Chadwick Healy
10 DEC 2009
ASEN 5050: Space Flight Dynamics
Aerospace Engineering Sciences
University of Colorado at Boulder
Tethered satellites provide the unique opportunity of creating thrust through its operation in the Earth’s magnetic field. The complex dynamics create challenges to design a suitable tethered satellite that can be stabilized and survive in the space environment, where risk of tether severance is a considerable factor. Recognizing that there have been no tethered satellites launched in over a decade, the necessity to reinvigorate research will require affordable opportunities for design until confidence is regained in the survivability of such a system. This paper conducts a preliminary design of a tethered small satellite suitable for the CubeSat missions, its potential role in science, and missions of future tethered satellites.
The purpose of this report is to portray an accurate model of an electrodynamic tethered satellite system, and the complexity in controlling it as a unit. A summary of past and current research efforts is provided as well as a suggested course of action, a preliminary design for a low budget electrodynamic tether, and future potential applications of a tethered satellite system.
The fact that increased mass means increased cost is common knowledge to every aerospace engineer. Laws developed years ago by visionaries such as Newton and Kepler still govern the how missions are designed and conducted. It is no wonder that alternatives to fuelbased propulsion are a constant focus of research. In this regard, the opportunities provided by the deployment of an electrodynamic tether extend into the realm not distant from the development of an ion engine. With respect to orbital maneuvering, an electrodynamic tether trumps all new propulsion techniques, in that as long as it is exposed to the sun, it will have adequate power to maneuver around Earth’s magnetic field. No longer restricted to Keplerian orbits, a tethered satellite system has caught the interest of military, scientists and government alike.
Unfortunately there hasn’t been a tethered satellite mission in over a decade, and those which were launched had limited success. Difficulties in the stability and survivability of a tethered system has discouraged institutions like NASA and NRL from funding future missions due to the high cost coupled with the high risk of failure. A viable solution is to take advantage of the various small satellite programs available to students and amateur satellite builders.
By conducting a preliminary design of a tethered satellite system which uses the CubeSat platform, one may see there are affordable opportunities to conduct the science and engineering necessary to ready a large tethered satellite mission. The opportunities and advantages provided through a tethered satellite system outweigh discarding it altogether. First, engineers must gain confidence in its safety and reliability, which can be accomplished through multiple small satellite missions.
The behavior of a tethered satellite system is a complex combination of forces with sources such as gravity, electromagnetics, and aerodynamics. With a complex, nonrigid body, stability of such a system becomes a primary concern. The simplest model would treat the system as two point masses connected by a massless, rigid rod. While for the purpose of some studies this model is sufficient, considering such concepts as the attitude control system, mass differentials and the distance between the masses are causes for variable accelerations to be experienced throughout the system. Each of the three main contributorsgravity, drag, and electromagnetic thrust, will be explored below and their implications for tethered satellite design.
Gravity gradient forces are paramount to tethered satellites. It acts as a means of stabilization, to keep each end mass separated, and maintain tether tension. Obtaining sufficient gravity gradient force becomes a driving factor in tether length and mass differential between ends. Consider Newton’s Law of Gravitation, shown below:
(Vallado) (1)
Where G is the Universal gravitational
constant (6.673 x 10^{11 }Nm^{2}/kg^{2}), M_{1}
and M_{2} are each object’s mass, and r is the distance between them. Observing the equation above, it is
clear that gravitational acceleration changes nonlinearly with the distance
from the center of the Earth. Applying
this to a tethered, earthpointing, satellite system, the subsatellite further
from the Earth experiences less gravitational acceleration than its sister
satellite below it. In other words, the
center of gravity may not correspond to the center of mass of the tethered
satellite. The gravitational force is
balanced by the centrifugal force, governed by the following equation:
(Vallado) (2)
A
model of forces acting on the satellite is shown below:
Figure 1. Model of Forces acting on a tether due to gravity and
rotation. (Cosmo)
While one may like to
assume that the tethered satellite is a simple dumbbell, the necessary gravity
gradient needs to be achieved in order to do so. From derivations, the gravity gradient force
is approximately equal to:
(Cosmo)
(3)
While the presence of gravity gradient
forces create stability in the vertical orientation of the tether system, one
must consider forces such as atmospheric drag, solar heating and electrodynamic
forces which cause the system to oscillate. According to the Tether in Space Handbook, if the
libration angle exceeds 65⁰, the tether can go slack. This would cause even greater difficulty in stabilizing
the tether and regaining tension.
Especially
important for the application of an electrodynamic tether is the magnetic
environment of Earth. The system can be
designed to either produce thrust, or electrical power. The electrodynamic tether produces this
thrust by using a Lorentz force, defined by the following equation:
(Stevens)
(4)
Where F is the force, i is the current, L is the length and
direction of the conductive tether, and B is the magnetic field vector. As the tether system moves in orbit through
the magnetic field, a current can be driven through the conductive tether,
creating a force vector along the direction of flight. This essentially converts the electrical
energy into orbital energy, raising the tether to a higher orbit. Conversely, a current can be induced in the
tether to counter the change in magnetic flux, trading orbital energy for
electrical power.
Approaching
the electrodynamics of the tether system as a circuit, in order for current to
flow it must be closed. Devices on each
end leave the tether open to the exchange electrons with the ambient
plasma. This creates a limitation on the
operational orbital altitudes of the satellite. It must be in the presence of plasma (in other
words, in the ionosphere) in order to conduct current. This sets the maximum allowable altitude at
about 2000 km. (Bitzer). This altitude
cap includes all manned missions and Low and Medium Earth Orbits.
In
order to accurately predict the amount of thrust the system can produce, one
must have an accurate model of the varying magnetic field. The Earth’s magnetic field can be modeled as
a tilted dipole shown below:
Figure 2. Earth’s Magnetic Field, A Tilted Dipole (Earth’s
Magnetic Field)
The magnetic field vector can be modeled
with the following equation:
(Stevens)
(4)
Where ϒ_{m
}is the Earth’s magnetic dipole moment, i is the inclination relative to the magnetic equator, and B_{r},
B_{t}, and B_{n} represent the magnetic flux density vector
components in the radial, transverse and normal directions. (Stevens). An important consideration in orbit selection
is low magnetic field values at high inclinations. This affects how much thrust
is available from a set amount of current running through the tether.
Due to the
length of the tether and the variable atmospheric density, differential drag on
the system will cause libration oscillations.
In order to compensate for drag using an electrodynamic tether, an
estimate of the drag force experienced is modeled as to determine the required
current. Force due to drag is calculated
using the following formula:
(Vallado)
(5)
Where ρ(r) is the average air density for radial
distance, r, and B* is the average ballistic coefficient of the tether system. The ballistic coefficient is defined as:
(Vallado)
(6)
Where C_{D} is
the average coefficient of drag, A is
the crosssectional area perpendicular to the velocity vector, and m is the mass. The atmosphere density can by modeled using
an exponential decay function based on a given scale height, h, and is shown below:
(Stevens)
(7)
A summary of
these forces can be seen in the figure below.
Figure 3. Force Diagram of the Electrodynamic Tether
An important consideration in the design of a tether is the material selection. The length of the tether will depend on factors such as the gravity gradient needed, or the amount of thrust required. The material of this tether must have a high enough tensile strength to withstand the full mass of each subsatellite, as well as have a low enough degradation rate to withstand heat generated by friction and exposure to elemental oxygen. One of the largest driving factors in selecting a tether length and diameter is its probability of severance due to micrometeoroids. This is dependent on the flux of orbital debris, the diameter of the debris and of the tether, and the mission duration. As we will discuss later, some tethered satellite missions have experienced this exact catastrophic failure.
Since the beginning of spacebased missions, there has been an interest in utilizing tethers for plethora of space applications. A tethered satellite’s mission ranges across the spectrum of scientific studies of the atmosphere, creating a space elevator, power and thrust generation, attitude stabilization of the International Space Station, and a rendezvous system. (Cosmo). Of this long list of applications, few have been launched, and even fewer have been successful. An overview of the tethered missions is found below with a summary of its intended mission and result.
The OEDIPUS (Observations of Electricfield Distribution in the Ionospheric Plasma – a Unique Strategy) series of satellites were launched in 1989 and 1995 with the objectives to make observations of the auroral ionosphere, measure the effects of the plasma on the satellite, and study radio frequency propagation in plasma. Both successfully deployed tethers of lengths 958 m and 1170 m, and had a short mission (around five days). (Cosmo).
The Tethered Satellite System1 (TSS1) was launched July 31, 1992 on STS46. The mission of the satellite was to demonstrate the capability to deploy a long gravitygradient stabilized tether from the shuttle, where it would conduct investigations in space physics and plasma electrodynamics. The mission was a partial success. Its tether deployed only 268 m before it jammed, but the satellite was still able to provide over 20 hours of stable deployment and demonstrate the concept of a gravitygradient stabilized tether. This allowed for the follow on mission TSS1R.
TSS1R planned to conduct investigations in the same categories as its predecessor, but this time be deployed over 20 km away from the shuttle. The mission was launched February 22, 1996 where it successfully deployed 19.7 km away from the shuttle before the tether failed due to an electrical arc. The satellite was still able to extract currents from the ionosphere, and demonstrate the potential for power conduction using an electrodynamic tether. (Cosmo).
The Small Expendable Deployer System1 (SEDS1) was launched March 29, 1993, and sought to demonstrate the successful deployment of a 20 km tether. Once deployed, the tether would be severed and the reentry of the payloads could be studied. This mission was accomplished successfully, and provided calculable data regarding the dynamics of a tethered system. SEDS1 also had a follow up mission which launched two years later.
SEDS2 mission objectives were to test a closedloop control system and to study the motion of a tether in space over long periods of time. Unfortunately, SEDS2’s tether was cut by a micrometeoroid only five days into its mission. This highlighted the potential risk posed by a fragile tether operating in the space environment. Although its mission was short, it was able demonstrate proper detumbling and stabilization within 10⁰ of nadirpointing.
The Plasma Motor Generator (PMG) was engineered to test the ability of a tether to generate power or thrust through interactions with the plasma environment. PMG was launched June 26^{th}, 1993, where it successfully deployed its 500 m tether, and was able to demonstrate its capabilities as both a generator system and a propulsion system. (Cosmo).
The Tether Physics and Survivability (TiPS) was a satellite developed by Naval Research Laboratory, with the simple experiment of studying the long term dynamics and survivability of a tethered satellite system. The two subsatellites were of nearly equivalent mass (the first of its kind), and connected via 4 kmlong tether. This mission was the longest lasting tether in space. The tether finally broke in July 2006 after 10 years of operability, well surpassing its estimated life of two years. This mission has helped provide confidence in the tether technology, and also provided an abundance of information on tethered satellite dynamics and attitude control over a long mission duration. (NRL).
The follow on mission to TiPS was the Advanced Tether Experiment (ATEx) which was launched October 8, 1998. This satellite sought to demonstrate tether system stability and control. It also utilized retroreflector arrays to be sued for laser ranging. Unfortunately the deployment of the tether failed, thus the mission was not successful. (NRL).
The Young Engineers’ Satellite2 (YES2) was launched September 14, 1997, where it successfully deployed a 30 km long tether, then released a minisatellite and reentry vehicle. This mission was the longest tether to be successfully deployed.
While many tethered missions were launched throughout the 1990’s there has been a lull in activity the last decade. Many missions have been designed, but very few have launched. By observing the failure rate of the missions mentioned previously, one may speculate that the potential risk (and cost) compared to the probability of success has deterred scientists and engineers from pursuing this strategy. Recognizing all the potential science and engineering which can be accomplished by a tethered satellite system coupled with a lack of further research has provided with the inspiration for my research topic.
Fortunately for students like myself, there are affordable ways to launch a satellite. One example is the CubeSat Project (CubeSat). This project was developed by California Polytechnic State University, San Luis Obispo and Stanford University. It provides the opportunity for industry to fly payloads a low cost, while creating educational opportunity for students in the 60 participating universities.
Within the project there are strict specifications, and standard mass and volume requirements. Each CubeSat is 10 cm x 10 cm x 10 cm, and must not have a mass greater than 1.50 kg. They are launched in deployment mechanisms nicknamed PPODs that hold three CubeSats at a time. This provides an opportunity to build a “triple CubeSat” with dimensions 30 cm x 10 cm x 10 cm. CubeSats generally range in price between $30,000 and $40,000, but can be built for as little as $25,000. (Space.com)
The advantages of using a CubeSat platform to test tethered satellite dynamics are three fold. The relatively low cost allows a certain “freedom to fail,” or take engineering risks that may or may not pay off. Because CubeSat is a large enough program, there are frequent launch windows. In the situation where a program cannot make a deadline, there will be more opportunities to launch. And lastly, a large part of the structural design is already complete, creating a “plug and play” type scenario. With these considerations, it seems suitable to use a triple CubeSat platform to continue testing the stability of a tethered satellite system. Consider TetherSat the first installment of many small satellite tether missions.
The mission of TetherSat is to demonstrate stability of a low mass tether system, as well as successfully perform orbital maneuvers by using the electrodynamic tether.
The design of the TetherSat is driven by the standards set forth by the CubeSat program. There are strict regulations on volume, mass and the deployment mechanism. The table below lists all strict, inflexible limits to the design. All otherspower, communications, orbits, attitude control, etc. are flexible.
Table 1: Design Specifications and Limitations
Specification 

Mass 
4.5 kg 
Volume 
3000 cm^{3} 
Cost 
$50,000 
Lifetime 
2 years 
The mission sequence would be as follows:
1. Launch/Stationing
a. Launch into desired orbit
b. Deploy from PPOD
c. Detumble
d. Deploy solar cells
e. Deploy tether
f. Deploy emitters/collectors
2. Orbit and Operation
a. Begin thrusting (altitude raising)
b. Maintain stability
c. Transmit telemetry
d. Sever tether and decay into atmosphere
Preliminary structural design is set by the CubeSat program. One design possibility would be to make identical pairs of satellites, each 15 cm x 10 cm x 10 cm in volume. The middle area would be filled with a tightly wound tether. This tether length, as discussed earlier, needs to be long enough to provide enough gravity gradient force for each endmass to stay separate, creating tension in the tether, and also be able to provide an adequate amount of thrust to maneuver. All the while, the tether length must be minimized because the longer it is, the larger the risk of severance and larger volume it consumes.
In Figure 4 below, you can find a plot of various altitude orbits, their tether lengths, and the amount of tension due to gravity gradient (calculated using Formula 3). Assumptions made were that there was 1 mA of current running through the tether, a Δm of 0.5 kg between each subsatellite, and a tether mass of 2.5 kg. The tether was also assumed to be a dumbbell model with efficient thrusting.
Figure 4: Tether Tension v. Tether Length at Various Altitudes
By observing the plot above, one can see the minimal difference between 500 km and 700 km of altitude. A value around 10 x 10^{3} N (about the weight of a dollar bill) is sufficient to hold each satellite apart. This suggests that with a tether length around 1000 m within the range of 500 km to 700 km of altitude will sufficiently hold the tether in tension. MATLAB code for this plot can be found in Appendix A.
Using a tether length of 1000 m from the above calculation, a current value of 1 mA, and a tether diameter of 2 mm, the crosssectional area (with deployed solar arrays, including the tether) is 2.10 m^{2}. By calculating the Ballistic coefficient, given in Formula 6, and the atmospheric density model MSISE 2000, assuming an inclination of 28.5⁰, the plot on the right estimates the time to reentry assuming no thrusting. Using the design requirement of a two year lifetime, and assuming a worst case scenario (B*=0.2), one might assume that a 600 km altitude orbit is sufficient. Continuing to the plot on the left, a design parameter accounting for current, tether length and cross sectional area is plotted at various values, the higher the parameter, the better the performance. The current design parameter using the above assumptions is 0.48, and is plotted as a bold black line. To verify the acceptability of both the design parameter and the altitude, the thrusttodrag ratio must be greater than one. This ensures that the tether system can maneuver. With the current parameters operating at a 600 km altitude, the average thrust to drag ratio is about 1.5, thus it is a suitable orbit to operate in. MATLAB code for these plots can be found in Appendix A.
Figure 5. Orbital Altitude Determination Based on Design Parameters
The table below summarizes the potential mass configuration for the tether satellite system. The mass estimates were taken from Space Mission Analysis and Design, as well as from online sources for parts such as the electron collector.
Table 2: Mass Estimates
Part 
Mass (kg) 
1000 m Tether 
0.375 
Bus 
1.737 
Solar Panels 
1.504 
PowerAmp Chip 
0.039 
Battery Pack (x2) 
0.506 
Sensors/Connectors 
0.100 
Comm Chip 
0.035 
Electron Collector 
0.200 
Total Mass: 
4.496 
Mass Margin: 
0.089% 
One can see that the margin is extremely low, and thus it should be a focus in the next iteration of design to reduce this mass, as to add an extra room incase one of the items end up weighing more than estimated.
The power requirements will vary depending on which mode the satellite is operating in. One might consider five separate modes of operation:
Mode A: Partially sunlit, no maneuver, no communications
Mode B: Partially sunlit, no maneuver, with communications
Mode C: Partially sunlit, maneuvering, no communications
Mode D: Partially sunlit, maneuvering, with communications
Mode E: Partially sunlit, safe mode.
In modes A, B and E, when no maneuver is taking place, the satellite can also receive power from the electron collector to charge the batteries. Other power demands such as the attitude control system sensors and controllers, the CPU, and the various modes of communication must be considered. Solar panel configuration is determined by the power demands of the system, yet recognizing the nadirpointing attitude the satellite must maintain, as well as the unfixed orientation towards the sun, the optimal solar panel design would have four hinged doublesided panels the size of each side of the subsatellite which would swing out and lock to 45 degrees, then expose another panel underneath it. The configuration is shown below.
Figure 6: Solar Panel Configuration
Power demands must account for battery degradation, solar cell degradation, and orbit eclipse time, taking these considerations into account, the power margin can be determined. With the solar panels, tether, and communication systems fully deployed, TetherSat may look like the figure below.
Figure 7: Potential Satellite
Configuration, fully deployed.
Once the tether is deployed, its second objective is to successfully demonstrate a rendezvous orbit using the electrodynamic tether. As mentioned earlier the amount of thrust depends on the Lorentz force generated and drag force present. Using the design parameters above for tether length, available current, cross sectional area. In this maneuver the satellite is assumed to be rigid and move at the same speed of its center of mass.
The source of a nearly constant, and infinite source of propellantless thrust allows for a new type of orbital transfer to be explored—the Ward spiral. This flight path occurs as a result of the effects of drag on the orbit of a satellite, but also traces the path to a higher orbit, when constant thrust is available.
Initially the satellite is placed in a circular orbit at 600 km altitude. The first plot demonstrates its ability to spiral upward by applying a constant current through the tether. The net force that results depends on the atmospheric density and magnetic field strength.
Assuming a circular orbit, velocity can be found using the formula:
(Vallado)
(8)
Where r
is the orbital radius, and µ is the gravitational parameter. The net thrust is modeled as:
where the drag force and electrodynamic force are calculated using
Formulas ## and ##, respectively. The
climb rate can be found by using the equation for power, and is found below:
(Rimrott)
(9)
The results of a climb using a slightly higher current (1 Amp) to
demonstrate the potential
maneuverability of the spacecraft, which is able to raise itself 1077.5
km higher over the course of two revolutions.
Figure 8: Demonstration of Orbit Raising Using an Up Ward Spiral
A good process to demonstrate the efficiency of these orbits is to compare it to the well known Hohmann transfer. This is shown below in Figure 9.
Figure 9. Hohmann Transfer between Two Circular Orbits
The governing equations of a Hohmann transfer can be found below:
(Vallado) (9)
(Vallado)
(10)
(Vallado)
(11)
(Vallado)
(12)
(Vallado)
(13)
(Vallado)
(14)
For the following case
study, the initial orbital altitude is 600 km with the final altitude at 1000
km. The equations used to calculate the
time of flight, change in true anomaly, and change in velocity required for a
Ward spiral are as follows:
(Rimrott)
(15)
(Rimrott) (16)
(Rimrott) (17) (18)
A plot of the Ward spiral transfer is
found below in Figure 10. A comparison
of results between the Hohmann and Ward transfer is summarized in Table 3.
Figure 10: Ward Spiral Maneuver from 600 km to 1000 km altitude
Circular Orbits
Table 3: Comparison of Hohmann Transfer and Ward Spiral Transfer
Maneuver Type 
Time of Flight (min) 
Phase Change (degrees) 
Δv (km/s) 
Hohmann 
50.437 
180 
0.20769 
Ward Spiral 
66.174 
246.398 
0.20773 
% difference 
23.8% 
26.9% 
0.0193% 
By comparing the results above, the Hohmann transfer arrives at its target in a shorter period of time, (16 minutes before the Ward Spiral), yet both expend nearly the same amount of fuel to get into that orbit. This example demonstrates the interchangeability of the transfers. One important fact to recognize is that Hohmann transfers are accomplished through the use of finite propellant onboard the spacecraft. With the use of an electrodynamic tether this maneuver could be repeated infinitely. If time of flight is an issue, the best solution is to increase the available thrust on the tether. By increasing the available thrust by about a factor of 4 (1 N versus 0.235 N), time of flight is significantly decreased. The table below summarizes the new results.
Table 4: Comparison of Hohmann Transfer with Higher Thrust Ward
Transfer
Maneuver Type 
Time of Flight (min) 
Phase Change (degrees) 
Δv (km/s) 
Hohmann 
50.437 
180 
0.20769 
Ward Spiral 
15.580 
58.008 
0.20773 
% difference 
223.7% 
210.3% 
0.0193% 
One can see the significant changes caused by quadrupling the thrust. The Ward spiral arrives in less than a third of the total time of the Hohmann transfer, all the while this thrust is sustainable through the design life of the spacecraft. This might suggest running a higher current (thus having a higher power requirement) through the tether. In other tether designs, increasing the tether length could be an option. Unfortunately quadrupling the length (and thus the volume) of the tether is not an option on board the CubeSat platform.
Using the above information it is clear that a successful and affordable tether mission can be accomplished. While this paper outlines the preliminary design of such a satellite mission, many other factors need to be accounted for. The communications system, attitude dynamics systems, and thermal control systems have been disregarded for this stage of the design. In future iterations, an estimate of power demands and a link budget need to be developed. As the satellite nears its final configuration, limitations on the hardware used should be determined, and trade studies made regarding the various configurations of the satellite.
With the potential to reinvigorate the enthusiasm towards tethered satellites, there is no better place to start with student with fresh minds (and free labor). If TetherSat is successful (and even if it is not) it will put the space program one step closer to realizing the missions which only tethered satellites can accomplish.
Of all the potential applications of a tethered satellite system, the one which, in my opinion, is of the upmost importance, is the ability to remove orbital debris. After about 50 years of space exploration, a significant amount of space debris is cluttered in the operational environment around Earth. A vast majority of this debris was put there by uscomposed of objects such as dead satellites, and upperstage rockets. These large objects pose a risk of collision with other satellites still in operation. Institutions have sought to track a large amount of space debris. Accurate models allow satellite operators to predict any future collisions with enough warning time to maneuver out of the way. The prospect of being able to avoid this tedious process, through the removal of large space debris is attractive to all who operate any payload in the space environment.
Current debris statistics place almost half of the high mass objects between 81⁰ and 83⁰ inclination. Most LEO objects lie between 500 and 1010 km altitudes (292 objects weighing 340 metric tons) and 803 to 860 km altitudes (67 objects totaling 240 metric tons). 79% of the mass in space came from Russia. Because the US depends more greatly on low earth orbit, we have taken initiative in clearing this area up.
Figure 11: Orbital Debris in Low Earth Orbit (NASA Orbital Debris Office)
The process for debris capture would take advantage of the maneuverability and propellantless thrust force provided by the electrodynamic tether. A space debris mission has been outlined by the National Aerospace Laboratory of Japan, which goes through five phases of operation. These phases are summarized below, and can be visualized using Figure 12.
Figure 12: Phases of Orbital Debris Deorbiting (Ishige)
Phase I involves the tethered satellite maneuvering into the debris’ orbit using the electrodynamic tether. Phase II requires the tether to retrieve itself. Rendezvous with the debris is accomplished using a conventional propellant system because of the limited thrust which an electrodynamic tether can provide. After rendezvous (with the debris attached to the bottom subsatellite) the system redeploys its tether and descends by running a current in the proper direction relative to the magnetic field (Phase III). Once the system reaches a low enough altitude to guarantee the debris will deorbit in less than 25 years (as specified by NASA) it is released (Phase IV). The satellite runs current in the opposite direction to create thrust and raise its altitude (Phase V), then continues on to the next target debris (Phase VI). (Ishige).
While this process may seem straight forward, there are many considerations which must be made with respect to the mass of the satellite, the strength of the tether, and the capability of the system. Considerable experience has been gained with rendezvous and docking in orbit through missions like Gemini through Apollo, to the International Space Station, yet capturing objects which were not intended to be captured poses a new engineering challenge. One advantage of picking up debris is that one does not need to worry about damaging the hardware. The capture mechanism must be versatile enough to allow for capture of a wide range of hardware. (Carroll).
While there is a large amount of simulation and design still necessary to conduct such a mission as described above, the fact that there is already a strong acknowledgement that space debris is a problem will aid in the timeline for its removal. Because of the longevity of this mission, it seems clear that the orbital maneuverability enabled by the renewable use of electrodynamic thrust makes a tethered satellite system a clear choice to accomplish the mission of clearing up our space environment.
Based on the research and analysis presented, one can conclude that a tethered satellite system offers many new and exciting mission opportunities that traditional satellites cannot. While many of the previous missions provide a stepping stone into future tethered satellite designs, issues of stability, control and survivability must first be resolved before large industries invest in such a design. The design for TetherSat is an example of how to accomplish these paramount steps in engineering a sustainable and useful tethered satellite at an affordable rate. Multiple small satellite missions can accomplish even more than one large tethered satellite mission at the same or smaller cost. Once confidence is gained in the survivability of a tethered system, missions such as the removal of space debris may be realized.
Bitzer, Matthew
S. Optimal Electrodynamic Tether and Orbit Raising Maneuvers. Thesis.
Virginia Polytechnic Institute, 2009. Print.
Carroll, Joseph
A. Space Transport Development Using Orbital Debris. Tech. 2002. Print.
Cosmo, M. L. Tethers
in Space Handbook. 3rd ed. 1997. Print.
CubeSat
in the News.
Web. 10 Dec. 2009. <http://www.cubesat.org>.
"Earth's
Magnetic Field." Lecture. Web.
<http://www.bu.edu/cism/cismdx/ref/Labs/2005_AFWA_ShortCourse/Lab03/refs/EarthMagneticField.pdf>.
Ishige, Yuuki.
"Study on electrodynamic tether system for space debris removal." Acta
Astronautica 55.917 (2004). Print.
NASA Orbital
Debris Program Office.
Web. 10 Dec. 2009. <http://orbitaldebris.jsc.nasa.gov/>.
Rimrott, F.P. J.
"Orbit Transfer By Means of a Ward Spiral." Technische Mechanik
22.4 (2002). Print.
Space Mission
Analysis and Design, 3rd edition (Space Technology Library) (Space Technology
Library). New York:
Microcosm, 1999. Print.
"SPACE.com
 Cubesats: Tiny Spacecraft, Huge Payoffs." Learn More at Space.com.
From Satellites to Stars, NASA information, Astronomy, the Sun and the Planets,
we have your information here. Web. 10 Dec. 2009.
<http://www.space.com/businesstechnology/cube_sats_040908.html>.
Stevens, Robert
E. "Optimal Control of Electrodynamic Tether Satellites." Diss. Air
Force Institution of Technology, 2008. Print.
Vallado, David A.
Fundamentals of Astrodynamics and Applications. New York:
Microcosm/Springer, 2007. Print.
%% Ward Spiral
clear all
%Constants
mu=3.986004418e5;%km3/s2
R_earth=6378.137;%km
%Orbital Parameters
%circular, i=0, 600 km
altitude
h=600;%km
(altitude)
r=h+R_earth;
v=sqrt(mu/r);
%Tether Parameters
L=1000;%m
I=10000/1000;%mA
d=2/1000;%mm
diameter tether
A=.3*.1+1000*d;
CD=2.0;
mass=4.5;%kg
%Environment Parameters
dens=atmdens(h);%kg/m3
Bmag=MagFluxDens(r);
F_EDT=I*L*Bmag;
F_drag=0.5*CD*A*dens*(v*1000)^2;
F_tot=F_EDTF_drag;
P=F_tot*v*1000;%Power of
thrust
E=mu*mass/(2*r);
rdot=2*F_tot*r^(3/2)/(sqrt(mu)*mass);
%integrate with r=r1 and
t=0 for initial conditions
t=1:100;
radius=r./(1F_tot/mass*sqrt(r/mass)*t).^2;
%polar coords
c_prime=F_tot*r^2/(1000*mass*mu);
theta=0:.1:4*pi;
r_polar=r./(1c_prime*theta).^2;
figure(1)
polar(theta, r_polar)
r_polar2=r;
v=sqrt(mu/r);
for i=2:length(theta)
dens(i)=atmdens(r_polar2(i1)R_earth);%kg/m3
Bmag(i)=MagFluxDens(r_polar2(i1));
F_EDT(i)=I*L*Bmag(i);
F_drag(i)=0.5*CD*A*dens(i)*(v(i1)*1000)^2;
F_tot(i)=F_EDT(i)F_drag(i);
c_prime(i)=F_tot(i)*r_polar2(i1)^2/(1000*mass*mu);
r_polar2(i)=r./(1c_prime(i)*theta(i)).^2;
v(i)=sqrt(mu/r_polar2(i));
end
figure(2)
polar(theta, r_polar2),title('Ward Up
Spiral Maneuver of TetherSat'),...
xlabel('True Anomaly (degrees)'),ylabel('Orbital
Radius (km)')
%Available Thrust
figure(3)
plot(F_tot),title('Available
thrust vs. time'),xlabel('Time (s)'),...
ylabel('Thrust Available (N)')
%% Orbital Transfer Up Ward
Spiral
clear all
%Constants
mu=3.986004418e5;%km3/s2
R_earth=6378.137;%km
%Lets try a tranfer
(coplanar) up 1000 km
%Tether Parameters
L=1000;%m
I=10000/1000;%mA
d=2/1000;%mm
diameter tether
A=.3*.1+1000*d;
CD=2.0;
mass=4.5;%kg
%Orbit Parameters
h1=600;%km
h2=1000;%km
r1=R_earth+h1;
r2=R_earth+h2;
v1=sqrt(mu/r1);%km/s
v2=sqrt(mu/r2);%km/s
%Environment Parameters
dens1=atmdens(h1);%kg/m3
Bmag1=MagFluxDens(r1);
F_EDT1=I*L*Bmag1;
F_drag1=0.5*CD*A*dens1*(v1*1000)^2;
F_tot1=F_EDT1F_drag1;
tw=mass/F_tot1*(v1v2)*1000;
thetaw=mass/F_tot1*mu/r1^2*(1sqrt(r1/r2))*1000;
vdot=F_tot1/mass;
delta_v=vdot*tw;
full_circle=0:.01:2*pi;
Orbit1=r1*ones(1,length(full_circle));
Orbit2=r2*ones(1,length(full_circle));
r_polar3=r1;
v=v1;
F_EDT=F_EDT1;
F_drag=F_drag1;
F_tot=F_tot1;
theta=0:.1:thetaw;
dens=dens1;
Bmag=Bmag1;
c_prime=F_tot*r1^2/(1000*mass*mu);
for i=2:length(theta)
dens(i)=atmdens(r_polar3(i1)R_earth);%kg/m3
Bmag(i)=MagFluxDens(r_polar3(i1));
F_EDT(i)=I*L*Bmag(i);
F_drag(i)=0.5*CD*A*dens(i)*(v(i1)*1000)^2;
F_tot(i)=F_EDT(i)F_drag(i);
c_prime(i)=F_tot(i)*r_polar3(i1)^2/(1000*mass*mu);
r_polar3(i)=r1./(1c_prime(i)*theta(i)).^2;
v(i)=sqrt(mu/r_polar3(i));
end
hold on
figure(4)
polar(theta,r_polar3,'b')
polar(full_circle,Orbit1,'g')
polar(full_circle,Orbit2,'r')
polar(theta(end),r_polar3(end),'bx')
polar(theta(1),r_polar3(1),'bx')
polar(theta(end),r_polar3(end),'bo')
polar(theta(1),r_polar3(1),'bo')
hold off
title('Orbit
Manuever from 600 km to 1000 km altitude')
disp(['Transfer Time
= ',num2str(tw/3600),' hours'])
disp(['Phase Change
= ',num2str(thetaw*180/pi),' degrees'])
disp(['Delta V
Required = ',num2str(delta_v/1000),' km/s'])
%% Calculate Necessary
Delta V for Hohmann Transfer
clear all
%Constants
mu=3.986004418e5;%km3/s2
R_earth=6378.137;%km
%Lets try a tranfer
(coplanar) up 1000 km
%Tether Parameters
L=1000;%m
I=10000/1000;%mA
d=2/1000;%mm
diameter tether
A=.3*.1+1000*d;
CD=2.0;
mass=4.5;%kg
%Orbit Parameters
h1=600;%km
h2=1000;%km
r1=R_earth+h1;
r2=R_earth+h2;
v1=sqrt(mu/r1);%km/s
v2=sqrt(mu/r2);%km/s
a_trans=(r1+r2)/2;
v1a=sqrt(2*mu/r1mu/a_trans);
v2a=sqrt(2*mu/r2mu/a_trans);
dv1=abs(v1av1);
dv2=abs(v2v2a);
delta_v_tot=dv1+dv2;
t_trans=pi*sqrt(a_trans^3/mu);
disp('For Hohmann
Transfer:')
disp(['Transfer Time
= ',num2str(t_trans/3600),' hours'])
disp(['Delta V
Required = ',num2str(delta_v_tot),' km/s'])
%% Orbital Transfer Up Ward
Spiral
%Case Study, More Thrust
clear all
%Constants
mu=3.986004418e5;%km3/s2
R_earth=6378.137;%km
%Lets try a tranfer
(coplanar) up 1000 km
%Tether Parameters
L=1000;%m
I=10000/1000;%mA
d=2/1000;%mm
diameter tether
A=.3*.1+1000*d;
CD=2.0;
mass=4.5;%kg
%Orbit Parameters
h1=600;%km
h2=1000;%km
r1=R_earth+h1;
r2=R_earth+h2;
v1=sqrt(mu/r1);%km/s
v2=sqrt(mu/r2);%km/s
%Environment Parameters
dens1=atmdens(h1);%kg/m3
Bmag1=MagFluxDens(r1);
F_EDT1=I*L*Bmag1;
F_drag1=0.5*CD*A*dens1*(v1*1000)^2;
%F_tot1=F_EDT1F_drag1;
F_tot1=1;
tw=mass/F_tot1*(v1v2)*1000;
thetaw=mass/F_tot1*mu/r1^2*(1sqrt(r1/r2))*1000;
vdot=F_tot1/mass;
delta_v=vdot*tw;
full_circle=0:.01:2*pi;
Orbit1=r1*ones(1,length(full_circle));
Orbit2=r2*ones(1,length(full_circle));
r_polar3=r1;
v=v1;
F_EDT=F_EDT1;
F_drag=F_drag1;
F_tot=F_tot1;
theta=0:.1:thetaw;
dens=dens1;
Bmag=Bmag1;
c_prime=F_tot*r1^2/(1000*mass*mu);
for i=2:length(theta)
dens(i)=atmdens(r_polar3(i1)R_earth);%kg/m3
Bmag(i)=MagFluxDens(r_polar3(i1));
% F_EDT(i)=I*L*Bmag(i);
% F_drag(i)=0.5*CD*A*dens(i)*(v(i1)*1000)^2;
% F_tot(i)=F_EDT(i)F_drag(i);
F_tot(i)=F_tot1;
c_prime(i)=F_tot(i)*r_polar3(i1)^2/(1000*mass*mu);
r_polar3(i)=r1./(1c_prime(i)*theta(i)).^2;
v(i)=sqrt(mu/r_polar3(i));
end
hold on
polar(theta,r_polar3)%full_circle,Orbit1,full_circle,Orbit2)
polar(full_circle,Orbit1,'r')
polar(full_circle,Orbit2,'g')
hold off
disp(['Transfer Time
= ',num2str(tw/3600),' hours'])
disp(['Phase Change
= ',num2str(thetaw*180/pi),' degrees'])
disp(['Delta V
Required = ',num2str(delta_v/1000),' km/s'])
%% Calculate Necessary
Delta V for Hohmann Transfer
clear all
%Constants
mu=3.986004418e5;%km3/s2
R_earth=6378.137;%km
%Lets try a tranfer
(coplanar) up 1000 km
%Tether Parameters
L=1000;%m
I=10000/1000;%mA
d=2/1000;%mm
diameter tether
A=.3*.1+1000*d;
CD=2.0;
mass=4.5;%kg
%Orbit Parameters
h1=600;%km
h2=1000;%km
r1=R_earth+h1;
r2=R_earth+h2;
v1=sqrt(mu/r1);%km/s
v2=sqrt(mu/r2);%km/s
a_trans=(r1+r2)/2;
v1a=sqrt(2*mu/r1mu/a_trans);
v2a=sqrt(2*mu/r2mu/a_trans);
dv1=abs(v1av1);
dv2=abs(v2v2a);
delta_v_tot=dv1+dv2;
t_trans=pi*sqrt(a_trans^3/mu);
disp('For Hohmann
Transfer:')
disp(['Transfer Time
= ',num2str(t_trans/3600),' hours'])
disp(['Delta V
Required = ',num2str(delta_v_tot),' km/s'])
function [dens_kg_m3] =
atmdens(alt_km)
% [density] = atmdens(alt)
outputs a density in kg/m^3 given an alt in km.
% Ref: SMAD back cover by
Wertz
if alt_km>750
dens_kg_m3=0;
else
alt=[0 100 150 200 250 300 350
400 450 500 550 600 650 700 750];
den=[1.2 5.25e7 1.73e9 2.41e10
5.97e11 1.87e11 6.67e12 2.62e12 ...
1.09e12 4.76e13 2.14e13 9.89e14
4.73e14 2.36e14 1.24e14];
dens_kg_m3 =
interp1(alt,den,alt_km);
end
function [Bmag] =
MagFluxDens(a_km)
% MagFluxDens(a) determines
the Mag Field Intensity [N/(Am)] at distance
% a km from the center of
the Earth.
Bmag = 8e6/a_km^3;
% Mag2DragCubesatPlot
% Plots the ratio of accel
due to electrodynamic forces to that due to
% drag forces in circular
orbit at ref inclination for various altitudes.
% %%% Parameters %%%
clear all
%% reference params (250 km
is ref altitude)
AltRef = 600; %
Reference alt [km]
IncRef = 28; %
Reference inclination [rad] (28 deg @ cape)
s2i = sin(IncRef*pi/180)^2; c2i =
cos(IncRef*pi/180)^2;
Ro_km = 6378+AltRef; Bo_N_Am =
MagFluxDens(Ro_km); mu_km3_s2 = 3.986e5;
delta = 11.5*pi/180; % Earths
tilted dipole = 11.5 deg
c2d = cos(delta)^2; s2d =
sin(delta)^2;
K = sqrt(c2i*c2d +
s2i*s2d/2); % Average
B field adjustment. Note when d=0, this
is cos(i)
vo2 = mu_km3_s2/Ro_km*1e6; % Ref velocity^2 [m^2/s^2]
altLow_km=AltRef150;
altHi_km=AltRef+30;
alt =
altLow_km:(altHi_kmaltLow_km)/50:altHi_km; %[km]
pref=atmdens(AltRef); hs_km =
atmcharht(AltRef);
ro_r = Ro_km./(6378+alt);
%Changed AltREf to 700 from
550 km
%% Tether params
ImL_A = .1*[2 ; 4; 6; 8; 16;
32]; % Current*Length/Area ratio
[Amp/m]
% Tether design point
diam_m=.002; L_m = 1000;
Im=.001; % tether
diam, length and max avg current [A]
A1=.051; A2=.051; At=L_m*diam_m;
A = A1 + A2 + At; % Area in m^2
ImL_A_design = Im*L_m/A;
disp(sprintf('Design Point:
Im*L/A = %u [A/m]', ImL_A_design));
%Changed ImL_A to
multiplied by .01 vice .1, changed back
%Changed L to 1000 m vice 60
m
%Changed diam_m=.0015 vice
.002 (Tape is 1mm by 2mm, averaged), changed to
% .002 (worst case scenario)
%Changed A1 and A2 to .051
from .02 to add solar wings
% worst case scenario = (2+sqrt(2))*A
%Changed max avg current to
.001 vice .0016
%Still uses oversimplified
model of Bfield
%% Plot F/D curves REF
Stevens Dissertation p 107
colors=['b','g','r','c','m','y','k','w'];
subplot(1,2,1)
hold on
for i = 1:6
k=colors(i);
F_D =
Bo_N_Am/(pref*vo2)*K*ImL_A(i)*ro_r.^2.*exp(Ro_km/hs_km*(1./ro_r1)); % Assumes
Cd=2
plot(F_D,alt,k)
end
line([1,1],[altLow_km,altHi_km])
plot(Im*Bo_N_Am/(pref*vo2)*K*L_m/A*1*exp(0),AltRef,'ks')
hold off
axis([0,4,altLow_km,altHi_km])
xlabel('Avg
Thrust/Drag'); ylabel('Altitude [km]');
legend('Im*L/A=0.2','Im*L/A=0.4','Im*L/A=0.6','Im*L/A=0.8','Im*L/A=1.6','Im*L/A=3.2')
msg = sprintf('Max Avg
Electrodynamic to Drag Force ratio for i = %u deg', IncRef);
title(msg);
grid on
%% Plot time for inert
tether to decay output from STK "lifetime" function
% % using NRLMSISE 2000
model i=28 deg. Cd=1. A=1.6 m^2. m=3kg => B* = 0.053
% % Also did for B*=0.107
and B*=0.160
h = 300:25:600;
td1 = [1/365*[19 36 66 113 208
331] [1.3 1.9 2.5 3.3 4.2 5.7 10.7]]; % For B* = 0.053
td2 = [1/365*[9 17 33 58 98 167
287] 1.1 1.6 2.2 2.8 3.5 4.2]; % For
B*=0.107
td3 = [1/365*[5 9 17 30 52 87 142
251 360 ] 1.4 1.9 2.4 3.1]; % For
B*=0.16
subplot(1,2,2)
plot(td1,h,'r',td2,h,'b',td3,h,'k')
legend('B*=0.05','B*=0.1','B*=0.2')
title('Satellite
lifetime based on drag only model NRLMSISE 2000, i=28 deg')
xlabel('Time to
Reentry [years]'); ylabel('Altitude [km]')
axis([0,6,altLow_km,altHi_km])
grid on
%I need this information
from STK to properly update the lifetime estimate