1. Modelling a satellite exiting Earth’s Sphere of influence from low Earth orbit
Ricardo Fritzke
Department of Physics, Whitworth University, 300 W. Hawthorne Rd., Spokane WA, 99251, USA
(Dated: October 24, 2015)
A program was developed to model the trajectory of a satellite leaving a stable orbit until it
exited the Earth’s sphere of influence. The program allowed for the user to input various initial
orbit heights and thruster angle, while outputting a plot of the trajectory and the time taken to
leave earth’s sphere of influence. Comparison of these models revealed that a thrust in the direction
of the satellite’s velocity offered the fastest escape of the earth’s sphere of influence.
I. INTRODUCTION
For the majority of my time in college, my focus has
been on learning physics theory. Recently I have been
able to expand and learn some of the mathematical meth-
ods used to apply physics to more practical, real life situ-
ations. Taking Computational Physics has allowed me to
take it a step further and develop a program that provides
a relevant application by combining my coding experi-
ence with physics theory. While people have looking to
the stars for millenia, Newton was the first to accurately
model the gravitational forces between two objects. For
my project, I used Newton’s gravitational law to model
a satellite exiting the earth’s sphere of influence from a
stable orbit. The program allows for variations in initial
orbital height and thruster angle and outputs the trajec-
tory of the satellite along with the time taken to exit the
earth’s sphere of influence.
II. METHODOLOGY
A. Analytical Methods
The underlying physics behind the interactions be-
tween a satellite and the earth can be reduced to New-
ton’s second law and Newton’s law of universal gravita-
tion, given by
Fg = G
memsat
r2
(1)
where Fg is the gravitational force on the satellite, G is
the gravitational constant, me and msat are the masses
of the earth and the satellite, and r is the distance be-
tween the two. The total mass of the satellite (msat) was
approximated by summing the fixed mass and the pro-
pellant mass.The rate at which mass decreased over time
due to lost propellant (δm) was calculated by dividing
the power of the thruster (Pth) by the exhaust velocity
(vex).
δm = 2
Pth
vex
(2)
All calculatios were done with the assumption that the
satellite was only under the influence of the its thrust,
Earth’s gravitational field, and atmospheric drag. The
acceleration of the satellite can be modelled with:
a(t) =
T − Fd
msat − δm ∗ t
(3)
where T is the force of the thruster and Fd is the force
of drag. The angle of the thruster (φ) was calculated
by adding the angle input by the user (φ0) to the angle
tangent to velocity vector, given by:
φ = φ0 + tan−1
(
ru
h
) (4)
By analyzing the forces on the satellite, four first or-
der differential equations can be integrated to model the
trajectory of the satellite:
˙r = u (5)
˙θ =
h
r2
(6)
˙u =
h2
r3
−
Gme
r2
+ a(t)sin(φ) (7)
˙h = ra(t)cos(φ) (8)
where r is the radial distance from Earth’s center of
mass and θ is the angle from the x-axis. u is the radial
velocity and h is the specific angular momentum of the
satellite. To find the initial velocity of a stable orbit, the
equation for centripetal force was also needed, given by
Fc = ms
v2
s
r
(9)
where vs is the velocity of the satellite in the inital
stable orbit.

2. 2
2
4
6
8
10
30
210
60
240
90
270
120
300
150
330
180 0
FIG. 1. This satellite had an initial stable orbit at 250 km
above the earth with a constant thrust of 3 N tangent to its
velocity. It was able to escape the earth’s sphere of influence
after 123 days.
B. Numerical Methods
In order to translate the analytical equations into a
program, it was necessary to transition from a continuous
domain to a discrete domain. This allows the program to
compute results quickly as an approximation. The pro-
gram begins by asking the user to input the initial height
of the satellite’s stable orbit as well as the angle of the
thruster. The program will then run several calcluations
to determine the rest of the initial conditions. It will then
send the intial values of r, θ, u, and h through a 4th order
Runge-Kutta approximation to solve the four differential
equations. The program will automatically cut off after
the satellite has reached the edge of the earth’s sphere of
influence and will then plot its trajectory along with the
time taken.
III. RESULTS
There are two main results that occur from initializing
the program with different parameters. In one scenario,
the satellite will be able to accumulate enough velocity to
exit the earth’s sphere of influence (see Fig. 1). A second
possibility is that the thruster is placed at an angle that
does not allow for the satellite to gain enough speed to
escape the sphere of influence, and it ends up orbiting the
earth indefinitely (see Fig. 2). When starting the satellite
from an initial orbit with a larger radius, the amount of
time it take to leave the sphere of influence is reduced,
agreeing with what we would expect (see Fig. 3).
After several trials, it was easy to see that the most
efficent angle of thrust was 0 degrees and that the higher
the initial orbit, the faster the satellite was able to escape
earth’s sphere of influence. (see Table I). Interestingly,
the force of drag had extremely small effects on the tra-
jectory of the satellite. Depending on the inital height of
0.02
0.04
0.06
0.08
0.1
30
210
60
240
90
270
120
300
150
330
180 0
FIG. 2. This satellite had an initial stable orbit at 250 km
above the earth with a constant thrust of 3 N 70 degrees above
the angle tangent to its velocity. It was unable to escape the
earth’s sphere of influence and continued to orbit indefinitely.
2
4
6
8
10
30
210
60
240
90
270
120
300
150
330
180 0
FIG. 3. This satellite had an initial stable orbit at 35786 km
above the earth (geosynchonous) with a constant thrust of 3
N 0 degrees above the angle tangent to its velocity. It escaped
the earth’s sphere of influence after 46.430 day.
the satellite, drag would only slow down the satellite by
several minutes over approximately a 3 month time pe-
riod. This is a result of the exremely low density of par-
ticles in space, again in agreement with what we would
expect.
IV. CONCLUSION AND FINAL THOUGHTS
A. Conclusion
This program was able to sucessfully model the tra-
jectory of a satellite exiting earth’s sphere of influence
from a stable orbit. The initial height or the orbit was a
large factor in travel time, and the angle of the thruster
was a large factor on the chance of escaping the sphere of
influence. From ( Table I) we can see that as the initial
orbit height increases, the exit time decreases. Likewise,

3. 3
TABLE I. Time taken to leave Earth’s SOI for differnet initial
conditions
Orbit Height Thruster Angle Time
250 0 123.008
250 30 134.722
250 70 N/A
1500 0 113.481
1500 45 155.581
35786 0 46.430
35786 30 52.316
35786 70 120.353
the lower the thruster angle, the faster the exit time.
While this project was fairly simple in nature, it was a
very valuable learning tool for MATLAB programming as
well as problem solving and troubleshooting. However,
there was a persistant issue with my code, and that was
the drag function. It would seem to work correctly under
certain conditions, but not during others. For whatever
reason, it would occasionally decrease the time to exit the
sphere of influence instead of increase. The drag function
was a last minute addition and I was unfortuanely not
able to rigorously verify it.
B. Final Thoughts
They say that hindsight is 20/20, and I am certainly
glad that I took this course. The learning process was
not simple, easy, or fast, but it was worth it. I knew that
this course would be harder than any Gen-Ed, but I took
it because I wanted to learn something useful. Coming
from nearly no coding experience, I feel like now I have a
very good understanding of MATLAB. This will prove to
be an important skill for any future job or internship, as
there are many situations where computational solutions
are the only way to solve a problem. An important lesson
that I have learned from this class is to be persistant,
precise, and tp utilize my resources. Almost every code
written this Jan-Term felt like it left me in the deep end
of the pool, and it required patience and resourcefulness
to solve, but that is how I needed to learn these skills
that I will be able to use for the rest of my engineering
career.