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Reentry of Satellites into the Atmosphere
Math 238 – Spring 2015
Rick Snell
Edgar Mendoza-Seclen
Hannah Carson

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Abstract
The reentry of a satellite can be modeled using Newton’s Second Law of motion. The
equation used in our model was based on the famous Russian satellite, Sputnik 1. The model
assumes the mass is constant, the Earth is flat and non-rotating, and neglects solar and weather
activity.
With this model, we are able to find the impact velocities and determine the time at
which the space debris, or in our case a satellite, will hit the ground. Assuming the satellite does
not burn up in the atmosphere, we can predict how long a populated area has in order to
evacuate and predict how much damage the space debris or satellite will cause, based on the
speed.
Introduction
There is an old saying, “what goes up, must also come down.” Today, scientists are
facing the issue of how man-made objects, such as satellites, will make it safely back to the
ground. Over the past several decades, satellites and other spacecraft have scattered debris
totaling to about 6,700 tons into space around the Earth. All of this space debris is merely
orbiting our planet, waiting to come back down.
Reentry into the atmosphere can cause an object to travel hundreds of thousands of
kilometers in a matter of minutes. Because of the vast uninhabited regions of the planet, space
debris tends to land in either the ocean or a desert area, causing no damage to human life or
property. Also, some objects are small enough, or structurally weak enough to simply burn up
upon reentry. But if a satellite or other piece of space debris lands in a populated area, it could

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potentially cause immense damage to buildings or people. This debris is neither controlled nor
stable, so it is difficult for scientists to predict the exact location of impact. But, with both lives
and structures at risk, it is imperative to try to model their trajectories.
The main concern this project will focus on is finding the impact velocities and
determining the time at which the space debris or in our case, a satellite, will hit the ground.
With these two results, we will be able to predict how much time a populated area may have to
evacuate and how much damage the space debris or satellite will cause.
Explanation of Model
A mathematical model of a satellite reentering Earth’s atmosphere and falling toward
the ground, same as any physical body in motion, is best described using Newton’s 2nd law of
motion:
∑ 𝐹 =
𝑑
𝑑𝑡
(𝑚𝑣)
For our purposes, we have chosen to focus on two forces acting on the satellite i.e. the forces
due to drag and gravity. For gravity, we once again turn to Newton for a description. The law
of universal gravitation states:
𝐹𝐺 = 𝐺
𝑚1 𝑚2
𝑟2
where G is the universal gravitational constant, m1 and m2 are the masses of the respective
bodies acting on each other, and r is the distance between the centers of mass of the objects.
For drag, we rely on information gathered from the University of Texas website. This resource
describes the force of drag on artificial satellites as follows1:
1. Equation fromsource is givenperunit mass.

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𝐹𝐷 = −
1
2
𝜌0 𝑒
−
𝑟−𝑅
𝐻 𝐶 𝐷 𝐴𝑣| 𝑣|
where ρ0 is the atmospheric density at ground level (approx. 1.3 kg/m3), r is the distance of the
satellite from the Earth’s center, R is the radius of the Earth (6.4x106 m), H is the atmospheric
scale height (approx. 8.5x103 m), CD is coefficient of drag, A is the cross-sectional area of the
satellite perpendicular to the direction of motion, and v is the velocity of the object. The
exponential term in this equation allows us to model the air density based on altitude, as
density decreases roughly exponentially with height. Inserting these equations into Newton’s
2nd Law of Motion and assigning up as the positive direction, we have:
−𝐺
𝑚1 𝑚2
𝑟2
−
1
2
𝜌0 𝑒
−
𝑟−𝑅
𝐻 𝐶 𝐷 𝐴𝑣| 𝑣| =
𝑑
𝑑𝑡
(𝑚𝑣)
Our next step is to substitute the appropriate symbols and dependent variables, setting the
ground as the origin and remembering that we have oriented up as positive:
−𝐺
𝑚𝑀
(𝑅 + 𝑦)2
−
1
2
𝜌0 𝐶 𝐷 𝐴𝑒
−
𝑦
𝐻 𝑦̇| 𝑦̇| = 𝑚𝑦̈
having m as the mass of the satellite, M as the mass of the Earth, y as the height above the
Earth’s surface, 𝑦̇ as velocity, and 𝑦̈ as acceleration.

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We selected a convenient historical example, Sputnik 1, as the satellite in question. The
convenience lies in the fact that Sputnik was spherical. This means that the value A in our
equation will be the same for all orientations of the satellite, and according to the same
University of Texas source, CD is approximately 2.2 for spherical satellites. Using the
specifications of Sputnik we can determine:
𝑚 = 83.6 kg , 𝐴 = 0.0841𝜋 m2
This leaves initial height and velocity as the only remaining values to be determined arbitrarily.
First, the boundary between Earth’s atmosphere and “outer space” is accepted by
international standard to be the Kármán line, or 100 km above the Earth’s surface and our
problem models the reentry of a satellite into Earth’s atmosphere. So we have:
𝑦0 = 100,000 m
Second, according to Tom Henderson of the Physics Classroom, a satellite’s orbital speed
is inversely proportional to the square root of its distance from the Earth’s center. Using the
values we already know, we have:
𝑣 = √
𝐺𝑀
𝑅 + 𝑦0
≈ 7835.3 m/s
From an online article posted by the Massachusetts Institute of Technology, the controlled
reentry angle of Space Shuttles is typically 40 degrees. This angle is calculated to balance the
risks of burning up in the atmosphere and bouncing back into orbit (bouncing is caused by the
force of lift due to the Space Shuttles wings). Because our satellite lacks wings, we need not be
concerned with bouncing, therefore any angle less than 40 degrees will suit our needs. Taking

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10 degrees below the horizontal as the angle of entry into Earth’s atmosphere, we can find the
initial vertical velocity by:
|𝑣 𝑦| = | 𝑦0̇ | = 7835.3sin(10) ≈ 1360.6 m/s
Our model is now complete, bearing in mind the following assumptions. (1) Solar
activity affects the density and thickness of the Earth’s atmosphere, so for our model we must
assume no solar activity. (2) Meteorological factors, such as jet streams and high/low-pressure
systems can impact the trajectory of an object falling to Earth; therefore, we must assume no
meteorological disturbances. And (3) the curvature, rotation, and topography of the Earth
must all be taken into account for a realistic model of satellite reentry; however, the
mathematics required to incorporate these factors is very advanced. Thus we must assume a
perfectly flat, non-rotating Earth in order for our one-dimensional model to make any sense.
Derivation of the Differential Equation
Our differential equation is an ordinary, second order, non-linear differential equation.
Because our equation is non-linear, we used a computer algebra system, Maple, to solve the
equation numerically. The dependent variables are y(t), y’(t), and y’’(t). These variables stand
for position, velocity, and acceleration respectively. Our independent variable is t, representing
time measured in seconds. Our equation contained many constants, such as G, m, M, R, ρ0, CD,
A, H. These stand for the gravitational constant, mass of the satellite, mass of the earth, radius
of the earth, density of the atmosphere at sea level, the coefficient of drag, cross-sectional area
of the satellite perpendicular to the direction of motion, and the atmospheric scale height,
respectively.

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The solution was produced by the Fehlberg fourth-fifth order Runge-Kutta method. With
this approximation, we were able to plot the satellite’s velocity versus time and the height
above the Earth’s surface versus time.
A few assumptions we made were a constant mass and density of the satellite, the
satellite doesn’t burn up, no solar or weather activity, and the Earth is a flat, non-rotating
object. These assumptions overall simplify the differential equation. Assuming the satellite does
not burn up or break apart allows us to hold mass as a constant, instead of another dependent
variable. Solar and weather activity could potentially change the force of air resistance changing
the density of the air and thickness of the atmosphere. Finally, assuming the Earth is flat and
not rotating, we are able to use a one dimensional model instead of a three dimensional model
with spherical coordinates.
Solution to Differential Equation
As mentioned above, this non-linear equation was solved numerically. The graph below
represents the height of the satellite versus time as it falls to the ground. As can be seen from
the graph, the satellite hits the ground at 330.27 seconds, or about 5.5 minutes. If the satellite
is uncontrolled this leaves less than adequate time for an area to evacuate, especially if it were
over a populated city.

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The second graph represents the velocity versus time. It shows that the velocity initially
increases but then decreases due to air resistance because the air density increases as height
decreases. The force of gravity is stronger than the force of air resistance for about the first 27
seconds. After that, the drag increases, causing the satellite to start slowing down. Because the
density of the air is a function of height in our model, terminal velocity is dependent upon
altitude and therefore decreases as the satellite approaches the ground.

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As determined from the graph, the impact velocity is around 47 m/s downward. The
diameter of a crater is proportional to the kinetic energy of the impactor. In other words, if this
were a smaller satellite, such as Sputnik, this would not cause significant damage. But as the
mass or velocity of the satellite (or piece of space debris) increases, the size of the crater and
damage will also increase.
The two graphs below show varying initial conditions, holding mass constant. The first is
with different initial velocities. Whether the satellite starts at rest or from a speed greater than
our modeled satellite, the impact velocity, when the satellite hits the ground, is consistently
around 47 m/s.
The second graph compares different initial heights, while holding mass constant. The
same is true as above that the impact velocity is about 47 m/s, no matter where the initial
height of the satellite.

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The final two graphs both hold initial velocity and initial height constant. The first
represents the height versus velocity of satellites with different masses. As can be seen, with
increasing mass, the terminal velocity is reached at a lower height and this causes the impact
velocity to increase.

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The second graph represents the height versus velocity at different cross-sectional
areas. When the surface area increases, terminal velocity is reached at a higher altitude and
thus the impact velocity is decreased.
Conclusion
If a piece of space debris or a satellite does not burn up while reentering the
atmosphere, scientists must consider the landing. The reentry can cause space debris or
satellites to travel thousands of kilometers in a matter of minutes. If the satellite is projected to
land over a body of water or a deserted area, scientist do not have to be concerned about the
impact on the Earth. If the satellite is projected to land over a populated area or structure,
scientists must take into account the landing speed and the time of fall. This will allow them to
take precautions, such as having an area evacuate.
Newton’s Second Law of Motion is a way to model the velocity and time of impact.
Using Sputnik’s specifications as our main model, we can predict that the satellite will hit the

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ground shortly after 5.5 minutes at a speed close to 47 m/s. Because Sputnik is a fairly small
satellite, the impact on the Earth is relatively insignificant. If the size of the satellite increased,
scientists may need to be more considerate of the impact. Overall, initial conditions such as
height and velocity do not alter the speed at impact, however mass and surface area have a
significant effect on impact velocity.

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References
Deziel, C. (n.d.). Facts on reentry into the earth's atmosphere. Retrieved May 8, 2015, from
Synonym website: http://classroom.synonym.com/reentry-earths-atmosphere-
6679.html
Fitzpatrick, R. (2014, August 2). Effect of atmospheric drag on artificial satellite orbits. Retrieved
May 8, 2015, from
http://farside.ph.utexas.edu/teaching/celestial/Celestialhtml/node94.html
Gaposchkin, E. M., & Coster, A. J. (1988). Analysis of satellite drag. The Lincoln Laboratory
Journal, 1(2), 203-224. Retrieved from
http://www.ll.mit.edu/publications/journal/pdf/vol01_no2/1.2.6.satellitedrag.pdf
Garber, S. (2007, October 10). Sputnik and the dawn of the space age. Retrieved May 8, 2015,
from NASA website: http://history.nasa.gov/sputnik/
German Aerospace Center. (n.d.). The re-entry of the ROSAT satellite and the risks (H. Klinkrad,
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http://www.dlr.de/dlr/en/Portaldata/1/Resources/documents/ROSAT_klinkrad_en.pdf
Grayzeck, E., Dr. (2014, August 26). Sputnik 1. Retrieved May 8, 2015, from National
Aeronautics and Space Administration website:
http://nssdc.gsfc.nasa.gov/nmc/spacecraftDisplay.do?id=1957-001B
Henderson, T. (2015). Mathematics of Satellite Motion. Retrieved May 18, 2015, from The
physics classroomwebsite: http://www.physicsclassroom.com/class/circles/Lesson-
4/Mathematics-of-Satellite-Motion

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Maplesoft. (2015). dsolve/numeric/rkf45. Retrieved May 8, 2015, from Maplesoft
website: http://www.maplesoft.com/support/help/Maple/
view.aspx?path=dsolve%2frkf45
Ritter, M. (2014, March 28). How many man-made satellites are currently orbiting earth.
Retrieved May 8, 2015, from Talking Points Memo website:
http://talkingpointsmemo.com/idealab/satellites-earth-orbit