Definition of Low Earth Orbit slotting architectures using ...

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Advances in Space Research xxx (2020) xxx–xxx

Definition of Low Earth Orbit slotting architectures using 2Dlattice flower constellations

David Arnas a,⇑, Miles Lifson a, Richard Linares a, Martın E. Avendano b

aMassachusetts Institute of Technology, Cambridge, MA 02139, USAbCUD-AGM (Zaragoza), Crtra Huesca s/n, Zaragoza 50090, Spain

Received 27 January 2020; received in revised form 7 April 2020; accepted 11 April 2020

Abstract

This work proposes the use of 2D Lattice Flower Constellations (2D-LFCs) to facilitate the design of a Low Earth Orbit (LEO) slot-ting system to avoid collisions between compliant satellites and to optimize the available orbital volume. Specifically, this manuscriptproposes the use of concentric orbital shells of admissible ‘‘slots” with stacked intersecting orbits that preserve a minimum separationdistance between satellites at all times. The problem is formulated in mathematical terms and three approaches are explored: randomconstellations, single 2D-LFCs, and unions of 2D-LFCs. Each approach is evaluated in terms of several metrics including capacity, Earthcoverage, orbits per shell, and symmetries. Additionally, a rough estimate for the capacity of LEO is generated, subject to certain min-imum separation and station-keeping assumptions, and several trade-offs are identified to guide policy-makers interested in the adoptionof a LEO slotting scheme for space traffic management.� 2020 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: Space mechanics; Satellite constellations; Space traffic management; Orbit design; Orbital slotting

1. Introduction

SpaceX, OneWeb, Telesat, and other companies are cur-rently planning mega-constellations for space-based inter-net connectivity and other applications that wouldsignificantly increase the number of on-orbit active satel-lites across a variety of altitudes (SpaceX, 2019; OneWeb,2019; Telesat, 2019; Portillo et al., 2019; Alary et al.,2018). As these and other actors seek to make more inten-sive use of the global space commons, there is growing needto characterize the fundamental limits to the capacity ofLow Earth Orbit (LEO), particularly in high-demand

https://doi.org/10.1016/j.asr.2020.04.021

0273-1177/� 2020 COSPAR. Published by Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: [email protected] (D. Arnas), [email protected] (M.

Lifson), [email protected] (R. Linares), [email protected] (M.E.Avendano).

Please cite this article as: D. Arnas, M. Lifson, R. Linares et al., Definitionstellations, Advances in Space Research, https://doi.org/10.1016/j.asr.202

orbits and altitudes, and minimize the extent to whichuse by one space actor hampers use by other actors.

This manuscript contributes to this goal by proposing aslotting mechanism for LEO, based on the Flower Constel-lation (FC) Theory (Mortari et al., 2004; Mortari andWilkins, 2008; Wilkins and Mortari, 2008). In particular,the 2D Lattice Flower Constellation (2D-LFC)(Avendano et al., 2013) formulation is used to define setsof admissible satellite locations or ‘‘slots” while ensuringa minimum separation distance between all slots at alltimes. This slotting system would greatly reduce the riskof satellite vs. satellite collisions (eliminating it for conjunc-tions involving compliant satellites), decrease the analysisand coordination burden associated with such conjunc-tions, and potentially allow for the safe co-location ofspacecraft controlled by different operators at the samealtitudes as mega-constellations. Additionally, this pro-posed solution is compared with other approaches in order

of Low Earth Orbit slotting architectures using 2D lattice flower con-0.04.021


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2 D. Arnas et al. / Advances in Space Research xxx (2020) xxx–xxx

to determine the trade-offs of different designmethodologies.

Access to Earth orbit belongs to a class of goods knownas common pool resources (CPRs) (Weeden and Chow,2012). These resources are characterized by the ability ofmultiple actors to make use of them and the fact thatone actor’s use degrades that resource for others. Terres-trial examples of CPRs include public roads, aquifers,and fisheries (Gardner et al., 1990). In the case of the spaceenvironment, costs of CPR use include increased risk ofphysical collision and spectral interference, as well as limi-tations on use of particular physical regions and spectrumbands already in use by others. A CPR dilemma can be dis-tinguished from a CPR situation by the presence of twoadditional conditions: (1) less-than optimal outcomes asmeasured by those using the resource and (2) the existenceof at least one potential strategy that is both feasible andpareto-improving (Gardner et al., 1990, 6–7). ManagingCPR dilemmas to ensure sustainability is a classic publicpolicy problem.

Assignment systems (including slotting mechanisms) area standard CPR management strategy. An assignment sys-tem enforces coordination on those using the CPR throughtop-down allocation of the right to use the resource subjectto certain conditions, with the objective of preservingusability for all.1 Within the space domain, such systemsexist and are widely used for the allocation of electromag-netic spectrum for communications and remote sensing atboth national and international levels. Most notably, theInternational Telecommunications Union operates a sys-tem to allocate spectrum for geostationary orbital slots toprevent harmful signal interference (Jakhu, 2017).

The International Academy of Astronautics’ 2018report on Space Traffic Management (STM) describespathways towards the development of both a ‘‘technicalSTM system” and a ‘‘regulatory STM regime” andsketches out the elements necessary for either a top-downor bottom-up approach to STM (Alary et al., 2018). ALEO slotting mechanism is not mentioned explicitly, butcould be implemented under either approach. With abottom-up strategy, a slotting system could see admissibleslot locations defined on a voluntary consensus-basis, andactors build norms for satellite deployment into slots andstation-keeping adequate to stay within the relevantcontrol-volumes. With a top-down approach, slots mightbe allocated by an international intergovernmentalorganization.

Because LEO is so much larger than GeostationaryEarth Orbit (GEO), there has historically been less concernabout orbital access in LEO (although debris continues tobe a major concern and particular regions of LEO, includ-ing Sun-Synchronous Orbits (SSOs), are increasingly con-gested). Nevertheless, the emergence of multiple proposed

1 Other systems are also possible. For instance, coordination can evolveiteratively and organically by cooperation and voluntary modifications touse strategies by individual CPR users (Gardner et al., 1990, 9–10).


mega-constellations, each occupying a significant altitudeband on a potentially exclusive basis, has sparked discus-sions about apportionment of orbital volume in LEO ina manner that preserves access for emerging actors, bothstates and commercial entrants. In 2019, representativesof OneWeb, Iridium, and DigitalGlobe, a Maxar Tech-nologies company, authored an op-ed for SpaceNews that,among other recommendations, called for avoiding over-lapping altitudes for large constellations. The EuropeanSpace Agency has conducted work to develop a spacecapacity metric to understand the impact of a particularmission on the overall LEO environment (Letizia et al.,2019).

The LEO slotting problem is considerably more com-plex than GEO due to the need to accommodate multiplealtitudes, eccentricities, and overlapping orbits. The limitedwork to date on LEO slotting has primarily focused onSSO. In 2007, the International Space University SummerSession Program produced a set of technical traffic andenvironmental rules that included a proposal for zoningof SSOs (Anilkumar et al., 2007, 22–27) that was furtherdeveloped in Weeden and Shortt (2008). The authors men-tion the possibility of using specially-defined timing toallow for overlapping orbits, but do not present a technicalsystem to determine such slotting. Later, Bilimoria andKrieger (2011) presented a slot architecture for SSO,including a phasing rule to preserve minimum separationat near-polar crossing points and a proposal for slot sizingand dead-band control to preserve satellite separation inthe presence of various perturbations. In his Master’s the-sis, Watson (2012) conducted analysis to characterize therelative motions of sun-synchronous satellites about agiven slot location and developed an architecture and toolto generate SSO slots and control-volume geometry basedon user-selected values for parameters for slot sizingincluding semi-major axis, time between station-keepingmaneuvers, altitude loss rate, and relative motion oscilla-tion amplitude. In a Master’s thesis, Noyes (2013) builton Watson’s work with further relative motion analysisto detect and analyse satellite periods of conformance fordifferent altitudes and ballistic coefficients, as well as char-acterize the impact of an SSO slotting architecture on col-lisions and debris generation using an evolutionary debrismodel.

This work builds on these efforts to present a generalizedapproach to designing orbits that preserve minimum sepa-ration between participating satellites at all times, expand-ing consideration beyond SSO. However, the nature of theapproach requires certain decisions regarding how to allo-cate orbits, managing concerns like orbital density, cover-age, and the prioritization of certain orbital inclinations(Arnas et al., 2020). For the purpose of this paper, we focuson the technical design of a potential slotting system andleave questions such as allocation, orbital configuration,and other value trade-offs to subsequent work.

This manuscript makes four main contributions. First, itprovides a mathematical formulation of the LEO slotting



D. Arnas et al. / Advances in Space Research xxx (2020) xxx–xxx 3

problem. Second, it describes three approaches to LEO slotgeneration and the resultant maximum capacity per shell.Third, it defines four metrics for evaluating potential slotconfigurations. Fourth, it demonstrates how the numberof admissible slots per shell can be used as a metric forLEO capacity, both to characterize overall capacity andto help understand the trade-offs between different deci-sions regarding spacecraft placement.

This manuscript is organized as follows. Section 2includes a summary of the FC Theory and previous resultsapplying FC theory to this problem. In Section 3, we pro-pose what we believe is a precise, but approach-agnosticdefinition for the problem of generating sets of LEO slotsfree from self-conjunctions. We then narrow the problemto a particular altitude shell and describe what constitutesa valid solution to this sub-problem. Additionally, we pro-vide an analytical tool to determine the minimum distancebetween satellites when dealing with constellations definedat the same altitude and composed of only circular orbits.In Section 4, several versions of potential orbital slottingsystems are considered including random constellations,single 2D-LFCs, and several forms of unions 2D-LFCs.These are experimentally evaluated using computer simula-tion against a set of four metrics: capacity, Earth coverage,orbits per shell, and symmetries. Section 5 describes severalimplications for a policy-maker or system architect taskedwith selecting slot generation strategies across various orbi-tal shells. Subject to a set of assumptions for minimum sep-aration distance and other factors, we estimate a capacityof admissible slots in LEO under a notional designstrategy.

2. Preliminaries

2.1. Flower constellation theory

The Flower Constellation Theory is a set of constella-tion design models that focuses on the generation of uni-form distributions of satellites in the space. FlowerConstellation theory has been demonstrated in the litera-ture to design constellations for a wide variety of applica-tions including Earth observation, Earth observation withintersatellite links, telecommunication, telemedicine, globalnavigation satellite service design, and Lunar and Martianconstellations (Marzano et al., 2009; Lee and Mortari,2017; Mortari et al., 2011; De Sanctis et al., 2008; Parket al., 2005; Mortari et al., 2013; McManus and Schaub,2016; De Sanctis et al., 2007). The theory started with adesign model to distribute satellites along a relative trajec-tory that is defined in an arbitrary rotating frame of refer-ence (Mortari et al., 2004). This formulation allowed thegeneration of completely uniform distributions that pre-sented a series of patterns that were repeated over theconstellation.

Later, in order to simplify the formulation and extendthe design possibilities, the 2-D Lattice Flower Constella-tions (2D-LFC) (Avendano et al., 2013) were introduced.


The idea behind this methodology is to generate completelyuniform distributions of satellites using, as distributionvariables, the right ascension of the ascending node X,and the mean anomaly M, while the semi-major axis ofthe orbits a, the inclination i, the eccentricity e, and theargument of perigee x are fixed for all the satellites in theconfiguration. Accordingly, it is possible to define the con-stellation configuration using just three independent integerparameters: the number of orbital planes, No, the numberof satellites per orbit, Nso, and the configuration number,Nc (phasing parameter). In particular, the relative distribu-tion of each satellite of the constellation (Xij and Mij) isprovided by the following equation:

No 0

Nc Nso

� �Xij

Mij

� �¼ 2p

i� 1

j� 1

� �ð1Þ

where i ¼ 1; � � � ;No; j ¼ 1; � � � ;Nso names the j-th satelliteon the i-th orbital plane of the constellation. In order toavoid duplicates in the formulation, the configurationnumber is defined in the range Nc 2 ½0;No � 1�.

All satellite distributions generated using this formula-tion have the property of perfect uniformity, that is, the rel-ative distribution of the constellation is fixed no matterwhich satellite is selected as a reference. This also meansthat the number of symmetries in the configuration is max-imized. This property can be used, for instance, to reducethe computational effort to assess the minimum distancebetween satellite pairs. In particular, it is sufficient to eval-uate the minimum distance between the first satellite½X11;M11� of the configuration, and all the other satellites.This implies ðNoNso � 1Þ total checks instead ofNoNsoðNoNso � 1Þ=2 check of a general distribution. Thisgreatly simplifies the optimization process to find configu-rations that avoid satellite conjunctions.

Lattice Flower Constellations include, as a subset, themost widespread satellite constellation designs, includingWalker Delta Pattern Constellations (Walker, 1984),Streets of Coverage (Luders, 1961), Draim Constellations(Draim, 1987), and Dufour Constellations (Dufour,2003). In fact, the 2D-LFC methodology is able to generateall the possible uniform distributions performed in the rightascension of the ascending node and the mean anomaly(see Theorems 4, 5, 6 and 7 in Arnas (2018)). This providesa powerful tool to study all possible uniform distributionsof satellites with a unified theory. Therefore, we use 2D-LFCs as a basis to propose solutions for LEO spacecraftslotting.

2.2. Non-self-intersecting 2-D lattice flower constellations

A first approach to design mega-constellations using2D-LFC was introduced in Lee et al. (2015). The idea pro-posed in that work was to generate constellations made ofcircular orbits belonging to a unique repeating relative tra-jectory that presents no self-intersections. This relative tra-jectory was defined in a fictitious frame of reference that




was rotating at a constant speed about the axis of rotationof the Earth. That way, a relation can be defined betweenthis rotating frame and the inertial frame of reference:

NdT d ¼ NpT p; ð2Þwhere T d is the rotating period about the axis of rotationof the Earth of a fictitious rotating frame of reference, T p

is the revolution period of the satellite in the inertialframe, and Nd and Np are the minimum number of com-plete revolutions of the fictitious rotating frame and theorbits respectively until the dynamic is repeated in boththe inertial and the rotating frames of reference. Then,under this framework, it is guaranteed that no satelliteconjunctions will occur since all satellites follow the samerelative non self-intersecting path at any instant. Thisfacilitates a focus on maximizing the minimum distancebetween satellites.

In general, the 2D-LFC design formulation generatesconstellations that are distributed in different repeating rel-ative trajectories. However, it is possible to use this formu-lation to generate constellations belonging to the samerelative trajectory by imposing the following condition:

No 0

Nc Nso

Np Nd

264

375 Xij

Mij

� �¼

0

0

0

8><>:

9>=>; mod2p ð3Þ

Moreover, in order to assure the non self-intersection ofthe relative trajectory, two additional constraints arerequired. In particular, these constraints are provided bythe following Theorem (Lee et al., 2015) for progradeorbits: ‘‘The relative trajectory of a satellite has no self-intersections if and only if the inclination i < p=2 and:

if Nd ¼ Np � 1

! imax cos�1tanð/ Np

Nd� p

2NpÞ

tanð/Þ

!" #8/ 2 ½0; p=2�;

if Nd ¼ Np þ 1 ! i cos�1ðNp=NdÞ:

ð4Þ

This means that, in general, the range of possible inclina-tions is very limited. Following this process, Lee et al.(2015) were able to generate constellations of 400 satellites.In this work we will overcome this limitation and showother possibilities that 2D-LFCs can provide, both increas-ing the number of satellites and the range of availableinclinations.

3. Mathematical formulation of the LEO slotting problem

This section defines the slotting problem and admissiblesolutions in mathematical terms. In doing so, we hope toformulate the problem precisely and in a manner thatallows for direct comparison and evaluation of the variousapproaches discussed here, as well as for comparison withsolutions that might be generated elsewhere in thecommunity.


3.1. The orbital slot definition problem

We define a ‘‘slot” as a three-dimensional, moving,rotating, and possibly morphing region S ¼ SðtÞ of thespace. The distance between two slots is defined as the min-imum distance that, at any instant of time, a point in one ofthe slots is from a point of the other.

distðS1; S2Þ ¼ minfkp � qk : p 2 S1ðtÞ; q 2 S2ðtÞ; t 2 RgWe want to produce a list of slots S1; . . . ; Sn in the LEOspace (the spherical shell of space located below 2000kmof altitude) in such a way that:

ðP 1Þ The slots do not overlap at any time. Moreover, theyare separated at any time by a given minimum dis-tance dslot > 0.

distðSi; SjÞ P dslot 8 i– j

This requirement ensures that two adequately sized slotsremain separated by a certain minimum distance suffi-cient to ensure that a satellite maintaining itself withinSi will never collide with a satellite maintaining itself inSj.ðP 2Þ It is feasible, control-wise, to keep a satellite con-

tained within any of the slots. That is, there is no needto perform expensive maneuvering to prevent thesatellite from leaving a slot, even in presence of orbi-tal perturbations such as the atmospheric drag, thesolar radiation pressure or the J 2 effect. Naturally,there is no point in defining a set of slots within whichit is prohibitively costly for satellites to maintain theirpositions, or that imposes a significant restriction ontheir propulsion systems.

ðP 3Þ If a hypothetical mission were possible using a set ofsatellites in LEO orbits, then it should also be possi-ble using a similar (or at least not much larger) num-ber of satellites spread across a set of slots that avoidself-conjunctions. This requirement stems from theidea that a LEO slotting system should be minimallyburdensome.

Note that the minimum distance dslot required betweensatellites, as well as the maximum station-keeping costper satellite per day, are design parameters that should beclearly stated in any proposed solution.

A successful solution of the problem must describe theposition, orientation, and shape of the slots S1; . . . ; Sn asa function of time, and provide a proof that those regionssatisfy condition ðP 1Þ. Additionally, to evaluate whether aproposed solution satisfies ðP 2Þ, the initial conditions (forinstance, position and velocity) of n satellites must be pro-vided, each moving within its corresponding slot, as well asa computer simulation of the motion under a chosen per-turbation model during several periods of time. Thisshould include a study on the frequency of the requiredstation-keeping maneuvers and their cost.




At the moment, there is not a clear method to properlyevaluate condition ðP 3Þ, but we would expect a candidatesolution to demonstrate analysis showing that the pro-posed slotting can accommodate various mission arche-types and at what (if any) additional cost. In that sense,a reference set of design missions would be helpful toenable comparative evaluation of different slottingconcepts.

3.2. Proposed approach A: a mega-constellation

Since any solution of the problem must include the tra-jectories of the satellites moving inside the slots (and anystation-keeping maneuvering required), it makes sense todesign the slots starting from these trajectories. More pre-cisely, we envision that a potential solution can be obtainedby the following procedure:

ðA1Þ Find a constellation of satellites s1; . . . ; sn, under thekeplerian/J 2 model, moving within the LEO regionin such a way that they are always separated by atleast a distance dconst.

ðA2Þ Study the maximum deviation dpert that any satellitecan experience from its nominal trajectory in a givenreference time tpert due to all non-keplerian/J 2 forcesincluded in the model.

ðA3Þ Define the slots as S1; . . . ; Sn as spheres with center atthe position of the satellites s1; . . . ; sn and radius dpert.

If the value of dconst chosen in ðS1Þ satisfiesdconst P 2dpert þ dslot, then it can be easily shown that theslotting strategy defined satisfies ðP 1Þ. The minimum fre-quency of the station-keeping maneuvers is controlled bythe parameter tpert, which can be defined by the designeras a function of the known orbital perturbations, but thecost of each maneuver depends mostly on dpert, which is aderived quantity. Finding the right balance between fre-quency of maneuvering and cost of each maneuvers shouldbe included as part of the analysis to be done in ðA2Þ. Nospecial provisions are made concerning ðP 3Þ. However,we believe that a large value of n and certain regularityin the design of the constellation in ðA1Þ would be enoughto comply with ðP 3Þ.

Of course, many improvements can be made to theapproach above. One of the most promising options comesfrom the shape of the slots themselves. While the idea ofusing spheres is neat and reduces the mathematical proofsto the triangle inequality, computer simulations show thateffects of the non-keplerian/J 2 forces in the cross-trackdirection are lower than in the along-track direction.Accordingly, more slots could potentially be accommo-dated for a given level of station-keeping requirements ifspherical slots were replaced with a shape more elongatedin the along-track direction.

Finally, the decision to include the J 2 effect in ðA1Þ is dueto the fact that at LEO altitudes it is almost impossible to


compensate the effects of this perturbation with station-keeping maneuvers. This implies that for usual values oftpert in ðA2Þ, we would get a huge value of dpert.

3.3. Proposed approach B: a multi-layer megaconstellation

As a first study of step ðA1Þ, we propose a layer-by-layerapproach instead of trying to build the entire constellationat once. The vast majority of active satellites use circularand near circular orbits, and the forces mentioned abovechange significantly in magnitude depending on the alti-tude. While this approach does restrict the solution to hav-ing only circular orbits, it is possible to introduce, using therevolution time compatibility between orbits, a small num-ber of elliptical orbits on demand at the cost of allocatingseveral slots to guarantee that no collisions can occur.Another positive aspect of this type of solutions is that itis possible to open a launching window from any pointon Earth, every lcmðN 1; . . . ;NnÞ days (where Ni is the num-ber of days that each satellite requires to repeat its dynamicfrom the Earth Centered Earth Fixed frame of reference),by leaving empty a subset of specific slots in each layer.Whether this least common multiple can be made smallenough to be practical is still under analysis.

With a layer-by-layer approach, the final constellation isa collection of more simple constellations, each corre-sponding to a single altitude. In particular, each of thesesimpler constellations has all its satellites moving in circu-lar orbits of the same radius. With this new idea in mind,the recipe above can be reformulated as follows:

ðB1Þ Choose an altitude h within LEO and find a constel-lation of satellites s1; . . . ; sn, all moving in circularorbits at that altitude in the keplerian/J 2 model, insuch a way that the distance between satellites is atleast dconst at any time.

ðB2Þ Estimate the maximum deviation in along-track dalong

and cross-track dacross that any satellite at altitude h

can experience due to non-conservative forces fromits nominal orbit in a given fixed time tpert.

ðB3Þ Define the slots S1; . . . ; Sn as the three-dimensionalregions of space delimited by altitudes h� dacross

and hþ dacross, and the spheres of radius dalong cen-tered at the corresponding satellite.

ðB4Þ Return to ðB1Þ and choose another altitude h in sucha way that the selected altitude also guarantees thesafety of the configuration.

The collection of all the slots produced by the shownprocess satisfies conditions ðP 1Þ and ðP 2Þ if some care istaken when choosing the different altitudes. More precisely,within each layer we need to impose the conditiondconst P 2dalong þ dmin, and between any two consecutive

layers, at altitudes hð1Þ and hð2Þ, we need

jhð1Þ � hð2Þj P dmin þ dð1Þcross þ dð2Þ

cross. These two conditionsare sufficient to guarantee that the slotting produced is




valid. Nevertheless, validity alone is insufficient to deter-mine whether a particular solution obtained for the mainproblem using approach B is useful or optimal since thatanswer will depend on the precise missions under studyand the characteristics of future spacecrafts and payloads.

The J 2 effect must be included in ðB1Þ if station-keepingmaneuvering is to be kept within reasonable values. How-ever, a simpler approach based on a keplerian model is pos-sible when dealing with a layered slotting architecture. Theidea is that if the constellation used in ðB1Þ has all satellitesat the same inclination (as in a 2D-LFC), and the distancesbetween satellites is at least dconst as required, then includ-ing J 2 in the model will not change the relative distributionof the constellation. In particular, the rate of precession ofall the satellites will be the same, hence relative distanceswill be preserved. The perfectly spherical shells that wewould have in the keplerian model will become slightlydeformed, but the deformation is in the same direction inall the shells, so no correction to the separation betweenlayers is needed.

3.4. Criteria to evaluate and compare solutions of the

problem

It is relatively easy to produce constellations of satellitesthat satisfy the requirements of the problem. However, notall such constellations are actually useful. For instance, aconstellation with all satellites in a single inertial orbitmakes little sense, since the practical applications are verylimited and the overall capacity is much smaller than couldotherwise be achieved. Therefore, in order to define a com-mon framework of design and study of this kind of config-urations, we propose the use of four characteristics to helpdistinguish between the trade-offs of various approaches tosolution generation. These characteristics are:

1. Capacity (number of slots per shell),2. Earth coverage,3. Orbits per shell,4. Symmetries.

The first two criteria give quantities that should gener-ally be maximized. The third quantity is trickier: the moreinertial orbits the constellation has, the more expensive itwould be to deploy (or at least time-consuming if nodalprecession is used to space out inertial orbits). On the otherhand, it makes little sense to put all satellites in the sameinertial orbits for the reasons just described. Finally, appli-cations of LEO constellations generally require coordi-nated use of the satellites, meaning operators prefer tohave temporal and spatial symmetries across their constel-lations. Nevertheless, a particular shell might be shared bymany small constellations (demonstrating little shell-widesymmetry), occupied by a single large constellation (with


high temporal and spatial shell symmetry), or be some-where in the middle.

3.5. Determination of the minimum distance between

satellites

The study of minimum distance between satellites atany time in their dynamic is one of the most costly com-putations that must be done in order to assure the slottingrequirements. In that regard, Speckman et al. (1990) hasprovided an important analytical result to ease this calcu-lation. In that study the closest (non-dimensional) dis-tance, qmin, between two satellites in two circular orbitswith same altitude is analytically expressed by the follow-ing equation:

qmin ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ cos i1 cos i2 þ sin i1 sin i2 cosDX

2

rsin

DF2

� �;ð5Þ

where:

DF ¼ DM � 2 tan�1 � tanDX2

� �cos i1þi2

2

�cos i1�i2

2

�" #

; ð6Þ

i1 and i2 are the orbit inclinations of the two satellites, andDM and DX are the differences in mean anomaly and rightascension of ascending node of the two satellites, respec-tively. Note that qmin is a non-dimensional distance thatmust be scaled by the orbit radius to find the actual mini-mum approach distance value.

In this work, we will use a similar analytical expressionthat has been proven more computational efficient(Avendano et al., 2020) when dealing with pairs of satelliteslocated at any inclination:

amin ¼ 1

2Aþ Dþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðA� DÞ2 þ ðBþ CÞ2

q; ð7Þ

where:

A ¼ cosðDXÞ cosðDMÞ � sinðDXÞ cosði1Þ sinðDMÞ;B ¼ � cosðDXÞ sinðDMÞ � sinðDXÞ cosði1Þ cosðDMÞ;C ¼ cosði2Þ sinðDXÞ cosðDMÞ

þ cosði1Þ cosði2Þ cosðDXÞ sinðDMÞþsinði1Þ sinði2Þ sinðDMÞ;

D ¼ � cosði2Þ sinðDXÞ sinðDMÞþ cosði1Þ cosði2Þ cosðDXÞ cosðDMÞþsinði1Þ sinði2Þ cosðDMÞ;

ð8Þ

where amin is the minimum angular distance between bothsatellites at any given instant. As it can be seen, the mini-mum distance provided by this equation is also non-dimensional. These two equations are important resultsfor those designing satellite constellations with circular




orbits at the same altitude, since no propagation must becomputed to check the minimum distance between eachpair of satellites belonging to different orbital planes, whichdrastically reduces the computational time to assess thiscondition.

4. Proposed approaches to shell design and results

We proceed now to obtain solutions of ðB1Þ using threedifferent approaches. In all three cases, the goal is to max-imize the number of satellites and Earth coverage of thewhole configuration. To that end, we also discuss someinsights regarding the results obtained from these differentmethodologies and comment on some key features regard-ing station-keeping of the configuration and adaptability toparticular missions.

In our first approach, we propose a constellation ofsatellites chosen entirely at random that fulfills our slottingrequirements. This kind of distribution will disregard thenotion of symmetries altogether. Instead, this methodologyresults in a constellation with potentially as many satellitesas different orbits, maximizing the number of orbits pershell and minimizing symmetries.

In the second approach, we focus on generating com-pletely uniform distributions where satellites share thesame inclination. To that end, we base the design of thiskind of configuration on the 2D-LFC formulation. Thisway, we obtain highly symmetric solutions that have, ingeneral, fewer orbits, and which are constrained to a partic-ular inclination per altitude. It is important to note thatalthough being restricted to a common inclination per shellcould be seen as limiting from a design point of view, itminimizes the station-keeping necessary to maintain theoverall configuration in the presence of the J 2 perturbation.

Our last approach is midway between the other two. Theidea is to create the constellation as the union of differentFlower Constellations. Following this process, we obtainsolutions that share some of the characteristics of the pre-vious methodologies. In this approach, we explore constel-lations defined at different inclinations and with a differentnumbers of satellites per Flower Constellation to explorethe performance of this kind of distribution. A study onthe optimal overlapping of two large Flower Constellationsat different inclinations is also included.

In both the first and third approaches, we deal with slot-ting configurations whose satellites occupy different incli-nations at the same altitude. Therefore, if no constraint isimposed in the propulsion system of those spacecrafts,these kind of solutions lead to unacceptably high station-keeping costs to counter the J 2 effect (on the order of100s of m/s per day). This is produced by the differentialdrift in the orbital planes of satellites that have differentinclinations and limits the technical feasibility of this kindof solution. Nevertheless, it is important to note that theseexamples are useful in understanding limits to capacity inthe absence of the J 2 effect and help inform trade-offs for


other experiments involving a single or limited number ofinclinations per shell.

As a side note, the astute reader will notice that wedefine the slotting problem in terns of the position, orienta-tion, and shape of slots as a function of time but decline topresent this information in this section. While we have gen-erated these parameters, they are not particularly meaning-ful for the purpose of this paper, particularly for thestochastic approaches. Rather than the particular slots ina given solution, we are more interested in the performanceof each approach to solution generation with respect to thefour characteristics identified in the previous section.Table 1 shows a summary of the trade-offs of eachapproach studied in this work.

4.1. Assumptions

We make several assumptions across the experimentsrun in this section. Namely, we selected an altitudeh ¼ 700km and dconst ¼ 122km (which corresponds to aminimum angular separation of 1 degree at all times).These parameters were chosen to enable comparabilityacross the different methods. Nevertheless, h and dconst arefree parameters that can and should be adjusted, eitheron a per-layer or constellation-wide basis as part of thedesign and optimization process. This is described in moredetail in the current and next sections. For simplicity, wealso assume spherical slots, although other slot geometriesare possible and likely to be more efficient.

4.2. Random constellation

As a first case, we explore what would happen if satel-lites were placed into orbits at random within a shell, sub-ject to the minimum-separation constraint. This roughlyapproximates what might occur if a rule were imposed tospecify that orbits must be designed to avoid satellite vs.satellite conjunctions, but without other coordination.

Let us start by fixing some notation on orbital mechan-ics. We use the classical six orbital parametersO ¼ ða; e; i;x;X;M0Þ to uniquely identify the trajectoryfollowed by a satellite, assuming a keplerian model. Herea is the semi-major axis, e is the orbit eccentricity, i is theorbit inclination, x is the argument of the perigee, X isthe right ascension of the ascending node, and M0 is themean anomaly at time t ¼ 0. The first five parameters pro-vide the orbit of the satellite while M0 locates the satellitewithin its orbit.

We call a ‘‘random constellation” to a solution of ðB1Þobtained by the following simple procedure:

ðR1Þ Start with an empty constellation C ¼ £.ðR2Þ Find values of i 2 ½i0; i1�;X 2 ½0�; 360��, M0 2 ½0�;

180�� uniformly at random, and let s be the orbitwhose elements are Os ¼ ðRearth þ h; 0:0; i;0:0;X;M0Þ; where h is the altitude of the layer inwhich the constellation is to be placed.



Table 1Summary of approaches to shell design and trade-offs.

Number of Satellites Earth Coverage Different Orbits Symmetries

Random Constellation Low Targeted Many Few or NoneSingle 2D-LFC High Uniform Few ManyUnion of 2D-LFCs Medium Varies Intermediate Intermediate


ðR3Þ Check that s is always more than dconst apart from allthe satellites in C at any time.

ðR4Þ If the satellite fails the test, go back to ðR2Þ, unless wehad nattempt consecutive fails, in which case we stop.Otherwise, add the satellite to C and go back to ðR2Þ.

The parameters h and dconst are given as input, and theoutput is the list of orbital elements of satellites. The num-ber of trials, nattempt, before terminating was hard-codedarbitrarily at 5 million. The intervals for X and M0 wereset to ½0�; 360�].

Table 2 shows the results of this study. The first twoexperiments use a range of inclinations, but as explainedearlier satellites orbiting with different inclinations at a par-ticular altitude will experience different rates of secular driftin the right ascension of the ascending node, imposingunacceptable frequent high DV to preserve alignment.These examples are included only to allow for comparisonto the counter-factual case where J 2 was not present. Inparticular, the second experiment includes only progradeorbits, while the first includes both prograde and retro-grade orbits. Generating the random constellation usingjust prograde orbits allows us to define a larger numberof slots. This result is expected due to the different dynam-ics of prograde and retrograde orbits, which increase thepotential intersections between the slots of the constella-tion. On the other hand, fixing the inclination of the con-stellation also reduces the maximum number of slots thatthe algorithm is able to generate. This is due to the addi-tional constraint imposed in the configuration, which limitsthe freedom that the algorithm had originally to find emptylocations. Finally, using close to equatorial and polar incli-nations also reduces the number of available slots for theconstellation. In this regard, i ¼ 90� is the worst case sce-nario due to the fact that all orbits intersect in the samepoints, the poles. This means that no matter the configura-tion used, it is not possible to improve the result of dis-

Table 2Random constellation results.

Inclination (deg) Number of satellites

½0; 180� 1479½0; 90� 191015 70730 92745 101560 97075 85090 252


tributing all satellites uniformly in the same polar orbit,that is locating 360 satellites under the consideredconditions.

In order to show how these constellations are dis-tributed in the space, the ðX;MÞ-space (Avendano et al.,2013) (left) and ðX;MÞ-torus (Arnas et al., 2017c) (right)of one of these configurations are presented in Fig. 1.For this example we chose the solution for i ¼ 60� as a ref-erence for its comparison with the other methodologiespresented in this work. From Fig. 1 it can be observed thatsatellites are distributed quite randomly in the space, notgenerating clusters of satellites.

4.3. Single 2D-LFC

The idea behind this methodology is to assess all possi-ble 2D-LFCs that fulfil the given criteria. In this regard,and contrary to what was studied in the work of Leeet al. (2015), we do not impose any additional constraintsto the 2D-LFC. This means that the relative trajectorycould be different for all the satellites of the constellation,and also that these relative trajectories may have self-intersections. At a first glance this conditions might appearto be a step back compared with previous methodologies.However, it turns out that constellations defined in relativetrajectories with self-intersections provide the best solu-tions, not only in maximum number of satellites, but alsoin range of possible orbit inclinations.

Table 3 shows a summary of the results using 2D-LFCat various fixed inclinations. These results were obtainedby searching for all possible 2D-LFC that have the selectedinclination, whose number of orbits (No) and number ofsatellites per orbit (Nso) were within the range from 1 to360, and that presented a minimum distance between satel-lites of at least 1 degree (which is assessed using Eq. (7)).This means that an exhaustive search on the possibilitiesof configuration that 2D-LFC provides is performed,where we selected configurations are the ones with a largernumber of satellites for each inclination.

As can be seen in the results from Table 3, the maximumnumber of slots is heavily influenced by the inclination ofthe constellation. In fact, small variations in the inclinationcan produce noticeable changes in the maximum numberof slots that can be defined. Similar to the random case,constellations whose inclinations are closer to i ¼ 0� andi ¼ 90� have lower capacity.

In addition, a complete study of all 2D-LFC in the rangeof inclination i 2 ½0�; 90�� in 0:1� increments was performedin order to identify a best 2D-LFC candidate for this prob-



Table 3Single 2D-LFC results.

Inclination (deg) Number of satellites Best Flower Constellation

1 6 No 6 360 1 6 Nso 6 360 0 < Nc < No

15 1376 16 86 730 1656 184 9 13245 1869 267 7 24360 1722 246 7 22475 1414 101 14 4390 359 1 359 0

Fig. 1. Distribution of a Random Constellation for i ¼ 60� ðNsat ¼ 970Þ.


lem. To that end, no constraint in the parameters No or Nso

was imposed this time. Instead, the methodology used wasthe following ðLÞ:

ðL1Þ Select a new inclination.ðL2Þ Set a number of satellites for the constellation Nsat.ðL3Þ Find the all pairs fNo;Nsog such that Nsat ¼ NoNso.ðL4Þ For each pair fNo;Nsog found, generate all possible

satellite configurations varying Nc 2 f0; . . . ;No � 1g.ðL5Þ For each one of these constellation configurations,

check the constraint of minimum distance betweensatellites:

Pleasestellat

� If the constraint is fulfilled, a constellation candi-date for that number of satellites and inclinationis generated. Continue in ðL7Þ.

� Otherwise, continue in (L4) with a different valueof Nc.

ðL6Þ Once all configurations have been checked for a givennumber of satellites, and if no configuration is found,we are sure that no uniform distribution can bedefined that maintains the minimum distance con-straint with that number of satellites and inclination.Then:
� If the process has changed the number of satellites
more than 1000 times consecutively without find-ing a constellation that fulfills the minimum dis-tance constraint, the process is finished for thestudied inclination. Return to ðL1Þ.

� Otherwise continue in ðL7Þ.

cite this article as: D. Arnas, M. Lifson, R. Linares et al., Definitionions, Advances in Space Research, https://doi.org/10.1016/j.asr.202

ðL7Þ Increase in one the number of satellites and return toðL2Þ.

As a result of this study, the highest capacity was foundat i ¼ 46:2� with a constellation of 2132 satellites. This con-stellation has the following 2D-LFC parameters:No ¼ 2132;Nso ¼ 1, and Nc ¼ 1772. As can be seen, eachsatellite is located in a different orbital plane (Nso ¼ 1).

In general, the best solutions provided by the 2D-LFCformulation are defined in a large number of different orbi-tal planes. Although this could be seen as a disadvantage ofthis kind of distribution, it is important to note that it ispossible to distribute satellites in different right ascensionsof the ascending node by taking advantage of the J 2 pertur-bation. This allows a single launch to distribute satellites tomultiple planes while expending less fuel at the cost ofsome additional time after launch before reaching thedesired operational orbit.

Compared with the results of random generated constel-lations, we observe that 2D-LFC are able to generate a lar-ger number of slots for the same minimum distancebetween satellites and fixed inclinations. This effect is dueto the natural structure and symmetries present in a 2D-LFC. In particular, in order to assess the efficiency of2D-LFCs, the following experiment was performed. Firstan optimal 2D-LFC was generated, like the ones presentedin Table 3. Then, a numerical algorithm was created toinclude additional satellites in the configuration followingthe same process seen in Section 4.2. In all the tests per-formed at different inclinations and with different configu-



Fig. 2. Distribution of a Single Flower Constellation for i ¼ 60� ðNsat ¼ 1722Þ.


rations, the algorithm was unable to include additionalsatellites. This means that 2D-LFCs are able to distributesatellites very efficiently, not leaving any empty space intheir configuration.

Fig. 2 shows the ðX;MÞ-space distribution of a 2D-LFCwith parameters No ¼ 246;Nso ¼ 7, and Nc ¼ 224. This dis-tribution corresponds to the best 2D-LFC obtained ati ¼ 60� and shown in Table 3. As it can be seen, the config-uration is completely uniform. In particular, the relativedistribution of the constellation remains identical no mat-ter the satellite selected as the reference for theconfiguration.

Another interesting case of study is sun-synchronousconstellations. A study of SSOs was performed for the cho-sen study parameters (700 km of altitude implyingi ¼ 98:186� and a minimum distance between satellites of1�). Under these conditions, and using the methodologypresented before ðLÞ, the best 2D-LFC found contains1254 satellites, with constellation parameters:No ¼ 418;Nso ¼ 3, and Nc ¼ 160. This result provides aupper boundary for the number of slots that it is possibleto define for SSOs under the considered conditions.

Moreover, in order to evaluate the effect of varying theminimum distance between satellites in the overall capacityof the constellation, a more in-depth study was performedfor SSOs. In particular, using 2D-LFC we assessed themaximum number of slots that can be defined as a functionof the minimum distance between satellites. This computa-

Fig. 3. Maximum number of slots in sun-synchronous orbits (based on the 2D


tion was performed following the same proceduredescribed in ðLÞ. The results from this study can be seenin Fig. 3. From it, we can observe that the relation betweenminimum angular distance and maximum number of slotsis linear when expressing it in logarithmic scale. Therefore,it is reasonable to expect the maximum number of satellitesto scale, in order of magnitude, with a function of the formNsat ¼ Aa�B

m , where Nsat is the maximum number of slotsthat can be defined, am is the minimum distance betweensatellites, and A and B are two constants. This result seemsalso to be coherent with a uniform distribution of the areaof a sphere between Nsat points if we consider am � 1:

Nsat � 4pR2

2pR2ð1� cosðam2ÞÞ �

16

a2m þ oða3mÞ� 16

a2m;

where R is the radius of the sphere, and the two relatedareas are the complete surface of the sphere, and the areaof a spherical casket of radius am. Thus, we can concludethat the maximum number of satellites depends heavilyon the size of the control box and the inclination, withmore sensitivity to the size of the control box.

4.4. Union of 2D-LFCs

In this subsection we focus on the study of combiningdifferent 2D-LFCs in a single shell instead of just one2D-LFC. To that end, three different approaches have been

-LFCs) as a function of the minimum angular distance between satellites.




performed to cover a wide range of configuration possibil-ities. Since these solutions depend heavily on the randomconfigurations generated, the results obtained should beregarded as reference values for similar studies, not as acomplete general result.

4.4.1. Random union of orbits

In the first approach, the idea is to merge different 2D-LFCs that are generated at random. The 2D-LFCs selectedfor this approach are limited to constellations defined in asingle orbit, the most simple 2D-LFC that can be gener-ated. Therefore, the process is similar to the one presentedfor the random constellations (see Section 4.2), but insteadof just including one satellite in each iteration of the algo-rithm, a set of satellites uniformly distributed in the ran-domly generated orbit are computed and checked underthe minimum distance constraint. If all the satellites in thatorbit fulfil the condition, this subset of satellites is includedin the constellation. Otherwise, the process is repeated.This method could represent a scenario where satellitesare launched in groups to the same inertial orbit to reducelaunch costs.

Table 4 shows the results for this approach. As can beseen, all these configurations perform better than theirequivalents using a pure random approach (see Section 4.2).This is due to the fact that providing a structure for theconstellation, even if it is as simple as the one proposedin this approach, allows it to better optimize the configura-tion space. This effect is more noticeable when using puredistributions based on the Flower Constellations formula-

Table 4Random union of orbits results.

Satellites per Orbit Inclination (deg) Number of Satellites

2 ½0; 180� 15803 ½0; 180� 16384 ½0; 180� 16765 ½0; 180� 17706 ½0; 180� 169210 ½0; 180� 180015 ½0; 180� 166520 ½0; 180� 162030 ½0; 180� 1710

Fig. 4. Distribution of a Random Union


tion, since they represent the most structuredconfigurations.

Fig. 4 shows the ðX;MÞ-space (left) and the ðX;MÞ-torus(right) of the solution for 10 satellites per orbit and 1800maximum number of slots presented in Table 4. As it canbe seen, orbits (vertical lines in the figure) are deployedquite uniformly in the space, more than in the completelyrandom approach (Fig. 1) but much less than in the 2D-LFC approach (Fig. 2). Note that these orbits are definedat different inclinations, and thus, these figures only repre-sent the projection of the configuration in the ðX;MÞ-space.

4.4.2. Random union of 2D-LFCs

In this set of experiments, we explore generating sets ofoverlapping random 2D-LFCs, each including a subset ofsatellites uniformly distributed following the 2D-LFC for-mulation (see Eq. (1)). As in previous cases, satellites aregenerated and checked under the minimum distance con-straint in each algorithm iteration. If all the satellites inthat 2D-LFC fulfil the condition, this subset of satellitesis included in the constellation; otherwise, the process isrepeated. In this case, we study both the situation wheresatellites have a fixed inclination and when the inclinationis left as a free parameter for the search algorithm. A sum-mary of the results of both studies can be seen in Tables5,6.

From the results obtained we can see that this strategyhas less capacity than the single 2D-LFC. Interestingly,capacity was not monotonic with respect to number ofsatellites per Flower Constellation for most inclinations,meaning that there are situations where it is beneficial forthe algorithm to have more degrees of freedom duringthe searching process. During this study no condition wasfound to determine when these situations happen.

4.4.3. Combination of two 2D-LFC at different inclinations

In this third approach, we generated two large 2D-LFCs, each one with No ¼ 986;Nso ¼ 1, and Nc ¼ 478, atdifferent inclinations, i ¼ f40�; 60�g. Each FC has 986satellites for a total capacity of 1972 satellites. The chosenconstellations were computed with compatible phasing andchecked under the minimum distance constraint. As can beseen, this result achieves capacity comparable to a pure 2D-

of Orbits with Nso ¼ 10 ðNsat ¼ 1800Þ.



Table 5Union of 2D-LFCs Results (i 2 ½0; 180�).

Satellites per FlowerConstellation

Inclination(deg)

Number ofSatellites

2 ½0; 180� 11823 ½0; 180� 12874 ½0; 180� 8765 ½0; 180� 14606 ½0; 180� 106810 ½0; 180� 96015 ½0; 180� 96020 ½0; 180� 1080100 ½0; 180� 1000

Table 6Union of 2D-LFCs Results (fixed inclination).

Satellites per FlowerConstellation

Inclination(deg)

Number ofSatellites

5 30 84510 30 76020 30 8205 45 81510 45 77020 45 7805 60 75510 60 77020 60 8605 75 69510 75 66020 75 700


LFC. This shows the potential of this kind of configura-tion. Note that this kind of distribution has the problemof presenting a differential drift in the orbital planes dueto the different inclinations. This would require, in general,prohibitively large and frequent station-keeping to preservethis configuration.

Fig. 5 shows the resultant constellation. In it, two differ-ent subsets of satellites are presented corresponding withinclinations of i ¼ 40� and i ¼ 60� respectively. As can beseen, the overall configuration of the constellation in theright ascension of the ascending node and the mean anom-aly is in fact a 2D-LFC.

Fig. 5. Distribution of a Union of


5. Discussion

This section describes the various free parameters avail-able in the trade-space of our analysis model. These factorsare parameters that can be set based on both engineeringrequirements and policy guidance, with consequences forsystem design and capacity.

5.1. Minimum separation distance

As seen from the results obtained in Section 4, the min-imum distance between slots is the most important param-eter to determine the maximum number of slots in a givenshell, followed by the inclination of the orbits. This meansthat small changes in the minimum distance between satel-lites lead to large variations in the maximum number ofpossible slots of the configuration.

For the analysis in this paper, we have chosen a deliber-ately conservative 1 degree separation, which at the chosen700 km altitude corresponds to approximately 122 km.Similar studies have been made for other angular separa-tions. An example of that can be seen in Fig. 3. It is clearthat smaller values allow denser packing of traffic, butreduce distances between spacecraft and may requireimproved state knowledge on the part of operators andmore frequent maneuvering to resist perturbations. A dif-ferent value could be chosen either for the entirety ofLEO, or for a certain altitude. In particular, we shouldexpect to loosen the minimum distance requirement asthe altitude of the orbit increases. This is mainly due totwo effects. First, higher orbits are perturbed less by orbitalperturbations such as the atmospheric drag or the J 2 termof the Earth’s gravitational field. Second, the dynamic ofhigher orbits is slower when compared to lower orbits. Thismeans that we should expect a larger maximum number ofslots in shells at higher altitudes, and a lower number atlower altitudes.

In addition, it is important to note that using a slottingmethodology it is still possible to perform formation flyingbetween satellites. In these cases, multiple satellites areexpected to operate in a single or various slots, with theoperator being responsible for the control of the formationinside the slot boundaries. This concept is similar to how

Two 2D-LFCs ðNsat ¼ 1972Þ.




multiple satellites can share overlapping physical slots atGEO.

Another important topic to describe is the actions avail-able to an operator should a conjunction occur between aslotted satellite and a non-compliant satellite or debrisobject. This area is still a work in progress, but the poten-tial options that are being considered are maneuveringwithin one’s slot if possible, or using a set of empty ‘‘street”shells between parking orbits for inter or intra shell maneu-vers (see also next subsection).

5.2. Layer altitude separation

In general, layer altitude separation should take intoaccount the effects of at least the J 2 orbital perturbationand atmospheric drag. This means that layer altitude sepa-ration is heavily influenced by accuracy of satellite stateknowledge and size of the control box defined for eachshell. Therefore, a looser control box for layer altitude sep-aration would allow a longer period of time betweenstation-keeping maneuvers, but result in lower overallcapacity. A preliminary analysis performed by the authorshas shown that using a not very restrictive control strategyfor satellites at 700 km of altitude, it is possible to separatelayers at an approximate distance of 200 m with monthlystation-keeping. Drag effects are smaller at higher altitudesand may permit even closer spacing. At lower altitudes,wider spacing between slots would be required.

Additionally, and in order to allow flexibility and somemobility inside the slotting configuration, we propose toleave some shells completely empty. These shells areincluded to facilitate:

� The reconfiguration of the constellation,� Possible orbital maneuvers between slots,� Some parking capabilities,� Additional room in case collision avoidance maneuversare needed to mitigate conjunctions with non-compliant satellites or debris objects,

� And a limited adaptation of the configuration to accom-modate some elliptic orbits.

This means that, following this design, we should expectan empty layer between each two pairs of occupied shells.This idea reduces the capacity of the system, but it providesa lot of flexibility for the slotting configuration.

5.3. Inclination

Section 4 shows that the inclination of the orbits in ashell is a key parameter that affects the maximum numberof slots that can be defined. In that regard, we were able toobtain the maximum number of slots for a shell, under a 1degree minimum separation between satellites, at an incli-nation of i ¼ 46:2�. Note that this inclination is not unique.There is a range of inclinations that fulfill the previous con-dition, a situation that should be used to define a safe


dynamic under orbital perturbations. While this inclinationoffers maximum capacity among the experiments weexplored, it may or may not be suitable for a particularapplication due to a lack coverage of higher latitudes.Higher inclinations, including sun-synchronous orbits,can be accommodated, but at a reduction in the total num-ber of slots that can be supported in that shell. In particu-lar, orbital planes close to an equatorial or polarconfiguration significantly reduces the number of satellitesper shell, with i ¼ 0� and i ¼ 90� being the worst solutionsfor capacity.

We have already discussed the problem of defining shellswhose slots have different inclinations. Having shells withslots at different inclinations allows more flexibility in con-stellation design, but imposes very important limitations onsatellite propulsion systems, being technically unfeasible inmany cases. Shells with a common inclination in theirorbits have a reduced cost for orbit maintenance. For thatreason, a special focus has been made to explore single-inclination shells. Unfortunately, the need to enforce a sin-gle inclination per shell is the most burdensome implicationof this slotting system. Nevertheless, it can be somewhatmitigated by the use of large numbers of nested circularaltitude shells.

5.4. Shell generation strategy

In Section 4, we discuss three different potential strate-gies to generate orbits within a shell. Overall, the randomconstellation approach offers the most flexibility regardingsatellite placement, but is only about 50–70% efficient ascompared to a single Flower Constellation approach. Thisrandom strategy is probably the most similar to the currentapproach and future trend if no measures are taken,accommodating traffic on an ad-hoc basis without commit-ting to certain constellation structures.

On the other hand, a union of small Flower Constella-tions shows that multiple Flower Constellations performsignificantly worse than a single Flower Constellationand are only moderately sensitive to the number of satel-lites per orbit. Nevertheless, this kind of distribution showsthat having a structure in the constellation, even a basicone, provides an improvement in the capacity of thesystem.

All else being equal, a single 2D-LFC provides greatercapacity, better coverage, and a maximum number of sym-metries. In that regard, Fig. 6 shows an example of the dis-tribution of this kind of constellation from an Earthperspective. As can be seen, satellites uniformly cover allthe regions of the Earth between the latitudes defined bytheir inclinations (in this example 60�). Therefore, amongthe single-inclination shell generation approaches consid-ered in this paper, 2D-LFCs show the best packing. In par-ticular, a single 2D-LFC is especially appropriate whenactors have compatible mission requirements (Arnas,2018) and the chief objective is to maximize capacity, orif a shell’s primary user is a single large constellation. This



Fig. 6. Distribution of a 2D-LFC (i ¼ 60�;No ¼ 246;Nso ¼ 7, and Nc ¼ 224) from an Earth perspective.


shows that constellation design models based on theFlower Constellation Theory (Mortari et al., 2004;Avendano et al., 2013; Arnas et al., 2017a; Arnas et al.,2017b) are an adequate tool to study this problem.

5.5. Adaptation of the slotting systems to current satellites in

orbit

The prior analysis does not take into account that thereare many resident space objects already in orbit, someactive and many inactive. Ideally, an orbital slotting systemwould allow for the retroactive integration of these activesatellites into a proposed slotting system to minimize bur-den on their operations and maximize the benefits to exist-ing and future spacecraft.

Inactive objects do not station-keep and cannot be inte-grated. Conjunctions with non-compliant objects (bothactive and inactive) would need to be analyzed using proce-dures similar to those used today for maneuver coordina-tion for conjunctions between active objects. The emptyphasing shells could potentially be used for collision avoid-ance maneuvers, or collision avoidance maneuvers could beconducted within a particular parking slot if station knowl-edge is sufficiently precise for both objects. The authorsintend to explore both designing slots to optimize for exist-ing active spacecraft and defining conceptions of operationfor collision avoidance under an orbital slotting system infuture work. Integration of satellites in near circular orbits,the vast majority of active satellites, is significantly simplerthan satellites in eccentric orbits which would require fur-ther theoretical development and case by case study.

In general, the same checking procedure used for therandom generation example in Section 4.2 and the unionof small random 2D-LFCs in Section 4.4.2 could be usedto generate slots compliant with existing objects. Underthis checking approach, generating shells compatible withexisting objects is limited to objects that are already incompatible orbits with one another, i.e. sets of orbits thatdo not result in hazardous self-conjunctions. When multi-


ple objects are in overlapping orbits with non-compatibleorbits, the shell design can only be optimized for a compat-ible subset of such objects. At altitudes where a large con-stellation is designed to avoid self-conjunctions, additionalslots could be generated in compatible orbits. For altitudeswhere there are many satellites in orbits that are notdesigned to avoid conjunctions, the room for optimizationmay be more limited.

Some room to accommodate existing objects also existswhen using the 2D-LFC shells. 2D-LFCs are defined basedon a relative distribution of the slots. This means that it ispossible to define the slotting architecture in each shellbased on the objects already in orbit, selecting an appropri-ate slot distribution to avoid conflict with the dynamics ofprevious missions. In addition, the inclination of the 2D-LFC can be chosen to minimize interference with priortraffic or offer the best trade-off between interference withexisting traffic and demands for new satellites to be placedinto the shell.

These methods allow designers to adapt a proposedarchitecture to minimize interference with current activeobjects in orbit at a cost to the designer’s flexibility in slotdefinition and/or overall system capacity.

5.6. Overall capacity estimate

In this section we estimate a potential number of admis-sible slots in LEO. We caution that this estimate is heavilydependent on the assumptions for the above free parame-ters and the selection of which approach and parametersare used for each layer. In particular, including moresun-synchronous shells, or clearing additional slots toaccommodate elliptical orbits, ‘‘street” layers, or launchcorridors will reduce the total from the value we reach.Rather than the specific value obtained in this estimate, thissection is included primarily to demonstrate that it is pos-sible to generate large numbers of slots in LEO that avoidself-conjunction. Moreover, the process followed to obtainthis estimate illustrates a clear tool to conceptualize LEO




capacity and trade-offs between capacity and specific usesof orbital regions (for instance, the inclination selectedfor a particular shell).

For this estimate, we assume that the shells start at650 km and end at 2000 km, with occupied layers every1 km (this provides sufficient space for an empty layerbetween every two occupied layers and some additionalsafety margin). This gives us a total of 2700 layers, 1350of which are occupied. In addition, if we assume a globalminimum distance between satellites of 1 degree (that is,dconst does not depend on the altitude of the shell), we haveestimated an average of 1700 slots per shell. This meansthat under this conditions, it is possible to define a totalof 2.3 million admissible slots in the LEO region. Note thatthis value is a conservative estimate that is expected togreatly vary depending on the minimum distance allowedbetween satellites and the final configuration selected. Forinstance, if one were to assume every shell had the samecapacity as a 700 km SSO shell,2 this would yield nearly1.7 millions satellites. In contrast, if a per-satellite separa-tion of 0.13 degrees were used, this number would jumpto nearly 18.7 million admissible slots.

6. Conclusion

In this paper we briefly described the motivation for aLEO slotting system, presented a formulation of the prob-lem, and showed three approaches for slot generation andtheir performance across a set of four metrics. We thendescribed several of the parameters available to thedesigner of a potential Space Traffic Management LEOslotting system and studied their impact on the perfor-mance of the system, with a special focus on systemcapacity.

The results obtained in this study show that if no policymeasure is defined or norm emerges for the LEO STM slot-ting problem, the resultant distribution of the space in LEOwill be highly sub-optimal, clearly benefiting the first oper-ator to launch a significant number of satellites at a givenaltitude. Even with very simple slotting strategies, thecapacity of the system, as well as its long term mainte-nance, improves significantly. For instance, we show thatusing random 2D-LFCs instead of a set of randomly-placed single slots provides benefits not only in overallcapacity, but also in orbit maintenance. The upper limitof this structured configurations is set by the 2D-LFCs.This formulation allows the generation of completely uni-form distributions that present the maximum number ofsymmetries in their configuration. 2D-LFCs support thelargest number of satellites among the methodologies stud-ied, and also allow the easiest control under the effect oforbital perturbations.

2 Even if only SSO orbits were used, inclination and thus capacity willvary slightly with altitude.


Additionally, we carried out a first study on the overallcapacity of the slotting system. In particular, given a partic-ular altitude, the authors were able to locate 2132 satellitesin circular orbits while maintaining a minimum separationbetween satellites of 1 degree during their whole dynamic.In general, and for shells of satellites at the same altitudeand inclination, the maximum number of slots that canbe defined is a direct function of the minimum distanceallowed between satellites and the inclination of the orbits,where the minimum distance allowed between satellites isthe most decisive parameter.

The concept proposed in this work is still in its earlystages of analysis. Some areas for additional technical workinclude theoretical expansions (demonstration of the inte-gration of high-value elliptical orbits, other non-sphericalEarth effects, incorporation of corridors for launch andorbit phasing, control-box analysis and sizing); develop-ment and simulation of concepts of operation for commonorbital behaviors within the slotting structure (phasing,rendezvous and proximity operations, transfer orbits, re-entry, servicing, etc.); optimization (alignment of shells tominimize the impact of current debris and satellite activi-ties, analysis of different allocation policies, resiliency toorbital failures and non-compliant satellites); and furthermetrics development and cost-benefit analysis (includingdesign reference missions and evaluation criteria, compar-ison of station-keeping costs and capacity against constel-lation designs that do not guarantee non-intersection,integrated capacity-risk metrics). We actively invite otherswithin the community to develop other potentialapproaches and solutions that satisfy the LEO slottingproblem.

We also invite feedback from those within the policycommunity. The technical feasibility of LEO slotting is anecessary, but not sufficient condition for the implementa-tion of such a system. Considerable thinking is required todecide if such a system is desirable. We acknowledge thatthere are important considerations including slot alloca-tion, shell-design optimization strategies, registration, andcoordination/ enforcement that we have not addressed inthis paper. Ongoing discourse between technical and policyexperts will be critical to help mature the proposed concept,characterize stakeholder requirements and preferences, andensure any such slotting system is fair, equitable, efficient,and responsive to stakeholder needs.

Acknowledgments

The authors wish to acknowledge useful conversationswith Prof. Daniele Mortari at Texas A&M University onFlower Constellation Theory.

David Arnas was partially supported by grant ESP2017-87113-R (Spanish Ministry of Economy and Competitive-ness). Martın Avendano was partially supported by grantMTM2016-76868-C2-2-P (Ministry of Science and Innova-tion, Spain). All authors were also funded by MISTI Glo-bal Seed Funds (La Caixa Foundation).




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