small Rocket for

Microgravity Experiments.

Ogive Recovery

System

Experiment Experiment

Service

Module Measurement

Module

Umbilical Adaptor

Ring

Experiment

Experiment

Experiment

Solid Propellant

Rocket Motor

Rate Control System Telemetry

Igniter

Housing

Experiment
11

The Mathematics of Microgravity

Questions for Discussion

• How does the shuttle stay in orbit? Use the

following two equations that describe the

force acting on an object. The first equation

represents the force of gravity acting on the

shuttle.

Where:

F1 = Force of gravity acting on the shuttle

G = Universal gravitational constant

me = Mass of Earth

ms = Mass of the shuttle

r = Distance from center of Earth to the shuttle

The second equation represents the force acting

on the shuttle that causes a centripetal

acceleration, v2/r.

This is an expression of Newton's second law,

F=ma.

F2 = Force acting on the shuttle that causes

uniform circular motion (with centripetal

acceleration)

v = Velocity of the shuttle

These two forces are equal: F1

 = F2

In order to stay in a circular orbit at a given

distance from the center of Earth, r, the shuttle

must travel at a precise velocity, v.

Orbiting Spacecraft

Although airplanes, drop facilities, and small rockets

can establish a microgravity environment, all of these

laboratories share a common problem. After a few

seconds or minutes of low-g, Earth gets in the way

and the free fall stops. In spite of this limitation, we

can learn much about fluid dynamics and mixing,

liquid-gas surface interactions, and crystallization and

macromolecular structure. To conduct longer term

experiments (days, weeks, months, and years), we

must travel into space and orbit Earth. Having more

time available for experiments allows scientists to

investigate slower processes and more subtle effects.

To establish microgravity conditions for long periods

of time, one must first understand what keeps a

spacecraft in orbit. Ask any group of students or

adults what keeps satellites and Space Shuttles in

orbit and you will probably get a variety of answers.

Two common answers are: “The rocket engines keep

firing to hold it up.” and “There is no gravity in space.”

Although the first answer is theoretically possible, the

path followed by the spacecraft would technically not

be an orbit. Other than the altitude involved and the

specific means of exerting an upward force, little

difference exists between a spacecraft with its engines constantly firing and an airplane flying around

the world. A satellite could not carry enough fuel to

maintain its altitude for more than a few minutes.

The second answer is also wrong. In a previous

section, we discussed that Isaac Newton proved that

the circular paths of the planets through space were

due to gravity’s presence, not its absence.

Newton expanded on his conclusions about gravity

and hypothesized how an artificial satellite could be

made to orbit Earth. He envisioned a very tall mountain extending above Earth’s atmosphere so that

friction with the air would not be a factor. He then

imagined a cannon at the top of that mountain firing

cannonballs parallel to the ground. Two forces acted

upon each cannonball as it was fired. One force, the

Mathematics Standards

Grades 5-8 (∆); Grades 9-12 (❑)

∆ ❑ Algebra

∆ Computation and Estimation

❑ Conceptual Underpinnings of Calculus

❑ Discrete Mathematics

∆ ❑ Mathematical Connections

∆ ❑ Mathematics as Problem Solving

∆ ❑ Mathematics as Reasoning

∆ Number and Number Relationships

Gmems

r 2 = msv2

r

 v2 = Gme

r

 v = Gme

r

F1= G

 me

ms

r2

F2 = ms

 v2

r

r2 r
The Mathematics of Microgravity

12

• How does the shuttle change its altitude?

From a detailed equation relating the

shuttle velocity with the shuttle altitude,

one can obtain the following simple

relationship for a circular orbit. Certain

simplifying assumptions are made in

developing this equation: 1) the radius of

the shuttle orbit is nearly the same as the

radius of Earth, and 2) the total energy of

the shuttle in orbit is due to its kinetic

energy, 1/2 mv2

; the change in potential

energy associated with the launch is

neglected.

∆r = τ

π ∆v

 τ = orbital period, the time it takes the

 shuttle to complete one revolution

 around Earth

 

=

2πr

3 2

(Gme )

1 2

∆v = the change in shuttle velocity

∆r = the change in shuttle altitude

For example:

Consider a shuttle in a circular orbit at

160 nm (296.3 km) altitude. Determine the

new altitude caused by the shuttle firing a

thruster that increases its velocity by 1m/s.

First, calculate the orbital period, τ , from the

above equation.

 

τ =

2π(re + 2.96x105m)3 2

(Gme )

1 2

 = 2π(6.37x106m + 2.96x105m)3 2

(6.67x10−11 m3

s

2

kg

x5.98x1024 kg)1 2

 = 5.41 x 103s 

Next, use the period and the applied velocity

change to calculate the altitude change.

 

∆r = τ

π

∆v

 = 5.41x103

s

π

(1m / s)

 = 1.72x103m

In order to make the orbit circular at the new

altitude, the shuttle needs to apply the

same ∆v at the other side of the orbit.

Figure 7. Illustration from Isaac Newton,

Principia, VII, Book III, p. 551.

∆v ∆v

explosion of the black powder, propelled the cannonball

straight outward. If no other force were to act on the

cannon ball, the shot would travel in a straight line and at

a constant velocity. But Newton knew that a second

force would act on the cannonball: the presence of

gravity would cause the path of the cannonball to bend

into an arc ending at Earth's surface (Figure 7).

Newton demonstrated how additional cannonballs would

travel farther from the mountain each time if the cannon

fired using more black powder. With each shot, the path

would lengthen and soon the cannonballs would disappear over the horizon. Eventually, if one fired a cannon

with enough energy, the cannonball would fall entirely

around Earth and come back to its starting point. The

cannonball would begin to orbit Earth. Provided no force

other than gravity interfered with the cannonball's

motion, it would continue circling Earth in that orbit.

This is how the Space Shuttle stays in orbit. It launches

into a trajectory that arcs above Earth so that the orbiter

travels at the right speed to keep it falling while maintaining a constant altitude above the surface. For example, if

the Shuttle climbs to a 320 kilometer high orbit, it must

travel at a speed of about 27,740 kilometers per hour to

achieve a stable orbit. At that speed and altitude, the

Shuttle executes a falling path parallel to the curvature of

Earth. Because the Space Shuttle travels in a state of

free fall around Earth and due to the extremely low

friction of the upper atmosphere, the Shuttle

establishes a microgravity environment.

2πr

3 2

(Gme )

1 2 =

∆vr

∆vr

=
13

The Mathematics of Microgravity

Orbiting spacecraft provide ideal laboratories for

microgravity research. As on airplanes, scientists can

fly with the experiments that are on the spacecraft.

Because scientists are there to tend the experiments,

the experiments do not have to be fully automatic in

operation. A malfunction in an experiment conducted

with a drop tower or small rocket means a loss of data

or complete failure. In orbiting spacecraft, crew

members can make repairs so that there is little or no

loss of data. They can also make on-orbit modifications in experiments to gather more diverse data.

Perhaps the greatest advantage of orbiting spacecraft

for microgravity research is the amount of time during

which microgravity conditions can be achieved.

Experiments lasting for more than two weeks are

possible with the Space Shuttle. When the International Space Station becomes operational, the time

available for experiments will stretch to months. The

International Space Station will provide a manned

microgravity laboratory facility unrivaled by any on

Earth.

In the discussion and example just given, we

state that the equations given are simple

approximations of more complex relationships

between shuttle velocity and altitude. The more

complex equations are used by the shuttle

guidance and navigation teams who track the

shuttles’ flights. But the equations given here

can be used for quick approximations of the

types of thruster firings needed to achieve

certain altitude changes. This is helpful when an

experiment team may want to request an altitude

change. Engineers supporting the experiment

teams can determine approximately how much

propellant would be required for such an altitude

change and whether enough would be left for

the required de-orbit burns. In this way, the

engineers and experiment teams can see if their

request is realistic and if it has any possibility of

being implemented.

Mathematics Standards

Grades 5-8 (∆); Grades 9-12 (❑)

❑ Functions

∆ ❑ Mathematical Connections

∆ ❑ Mathematics as Communication

∆ Patterns and Functions

When microgravity researchers study the effects

of atmospheric friction, shuttle motion, and

other related acceleration phenomena on their

experiments, three concepts play an important

role: Right-hand rule, Coordinate systems, and

Frames of reference.

"Microgravity Room"

One of the common questions asked by visitors to

the NASA Johnson Space Center in Houston, Texas

is, “Where is the room where a button is pushed

and gravity goes away so that astronauts float?”

No such room exists because gravity can never be

made to go away. The misconception comes from

the television pictures that NASA takes of

astronauts training in the KC-135 and from

underwater training pictures. Astronauts

scheduled to wear spacesuits for extravehicular

activities train in the Weightless Environment

Training Facility (WET F). The WET F is a

swimming pool large enough to hold a Space

Shuttle payload bay mock-up and mock-ups of

satellites and experiments. Astronauts add heavy

weights to their spacesuits, which are filled with

air, to achieve neutral buoyancy in the water. The

facility provides an excellent simulation of what it

is like to work in space, with two exceptions: in the

pool astronauts can swim with hand and leg

motions, and if they drop a hand tool, it falls to the

bottom.
The Mathematics of Microgravity

14

NASA Resources for Educators

NASA’s Central Operation of Resources for Educators (CORE) was established for the national and international distribution of NASA-produced educational materials in

audiovisual format. Educators can obtain a catalog and an

order form by one of the following methods:

• NASA CORE

Lorain County Joint Vocational School

15181 Route 58 South

Oberlin, OH 44074

• Phone (216) 774-1051, Ext. 249 or 293

• Fax (216) 774-2144

• E-mail: nasaco@leeca8.leeca.ohio.gov

• Home Page: http://spacelink.msfc.nasa.gov/CORE

Educator Resource Center Network

To make additional information available to the education

community, the NASA Education Division has created the

NASA Educator Resource Center (ERC) network. ERCs

contain a wealth of information for educators: publications,

reference books, slide sets, audio cassettes, videotapes,

telelecture programs, computer programs, lesson plans, and

teacher guides with activities. Educators may preview, copy, or

receive NASA materials at these sites. Because each NASA

Field Center has its own areas of expertise, no two ERCs are

exactly alike. Phone calls are welcome if you are unable to visit

the ERC that serves your geographic area. A list of the centers

and the regions they serve includes:

NASA On-line Resources for Educators provide current

educational information and instructional resource materials to

teachers, faculty, and students. A wide range of information is

available, including science, mathematics, engineering, and

technology education lesson plans, historical information related

to the aeronautics and space program, current status reports on

NASA projects, news releases, information on NASA educational

programs, useful software and graphics files. Educators and

students can also use NASA resources as learning tools to

explore the Internet, to access information about educational

grants, to interact with other schools that are already on-line, to

participate in on-line interactive projects, and to communicate

with NASA scientists, engineers, and other team members to

experience the excitement of real NASA projects.

Access these resources through the NASA Education Home

Page: http://www.hq.nasa.gov/education

or, for more infromation, send an e-mail to:

comments@spacelink.msfc.nasa.gov

NASA Television (NTV)is the Agency's distribution system for

live and taped programs. It offers the public a front-row seat for

launches and missions, as well as informational and educational

programming, historical documentaries, and updates on the latest

developments in aeronautics and space science. NTV is transmitted

on GE-2, Transponder 9C at 85 degrees West longitude, vertical

polarization, with a frequency of 3,880 megahertz, audio on 6.8

megahertz, or through collaborating distance learning networks

and local cable providers.

Apart from live mission coverage, regular NASA Television

programming includes a News Video File from noon to 1:00 pm,

a NASA History File from 1:00 to 2:00 pm, and an Education File

from 2:00 to 3:00 pm (all times Eastern). This sequence is

repeated at 3:00 pm, 6:00 pm, and 9:00 pm, Monday through

Friday. The NTV Education File features programming for teachers

and students on science, mathematics, and technology, including

the NASA. . .On the Cutting Edge Education Satellite

Videoconference Series. The videoconferences include NASA

scientists, astronauts, and education specialists presenting

aeronautics and Earth and space science topics of interest to

teachers and students of grades 5-12. The series is free to

registered educational institutions. The videoconferences and all

NASA Television programming may be videotaped for later use.

For more information on NASA Television, contact:

NASA Headquarters, Code P-2, NASA TV, Washington, DC

20546-0001 Phone: (202) 358-3572

Home Page: http://www.hq.nasa.gov/office/pao/ntv.html

For more information about the Education Satellite Videoconference

Series, contact: Videoconference Producer, NASA Teaching From

Space Program, 308 CITD, Room A, Oklahoma State University,

Stillwater, OK 74078-8089

E-mail: edge@aesp.nasa.okstate.edu

Home Page: http://www.okstate.edu/aesp/VC.html

How to Access NASA's Education Materials and Services,

EP-1996-11-345-HQ This brochure serves as a guide to accessing a variety of NASA materials and services for educators.

Copies are available through the ERC network, or electronically

via NASA Spacelink.

Regional Educator Resource Centers (RERCs) offer

more educators access to NASA educational materials.

NASA has formed partnerships with universities, museums,

and other educational institutions to serve as RERCs in

many states. A complete list of RERCs is available through

CORE, or electronically via NASA Spacelink.

AL, AR, IA, LA, MO,TN

U.S. Space and Rocket Center

NASA Educator Resource Center for

NASA Marshall Space Flight Center

P.O. Box 070015

Huntsville, AL 35807-7015

Phone: (205) 544-5812

MS

NASA Educator Resource Center

Building 1200

NASA John C. Stennis Space Center

Stennis Space Center, MS 39529-6000

Phone: (601) 688-3338

NASA Educator Resource Center

JPL Educational Outreach

Mail Stop CS-530

NASA Jet Propulsion Laboratory

4800 Oak Grove Drive

Pasadena, CA 91109-8099

Phone: (818) 354-6916

CA cities near the center

NASA Educator Resource Center for

NASA Dryden Flight Research Center

45108 N. 3rd. Street East

Lancaster, CA 93535

Phone: (805) 948-7347

VA and MD's Eastern Shores

NASA Educator Resource Lab

Education Complex - Visitor Center

Building J-1

NASA Wallops Flight Facility

Wallops Island, VA 23337-5099

Phone: (757) 824-2297/2298

AK, AZ, CA, HI, ID, MT, NV, OR,

UT, WA, WY

NASA Educator Resource Center

Mail Stop 253-2

NASA Ames Research Center

Moffett Field, CA 94035-1000

Phone: (415) 604-3574

CT, DE, DC, ME, MD, MA, NH,

NJ, NY, PA, RI, VT

NASA Educator Resource Laboratory

Mail Code 130.3

NASA Goddard Space Flight Center

Greenbelt, MD 20771-0001

Phone: (301) 286-8570

CO, KS, NE, NM, ND, OK, SD, TX

NASA Educator Resource Room

Mail Code AP-2

NASA Johnson Space Center

2101 NASA Road One

Houston, TX 77058-3696

Phone: (281) 483-8696

FL, GA, PR, VI

NASA Educator Resource Laboratory

Mail Code ERL

NASA Kennedy Space Center

Kennedy Space Center, FL 32899-0001

Phone: (407) 867-4090

KY, NC, SC, VA, WV

Virginia Air and Space Museum

NASA Educator Resource Center for

NASA Langley Research Center

600 Settler's Landing Road

Hampton, VA 23669-4033

Phone: (757) 727-0900 x 757

IL, IN, MI, MN, OH, WI

NASA Educator Resource Center

Mail Stop 8-1

NASA Lewis Research Center

21000 Brookpark Road

Cleveland, OH 44135-3191

Phone: (216) 433-2017
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The Mathematics of Microgravity

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