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Paper text:
collisions in orbit pose a threat to spacecraft , to astronauts and to the global commons of near - earth space .
several collisions have already occurred between spacecraft and debris , while the iridium 33/cosmos 2251 collision of january 2009 represented the first documented satellite - satellite collision .
unfortunately , the cumulative number of collisions thus far is consistent with the prescient predictions of a runaway chain reaction @xcite .    reducing debris from current and future space missions ( debris mitigation )
is an important measure .
however , even the unrealistic case of  no new launches " is inadequate to curb runaway growth @xcite .
there are two remaining ways to alleviate the problem : debris removal and collision avoidance . for the latter
, one needs prediction precision to be sufficiently accurate so as not to imply an unwieldy number of collision avoidance maneuvers .
in addition , one needs a means of doing debris - debris collision avoidance since applying collision avoidance only to the subset of conjunctions involving maneuverable spacecraft does not suffice to curb the growth .
this paper describes a method to increase the orbit prediction accuracy based on publicly available tles .
many satellite owner operations have inadequate ( if any ) processes for conjunction assessment and collision avoidance , since they would have to screen their asset(s ) against all other space objects .
the only source of knowledge at their disposal for the majority of other objects is the publicly available two - line element ( tle ) sets . but
predictions based on tles using the associated analytic propagator ( sgp4 ) are not sufficiently accurate to warrant maneuvering to avoid potential collisions : they imply an unacceptably large number of potential collisions per space object , each of which has very low probability .
the problem is similar for debris - debris conjunctions except then both objects , not just one , are subject to these imprecisions .    to address this tle / sgp4 accuracy problem
, we investigated several methods to improve the propagation errors for non - maneuvering orbital objects whilst using only tles as input data .
the following research target was set : `` increase the predictive accuracy for orbital objects , using only historical tle data , such that it enables operational conjunction assessment for collision avoidance '' .    in the following sections , we describe one approach to this target , assess its accuracy , and discuss the requirements for extension to the entire space object catalog .
finally , we propose a new method of debris - debris collision avoidance enabled by long - term high - accuracy conjunction assessment .
esa uses tles from the publicly available catalog to initially screen their sun - synchronous orbit ( sso ) spacecraft ers2 and envisat for conjunctions @xcite .
telemetry from their operational spacecraft provide precision orbital ephemeredes ( poes ) for those spacecraft . for screening against all other potentially conjuncting objects ,
only tles are used .
@xcite describe a method to estimate error covariances of tles in order to quantify collision probability assessments .
their method provided inspiration for the present work . here
we extend and adapt their approach : based solely on the object s historical tles , we improve the accuracy of the object s predicted position , as opposed to quantifying the accuracy of the object s sgp4 propagation errors .
our method is essentially to use tle data as `` pseudo - observations '' and to fit an orbit to these pseudo - observations using a high - precision special perturbations propagator and traditional batch least - squares differential correction .
the fitted orbit is then propagated into the future using the same high - precision orbit propagator .
the prediction accuracy is assessed by comparison with precision orbital ephemeris ( poe ) data from the international laser ranging service @xcite .    for each object
we wish to analyze we choose a time window with two sections : a fitting period and a subsequent prediction period .
the fitting period is initially set to ten days , typical of the period over which u.s .
space surveillance network observations are fitted when generating tles for leo objects @xcite .
section @xmath0 details a more principled approach to determining the fitting period .
the prediction period is 30 days .    for a given object ,
all tles with epochs within its entire window are obtained .
however , only tles with epochs within the fitting period are used for the fitting process ; those in the prediction period are only used for validation and test .    using all tles from the fitting period and interpolating using sgp4
, we generate a series of order @xmath1 state vectors ( `` pseudo - observations '' ) equally spaced in time within the fitting period .
we initialize the differential correction with a state vector derived from the first tle in the fitting period , and then propagate that using a high precision propagator .
we apply batch least - squares differential correction to minimize the rms error in the radial , in track and cross track ( ric ) components of relative positions between the fitted orbit and the pseudo - observations .
the trajectory obtained from the converged differential corrector is our fit .
the high precision propagator incorporates a @xmath2 egm2008 gravity field including solid earth tides , point masses for the solar and lunar gravity fields and a nrl - msise @xmath3 atmospheric drag model .
we assume constant values for solar f@xmath4 and ap equal to their averages over the fitting period .
solar radiation pressure is modelled using a simple the biconic approximation with earth and moon as eclipsing bodies .
[ figure1 ]     days .
the fitting period is @xmath5 days for stella and @xmath6 days for etalon-@xmath7 .
all fitting uses only publicly available tles .
truth data are only used for plotting .
selection of the length of the fitting period is discussed in section 4 .
also shown are the updated tles ( no prediction ) in blue .
, width=529 ]    we applied our method to four non - manoeuvring spacecraft for which poe data were readily available @xcite : stella , starlette , ajisai and etalon-@xmath7 .
the summary of basic orbital and physical properties for these spacecraft are shown in table @xmath8 .
they were picked in part for their range of orbital altitudes .
.key parameters of geodesic spacecraft used [ cols="<,^,^,^,^,^,^ " , ]     [ table1 ]
[ figure4 ]     runs corresponding to @xmath9 random starting dates in @xmath10 .
, width=377 ]    the majority of tracked space objects ( i.e debris ) do not have precision orbital ephemerides  they are not tracked by the ilrs and they transmit nothing . many are , however , tracked by the united states space surveillance network ( ssn ) .
unfortunately , the only publicly available source of data derived from these ssn observations are , at present , the tles accessible at space-track.org ( and archived e.g. at celestrak.com ) .
while there is currently no publicly available source of high - accuracy trajectory for debris objects , we believe the method described above could be applied to refine the predictions of debris objects future positions based only upon comparatively inaccurate tles , facilitating conjunction assessment and collision avoidance .
however , there are several issues which complicate the application of our procedure to debris :    the first set of issues arise when tles are the _ only _ source of data - we have no `` truth '' in order to perform validation and test .
this is solved by the use of tles from beyond the fitting period as a proxy for truth data .
our initial manual experiments suggested that different fitting periods were optimal for different orbit categories .
these results are consistent with @xcite .
we made a more principled investigation , utilizing cross - validation , to optimize the length of the fitting interval on a per - object basis using only tles for both fitting and validation .
figure 4 shows how a fitting period of @xmath11 days was determined to be optimal for satellite ajisai .
the important point shown in the figure is that a @xmath11 day fitting period is optimal both when validating predictions with truth data _ and _ when validating predictions with future tles . and
the @xmath11 day fitting period is optimal whether predicting @xmath12 , @xmath11 or @xmath13 days into the future .
thus it appears `` future '' tles can be used as a proxy for truth data when optimizing the length of the fitting period on a per object basis .
the second set of issues is that for debris objects one must also solve for area to mass ratio ( drag ) and validate this as well using only future tles .
we are currently performing these experiments .
further accuracy improvements could come from a principled analysis of the number of fitting points and their weighting as a function of location within the fitting period , the removal of outlier data ( i.e. filtering ) and other techniques from statistical orbit determination , treating tles as observations . also , recent work @xcite suggests that there may be simple global transformations of tle data that can improve tle / sgp@xmath0 accuracy .
these systematic corrections might also be further refined on the basis of orbital parameters , resulting in a look - up - table used to correct tles as a function of their orbital regimes .
in addition , the scheme could be tested on actual known conjunctions in order to analyse its efficiency for conjunction assessment .
these issues will be the subject of a future paper .
as an initial assessment of the computational requirements needed to perform conjunction analysis for all space objects in the publicly available catalog , a simple conjunction analysis system was parallelized on the pleiades supercomputer at nasa ames research center . using approximately @xmath14 cpu cores , checking all objects against all objects for conjunctions @xmath15 days into the future takes about a minute .
the conjunctions involving spacecraft precisely match the results from socrates @xcite .
the same system takes about @xmath16 minutes to perform 7 days of all - on - all conjunction assessment of a `` simulated s - band fence '' catalog ( obtained from the nasa orbital debris office ) containing `` pseudo - tles '' for approximately @xmath17 million objects of size @xmath18 cm .
this demonstrates the feasibility of doing all - on - all conjunction assessment on the present and expected near - future catalogs .
scaling the fitting method discussed above to the future catalog will be the subject of a later paper .
an along track @xmath19 of @xmath8 cm / s provides of order 1 km displacement per day .
since error growth for our method is @xmath20 m per day , such small manoeuvres might suffice for collision avoidance . for manoeuvres this small one could use radiation pressure to impart the necessary momentum . to verify the conjecture
, we increased the flux of radiation by a factor of 10x the solar radiation constant for 10 minutes on a typical debris object ( area to mass ratio @xmath21 m@xmath22/kg ) in sso .
we then propagated that for a further 5 days .
the displacement was @xmath23 km / day along track compared to a reference object that was not so irradiated .
two cumulative such events , if one ensured appropriate geometry , would thus lead to the necessary along - track displacement .
a factor of 10x the solar rational constant could conceivably be applied from the ground .
for example , a 10kw class laser attached to a 1 m class telescope with appropriate tracking , could , ( considering diffraction limits only ) provide the necessary radiation pressure for a debris object of area @xmath241m@xmath22 in sso .
one would need to perform a detailed engineering analysis of the feasibility of such a device , particularly the viability of maintaining small beam divergence and tracking .
debris - debris collision avoidance could eliminate the need for an active debris removal program .
the results of @xcite imply , to first order , that performing debris - debris collision avoidance for 5 additional carefully selected objects per annum would curb debris field growth .
furthermore , any additional debris - debris collision avoidance would start reducing the net debris density , eventually to below a critical level , because debris creating collisions are being nulled whereas atmospheric drag continues to cause objects re - entry .
thus this method could provide not just a stop gap measure , but a permanent solution .
this fitting and propagation method , based solely on tles , could potentially provide initial conjunction analysis sufficiently accurate for an operationally viable collision avoidance system .
if similar improvements are possible for debris objects , as we expect , then it could be used for all - on - all conjunction assessment . however , to ensure that the scheme works in an operational setting , it would be necessary to have additional data : the so - called `` uncorrelated objects '' ( a.k.a . 
analysts set ) which account for approximately 30 percent of potential conjunctions in leo @xcite .
further , we show that all - on - all conjunction assessment is possible with moderate computer infrastructure , even with the large increase in size of the catalog of tracked objects that is expected in the next few years .    finally , given accurate predictions
, we claim _ debris - debris collision avoidance may be possible _ by externally inducing small manoeuvres using radiation pressure from @xmath11 kw class power density ground - based lasers . if feasible , this could negate the need for a large scale debris removal program .
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the density of orbital space debris constitutes an increasing environmental challenge . 
 there are three ways to alleviate the problem : debris mitigation , debris removal and collision avoidance . 
 this paper addresses collision avoidance , by describing a method that contributes to achieving a requisite increase in orbit prediction accuracy . 
 batch least - squares differential correction is applied to the publicly available two - line element ( tle ) catalog of space objects . using a high - precision numerical propagator 
 , we fit an orbit to state vectors derived from successive tles . 
 we then propagate the fitted orbit further forward in time . 
 these predictions are compared to precision ephemeris data derived from the international laser ranging service ( ilrs ) for several satellites , including objects in the congested sun - synchronous orbital region . 
 the method leads to a predicted range error that increases at a typical rate of 100 meters per day , approximately a 10-fold improvement over tle s propagated with their associated analytic propagator ( sgp4 ) . 
 corresponding improvements for debris trajectories could potentially provide initial conjunction analysis sufficiently accurate for an operationally viable collision avoidance system .    
 we discuss additional optimization and the computational requirements for applying all - on - all conjunction analysis to the whole tle catalog , present and near future . 
 finally , we outline a scheme for debris - debris collision avoidance that may become practicable given these developments .    space debris , conjunction analysis , orbit determination    = 0.5 cm