December
17, 2003

ASEN
5050

Use of Differential Drag as a Satellite Constellation Stationkeeping Strategy

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.

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 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:

_{}.

C_{d} 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. *v _{rel}* 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 *r** _{p}* is the atmospheric density
at perigee and

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.

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

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.

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.

**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

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, 5.5m^{2}. 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 13.5m^{2}.

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 10m^{2} 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
13.5m^{2} in eclipse, and satellite 2 flies with a minimum drag area of
5.5m^{2}, 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.

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).

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 2^{nd}
or 4^{th} quadrant, the phase angle is coming closer to ideal. However, if the satellite lies in the 1^{st}
or 3^{rd} 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

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.

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.

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

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