Aerodynamic Lift and
Drag Effects on the Orbital Lifetime Low Earth Orbit (LEO) Satellites
Carlos L. Pulido
Department of Aerospace Engineering Sciences
University of Colorado Boulder
Abstract
The effects of aerodynamic perturbation
forces on the orbital life of low altitude Low Earth Orbit (LEO) satellites are
investigated, with a focus on the secular perturbations caused by lift and drag.
The secular effects can be divided into the circularization of the orbit for
both lift and drag perturbation forces, and the decrease of the orbit
semi-major axis for drag perturbation. A study of these effects on an attitude
stabilized satellite with an orbit of initial perigee altitude of 300 km and initial
eccentricity of 0.12 is performed. The objective is to determine whether the
effects of aerodynamic forces (i.e., lift and drag) can be effectively used to
control the orbital lifetime of a satellite.
I.
Introduction
a. Background
The analysis of a satellite’s motion around the
Earth can be approximated by assuming a Keplerian orbit, where Earth is assumed
to be a perfect sphere with a mass much greater than that of the satellite, and
where gravity is the only force acting on the orbiting body [1]. The actual
motion of the satellite is then perturbed from this “reference” orbit by other
external forces, such as aerodynamic forces, solar radiation pressure and
non-spherical mass effects [2]. In low-Earth orbits (LEO) the second major
perturbation force acting on a satellite is aerodynamic drag. To a lesser
extent, the aerodynamic lift force may also perturb the orbit of the satellite
if it is attitude stabilized. These forces play a major role into a satellite’s
orbital life and eventual decay. For LEO missions of perigee altitudes of less
than 500 km, the secular effects due to aerodynamic forces are of great
importance as the orbital lifetime has to be long enough to satisfy mission
requirements [3].
b. Foundation
At low altitude LEO orbits, aerodynamic forces are
the principal non-gravitational forces acting on a satellite. The aerodynamic
drag acceleration, f_{D}, is defined in Equation 1, where ρ is the atmospheric density, is the
coefficient of drag, A is the
cross-sectional area, m is the mass
of the satellite and V is the
magnitude of the velocity. Drag acts opposite to the velocity vector and
continuously slows down and removes energy from the satellite [3]. Aerodynamic
lift acceleration, f_{L},
acts perpendicular to the velocity vector and is defined in Equation 2, where ρ is the atmospheric density, is the
coefficient of lift, A is the
cross-sectional area, m is the mass
of the satellite and V is the
magnitude of the velocity. Lift is mainly achieved through the use of lifting
surfaces, such as airfoils. These forces are highly dependent on the
atmospheric density (varies with altitude and epoch) and the cross-sectional
area normal to the velocity vector (varies with attitude) [3].
To better understand the aerodynamic forces acting
on a satellite, the flow of atmospheric air around the satellite has to be
characterized. At very low altitudes, the density of the atmospheric air is
high enough to consider the air flow a continuum or bulk of air particles
moving over a body. However, at altitudes above 75 km the flow of air cannot be
assumed to be a continuum as the mean free path, λ ,between air molecules
becomes large (on the order of 1cm), and kinetic theory of gases has to be used
instead to describe the airflow [4]. The Aerodynamic drag has an appreciable effect
up to an altitude of around 1000 km, where there is a lack of collisions between
the satellite’s surface and air molecules [4]. Similarly, the aerodynamic lift only
has a significant effect on the orbit for perigee altitudes up to 500 km [5].
The presence of the aerodynamic forces in LEO, will cause perturbations to the Keplerian orbital
elements. These perturbations can be categorized as secular (i.e.,
non-periodic), long period (i.e., longer than the orbit’s period), or short
period (i.e., shorter than the orbit’s period) [1]. Most of the satellites will
exhibit a constant tumbling or rotational motion during the period of their
orbit, which will cause the lift vector that acts
normal to the velocity to average out over their life time [5]. For these cases,
only the perturbations due to atmospheric drag are considered. However, when a
satellite’s attitude is stabilized at a particular angle of incidence to the
airflow for a long period of the orbit, particularly near the perigee, it is
expected that aerodynamic lift will have an effect on the orbit [5]. Figure 1.1
shows the direction of the forces acting on the satellite for this case. The
perturbation to the orbital elements will affect the satellite’s lifetime and
will have to be taken into account for mission planning. This study will focus
on the secular perturbations to the orbit caused by aerodynamic lift and drag.
Figure 1.1 – Direction of Lift, L, and Drag, D, forces with respect to the velocity vector, V, for a generic attitude stabilized satellite.
c. Fundamental Concepts
The perturbations caused by the aerodynamic forces
can be analyzed starting with the equations of motion for a Keplerian orbit and
introducing a perturbing acceleration f,
as defined by Equation 3, where r
is the satellite’s position vector and μ
is the gravitational parameter [5]. To simplify the initial analysis, only one
perturbing force will be analyzed at a given time.
The perturbing forces f_{D}
and f_{L}
due to lift and drag are defined in Equations 4 and 5. We can define the
direction of the perturbing accelerations by introducing the unit vectors P, Q, W from the
perifocal coordinate system [3], where P
is in the orbital plane in the direction of perigee, W in the direction of the angular momentum vector h, Q is the cross product between P and W.
For perturbed motion, the semi-major axis a, the specific angular momentum vector h and Hamilton’s integral e are not
constant, and their rates of change are defined by [5]:
From these equations, we can observe that the perturbing
force due to lift will not have an effect in the change in semi-major axis, as
it is normal to the velocity vector [5]. In addition, we can
see that the perturbing force due to drag will have an effect in both the
semi-major axis and the eccentricity. However, to provide a true analysis of
the orbit’s perturbation the parameters that make up the lift and drag forces
have to be understood.
Atmospheric
Density
The atmospheric density is imperative in the
calculations of the aerodynamic forces, yet it is a parameter that varies with
altitude and solar activity, principally over the course of a solar cycle. The
mean daily F10.7 index of radio flux from the Sun, is a measurement of the
radio flux at a wavelength of 10.7 cm in units of 10^{-22} watts/m^{2}/Hz,
and is a tool used to determine solar maxima and minima (Figure 1.2) [3]. In
addition, to the radio flux index, geomagnetic indices, such as the AP index
can be used to model the environment due to solar variations. These variations
in solar activity can have a big impact in satellite orbital life. As seen in
Wertz, satellites decay very rapidly during solar maxima and very slowly during
solar minima [3]. The National Oceanic and Atmospheric Administration is in charge of these measurements, and their values for any
particular past day can be found on their website (http://www.noaa.gov). Using
this atmospheric information, robust atmospheric models can be used such as the
2001 United States Naval Research Laboratory Mass Spectrometer and Incoherent
Scatter Radar Exosphere (NRLMSISE-00 Model 2001), which maps out the atmosphere
from sea level to 1000 km [6]. This
project will use the NRLMSISE-00 Model 2001 for density calculations.
Figure 1.2 – Observed daily mean radio flux at 10.7 cm.
(http://vademecum.brandenberger.eu/grafiken/klima/solar_flux.png)
Coefficients
of Lift and Drag
The spacecraft’s lift and drag coefficients are
primarily affected by the satellite’s shape, attitude, atmospheric density and
surface conditions (i.e., smooth, rough, or sticky) [3]. The surface condition
greatly affects the manner which the molecules reflect from the surface of the
satellite. Equations 9 through 12 show how the coefficients of lift and drag
can be calculated for specular and diffuse cases, with the angle of incidence ψ, surface area S, incoming velocity v_{i} and reflecting velocity v_{r}_{
}[3] [7]. The coefficient of lift can be assumed constant over the orbit,
or it can vary by the eccentric anomaly [5]. For an eccentric anomaly is
negative in the range 0 < E <
π, and is positive in the range π < E < 2π. For this project, the coefficient of lift will vary
over the orbit and will be calculated using only specular reflection, as described
by Equation 10, at an angle of incidence ψ
= 45° to match the work done by G.E. Cook [5]. A baseline coefficient of
drag C_{D }of 2.2 will be
selected, to match the work done by K. Moe and M. Moe [8].
·
Specular Reflection:
·
Diffuse Reflection:
Ballistic
Coefficient
The coefficient of drag C_{D}, the cross-sectional area A, and the mass of the satellite m can be combined into a single parameter: the ballistic
coefficient β, as defined in
Equation 13. Similarly, G.E. Cook defines a parameter Φ, by grouping the absolute value of the coefficient of lift C_{L}, the cross-sectional area A, and the mass of
the satellite m, as defined in
Equation 14 [5]. Inspection of Equations 13 and 12, show that a satellite with
the same cross-sectional area and higher mass will be more resistant to the
aerodynamic forces, as expected due to the inertial properties of mass [3]. The area to mass ratio for this project will
be 0.01 m^{2}/kg, in order to compare to the
work performed by G.E. Cook [5].
Secular
Perturbations Due to Lift and Drag
To find the secular perturbations due to lift and drag,
start with the equation of motion of Equation 3, and decomposing the vector
equations into can be split into radial, cross-radial, and perpendicular
components. This method generates the Gaussian variational equations (GVE).
Walter shows that through these equations one can determine that drag will have
secular effects on semi-major axis and eccentricity [9]. Through a similar
approach, Cook determines that lift will only have secular effect on
eccentricity, and will not act once the orbit is circularized
[5]. Following the work of Walter and Cook, the decoupled secular effects of
lift and drag on the orbit can be found for elliptical and circular orbits, as
shown in Equations 15-19 [9] [5]. In these equations, H_{per} is the scale
height at the perigee, r_{per}
is the radius at perigee and ρ_{per} is the density at the perigee.
·
Elliptic Orbits
For the following equations, an orbit is defined as
circular when . The following equations are only valid when this
is true.
o
Secular Effects Caused
by Atmospheric Drag
o
Secular Effects Caused
by Atmospheric Lift
·
Circular Orbits
For the following equations, an orbit is defined as
circular when . The following equations are only valid for .
o
Secular Effects Caused
by Atmospheric Drag
II.
Literature Search
[1] D.A. Vallado, Fundamentals of Astrodynamics and
Applications, 3rd. Edition.
Hawthorne: Microcosm Press and
New York: Springer, 2007.
[2] B. Wie, Space
Vehicle Dynamics and Control, AIAA, 2008
[3] J. Wertz, Space
Mission Engineering: The New SMAD, Microcosm Press, 2011.
[4] C. Shen, Rarefied Gas Dynamics: Fundamentals, Simulations
and Micro Flows, Springer, 2005 .
[5] G.E. Cook, “The Effect of Aerodynamic Lift on Satellite
Orbits”, Planetary and Space Science, Vol. 12, pp. 1009-1020, 1964.
[6]
J. Picone, "NRLMSISE-00: A New Empirical Model of the
Atmosphere," 2003. [Online]. Available:
http://www.nrl.navy.mil/research/nrl-review/2003/atmospheric-science/picone/.
[Accessed 10 12 2012].
[7] P. Moore, “The Effect
of Aerodynamic Lift on Near Circular Satellite Orbits”, Planetary and Space Science,
Vol. 33, No. 5, pp. 479-491, 1985.
[8] K. Moe and M. Moe,
"Gas-Surface Interactions and Satellite Drag Coefficients", Planetary
and Space Science, 2005.
[9] U. Walter, Astronautics, Wiley-VCH, 2008.
[10] M.F. Storz
et al., “High accuracy satellite drag model (HASDM)”, Advances in Space Research, Vol 36, Issue 12, 2005.
[11] D.G. King-Hele, D.M.C. Walker, “Predicting the orbital
lifetimes of Earth satellites”, Acta
Astronautica, Vol. 18, pp. 123-131, 1988.
III.
Problem of Interest
a. Importance of Aspect to
the Study
The main objective of this
study is to assess the how aerodynamic forces perturb the orbit of a satellite
and how they affect the orbital lifetime of a satellite in LEO. This section will take a look at how the
acceleration due to atmospheric drag circularizes the orbit of an attitude
stabilized satellite in an initial baseline orbit of perigee altitude of 300 km
and eccentricity of 0.12. The mission will be flown for three different values
of C_{D}, starting with a
baseline value of 2.2. The difference in coefficient of drag, will lead to a
modification in ballistic coefficient which translates to a different satellite
design. The circularization times of these three different designs will be
compared.
b. Importance of Problem to
the Field of Astrodynamics
Since aerodynamic forces are the
principal non-gravitational forces acting on a satellite at low altitude LEO
orbits, it is important to understand how different satellite designs will
affect the orbital lifetime. Understanding the rate of orbital decay attributed
to atmospheric drag can allow a mission designer to allocate the appropriate
propulsion requirements to counter act drag in order for the satellite to stay
in orbit for the duration of the mission. This study can also have implications
in orbital debris removal, since a mission can be designed to decrease the
orbital lifetime of debris taking advantage of the decay due to atmospheric
drag.
c. Development of Solution
Method
The following steps are followed in the
development of the solution:
1. Initialize all of the orbit parameters for the
baseline mission of perigee altitude of 300 km and eccentricity of 0.12, using
the information in the Fundamental Concepts section. The satellite will begin
its orbit above Boulder, on Jan. 8^{th}, 2009, and will use this date
to calculate the atmospheric data for density calculations.
2. Set the coefficient of drag to the desired value and
calculate the ballistic coefficient given the baseline area to mass ratio of 0.01 m^{2}/kg. The baseline mission uses a C_{D }of 2.2, and comparison
missions use a C_{D }of 3 and
4.
3. Using numerical methods propagate the orbit in time,
doing a time step every hour. The secular changes in semi-major axis and
eccentricity are calculated at each time step using the equations developed in
the Fundamental Concepts section. The time for circularization is recorded, and
the orbital lifetime is estimated.
d. Analysis of Method
The following circularization times were
obtained for the missions using the method described in the Description of the
Solution Method Section:
Mission |
Time of Circularization |
Mission I (Baseline) |
706.9 days |
Mission II (C_{D}
= 3) |
605.3 days |
Mission III (C_{D}
= 4) |
524.3 days |
It is observed that as the coefficient
of drag and ballistic coefficient increase, the time it takes for the elliptic
orbit to become circular decreases. This is expected, from the discussion in
the Fundamental Concept section. The mission profiles are shown below.
·
Baseline Mission: C_{D} = 2.2
·
Mission II: C_{D} = 3
·
Mission III: C_{D} = 4
·
Mission Comparison
Further analysis on the orbital decay of
the satellite was planned, but due to numerical problems in the propagation
algorithm an analysis of the decay from a circular orbit could not be achieved.
The time for circularization, however, does provide some insight on the effects
of atmospheric drag on the orbital lifetime. From Walter, we know that the
ratio between the time of circularization of an elliptical orbit and the time
of circular orbit lifetime is equal to 1.7 [9]. Using this information we can
provide an estimate for the total orbital lifetime of the satellites for the
missions:
Mission |
Estimated Orbital Lifetime |
Mission I (Baseline) |
1122.72 days |
Mission II (C_{D}
= 3) |
961.36 days |
Mission III (C_{D}
= 4) |
832.71 days |
IV.
Extension
a. Extension to the Work
Presented in the Literature
This project will build on the baseline
mission, and include the secular perturbations due to aerodynamic lift in
addition to the secular perturbation due to drag. Only the baseline case of a
coefficient of drag equal to 2.2 will be considered. The time to circularize
the orbit will be compared to the case where only the drag is present. This
study will make a case to whether the aerodynamic lift is important in the
modeling of perturbations to an attitude stabilized satellite in a low LEO
orbit.
b. Analyze the Extension
The following circularization times were
obtained for the missions using the method described in the Description of the
Solution Method Section:
Mission |
Time of Circularization |
Mission I (Baseline) |
706.9 days |
Extension Mission (Lift
Effects) |
460.7 days |
Adding the secular perturbation due to
lift to the baseline mission significantly reduces the time of circularization
by 34.82 %. This effect is greater than that exhibited by missions II and III,
which increased the coefficient of drag by a significant amount. In addition,
looking at the mission profiles below, we can see that the circularization is
faster, but the final circular orbit is of greater radius than that of the
baseline design. This is an interesting effect that was not expected. Again,
due to numerical
problems in the propagation algorithm an analysis of the decay from a circular
orbit could not be achieved, and is something that should be recommended for
future work in order to investigate the larger radius of the final circular
orbit.
·
Baseline Including Aerodynamic Lift Effects
·
Comparison with Baseline Mission
c. Recommendations Based on
Analysis
From the results above, we can see that
including the secular effects due to aerodynamic lift significantly reduces the
circularization time. Therefore, based on the analysis it is recommended to
include the effects of aerodynamic lift for an attitude stabilized satellite in
LEO.
V.
Summary and Conclusions
The effects of aerodynamic perturbation
forces on the orbital life of low altitude Low Earth Orbit (LEO) satellites were
investigated, with a focus on the secular perturbations caused by lift and drag.
The secular effects can be divided into the circularization of the orbit for
both lift and drag perturbation forces, and the decrease of the orbit
semi-major axis for drag perturbation. A study of these effects on an attitude
stabilized satellite with an orbit of initial perigee altitude of 300 km and initial
eccentricity of 0.12 is performed. It was determined that aerodynamic lift does
contribute to the orbital decay of the satellite and must be included in the
modeling of attitude stabilized satellites. However, further analysis has to be
carried out to fully understand these effects, as numerical problems with the
algorithm did not allow for an extensive analysis of the subject. From the
current analysis, one can infer that a combination of lift and drag secular
effects could be used in order to decrease the orbital lifetime of a satellite
faster than just increasing the ballistic coefficient. This could prove useful
in the field of orbital debris mitigation and collection.