UNIVERSITY OF CINCINNATI
______, 20 _____
I,______, hereby submit this as part of the requirements for the degree of:
______in: ______It is entitled: ______
Approved by: ______
Use of Near-Frozen Orbits for Satellite Formation Flying
A thesis submitted to the
Division of Research and Advanced Studies of the University of Cincinnati
in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE (M.S.) in the Department of Aerospace Engineering and Engineering Mechanics of the College of Engineering
2001
by
Heidi L. Davidz
B.S., The Ohio State University, 1997
Committee Chair: Dr. Trevor Williams
Abstract
There is growing interest in flying coordinated clusters of small spacecraft to perform missions once accomplished by single, larger spacecraft. Using these satellite clusters reduces cost, improves survivability, and increases the flexibility of the mission.
One challenge in implementing these satellite clusters is maintaining the formation as it experiences orbital perturbations, most notably due to the non-spherical Earth.
Certain aspects of the orbital geometry can remain virtually fixed over extended periods of time due to a natural phenomenon called a frozen orbit. Specifically, the elements of the orbital geometry that can remain fixed are the argument of perigee (a measure of where the orbit is closest to the Earth) and the eccentricity (a measure of how circular or elliptical the orbit is). For satellite formations, using this frozen orbit phenomenon results in considerable propellant savings. In this study, a discussion of current literature on this topic is given. Some examples of formations at near-frozen conditions are shown. There is also a discussion of the propellant impact of near-frozen conditions.
If two orbits meet a certain set of initial conditions, the orbits will naturally stay in the vicinity of each other. These orbits are sometimes called “Hill’s orbits”. An algorithm is developed here that determines if the Hill’s orbit conditions will be met given the initial differences in eccentricity and argument of perigee for two satellites.
Acknowledgments
I would like to thank Professor Trevor Williams for his guidance in conducting this research. Also, I would like to thank my husband Michael Cluff for his patience and support as I completed this work. Table of Contents
1 Introduction ...... 9
1.1 Introduction to Satellite Formation Flight...... 9
1.2 Propulsion for Formation Flight...... 11
1.3 Proposed Work...... 13
2 Satellite Formation Flight Dynamics ...... 13
2.1 Hill’s Orbits...... 13
2.2 Perturbations...... 17
3 Frozen Orbits...... 23
3.1 Definition of Frozen Orbit...... 23
3.2 Equation Development...... 27
3.3 Experience with Frozen Orbits...... 28
4 Near-Frozen Orbit Test Cases...... 32
4.1 Introduction ...... 32
4.2 Development of Parameter Values...... 33
4.3 Development of Difference Equations...... 34
4.4 Simulation Results...... 36
4.5 Case 4 Motion ...... 49
5 Application of Cases to the Clohessy-Wiltshire/Hill’s Equations ...... 50
5.1 Introduction ...... 50
5.2 Review of the CW/Hill’s Equations...... 50
5.3 Coordinate Systems...... 51
5.4 “Algorithm 6” from Vallado and McClain ...... 51
Page 1 5.5 Application of “Algorithm 6” to Find PQW Radius and Velocity Vectors.... 52
5.6 Find IJK Radius and Velocity Vectors...... 54
5.7 Find RSW Radius and Velocity Vectors...... 54
5.8 Identify the CW/Hill’s Initial Conditions ...... 56
5.9 Numerical Example...... 57
6 Propellant Calculations...... 64
6.1 Derivation of the Tangential Thrust ∆V Equation ...... 65
6.2 Propellant Calculation Results ...... 70
7 Summary ...... 78
8 Suggestions for Future Work ...... 79
9 Bibliography...... 80
10 Appendix ...... 83
10.1 Matlab Code to Generate Frozen Orbit Circulations ...... 83
10.2 Matlab Code for Running Cases, Pseudo State Space Program ...... 84
10.3 Matlab Code for Running Cases, ODE Program ...... 85
10.4 Matlab Code to Generate Propellant Calculation Graphs ...... 86
Page 2 Table of Figures
Figure 1: Zonal Harmonics ...... 17
Figure 2: Sectorial Harmonics...... 17
Figure 3: Tesseral Harmonics ...... 17
Figure 4: Circulations Around the Frozen Orbit Conditions...... 26
Figure 5: Case 1 Eccentricity ...... 37
Figure 6: Case 2 Eccentricity ...... 37
Figure 7: Case 3 Eccentricity ...... 38
Figure 8: Case 4 Eccentricity ...... 38
Figure 9: Case 1 Argument of Perigee...... 39
Figure 10: Case 2 Argument of Perigee...... 40
Figure 11: Case 3 Argument of Perigee...... 40
Figure 12: Case 4 Argument of Perigee...... 41
Figure 13: Case 1 Eccentricity vs. Argument of Perigee ...... 42
Figure 14: Case 2 Eccentricity vs. Argument of Perigee ...... 42
Figure 15: Case 3 Eccentricity vs. Argument of Perigee ...... 43
Figure 16: Case 4 Eccentricity vs. Argument of Perigee ...... 43
Figure 17: Case 1 Eccentricity Difference ...... 44
Figure 18: Case 2 Eccentricity Difference ...... 45
Figure 19: Case 3 Eccentricity Difference ...... 45
Figure 20: Case 4 Eccentricity Difference ...... 46
Figure 21: Case 1 Argument of Perigee Difference...... 47
Figure 22: Case 2 Argument of Perigee Difference...... 47
Page 3 Figure 23: Case 3 Argument of Perigee Difference...... 48
Figure 24: Case 4 Argument of Perigee Difference...... 48
Figure 25: Along-Track Motion...... 62
Figure 26: In-Plane Motion Approximately a 2:1 Ellipse...... 63
Figure 27: Annual ∆V Requirements for i=90 deg and a=7714km ...... 71
Figure 28: Annual ∆V Requirements with Lines of Constant ∆V ...... 72
Figure 29: Annual Propellant Requirements for an Expanded Range of Eccentricities ...73
Figure 30: Annual ∆V Requirements for i=30 deg ...... 74
Figure 31: Annual ∆V Requirements for i=30 deg, with Lines of Constant ∆V ...... 75
Figure 32: Annual ∆V Requirements a=7000 km...... 76
Figure 33: Annual ∆V Requirements for a=7000 km, with Lines of Constant ∆V ...... 77
Page 4 List of Symbols
Greek
β' Angle between the orbit plane and the geocentric direction to the sun
∆V Delta-v, change in velocity needed
δe Deviation from the original eccentricity, change in eccentricity resulting
from burn
δω Deviation from the original argument of perigee, change in argument of
perigee resulting from burn
Φ Arbitrary angle used in derivation
km3 µ Gravitational parameter (for the Earth µ = 3.986x105 ) s 2
θ Angle between Sun line and the outward normal to the surface
ρ Atmospheric density
Ω Longitude of the ascending node
ω Argument of perigee
ω0 Argument of perigee before the burn in the ∆V calculations
ω1 Argument of perigee after the burn in the ∆V calculations
English
A Arbitrary variable used in derivation, area a Semi-major axis adrag Atmospheric drag
Page 5 cD Drag coefficient
Cs Specular reflection coefficient
E Energy e Eccentricity e0 Eccentricity before the burn in the ∆V calculations e1 Eccentricity after the burn in the ∆V calculations
Fthrust Force required for rendezvous fi Vector components of force
H Angular momentum i Inclination
J2, J3 Zonal harmonic parameters m Mass
Nˆ Outward normal to the surface n Mean motion, angular rate
P,Q,W Coordinate system where the P axis points to perigee p Semiparameter, solar radiation pressure
RE Earth’s mean equatorial radius
R,S,W Rotating coordinate system fixed to the reference satellite r Radius r Derivative of the radius r, which is the velocity in the radial direction L rint Vector radius from Earth center to interceptor satellite L rrel Vector difference between the interceptor and target satellites L rtgt Vector radius from Earth center to target satellite
Page 6 u Argument of latitude v Velocity x,y General variables used in trigonometric definitions x,y,z Position variables x, y, z Velocity variables
Page 7 List of Acronyms
AU Astronomical Unit (mean distance of the Earth from the Sun)
CW Clohessy-Wiltshire Equations
DSS Distributed Satellite Systems
GPS Global Positioning System
MEMS Micro-Electro-Mechanical Systems
MSFF Multiple Spacecraft Formation Flying
OMM Orbital Maintenance Maneuver
SRP Solar Radiation Pressure
Page 8 1 Introduction
1.1 Introduction to Satellite Formation Flight
Currently, there is growing interest in flying coordinated clusters of small spacecraft to perform missions once accomplished by single, larger spacecraft. In the past, many satellites were large, complex and expensive to launch1. Using a cluster of smaller, less complicated spacecraft decreases cost, improves survivability, and increases flexibility. This technology has many names, such as satellite formation flying, satellite clusters, satellite swarms, multiple spacecraft formation flying (MSFF), and Distributed
Satellite Systems (DSS). Government agencies, such as the Department of Defense and
NASA, and the commercial sector are considering this concept. The Air Force Scientific
Advisory Board Space Technology Panel says that this is “the technology that will lead to a new exploitation of capabilities in space.”
It is important to understand that, if spacecraft were flown side by side as aircraft are flown in formation, an exorbitant and prohibitive amount of propellant would be consumed. To minimize propellant needs, the natural orbital motion of the satellites in formation is used. Also, current concepts have the formations 1-10 km in diameter2.
This is not the same as constellations of satellites such as GPS, which are separated by thousands of kilometers.
There are three categories of small spacecraft. Microsatellites have mass less than
100 kg, nanosatellites are less than 10 kg, and picosatellites are less than 1 kg.
Capabilities have improved so that these small satellites can be effective. Micro-Electro-
Mechanical Systems (MEMS) are part of this enabling technology. Low-weight, high- performance thrusters like Hall thrusters and plasma thrusters can help meet the
Page 9 propulsion needs of these missions.
These satellite formations must stay together for long periods of time, while influenced by solar pressure, gravitational perturbations and differential drag. This is very different from rendezvous and docking missions, which only last for a matter of days. In addition, MSFF was previously accomplished for a one-leader-one-follower system with manual control. A ground-based control system may not be quick enough for collision avoidance and formation reconfiguration, so autonomous control may be needed.
Sabol, Burns and McLaughlin describe the enhanced survivability of satellite clusters3. “If a singe/large satellite has a system failure, the entire mission is at risk. If a single satellite in a cluster fails, the remaining satellites in the cluster may continue to perform the mission at a lower performance level. The cluster could then be brought back up to mission design specifications or even improved with the addition of another inexpensive replacement satellite.”
Satellite clusters are less expensive for multiple reasons. Manufacturing costs can be reduced since the satellites could be mass-produced. Since the satellites are small, they can be secondary payloads. Or, the launch vehicle’s cargo capacity could be optimized to reduce launch costs. Once in orbit, if there is a problem with a satellite, replacing a single, small satellite could be cheaper than replacing the entire satellite cluster or replacing a single, monolithic satellite. Also, if an upgrade is desired, improved satellites could be launched to join the cluster and augment the cluster’s capability instead of replacing the entire cluster. This would reduce the cost of upgrading a satellite system.
Page 10 Satellite clusters also add flexibility. Some of the missions proposed for satellite formation flying accomplish goals very difficult for a single satellite. By using a cluster of satellites, a large aperture can be synthesized. Larger than the aperture any one satellite could provide, this synthesized aperture could provide improved image resolution through interferometry. Also, a ground observing space-based sensor using a cluster of satellites may adjust the formation in orbit to adjust aperture size and orientation4. Since the cluster is a “virtual satellite”, it could potentially be reconfigured and optimized for different mission applications.
Some satellite formation flight missions are scheduled. The NASA Goddard spacecraft, Landsat-7 and EO-1 are now flying in an in-line formation. The Air Force
Research Laboratory TechSat 21 (Technology Satellite of the 21st Century) will launch in
2003 for a space-based radar mission. Also scheduled for a 2003 launch, the ST-5 NASA
Goddard mission will have three 22 kg satellites. The NASA JPL ST-3 deep space interferometry mission will have two satellites, with launch scheduled for 2005.
1.2 Propulsion for Formation Flight
Various low-weight propulsion methods exist for formation flight. Among the electric propulsion devices considered are the resistojet and ion propulsion. The Surrey
Space Centre used the electric propulsion resistojet on the minisatellite UoSAT-125.
According to Surrey this is “a form of electric propulsion where a fluid, such as water or nitrous oxide, is super-heated over an electrically-heated element and the resulting hot gas is expelled through a nozzle to produce low-level thrust.” In this application, nitrous oxide is used to produce 93 mN of thrust at 90 watts of input power, with a specific
Page 11 impulse of 127 seconds, for a total ∆V of 10.4 m/s. On the company website, Surrey
Satellite Technology, Ltd. describes both their Water Resistojets and their Nitrous Oxide
Resistojets.
In addition, Surrey Satellite Technology, Ltd. has a cold-gas propulsion system that was used on SNAP-1 spacecraft. This is different from the resistojet, which results in hot gas. Siegel states that Surrey Satellite Technology, Ltd used the smallest propulsion system to ever fly in space6. Used aboard the SNAP-1 (Surrey Nanosatellite
Applications Platform) satellite, this butane fueled propulsion system was built with off- the-shelf parts. The butane is not ignited, but vaporizes when it is expelled from the system. The entire propulsion system carries only 1.15 ounces of butane. SNAP-1 is the size of a soccer ball and was designed, built and launched in only seven months.
According to the article, the purpose of the tiny satellite was “to demonstrate how nanosatellites can be used to rendezvous with and inspect other satellites, and to test technologies for rendezvous and for swarms of nanosatellites to fly in formation.”
Butane was chosen for its safety and its ability to be used at low pressures.
Hydrazine and ammonia were too toxic, and nitrogen and xenon gas required high pressures. Jeffrey Ward who is Managing Director of SSTL said that ammonia might be used in future systems.
Hoversten states that ion propulsion has set a record while on NASA’s Deep
Space 1 probe7. This probe has now logged a record of 200 days operating its ion-driven engine. The article says that it has “more accumulated engine time than any other propulsion system in the history of the space program.” The ion propulsion system runs on atomic particles instead of chemical fuel. Built by Hughes Space and
Page 12 Communications, the system allows a spacecraft to “attain great speed with a minimum amount of fuel”. This form of propulsion is very efficient. Very little propellant is used, so the weight is less. Less weight means lower launch cost and a faster vehicle speed. JPL is planning on using ion propulsion on a number of upcoming missions. Ion propulsion is rather low-thrust though. Eleven Hughes-built satellites and six Russian satellites use ion propulsion for station keeping, but Deep Space 1 is the first spacecraft to use the technology as a primary means of propulsion.
1.3 Proposed Work
As an initial step, this study seeks to clarify some conflicting information in the literature regarding frozen orbits. Then some examples of formations at near-frozen conditions show behavior in this region. An algorithm is developed here that determines if the Hill’s orbit conditions will be met given the initial differences in eccentricity and argument of perigee for two satellites. With little or no propellant capability, the small satellites used in formation flying might be able to maintain a formation if the benefits of frozen orbit conditions are utilized. This study concludes with a discussion of the propellant impact of near-frozen conditions.
2 Satellite Formation Flight Dynamics
2.1 Hill’s Orbits
One technology that makes satellite formation flight feasible is an idea called
“Hill’s Orbits.” The Hill’s equations, otherwise known as the Clohessy-Wiltshire (CW) equations, have historically been used in rendezvous and docking missions. Written in a
Page 13 rotating coordinate system that remains fixed to a reference satellite, these equations describe the relative motion of a second satellite in terms of this reference satellite. The reference satellite is also known as the “master”, “leader”, “target”, “centersat” or
“primary” satellite, while the second satellite is also known as the “slave”, “follower”,
“interceptor”, “subsat” or “secondary” satellite.
The full development of these equations can be found in multiple texts including
Vallado and McClain8, as followed here. The derivation begins with the orbital two-body equation of motion of the reference satellite.
L µ L rtgt = − [1] rtgt 3 rtgt L Here, rtgt is the vector from the center of the Earth to the target satellite, rtgt is the magnitude of this vector, and µ is the gravitational parameter of the Earth.
The orbital two-body equation of motion of the interceptor satellite is similar, but with the addition of a force to maneuver for rendezvous, Fthrust.
L L µ L rint = − + [2] rint 3 Fthrust rint L Here, rint is the vector from the center of the Earth to the interceptor satellite and rint is the magnitude of this vector.
Next, define the relative vector as the difference between the interceptor and the target satellite.
L L L = − rrel rint rtgt [3] Differentiate this equation twice, so Equations [1] and [2] can be substituted in. The equations are rearranged and linearized about the reference orbit.
Page 14 The RSW coordinate system is used, where the origin of the RSW coordinate system remains fixed on the reference satellite. The positive R-axis is directed away from the Earth in the radial direction. The W-axis is in the out-of-plane direction along the positive orbit normal. Normal to the position vector, the S-axis is in the along-track direction, directed such that R x S = W. The Hill’s Equations are then expressed in terms of x, y, z, where x, y, z correspond to the relative position in the R, S, W directions, respectively.
− − 2 = x 2ny 3n x f x [4] + = y 2nx f y [5]
+ 2 = z n z f z [6]
Here, fi represents the vector components of the thrust Fthrust. Since the orbits are circular, the angular rate, n, of the rotating RSW frame is the same as the target satellite’s mean motion.
µ = [7] n 3 rtgt These equations are for close-orbiting satellites with a reference satellite that is in a near-circular orbit. Since the development of these equations neglects non-linear terms, the CW/Hill’s linearization breaks down over long durations and at large distances. Also, the equation development is for the two-body problem with a perfectly spherical Earth and no other perturbations. Nonetheless, these equations do help explain, for a first approximation, how the orbits in the formation will naturally behave.
To solve Hill’s equations for coasting or unforced motion, set Fthrust equal to 0.
Any applied thruster firings are assumed to be impulsive, changing the initial difference in velocity that is used as an initial condition. Again, a detailed derivation of the solution
Page 15 can be found in Vallado and McClain8 or other texts. Given the initial relative difference in position between the two satellites (x0, y0, z0), the initial relative difference in velocity
between the two satellites ( x0 , y0 , z0 ), the mean motion of the reference satellite (n), and the time interval of interest, the Hill’s equations give the relative motion described by the following equations.
x 2y 2y x(t) = 0 sin(n ⋅t) − (3x + 0 )cos(n ⋅t) + (4x + 0 ) [8] n 0 n 0 n 4y 2x 2x y(t) = (6x + 0 )sin(n ⋅ t) + 0 cos(n ⋅ t) − (6nx + 3y )t + (y − 0 ) [9] 0 n n 0 0 0 n z z(t) = z cos(n ⋅ t) + 0 sin(n ⋅ t) [10] 0 n The corresponding rates are then,
= ⋅ + + ⋅ x(t) x0 cos(n t) (3nx0 2y0 )sin(n t) [11] = + ⋅ − ⋅ − + y(t) (6nx0 4y0 )cos(n t) 2x0 sin(n t) (6nx0 3y0 ) [12] = − ⋅ + ⋅ z(t) z0 nsin(n t) z0 cos(n t) [13] The third group of terms in the solution for the along-track position, y(t), is a secular term that grows with time. This secular term becomes zero under the condition,
= − y0 2nx0 [14] Using this condition in the Hill’s solutions above describes orbit geometries that stay together naturally. The resulting in-plane relative motion can be shown to be a 2:1 ellipse, with the semimajor axis twice the semiminor axis. These orbits are sometimes called Hill’s orbits. This cancellation of the secular growth of the relative position in the along-track direction is one of the fundamental reasons that formation flight is feasible.
Page 16 2.2 Perturbations
2.2.1 J2 and J3 The Earth is not a perfect sphere. For example, there is a difference between the equatorial and polar radii of 21.4 km. The non-spherical Earth can be mathematically described using an aspherical-potential function with spherical harmonics. There are three types of spherical harmonics that account for the extra mass distributions. First, the zonal harmonics are symmetrical about the polar axis and depend on
latitude only. These types of bands are Figure 1: Zonal Harmonics shown in Figure 1. Second, the sectorial
harmonics account for the extra mass in longitudinal regions, as
Figure 2 illustrates. As Figure 3 shows, the tesseral harmonics
depend on both latitude and longitude. Figure 2: Sectorial Harmonics The most important zonal harmonic is J2, which is three orders of magnitude larger than the next largest coefficient J3. J2 accounts for the oblateness of
the Earth, while J3 accounts for the “pear shaped” bulge of the Figure 3: Tesseral Harmonics Earth. Table 1 shows the various values found for J2 and J3.
Table 1: Various J2 and J3 Values
Source J2 J3
Vallado and McClain8, p. 878 0.0010826269 -0.0000025323 Vallado and McClain8, p. 779 -0.108265x10-2 -0.254503x10-5 Chobotov9 1.08263x10-3 -2.53215x10-6 Values Used in This Analysis 1.0826269x10-3 -2.5323x10-6
Page 17 2.2.2 Orbital Elements
There are six classical orbital elements: a, semimajor axis; e, eccentricity; i, inclination; ω, argument of perigee; Ω, longitude of the ascending node; and ν, true anomaly. The semimajor axis (a) describes the size of the orbit. The eccentricity (e) is a measure of how circular or elliptical the orbit is. The tilt of the orbit plane is the inclination (i). The ascending node is the location in the equatorial plane where the satellite crosses the equator from the south to the north. The longitude of the ascending node (Ω) is a measure from the I unit vector to the location of the ascending node.
Perigee is where the orbit is closest to the Earth. The argument of perigee (ω) is the angle between the location of the ascending node and perigee. The true anomaly (ν) gives the satellite’s current position relative to perigee.
2.2.3 Nodal Regression
The oblateness of the Earth causes both nodal regression and apsidal rotation.
The term nodal regression refers to how the extra material around the Earth’s equator exerts a gravitational pull on the orbit, which applies a torque to the orbit, so causing the angular momentum vector to precess over time. As a result, the longitude of the ascending node (Ω) shifts at the nodal regression rate of,
æ ö 2 3nJ 2 RE Ω = − ç ÷ cos(i) [15] 2 è p ø
Here, n is the orbital angular rate, RE the radius of the Earth, p the semiparameter of the orbit, and i the inclination of the orbit.
Apsidal rotation refers to how the location of perigee, which is the argument of perigee (ω), shows a secular drift over time. Using the same variable descriptions as in
Page 18 Equation 15, the equation for the apsidal rotation rate is
2 3nJ æ R ö ω = 2 ç E ÷ {}4 − 5sin 2 (i) [16] 4 è p ø There are three classifications of perturbation effects on an orbit: short-periodic, long-periodic, and secular. Short-periodic is on the order of the satellite’s period or less, while long-periodic is on the order of days or weeks. Vallado and McClain8 have an excellent figure illustrating these effects. Osculating (instantaneous) elements are the orbital element values that include all three of these variations. Mean elements are the orbital element values averaged over some time intervals and do not include the short- periodic variations. However, since there are different methods for averaging and different types of mean elements, it is important to understand the definition and development of the mean elements used in a particular case. The time interval used in the averaging is also important.
In order to have an out-of-plane spread in a formation at all points in the orbit, the satellites in a formation will need to have different inclinations. Otherwise, the formation has zero out-of-plane spread at higher latitudes. Assume that the inclination for the first satellite is i0, such that the nodal regression rate from Equation 15 is
æ ö 2 3nJ 2 RE Ω = − ç ÷ cos(i ) [17] 2 è p ø 0
If the second satellite has an inclination (i0 + ∆i), its nodal regression rate is,
æ ö 2 3nJ 2 RE Ω = − ç ÷ cos(i + ∆i) [18] 2 è p ø 0 The differential nodal regression rate is then,
Page 19 æ ö 2 3nJ 2 RE ∆Ω = − ç ÷ []cos(i + ∆i) − cos(i ) [19] 2 è p ø 0 0 If uncorrected, this differential nodal regression rate will steadily increase the satellite formation out-of-plane size. Correcting for this differential rate is the largest term in the propellant budget.
10 Schaub and Alfriend describe a method to obtain J2 invariant relative orbits.
These orbits have equal nodal regression rates, so they will not pull apart over time due to the influence of J2. In order to accomplish this, the secular drift of the longitude of the ascending node and the sum of the argument of perigee and mean anomaly are set equal between the two orbits. One of the criteria is that,
(1− e2 ) tan(i) δe = δi [20] 4e A problem arises in this method for near-polar and near-circular orbits. As the inclination i approaches 90°, the tan(i) term grows very large. Even a small δi results in a large δe. Since eccentricity controls the in-plane size of the formation, a large δe means that the in-plane relative motion of the satellites will be too large for practical use.
Likewise, for near-circular orbits, the eccentricity e approaches zero such that Equation
20 approaches infinity.
Another key finding in this paper is that it is desirable to use mean elements and not osculating elements to set the orbit geometry. Various graphs are shown to illustrate this effect. In one case, the ∆v required per year is ~40.15 m/s when the osculating elements are used to set the relative orbit geometry. On the contrary, using mean orbit elements to set the relative orbit geometry reduces the ∆v required per year to 0.145 m/s.
Page 20 2.2.4 Drag For low Earth orbits, atmospheric drag can be a significant perturbation to a satellite formation. The equation for atmospheric drag is8,11
L L c A v = − 1 D ρ 2 rel adrag vrel L [21] 2 m vrel
Here, cD is the drag coefficient, which is often estimated as 2 for orbital applications. A is the cross-sectional area normal to the satellite’s velocity vector, m is the satellite’s mass, ρ is the atmospheric density, and vrel is the satellite’s velocity vector relative to the atmosphere. The term m/(cDA) is called the ballistic coefficient. A typical acceleration due to drag for a satellite in low Earth orbit2 is –4x10-6 m/s2.
As with the other perturbations, it is the differential, not absolute, drag effects that will cause the satellite formation to be perturbed. To minimize this, most formations consist of individual satellites that are identical, so that they have the same ballistic coefficient. By contrast, drag can also be helpful when used to perform along-track maneuvering within the formation if the ballistic coefficient of individual satellites can be adjusted at will. This could be particularly useful in nanosatellites, where propulsion is limited. One approach would be to shift a ballast mass in the satellite to change the attitude of the spacecraft. This changes the cross-sectional area normal to the satellite’s velocity vector, which then alters the amount of drag2. The ballast mass in the satellite could be existing parts, such as a battery. Adding ridges to the front face of the satellite could enhance this drag modulation technique.
2.2.5 Solar Radiation Pressure Solar radiation pressure is produced when the photons from the Sun hit a surface,
Page 21 resulting in an exchange of momentum. For a distance of 1 AU from the Sun, the solar radiation pressure p≅5x10-6 N/m2. This is not the same as the solar wind, which consists of charged particles from the Sun. The equation for the solar radiation force on a specular reflection surface is2,
= − θ θ ˆ f s 2 pCs Acos( ) cos( ) N [22] Here, A is the surface area, Cs is the specular reflection coefficient, θ is the angle between the Sun line (unit vector directed towards the Sun) and the outward normal to the surface (unit vector Nˆ ).
Solar radiation pressure acts as a perturbation on a formation2. However, this perturbation can also be used to perform orbital maneuvers in the formation. Williams and Wang12 suggest using a solar wing instead of propulsive means to cancel the differential J2 nodal regression rate between satellites flying in formation. Since the wing would be a specular reflector, the solar force could be steered. In comparison, solar sails are used for large orbital changes. Solar flaps, which are surfaces designed for disturbance torque balancing, are typically even smaller than a solar wing.
Page 22 3 Frozen Orbits
3.1 Definition of Frozen Orbit
Chobotov describes a “frozen orbit” as one for which the mean elements are chosen to produce constant, or nearly constant, values of eccentricity (e) and argument of perigee (ω) with time9. This stops the rotation of perigee. The eccentricity of the frozen orbit will remain constant for years if the solar radiation pressure and the atmospheric drag are not too influential. Active maneuvering to maintain a frozen orbit is also possible. Chobotov says the first mention in the literature of the term “frozen orbit” was in a 1978 in a paper by Cutting, et al.13
The variational rate equations for eccentricity and argument of perigee that incorporate only zonal harmonics J2 and J3 are as follows. de 3J n æ R ö3 æ 5 ö = − 3 ç E ÷ sin(i)ç1− sin 2 i÷cos(ω) [23] dt 2(1− e2 )2 è a ø è 4 ø dω 3J n æ R ö2 æ 5 öé J æ R ö sin(i)sin(ω)ù = 2 ç E ÷ ç1− sin 2 i÷ê1+ 3 × ç E ÷ ú [24] − 2 2 − 2 dt (1 e ) è a ø è 4 øë 2J2 (1 e ) è a ø e û
Here, n is the satellite mean motion.
µ n = [25] a 3
RE is the Earth’s mean equatorial radius. The elements are mean elements, i.e. the osculating elements with the short-period oscillations averaged out.
At the critical inclination of i=63.4 deg or 116.6 deg, the term (1-5/4*sin2i) becomes zero in both Equations 23 and 24, resulting in solutions with nearly constant values of eccentricity and argument of perigee. The Russian Molniya orbits are near the
Page 23 critical inclination of 63.4° and have this characteristic.
The eccentricity rate de/dt is zero if i is 0 or the critical inclination, or if ω=90 or
270 degrees. Since most missions do not fly in the equatorial plane or at critical inclination, ω must be 90 or 270 degrees for this condition to occur.
The argument of perigee rate expression dω/dt is zero at the critical inclination or when the square bracketed term in Equation 24 is equal to zero. By setting this term equal to zero, for a given value of a and i, the mean “frozen eccentricity” can thus be found. For ω=90 degrees, the frozen eccentricity is,
1 J æ R ö ≈ − 3 ç E ÷ e f sin(i) [26] 2 J2 è a ø -3 This eccentricity is of the order of 10 because J3 is three orders of magnitude less than
J2.
In the Chobotov9 text, there is a graph (Figure 11.22) of frozen-orbit eccentricity solutions plotted against inclination. Using a semimajor axis of 7041.1 km, this graph shows the difference between using only zonal harmonics J2 and J3 and incorporating the higher-order zonal harmonics J2-J12. The graphs are drastically different for inclinations between 50 and 75 degrees. In the graph incorporating harmonics J2-J12, the frozen eccentricity solutions fall into three inclination regions. For i<63.4 deg, the frozen eccentricity is at ω=90 deg. The frozen eccentricity increases drastically as the inclination approaches the critical inclination. Between inclinations of 63.0 to 63.435 deg, the frozen-orbit solution shows comparatively large eccentricities. For the second inclination region of 63.4 deg < i < 66.9 deg, inverted frozen-orbit solutions occur.
Inverted frozen-orbit solutions refer to when ω =270 deg instead of ω =90 deg.
Page 24 Eccentricity decreases dramatically in this region. The third inclination region is 66.9 deg < i ≤ 90 deg. Solutions for ω =90 deg again occur.
Chobotov states, “for a narrow range of inclinations no frozen-orbit solutions are found.” From looking at the numbers quoted and looking at the graph, this region must
-3 occur somewhere between i=66.8 deg and i=67.1 deg. At i=66.6 deg, ef=0.124x10 ,
-3 -3 ω=90 deg; for i =66.8 deg, ef=0.041x10 , ω=90 deg; for i =67.1 deg, ef=0.068x10 and
ω=270 deg. In addition, there is discussion of how ω=270 inverted frozen orbits also occur in a low-altitude (a<1.09 Earth radii), low-inclination (i<10deg) region, as well as in the critical inclination region.
Chobotov also states that, “For initial conditions that are near, but not at, the frozen point, e and ω will move counterclockwise in closed contours. For inclinations less than 63.435 deg or greater than 116.565 deg, the motion is clockwise.” For conditions farther from the frozen point, the contours do not close. Figure 4 shows an example of these circulations or closed contours about the frozen-orbit conditions.
Chobotov also has a plot of these circulations.
Although it is not the standard terminology, Vallado and McClain8 have a slightly different definition of a frozen orbit. This definition of frozen orbits is, “specialized orbits that try to fix one or more orbital elements in the presence of perturbations. The
Sun-synchronous and minimum altitude variation orbits are examples of frozen orbits.”
Vallado and McClain define a “frozen-eccentricity” orbit as an orbit that minimizes global variations in altitude. Vallado and McClain also describes the phenomenon that plotting e*cos(ω) versus e*sin(ω) produces closed-circle trajectories centered on the frozen-orbit values.
Page 25 Frauenholz, Bhat, and Shapiro14 describe the frozen orbit condition as, “Frozen orbit conditions are realized through the balancing of the secular perturbations of the even zonal harmonics with the long-period perturbations of the odd zonal harmonics.”
They also say that the closed contours of the frozen orbit can remain closed even under the influence of non-gravitational perturbations such as drag and solar radiation pressure.
Figure 4: Circulations Around the Frozen Orbit Conditions
Page 26 3.2 Equation Development
There are multiple expressions for the long-periodic and secular change in eccentricity and argument of perigee for the frozen eccentricity orbit. Chobotov9 uses the following variational rate equations for eccentricity and argument of perigee.
3 de 3J n æ R ö æ 5 ö = 3 ç E ÷ sin(i)ç1− sin 2 i÷cos(ω) [27] dt 2(1− e 2 ) 2 è a ø è 4 ø
2 dω 3J n æ R ö æ 5 öé J æ R ö sin(i)sin(ω)ù = 2 ç E ÷ ç1− sin 2 i÷ê1+ 3 × ç E ÷ ú [28] − 2 2 − 2 dt (1 e ) è a ø è 4 øë 2J 2 (1 e ) è a ø e û
The equations given in the Vallado and McClain text are almost identical to the
Chobotov equations. The long-periodic argument of perigee rate is given as,
2 dω 3n æ R ö æ 5 ö = J ç E ÷ ç1− sin 2 i÷θ [29] dt (1− e 2 ) 2 2 è a ø è 4 ø
é J 1 æ R öæ sin 2 i − e *cos 2 i ö sin(ω)ù where, θ = ê1+ 3 ç E ÷ç ÷ ú [30] − 2 ç ÷ ë J 2 (1 e ) è a øè sin i ø e û No physical interpretation of θ is given. The long-periodic eccentricity rate is,
3 de − 3 n æ R ö æ 5 ö = J ç E ÷ sin(i)ç1− sin 2 i÷cos(ω) [31] dt 2 (1− e2 ) 2 3 è a ø è 4 ø Here the variables used are all as defined before.
In a paper discussing the orbit analysis for SEASAT-A (a NASA Earth satellite for measuring global ocean dynamics), Cutting, Born and Frautnick13 differentiate
Brouwer’s equations of motion to get the following expressions for the variation in eccentricity and argument of perigee.
3 de − 3J n æ R ö æ 5 ö = 3 ç E ÷ sin(i)ç1− sin 2 i÷cos(ω) [32] dt 2(1− e 2 ) 2 è a ø è 4 ø
Page 27 2 dω 3J n æ R ö æ 5 öé J æ R öæ sin 2 i − e *cos 2 i ö sin(ω)ù = 2 ç E ÷ ç1− sin 2 i÷ê1+ 3 × ç E ÷ç ÷ ú − 2 2 − 2 ç ÷ dt (1 e ) è a ø è 4 øë 2J 2 (1 e ) è a øè sin i ø e û
[33] This expression differs from the two previously quoted. If the eccentricity e is considered to be small, the term,
sin 2 (i) − ecos2 (i)
sin(i) reduces to sin(i) as in the Chobotov expression.
It is also known that the Chobotov expression for de/dt is missing a negative sign.
Also, in the Vallado θ expression, a “2” is missing in the denominator. So the expressions used in this analysis will be as given in Equations 23 and 24.
3 de 3J n æ R ö æ 5 ö = − 3 ç E ÷ sin(i)ç1− sin 2 i÷cos(ω) [34] dt 2(1− e2 ) 2 è a ø è 4 ø
2 dω 3J n æ R ö æ 5 öé J æ R ö sin(i)sin(ω)ù = 2 ç E ÷ ç1− sin 2 i÷ê1+ 3 × ç E ÷ ú [35] − 2 2 − 2 dt (1 e ) è a ø è 4 øë 2J 2 (1 e ) è a ø e û
3.3 Experience with Frozen Orbits
One mission where a frozen orbit was used is the TOPEX/Poseidon mission.
Frauenholz, Bhat, and Shapiro14 discuss the observed behavior over three years of the semi-major axis, inclination and the frozen eccentricity vector in this mission. The precision orbit determination used on this project produced a radial accuracy of ~4 cm rms. The orbit was a near-circular frozen orbit at a mean altitude of ~1336 km and an orbital period of ~112 min. The reference mean elements were: semimajor axis of
7714.42938 km, inclination of 66.0408 degrees, eccentricity of 95 ppm, argument of periapse of 90 degrees. For this mission, the osculating-to-mean element conversion
Page 28 technique was a method developed by Guinn. Here, the mean orbital elements remove all central and third-body perturbations acting over ten days
The predicted decay of the semi-major axis was almost entirely due to drag. As observed, other forces influencing the semi-major axis include solar radiation, thermal gradients and molecular outgassing. These body-fixed forces either boost or deboost the orbit, depending on the orientation of the satellite.
As for the inclination, the observed variations in inclination corresponded very well to the predictions. The primary perturbing factors were the third-body forces of lunar and solar gravity. Small inclination variations were also caused by tidal forces, which are induced by lunar and solar gravity. Nonetheless, these effects were an order of magnitude smaller than third-body perturbations.
As stated previously, the target values to obtain frozen conditions were for an eccentricity (e) of 95 ppm and an argument of periapse (ω) of 90 degrees. The conditions actually achieved were an eccentricity of 142.9 ppm and an argument of periapse 90.6 degrees. The observed (e, ω) vectors were grouped together after each orbital maintenance maneuver (OMM). As expected, this data generally moves counterclockwise about the design point in a closed contour. Even though the counterclockwise movement could be seen, it is hard to distinguish which perturbations other than earth gravity are affecting the (e, ω) values. When the frozen predictions are updated with current e and ω estimates, the agreement between observed and predicted values is better.
The maximum differences between the observed and predicted e and ω values correspond to the β’ angle variations. The authors define β’ as the angle between the
Page 29 orbit plane and the geocentric direction to the sun. (The authors use β’ rather than the more standard β.) The authors state that, “During periods of peak β’ when the orbit is in full sun, the observed mean eccentricity is always less than the frozen value (β’>0); this trend reverses when β’<0. This behavior is caused by solar radiation pressure...” It is also stated that, “The argument of periapse exhibits maximum deviations from the updated frozen values near β’=0 when solar radiation pressure has the greatest effect during the longest earth occultation intervals.” Without any dedicated maneuvers, the frozen orbit has been maintained throughout the three-year period discussed. During each OMM, efforts were made not to increase the mean eccentricity.
Chao, Pollard, and Janson15 describe a formation that uses a frozen orbit as the reference orbit of the centersat. This strategy also uses differential GPS measurements,
Micro-Electro-Mechanical Systems and auto-feedback control. Station-keeping maneuvers are necessary to keep the centersat in a frozen orbit. Formation-keeping maneuvers are required to have each subsat follow its reference orbit. The results of the analysis show only 10 to 30 m/s per year per satellite in ∆V are required to control a 1 km radius cluster for LEO. Raising the orbit altitude or reducing the area-to-mass ratio reduces the ∆V of the centersat.
Shapiro16 says that the eccentricity frozen orbit was first described for SEASAT, but has been studied for the Earth-orbiting missions of the Atmospheric Explorer, the
Heat Capacity mapping Mission, LANDSAT, GEOSAT, and TOPEX/Poseidon and for
Martian, Venusian and Lunar orbiters. The author states that in an abstract perspective,
“frozen orbits arise from bifurcations or singularities in the relevant system of differential equations obtained via the appropriate Hamiltonian or Lagrangian formulation.” The
Page 30 Hopf bifurcation is of particular interest since in the vicinity of a Hopf bifurcation under certain conditions stable limit cycles exist (e.g. closed orbits).
Shapiro gives equations for de/dt and dω/dt where all harmonics can be included.
The author states that “Drag is stabilizing and thrust may be either stabilizing or non- stabilizing. A saddle/node bifurcation is introduced by SRP. Variations due to shadowing and solar geometry cause the steady state to significantly depart from its unperturbed value. This leads to a complicated phase portrait which can fold back on itself as the bifurcation parameters change dynamically.” In a hypothetical case of a satellite with half the mass of TOPEX/Poseidon, the rate of variation would double such that “the system passes through all three regions of the parameter plane, stable/unstable spiral, saddle, and total non-existence of a stationary point.” This might be something worth examining for the case of tiny nanosatellites.
Nickerson, Herder, Glass and Cooley17 also discuss frozen orbits. They say that the effects of solar radiation and atmospheric drag greatly affect the oscillation pattern, while the gravitational effects of the Sun and Moon only slightly affect this pattern. This forces correction maneuvers to maintain the frozen orbit.
Page 31 4 Near-Frozen Orbit Test Cases
4.1 Introduction
Questions about the applications of frozen orbits in satellite formation flying arose. Is it better to have one satellite in the formation at the frozen eccentricity or is it better to have the satellites equidistant from the frozen value? How does the initial selection of the eccentricity affect the final values of eccentricity and argument of perigee? Is it best to have the formation in near-frozen orbit? If so, how much delta-v does this save?
Four test cases were examined, with each case consisting of a formation of two satellites. The eccentricity and argument of perigee values used in these cases are described in Table 2.
Table 2: Near-Frozen Orbit Test Cases Eccentricity and Argument of Perigee Values Satellite 1 Satellite 2 Satellite 1 Satellite 2 Eccentricity Eccentricity Argument of Argument of Perigee (deg) Perigee (deg) Case 1 0.000967 0.000977 90 90 (frozen) (frozen + 1x10-5) Case 2 0.000962 0.000972 90 90 (frozen – 0.5x10-5) (frozen + 0.5x10-5) Case 3 0.0001 0.00011 90 90 Case 4 0.0001 0.00011 60 120
Case 1 has one satellite at the frozen eccentricity value and one satellite with an eccentricity 10-5 greater than this value. In Case 2, both satellites have an eccentricity equidistant from the frozen value. Case 3 has two satellites in near-frozen conditions, with an eccentricity difference of 10-5. Similar to Case 3, Case 4 has the same eccentricity values, but an argument of perigee of 60 and 120 degrees.
Page 32 4.2 Development of Parameter Values
For both satellites, the values of RE, a and i must be defined. In Vallado and
8 McClain on p. 779 and 833, the RE in these formulas is listed as the "Radius of Earth".
In Chobotov, the RE is "the Earth's mean equatorial radius". This analysis uses the mean equatorial radius of the Earth from Vallado p. 878 of RE = 6378.1363 km.
The value of the semimajor axis is arbitrary here. In the TOPEX/Poseidon mission quoted by Shapiro18, the altitude is ~1336 km and the semimajor axis is 7714 km
9 (a = RE+1336 km = 7714 km). In Chobotov (p. 262), a semimajor axis of 7198.7 km is used, so the value of 7714 km seems reasonable for comparing results.
The inclination is set at 90 degrees, since a polar orbit is being used. The gravitational parameter of the Earth (µ) is defined as 3.986004415x105 km3/s2 in Vallado and McClain p. 878. Using the defined values, the mean motion is then n=9.318x10-4 rad/sec.
The long-periodic eccentricity rate vanishes if i=0, i=critical inclination of 63.4 deg,
ω=90 deg, or ω=270 deg. Since the desired inclination is already given, the value of
ω=90 deg will be used. This corresponds to the value used in the Chobotov9 examples and the TOPEX/Poseidon mission14, 18. The frozen eccentricity at these conditions must be calculated. The equation from Chobotov p. 261 is used to find an approximate eccentricity value that will maintain frozen conditions.
1 J æ R ö = − 3 ç E ÷ e f sin(i) [36] 2 J 2 è a ø -4 So, for the first satellite, ef = 9.67 x 10 .
For Case 2, assume that both satellites are equidistant from the frozen value. The two
Page 33 eccentricity values are thus,
∆e e = e − [37] 1 f 2 ∆e e = e + [38] 2 f 2
4.3 Development of Difference Equations
In order to directly find the time history of the differences in eccentricity and argument of perigee between the satellites, expressions for d(∆e)/dt and d(∆ω)/dt are needed. The following definitions are used.
DEFINITIONS:
e1 = eccentricity of satellite 1
∆e = the difference in eccentricity between the satellites
e2 = e1 + ∆e = eccentricity of satellite 2
ω1 = argument of perigee of satellite 1
∆ω = the difference in argument of perigee between the satellites
ω2 = ω1 + ∆ω = argument of perigee of satellite 2
de/dt = change in eccentricity with time
dω/dt = change in argument of perigee with time
d(∆e)/dt = change in the difference in eccentricity between the two satellites with
time
d(∆ω)/dt = change in the difference in the argument of perigee between the two
satellites with time
As developed previously, the equations for the change in eccentricity and argument of
Page 34 perigee with time are
3 de 3J n æ R ö æ 5 ö 1 = − 3 ç E ÷ sin(i)ç1− sin 2 i÷cos(ω ) [39] − 2 2 è ø è ø 1 dt 2(1 e1 ) a 4
2 dω 3J n æ R ö æ 5 öé J æ R ö sin(i)sin(ω )ù 1 = 2 ç E ÷ ç1− sin 2 i÷ê1+ 3 ç E ÷ 1 ú [40] dt − 2 2 è a ø è 4 ø − 2 è a ø e (1 e1 ) ë 2J 2 (1 e1 ) 1 û Using the delta functions above with Equations 39 and 40, it follows that,
3 d(e + ∆e) 3J n æ R ö æ 5 ö 1 = − 3 ç E ÷ sin(i)ç1− sin 2 i÷cos(ω + ∆ω) [41] − + ∆ 2 2 1 dt 2(1 (e1 e) ) è a ø è 4 ø
2 d(ω + ∆ω) 3J n æ R ö æ 5 öé J æ R ö sin(i)sin(ω + ∆ω))ù 1 = 2 ç E ÷ ç1− sin 2 i÷ê1+ 3 ç E ÷ 1 ú − + ∆ 2 2 []− + ∆ 2 + ∆ dt (1 (e1 e) ) è a ø è 4 øë 2J 2 1 (e1 e) è a ø (e1 e) û [42] Remember that
de d(e + ∆e) 2 = 1 [43] dt dt dω d(ω + ∆ω) 2 = 1 [44] dt dt Then,
d(∆e) d()e + ∆e de = 1 − 1 [45] dt dt dt
d(∆ω) d()ω + ∆ω dω = 1 − 1 [46] dt dt dt
In order to simplify Equations 45 and 46, one might want to make a few assumptions using small angle approximations. For Case 1 that was just considered, the
-4 2 eccentricity is O(10 ). The e1 term used in Equation 39 is then squared, resulting in a
4 -4 4 -4 4 -16 term of O(e1 ). If e1 is 9.67x10 , the term e1 is O(10 ) =O(10 ) is included in the
-4 -5 -9 2 -10 original equation. If e1 is 9.67x10 and ∆e is 10 , e∆e is O(10 ) and ∆e is O(10 ).
These terms will not be neglected since they are of higher order than O(10-16).
Page 35
4.4 Simulation Results
When trying to find ∆e and ∆ω, e and ω cannot remain constant, since these values are changing with time. Due to the very small numbers used in this analysis, subtracting or adding combinations of independent states resulted in numerical errors.
Therefore, the redundant variables ∆e, ∆ω, e1, ω1, e2 and ω2 were used directly in the
Matlab ode routine. The simulations show the precession of the eccentricity and argument of perigee for Cases 1-4. Figures 5-24 are the results for a two-year time duration. The Matlab scripts used to produce these graphs are given in the Appendix.
Figures 5-8 show the eccentricity progression over time for Cases 1-4. For all the cases, the eccentricity is periodic as expected. In Figure 1, the frozen eccentricity remains constant over time as expected. In Figure 4, the eccentricity spends more time above the frozen eccentricity value of 9.67x10-4 than below it. This agrees with the
Chobotov text. Note that the variances in eccentricity in Figures 7 and 8 (for Cases 3 and
4) are an order of magnitude larger than in Figures 5 and 6 (for Cases 1 and 2).
Page 36
Figure 5: Case 1 Eccentricity
Figure 6: Case 2 Eccentricity
Page 37
Figure 7: Case 3 Eccentricity
Figure 8: Case 4 Eccentricity
Page 38 The progressions of argument of perigee over time are shown in Figures 9-12.
Again, graphs of all cases are periodic. The Case 1 Satellite 1 that begins at the frozen conditions has a constant argument of perigee value, as expected. Figures 11 and 12 show that the satellites in Cases 3 and 4 move very quickly from the minimum argument of perigee value to the maximum argument of perigee value.
Figure 9: Case 1 Argument of Perigee
Page 39
Figure 10: Case 2 Argument of Perigee
Figure 11: Case 3 Argument of Perigee
Page 40
Figure 12: Case 4 Argument of Perigee
The closed contours that occur when the eccentricity is plotted against the argument of perigee near the frozen condition are evident in Figures 13-16. Shown in
Figures 13 and 14, Cases 1 and 2 are closer to the frozen condition and circular-type contours result. Cases 3 and 4 are farther from the frozen conditions, so pear-shaped closed contours result.
Page 41
Figure 13: Case 1 Eccentricity vs. Argument of Perigee
Figure 14: Case 2 Eccentricity vs. Argument of Perigee
Page 42
Figure 15: Case 3 Eccentricity vs. Argument of Perigee
Figure 16: Case 4 Eccentricity vs. Argument of Perigee
Page 43 The differences in eccentricity between Satellites 1 and 2 are shown in Figures
17-20. Again, the graphs repeat periodically to the same magnitudes. The magnitudes for Cases 1-3 vary between +/- (∆e)o, the initial eccentricity difference. However, for
Case 4, the magnitude of this variation is an order of magnitude larger than the initial difference in eccentricity. Also, in Case 4 the difference in eccentricity rises sharply from the minimum eccentricity difference to the maximum eccentricity difference.
Figure 17: Case 1 Eccentricity Difference
Page 44
Figure 18: Case 2 Eccentricity Difference
Figure 19: Case 3 Eccentricity Difference
Page 45
Figure 20: Case 4 Eccentricity Difference
The differences in argument of perigee between Satellites 1 and 2 are shown in
Figures 21-24. The graphs are again periodic in magnitude. For Case 4, even though the maximum difference in argument of perigee is two orders of magnitude larger than the other cases, this maximum is passed through very quickly. This means that for the majority of the time, the two satellites have similar argument of perigee values.
Page 46
Figure 21: Case 1 Argument of Perigee Difference
Figure 22: Case 2 Argument of Perigee Difference
Page 47
Figure 23: Case 3 Argument of Perigee Difference
Figure 24: Case 4 Argument of Perigee Difference
Page 48 4.5 Case 4 Motion
Looking at the closed contour of the Case 4 Eccentricity vs. Argument of Perigee in Figure 16, one might worry about the two satellites being at the far left and far right bottom corners of the pear shape, resulting in the maximum argument of perigee difference. It is important to note that the motion is not at a constant rate along this closed contour. In this case, the time spent at the base of the pear-shape is minimal compared to that at the top. This is in agreement with the Chobotov text. Thus, more time is spent at eccentricity values above the frozen eccentricity value than below. For these orbits that are near-frozen (not frozen), the satellites follow each other closely and proceed quickly through the points of maximum difference.
Page 49 5 Application of Cases to the Clohessy-Wiltshire/Hill’s Equations
5.1 Introduction
In the analysis discussed so far, the orbital elements have been used to describe the orbits of the reference and interceptor satellites. To understand the general behavior of these orbits, the CW/Hill’s equations are used. To study the relative motion of these near-frozen orbits, a translation between the orbital elements and the CW/Hill’s initial conditions is desired. An algorithm is developed here to take the orbital elements of the interceptor and reference satellites and translate them into initial conditions for the
CW/Hill’s equations.
An algorithm from p. 151 of Vallado and McClain8 is used to produce the radius and velocity vectors given the orbital elements. The resulting radius and velocity vectors are in the PQW coordinate system. The PQW coordinates are transformed to IJK coordinates, then these IJK coordinates are transformed to RSW coordinates. The relative RSW coordinates are used to determine the CW/Hill’s initial conditions. As an example, this algorithm will be applied to Case 1.
5.2 Review of the CW/Hill’s Equations
As discussed in Section 2.1, the CW/Hill’s equations for near-circular orbits are as follows.
− − 2 = x 2ny 3n x f x [47] + = y 2nx f y [48]
+ 2 = z n z f z [49] Again, the solutions for the no-thrust case are given as,
Page 50 x 2y 2y x(t) = 0 sin(n ⋅t) − (3x + 0 )cos(n ⋅t) + (4x + 0 ) [50] n 0 n 0 n 4y 2x 2x y(t) = (6x + 0 )sin(n ⋅ t) + 0 cos(n ⋅ t) − (6nx + 3y )t + (y − 0 ) [51] 0 n n 0 0 0 n z z(t) = z cos(n ⋅t) + 0 sin(n ⋅ t) [52] 0 n = ⋅ + + ⋅ x(t) x0 cos(n t) (3nx0 2y0 )sin(n t) [53] = + ⋅ − ⋅ − + y(t) (6nx0 4y0 )cos(n t) 2x0 sin(n t) (6nx0 3y0 ) [54] = − ⋅ + ⋅ z(t) z0 nsin(n t) z0 cos(n t) [55]
5.3 Coordinate Systems
Three coordinate systems are used here. The definitions correspond to those used in Vallado and McClain8 on pages 36-43. Also called the Geocentric Equatorial System, the IJK coordinate system is an inertial coordinate system. The I-axis points towards the vernal equinox, the J-axis is 90 degrees to the east of this in the equatorial plane, and the
K-axis points towards the North Pole. For the PQW coordinate system, the P-axis points to perigee, the Q-axis is 90 degrees from P in the direction of satellite motion, and the W- axis is normal to the orbit. The PQW system is used in the Vallado and McClain algorithm that will be discussed shortly. For the RSW system, the R-axis extends from the center of the Earth to the satellite in the radial direction. The S-axis is in the satellite along-track direction, and the W-axis is normal to the orbital plane in the cross-track direction. The RSW system is the coordinate system that the CW/Hill’s equations are defined in for this analysis.
5.4 “Algorithm 6” from Vallado and McClain
On page 151 of Vallado and McClain8, there is an algorithm that produces the
Page 51 radius and velocity vectors given the orbital elements. The resulting radius and velocity vectors are in the PQW coordinate system. There are special conditions if the orbit is circular equatorial, circular inclined or elliptical equatorial. In particular, if the orbit is circular inclined, then set ω=0 and ν=u, where u is the argument of latitude defined as u=ω+ν. The general algorithm is as follows. All variables are as defined previously.
é pcos(ν ) ù ê + ν ú ê1 ecos( )ú ê psin(ν ) ú r = [56] PQW ê + ν ú ê1 ecos( )ú ê0 ú ê ú ë û
é µ ù ê− sin(ν ) ú ê p ú ê µ ú V = ê (e + cos(ν ))ú [57] PQW ê p ú ê ú ê0 ú ê ú ë û
écos(Ω)cos()ω − sin ()Ω sin ()ω cos ()i − cos ()Ω sin ()ω − sin ()Ω cos ()ω cos ()i sin ()Ω sin ()i ù é IJK ù ê ú ê ú = êsin()Ω cos ()ω + cos ()Ω sin ()ω cos ()i − sin ()Ω sin ()ω + cos ()Ω cos ()ω cos ()i − cos ()Ω sin ()i ú ë PQW û ëê sin(ω )sin ()i cos (ω )sin ()i cos ()i ûú
[58] L é IJK ùL r = ê úr [59] IJK ë PQW û PQW
L é IJK ù L V = ê úV [60] IJK ë PQW û PQW
5.5 Application of “Algorithm 6” to Find PQW Radius and Velocity Vectors
This algorithm can now be applied. First, the PQW radius and velocity vectors
Page 52 are found from the orbital elements. For the reference satellite,
= − 2 pref aref (1 eref ) [61] é p cos(ν ) ù ê ref ref ú + ν ê1 eref cos( ref )ú ê ν ú pref sin( ref ) r = ê ú [62] ref ,PQW + ν ê1 eref cos( ref )ú ê ú ê0 ú ëê ûú
é µ ù − ν ê sin( ref ) ú ê pref ú ê ú ê µ ú V = (e + cos(ν )) [63] ref ,PQW ê p ref ref ú ê ref ú ê0 ú ê ú ëê ûú
For the interceptor satellite,
= − 2 p1 a1(1 e1 ) [64] é ν ù p1 cos( 1) ê + ν ú ê1 e1cos( 1)ú ê p sin(ν ) ú r = 1 1 [65] 1,PQW ê + ν ú ê1 e1cos( 1)ú ê0 ú ê ú ë û
é µ ù − ν ê sin( 1) ú ê p1 ú ú ê µ V = ê (e + cos(ν ))ú [66] 1,PQW ê p 1 1 ú ê 1 ú ê0 ú ê ú ë û
Page 53 5.6 Find IJK Radius and Velocity Vectors
Next, the IJK radius and velocity vectors can be found from these PQW radius and velocity vectors. For the reference satellite, let
Ω = Ω ref [67]
ω = ω ref [68]
= i iref [69]
é ù = IJK rref ,IJK ê ú rref ,PQW [70] ë PQW ûref
é ù = IJK Vref ,IJK ê ú Vref ,PQW [71] ë PQW ûref
For the interceptor satellite,
Ω = Ω 1 [72] ω = ω 1 [73] = i i1 [74]
é ù = IJK r1,IJK ê ú r1,PQW [75] ë PQW û1
é ù = IJK V1,IJK ê ú V1,PQW [76] ë PQW û1
The relative distance between the interceptor and reference satellites in the IJK system can now be found.
= − rrel,IJK r1,IJK rref ,IJK
5.7 Find RSW Radius and Velocity Vectors
These IJK vectors must now be converted to the RSW system. From Vallado and Page 54 McClain8, the RSW coordinate system is defined as,
é0ù rref ,IJK ê ú R = = ê0ú [77] r ref ,IJK ëê1ûú
é1ù × rref ,IJK Vref ,IJK ê ú W = = ê0ú [78] r ×V ref ,IJK ref ,IJK ëê0ûú
é 0 ù ê ú S = W × R = ê−1ú [79] ëê 0 ûú
é RT ù é0 0 1ù é RSW ù ê T ú ê ú ê ú = ê S ú = 0 −1 0 [80] ë û ê ú IJK ê T ú ëW û ëê1 0 0ûú
The relative radius vector in the RSW coordinate system is then,
é ù = RSW rrel,RSW ê úrrel,IJK [81] ë IJK û
In order to find the relative velocity vector, the angular rate of the RSW system must be taken into account. From Vallado and McClain8 p. 52, the inertial and rotational velocities are related by,
vinertial = vrot + ωIJK x rrot + vorg [82] So,
vrot = vinertial - ωIJK x rrot - vorg [83]
Assume that the IJK coordinate system is the inertial system in this case. The rotating velocity vector vrot is VRSW here. vinertial is VIJK. However, in order to properly combine
Page 55 this vector with the other vectors, it must be expressed in RSW coordinates. ωIJK is the moving system’s rate of rotation with respect to the inertial system, which is the RSW coordinate system’s rate of rotation with respect to the IJK system. In this case, vorg is the velocity of the origin, which is vref here. Again, this must be expressed in RSW coordinates. The RSW coordinate system’s rate of rotation with respect to the IJK system is,
é ù ê ú ê 0 ú ω = ê ú IJK 0 [84] ê µ ú ê ú ê 3 ú ë rref ,IJK û
The RSW velocity vectors are then,
é ù é ù = RSW − ω × − RSW Vref ,RSW ê úVref ,IJK IJK rref ,RSW ê úVref ,IJK [85] ë IJK û ë IJK û
é ù é ù = RSW −ω × − RSW V1,RSW ê úV1,IJK IJK rref ,RSW ê úVref ,IJK [86] ë IJK û ë IJK û
5.8 Identify the CW/Hill’s Initial Conditions
The RSW relative radius vector was found in Equation 81.
The RSW relative velocity vector is,
= − Vrel,RSW V1,RSW Vref ,RSW [87] These are then the CW/Hills initial conditions.
x = r [88] 0 1,RSW ,relative0
y = r [89] 0 1,RSW ,relative1
z = r [90] 0 1,RSW ,relative2
Page 56 x = V [91] 0 1,RSW0
y = V [92] 0 1,RSW1
z = V [93] 0 1,RSW2
5.9 Numerical Example
5.9.1 Orbital Element Definition First, the six orbital elements for the reference satellite will be defined. As in
Case 1 above, the semi-major axis value is set to 7714 km. The inclination is set at 90 degrees, since a polar orbit is being used. The gravitational parameter of the Earth (µ) is defined as 3.986004415 km3/s2. Differing from Case 1, the reference orbit will be circular. The longitude of the ascending node (Ω) is arbitrarily chosen. Since the reference orbit is circular inclined, the orbit is in one of the special conditions for using the Vallado and McClain algorithm. For a circular inclined orbit, the algorithm says that the argument of perigee (ω) must be set to 0 and the true anomaly (ν) is equal to the argument of latitude (u). Assuming the argument of latitude is 90 degrees for now, the six orbital elements are then as follows.
aref = 7714 km Ωref = 90 deg νref = u_ref
eref = 0 ωref = 0 deg
i = 90 deg ref uref = 90 deg
The six orbital elements of the interceptor satellite must now be identified. Comparing the orbital elements of the reference and the interceptor satellite, assume for now that the difference in eccentricity between the two orbits is 9.67x10-4 and all other parameters are equal.
Page 57 5.9.2 Application of “Algorithm 6” to Find PQW Radius and Velocity Vectors
The Vallado and McClain “Algorithm 6” can now be applied. First, the PQW radius and velocity vectors are found from the orbital elements. For the reference satellite,
= − 2 = pref aref (1 eref ) 7714km [94] é p cos(ν ) ù ê ref ref ú + ν ê1 e ref cos( ref )ú é0 ù ê p sin(ν ) ú = ê ref ref ú = ê ú rref ,PQW ê7714úkm [95] ê1+e cos(ν )ú ê ref ref ú ëê0 ûú ê0 ú ê ú ë û
é µ ù − ν ê sin( ref ) ú ê pref ú ê ú é− 7188ù µ = ê + ν ú = ê ú m Vref ,PQW (eref cos( ref )) ê0 ú [96] ê p ú s ê ref ú ëê0 ûú ê0 ú ê ú ë û
For the interceptor satellite,
= − 2 = p1 a1 (1 e1 ) 7713.99km [97] é p cos(ν ) ù ê 1 1 ú 1+e cos(ν ) ê 1 1 ú é0 ù ê p sin(ν ) ú ê ú r = 1 1 = 7713.99 km [98] 1,PQW ê + ν ú ê ú ê1 e1 cos( 1 )ú ëê0 ûú ê0 ú ê ú ë û
Page 58 é µ ù − ν ê sin( 1 ) ú ê p1 ú é− 7188ù ê µ ú = ê + ν ú = ê ú m V1,PQW (e1 cos( 1 )) ê6.951 ú [99] ê p ú s ê 1 ú ëê0 ûú ê0 ú ê ú ë û
5.9.3 Find IJK Radius and Velocity Vectors
Next, the IJK radius and velocity vectors can be found from these PQW radius and velocity vectors. For the reference satellite, let
Ω = Ω ref [100]
ω = ω ref [101]
= i iref [102]
é 0 ù é ù = IJK = ê ú rref ,IJK ê ú rref ,PQW ê 0 úkm [103] ë PQW û ref ëê7714ûú
é 0 ù é ù = IJK = ê− ú m Vref ,IJK ê ú Vref ,PQW ê 7188ú [104] ë PQW û s ref ëê 0 ûú
For the interceptor satellite,
Ω = Ω 1 [105] ω = ω 1 [106]
= i i1 [107]
é 0 ù é ù = IJK = ê ú r1,IJK ê ú r1,PQW ê 0 úkm [108] ë PQW û 1 ëê7714ûú
Page 59 é 0 ù é ù = IJK = ê− ú m V1,IJK ê ú V1,PQW ê 7188ú [109] ë PQW û s 1 ëê 6.951 ûú
The relative distance between the interceptor and reference satellites in the IJK system can now be found.
é 0 ù = − = ê ú rrel,IJK r1,IJK rref ,IJK ê 0 úm [110] ëê− 7.213ûú
5.9.4 Find RSW Radius and Velocity Vectors
These IJK vectors must now be converted to the RSW system. As defined previously,
é RT ù é0 0 1ù é RSW ù ê T ú ê ú ê ú = ê S ú = 0 −1 0 [111] ë û ê ú IJK ê T ú ëW û ëê1 0 0ûú The relative radius vector in the RSW coordinate system is then,
é− 7.213x10−3 ù é ù ê ú = RSW = rrel,RSW ê úrrel,IJK ê 0 úkm [112] ë IJK û ê ú ë 0 û
The RSW coordinate system’s rate of rotation with respect to the IJK system is,
é ù ê ú ê 0 ú é 0 ù ê ú rad ω = ê 0 ú = 0 [113] IJK ê ú ê ú µ − s ê ú ëê9.319x10 4 ûú ê 3 ú ë rref ,IJK û
The RSW velocity vectors are then,
Page 60 é0ù é ù é ù = RSW −ω × − RSW = ê ú m Vref ,RSW ê úVref ,IJK IJK rref ,RSW ê úVref ,IJK ê0ú [114] ë IJK û ë IJK û s ëê0ûú
é 6.951 ù é ù é ù ê ú = RSW − ω × − RSW = −3 m V1,RSW ê úV1,IJK IJK rref ,RSW ê úVref ,IJK ê3.361x10 ú [115] ë IJK û ë IJK û s ëê 0 ûú
5.9.5 Identify the CW/Hill’s Initial Conditions
The RSW relative radius vector was found in Equation 112. The RSW relative velocity vector is,
é 6.951 ù ê ú = − = −3 m Vrel,RSW V1,RSW Vref ,RSW ê3.361x10 ú [116] s ëê 0 ûú These are then the CW/Hills initial conditions.
x = r = −7.213m [117] 0 1,RSW ,relative0
y = r = 0m [118] 0 1,RSW ,relative1
z = r = 0m [119] 0 1,RSW ,relative2
m x = V = 6.951 [120] 0 1,RSW0 s
− m y = V = 3.361x10 3 [121] 0 1,RSW1 s
m z = V = 0 [122] 0 1,RSW2 s = − The condition to achieve a Hill’s orbit is that y0 2nx0 . Here,
-4 = ≠ -2nx0=-2(9.319x10 )(-7.213)=0.01344. Sincey0 0.003361 0. 01344, the condition for a Hill’s orbit is not met. When the results are plotted, the along-track motion is close
Page 61 to sinusoidal as seen in Figure 25. Figure 26 shows that the resulting in-plane motion is roughly a 2:1 ellipse, but drifts slightly since the Hill’s orbit condition is not met.
Relative Along-Track Motion
5.00
0.00
-5.00
-10.00
-15.00
-20.00
-25.00 Relative Along-Track Position, y(t) [km] y(t) Position, Along-Track Relative
-30.00
-35.00 0123456 Time, t [hours]
Figure 25: Along-Track Motion
Page 62 In-Plane Relative Motion
10.00
8.00
6.00
4.00
2.00
0.00 -35.00 -30.00 -25.00 -20.00 -15.00 -10.00 -5.00 0.00 5.00
-2.00
-4.00 Relative Radial Position, x(t) [km] -6.00
-8.00
-10.00 Relative Along-Track Position, y(t) [km]
Figure 26: In-Plane Motion Approximately a 2:1 Ellipse
Page 63 6 Propellant Calculations
Since satellite formations will utilize smaller satellites with little or no propellant capability, the amount of propellant required to maintain a formation is of primary concern. An analysis was performed in order to understand the annual propellant requirements for orbits at near-frozen conditions.
This initial study quantifies the amount of propellant needed to correct the orbital perturbations over the course of a year. As discussed previously, the orbital perturbations of interest are quantified by the equations that describe the change in eccentricity and argument of perigee with time. These equations for de/dt and dω/dt are Equations 23 and
24, respectively. These values are then multiplied by the time interval of one year to get an approximation for the yearly deviation in eccentricity and argument of perigee. The resulting values will be referred to as δe and δω, the deviations in the eccentricity and argument of perigee. The amount of propellant (∆V) calculated is the amount needed to correct these δe and δω values. This is only a snapshot in time of a cyclic motion, but for comparison the same snapshot in time is made for a range of initial settings in eccentricity and argument of perigee near the frozen condition. The Matlab script for this
∆V program is included in the Appendix.
A purely tangential thrust is assumed, since this type of thrust is the most efficient in altering the eccentricity. The amount of propellant needed can be estimated as,
1 ∆V = V (δe) 2 + e 2 (δω) 2 [123] 2 C µ where V = [124] C r
Vc is the velocity of a circular orbit and r is the radius of the orbit. The derivation of this
Page 64 equation is quite extensive, as shown next.
6.1 Derivation of the Tangential Thrust ∆V Equation
Much of the derivation of Equation 123 is based on notes given by Professor
Trevor Williams. The derivation begins with the trajectory equation. All variables are as described previously. On p. 111 of Vallado and McClain8, the trajectory equation is given as
p r = [125] 1+ ecos(υ)
Differentiating gives the velocity in the radial direction Vr.
υ υ = = − + υ −2 − υ υ = pe (sin ) Vr r p(1 ecos ) ( e(sin ) ) [126] ()1+ ecos(υ) 2
Using the trajectory equation again
reυ(sinυ) r = [127] 1+ ecosυ
From p. 107 of Vallado and McClain8, the following equation is given.
H = r 2υ [128]
It is also known that
H = µp [129]
Using Equations 128 and 129 gives
H µp = rυ = [130] r r
Using the trajectory equation
µr(1+ ecosυ) µ(1+ ecosυ) rυ = = [131] r r
Page 65 Using this result in Equation 127 gives
esinυ µ(1+ ecosυ) r = [132] 1+ ecosυ r
Since the velocity of a circular orbit vc is
µ V = [133] c r
Equation 132 becomes
= = υ + υ −1/ 2 r Vr eVc (sin )(1 ecos ) [134]
The binomial theorem states that
n−2 2 − n(n −1)a x (a + x) n = a n + na n 1 x + + ... [135] 2!
Here
− 1 (1+ ecosυ) 1/ 2 = 1− ecosυ + ... [136] 2
Assuming that the second term is small relative to the first, this value approaches 1.
≅ υ Vr eVc (sin ) [137]
With a burn that is only tangential, there is no component in the radial direction.
∆ = Vr 0 [138]
Use this fact with Equation 137. To clarify, e0 is the eccentricity before the burn, e1 is the eccentricity after the burn, and δe is the change in eccentricity brought on by the burn.
This nomenclature also applies to the other variables.
υ = υ = + δ υ + δυ e0Vc,0 (sin 0 ) e1Vc,1 (sin 1 ) (e0 e)Vc,1 sin( 0 ) [139]
Note that Vc,0 = Vc,1 since the radius at the instant of the burn is the same.
Since
δυ = −δω [140]
Page 66 Equation 139 becomes
υ = + δ υ − δω e0 (sin 0 ) (e0 e)sin( 0 ) [141]
Using the trigonometric law that
sin(x − y) = sin x cos y − cos xsin y [142]
Equation 141 becomes
υ = + δ υ δω − υ δω e0 (sin 0 ) (e0 e)(sin 0 cos cos 0 sin ) [143]
Grouping terms,
υ − + δω + δ δω − υ δω + δ δω = sin 0 ( e0 e0 cos ecos ) cos 0 (e0 sin esin ) 0 [144]
Using the definition e1=e0+δe,
υ − + δω − υ δω = sin 0 ( e0 e1 cos ) cos 0 (e1 sin ) 0 [145]
Hypothetically, choose variables A and such that
Φ = − + δω Acos e0 e1 cos [146]
Φ = δω Asin e1 sin [147]
Equation 145 then becomes
υ Φ − υ Φ = υ Φ − υ Φ = sin 0 Acos cos 0 Asin A(sin 0 cos cos 0 sin ) 0 [148]
Using the trigonometric identity from above gives
υ − Φ = Asin( 0 ) 0 [149]
υ υ = Φ Φ + π If 0 is the position at the burn, let 0 , , etc.
sin Φ e sinδω tan Φ = = 1 [150] Φ δω − cos e1 cos e0
Then
− æ e sinδω ö Φ = tan 1 ç 1 ÷ [151] ç δω − ÷ è e1 cos e0 ø
Page 67 To define A, use a trigonometric identity
æ ()e cosδω − e 2 e 2 sin 2 δω ö 2 Φ + 2 Φ = = ç 1 0 + 1 ÷ [152] A(cos sin ) A Aç 2 2 ÷ è A A ø
= ()δω − 2 + 2 2 δω A e1 cos e0 e1 sin [153]
For small δe and δω,
cos(δω) → 1 and sin(δω) → δω [154]
So
→ ()− 2 + 2 δω 2 = ()δ 2 + 2 δω 2 A e1 e0 e1 ( ) e e1 ( ) [155]
Since the entire magnitude of the burn is tangential
∆ = − V Vtan,1 Vtan,0 [156]
Using the definitions for the angular momentum
H − H µ ∆V = 1 0 = ( p − p ) [157] r r 1 0
Since the instantaneous radius before and after the burn is the same, the trajectory equation can once again be used to say
p p r = 0 = 1 [158] + υ + υ 1 e0 cos 0 1 e1 cos 1
Rearranging,
p ()1+ e cos(υ − δω) p = 0 1 0 1 [159] 1 + υ 1 e0 cos 0
Equation 157 then can be expressed as
µ ïì ()1+ e cos(υ − δω) ïü ∆V = p í 1 0 −1ý [160] 0 ï + υ ï r î 1 e0 cos 0 þ
Again using the trajectory equation
Page 68 µ ∆V = { ()1+ e cos(υ − δω − 1+ e cosυ } [161] r 1 0 0 0
With the definition of Vc and the binomial theorem,
ì 1 1 ü ∆V = V í1 + e cos(υ − δω) + ... − (1 + e cosυ + ...)ý [162] c î 2 1 0 2 0 þ
Assume that the additional terms are small compared to those shown. The “1” terms cancel, leaving the following.
1 ∆V = V {}e cos(υ − δω) − e cosυ [163] 2 c 1 0 0 0
Using the trigonometric identity
cos(x − y) = cos xcos y + sin xsin y [164]
1 ∆V = V {}e cosυ cosδω + e sinυ sinδω − e cosυ [165] 2 c 1 0 1 0 0 0
Simplifying using the variables from before
1 ∆V = V {}cosυ AcosΦ + sinυ Asin Φ [166] 2 c 0 0
Using the trigonometric identity from before
1 ∆V = V Acos(υ − Φ) [167] 2 c 0 υ = Φ Φ + π Since 0 , , etc., the cosine term will be either +/-1. Thus the absolute value of
∆V is
1 ∆V = V A [168] 2 c
Using the A from Equation 155
1 2 2 ∆V ≅ V ()δe + e (δω) 2 [169] 2 c 1
Page 69 6.2 Propellant Calculation Results
Figures 27-33 show the results of this analysis. As all the figures show, the lowest ∆V value is at the frozen condition, as expected. The argument of perigee value requiring the least propellant is an ω of 90 degrees. This could be expected, since the cos(ω) term in the de/dt equation is zero for an ω of 90 degrees. Resulting in a de/dt value of zero, this means that the eccentricity value remains fixed even if the eccentricity is not the “frozen” value. It is interesting to note that the amount of propellant required is not symmetric about the frozen condition. Figure 29 shows an expanded range of eccentricity values. As the eccentricity values increase, the amount of propellant needed for corrections also increases. For Figures 30 and 31, the inclination is changed to 30 degrees. As expected, the location of the frozen condition changes. However, the general shape of the resulting contours remains the same. In Figures 32 and 33, a reduced semimajor axis is used. Again, the location of the frozen condition changes, but the general shape of the contours is the same.
Page 70
Figure 27: Annual ∆V Requirements for i=90 deg and a=7714km
Page 71
Figure 28: Annual ∆V Requirements with Lines of Constant ∆V
Page 72
Figure 29: Annual Propellant Requirements for an Expanded Range of Eccentricities
Page 73
Figure 30: Annual ∆V Requirements for i=30 deg
Page 74 Figure 31: Annual ∆V Requirements for i=30 deg, with Lines of Constant ∆V
Page 75
Figure 32: Annual ∆V Requirements a=7000 km
Page 76
Figure 33: Annual ∆V Requirements for a=7000 km, with Lines of Constant ∆V
Page 77 7 Summary
There is growing interest in flying coordinated clusters of small spacecraft to perform missions once accomplished by single, larger spacecraft. Using these satellite clusters reduces cost, improves survivability, and increases flexibility of the mission.
One challenge in implementing these satellite clusters is maintaining the formation as it experiences orbital perturbations due to the non-spherical Earth. A natural phenomenon exists called a frozen orbit, for which the orbital elements of argument of perigee and eccentricity remain virtually fixed over extended periods of time. Using this frozen orbit phenomenon results in considerable propellant savings.
A discussion of current literature on this topic was given. Some examples of formations at near-frozen conditions were shown. In addition, an algorithm was developed that determines if the Hill’s orbit conditions will be met given the initial differences in eccentricity and argument of perigee for two satellites. Finally, a study of the propellant impact of near-frozen conditions shows that the amount of propellant required is not linearly symmetric about the frozen condition.
Page 78 8 Suggestions for Future Work
This a preliminary study into the effects of near-frozen conditions on satellite formation flying. In the future, one could study the effects of other perturbations, such as drag, solar radiation pressure, and lunar and solar gravity on the frozen orbit results.
Perhaps these forces could be used to “push” the satellite back into the frozen contour if it does not have propellant. Also, one could study the effects of higher-level harmonics such as J4, J5, etc. In estimating the propellant needs, a more exact model could be used.
Also, it might be useful to study the propellant required to keep two satellites in formation relative to each other, and not the propellant required to maintain the near- frozen condition.
A limitation of this analysis is the use of the CW/Hill’s equations. The linearization of these equations breaks down over long durations. Also, these equations are for near- circular orbits. These equations also do not account for the case of continuous low thrust.
Currently, there are efforts to create alternate equations to the CW/Hill’s equation, both at
MIT and by Dr. William Wiesel at the Air Force Institute of Technology. Comparisons could be made between this work and a near-frozen analysis utilizing these new equations.
Page 79 9 Bibliography
1. http://www.vs.afrl.mil/factsheets/TechSat21.html, information current as of June
1998, Public Affairs Air Force Research Laboratory, Space Vehicles Directorate.
2. Williams, Trevor, “Satellite Formation Flying”, class notes from SP2001.
3. Sabol, Chris, Burns, Richard, and McLaughlin, Craig A., “Satellite Formation
Flying Design and Evolution,” AAS 99-121, 1999 AAS/AIAA Spaceflight
Mechanics Meeting, Breckenridge, CO, Advances in the Astronautical Sciences,
Volume 102, Part I, 1999, pp. 265-284.
4. Kong, Edmund M.C., Miller, David W., and Sedwick, Raymond J., “Exploiting
Orbital Dynamics for Aperture Synthesis Using Distributed Satellite Systems:
Applications to a Visible Earth Imager System,” AAS 99-122, 1999 AAS/AIAA
Spaceflight Mechanics Meeting, Breckenridge, CO, Advances in the
Astronautical Sciences, Volume 102, Part I, 1999, pp. 285-302.
5. http://www.sstl.co.uk/news/pr_940521060.html, “Another ‘First’ in Space for
Surrey, Electric Propulsion Thruster Firings in Orbit”, Oct. 21, 1999, further
information from Audrey Nice at www.sstl.co.uk.
6. Siegel, Lee, http://www.space.com/news/bic_fuel_000822.html, “Super Fuel’s
Little Secret”, August 22, 2000.
7. Hoversten, Paul,
http://www.space.com/scienceastronomy/solarsystem/deepspace_propulsion_000
816.html, “Ion Propulsion Sets a Record”, August 17, 2000.
8. Vallado, David A. and McClain, Wayne D., Fundamentals of Astrodynamics and
Applications, McGraw-Hill, 1997.
Page 80 9. Chobotov, Vladimir A., Orbital Mechanics, AIAA, 1996, 2nd ed.
10. Schaub, Hanspeter, and Alfriend, Kyle T., “J2 Invariant Relative Orbits for
Spacecraft Formations,” NASA/CP-1999-209235, 1999 Flight Mechanics
Symposium, May 1999, pp. 125-139.
11. Bate, Roger R., Mueller, Donald D., and White, Jerry E., Fundamentals of
Astrodynamics, Dover Publications, Inc., 1971.
12. Williams, Trevor, and Wang, Zhong-Sheng, “Potential Uses of Solar Radiation
Pressure in Satellite Formation Flight,” AAS 00-204, 10th AAS/AIAA Space
Flight Mechanics Meeting, Clearwater, FL, January 2000.
13. Cutting, E., Born, G.H., and Frautnick, J.C., “Orbit Analysis for SEASAT-A,”
The Journal of the Astronautical Sciences, Vol. XXVI, No. 4, pp. 315-342, Oct.-
Dec. 1978.
14. Frauenholz, R.B., Bhat, R.S. and Shapiro, B.E., “An Analysis of the
TOPEX/Poseidon Operational Orbit: Observed Variations and Why,” AAS 95-
366, Advances in the Astronauical Sciences, Vol. 90, pt. 2, 1996, p. 1127-1144.
15. Chao, C.C., Pollard, J.E., and Janson, S.W., “Dynamics and Control of Cluster
Orbits for Distributed Space Missions,” AAS 99-126, 1999 AAS/AIAA
Spaceflight Mechanics Meeting, Breckenridge, CO, Advances in the
Astronautical Sciences, Volume 102, Part I, 1999, pp. 355-374.
16. Shapiro, Bruce, “Phase Plane Analysis and Observed Frozen Orbit for the
TOPEX/POSEIDON Mission”, Sixth International Space Conference of Pacific
Basin Societies, Marina Del Rey, California, December 6-8, 1995.
17. Nickerson, K.G., Herder, R.W., Glass, A.B. and Cooley, J.L. “Application of
Page 81 Altitude Control Techniques for Low Altitude Earth Satellites,” The Journal of
the Astronautical Sciences, Vol. XXVI, No. 2, pp. 129-148, April-June 1978.
18. Shapiro, Bruce, “Phase Plane Analysis and Observed Frozen Orbit for the
TOPEX/POSEIDON Mission,” Sixth International Space Conference of Pacific
Basin Societies, Marina Del Rey, California, December 6-8, 1995.
Page 82 10 Appendix
10.1 Matlab Code to Generate Frozen Orbit Circulations
% Heidi Davidz % 03/18/01 % Program cases_run_frozen.m % This program is shows the circulations about the frozen orbit conditions
% Clear variables clear clg % Define initial conditions of the pseudo states y0=[0.001 90*pi/180 0.0012 90*pi/180 .0002 0]; % Setup ode function options=odeset('RelTol',1e-20,'AbsTol',1e-20); [t,y]=ode45('cases_froz',0,31556925.9747,y0,options); % 1 year format long; % Plot results plot(y(:,2)*180/pi,y(:,1),'b-',y(:,4)*180/pi,y(:,3),'b-'); axis([0 360 0 0.003]); title('Circulations Around the Frozen Orbit Condition'); xlabel('Argument of Perigee (degrees)'); ylabel('Eccentricity'); grid; hold on;
% RESET Y0 y0=[0.0014 90*pi/180 0.0016 90*pi/180 0.0002 0]; options=odeset('RelTol',1e-20,'AbsTol',1e-20); [t,y]=ode45('cases_froz',0,31556925.9747,y0,options); % 1 year format long; plot(y(:,2)*180/pi,y(:,1),'b-',y(:,4)*180/pi,y(:,3),'b-');
% RESET Y0 y0=[0.0018 90*pi/180 0.002 90*pi/180 0.0002 0]; options=odeset('RelTol',1e-20,'AbsTol',1e-20); [t,y]=ode45('cases_froz',0,31556925.9747,y0,options); % 1 year format long; plot(y(:,2)*180/pi,y(:,1),'b-',y(:,4)*180/pi+360,y(:,3),'b-');
% RESET Y0 y0=[0.0025 90*pi/180 0.003 90*pi/180 0.0005 0]; options=odeset('RelTol',1e-20,'AbsTol',1e-20); [t,y]=ode45('cases_froz',0,31556925.9747,y0,options); % 1 year format long;
Page 83 plot(y(:,2)*180/pi+360,y(:,1),'b-',y(:,4)*180/pi+360,y(:,3),'b-');
10.2 Matlab Code for Running Cases, Pseudo State Space Program
% Heidi Davidz % 02/22/2001 % Program cases.m % % This program is paired with cases_run.m to evaluate the precession. % % A pseudo state space will be set up as follows: % y(:,1) = e1 = eccentricity of first satellite % y(:,2) = w1 = argument of perigee of first satellite % y(:,3) = e2 = eccentricity of second satellite % y(:,4) = w2 = argument of perigee of second satellite % y(:,5) = del_e = eccentricity delta between satellites 1 & 2 % y(:,6) = del_w = argument of perigee delta between satellites 1 & 2 function ydot = cases(t,y)
% Variable values global J2 J3 R a i w n; J2=0.0010826269; J3=-0.0000025323; R=6378.1363; a=7714; i=90*pi/180; n=0.0009318;
% Description for e1,w1,e2,w2,dele,delw e1_dot=-3*J3*n/(2*(1-y(1)^2)^2)*(R/a)^3*sin(i)*(1-5/4*(sin(i))^2)*cos(y(2)); w1_dot=3*J2*n/((1-y(1)^2)^2)*(R/a)^2*(1- 5/4*(sin(i))^2)*(1+J3*R*sin(i)*sin(y(2))/(2*J2*a*y(1)*(1-y(1)^2))); e2_dot=-3*J3*n/(2*(1-y(3)^2)^2)*(R/a)^3*sin(i)*(1-5/4*(sin(i))^2)*cos(y(4)); w2_dot=3*J2*n/((1-y(3)^2)^2)*(R/a)^2*(1- 5/4*(sin(i))^2)*(1+J3*R*sin(i)*sin(y(4))/(2*J2*a*y(3)*(1-y(3)^2))); e_del_dot=(-3*J3*n/(2*(1-(y(1)+y(5))^2)^2)*(R/a)^3*sin(i)*(1- 5/4*(sin(i))^2)*cos(y(2)+y(6)))-e1_dot; aa=3*J2*n/((1-(y(1)+y(5))^2)^2)*(R/a)^2*(1-5/4*(sin(i))^2); bb=1+J3*R*sin(i)*sin(y(2)+y(6))/(2*J2*a*(y(1)+y(5))*(1-(y(1)+y(5))^2)); w_del_dot=(aa*bb)-w1_dot; ydot=[e1_dot w1_dot e2_dot w2_dot e_del_dot w_del_dot]';
Page 84 10.3 Matlab Code for Running Cases, ODE Program
% Heidi Davidz % 02/22/2001 % Program cases_run.m % % This program is paired with cases.m to evaluate the orbit precession. % % The pseudo state space will be set up as follows: % y(:,1) = e1 = eccentricity of first satellite % y(:,2) = w1 = argument of perigee of first satellite % y(:,3) = e2 = eccentricity of second satellite % y(:,4) = w2 = argument of perigee of second satellite % y(:,5) = del_e = eccentricity delta between satellites 1 & 2 % y(:,6) = del_w = argument of perigee delta between satellites 1 & 2 % % Clear variables clear % Define initial conditions %y0=[0.000967 90*pi/180 0.000977 90*pi/180 0.00001 0];%For Case 1 %y0=[0.000962 90*pi/180 0.000972 90*pi/180 0.00001 0];%For Case 2 y0=[0.00010 90*pi/180 0.00011 90*pi/180 0.00001 0];%For Case 3 %y0=[0.00010 60*pi/180 0.00011 120*pi/180 0.00001 60*pi/180];%For Case 4 % Setup ode function options=odeset('RelTol',1e-20,'AbsTol',1e-20); [t,y]=ode45('cases',0,63113852,y0,options); % 2 years format long; cases=y % Plots for e1,w1,e2,w2,del_e,del_w plot(t/(86400*365.25),y(:,1),'b-',t/(86400*365.25),y(:,3),'r-.'); legend('Sat. 1','Sat. 2',0) %title('Case 1 Satellite 1&2 Eccentricity'); % For Case 1 %title('Case 2 Satellite 1&2 Eccentricity'); % For Case 2 title('Case 3 Satellite 1&2 Eccentricity'); % For Case 3 %title('Case 4 Satellite 1&2 Eccentricity'); % For Case 4 xlabel('Simulated Time (years)'); ylabel('Eccentricity'); grid; pause plot(t/(86400*365.25),y(:,2),'b-',t/(86400*365.25),y(:,4),'r-.'); legend('Sat. 1','Sat. 2',0) %title('Case 1 Satellites 1&2 Argument of Perigee'); % For Case 1 %title('Case 2 Satellites 1&2 Argument of Perigee'); % For Case 2 title('Case 3 Satellites 1&2 Argument of Perigee'); % For Case 3 %title('Case 4 Satellites 1&2 Argument of Perigee'); % For Case 4 xlabel('Simulated Time (years)');
Page 85 ylabel('Argument of Perigee (rad)'); grid; pause plot(y(:,2),y(:,1),'b-',y(:,4),y(:,3),'r-.'); legend('Sat. 1','Sat. 2',0) %title('Case 1 Satellites 1&2 e versus w'); % For Case 1 %title('Case 2 Satellites 1&2 e versus w'); % For Case 2 title('Case 3 Satellites 1&2 e versus w'); % For Case 3 %title('Case 4 Satellites 1&2 e versus w'); % For Case 4 xlabel('Argument of Perigee (rad)'); ylabel('Eccentricity'); grid; pause plot(t/(86400*365.25),y(:,5),'b-'); %title('Case 1 Eccentricity Delta between Satellites 1&2'); % For Case 1 %title('Case 2 Eccentricity Delta between Satellites 1&2'); % For Case 2 title('Case 3 Eccentricity Delta between Satellites 1&2'); % For Case 3 %title('Case 4 Eccentricity Delta between Satellites 1&2'); % For Case 4 xlabel('Simulated Time (years)'); ylabel('Delta in Eccentricity'); grid; pause plot(t/(86400*365.25),y(:,6),'b-'); %title('Case 1 Argument of Perigee Delta between Satellites 1&2');% For Case 1 %title('Case 2 Argument of Perigee Delta between Satellites 1&2');% For Case 2 title('Case 3 Argument of Perigee Delta between Satellites 1&2');% For Case 3 %title('Case 4 Argument of Perigee Delta between Satellites 1&2');% For Case 4 xlabel('Simulated Time (years)'); ylabel('Delta in Argument of Perigee (rad)'); grid;
10.4 Matlab Code to Generate Propellant Calculation Graphs
% Heidi Davidz % 03/06/01 % Program delv.m % This program plots the amount of delta-v needed to correct % different initial of eccentricity & argument of perigee % combinations.
% Clear settings clear; clf;
% Given variable values
Page 86 J2=0.0010826269; J3=-0.0000025323; R=6378.1363; mu=3.986e5; %km^3/s^2 a=7714; %km n=(mu/(a^3))^0.5; %n=0.0009318; i=90*pi/180; %radians time=31556925.9747; % One year in seconds for jj=1:35 % for eccentricity for kk=1:36 % for argument of perigee e(jj,kk)=10^(-4)*jj; w(jj,kk)=kk*10*pi/180; %radians %Calculate e_dot and w_dot (which is de/dt and dw/dt, respectively) e_dot(jj,kk)=-3*J3*n/(2*(1-e(jj,kk)^2)^2)*(R/a)^3*sin(i)*(1- 5/4*(sin(i))^2)*cos(w(jj,kk)); w_dot(jj,kk)=3*J2*n/((1-e(jj,kk)^2)^2)*(R/a)^2*(1- 5/4*(sin(i))^2)*(1+J3*R*sin(i)*sin(w(jj,kk))/(2*J2*a*e(jj,kk)*(1-(e(jj,kk))^2))); del_e(jj,kk)=e_dot(jj,kk)*time; del_w(jj,kk)=w_dot(jj,kk)*time; v_c=(mu/a)^0.5; del_v(jj,kk)=1/2*v_c*((del_e(jj,kk))^2+(e(jj,kk))^2*(del_w(jj,kk))^2)^(1/2); end end
% Plots plot3(e,w*180/pi,del_v); title('Delta v Dependence on Eccentricity and Argument of Perigee'); xlabel('Eccentricity'); ylabel('Argument of Perigee (deg)'); zlabel('Delta v (m/s)'); grid;
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