The problem of space pollution caused by human-made debris in orbit is presented as a context for studying mathematical modeling. This unit should help students realize that not all problems are solvable with precise
measurements and exact answers but that mathematical models enable us to look at trends and to make predictions about probable outcomes. The ambiguity of some of these
activities may be frustrating to a few students, but it reflects the realities with which mathematicians and scientists must work.
Purposes
In this unit, students create and compare various mathematical models as a way to investigate some of the questions raised by the proliferation of orbital debris. Although the models are greatly simplified to make them understandable to high school students, they offer insight into the process of mathematical modeling and its importance. Throughout the unit it is assumed that your
students will work with appropriate technology. Depending on what you have available and what you prefer, these tools may include graphing calculators, computer graphing utilities, spreadsheets, or non-graphing calculators.
Introduction
One important outcome for this unit is helping students develop an appreciation of the power and limitations of mathematical modeling. They should realize that the two most basic expectations of models are (1) the ability to
account for or represent known phenomena and (2) the ability to predict future results. Thus, with the models that students develop in this unit, they should continually be asking such questions as, What will happen if this trend
continues? Or What if this element is change?
Other outcomes that we anticipate from this unit are a better understanding of the difference among linear, quadratic, and exponential functions (models) and the patterns of growth that arise from them; a more concrete realization of the vastness of space and the seeming paradox of having very large quantities of debris in orbit yet, at the same time, quite small probabilities of encountering any - although with potentially lethal results should an encounter happen; and an appreciation of the power of mathematics to help us "get our arms around "
seemingly unmanageable problems.
Data included in this unit are taken from various NASA sources, but you will want to be alert to updated information and current events related to the problem. For example, three events in early 1996 received extensive coverage in the media. One was the Space Shuttle Endeavour's maneuver to steer clear of a defunct satellite. A second was the breaking loose of a satellite and its 12-mile-long tether during an attempt to deploy it from the Shuttle. A third was the return to Earth of a Chinese satellite, which problem landed somewhere in the Atlantic Ocean. Such events help make more real the concern over the
problem of space pollution. Also, you will want to remind your students that such data as the amount of debris in orbit are estimates based on the best technology and information available at a given time. These estimates are
continually revised as new information becomes available. You should not be unduly concerned about whether the amount of debris or the rate of accumulations, for example,
represents the latest, most precise numbers; rather, you will want to focus on patterns of change and on how those quantities grow or decrease over time under various assumptions, such as linear versus quadratic versus exponential growth. When you find different estimates of some of these quantities, use this excellent opportunity to ask, How shall we adjust our models to account for this
new information?
Getting Started
In January 1996, two days into their mission, the crew of the Space Shuttle Endeavour performed a maneuver to slow its speed by 4 feet per second, thereby steering clear of a 350-pound piece of "space junk," a defunct Air Force
satellite that would have passed within 4/5 of a mile of Endeavour. Although the Shuttle was in no danger of collision, NASA flight safety rules call for at least 1.3 miles of separation between the Shuttle and any other orbiting object. The maneuver widened the distance to 6 miles.
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This computer-generated image shows the thousands of satellites, spent-rocket stages, and breakup debris in low-earth orbit. |
This computer-generated image illustrates Earth's rings formed from human-made orbital debris. |
The incident, however, called attention to the growing problem of space debris. In seventy-four Shuttle missions, only about six have had to perform maneuvers to avoid orbiting objects; yet the Air Force was catalogs nearly 8000 orbiting manufactured objects of grapefruit size and larger. As humans and their vehicles and probes prepare to spend longer and longer times in space, they must also prepare to confront the growing problem of space debris.
NASA scientists are just beginning to realize the enormity of this space-age form of pollution. And just as pollution is creating terrible problems for us on Earth, so, too, are we experiencing hazards with space pollution. Estimates made in 1990 stated that more than 4 million pounds of manufactured materials were in Earth orbit. Of that amount, only 5 percent represented operating payloads; the other 95 percent consisted of human-made debris: old rocket parts, nonfunctioning satellites, discarded tools, the by-products of explosions and collisions, and other odds and ends, as well as countless, numbers of smaller
objects, such as paint chips, and dust-sized particles. On the basis of the rate at which launches were occurring in 1990, we expected the nations in the space business to contribute nearly 2.7 million pounds per year by the year
2000. The prediction warns that if we don't change our ways, we will have 9.5 million pounds of human-made materials circling the Earth.
Rings are already forming around Earth - not the rings of rock and dust and ice that encircle other planets, but rings of human-made orbital debris - and their density is increasing. According to Don Kessler, a NASA scientist who has made a career of studying space debris, "Rings are nature's way of saying it doesn't like things in non-circular orbit out of Earth's equatorial plane. Nature wants to tear these objects apart and reform them into either a ring or a single object-it is just a question of when."
The North American Aerospace Defense Command (NORAD) reported in 1986 that 4488 of 6194 radar-trackable objects were orbital debris. The rest included 1582 payloads, 68 interplanetary probes, and 56 items of interplanetary probe
debris. A radar-trackable object in space is baseball size or larger, but during hypervelocity, which begins at 3 kilometers per second, particles as small as a paint flake can be damaging or even lethal. By the mid-1980's, ground based telescopes made it possible for scientists to see marble-sized pieces of orbital debris. From these observations, they concluded that the number of debris
objects was many times the number that NORAD had cataloged.
The first loss of a spacecraft part directly attributable to human-made orbital debris occurred during the Shuttle mission of STS-7 in 1983. The crew of the Challenger reported an impact crater on one of the orbiter's windows significant enough to require replacement despite the fact that the window was 5/8 inch thick and built to withstand pressures of 8600 pounds per square inch
and temperatures up to 482°C. By studying the traces found in the pitted window, NASA determined that the damage was caused by white paint specks about 0.2 millimeters in diameter traveling between 3 and 5 kilometers per second.
Cosmonauts on the Soviet spacecraft Salyut 7 reported a similar window incident just weeks later and even reported hearing the impact. The Solar Maximum Mission satellite had been in space for fifty months when the crew of STS-41C repaired it in space and returned to Earth with 15 square feet of the insulation blanket and 10 square feet of aluminum louvers showing thousands of pits and excessive wear and tear. The blanket showed thirty-two holes per
square foot and the louvers, six holes per square foot, many more than NASA scientists expected; analysis revealed that most of the pits were caused by paint flakes.
Collisions and breakups significantly increase the number of particles orbiting Earth, but they differ in fragmentation and the resulting hazards. Collisions produce smaller fragments and increased hazard. If a 10-pound mass
hits a 1000 pound stage at orbital speeds, a tremendous amount of energy is released that could result in 4 million particles and 10000 larger pieces. Breakups caused by explosions produce fewer small fragments, which give
scientists a greater likelihood that they can learn about the causes of explosions and thus prevent them. For example, when seven second-stage Delta rocket breakups occurred three years after launch, scientists were able to
determine that they had been caused by unspent hypergolic fuels; this knowledge resulted in launch changes that apparently corrected that problem.
Modeling Space-Debris Accumulation
Space Debris: Is It Really That Bad?
The information presented in the Modeling Orbital Debris Problems activity sheet includes numbers like 4 million pounds, 1.8 million pounds per year, and 9.5 million pounds. We have become accustomed to reading such numbers, but few of us have an intuitive sense of what they represent. At the beginning of the activity, students are asked to come up with some concrete models of just what such quantities mean. They should find it enjoyable to come up with examples like "9.5 million pounds of pennies would fill" and other original representation.
Modeling the Problem: Linear Growth
The NASA predictions reported in this activity afford an excellent opportunity to have students create and analyze simple mathematical models. Because mathematicians strive to find the simplest model that will adequately represent the problem, a good starting point for students is to create linear models for the amount of accumulated space debris on the basis of the number of pounds being added each year. In so doing, they will look at the 1990
accumulation rate of 1.8 million pounds pre year (rate of increase) and an initial amount of 4 million pounds. If we let the years be represent by t (with t = 1 in 1990) and the total pounds of debris by y (in units of 10^6 pounds),
our linear model is y = 1.8t + 4. Similarly, if we use the rate of .7 million pounds per year, we get y = 2.7t + 4. This result gives rise to two new questions: (1)Where does the "eventual" 9.5 million pounds come from? and (2)Does either model realistically describe the situation? In this activity, it is also suggested that you make the analogy to velocity. Because students are familiar with distance-rate-time problems, they should be able to see that increasing the amount of debris at a constant rate is equivalent to increasing the distance traveled when moving at a constant speed. Using the two linear equations or a table, students discover that 9.5 million pounds occur in just over three years at the first rate and in just over two years at the second rate (see table 5).
A good discussion should result regarding the fact that the prediction did not specify when the 9.5 million pounds would be reached; as students will see later, other actors must be considered when evaluating the prediction. The
students should also realize that neither model is accurate because neither rate, 1.8 million pounds per year and 2.7 million pounds per year, applies for the entire period from 1990 through 2000.
Refining the Model: Quadratic Growth
Once students realize that neither of their original attempts at a model truly represents the situation, they should try to adjust the model to account for the fact that neither the rate of 1.8 million pounds per year continues
throughout the period. Because scientists and mathematicians prefer to start with a simple model and then add complexities to refine it as needed, we make the simplest assumption to account for the changing rate of increase over the period, namely, that the rate of adding debris increases at a constant rate from 1.8 million pounds per year to 2.7 million pounds per year. That is, the rate
(velocity) of littering increased by 0.9 million pounds per year achieved in equal increments of 0.09 million pounds per year each year over a ten-year period. The students can then create a table and a graph of the rate of "dumping" versus the year by using the data points (0,1,8) and (10,2,7). This change of rate of dumping is given by the equation d = 0.09a + 1.8, where a = 0 in 1990. The increase in both the dumping rate and the total accumulation are
shown in table 6 and figure 3.
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Fig. 3. Graphs showing annual increases and total amounts of debris (in millions of pounds)
At this point emerge a number of important concepts to explore. Although both plots in figure 3, and those that your students generate, may appear to be linear over the ten-year range of the initial data, extending the domain to
twenty or thirty or fifty years, as suggested on the student page, makes it evident that the graph of the end-of-year total accumulation is, indeed, not linear. The fact that it is quadratic can be approached from several
perspectives:
- Students can manually create a best-fit line and check its predictions against those generated by the table patterns.
- Students can generate linear and quadratic regression equations on a graphing calculator and overlay both the linear-regression graph and the quadratic-regression graph on the scatterplot of the data.
- Students can then use the method of constant differences and the data from the tables to generate a quadratic equation. This method is summarized in the
next section.
Applying the Method of Constant Differences
- Review the fact that for all linear functions, the differences between successive values of f(x) are constant when x is an integer. We refer to these as "first differences." In the case of quadratic functions, the first
differences are not constant, but the differences between them, the "second differences," are constant. This pattern can be extended to higher-order polynomial functions: cubic functions are constant in the third differences, quartic equations are constant in the fourth differences, and so on. Table 7 shows the pattern for the general linear and quadratic cases and for two specific examples from our debris problem.
- Use the data in table 7 to generate the following equations. Solve them for a, b, and c.
|
a + b + c =
5.8 |
c = 4 |
|
3a + b = 1.89 |
b = 1.755 |
|
2a = 0.09 |
a = 0.045 |
The desired function is f(x) = 0.045x2 + 1.755x + 4, where f(x) gives the amount of debris in orbit at the end of year x and x
equals 1 in 1990. Overlay this equation on the earlier scatterplot.
| Table
7
Pattern for General Linear and Quadratic Cases
with Examples |
|
Linear Function: f(x) =
ax + b |
|
General Case |
f(x) = 1.8x + 4
|
| x= |
f(x) |
First difference
|
f(x) |
First difference
|
|
1 |
a + b |
|
5.8 |
|
| |
|
a |
|
1.8 |
|
2 |
2a + b
|
|
7.6 |
|
| |
|
a |
|
1.8 |
|
3 |
3a + b
|
|
9.4 |
|
| |
|
a |
|
1.8 |
|
4 |
4a + b
|
|
11.2 |
|
|
Quadratic function: f(x) =
ax2 + bx + c |
|
General Case |
Debris Example
|
|
x= |
f(x) |
First differ- ence
|
Second
differ-
ence |
f(x) |
First differ- ence
|
Second differ-
ence |
|
1 |
a + b + c
|
|
|
5.8 |
|
|
| |
|
3a + b
|
|
|
1.89 |
|
|
2 |
4a +2b +c
|
|
2a |
7.69 |
|
0.09 |
| |
|
5a + b
|
|
|
1.98 |
|
|
3 |
9a + 3b +c
|
|
2a |
9.67 |
|
0.09 |
| |
|
7a + b
|
|
|
2.07 |
|
|
4 |
16a + 4b+c
|
|
|
11.74 |
|
|
Velocity-Acceleration Analogy
The student pages compare the quadratic model for debris with the more familiar case of constant acceleration. Students who have studied motion in a physics class should make the connection between examples of uniformly
accelerated motion, such as a free fall, a ball rolling down an incline, or a car accelerating at a constant rate, and the "uniform acceleration of space dumping." With your more advanced mathematics students, you will want to pursue
the fact that the linear equation d = 0.09x + 1.8, where
x is the number of years since 1990, is the equation of the tangent to the curve y = 0.045x^2 + 1.755x + 4, the equation previously derived. That is, the first equation is the derivative of the second.
In this section, students are likely to have the greatest difficulty differentiating among the "rate (velocity) of littering" (e.g., 1.8 million pounds per year in 1990), the "total change in the littering rate over then
years" (i.e., an increase of 0.9 million pounds per year between the 1990 rate of 1.8 million pounds per year and the 2000 rate of 2.7 million pounds per year), and the "rate of change (acceleration) of littering velocity" (an increase of 0.09 million pounds per year each year). You will need to pay special attention to these concepts.
In comparing the quadratic model, y = 0.045x^2 +1.755 + 4, with the two linear models, y = 1.8x + 4 and y = 2.7x + 4, students will find that the graph of the quadratic lie between the two lines. They should be able to interpret
this result in terms of what would happen if the rate of adding debris stayed at 1.8 million pounds per year, was always 2.7 million pounds per year, or gradually changed from 1.8 to 2.7 million pounds per year. They should also
determine when the value of the quadratic will surpass the second linear function and recognize that thereafter the quadratic will always produce greater values.
One More Perspective: Exponential Growth
The third model proposed in this unit is an exponential one: What will happen if the debris accumulates as a fixed percent of what is already in orbit? This question is, of course, analogous to the familiar problem of compound interest, and it gives an alternative context for exploring the concept of exponential growth. It is a setting in which you will almost surely want students to use a spreadsheet.
The major objective is to help students realize the impact of exponential growth. With a spreadsheet they can hypothesize different rates of increase and explore what would happen if debris were to be added at those rates. The
findings can be compared with the patterns of linear and quadratic growth. Some examples are shown in table 8 and the graphs in figure 4; the amounts are in millions of pounds.
Fig. 4
What Goes Up Might Come Down: Extending the Models
Students should realize, however, that not everything in orbit stays there forever. Orbiting objects will also be pulled back into Earth's atmosphere, where they will burn up or, on occasion, return to Earth. To be realistic, the models that students develop should take into account both the addition to, and the destruction of, orbital debris.
We do not provide any particular data for this section so that students will have the opportunity to form and test hypotheses of their own. They should try modifications that assume destruction at a fixed rate per year (linear) as well as at a fixed percent (exponential decay). They are looking for models that might predict an eventual decrease in the amount of orbital debris even though we assume that NASA and others will continue to launch payloads and thereby continue to add to the orbital debris. These open-ended investigations are designed to let students experience the search for models that might lead to
desirable consequences - although, in reality, we may need to invent the necessary technology to implement the model.
Putting Your Models to Work
Finally, students are encouraged to generate "What if" questions and to use their models to help generate answers. Before you ask them to formulate their questions, you might want to pose some of your own. The following are examples:
What if, at the end of 1995, we were able to stop adding to the debris and were able to decrease the existing debris at a rate of 0.35 million pounds per year for the next three years? What would be the total poundage
remaining at the end of 1998? What would be the percent of decrease? Note that the answers will vary depending on what assumptions are made about the rate of increase during
the years 1990-1995.
In addition, what if, at the beginning of 1999, new technology would allow us to improve the rate of decrease to 0.65 million pounds per year for a two-year period? What would be the total debris remaining at the end of the century?
Modeling Collision Effects
The photo shows the "energy flash" when a projectile launched at speeds up to 17,000 mph impacts a solid surface at the Hypervelocity Ballistic Range at NASA's Ames Research Center, Mountain View, California. This test is used to simulate what happens when a piece of
orbital debris
hits a spacecraft in orbit.
The Modeling Collision Effects activity sheet presents a problem of major significance to the aerospace community: the consequences of impact at hypervelocity. Tests done in the NASA laboratories as well as evidence from returned spacecraft confirm the enormous
damage that can be rendered by even very small particles striking at orbital speeds. This activity introduces the concept of kinetic energy. If your students are studying physics or physical science, they should be familiar with
this form of energy. The unit of a joule will be less familiar, since it is a "highbred" unit and not something for which we have everyday references. Rather than dwell
on what one joule might be, it is more useful to calculate the kinetic energy of some everyday examples, such as a truck traveling at highway speeds or a hard-hit ball, and then relate other energies, such as the kinetic energy of
impact from a point flake in orbit, as some multiple or fraction of the more familiar examples. You may be surprised to discover that a 1-gram paint chip traveling at orbital speed has nearly 1/10 the kinetic energy of a 3500-pound truck moving at 60 miles per hour and that a 350-pound satellite at orbital speed has 10000 times the kinetic energy of the truck.
In this section, students must pay close attention to units, and you will want to be sure that they recognize that to be meaningful, their calculations of such quantities as mass, velocity, and energy must be expressed in appropriate units. You will note that the units used in the activity are mixed - pounds, kilograms, grams, miles per hour, kilometers per second, and so on - because they are the units commonly used in measuring the associated phenomena. This situation is realistic, and with spreadsheets and calculators, conversions should pose no particular problems. Cautious attention to units, however, is required. Also, some measurements are left for students to find or approximate, such as the mass of a baseball or the velocity at which it may be hit. Sports enthusiasts sometimes carry this sort of information in their heads; others may prefer to use data from their own sports exploits; still others will look up the data in a book of sports statistics.
Again, the purpose of the activity is to foster open-ended investigation and discussion of the relative effects of the linear factor (mass) compared with the quadratic component (velocity). We are less concerned with precise answers to particular questions than we are with the larger concept of how these functions behave.
Modeling Collision Probability
Throughout this unit, we have looked at enormous quantities - millions of pounds - of debris. But we have not yet related those quantities to the vastness of space, even the relatively "close fitting" sphere in which payloads orbit Earth. This activity has students calculate the volume of a "shell" 200 kilometers thick extending from an altitude of 200 to 400 kilometers above Earth - a volume of just over 1.1(1011) cubic kilometers. If we
assume that 4 million pounds (1.8 million kilograms) of debris are evenly distributed throughout this volume, we find the density of debris to be on the order of 1.6(10-5) kilogram, or 0.016 gram, per cubic kilometer.
In the Modeling Collision Probability activity sheet, we assume that the debris has the density of aluminum, 2.7 grams per cubic centimeter. By making this assumption, we can estimate that the 0.016 gram per cubic kilometer represents approximately 0.006 cubic centimeter of debris per cubic kilometer of the shell. The probability of hitting any of that debris is the ratio of the volume of debris to the volume of the shell; hence, we can estimate the probability of hitting the debris to be on the order of 6(10-12). Removing the simplifying assumption of a 200-kilometer thick shell does not change the order of magnitude of the probability.
