A preliminary formation flying orbit dynamics analysis for Leonardo-BRDF

Leonardo-BRDF is a new NASA mission concept proposed to allow the investigation of radiative transfer and its effect on the Earth's climate and atmospheric phenomenon. Enabled by the recent developments in small-satellite and formation flying technology, the mission is envisioned to be composed of an array of spacecraft in carefully designed orbits. The different perspectives provided by a distributed array of spacecraft offer a unique advantage to study the Earth's albedo. This paper presents the flight dynamics analysis performed in the context of the Leonardo-BRDF science requirements. First, the albedo integral is investigated and the effect of viewing geometry on science return is studied. The method used in this paper, based on Gauss quadrature, provides the optimal formation geometry to ensure that the value of the integral is accurately approximated. An orbit design approach is presented to achieve specific relative orbit geometries while simultaneously satisfying orbit dynamics constraints to reduce formation-keeping fuel expenditure. The relative geometry afforded by the design is discussed in terms of mission requirements. An optimal Lambert initialization scheme is presented with the required /spl Delta/V to distribute all spacecraft from a common parking orbit into their appropriate orbits in the formation. Finally, formation-keeping strategies are developed and the associated /spl Delta/V's are calculated to maintain the formation in the presence of perturbations.

requirements imposed by the science mission. defirution of radiative forcing uscd in this analysis. First. we present a conceptual description of BRDF. The coordinate sys-This paper presents the preliminary flight dynamics analysis tem and the vlulables uscd to describe radiative forcing are deperformed in context of the science requirements to effectively fined. A mathematical model is presented along with a Gauss estimate the albedo effect. We begin with an introduction quadrature technique for obtaining accurate approximate soluto the science of the Bi-Directional Reflectance Distribution tions. Based on the solutions. we present orbit requirements to Function(BRDF). The reference frames chosen for the anal-meet the science objectives. ysis aredescribed and the equation for the radiative forcing is presented. Subsequently, we present a method for estimat-, ing the albedo using a finite number of measurements. The What is BRDF?
technique is based on Gauss quadrature. From the study of me Bi-~kectional Reflectance ~i~h b~t i~~ ~~~~~i~~ (BRDF) the albedo integral. we determine the orbit dynamics require-characterizes the variation of reflected sunlight from a given menb to accomplish the science objectives. These include the surface. The BRDFofa surface varies with its smctural (smoothrelative geometry of the spacecraft in formation as well % the ness eu.1 and optical properties (transmission. reflection. abreference orbit characteristics. sorption, emission etc,). and the wavelength of radiated light. The BRDF also depends on the angle at which the light im-A significant portion of the paper is devoted to orbit design. pacts the surface (illumination garnew) and on the angle at Reference [2] presented a strategy to design relative orbits that which you look at the surface (viewing geometry).
intuare " J2 invariant" by requiring that the node rates and arguitive example of BRDF is provided by considering a recently ment of latitude rates for the reftXence and companion orbits mowed field of grass. The blades of grass tend to lie in the dibe equal. ~eference 131 developed an algorithm to design ring rectjon of the motion of the mower. The affect of the mowing formations anddetermined a correction technique to lessen the direction is easily observed with the human eye. ~h~ svipes degeneration of the geometry due to along-track secular effects in he field appear because the grass is leaning in the dkecof J2. Yet, little work is found in the literature on methods that tion in which it was mowed. iff^^^^^ amounts of light are mitigate perturbation effects through careful orbit design and reflected depending upon the direction of the blades of grass. achieve a specific formation geometry. Although h e f~rmation However, the complexity of BRDF is not completely portrayed geometry cannot be maintained in the presence of disturbing in this simple example. The effects of BRDF extend into the perturbations without an active control system, it is possible non-visible spectrum and depend on the position of the sun, atto design the orbits to reduce the amount of control effort re-mospheric conditions, and the type of surface. on large scales, quked. The primary perturbation causing formationdecom~o-BRDF has a significant effect on climate phenomenon because sition is J2. The effects of J2 can be broken down into specific it determines how much energy is radiated back into space and effects on the orbital motion. In this work, we identify which how much is retained in the atmosphere. ln this study, we are dfects cause a change in the formation geometry and which interested in the total net radiation flux from a given region. affects, assuming a careful orbit design, will not degrade the In other words, we are interested in the integral of^^^^ over relative motion. We present a suategy to diminish the effects the entire viewing geometry space for a particular sub,ect. we of J2 on orbit evolution while simultaneously satisfying con-will this integral the forcingw, although in the straints on the formation geometry imposed by the science rescience community, the term is used slightly differently. quirements.
For Leonardo-BRDF, we are interested ih the radiative forc-1n order to ascertain the fasibility of the approach* we Present ing of targets of interest such as, clouds, ocean surfaces, and a preliminary AV budget We assume that aH space-various types of land surfaces. Although considerable efforts craft will be launchedon a single vehicle. Therefore, the space-have been spent on radiative flux measurements, the errors are craft must be relocated from a common parking orbit into their too large for many climate applications. ~~~~ to wielicki appropriate final orbit. This is further complicated by timel a[.[53, the biggest instantaneous flux errors in past and curing constraints to insure that the desired relative positions are rent satellite measurements come from poor knowledge of the achieved. The AV'S are obtained by finding the optimal twoangular variation of Earth-leaving radiance. These angulat erimpubive-burn transfer orbit that meets the timing constraints rors range from 12.5 wlm2 for the ideal clouds and the Earth's to achieve the desired relative positions. A similar strategy, Radiant E~~~~ system (CERES) to 37.5 wfm2 for the Earth based On work by Chao et.al.E41, is used to determine the A v ~~d i~t i~~ Budget Experiment (ERBE), and account for over budget for fomation-kee~ing in the presence perturbations. 90% of the total error from all sources. These errors are an order of magnitude larger than typical climate forcings (e.g. 4

SCIENCE REQUIREMENTS
W/m2 for doubled C02). In order to improve climate model- In this section we introduce the science of BRDF and the ing with radiative forcing effects the albedo must be measured to within 0.5% accuracy.
Before introducing the physics of BRDF and radiative forcing. we define the reference frame for the BRDF analysis. The origin of the BRDF frame, FB. is a target located at the top of the atmosphere (TOA). An illustration of FB is shown in Fig. 1.
We assume TOA to be at an altitude of 30-km. The location of the target is defined as the nadir point of the reference spacecraft anQwill be discussed in more depth in a later section. We define the z-axis as the normal to the Earth's surface in the radial direction of the reference spacecraft. The principal plane is defined as the plane formed by the Sun, the point being viewed and the reference spacecraft. For convenience, we choose the x-axis to lie in the principal plane and the y-axis completes the right hand set. Similarly, the sun zenith angle, n is defined as the angle between the sun vector and the zB-axis. The view azimuth angle, 4. is defined as the angle between the projection of the line of sight in the xy-plane and the x-axis. The BRDF frame rotates according to the position of the Sun with respect to the target. Therefore, the azimuth angle evolution in the BRDF frame is highly non-intuitive, which will affect our future approach to the orbit design. The orbit design has been performed in an intermediate reference frame discussed later. We show in a later section that only the relative spacing of the spacecraft in 4 is important. Thus, we can examine. without loss of generality, the evolution of the azimuth angle in a frame that does not rotate about the zB-axis.
With the definitions of the coordinate systems and variables in hand, we can now investigate the mathematical model for the radiative forcing.
The radiative forcing is the integral of BRDF over the entire viewing geometry space of q5 and 8. Furthermore, the albedo is also dependent on the area of the pixel. A. and the sun zenith angle. n. To simplify the expression for the flux integral we define a new variable p, given by The exact expression for the instantaneous flux for a pixel of area A is expressed as where IA is the radiance from the TOA pixel in the direction of 8 and 4. In this expression n and X have been included in the flux expression as a reminder that the flux is dependent on the illuminating geometry and the wavelength of light being investigated. In subsequent expressions, n and X are left out of the expression without loss of generality. However, it must be remembered that although there is no integration over n and X a particular flux value is dependent on them. The flux value is assumed to be for a particular wavelength of light and a specific illuminating geometry.
To obtain the exact measure of the flux, measurements must be available from all possible viewing perspectives. In practice, such information is not available. Therefore, methods must be developed to accurately estimate the radiance integral in Eq.
(2) using a finite number of measurements. Generally, the more spacecraft in the array. the more accurate the approximation will be. Yet, we need to minimize the number of spacecraft for practical reasons. In order to estimate the flux integral, we separate the 8 and q5 dependency using common radiative transfer theory and expand the radiance IA in Fourier series: For 1-Dimensional radiative transfer, the sine terms vanish. Furthermore. when the right-hand side is integrated over q5 the only remaining term is I,$' Thc current formation design attempts to demonstrate the feasibility of selecting spacecraft orbits that follow a set of quadrature rules. The abscissae values depend on the choice of quadrature rul<chosen. In this work, we have investigated several types but continue the analysis without reference to a specific set of abscissae. Yet. it is important to discuss some trends seen in the solutions using the Gauss and Chebychev equal weight quadrature approaches. Most importantly, both techniques require abscissae values. pi, near 0 and near 1. From Eq.
(1) we see that this results in zenith angles near, but not equal to zero, and near, but not equal to a/2. Therefore, we must design the formation tb provide these extremes in viewing geometry. Generally. a formation will not be able to provide viewing geometries that exactly satisfy the quadrature analysis. The following analysis attempts to ascertain to what degree we can achieve a quadrature array formation.
With a basic understanding of BRDE we are now prepared to discuss the specific orbit requiremenrs in light of the science objectives and the formation geometry required to estimate the flux integral.

What are the Orbit Requirements?
Thcrctbre. data points on one side of the princ~pal planc can be reflcctcd to the othcr side. By using a Sun-synchronous reference orbit with a noon-equator crossing. the reference orbit is in the principal plane, which can greatly reduce the amount of post-processing required to perform There are further considerations for using a Sun-synchronous reference orbit. By definition. the orbit plane rotates about the Earth at the same rate as the Earth rotates about the Sun. Therefore, the reference plane maintains a constant geometry with respect to the Sun. The result is that only half of the orbit can view the sun-lit side of the Earth. This condition will be maintained for the entire mission lifetime. Therefore. we are free to design the orbits such that only the portion that experiences daylight provides a useful geometry. This will prove to yield some interesting results. On the other hand, for Sunsynchronous orbits there is a coupling between the sun position and latitude. A given latitude will always experience the same illuminating geometry. This limits the amount of understanding that can be gained on the albedo effect for different illuminating geometries. However, if the orbit is not chosen to be Sun-synchronous then the orbit planes will rotate with respect to the sun. Eventually the portion of the orbit that was on the dark side of the Earth will rotate into the Sun-lit side. Therefore, the entire orbit must be designed to provide a useful geometry. This will prove to be very difficult.
The nominal mission for Leonardo-BRDF assumes that the The orbital requirements originate from three main areas: the "target" or point of interest lies at the nadir of the reference quadrature analysis, BRDF symmetry properties and general spacecraft. Yet, it is likely that there will often be targets of formation flying dynamics constraints. me G~~~~ quadra-interest that are within the formation field of view that do not ture analysis requires specific viewing geometries, which place lie On the reference nadir.
we need to develop an constrain~s on the relative positions of the spacecraft in forma-approach to determine the ability of the formation to take usetion. symrneuy of the BRDF integral can be used to ful measurements of off-nadir targets. Yet, time constraints our advantage imposing further on the reference have not permitted a rigorous pursuit of this approach and it is orbit. Finally, there are general formation flying concerns that a topic future researchare addressed in Section 3.
Perhaps the most challenging constraints in the context of Leonardo are those resulting from the quadrature analysis. To accurately estimate the radiative transfer. measurements must be taken at specific positions with respect to the BRDF reference frame. This places stringent requirements on the relative positions of spacecraft in formation. First, to obtain an accurate average of the azimuthal dependence, the spacecraft must be spaced nearly equal in azimuth. The zenith angle requirement is even more stringent. Depending on the type of quadrature solution, the spacecraft must be permanently located at specific values of the view zenith angle.
By taking advantage of symmetry principles, we can obtain "extra" data points to increase the accuracy of the approximation. If the surface being investigated is assumed homogeneous, the BRDF is symmetric about the principal plane.
In summary, the orbit requirements originate from three main areas: the quadrature analysis, the BRDFsymmetry properties and general formation flying constraints. The quadrature approach requires specific viewing geometries, which place constraints on the relative positions of the spacecraft in formation. Further complications arise when off-nadir targets are of interest. Symmetry properties of the BRDF integral can be used to our advantage by imposing further constraints on the reference orbit. Choosing a Sun-synchronous orbit has numerous advantages. Finally, there are general formation flying concerns that are addressed in Section 3. In thc next section. we discuss some general formation flying dynamics concerns and develop the orbits that provide the necessary relative dynamics to perform BRDF measurements.
We begin this section by presenting some convenient coordinate syssms and some challenges of orbit design to minimize the adverse effects of perturbations. Next. we develop the formation dynamics. The approach allows the analyst to specify a specific formation geometry and the resulting formation satisfies dynamical constraints to reduce J2 perturbation effects. Finally, we discuss the physics of the allowable solutions and compare the non-linear method with a Hill'sequations approach.
Three coordinate systems are used in the development of the formation dynamics. An illustration of the frames is shown in Fig. 2. The first frame is similar to the ones used by Hill 173 and later by Clohessy and Wiltshire [8] in their original studies of relative motion. We denote the coordinate system FH. The origin of FH is located at the reference spacecraft. The XHaxis is in the radial direction, the zH-axis is in the orbit normal direction, and the ys-axis completes the right handed set. Note that the frame is not inertial because it rotates depending on the orbit of the reference spacecraft. The term "in-plane" refers to the projection of the relative motion in the xy-plane. Similarly, the terms "along-track" and "cross-track" are used to describe motion in the y~ and ZH directions respectively.
The second frame used in the analysis is the common Earthcentered inertial frame, FI. The origin of FI is the center of the Earth. The xr -axis point.. towards the vernal equinox and the at-axis extends through the North Pole. The yI-axis is chosen to complete the right-handed set.
The third and final coordinate system is denoted FR. The frame is fixed in inertial space with its origin at the center of the Earth. However, the X R -y~ plane is not the Earth's equatorial plane, it is the orbital plane of the reference circular orbit. The XR-axis is defined by the line of relative nodes of the reference and a particular companion orbit. The zR-axis is normal to the reference plane of motion and the YR-S~S completes the right-handed set. Defining TR in this manner allows some useful algebraic simplification that is demonstrated in a later section. Properties associated with the reference orbit are given with the subscript "r". The remaining orbits composing the formation are termed "companion" orbits and there associated properties are shown with the subscript "c" .
Many formidable complications are caused by orbit perturbations. The dominant perturbation, and the only one considered in this study. is J2. In general, J2 causes periodic changes in all of the orbital elements. However. only the right ascension of the ascending node, fl. the argument of periapsis. w , and the mean anomaly, M. experience secular drifts. To first order in J2, the secular drift rates are given by 3nRz J2

2P2 cos i
Foremost, we must design the formation to provide the necessary relative geometry to accomplish the science. However, the J2 perturbation will cause the formation geometry to degenerate rapidly. Although formation geometry cannot be maintained in the presence of disturbing perturbations without an active control system, it is possible to design the orbits to reduce the amount of control effort required. In the next few subsections, we explore some techniques that simultaneously provide a desired formation geometry and mitigate the effects of the secular J2 perturbations. For reasons to be explained later, we enforce the constraints The results are applicable to problems beyond the scope of Leonardo. Therefore, the solutions are presented in a general setting. Their implications for Leonardo are presented in a separate section.

Rotating Formation Design
The formation is developed by considering the projection of the actual motion onto the 3 j y -a~ plane. A circular reference orbit is assumed. We investigate both linear and non-linear techniques to develop the orbits. For the non-linear approach, we characterize the size of the formation by using the angular. as opposed to the spatial. separations of the spacecraft where the vertex of the angular separation is the center of the Earth. A technique is developed to produce an ellipse-like projection of the relative motion in the YH-ZH plane. An illustration of several different types of projections is shown in Fig. 3. We wish to obtain relative motion projections of Type 1. This can be achieved by applying the simple dynamical constraints: We refer to these constraints as the turn-around conditions.
We begin the investigation with some preliminary relative motion considerations. The dimensions of the relative motion are determined in terms of the eccentricity and relative inclination be solved numerically to higher order using the second order approximation as an initial guess. The section is concluded --with a discussion of the four solutions that satisfy all of the constrain& and a linear approach based on Hill's equations [7] is developed and compared to the non-linear approach.
The angles a, and a, are defined as the maximum angular separations in the YH and Z H directions, respectively. They are the angular equivalents to the amplitudes in Hill's equations. The vertex of the angles is the center of the Earth. The out of plane angular amplitude, a,, is entirely dependent on . -the relative inclination, Ai, of the two planes of motion. The  --_ _ _ -* relative inclination is defined to be the angle between the orbit normal's of the reference and a particular companion orbit. To satisfy the orbit constraints presented later, we must posi-We wish to determine the dimensions of the relative orbit in terms of the orbital elemenc of the companion orbit. The plane of motion of the reference orbit is assumed known and the goal is to calculate the elements of the companion orbit to meet the geometry and orbit constraints. The argument of latitude, u,, of the reference spacecraft in the reference plane of motion is not yet specified. By choosing u, carefully the formatieo design can be greatly simplified.
A view of the relative motion geometry is shown in Fig. 4. By definition, the reference spacecraft is always at the origin of FH and is defined by the vector r,. The vector r , describes the position of an arbitrary companion spacecraft. The quantity + is the along-track component of a n g u l~ separation of the spacecraft in 3 H . i.e., $ is defined as an angle and not a distance. From inspection of Fig. 4 we see that a, is solely dependent on the relative inclination, Ai, of the companion and reference orbits. The expression for a, in terms of Ai is simply For close formations the along-track relative motion is almost entirely dependent on the eccentricity of the companion orbit. The relative inclination of the planes of motion does have a small effect on the along-track relative motion. It was shown by Hughes and Hall [9] that to first order, the angular amplitude in the ye-direction, a,, can be expressed in terms of the eccentricity only as ail = 2ec (12) Chichka [lo] demonstrated inkpendently that Eq. (12) is accurate to second order in e, and that the effects of Ai on a, are smaller than O(Ai 2 ) .
Although Eqs. (1 1-12) determine the dimensions of the relative motion. the shape is still undefined. The turn-around conditions, seen in Eq.(lO), are a convenient way to ensure that Type 1 motion is achieved. Type 2 motion warrants further study to determine if it can be of use to Leonardo. However, due to time constrain& it has not yet been investigated.
The turn-around conditions impose constraints on the m e anomaly, u,, and the argument of periapsis, w,, of the companion orbit. Before proceeding to develop the equations for u, and w,, we define some variables. Refer to Fig. 4 for an illustration. The argument of latitude of the companion spacecraft in FR is termed u,,. The argument of latitude of the reference spacecraft is denoted M, because the reference orbit is assumed to be circular. The along-track angular displacement of the companion with respect to the reference is denoted + and the angle between the planes of motion is denoted Ai.
and substituting into Eq.(13) yields The turn-around conditions of Eq. (10) are expressed in terms of the spatial separations. In terms of the angular separations they are simply Taking the derivative of + with respect to time and making use of that fact that u, = w,, + u, we obtain Now, applying the turn-around conditions to ensure that the desired shape is obtained, we obtain the set of simultaneous equations w,, + uc = k7r k = 0,1,2 ...
To ensure that the desired motion is obtained in the presence of J2, we must obtain aperiodic solution to Eqs. (18-19) while simultaneously satisfying the relation The reasoning for choosing these orbit constraints is explained momentarily.
To explore the solutions to these equations, first consider their functional dependence without considering the actual form.
Recall that we are neglecting the periodic variations in w,,. O,, and u, and only considering the secular drifts to first order in J 2 . The values of kc, d,. and A& are dependent on a,, e,. and i,. However, from Eq. (12). to achieve the desired alongtrack dimension. the eccentricity is fixed. Therefore, only a,. and i , can be varied. Furthermore. to obtain a desired across track dimension we must have a specific Ai. Fortunately, be- , i,, AR), we can vary i , and still maintain the required Ai by choosing the appropriate AR. We are in- terested? finding a,, i,, w,,, and u, where the subscript "t" is used denote the value satisfying the turn-aroundconditions. Substituting Eq. (29) into Eq. (28) and From Appendix 1 we know Aa and Ai to satisfy the mean solving for cos Mt we obtain anomaly and node rate constraints simultaneously.
-2 It J50ez + 20/ cos Ai -16 Note that Ai, is the difference in inclination between the reference and companion orbits and is not to be confused with hi. the angle between the reference and companion orbit normal vectors.
From inspection of Fig. 5

52, = R , + A R (38)
where cos a,cos i, cos i, The two solutions for AR provide two unique relative motion solutions. However, for the remainder of this work the firstquadrant solution is assumed. Although the approach is the same regardless of which soiution is used, some of the following expressions are affected by the choice.
From inspection of Fig. 5 we can write where 4, is the arclength, in the companion plane of motion, between the Earth's equator and the line of relative nodes: where C is given by C = (a, +i, -ic+a)/2 (42) and the first or second quadrant solution for 4, must be used in Eq. (40).

Figure 5: Geometry of the Planes of Motion
This completes the solution of the companion orbital elements. However. to simplify the analysis we have assumed that the companion spacecraft is at the point specified by the turnaround conditions. To complete the analysis, we must ensure that the reference spacecraft is at the appropriate point in its orbit to obtain the desired relative motion. From inspection of In this section we developed an algorithm to determine the state of a companion orbit, given the plane of motion of the reference circular orbit, to satisfy geometry constraints given by a, and a, and the relations d, = 6, and & = A&. To do so we chose a point in the reference plane that greatly simplifies the algebra. The point in the reference plane will change, although only slightly, with the dimensions of the relative orbit. .Therefore, if relative orbits of differing dimensions are desired, care must be taken to propagate the states to a common point in the reference orbit to obtain the correct states for the formation.

Physics of the Relative Morion Solz$tions
Recall that for a given plane of motion for the companion orbit, there are two solutions for the true anomaly at turn around ut. Also, to satisfy the orbit constraints, for a given reference orbit, there are only two allowable planes of motion for the companion orbit. Therefore, at a given instant there are only four solutions that provide the desired relative geometry while simultanwusly satisfying the mean anomaly and node rate constraints.
The four solutions are defined using Table 1. A three dimensional plot has been included of the relative motion paths of all four solutions for one orbit period (Gig. 6). The plot was created using a, = lo and a, = lo . Note that there are only two distinct trajectories and two of the solutions share one trajectory while the other two share the second trajectory.

Hill's Equations Approach
To get a more intuitive understanding of the problem, we investigate another method based on Hill's equations as a first guess of the solution. We show that the results are in good agreement with the non-linear approach developed above, even for large formations.
A,, and A, characterize the amplitude or size of the relative motion. A and B define the "center" of the relative motion in FH. For this study, A and B are chosen to be zero so that the relative motion is centered on the reference.
For small amplitudes of the relative motion, the following relations between the HCW' s amplitudes and the Keplerian orbital elements can be established 1131: Knowing the desired amplitudes and the reference inclination.
we can determine AR and e, from Eqs. (51-52). Next. we need to relate the phases 4,, and 4, to the node rate constraint and the turn-around conditions. The z-phase controls the initial out-of-plane distance. For nearly polar orbits, 1 ARI is mainly determined by A,. Thus, the z-phase primarily determines the sign of AR and the inclination.    The xy-phase primarily determines the shape of the projection of the relative motion in the yz-plane by changing the location at which ~( t ) = 0 occurs as shown in Fig. 8. In terms of Keplerian elements, we observe that for nearly polar orbits, a change in xy-phase induces a change in argument of perigee and true anomaly. Therefore, we can infer that a change in argument of perigee will lead to a change of shape.
We wish to achieve a Type 1 relative motion described in Fig.  3. To obtain the desired relative motion geometry, we must ensure that the turn around conditions in Eq. (10) are satisfied. Hence, we must simultaneously satisfy the equations $(t) = -2 . n -A,, cos(nt, + 4,,) = 0 (53) From Eqs. (53-54), the xy-phase and the z-phase can be related by #,,=n.n++, n E Z (55) To find the proper phasing, we perform a parametric scan of the z-phase so that a Sun-synchronous node rate is achieved and use Eq. (55) as a first-guess for the xy-phase. For large amplitudes, we observe a non-centered z-motion not predicted by the HCW model. Consequently, the value of +,, obtained from Eq. 55 . is refined to obtain a relative motion as close as possible to a Type 1 solution. To conclude, the HCW approach provides results in agreement with the non-linear method. The HCW equations are expressed directly in terms of the relative motion geometry. They offer an insight into the relationship between the Keplerian elements and the relative motion parameters. Although the HCW equations provide a good first-guess of the solutions,the linear approach is used here as a validation tool rather than a rigorous and reliable method. However, it works surprisingly well for large formations.
In this section we examine the performance the of the formation in light of the science requirements. The allowable formation solutions determined in section 3 are used to find specific orbits that simultaneously satisfy the science and orbit dynamics constraints. We begin by determining a reference orbit for the formation. Then, the formation geometry that best satisfies the science requirements is determined. Interestingly, if the aspect ratio of the relative motion a,/az or A,,/A, is carefully chosen. then constant view zenith angles can be maintained over the regions of interest. Finally, we investigate the strengths and weaknesses of the formation in light of the BRDF science requirements.

Orbit Design to Meer Science Requirements
To satisfy the science requirements. the reference orbit must be Sun-synchronous with a noon equator-crossing. The orbit dcsign approach assumes a circular reference orbit. In order to have a noon equator crossing. an epoch must be chosen.  Recall from Eq. (32). that the along-track dimensions of the relative motion are proportional to the eccentricity. We can write the altitude variation, Ah, of a particular orbit in formation as Ah=ra -r p = aa, where r, and r, are the radii of apoapsis and periapsis respectively. From Eq. (56). we see that any formation with non-zero along-track motion will experience a variation in altitude over the course of an orbit. Moreover, the larger the relative orbit, the larger the altitude variation. Fig. 9 illustrates the altitude variation and its affect on viewing geometry. Due to the altitude variation, the view zenith angle is not constant for a circular formation defined by a, = a,. This is because the angular dimensions a, and a, are defined with the vertex at the center of the Earth. The vertex for the view zenith angle is an object at TOA. However, it is possible to adjust the aspect ratio a,/a, to obtain a nearly constant q5 for a portion of the orbit. The approach taken here is to design the formation so that + , , , x $mid. This is done using a simple trial and error approach. Further study is required to determine a functional relationship between a, and a, to obtain the &sired aspect ratio.
The altitude variation described by Eq. (56) causes some other difficulties in the orbit design. To obtain a set of view zenith angles, we must have several spacecraft in concentric relative orbits of differing dimensions. Since a, is different for the different relative orbits, they have different altitude variations and hence experience significantly different drag forces. For small formations, the differential drag is not as significant as the differential affects of Jz. However, to achieve large view zenith angles, on the order of 500, the altitude variation is on the order of 1300 km for a formation with a = 7600 km! Hence, the reference semi-major axis must be chosen so that the formation is either entirely above the existing atmosphere or only in the outermost region. Otherwise, there will be a significant relative drift due to the differing drag forces experienced by different spacecraft. For the preliminary analysis, we require that all spacecraft remain above 500 km. The overall design is a complicated trade-off between the desired view zenith angles. the required formation aspect ratio, and the semimajor axis. The choice of semimajor axis is driven by the largest relative orbit. To achieve a view zenith angle of 460 and have a periapsis above 500 km, the semimajor axis should be at least 7600 km. With the reference orbit completely defined, the companion orbits can be determined to achieve the desired view zenith angles. Tables 4 and 5 summarize respectively the orbit properties for the non-linear and linear methods. The orbits are labeled "Leol", "LeoT. and "Leo3' and are designed to achieve zenith angles of 26'. 37". and 46' respectively.
A plot of the relative motion is shown in Fig. 10. The solid black lines represent the actual relative orbits of the companion spacecraft. The dashed lines represent the projection of the motion into the local horizontal plane. The enormous size of the relative orbits is apparent by looking at the scale of the axes. The largest relative orbit has a diameter of over 3000

km!
Results-Recall that we are not concerned with the formation geometry on the dark side of the Earth. Because a Sunsynchronous orbit maintains a constant orientation with respect to the Sun. the formation can be designed to provide useful geometry only on the Sun-lit side. Furthermore, we are only interested in latitudes within f 60' . In the following plots, the viewing angle is set to zero if the formation is outside of f 600 latitude. The time has been normalized on the orbit period.
The view zenith angle, 8. is defined as the angle between the line of sight and the local vertical of the target. Recall that according to the quadrature approach. we need to maintain specific values of the view zenith angle. Although we cannot maintain a view zenith angle over an entire orbit. it is possible to maintain the angle within the latitude band of &6O0 as seen in Fig. 11. This is accomplished by adjusting the aspect ratio of the relative motion projection. Although there are four relative motion solutions that provide a specific value of 8 and satisfy the orbit dynamics constraints, only two of the solutions can be used to obtain the desired 8 on the Sun-lit portion of the Earth. The difference in the two configurations is that they are 180" out of phase in the view azimuth angle. A significant problem with the formation is that initializing and maintaining high view zenith angles results in prohibitively large fuel ex- The view azimuth angle, 4, is defined as the angle between 9,-axis and the projection of the line of sight into the local horizontal plane. Recall that according to the quadrature approach the spacecraft must be spaced somewhat evenly in 4.
This is because only the average dependence of the albedo on view azimuth is used to calculate the flux integral. The azimuth angle evolution for the orbits is illustrated in Fig. 12. The azimuth angle rotates through 3600 over one orbit. However, as discussed earlier, for a relative orbit of given dimensions there are only four unique relative motion solutions that satisfy the geometry and node rate constraints. Because we are only interested in the Sun-lit portion of the Earth, we can only use one pair of the solutions. The spacecraft within each pair are separated by approximately 180 degrees in 4. Therefore, it is only possible to have two 4 values that are equally spaced for a ring of given dimensions.
In the next section we investigate strategies to initialize and maintain the formation. The approach is a simple impulsivebum method intended to give first estimates of the AV's.
The total AV is a significant measure of the feasibility of the mission. The spacecraft are envisioned to be small, light. We assume all spacecraft will be launched on the same vehicle into a common orbit. First, a common launch orbit is chosen so that the initialization maneuvers are distributed among the spacecraft within the array. Then, we develop an optimal Lambert scheme to find the minimum two-impulsive-bum initialization AV. Once the array is initialized, frequent, small maneuvers are necessary to maintain the desired formation geometry. Perturbations such as the Earth's oblateness (mainly 52) and atmospheric drag induce secular drifts between the spacecraft which, if not corrected, will cause formation separation. Consequently, we develop maintenance strategies based on work by Chao 141.

Formation Initialization
In this section we investigate a general two-impulse orbit initialization maneuver strategy based on Lambert's problem. For single-satellite missions, the main concern is to achieve the final desired orbit. For multiple-satellite missions the spacecraft must reach the desired final position at the correct time so that the desired array configuration is achieved. In general, a simple satellite rendezvous sequence depends on three parameters: the initial orbit state at the first maneuver, the time of flight. and the target state. For a formation of spacecraft. the final state is completely determined by the initial orbit state and the time of flight. Hence, we lose a degree of freedom and can expect to have larger AV's for initializing formation flying missions. To minimize the fuel consumption, we consider the orbit initialization sequence of each spacecraft independently.
Consequently. the transfer from the parking orbit to the final orbit will occur at different times for different spacecraft. The formation is not considered "initialized" until the last space-craft joins the array.
An illustration of the orbit geometry of the initial and final states is shown in Fig. 13. In general, the orbits are not coplanar but they have been drawn as coplanar orbits for simplicity.

A l l spacecraft are assumed to be launched into the Leo2 orbit.
This allows a sharing of the AV between the spacecraft Leo1 and Leo3. Leo2 will require no maneuver to achieve the de-  Table 6 .
The maneuvers to initialize the formation are prohibitively high for the scenarios chosen. The estimates may be lowered by investigating alternative parking orbits and using more than two impulses. However, it is probable that rotating formations providing view zenith angles of more than 25" are infeasible within the constraints of Leonardo.

Formarion Mainrenance
In general, perturbing forces from atmospheric drag, non-spher- Figure 13: Initialization Maneuver Approach The science requirements for Leonardo favor a Sun-synchronous ics of the problem presented in a previous section suggest that Leo3 536.67 considerable fuel savings may be afforded by using orbits at the critical inclination. For critically inclined orbits the secuical affects, solar radiation pressure and third body effects will lar argument of periapsis drift is zero. The impact of the new cause dramatic changes in the formation geometry over short formation geometry on science has not been investigated. The time scales. Control strategies to maintain the formation in the dynamics and control of a critically inclined formation is prepresence of these perturbations can take on many forms. In sented in an attempt to explore more realistic possibilities for this work we investigate a simple impulsive burn strategy to formations on the scale of Leonardo. obtain some estimates. More sophisticated-itate feedback techniques and primer vector theory are topics of fu-The states for the critical formation are presented in Table 8. ture research and are expected to reduce the AV required to The values for a, and a, to achieve the desired viewing angles maintain the formation.
were determined by trial and error.
In section 3, we investigate a formation design approach that suggests if the periapsis of the companion orbit drifts from its initial value. the shape of the relative orbit will change. The affect of the mean anomaly drift of the companion orbit is not as problematic because the reference mean anomaly drifts as well. We are only concerned with the relative mean anomaly drift. Therefore, the maneuver strategy focuses on maintaining the argument of periapsis. Chao et al. 141 developed a method to find the optimal locations of two symmetrical tangential bums to change the argument of periapsis without changing the other orbital elements. The true anomaly to provide an optimal maneuver depends on the desired change in argument of periapsis, and the eccentricity: These corrections arc applied at the initial mean anomaly and the reference is set back to its initial state. This technique only provides a rough estimation of the necessary maintenance AV. 13.8 Table 7 shows that the maintenance AV budget for the formation is highly expensive. Again. the technique used is not optimal and other techniques. such a. Primer Vector Theory and Linear Quadratic Regulator, are yet to be investigated. but are expected to improve the maintenance budget. As expected the 4 V to initialize the critically inclined formation are still considerably high. The values are shown in Table 9. The same initialization strategy is used for the Sunsynchronous formation is used here. The spacecraft are as-Maintenance maneuvers are performed daily to correct for the sumed to be launched into the Leo2 orbit form a common argument of perigee. semi-major axis and eccentricity drift. launch vchiclc. Lcol and Leo3 are maneuvered into their appropriate orbits from the Leo2 orbit. For this approach no fuel expense is required to initialize Leo2. As explained earlier, the argument of perigee drift is the main driver of the maintenance AV cost. Using a critically inclined reference orbit improves the maintenance AV considerably. Table 8 presents the formation orbit properties for one possible solution. The improvement is on the order of 70 times less fuel to maintain the Leol orbit and 30 times less fuel to maintain the Leo3 orbit. Critically inclined formations suffer from one difficulty that is avoided by using Sun-synchronous formations. A Sun-synchronc orbit, by definition, rotates at the same rate as the Earth rotates about the Sun. Hence a Sun-synchronous formation will maintain a constant geometry with respect to the Sun. However, a critically inclined formation will change its orientation with respect to the Sun. For Leonardo we wish to make measurements of the reflected Sunlight and these measurements can only be made on the side of the orbit that can provide the appropriate geometry. Therefore two spacecraft will be needed to ensure a specific viewing geometry is maintained over the life of the mission. However. there are some scientific advantages in the critically inclined formation. The Sun position will change with respect to the reference plane and a much more general understanding of the effect of illumination geometry can be obtained.
A preliminary flight dynamics analysis was presented for Leonardo-BRDF. The orbit dynamics constraints, driven by the science requirements, are extremely difficult to achieve in a fuel efficient manner. To accurately estimate the radiative forcing, the formation must provide specific. constant view zenith angles and an nearly equal separation in view azimuth. Furthermore, the reference orbit must be Sun-synchronous and noon equator-crossing. To achieve these requirements we developed a design approach using orbital elements that simultaneously satisfies geometry and orbital constraints. We showed that it is possible to maintain specific view zenith angles for at least half of an orbit. However, there are only two solutions among four possible solutions that provide the desired view zenith angles on the Sun-lit portion of the Earth. The azimuth angle constraint is not satisfied. Indeed. the azimuth angle history is fully determined by the view zenith angle constraints and the reference orbit. We foresee that a combination of the rotating formation and alternative formation geometries c&ld provide better radiative forcing approximations. The AV's necessary to establish and maintain the formation were estimated using some preliminary impulsive maneuver strategies. The fuel expenditure, especially for maintenance. was found to be prohibitively high for a Sun-synchronous formation. Consequently, we investigated a formation designed using a critically-inclined reference orbit. The new maintenance AV'S were on the order of 70 times lower for certain orbits in the formation. Future designs will focus on lowering both the initialization and maintenance AVs by carefully choosing the launch orbit and reference orbit. In addition to improvements in the orbit design, the maneuver strategies could be improved by using optimal control approaches.
are equal for the reference and companion orbits in the presence of first order J2 effects. Note that Ai, is the differnce in inclination of the reference and companion orbits and not to be confused with Ai, the angle between the orbit normal vectors of the reference and companion planes of motion. In general f (a, e, i). Recall that we assume a circular reference orbit. Also, for a given formation dimension the eccentricity of the companion orbit is defined. to second order, by equation Eq. (12). So the following is a system of two second order equations in two unknowns. The exact expressions for M and ah, da -2 O(JZ) -, Therefore in Eq. (58) we neglect the terms involving derivatives with respect to a because they are higher order terms. The resulting expression is There are two distinct advantages to this simplification. Fmt the algebraic form of the solutions will likely be much simpler. Secondly, we have decoupled the node rate equation from the mean anomaly rate equation and the system of second order equations can now be solved in sequence rather than simultaneously. Because we have assumed a circular reference and e, Again the physically meaningful solution of the quadratic equation is used.
Determining the range of validity of these approximate solutions is non-trivial and will depend on the size of the relative orbit and properties of the reference orbit. However, the approximations are excellent for the formations investigated here which are on the order of 3000 krn in diameter.

Appendix 2
Appendix 2 presents a summary of the four relative motion solutions equations discussed in section 3. The solutions are generated for a given reference plane of motion and the desired angulardimensions, a, and a,, of the relative orbit. The quantities, a,, i,, and R, of the circular reference orbit are assumed known. However, the position of the reference spacecraft in the reference orbit is chosen to simplify the algebra. Therefore u,, the argument of latitude of the reference orbit, is determined below. Care must be taken to propagate the the states of all spacecraft in formation to a common point in the reference orbit.