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, fD, 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, fL, 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 fD and fL 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/m2/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.

http://vademecum.brandenberger.eu/grafiken/klima/solar_flux.png

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 vi and reflecting velocity vr [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 CD 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 CD, 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 CL, 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 m2/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, Hper is the scale height at the perigee, rper is the radius at perigee and ρper 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 CD, 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. 8th, 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 m2/kg. The baseline mission uses a CD of 2.2, and comparison missions use a CD 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 (CD = 3)

605.3 days

Mission III (CD = 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: CD = 2.2


·        Mission II: CD = 3

  

 

·        Mission III: CD = 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 (CD = 3)

961.36 days

Mission III (CD = 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.

 

 

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