Jim Franconeri

December 17, 2003

ASEN 5050

 

 

Use of Differential Drag as a Satellite Constellation Stationkeeping Strategy

 

Introduction and Theory

 

There is a growing trend in the commercial satellite industry to use constellations of satellites working together to accomplish complex mission objectives (Orbcomm, Iridium, Globalstar, Teledesic).  Many of these new constellations are consist of several small satellites orbiting at LEO (Low-Earth-Orbit) altitudes in precise lattice formations.  Building and launching a cluster of small satellites can be cheaper and can offer more versatility than a single large one.  Additionally, a cluster can be more robust than a single large satellite; if one part fails, another can be maneuvered into its place and take over its function.

 

One of the challenges of operating small satellites flying in formation is that they require periodic orbit maintenance, or "stationkeeping", to maximize their utility.  Stationkeeping refers to the adjustment of a satellite’s orbital characteristics in an effort to maintain a certain optimal position relative to the other satellites in the constellation. 

 

This paper discusses the advantages and risks of this technology as they pertain to the orbit maintenance requirements of a constellation of low-earth-orbit satellites.  A hypothetical LEO constellation will be postulated, along with an example of how differential drag calculations would be performed in support of the mission design and operations stages of the constellation.

 

 

The Need for Stationkeeping

 

It is in the interest of satellite constellation companies to maintain certain relative positions between the satellites in their constellations.  Ideally, under two-body dynamics, a group of satellites orbiting a planet in the same orbital plane would experience no force variations, which would cause them to stray from their optimal relative positions within the constellation.  But in fact, variations in Earth's geopotential forces (due to unique satellite groundtracks), variations in the density of the Earth's atmosphere, variations in solar radiation pressure, and variations in spacecraft attitude cause differing orbit perturbations, resulting in relative drift among satellites in a plane.  If left uncorrected, these perturbations can accumulate and significantly change the relative position of satellites, destroying the overall structure of the constellation. 

 

The term "stationkeeping" refers to efforts undertaken to ensure that satellites are in their proper orbits, which maximizes the usefulness of the constellation as a whole.  Typically, stationkeeping is performed by firing a satellite's thrusters to raise or lower an orbit.  This paper offers an explanation of an alternative stationkeeping method, called differential drag.

 

Proper stationkeeping provides optimal satellite coverage for customers, minimizing coverage gaps and providing satellite contacts at regular intervals.  Some constellations require high-fidelity stationkeeping to maintain communication cross-links between satellites, maintaining the integrity of the satellite network.  For satellites flying in very close proximity, stationkeeping is needed to ensure against collision.

 

The two orbit characteristics that are managed with stationkeeping efforts are the Argument of Latitude and the Period.  The Argument of Latitude, (or ArgLat,) is defined as the Argument of Perigee plus the True Anomaly, and is therefore a measure of the number of degrees subtended by the satellite since it last crossed the equator heading north.  Period is defined as the time the satellite takes to complete one orbit (seconds per orbit).  (Another useful representation of the period is as a number of degrees subtended per day, obtained by dividing the number of seconds in a day by the orbital period, and then multiplying by 360 degrees.)  Comparing these two characteristics between satellites in a plane allows for a measurement of stationkeeping success.  Perfect stationkeeping in a plane would mean that satellites would orbit at evenly spaced Arguments of Latitude, and would all have the same period. 

 

Two convenient terms used in describing the stationkeeping status of satellites in a plane are "phase error" and "relative drift rate."  The phase error between two satellites is defined as the actual ArgLat difference between the two minus their desired ArgLat difference.  The relative drift rate is defined as the difference between two satellites' periods, most conveniently measured in degrees of ArgLat per day.

 

 

The Differential Drag Technique

 

The concept of using differential drag as a stationkeeping strategy is a relatively new idea in the field of constellation satellite operations, and there are very few companies currently making use of it.  One of these companies is Orbcomm, whose constellation of 35 LEO satellites uses differential drag as a supplement to thrust-based stationkeeping.  Differential drag has been proposed as a means of maintaining spacecraft proximity to the International Space Station.  Use of intentional differential drag has also been proposed as a perturbing mechanism to be used to in demonstration missions verifying the feasibility of propellant-based stationkeeping systems.

 

The differential drag stationkeeping technique requires attitude or geometry changes to maximize or minimize the amount of atmospheric drag a satellite encounters, in order to maintain or speed up its orbital velocity.  This is accomplished by varying the amount of drag area presented to the atmosphere between satellites of the same plane. 

 

The acceleration due to drag on a satellite is given by:

.

Cd indicates the coefficient of drag, a dimensionless quantity that represents the extent to which the satellite is susceptible to atmospheric drag.  It depends upon the material out of which the satellite is made, and upon the aerodynamic shape of the satellite.  M is the satellite mass, and A is the cross-sectional area presented in the velocity direction.  r  is the density of the atmosphere through which the satellite is flying, and is difficult to determine accurately.  vrel is not the orbital velocity of the satellite, but rather the velocity of the satellite relative to the rotating atmosphere.  The negative sign indicates that the acceleration of drag is always in the anti-velocity direction.  For identical satellites flying in the same formation and in the same plane, all of these parameters will be approximately equal, except the atmospheric density.  It is primarily the density fluctuations in the Earth's atmosphere that cause acceleration differences between satellites flying in formation.

 

The density of the upper atmosphere is subject to variations caused by three main factors.  Imperfect homogeneity of the molecules composing the atmosphere has a strong effect on its density.  Radiation from the Sun heats up the Earth's atmosphere, causing small local density differences and large density differences between the day and night sides of the Earth.  And finally, the Earth's own geomagnetic activity heats atmospheric particles by causing high-speed collisions with charged energetic particles from the Sun, so variations in the Earth's magnetic field translate to variations in the atmospheric density.

 

Control of the varying drag accelerations could be performed by altering the drag coefficients, masses, velocities of the satellites, but changing these parameters is inefficient and impractical.  The velocity differences between satellites in the same plane will be very small, and the resulting effects on drag acceleration are negligible compared to the effects of differing areas.  Assuming that satellites in the constellation all are designed to have the same mass and coefficient of drag, the only option available for actively controlling drag accelerations is the presented area.  The orbits of satellites with a larger presented area will experience greater drag acceleration, and thus decay more rapidly than those with a smaller presented area.  As seen in the above equation, the drag acceleration is proportional to the presented area.

 

The rate of period change is given by:

where rp is the atmospheric density at perigee and s is the satellite's path.  For circular or near-circular orbits, this equation shows the directly proportional relationship between the rate of change of the period and the presented area.  This is the fundamental idea behind differential drag: differing the presented areas of two satellites to create an unequal rate of period change between them.

 

 

Practical Implementation

 

Achieving an unequal presented area requires variable geometry on the satellite.  The area presented to the velocity direction must be adjustable, to provide a proportional change in the drag force.  Possible options for altering area profiles include the angling of solar panels, yaw/pitch/roll angle changes, or deploying a specifically designed drag sail.  (Higher altitude orbits for which atmospheric drag plays little role may use a solar sail in a similar way, taking advantage of solar radiation pressure.) 

 

Differential drag maneuvers must be planned so as to avoid impact on normal satellite operations.  Spacecraft need not hold their differential drag orientations throughout an entire orbit.  For instance, if solar panels are being used to achieve the change in presented area, the maneuvers could occur during eclipse, in order to minimize the effect of solar array pointing efficiency loss.  The differential drag operations should be performed during as large a portion of the orbit as possible without adversely impacting operations.

 

 

Differential Drag Planning for a Hypothetical Constellation

 

This section will demonstrate how stationkeeping via differential drag is planned before the launch of a LEO satellite constellation. 

 

 

Satellite Orbital Characteristics

 

Imagine that we are responsible for the stationkeeping of a satellite constellation planned to consist of 6 satellites orbiting in the same plane.  The satellites' target altitude will be 600 kilometers, and their target eccentricity will be 0.  To optimize coverage and minimize gaps, we would like these satellites to orbit at a separation of 60º in Argument of Latitude.  Their inclination will be chosen such that the satellites will be in a sun-synchronous orbit. 

 

The Earth's sun-synchronous orbit RAAN rate is:

 

We will be in a circular orbit, so

 

 

 

 

 

 

The satellites must be launched at an inclination of 97.79º in order to achieve a sun-synchronous orbit. 

 

 

Satellite Physical Characteristics

 

Assume that the main body of the satellite is cylindrical, and rectangular solar panels are attached at right angles to the body.  The solar panels are free to rotate about their long axis (the x-axis).  Also assume that there is a cylindrical antenna connected to the bottom of the main body, pointing towards the Earth (the negative z-axis).  The satellite travels at an angle normal to the solar panel and antenna directions (along the y-axis).  The satellite maintains the antenna pointing to nadir at all times, but the vehicle is able to yaw (rotate about the z-axis) freely and articulate the solar panels in order to maximize solar power collection.

 

 

 

 

Dimensions

 

Solar panels: Rectangular, 1 meter x 4 meters, negligible thickness

 

Main body: Cylindrical, 1-meter diameter, 5 meters long

 

Antenna: Cylindrical, 0.1 meters diameter, 5 meters long

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Calculating presented area

 

Since the vehicle is always nadir-pointing, and the main body and the antenna are cylindrical, their contribution to the satellite's presented area remains constant for all yaw angles. Their contribution can be calculated simply by multiplying their diameters by their lengths:

 

The presented area contribution of the solar panels is dependent upon the solar panel rotation angle and the yaw angle. Let us define a 0º yaw angle to be the configuration shown in the diagram above, with the long axis of the solar panels being perpendicular to the velocity vector. A 90º yaw angle would mean that the long axis of the solar panels would lie parallel to the velocity vector.

 

For a yaw angle of zero, if the solar panels are edge-on to the velocity direction (solar panel angle = 0º), they contribute nothing to the presented area because they are of negligible thickness. If the solar panels are face-on to the velocity direction (solar panel angle = 90º), their contribution to the presented area is calculated by multiplying their length by their width. The presented area calculation becomes more complicated at different yaw angles and solar panel angles. Keeping in mind that we must multiply by two since there are two panels, and designating the yaw angle as γ and the solar panel angle as θ, we have:

 

 

The area drops off sinusoidally as the yaw angle approaches 90º and the solar panel angle approaches 0º.

 

 

The chart above shows the calculated satellite presented area as a function of solar panel angle and yaw angle.  At a yaw angle of 90º or a solar panel angle of 0º, the solar panels contribute nothing to the presented drag area.  In that case, the presented drag area is simply the baseline sum of the body and antenna areas, 5.5m2.  But as the yaw angle decreases and the solar panel angle increases, the drag area contribution from the solar panels increases.  The panel contribution (and thus the total presented area) reaches a maximum at a yaw angle of 0º and a solar panel angle of 90º.  In this configuration, the total area that the satellite presents to atmospheric drag is 13.5m2.

 

These calculations show that articulating the solar panels allows us to drastically change the satellite's drag area.  When flying at a yaw angle of 0º, changing from a solar panel angle of 0º to a solar panel angle of 90º allows us to increase the presented area by 245%. 

 

Of course, we are not free to move the solar panels and yaw to any angle we would like at any time.  The satellites must use their solar panels for power collection, and so must fly with the specific yaw and solar panel angles that point the panels at the Sun to allow for optimal power collection.  But during the segment of the orbit while the satellite is flying through the Earth's eclipse, power collection concerns do not apply.  During the eclipse, we are free to maneuver the yaw angle and the solar panel angle to any configuration we choose without fear that we are missing out on power collection.

 

Since we have chosen a circular sun-synchronous orbit , each satellite will experience the same eclipse duration each day.  The following diagram illustrates the percentage of each orbit that falls in eclipse.  The view is from a perspective perpendicular to the plane of the satellites' orbit, and is not to scale.

To calculate the size of the eclipse region, determine the angle a:

 

 

The satellite is in eclipse for 132.1º out of  360º, or 37% of each orbit.  This time can be used to perform differential drag operations without impacting the satellites' power generation requirements.

 

We must make some simplifying assumptions in order to calculate the effect that these maneuvers will have on the drag accelerations.  We will assume that the satellites in the plane will be flying with similar yaw and solar panel profiles while they are in sunlight, meaning that the only differential drag they will experience in sunlight will be caused by the atmospheric density variations.  We assume that all vehicles in the plane have the same drag coefficients, velocities, and masses.  We will perform all of our differential yaw and solar panel maneuvers while in eclipse, and thus we can effect a change in the presented area of 245% for 37% of each orbit.  Let us assume that in sunlight, the satellites will fly with an average of 10m2 presented area.  We may then compute the effectiveness of the maneuvers:

For any two satellites in the plane, A and B,

                        

Since our objective is to equalize the accelerations on these two vehicles, we set these two equations equal to one another:

 

 

If satellite 1 flies with a maximum drag area of 13.5m2 in eclipse, and satellite 2 flies with a minimum drag area of 5.5m2, then averaged over the whole orbit,

 

Therefore, our solar panel and yaw maneuvers should be able to compensate for the differential accelerations caused by a 36% variation in the atmospheric density encountered by two satellites.

 

 

An Operational Example of the Technique

 

To demonstrate the differential drag technique operationally, consider the two satellites in our plane designed to be spaced at 180 degrees apart in Argument of Latitude from one another.  Assume that perturbations have caused the two to have slightly different semi-major axes (and hence, slightly different orbital periods).  The satellite with the larger semi-major axis (Satellite A) will have the larger orbital period (figure 1).  Since Satellite A’s period is longer than that of Satellite B, Satellite A will start to lag behind Satellite B in the orbit (figure 2).  The Argument of Latitude difference between the two will increase from the optimum of 180 degrees.

 

Differential drag will be used to remedy the situation by placing Satellite A into a maximum drag configuration and Satellite B into a minimum drag configuration.  This will have the effect that Satellite A’s semi-major axis will decay more rapidly than that of Satellite B.  Lowering a satellite’s orbit decreases its period.  Satellite A’s period will decrease more rapidly than Satellite B’s (figure 3).  It is important to note that at some point as the satellites return to the optimal 180-degree phasing, the differential drag configurations must be reversed so that when they reach the correct phase, they also have the same period.  The satellites will maintain differing area profiles until the two periods are matched and the phase angle is the desired 180 degrees, at which time the two satellites will be returned to their nominal attitude configurations (figure 4).

 

Phase plots

 

To visualize the relative drift situation in a satellite plane, it is useful to construct a phase plot on which each satellite is represented according to its phase angle and drift rate relative to some reference orbit.  For convenience in discussing phase plots, it is useful to define the terms "flare" and "feather".  The flare mode will refer to the maximum presented area configuration, and the feather mode will refer to the minimum presented area configuration.  The reference orbit should be picked so as to be convenient for assessing which satellites should flare and which should feather.  The orbit of a particular disabled satellite that cannot change attitude may be a convenient choice as a reference.  If all satellites in a plane are equally healthy, a convenient reference vehicle could be selected by determining which selection would require the least amount of correction to achieve perfect stationkeeping.  The reference need not even necessarily coincide with the orbit of one of the satellites, but may rather be some computed mean of all the satellites' orbits.

 

The reference satellite (or the reference orbit) lies at the origin of the plot.  The remaining satellites in the plane are then plotted according to their phase error and drift rate with respect to the reference satellite.  The phase plot may then be used in making differential drag decisions.

 

A sample phase plot is shown below, for our plane of six satellites flying in formation.  The ideal situation for our plane would be a perfect 60 degrees of ArgLat between each satellite, and each satellite orbiting with the same identical period.  In this example, Satellite 1 has been designated as the reference vehicle, so it lies at the origin. 

 

 

We desire all the satellites to have the same period as Satellite 1 – this means every satellite will ideally have a zero relative drift rate with respect to Satellite 1.  Satellites with positive drift rates are in lower orbits than the reference and have shorter periods than the reference.  Satellites with negative drift rates are in higher orbits than the reference and have longer periods than the reference.  We also desire all the satellites to have perfect phase separation from one another – this means every satellite will ideally have zero ArgLat phase error with respect to Satellite 1.

 

A non-zero drift rate means that the satellite’s phase angle with respect to the reference vehicle is changing.  If the satellite lies in the 2nd or 4th quadrant, the phase angle is coming closer to ideal.  However, if the satellite lies in the 1st or 3rd quadrant, the phase angle is growing further from ideal.

 

As seen in the phase plot above, three satellites are drifting in the correct direction (Satellites 3, 4, and 5), and two are drifting in the wrong direction (Satellites 2 and 6).  Satellite 2 is two degrees away from its ideal position, and that spacing is growing worse by 0.03 degrees every day.  As time passes, Satellite 2 will move further and further left on this plot unless action is taken to return it to its ideal orbit.

 

A decelerate thrust (a thrust in the anti-velocity direction) could accomplish the correction, lowering Satellite 2’s orbit and reversing the drift direction.  However, since our constellation is utilizing differential drag, may be able to remedy the situation without expending valuable propellant.  If the presented area of Satellite 2 is maximized so that it exceeds the presented area of the other satellites in the plane, it will experience a greater drag force, and its orbit will be lowered faster than the others.  While it is true that this method will take an extended period of time compared to an instantaneous correction from a thrust, the propellant savings makes the differential drag method more efficient.  Even if differential drag is not fully successful on its own, if it is employed until the point where stationkeeping limits are about to be violated, then the corrective thrust will be smaller in magnitude than it would have been if differential drag was not attempted.  This still represents a propellant savings over the solely thrust-based correction.

 

 

Advantages and Limitations of Differential Drag

 

Advantages

 

The major advantage of the differential drag technique is the spacecraft mass savings that it affords.  A more massive satellite is more expensive to launch, therefore satellite manufacturers try to conserve mass wherever possible.  If a satellite can perform all or part of its stationkeeping without propellant, the propellant mass savings is translated into launch cost savings.  If a satellite can perform all its stationkeeping using differential drag, it not only saves propellant mass, but also thruster instrumentation mass (thruster valves and tubing, propellant tank, temperature and pressure sensors, etc.).

 

For constellations using both differential drag and propulsive thrusting as stationkeeping mechanisms, prudent use of differential drag can help conserve propellant, which may be valuable over the life of the satellite to accomplish other larger-scale orbit changes, such as inter-plane re-spacing or changing ascending node drift rates.

 

If designers do not want to rely on differential drag for stationkeeping, it may be used as a backup plan in the event that there is damage to the propulsion system.

 

Using differential drag is less disruptive to the attitude control system than a propulsive thrust.  Differential drag allows the orbit-changing impulse to be spread out over a long period of time, meaning that the attitude control system doesn't have to work as hard to maintain the proper satellite attitude.  An instantaneous thrust may jar the satellite violently enough that attitude control may be temporarily lost.  In this way, differential drag is gentler on the other satellite subsystems when compared to impulsive thrusting.

 

Atmospheric drag eventually causes the orbits of LEO satellites to decay to the point that they de-orbit.  The capability to minimize the presented area allows a satellite operator to delay this de-orbit for as long as possible.  If a satellite were nearing its de-orbit altitude, the satellite could be switched to its minimum drag mode.  This mode would postpone the de-orbit for as long as possible, maximizing the satellite's service-providing lifetime, and allowing for an increased measure of control over the location of the eventual de-orbit.

 

Differential drag may be used to achieve and maintain orbit circularity.  Satellites in elliptical orbits will experience greater atmospheric drag while near periapse than they will near apoapse.  If a certain percentage of an orbit were allotted to differential drag operations, the segment of the orbit near periapse could be chosen to operate in a maximum drag configuration, resulting in a more rapid lowering of the apoapse and a circularization of the orbit.

 

 

Limitations

 

Differential drag is more effective when the density of the atmosphere is higher.  Therefore, differential drag is more efficient for constellations operating at lower altitudes.  Differential drag may not be feasible for constellations whose satellites orbit at high altitude.  At altitudes around 800 kilometers, the perturbing effects of solar radiation pressure are approximately equivalent to those of atmospheric drag.  For orbits operating above this approximate threshold, it would be more efficient to design a system that would make use of solar radiation pressure differentials rather than atmospheric drag differentials.  

 

The solar cycle should also be accounted for when planning a mission where differential drag will be used.  The density of the atmosphere is greater during solar maximum, and therefore differential drag will be more effective at solar maximum.

 

A thorough analysis of the atmospheric density that the satellites are expected to encounter over their designed lifetimes should be performed prior to adopting the differential drag strategy.  Since prediction of atmospheric density is very difficult, a significant margin should be built in to the stationkeeping budget to ensure control even in unexpectedly low-density atmospheric conditions.

 

If differential drag is to be the sole method of stationkeeping, then mission designers must be able to show that it will be sufficient even throughout the minimum atmospheric density conditions expected to be encountered during the mission.  For satellite missions planned to operate primarily during years in which the solar activity is near a minimum, the resulting smaller benefits of differential drag might be counterbalanced by the increased operational complexity required to perform the maneuvers themselves, to the point that the strategy is not worthwhile.

 

An important consideration of using differential drag is that orbits can never be raised using the technique.  Differential drag merely gives a measure of control over how fast the orbits decay.  Stationkeeping is maintained by creating different decay rates among satellites, rather than by impulsively raising or lowering orbits using thrust maneuvers.  In fact, it results in increased drag effects and an increase in the altitude loss over time when compared to a satellite flying at a minimum drag configuration throughout the duration of its mission lifetime.  Therefore differential drag would not be effective for a mission that requires satellites to remain at a fixed altitude.

 

The differential drag operations may adversely impact other satellite subsystems.  Changing the satellite's attitude may result in decreased power generation capability, or decreased attitude control.  These risks may be mitigated by performing the maneuvers during times in the orbit that the satellite has surplus power, or when attitude changes will have no effect on power generation (such as eclipse periods).

 

Another risk is malfunction of the area change actuator.  If solar panels are designed to be used to aid in stationkeeping, but as a result of some anomaly become immobile, or are restricted as a result of unforeseen operational constraints, the stationkeeping strategy would be sabotaged.  Of course, the malfunction risk always exists for any satellite subsystem, so it is not necessarily a strike against differential drag when comparing it to other methods.  A satellite relying on a thruster system for stationkeeping faces risks that that system may malfunction.  Whatever method is chosen, it must be designed to be as robust as possible.  It is wise design practice to build in redundancy wherever possible, including the stationkeeping mechanism.

 

Design teams must weigh each of these risks and design the spacecraft to function with an amount of risk/performance tradeoff they are comfortable with.

 

 

Conclusion

 

The option of differential drag is an inexpensive stationkeeping aid for satellite companies.  It should be considered in the planning of any low-earth-orbit constellation that has stationkeeping requirements.  Its use may not be practical for every venture, due to its various limitations, but for missions where it is feasible, it can prove to be an efficient propellant-saving technique.

 

 

References

 

Fundamentals of Astrodynamics and Applications (Second Edition)

David A. Vallado

 

Autonomous Constellation Maintenance

James R. Wertz, John T. Collins, Simon Dawson, Hans J. Koenigsmann, Curtis W. Potterveld

http://www.smad.com/analysis/Autonomous.pdf

 

Autonomous Formation Flying Control for Multiple Satellite Constellations

http://www.aa.washington.edu/faculty/campbell/ff/ff.htm

 

Low-Cost, Minimum-Size Satellites for Demonstration of Formation Flying Modes at Small, Kilometer-Size Distances

Hans F. Meissinger, John Collins, Gwynne Gurevich, Simon Dawson

http://www.smad.com/analysis/TS-VI-3.pdf

 

 

MEMS Mega-pixel Micro-thruster Arrays for Small Satellite Stationkeeping

Daniel W. Youngner, Son Thai Lu, Edgar Choueiri, Jamie B. Neidert, Robert E. Black III, Kenneth J. Graham, Dave Fahey, Rodney Lucus, Xiaoyang Zhu

http://alfven.princeton.edu/papers/MMMA.pdf