TAXONOMY AND CLASSIFICATION SCHEME FOR ARTIFICIAL SPACE OBJECTS

C. Fruh(1), M. Jah(2), E. Valdez(3), P. Kervin(4), and T. Kelecy(5)

(1)Mechanical Engineering Department, University of New Mexico / Air Force Research Laboratory, Space VehiclesDirectorate, NM 87117, USA

(2)Air Force Research Laboratory, Space Vehicles Directorate, Albuquerque, NM 87117, USA(3)United States Geological Survey, Dept. of Biology, University of New Mexico, NM 87117, USA

(4)Air Force Maui Optical & Supercomputing (AMOS), Kihei, HI, 96753, USA(5)The Boeing Company, Colorado Springs, CO, 80919, USA, Email: thomas.m.kelecy@boeing.com

ABSTRACT

As space gets more and more populated, a classificationscheme based upon scientific taxonomy is needed toproperly identify, group, and discriminate space objects.Using artificial space object taxonomy also allows forscientific understanding of the nature of the space objectpopulation and the processes, natural or not, that drivechanges of an artificial space object class from one toanother.

In a first step, an ancestral-dynamic hierarchicaltree based on a priori knowledge is established, mo-tivated by taxonomy schemes used in biology. In asecond step, available orbital element data has beenclustered. Therefore, a normalization of a reducedorbital element space has been established to provide aweighting of the input values. The clustering in the fivedimensional normalized parameter space is divided intwo sub-steps. In a first sub-step, a pre-clustering in amodified cluster-feature tree has been applied, to initiallygroup the objects and reduce the sheer number of singleentities, which need to be clustered. In a second sub-step,a Euclidean minimal tree algorithm has been applied, todetermine arbitrarily shaped clusters. The clusters alsoallow determination of a passive hazard value for thesingle clusters, making use of their closest neighbors inthe minimal tree and the radar cross section of the clusterin question.

Key words: taxonomy; classification; space debris.

1. INTRODUCTION

A study of a set of different objects leads to a specificset of parameters describing the characteristics of thoseobjects. With the increasing number of objects andaccuracy to capture all possible characteristics, a largeparameter space is utilized. In order to make the dataset accessible and manageable, it is desired to reduce

the parameter space to significant quantities whichallow determination of differences and similaritiesbetween different objects and to group and classify themaccordingly. However, the aim is not to introduce arandom grouping, but to find a taxonomy of significantparameters corresponding to an actual physical andbehavioral (e.g. dynamic) attributes.

Currently, about 20,000 objects are cataloged in thepublicly available USSTRATCOM catalog, whereas insitu measurements suggest around 300,000 objects tobe in orbit around the Earth. So far, only a very broadtaxonomy has been applied in different orbital regions,such as the orbital classifications of geostationary,geostationary transfer orbits, Molniya, low Earth orbits,which have been applied ad hoc. Another classificationscheme is based subsets of objects, such as the ESAClassification of Geosynchronous objects, sorting objectsby their orbital evolution, such as objects in drift orbits,around libration points, or controlled orbits. Anotherclassification that has been readily adopted is the dis-crimination in classified and unclassified objects. If inthe following the term classification is used, it prescribesthe scientific terminology and shall not be confused witha security relevant grouping of objects. But discussionsare ongoing about that a refinement of this structure isneeded. In the following, only unclassified objects aretaken into account. In this paper the focus is on orbitalelement classification. The important topic of furthermeans of characterization and classification based on theinclusion of spectral and light curve measurements is notdiscussed here.

The oldest taxonomic systems are rooted in biologyprimarily established by Aristotle. Biological taxonomyorders plants and animals into an organized system thatincludes species, genera, families and higher forms oftaxonomy. The system as applied to biology also showsthat taxonomies are not static, but subject to changeover time as new knowledge arises. Mayr defines thecrucial steps [11] in building a taxonomy of any kind:The first step is (1) the collection of possible data, asa second step he defines (2) the identification. At the

identification step the individual objects are sorted ingroups. The challenge is to select the relevant groups,which are as broad as possible, while not overlookingdistinguishing features; the identification is, in general,the analytical taxonomy step. The identification alsoincludes the process of naming the groups that havebeen identified with a useful term that is precise enoughto represent the group, but also short enough to beuseful. As a third step, (3) the classification followsas a synthetic taxonomy categorization. In this stepthe different identified genera and species are ordered.Available a priori knowledge can be fed in, in addition toassessing the physical reality of the defined classes. Theaim is to find an ancestral descent of the different generaand species, and their interrelations. Using traditionalmorphological taxonomy, convergence to a habitual stateis sought, e.g.. As it is easily conceived, the three stepsare highly interdependent. The data at hand determinesand limits the identification that is actually possible, andidentification is reiterated depending on the classificationstep. As new knowledge independent of the initial dataset is added, classification can change, which traces backto the identification step. This complete interdependentsystem of data, identification and classification is namedthe taxonomy.

The taxonomic classification used in astronomy hasthe most overlap with the problem of artificial spaceobjects are perhaps asteroids. Taxonomy systems ofasteroids are traditionally based on color measurements(filter UVB and spectroscopic measurements) andalbedo (including polarimetry). Earliest classificationsof asteroids [17] were based on the filter similarities ofthe asteroid colors to K0 to K2V stars. The first morecomplete asteroid taxonomy was based on a synthesis ofpolarimetry, radiometry, and spectrophotometry, usinga survey of 110 asteroids [1]. The defined a class C fordark carbonaceous objects, a class with the label S forsilicaceous objects, and U for objects that did not fiteither class. This system has the disadvantage that it wasnot detailed enough and is based on the exclusion princi-ple. The groundwork for the most complete taxonomy,that significantly expands the previous taxonomy hasbeen proposed by Tholen [16]. This taxonomy is, withsmall modifications, still in use today. Tholen establishedseveral asteroid classes, which are based on the albedo aswell as eight channel color indices. The overall albedo,as well as the spread and inclination of the color values,distinguishes the classes. Tholen based his taxonomyon the cluster analysis of a normalized set of the colorand albedo indices. The aim of asteroid taxonomy isto link those classes to heliocentric distance, diameter,and rotation rate, but also to the evolution, creation anddynamic (orbit-attitude) long term behavior of asteroids.Relatively independently a second asteroid classificationhas been induced recently. These second classificationis a risk assessment of asteroids, with respect of theirmiss-distance to the Earth.

A second increasingly relevant task of an asteroidtaxonomy is the quantification of the impact risk. Theimpact risk according to the so-called Palermo scale [2]

is based on the expected energy of in impact, normalizedwith the background risk and the time frame of thecollision. The energy of the impact is correlated to themass and the impact velocity. The impact mass is tracedback to the albedo and density (material properties) of theobject and hence correlates to the Tholen albedo-colortaxonomy. This is also a hint that the spectral classescorrespond to a physical reality.

In the case of artificial objects, a historically newsituation arises. For the first time the ancestral stateis theoretically and a priori fully known as the objectsoriginate from known man-made objects and materials.This is a state that is normally determined at the end ofthe development of a taxonomy. However, it does noteliminate the task of identification as derived from thedata that is collected with the means at hand. Hence, inthis case the taxonomy is done in reverse as comparedto other applications in science or other disciplines. Thequestion is, how does the physical reality influence theobserved dynamical state the objects are in, nominalorbital state, as well as quantities that are of interest ofus, such as the potential hazard those objects bear for thepreservation of space environment.

The topic of finding a taxonomic system for artifi-cial space objects is relatively young [18, 5]. Wilkinssuggests to make use a Linean system, dividing the spaceobjects in kingdoms, geni, families, orders, classes, lhylaand kingdoms. The Linean system is historically basedon similarity criteria and obvious physical traits. Inmodern biologoly, with the rise of Darwins biologicalsystem and even more nowadays with modern genetics,the Linean system has been replaced by the Phylogeniticsystem. In the Phylogenetic system, an ancestral tree isbuild, each node in the tree defines a split in the commonancestor. A similar first approach to this has been madein [5], but is missing the rigorousness that is required forthe task.

In the current paper a rigorous development of aPhylogenetic system of space objects is been made in afirst step, defining the framework and analogies of thesystem, which allows to trace back all object to theirorigin of creation in principle. This is the so-called apriori step. This is followed by two empirical steps: Ithe first step, the connection of the a priori classes andthe probability of detection is shown. In the second step,a reduction on the physical characteristic of the orbit ispursued. In a weighted minimal tree approach a groupingof the known space objects is done, in order to draw theanalogy to the a priori taxonomic system. In a last step ahazard value is established based to establish a measurefor the passive hazard a number for the groups of spaceobjects according the risk they pose for crossing spaceobjects.

Table 1. Key Physical Characteristics serving as the ”genetics”of the taxonomy.The orbital regime itself serves as a thirddimension to the whole scheme.

trait a priori established discriminators

orbit controlled (C) – uncontrolled(U)attitude controlled stable (S) – regularly spinning/rotating (R)– tumbling/irregular(T)materials composite of many materials (M)– few materials (F)– single material (S)shape regular convex (X) – regular (with concavities) (R)– irregular (I)size large (¡ 1.5m) (L)– medium (M) – small ( 10cm) (S)AMR HAMR(¡ 2m2kg) (hi)– medium AMR (me)– low AMR ( 0.8m2kg)(lo)

Figure 1. Classes based on key characteristics.

Table 2. Classes based on the physical characteristics combination.

name Example member

operationalCRMRLlo spin stabilized maneuvering satelliteCSMRLlo stabilized maneuvering satelliteUSMRLlo stabalized satelliteUSMRMlo stabalized (multi) cubesatpassively operational /debris - mission related/decommissioned intactURFXLlo upper stages/ cal spheres /decommissioned satellites with initial spinURFXMlo smaller upper stages/ cal spheres with initial spinURSXLme e.g. covers or other mission related objects with initial spinURSXMme smaller covers or other mission related objects with initial spinDebris-decommissioned/mission related intactUTMRLlo decomissioned satellitesUTMRMlo decommissioned (multi)cubesatsUTFXLlo tumbling upper stagesUTFXMlo tumbling smaller mission related objects (bolts)UTSXLme tumbling mission related objects (covers)UTSXMme tumbling mission related objects (covers)Debris-fragments, explosion pieces, delamination remnantsUTFIMlo medium sized composite compact pieceUTFIMme medium sized composite large area pieceUTFISlo small sized composite compact pieceUTFISme small sized composite large area pieceUTSIMlo medium sized single material compact pieceUTSIMme medium sized single material large area pieceUTSIMhi medium sized single material large area pieceUTSISlo small sized single material compact pieceUTSISme small sized single material large area pieceUTSIShi small sized single material very large area piece

Figure 2. Association of the classes with the class transition processes.

2. KEY PHYSICAL CHARACTERISTICS CLAS-SIFICATION

If we want to adopt a Phylogenetic system for spaceobjects, it has to be clear, which our analog to thegenetics is. Of course we are only based on the physicalcharacteristics of the objects. Inspired by the charac-teristics Tholen was investigating, the following fivecharacteristics have been established: Orbit (state, re-gion), Attitude state, materials, shape, size, area-to-massratio (AMR). The AMR is more a replacement of theweight of the body, which is not directly accessiblevia the measurements, but is readily determined fromthe AMR if size information is available. The orbitincludes the orbital state, especially if it is a controlled oruncontrolled orbit. However, as we will see later, the dis-tinction between orbital regions is like a third dimensionto the tree, which is in itself beyond orbital limitations,but could be build also in each orbital region separately,with different numbers of inhabitants in the differentcategories. The attitude state is the characteristic howthe attitude of the object is evolving over time, is itactively stabilized, naturally stable, spinning moderatelyaround one axis, spinning in irregular manner. Thematerial parameter describes the materials of the object,it is a single material, a few or a composite of manydifferent materials. The materials parameter does includeinner and outer materials of the object. Although innermaterials are not observable, they can become relevantin a collision. The shape parameters is describing theshape of the object, in general one has to distinguishbetween a well defined shape which can possibly be

known a priori and a irregular shape, as it is the resultin collisions. The size is distinguished in three differentkinds, small, which is below 10cm, medium, which isbetween 10cm and 1.5 meter and larger than 1.5 meterin diameter. The threshold values are oriented towardsbelow cube sat size, cubesat size made of multiple cubes,and sizes of smaller mission related objects, and largeobjects, which are whole satellites. The AMR valuesplits in three categories, HAMR, which AMR valueslarger than 2, medium AMR, between 0.8 and 2, and lowAMR below 0.8 m2kg . An overview can be found inTab. 1. The characters in paranthesis are used to definethe characteristics of the categories that are formed.

Theoretically, evaluating all permutations of thecharacteristics, 468 classes would have to be formed.Fortunately, not all of them are populated, and the major-ity of classes can be excluded from the start. Fig.2 showsthe explication of the permutations that are realized.The order at which the tree is split is not relevant. Itfollows the intuitive order that is listed in Tab.1. Classesare excluded by the following criteria. Objects that areboth orbit and attitude controlled require a full satellite,which consists of many materials and must be largerthan cube sat size. Those objects, necessarily have a lowAMR value. High and medium AMR values, can only beachieved using few or single materials. Fragmentationpieces are the smallest artificial space objects, they havealso necessarily an irregular shape, and can have allkinds of AMR values. Tab.2 list all 24 classes that havebeen found and lists objects that are in these classes asexamples.

Table 3. Processes inducing class associations and classchanges.

1 launch2 decommission/system failure3 fragmentation (collision, explosion, break-up)4 delamination5 reentry

These classes are build from the combination ofphysical characteristics or key aspects. With this back-ground we are now able to fit those classes in the knowncreation processes. The known creation processes thatplace objects in their initial class but are also responsiblefor them to change their class are the following: launch,decommission/system failure, fragmentation (explosion,collision), delamination, breakup, reentry. We neglectreentry here, as this leads to the destruction of the object,and hence no classification is needed any more. Takingprocesses as the Kessler Syndrome, all objects have thetendency to converge to classes that are listed under theUTFI and UTSI classes. They are also the ones thatare underrepresented in the official catalog, as they aresmall sized and hard to detect. The methods that induceclass changes are listed in Tab.3. In Fig.2 one can see theclasses that are listed together with the processes of classchanges. The figure clearly shows, that the classes are ondifferent hierarchical levels. Some equal to the status theobject have directly after the launch, others underwentfurther changes, and have transitioned between classes totheir current class.

3. DERIVATIONS FROM THE CLASSIFICA-TION

3.1. Connection with the Probability of Detection

The probability of detection is directly linked to the phys-ical properties and hence to the classes that are laid out.For the overall probability of detection Pi of a class ofobjects i, three things have to be taken into account. Theprobability of detection of the class in itself, based onits physical key properties Pphys, that define the class,the observation scenario and sensor characteristics thatis used Psensor, and the interclass probability Prel.. Thelatter states the amount of objects, that are in the certainclasses, that are theoretically detectable by the specificsensor. Taking only the latter probability into accountnecessarily leads to wrong results, e.g. when used as in-put in a Bayesian filtering[18]. The probability of detec-tion for a specific class is hence defined as:

Pi Pphys,i Psensor Prel.,i (1)

This is especially important when we imagine, consider-ing the probability of detection for small fragments, ofthe classes UTSIS and UTFIS . The majority of the

objects can be found in that class. But, they are in the ma-jority of cases beyond the detection limit of the sensor,hence have a very low probability of detection overall.For the classification in a filter setup, one might considerto renormalize the probability expression, so all possibleclasses add up to probability of one again:

Pi Pphys,i Psensor Prel.,i°24i1 Pi

(2)

Prel.,i is highly dependent on the specific model that isused, e.g. does one focus on the publicly available spaceobject catalog, or on in situ measurements, or on datafrom independent sensor networks, such as ISON. Theevaluation of this quantity is subject to a separate pa-per. Here, we will focus on the evaluation of the first part.

For optical sensors, the detection probability is de-termined via the signal to noise ratio (SNR). For adetailed assessment of the signal to noise ratio back-ground sources, detector capabilities, atmosphericconditions and visibility constraints have to be taken intoaccount [7]. For the connection to taxonomy, we choosea simplified approach. The signal to noise ratio can beapproximated, when we only take the object signal itselfinto account:

SN S?S?S (3)

The signal has two parts, the detector dependent part, andthe object dependent part. The signal can be expressed asthe following, averaging over the input spectra [7]:

S const. Csensor Cobsscenario Iobject (4)

const π

4

λ

hc∆λI@ 1.9457 105 (5)

Csensor pD dqQEpλq (6)

Cobsscenario F eτpλq secpπεeleq

p∆t (7)

Iobject Csensloc A Ψ 1

x2topo

(8)

(9)

The signal can be divided into four parts. A constant partconst, a sensor part Csensor and a object part, which iscoupled with the sensor location part Iobject Csensloc. Dis the sensor aperture, d the amount of the aperture whichis obstructed. D is the aperture of the measurementdevice, d is the area of the aperture which is blocked bythe construction e.g. of secondary reflection mirrors, Iis the intensity of the source, that is measured, QE isthe quantum efficiency of the detector, the exponentialfunction accounts for the atmospheric extinction, τ isthe wavelength dependent extinction coefficient. Here azenith angle dependent wavelength dependent attenua-tion is assume, c is the speed of light and h is Planck’sconstant, and ∆t is the integration time, and p is thenumber of pixels over which the signal is spread. F is thevisibility condition, which evaluates, if the object is in

the field of view, above horizon, and not in Earth shadow.The function Ψ is the reflection function, it dependson both the sun-object-observer geometry and on theobject properties. Ideally, we would like to decouple theobject and the sensor location part, as our classificationcategories only depend on the object itself. however, aswe wiill see below, this is not possible. Tab.4 lists theclasses that have a different probability of detection. Theunderscores indicate a placeholder, all possible values inthat part, cannot be discriminated from each other usingthe probability of detection. In the first part of the table,the S/R for the different classes are listed. In the secondpart of the table a standardized S/R for standard object ofone meter size. This allows to multiply the S/R just withthe appropriate object size in case it is given to a higheraccuracy than just the limits of a specific class. We arederiving now averaged expressions for each of the poolof classes that are listed in the second part of Tab.4.

For a flat plate Iobject Csensloc will take the fol-lowing form [7]:

Iobject Csensloc (10)A

x2topo

cosβCd

πcos γ, (11)

δ arccos ~O~S|~O~S|

~N, whereas ~O is the direction of the

observer, ~S is the direction to the Sun, and ~N is the nor-mal vector. The dimension of the sun has been taken intoaccount, where a@ is the radius of the Sun, and x@ is thedistance from the object to the Sun. Cs, Cd are the diffuseand speculat reflection components. No limb darkeningeffects have been accounted for. For a spherical object,the function reads as:

Iobject Csensloc (12)A

x2topo

Cd

πpsinpαq pπ αq cosαq,

where the function is scaled with the diffuse reflectioncoefficient Cd. α is the phase angle between the directionand the direction of the observer.

S R which can come in the size medium to large, areactively stabilized. Their reflection can be assumed to bedominated by the solar panels, which are flat surfaces,which keep the same angle to the sun, zero or close tozero angle between the surface normal vectors and thevector to the sun. This simplifies Eq. 10 to:

Iobject Csensloc ¡ S R (13)A

x2topo

cosβCd

π τ Cs x2

@

a2@

, (14)

which then equals a simple cosβ phase angle depen-dence. For R R and R R a high symmetry of theobject can be assumed. Most of the times, the objectsin this class have some kind of cylindrical or half-roundshape. For simplicity as we do not know the exact orien-tation of the cylinder we assume, it can be approximated

with a spherical surface, exposing most light from somekind of round surface as the cylinder cover, and/or halfround upper and lower parts of the cylinder. We also as-sume, as the object is stabilized in some sense, the rota-tion axis is nearly perpendicular to the line of sight. Thetwo classes are discriminated by their shape, that is doconcavities occur or not. Fruh et al [6] showed, that self-shadowing can be approximated by assigning differentshadow probability values which can be used to scale thereflectivity. For R X no self shadowing occurs, it canbe approximated with Eq.13. For R R a diminishingfactor has to be calculated.

Iobject Csensloc ¡ R X (15)r Iobject Csensloc ¡ R R ,

where r is the self-shadowing probability coefficient. itcan reach values of zero, total self-shadowing to 1, noself-shadowing. For a given surface r is calculated as[6]:

r 1

2πparccospe1 e2q arccospe3 e4qq, (16)

where e1234 are the unit vectors connecting the positionof the sub-facet midpoint with the edges of the concaveplane. See Fruh et al [6] for further details. In theabsence of the concrete shape of the object, an exactvalue for r cannot be readily determined. The concretevalue depends on how many surfaces have a convexrelation. For members of the R R group, only smallconcavities are expected, self-shadowing may be causedby antenna, or small concavities only, hence a smallvalue of nearly one, 0.9 is assumed for r.

The classes T R , T X , T I all deal withfreely tumbling objects with different shape properties.Schildknecht [14] showed, that uniformly tumblingdiffuse reflective flat plates, maybe be approximatedby a sphere with a diminished radius (factor 1/3),however, the integration assumes, that all angles areswept uniformly. Simulations of objects have shown thatthis is not the case [8, 6]. Angles that are not swept arelike self-shadowed concavities, which do not reflect anylight in the direction of the observer. This leads to thefollowing equations:

Iobject Csensloc ¡ T R (17)A

3 x2topo

Cd

πpsinpαq pπ αq cosαq q

Iobject Csensloc ¡ T X (18) Iobject Csensloc ¡ T R r1

Iobject Csensloc ¡ T I (19) Iobject Csensloc ¡ T R r2

where q is the factor that accounts for the non-regulartumbling, and r1 and r2 are the factors accounting for theself-shadowing in the regular shape with concavities andirregular shape. A value of q 0.8 is taken into account,which is consistent with the simulations in [8], r1 has avalue of 0.7, and r2 of 0.6, as more self-shadowing isexpected for irregular random shapes. Putting it back

Table 4. Probability of Detection and S/R.

name S/R standardized factors for non-standardized versions

S RL 49.8bCsensorCobscenario 106 x2

topocosβ

S RM 25.7bCsensorCobscenario 106 x2

topocosβ

R XL 49.8bCsensorCobscenario 106x2

topopsinpαq pπ αq cosαqR XM 25.7

bCsensorCobscenario 106x2

topopsinpαq pπ αq cosαqR RL 44.8

bCsensorCobscenario 106x2

topopsinpαq pπ αq cosαqT RL 44.5

bCsensorCobscenario 106x2

topopsinpαq pπ αq cosαqT RM 23.0

bCsensorCobscenario 106x2

topopsinpαq pπ αq cosαqT XL 37.2

bCsensorCobscenario 106x2

topopsinpαq pπ αq cosαqT XM 19.2

bCsensorCobscenario 106x2

topopsinpαq pπ αq cosαqT IM 17.8

bCsensorCobscenario 106x2

topopsinpαq pπ αq cosαqT IS 4.45

bCsensorCobscenario 106x2

topopsinpαq pπ αq cosαq

S R 28.7bA CsensorCobscenario 106 x2

topocosβ

R X 28.7bA CsensorCobscenario 106x2

topopsinpαq pπ αq cosαqR R 27.6

bA CsensorCobscenario 106x2

topopsinpαq pπ αq cosαqT R 27.3

bA CsensorCobscenario 106x2

topopsinpαq pπ αq cosαqT X 21.5

bA CsensorCobscenario 106x2

topopsinpαq pπ αq cosαqT I 19.9

bA CsensorCobscenario 106x2

topopsinpαq pπ αq cosαq

together, Tab.4 shows the signal to noise ratios. The firstentry is for a standardized setup. A phase angle of zerois assumed, a standard distance of 10 000 km, a onemeter aperture with d 0, a quantum efficiency of 1.0and visibility constraints are neglected F 1. For themean wavelength a value of 600nm has been assumed.For the second part of the table, where the discriminationin size classes has been neglected, an area of 1 has beenassumed. For the sizes, 3m, 0.8m and 0.05 meters hasbeen used. This just allows to multiply all quantities tothe standard value in case they are shifted.

3.2. A Hazard Scale

A second application of the classes is the hazard scale.A hazard scale is similar to the Palermo scale in asteroidphysics. In general the severity of a collision is linked tothe energy and impulse conservation of the participantsinvolved in a possible collision process. That is the massof the objects involved, material properties, and theirrelative velocities.

A hazard scale can be defined. As before in theconnection to the it is only concerned with a subset ofthe physical keys that are established in the taxonomy.However, of course. Those are the size and the AMRonly, and possibly if the object is in an controlled orbit ornot, which would allow for possible collision avoidanceor not. Attitude, material and shape however, are relevantwhen it comes to the accuracy of the predicted orbits,which eventually allow for collision avoidance measures.A hazard value h can be defined via the energies that areinvolved:

h 1

2

s

AMR v2 Cclass Corbit (20)

s is the size of the object, and v its velocity. The velocityof course depends on the orbit the object it in. The veloc-ity is approximated as the arithmetic mean of the apogeeand perigee velocity, leading to the following equation:

Corbit ra n 1 e2

p1 eqp1 eq s2 (21)

where a is the semi-major axis, n the mean velocity and ethe eccentricity. This leads to the following hazard valuesas displayed in Tab.5. For the AMR values, the valuesof 4m2kg, 1.4m2kg and 0.4m2kg has been used, forthe sizes, 3m, 0.8m and 0.05 meters. For the orbit a meanvelocity of 1 km/s has been used to scale the values.

3.3. Clustering of Orbital Space Level

The orbital space same as the creation, is a level to thetaxonomy. Different classes appear at different levelsonly, some may appear in all levels. A cluster analysis

groups data by means of a similarity criterion in aselected feature space. The cluster analysis allows thedetermination of interrelationships within sample data.In general, one distinguishes between hierarchical clus-tering, heuristic segmentation methods and partitioningmethods involving objective functions. Whereas thelatter two methods produce clustered data on the samelevel, hierarchical clustering methods arrange data innested sequences of groups, and can be displayed in adendrogram, or tree structure.

Cluster analyses have a long tradition in taxonomyof natural objects. In biological taxonomy of bacteria,minimum distance sorting in a equal weighted featurespace has been determined to lead to a classification[15]. A similar approach has been applied to a smallset of asteroid spectra [3]. Tholen has used minimalspanning trees to establish an asteroid taxonomy [16].The results of Tholen were confirmed using the samedata and artificial neural network clustering [10]. Zahncompared different nearest neighbor algorithms tothe minimal tree structure for clustering, proving theefficiency for cluster detection and pattern recognition,especially for irregular patterns [19]. Extensive researchhas been done on minimum spanning Euclidean trees,where the distance in a Euclidean parameter space isused as the measure of similarity [9, 12]. A greedyalgorithm for finding a minimal tree is also being used[13], but comes with a large computational burden.In two dimensions, Delaunay triangulation provides amean to supplement Prim’s algorithm [4], though thisis not feasible for higher dimensions. In this paper thefocus is on hierarchical methods for clustering. Further-more it is assumed that the number of clusters is nota priori known, and that a priori means have been defined.

The sheer amount of space resident objects, evenwhen only cataloged objects are taken into accountis very large, which makes the direct application of aminimal tree for clustering not very attractive. A modi-fication of the Iterative Reducing and Clustering UsingHierarchies (BIRCH) algorithm is applied [20]. BIRCHis a very effective algorithm, leading to a hierarchicalclustering in a single run. However, only the first stepis used, ordering the data in a so-called CF (ClusterFeature) tree. A primary CF tree is created in a singlerun. The classical BIRCH algorithm then continuesby slimming down the tree, with different resortingand merging techniques used to overcome some of theshortcomings of the initial CF tree. In the work presentedhere, a minimal tree is implemented as a second step,deviating from other CF combined approaches. Clusterfeatures are a triple, consisting of the number of datapoints, which have been merged in the cluster, the linearand quadratic sum of the data points. This allows for aconvenient adding of new data points into existing clus-ters. In the current approach the last entry is substitutedby the sum of the radar cross sections (RCS). The featureconsists of the three following entries:

CF pN,¸~xi,¸

RCSiq, (22)

Table 5. Taxonomy based hazard values.

name h standardized [J] factor for non-standard [m/s]

C Llo C3.750 v2

U Llo 3.750 v2

U Mlo 1.000 v2

U Slo 0.063 v2

U Lme 1.071 v2

U Mme 0.288 v2

U Sme 0.018 v2

U Mhi 0.100 v2

U Shi 0.006 v2

N is the number of objects in the cluster, ~xi is the fivedimensional vector of to the object in normalized andcropped orbital element space, and RCS is the radar crosssection of the single objects that have joined the cluster.The distance, which is used in determining the threshold,between two clusters or between an existing cluster and anew data point is determined as the following

dCF1CF2dp°~x1i

N1°~x2i

N2q2, (23)

which makes direct use of the CF structure, which alsomakes it easy to join two clusters together, by simplevector addition. Two different algorithms have beenimplemented to group the elements. The first step isidentical in both, a threshold T is determined on theleaf level for the maximum radius of a leaf-cluster.Thismeans, these clusters have a circular structure. For bothalgorithms, a threshold value for the number of leafs thatcan be combined in one node, cannot be larger than amaximum number B , and similarly a maximum numberof nodes in a root of L. This allows for deviations fromthe circular shape. Nodes and roots are split, when themaximum numbers are reached, sorting out the leafor node, respectively, that is furthest from the mean,determined by the values stored in the node or rootrespectively. The difference between the algorithmsis the following. In the classical BIRCH approach,the first node and root are filled until their thresholdvalues are reached and then split according to the rule.This is computationally very efficient, one of the greatadvantages of the method. In the next step the next leaf isadded to the node to which it is closest, and so on. In thereal of a limited number of entries, this can lead to wrongresults; the method relies on a sufficient amount of datapoints, that lead to enough splits in the nodes or roots,respectively, so no nodes and roots are kept, combiningvery diverse data points. Sufficient is determined by theamount of data in combination with the threshold values,but also depends on the diversity of the data, e.g. thenumber of outliers. That is the cost at which the compu-tational savings are achieved. If the distribution wouldbe perfectly uniform, the number of numberleafs/B andnumbernodes/L are built. The number of data is nearly 20000 objects, threshold values are chosen as low as 10 for

both roots and nodes. Nevertheless, the CF tree is crosschecked with a different algorithm. In that algorithm, theleafs and on the next level the nodes are combined to theabsolute closest node and root, respectively. Nodes androots are split at the threshold values at the same criterionas in the BIRCH algorithm. The modified algorithmallows a larger number of nodes and roots that are builtautomatically when the data is very diverse, no limits onthe size of the data that is analyzed is enforced. For auniform distribution of leafs, a number of numberleafs/numberclosest neighbours is built, that is in a perfectlyuniform distribution numbercloses neighbours is equalto two. If this number is larger than the threshold, thisnumber is replaced by the threshold value and hence isequal to the BIRCH approach. To explain this further.BIRCH relies on the fact, that the nodes and rootsare overpopulated in order to find meaningful nodesand roots, that represent the lower tree structures well,the alternative approach, does equal to BIRCH in theoverpopulated case, but provides a more meaningfulstructure, in sparse regions, where overpopulation is notreached to enforce splitting in the BIRCH algorithm. Itis hence more flexible to data which is not circularlyshaped.

In a second step all leaf-nodes are taken as initialinput data points in the minimal tree, with their centroidposition as their new nominal position of a mean object.For those nodes, a minimal tree is fitted [13], which leadsto the final clustering in cutting the longest links. Cuttingthe longest links sacrifices some of the hierarchicalstructure and leads to a flat clustering into same-levelclouds.

For the analysis, two line elements (TLE) have been usedas the source of orbital information. Radar cross sectionsfrom the satcat catalog have been used to supplementthe data. In the case where no radar cross sections wereavailable, a default value of 0.3 m2 has been used. Itis the very aim of the work to expand the classificationto include light curve and spectral measurements, tosupplement the orbital element classification. However,the present lack of light curves for the majority ofobjects made this intractable. A crucial point in evenstarting the clustering analysis is to express the quantities

in a comparable manner, which expresses the valuescommon units and hence makes them comparablein the first place. The second step is to weight thedifferent orbital elements in a physical meaningful way.To find a common scale, the semi-major axis is usedas a scaling factor and eccentricity, inclination, rightascension of the ascending node (RAAN), and argumentof perigee are expressed in units of lengths as well, forthe specific orbit. Orbital anomalies are not included forthe clustering analysis. For the scaling of RAAN andargument of perigee using the approximated value fromthe circumference of the ellipse u, independently of thespecific anomaly, the system is assumed to be locallyflat. Because the accepted distances are limited by thethreshold T the locally flat approach is justified. Theinclination i, RAAN Ω and argument of perigee ω arescaled as the following:

i1, ω1,Ω1 i, ω,Ω

2π u, (24)

The circumference u is approximated by the following:

u πpa bq1 3λ2

10 ?4 3λ2

q (25)

λ a b

a b, b

aa2p1 e2q, (26)

where a is the semi-major axis. The eccentricity e isscaled to the linear eccentricity ε:

ε e a. (27)

The absolute scales are necessary to determine thedistance in the five dimensional space, so only therelative values enter the clustering process. One way toweight the orbital elements would be an equal weightfor all elements. The topic of normalization cannotbe overemphasized, since it directly determines andshapes the clustering, as it has the effect of definingwhat is regarded as dense region and close neighbors.The scaling and weighting is done a priori, to savecomputational time.

In the initial run for building the CF tree in theclassical approach, a total number of 575 roots werefound with a total number of 1206 nodes holding 5229leafs, 1964 leafs hold more than one element. With themodified algorithm 599 roots were found, with a numberof 1782 nodes. On the root level, the algorithms producenearly the same roots, on the node level, a significantlylarger amount of nodes are created with the modifiedapproach. The orbital elements of the six cluster leafswith the largest amount of data points are listed in Tab.6.Those are all high inclination low Earth orbits with smalleccentricities. The RCS sum of the clusters is of theorder of eight to six meters. Similar orbital regions arecovered by the roots containing the non-leaf nodes andleaf nodes. The orbital elements and radar cross sectionof the roots with the highest number of member objectscreated in the modified approach is displayed in Tab.7.

Table 6. Six most populated leafs in the circular clus-ter feature tree and their mid points: number of objects,semi-major axis (km), eccentricity, inclination (deg), ar-gument of perigee (deg), RAAN (deg), radar cross sectionof the whole cluster (m)

# obj a e i ω Ω RCS

30 7177.6 0.009 98.72 85.73 40.57 9.0029 7196.4 0.008 73.25 158.03 279.60 8.8529 7152.8 0.004 74.04 145.01 286.79 8.7027 7201.4 0.008 72.15 141.45 271.10 7.2227 7144.8 0.004 74.04 143.10 299.69 8.1026 7203.6 0.008 98.82 29.18 84.507 6.87

Table 7. Six most populated root nodes in the circu-lar cluster feature tree and their mid points: numberof objects, semi-major axis (km), eccentricity, inclina-tion (deg), argument of perigee (deg), RAAN (deg), radarcross section of the whole cluster (m)

# obj a e i ω Ω RCS

197 7276.5 0.012 98.55 127.75 90.21 57.38133 7546.6 0.056 77.07 81.70 97.40 83.75131 7256.7 0.011 98.73 94.07 290.27 38.15130 7361.2 0.010 97.16 73.26 341.12 41.81129 7172.1 0.008 73.03 97.40 298.99 44.62

Fig.3 shows the inclination as a function of the semi-major axis and the eccentricity as a function of theright ascension of the ascending node for the leaf nodesdetermined in the pre-clustering step. In Fig.3(a) thedensely populated area is clearly visible is between 7000and 9000 km, which contains the leafs with the largestnumber of objects. The geosynchronous region around42000 km is also clearly discernible. Fig.3(b) showsthe accumulation of the objects at low eccentricities andaround 0.7 for all right ascension values. The modifiedalgorithm tends to provides a stronger focus on thedensely populated areas and seems to captures better theoverall structure, which is discernible in the data.

4. CONCLUSIONS

In this paper an initial taxonomy system for space objectshas been established. The Taxonomy is based on thebiological phylogenetic system. This lead to the defini-tion of physical key features. Those features are orbitalstate, attitude, amount of different materials, shape, sizeand area-to-mass ratio. Each of the key features comeswith different discriminators, at least two, at most three.All permutations of the discriminators would lead to alarge number of classes. However, not all permutationhave an object that corresponds to it in reality, whichhence lead to the establishment of 24 object classes.

Figure 3. Cataloged space objects clustered in leafs, nodes and roots with the modified algorithm.

Figure 4. Leafs of clustered space objects with more than one member.

They classes are labeled defined by five capital letters,for the five first key features, and two lower case latterfor the discriminators of the last category. If one doesnot want to distinguish between the discriminators of onespecific feature and wants to include all, a lower spacecan replace the discriminator at that position.

Two levels to the taxonomy have been defined. Levelscan hold one or more classes, classes can belong tomore than one level. One level is the creation processes,another level the orbital region. For the clustering oforbital region a modified version of the BIRCH approachhas been established. The densest orbital regions arein the sun-synchronous region. In total 599 roots werefound, populated with 1782 nodes have been found.

The taxonomic classes and their combination withthe orbit level allows to determine, averaged probabilitiesof detection using optical sensors for the differentclasses. Standardized values have been calculated, whichcan be multiplied with a factor that takes the concretesensor and observation scenario into account. Similarly,hazard values for all classes have been defined, based onthe kinetic energy of the different classes. The hazardvalues may be combined with the density of the orbitalclusters that have been found.

ACKNOWLEDGMENTS

The first author would like to thank the National ScienceFoundation for providing the funding that supported thiswork.

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