Ballistic Coefficient Estimation for Reentry Prediction of ...

Research ArticleBallistic Coefficient Estimation for Reentry Prediction of RocketBodies in Eccentric Orbits Based on TLE Data

David J Gondelach12 Roberto Armellin2 and Aleksander A Lidtke3

1Astronautics Research Group University of Southampton Highfield Campus Southampton SO17 1BJ UK2Surrey Space Centre University of Surrey Guildford GU2 7XH UK3Department of Integrated System Engineering Kyushu Institute of Technology Kitakyushu Japan

Correspondence should be addressed to David J Gondelach davidgondelachgmailcom

Received 30 June 2017 Accepted 14 November 2017 Published 10 December 2017

Academic Editor Alessandro Gasparetto

Copyright copy 2017 David J Gondelach et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Spent rocket bodies in geostationary transfer orbit (GTO) pose impact risks to the Earthrsquos surface when they reenter the Earthrsquosatmosphere To mitigate these risks reentry prediction of GTO rocket bodies is required In this paper the reentry prediction ofrocket bodies in eccentric orbits based on only Two-Line Element (TLE) data and using only ballistic coefficient (BC) estimation isassessedThe TLEs are preprocessed to filter out outliers and the BC is estimated using only semimajor axis dataThe BC estimationand reentry prediction accuracy are analyzed by performing predictions for 101 rocket bodies initially in GTO and comparing withthe actual reentry epoch at different times before reentry Predictions using a single and multiple BC estimates and using stateestimation by orbit determination are quantitatively compared with each other for the 101 upper stages

1 Introduction

Rocket bodies in geostationary transfer orbits (GTOs) havetheir apogee near geosynchronous altitude and their perigeewithin the Earthrsquos atmosphereThe atmospheric drag reducesthe orbital energy of the rocket bodies and lowers the orbituntil reentry occurs Lunisolar perturbations speed up orslow down this process by changing the eccentricity of theorbit and raising or lowering the perigee altitude whichin extreme cases results in direct reentry without drag-induced decayThe reentry of spent rocket bodies is desirablebecause the deorbiting of these uncontrolled bodies preventscollisions with functional spacecraft and potential generationof new space debris However the reentry poses a riskto the Earthrsquos population because rocket bodies consist ofcomponents likely to survive the reentry and impact theEarthrsquos surface (such as propellant tanks) [1] Therefore tobe able to mitigate any risks due to deorbiting the reentryof rocket bodies needs to be predicted

The major source of error in orbit prediction is thecomputation of the atmospheric drag [2] The perturbingacceleration due to drag 119903drag depends on the spacecraftrsquos

drag coefficient 119862119889 area-to-mass ratio 119860119898 velocity withrespect to the atmosphere V and the atmospheric density 120588

119903drag = 12119862119889119860119898120588V2 (1)

The drag coefficient and the effective area-to-mass ratiodepend on the objectrsquos attitude which is generally uncertainThe local atmospheric density on the other hand depends onthe solar and geomagnetic activity for which future values areunknown [3 4] In addition the drag calculation is subject toinaccuracies in the atmospheric density model and possiblemismodeling of the drag coefficient [1] Finally the velocitywith respect to the atmosphere is uncertain because the localwind speed is unknown

For state-of-the-art reentry prediction the accuracy ofatmospheric density calculations can be improved by cal-ibrating the density models using near real-time satellitetracking data [5ndash7] In addition the effective area can becomputed by performing six degrees-of-freedom (6DoF)propagation to calculate the attitude of the rocket body [8]Moreover using the attitude and a physical model of the

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 7309637 13 pageshttpsdoiorg10115520177309637

2 Mathematical Problems in Engineering

rocket body the drag coefficient can be computed [8 9]Furthermore a wind model can be used to compute thehorizontal wind speeds in the atmosphere [10]

When density correction models and 6DoF propagationtechniques are not available (eg because the object detailsare unknown or the measurements necessary for densitycorrections are unavailable) the drag coefficient119862119889 and area-to-mass ratio 119860119898 can be combined into one parametercalled the ballistic coefficient (BC = 119862119889119860119898) that can beestimated from orbital data Such an estimated BC dependson the actual 119862119889 and area-to-mass ratio but also soaks upatmospheric density model errors and possibly other errorsfor example orbital data inaccuracies More accurate orbitaldata and dynamical models therefore result in estimated BCsthat are closer to the true BC [6]

The application of highly accurate models and orbitaldata is required for accurately predicting the impact pointof reentering objects Sufficiently accurate orbital data ishowever often not available and Two-Line Element sets(TLEs) provided by the United States Strategic Commandare the only available data to perform reentry predictionThe accuracy of TLE data is however limited due to theapplication of simplified perturbation models (SGP4 andSDP4) [11 12] especially for objects in GTOs [13 14] and inorbits with high energy dissipation rates [15]

In this paper the reentry prediction of rocket bodies ineccentric orbits based on only TLE data is assessed Becauseattitude and density correction data are not directly availablefrom TLEs the predictions are carried out using 3DOFpropagation and a standard empirical atmospheric densitymodel Different methods have been developed in the pastto improve TLE-based reentry prediction by preprocessingTLE data and by estimating the BC solar radiation pressurecoefficient (SRPC) object state vector or a combination ofthese In this paper reentry predictions using only an estimatefor the BC are investigated This approach is straightforwardand can be used to obtain a first-order guess of the reentrydate several weeks or months before reentry when accurateprediction of the impact point is not feasible due to uncertain-ties in future space weather predictions In addition reentrypredictions using only BC estimates can easily be automatedto perform daily predictions for many objects Within thisassumption (only BC estimation) the goal of this paper is toprovide guidelines on how to estimate the BC to obtain themost accurate reentry predictions

Ballistic Coefficient Estimation For the estimation of the BCbased on TLEs several methods have been developed [16ndash20] Saunders et al [17] and Sang et al [18] estimate the BCby comparing the change in semimajor axis according to TLEdata with the change in semimajor axis due to drag computedby propagation using an initial state from TLEsThis methodis straightforward and uses semimajor axis data from TLEswhich are generally accurate The methods by Saunders andSang are almost equivalent the main difference is that Sangcomputes a single BC estimate directly where Saunders findsimproved estimates by iteration Gupta and Anilkumar [20]on the other hand estimate the BC by minimizing thedifference between apogee and perigee altitudes according

to TLEs and propagation This method is said to performwell for reentry prediction during the last phase of orbitaldecay It is however more complex and requires the use ofthe eccentricity from TLEs which is generally less accuratethan semimajor axis data A method for estimating boththe BC and initial eccentricity was developed by Sharmaet al [16] to improve reentry prediction of upper stages inGTO [21ndash23] Here the eccentricity and BC are estimatedby fitting the apogee altitude according to propagation toTLE apogee data using the response surface methodologyFinally Dolado-Perez et al [19] developed a method forestimating the BC and SRPC simultaneously This is carriedout by comparing the rate of change of the semimajor axisand eccentricity according to TLE data and propagationThemethod assumes that the change in semimajor axis is due toboth drag and SRP which should improve the BC estimateHowever again less accurate eccentricity data from TLEs areused for the estimation In addition because the eccentricityis strongly affected by lunisolar perturbations the changes ineccentricity due to drag and SRP are hard to observe Finallythe methods by Sharma et al [16] and Gupta and Anilkumar[20] estimate a single BC that is used for the purpose ofreentry prediction Saunders Sang andDolado-Perez on theother hand estimate multiple BCs and subsequently take astatistical measure of the set as final estimate

It should be noted that all thesemethods estimate a singleand thus fixed ballistic coefficient In reality the BC howevervaries over time due to for example rotation of the object orchanges in119862119889 due to altering atmospheric conditions Effortscan be made to predict the future variation of the BC [24] orassume a relation between the drag coefficient and the orbitalregime [25] but this is beyond the scope of this paper

State Estimation To obtain an accurate state of the objectfor reentry prediction state estimation can be carried out byorbit determination using pseudo-observations derived fromTLE data This approach is widely used and is described byfor example Levit and Marshall [26] Vallado et al [14] andDolado-Perez et al [19] In this paper state estimation willonly be utilized for comparison

TLE Preprocessing TLE data is used for estimating theBC and state of an object however the quality of TLEsassociatedwith an object is not homogeneous sometimes lowquality or even wrong TLEs are distributed For this reasonpreprocessing of TLEs is needed to identify outliers and TLEsof poor quality [27]

TLE Based Reentry Prediction Approach The goal of thispaper is to obtain accurate reentry predictions of decayingGTO rocket bodies using only an estimate for the BC andirrespective of TLE quality and availability This is achievedby TLE preprocessing (see Lidtke et al [27]) and enhancingthe BC estimation for the purpose of reentry prediction Themain contributions of this work are as follows

(i) The estimation of the BC is tailored for reentry predic-tions by comparing the decay of the mean semimajoraxis according to TLE data and according to a high-fidelity propagator considering all perturbations

Mathematical Problems in Engineering 3

(ii) The impact of the initial state used for BC estimationon the reentry prediction is shown

(iii) The performance of the method is assessed andimproved based on predicting the reentry dates of 101upper stages in highly eccentric orbits (all initially inGTO) and the sources of inaccurate predictions areanalyzed

(iv) The good performance of using a single BC estimateversus the use of a median BC estimate and versus BCand state estimation is shown

Because the considered rocket bodies are in highly eccentricorbits all relevant perturbations (geopotential lunisolardrag and SRP) are always considered during orbit propaga-tion

The methods used in this approach are discussed in thefollowing section After that the BC estimation and reentryprediction results using a single and multiple BC estimatesare discussed

2 Methods

The orbital propagator and BC and state estimation andTLE preprocessing methods used for TLE-based reentryprediction are discussed in the following

21 Propagation Method The orbital propagator used inthis study is the Accurate Integrator for Debris Analysis(AIDA) a high-precision numerical propagator tailored forthe analysis of space debris dynamics using up-to-dateperturbation models AIDA includes the following forcemodels [28] geopotential acceleration computed using theEGM2008 model (10 times 10) atmospheric drag modeled usingtheNRLMSISE-00 air densitymodel solar radiation pressurewith dual-cone shadow model and third body perturbationsfrom Sun and Moon

NASArsquos SPICE toolbox (httpsnaifjplnasagovnaifin-dexhtml) is used both for Moon and Sun ephemerides(DE405 kernels) and for reference frame and time trans-formations (ITRF93 and J2000 reference frames and leap-seconds kernel) Solar and geomagnetic activity data (F107and Ap indexes) are obtained from CelesTrak (httpwwwcelestrakcomSpaceDatasw19571001txt) and Earth orienta-tion parameters from IERS (ftpftpiersorgproductseoprapidstandardfinalsdata) A wind model is not usedbecause the effect of wind generally cancels out over oneorbital revolution [29] and the impact of neglecting wind issmall compared to the effect of inaccuracies in atmosphericdensity modeling

22 Ballistic Coefficient Estimation Method The approachused for the estimation of the BC is based on the methodfor deriving accurate satellite BCs from TLEs proposed bySaunders et al [17] Several modifications were made toimprove the method for the reentry prediction purpose TheBC estimation algorithm uses the data of two TLEs TheBC is estimated by comparing the change in semimajor axisaccording to two TLEs to the change in semimajor axis due todrag computed by accurate orbit propagation using an initial

state derived from the first TLE (if not stated otherwise statesare obtained from TLEs using SGP4 to convert the TLE toan osculating state at the desired epoch and subsequentlyconverting the state from the TEME to J2000 referenceframe) Since short-periodic changes are removed from TLEdata the change in semimajor axis according to TLEs canbe assumed to be purely the secular change caused byatmospheric drag (long-periodic variation of semimajor axisdue to gravitational terms and SRP may be included in TLEdata but are generally small compared to changes due todrag [30]) Therefore any difference between the changein semimajor axis according to TLE data and due to dragcomputed by orbit propagation can be assumed to be causedby a wrong guess for the BC The BC that gives the correctchange in semimajor axis is obtained as follows

(1) Compute the change in semimajor axis between thetwo TLEs Δ119886TLE using the ldquomeanrdquo mean motion 119899119900available in a TLE

119886TLE = (120583 sdot 864002

12058721198992119900 )13

Δ119886TLE = 119886TLE2 minus 119886TLE1(2)

(2) Take guess for value of the BC(3) Propagate the orbit with the full dynamical model

between the two TLE epochs and simultaneouslycompute

11988911988611988911990510038161003816100381610038161003816100381610038161003816drag = 2

1198862radic120583119901 [119891119903drag119890 sin 120579 + 119891119905drag

119901119903 ] (3)

where 119901 is the semilatus rectum 120579 the true anomalyand 119891119903drag and 119891119905drag the acceleration due to drag inradial and transverse direction respectively

(4) Integrate (119889119886119889119905)|drag over time to obtain the changein semimajor axis due to drag only Δ119886PROP

Δ119886PROP = intTLE2

TLE1

11988911988611988911990510038161003816100381610038161003816100381610038161003816drag 119889119905 (4)

(5) Update the BC estimate value using the Secantmethod

BC119899 = BC119899minus1 minus Δ119886DIFF (BC119899minus1)sdot BC119899minus1 minus BC119899minus2Δ119886DIFF (BC119899minus1) minus Δ119886DIFF (BC119899minus2)

(5)

where BC119899 is the 119899th BC estimate and Δ119886DIFF =Δ119886TLE minus Δ119886PROP(6) Repeat the procedure from step 3 until convergence is

reached

The first guess BC1 for this method is taken from 119861lowast ofthe first TLEThe 119861lowast parameter in TLEs is an SGP4 drag-likecoefficient and a BC value can be recovered from it BC =

4 Mathematical Problems in Engineering

12741621 sdot 119861lowast [31] The second guess BC2 needed for theSecant method is computed by performing one propagationusing the first guess and assuming a linear relation betweenthe BC and Δ119886PROP

BC2 = Δ119886TLEΔ119886PROP (BC1)BC1 (6)

The convergence criterion is met when Δ119886DIFF is less than10minus4 kmSeveral changes were made to the original method by

Saunders First during the BC estimation process it mayhappen that the object unexpectedly reenters during prop-agation Such a reentry is generally the result of a too-high estimate for the BC Therefore the propagation is thenrepeated assuming a smaller value for BC namely 90 ofthe initial value This prevents failure of BC estimation dueto reentry but may require several iterations to sufficientlyreduce the BC value

By default forward propagation is applied for BC esti-mation that is taking the state at the earliest TLE andpropagating it until the epoch of the latest TLE In additionalso backward propagation was implemented starting fromthe latest TLE and propagating backward until the prior oneBy propagating backward one prevents reentry occurringduring propagationThis is especially useful when estimatingthe BC close to reentry where an inaccurate BC guess caneasily cause unexpected reentry

Furthermore the change in semimajor axis due to drag(see (3)) is computed considering all perturbations duringpropagation This is important because the effect of couplingbetween different perturbations cannot be neglected

Finally the average semimajor axis is computed fromosculating data from AIDA to compare the change in semi-major axis with TLE data This improves the estimationbecause the osculating data includes short-periodic varia-tions whereas the mean TLE data does not [30]

Besides estimating the BC also the SRPC can be esti-mated Dolado-Perez et al [19] developed a method wherethe BC and SRPC are estimated simultaneously by comparingsemimajor axis and eccentricity data from TLEs with thechanges in semimajor axis and eccentricity due to drag SRPand conservative forces This method was implemented andtested but was found to give aberrant results because in alltest cases the effect of SRP was at least an order of magnitudesmaller than the effect of drag This resulted in an ill-conditioned system of equations and consequently aberrantSRPC estimates Therefore SRPC estimation was omittedand known area-to-mass ratio data was used to computethe SRPC for SRP perturbation computation assuming thetypical reflectivity coefficient value of 119862119877 = 1423 State Estimation The state estimation performed in thiswork is carried out by fitting accurate orbit propagation statesto pseudo-observations derived from TLEs using nonlinearleast-squares This is a consolidated method widely usedfor offline (ground-based) orbit determination (OD) [32] Afive-day observation window with 21 pseudo-observationsis used to estimate the state together with the BC The

initial state is located at the end of the observation periodand is expressed in modified equinoctial elements [33] Theresiduals minimized during least-squares optimization areexpressed in Cartesian coordinates aligned with satellitecoordinate system in radial transverse normal directionsMore details on the algorithm and settings can be found inGondelach et al [34]

24 TLE Preprocessing The TLEs have to be filtered becauseincorrect outlying TLEs and entire sequences thereof couldbe present in the data from Space-Track and using suchaberrant TLEs in subsequent analyses would deteriorate theaccuracy of the results Filtering out aberrant or incorrectTLEs consists of a number of stages [27] namely

(1) filter out TLEs that were published but subsequentlycorrected

(2) find large time gaps between TLEs because theyhinder proper checking of TLE consistency

(3) identify single TLEs with inconsistent mean motionas well as entire sequences thereof using a slidingwindow approach

(4) filter out TLEs outlying in perigee radius(5) filter out TLEs outlying in inclination(6) filter out TLEs with negative 119861lowast as they cause incor-

rect SGP4 propagation

TLEs with negative 119861lowast are filtered out because they pro-duce SGP4 propagations where the semimajor axis increaseswhich is not realistic for decaying orbits More details on theapplied filtering methods and results are discussed by Lidtkeet al [27]

3 Test Cases

To determine the quality of the BC estimates the estimateswere compared with BC values derived from 119861lowast in TLEs andwith real object data In addition to measure accuracy ofthe reentry predictions the error between the predicted andactual reentry date is computedThis error with respect to thetime to reentry is calculated as follows

Error = 10038161003816100381610038161003816100381610038161003816119905predicted minus 119905actual119905actual minus 119905lastTLE

10038161003816100381610038161003816100381610038161003816 times 100 (7)

where 119905predicted is the predicted reentry date 119905actual the actualreentry date and 119905lastTLE the epoch of the last TLE used forthe prediction

To test the reentry prediction performance a set of 101rocket bodies that reentered in the past 50 years was selectedThis makes it possible to compare the predicted reentrydate with the real one The reentry dates were taken fromsatellite decay messages from the Space-Trackorg website(httpswwwspace-trackorg) that provides the decay date ofspace objects It is worth mentioning that the exact reentrytime is not known because all decay times are at midnight(this can produce a bias in the calculated reentry predictionerror when predictions are made close to the actual reentry)

Mathematical Problems in Engineering 5

Filtered on mean motion

BC estimateBC from Blowast

0

002

004

006

BC (Ｇ

2k

g)

minus150 minus100 minus50 0minus200Days before reentry

(a)

Filtered on mean motion and perigee radius

BC estimate

0

002

004

006

BC (Ｇ

2k

g)

minus150 minus100 minus50 0minus200Days before reentry

BC from Blowast

(b)

TLE

Filtered on mean motion

6480

6500

6520

6540

6560

Perig

ee ra

dius

(km

)

minus150 minus100 minus50 0minus200Days before reentry

(c)

Filtered on mean motion and perigee radius

TLE

6480

6500

6520

6540

6560Pe

rigee

radi

us (k

m)

minus150 minus100 minus50 0minus200Days before reentry

(d)

Figure 1 BC estimates and BC from 119861lowast from TLE data (a b) and the mean perigee radius according to TLEs (c d) for object 28452 in the180 days before reentry In (a c) the TLEs have been filtered on mean motion only and in (b d) on mean motion and perigee radius

All upper stages were initially in GTOs but their reentrydates lifetimes inclinations and area-to-mass ratios differsignificantly To give an indication the perigee altitude 180days before reentry lies between 131 and 259 km and theeccentricity between 01 and 073 The number of TLEsavailable in the last 180 days before reentry varies from 45 to543 and the area-to-mass ratio according to object data liesbetween 0002 and 003m2kg

In addition all objects have been used to predict thereentry 10 20 30 60 90 and 180 days before the actualreentry date Some of the 101 objects were not suitable forseveral reentry prediction tests because they had no TLEswithin a specific number of days before the reentry (eg lastTLE is 90 days before reentry)

In real reentry prediction cases the actual reentry dateof the object is of course not known Analyzing the resultshas therefore not only the goal to examine the quality of thereentry predictions but also the goal to define guidelines forreal reentry prediction scenarios

4 Results

41 Ballistic Coefficient Estimation Figure 1 shows BC esti-mates and BCs from 119861lowast for object 28452 together with theperigee radius according to TLE data in the 180 days beforereentry For the left plots TLEs filtered on mean motion wereused whereas for the right plots the TLEs were filtered onmean motion and perigee radius First of all the trend ofthe BC estimates is similar to the trend of the BC from 119861lowastbut with an offset (note that in general it is however nottrue that BC estimates and BC from 119861lowast follow the sametrend) This proves that a BC estimate is required to performreentry prediction with a dynamical model different fromSGP4SDP4

Besides there is a clear relation between outliers in TLEperigee radius and estimated BC an outlier in perigee radiusresults in an outlier in the BC estimates More precisely ofthe two TLEs that are used for BC estimation the outlyingTLE that is used to obtain the initial state for propagation

6 Mathematical Problems in Engineering

0

001

002

003

BC (Ｇ

2k

g)

minus160 minus140 minus120 minus100 minus80 minus60 minus40 minus20 0minus180Days before reentry

(a)

6480

6500

6520

6540

Perig

ee ra

dius

(km

)

minus160 minus140 minus120 minus100 minus80 minus60 minus40 minus20 0minus180Days before reentry

(b)

0

001

002

003

BC (Ｇ

2k

g)

6490 6495 6500 6505 6510 6515 6520 65256485Perigee radius (km)

minus150

minus100

minus50D

ays b

efor

e ree

ntry

(c)

Figure 2 BC estimates (a) the osculating perigee radius accordingto TLE data (b) and BC estimates against perigee radius (c) forobject 27808 in the 180 days before reentry

results in an outlier in BC estimateTheotherTLE is only usedto compute the change in semimajor axis according to theTLEs and does not have such a strong effect Therefore it canbe concluded that the BC estimate strongly depends on theinitial state used in the estimation Because the atmosphericdrag depends largely on altitude an incorrect value of theinitial state that translates in an aberrant perigee height resultsin a poor BC estimate The BC estimate compensates for theincorrect initial state such that the state and BC together givethe correct decay in the estimation period 119861lowast is stronglycorrelated to the perigee height and thus both BC estimateand 119861lowast depend on the initial state This may explain why theBC estimate and 119861lowast in Figure 1 follow the same trend

Figures 1(b) and 1(d) show the BC estimates and perigeeradius after filtering the TLEs on outliers in perigee radiusThe BC estimates improve because outliers in BC estimatedisappear when TLE outliers in perigee radius are removedNevertheless there are still outliers in the BC estimateswhich may be removed when also smaller outliers in perigeeradius are filtered out

To have a closer look at the dependency of the BCestimate on the perigee radius the BC estimates are plottedagainst perigee radius according to TLE data for object 27808in Figure 2 where the color indicates the epoch of theBC estimate In Figure 2(c) one can observe a correlationbetween the BC estimates and perigee radii for estimates at

similar epochs For a set of BC estimates with similar epochsthe BC varies almost linearly with changing perigee radiusFigures 2(a) and 2(b) show that this relation is mainly due tonoise in the perigee radius that is compensated by the BCestimates If the TLE data were more accurate then the BCestimates would not vary as much and would be closer to thereal BC

This proves that to obtain a good single BC estimatethe TLEs should be filtered on perigee radius or on bothsemimajor axis and eccentricity Another option to reduce theimpact of outliers on the estimate is to compute multiple BCestimates and take themedian of the estimates as the final BCestimate The reentry prediction results using a single and amedian BC estimate are discussed in the next two sections

Besides different epoch separations between the twoTLEs used for BC estimation have been tested namely 2 510 and 20 days A TLE separation of 10 days was found tobe least sensitive to outliers and short-period effects becausethe difference between mean and median of the estimateswas the smallest and the dispersion in terms of standarddeviation and median absolute deviation was small as wellTherefore 10-day separation is used for BC estimation whichis in agreement with Saunders et al [17]

Finally BCs were estimated for the 101 test objects in the180 days before reentry It was found that 80 of the mediansof the BC estimates were within the range of possible area-to-mass ratio (assuming 119862119889 = 22) according to physicalobject data taken from European Space Agencyrsquos DISCOSdatabase (httpsdiscoswebesocesaint) see Figure 3 Thisgives confidence that the estimation method provides goodresults

411 Reentry Prediction Using Single BC Estimate The objec-tive of this section is to show that for reentry prediction usingonly a BC estimate it is of fundamental importance to runthe reentry predictions using the same state that is used forBC estimation

As described in Section 22 two TLEs are needed for esti-mating the BC thus to run the subsequent reentry predictionone can use the state of either one of the two TLEs Nowconsider the test case of predicting the reentry for 91 rocketbodies 30 days before reentry that is all reentry predictionsstart from the state of the TLE at 30 days (TLEstart) In onecase TLEstart and an older TLE (TLEolder) are used for BCestimation BC is estimated by propagating from the state ofTLEstart backward to TLEolder and the state of TLEstart is alsoused for the reentry prediction This case is labeled ldquoolderTLE same staterdquo In the second case the BC estimation isperformed using TLEstart and a newer TLE (TLEnewer) bypropagating backward from TLEnewer to TLEstart Here thestate (of TLEnewer) that is used for BC estimation is notequal to the state (of TLEstart) that is used for the reentryprediction This case is called ldquonewer TLE different staterdquoFigure 4 shows the cumulative distributions of the reentryprediction errors and their 90-confidence regions (the 90-confidence region is the interval where the true cumulativedistribution is located with 90 probability The width of theinterval depends on the number of samples and is computedusing the Dvoretzky-Kiefer-Wolfowitz inequality [35]) for

Mathematical Problems in Engineering 7

625

7252

9017

8479

7794

9859

9787

2609

2780

839

499

2870

326

579

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912

810

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015

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1428

711

718

2862

3

2379

725

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2845

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536

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776

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1308

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332

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997

2479

9

1394

0

3723

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1368

4

2949

7

2825

3

2841

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2537

225

496

2466

621

990

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240

2391

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770

2531

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051

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2165

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2077

821

057

2004

219

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1987

721

766

2114

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932

2211

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2771

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025

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1313

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Objects

000

001

002

003

004

005

006

007

008

009BC

(Ｇ2k

g)

Min BC (data)Median BC estimate

Max BC (data)Median BC outside minndashmax range

Figure 3 Median of the BC estimates and the minimum and maximum BC according to object data for all 101 objects Median BC estimatesoutside the BC range according to data are indicated with an orange dot (Objects are sorted on increasing average area-to-mass ratio)

0

01

02

03

04

05

06

07

08

09

1

CDF

of re

entr

y pr

edic

tion

erro

r (-)

10 20 30 40 500Reentry prediction error ()

Older TLE diff state(CDF)Older TLE diff state(conf reg)Newer TLE diff state(CDF)Newer TLE diff state(conf reg)

Older TLE same state(CDF)Older TLE same state(conf reg)Newer TLE same state(CDF)Newer TLE same state(conf reg)

(a) Cumulative distributions and 90-confidence regions of reentryprediction errors using only an estimate for BC for 91 objects 30 daysbefore reentry

a

reentry prediction)

BC estimationfrom tofrom tofrom tofrom toReentry prediction from

Older TLE (only used for BC estimation)Start TLE (used for both BC estimation and

Newer TLE (only used for BC estimation)

tＬＨＮＬＳt+t0tminus

(b) Schematic diagram of BC estimation

Figure 4 Reentry predictions 30 days before reentry using an older or newer TLE for BC estimation and the same or a different state for BCestimation and reentry prediction All reentry predictions start from the TLE at 30 days before reentry BC estimation starts from the sameTLE (orange and blue lines) or ends there and starts at a different TLE (yellow and green lines)The other TLE used of BC estimation is eitheran older or a newer TLE with respect to the TLE at 30 days (The colors of the plots in (a) and the arrows in (b) correspond)

8 Mathematical Problems in Engineering

0

01

02

03

04

05

06

07

08

09

1CD

F of

reen

try

pred

ictio

n er

ror (

-)

10 20 30 40 500Reentry prediction error ()

Single BC (CDF)Single BC(confidence region)

Median BC (CDF)Median BC(confidence region)

(a) 30 days before reentry median taken from BC estimates between 90and 30 days before reentry

0 10 20 30 40 500Reentry prediction error ()

0

01

02

03

04

05

06

07

08

09

1

CDF

of re

entr

y pr

edic

tion

erro

r (-)

Single BC (CDF)Single BC(confidence region)

Median BC (CDF)Median BC(confidence region)

(b) 60 days before reentry median taken from BC estimates between 120and 60 days before reentry

Figure 5 Cumulative distribution and 90-confidence region of reentry prediction error using a single BC estimate (orange) or the medianBC (blue) for (a) 91 objects 30 days before reentry and (b) 93 objects 60 days before reentry

both cases One can see that although newer information isused in the second case the first case which uses an olderTLE but the same state results in more accurate reentrypredictions The difference between the prediction resultsof the two cases is significant because the corresponding90-confidence intervals only overlap for small predictionerrors The use of the newer TLE only gives more accuratereentry predictions if the same state is used for BC estimationand reentry prediction see case ldquonewer TLE same staterdquoin Figure 4 For completeness Figure 4 also shows thecase ldquoolder TLE different staterdquo that results in less accuratepredictions compared to using the ldquosame staterdquo

Using the same state for BC estimation and reentryprediction gives better results because the BC estimate iscomputed such that together with the state it gives the correctdecay rate of the semimajor axis in the estimation periodUsing that BC estimate with another state will generally notresult in the correct decay rate and the reentry predictionis thus more likely to be less accurate Therefore the sameinitial state for BC estimation and reentry prediction shouldbe applied

The reentry predictions using a single BC estimate that arepresented in the following sections are computed using theldquoolder TLE same staterdquo approach such that the latest availableTLE is used for the initial state

412 Reentry PredictionUsingMultiple BCEstimates Insteadof using a single estimate one can computemultiple estimatesand take the mean or median of the set that may better

represent the average BC behavior This approach was testedby estimating the BC for every TLE between 90 and 30days and from 180 to 60 before reentry and use the medianof the estimates for reentry prediction at 30 and 60 daysbefore reentry respectively The prediction errors are shownin Figure 5 Compared with the predictions based on a singleBC the results are significantly worse the majority of themedian-BC samples is outside the 90-confidence intervalof the single-BC error distribution On average the reentrypredictions are 8 and 6 less accurate at 30 and 60 daysbefore reentry respectively

It was found that especially for orbits with a high eccen-tricity and low inclination the predictions with median BCare less accurate Figure 6 shows the prediction error againsteccentricity with different markers for different inclinationsat 60 days before reentry (similar results were found for 30days)The resultswithmedianBC showa correlation betweenincreasing eccentricity and increasing error whereas with asingle BC estimate this correlation is less strong In additionthe majority of the inaccurate predictions with median BCat lower eccentricity corresponds to low inclination orbits(119894 lt 12 deg) A possible cause for this is the TLE accuracybecause the accuracy of TLEs for objects in HEO GTO andorbits with low inclination is less than for other objects [36]This is also shown in Figure 7 that shows the dispersionof the mean perigee data (the median absolute deviation ofdetrended perigee data (the mean perigee radius data wasdetrended by subtracting the moving median from the datasee Lidtke et al [27])) against eccentricity The dispersion of

Mathematical Problems in Engineering 9

02 04 06 080Eccentricity (-)

0

10

20

30

40

50

Reen

try

pred

ictio

n er

ror (

)

Single BC - i = 0ndash12∘

Single BC - i = 18ndash32∘

Single BC - i = 48ndash55∘

Median BC - i = 0ndash12∘

Median BC - i = 18ndash32∘

Median BC - i = 48ndash55∘

Figure 6 Reentry prediction error 60 days before reentry using asingle BC (orange) or median BC (blue) plotted against eccentricitywith different markers for different inclination ranges

0

1

2

3

4

5

6

MA

D p

erig

ee ra

dius

01 02 03 04 05 06 07 080Eccentricity (-)

Figure 7 Median absolute deviation (MAD) of detrended meanperigee radius data in 180 days before reentry against eccentricityat 60 days before reentry

the perigee data that is the noise increases with increasingeccentricity A single BC estimate can compensate for suchinaccuracies by soaking up the error However when using amedian BC the individual TLE errors are averaged out andnot compensated for except for possible biases

These results suggest that estimation of the perigeealtitude or eccentricity is required in order to improvethe perigee data and thus the BC estimation and reentryprediction Indeed Sharma et al [16] developed amethod forestimating both the BC and eccentricity with good reentryprediction results for upper stages in GTO

413 Only BC versus Full State Estimation The reentrypredictions using only BC estimates are compared with thoseafter full state estimation using OD Figure 8(a) shows thereentry prediction results for 30 days before reentry after

only BC estimation (orange) and after full state estimation(blue) Surprisingly the results obtained after OD are notbetter than the predictions using only an estimate for the BCTheBC-only predictions are on average 06 better howeverthis difference is not significant for the number of samples(notice that the cumulative distributions are well within eachothers 90-confidence intervals) This outcome is oppositeto what one would expect because a state estimated usingOD is supposed to be a better starting point for accurateorbit propagation than a state taken directly from TLE datausing SGP4 To check if state estimation improves reentrypredictions at all a test was performed where after thestate estimation the BC is reestimated using the new stateestimate The results are shown in Figure 8(b) and they areon average 04 better than using only an estimate for theBC however again this difference is not significant for thenumber of samples used This indicates that state estimationhas less impact on the reentry prediction accuracy than BCestimation

To assess whether an accurate state and BC estimateresult in an accurate reentry prediction the six objects withthe lowest position residuals after state and BC estimationusing OD at 30 days before reentry were analyzed Table 1shows their mean position residuals and reentry predictionerrors before OD (ie only BC estimation) and after ODThe residuals after OD are all two orders of magnitudesmaller than before OD The state estimation thus improvedthe accuracy of the orbit in the 5-day observation periodsignificantly with respect to only estimating the BCHoweverjust half of the corresponding reentry predictions improvedand the highest prediction error is still 166This shows thata state and BC that give an accurate orbit in the past do notnecessarily give an accurate reentry prediction

This outcome may be the consequence of taking a fixedBC for prediction Figures 1 and 2 show that the BC changesover time (possibly due to object attitude variation changingdrag coefficient [25] and atmospheric modeling errors [6])These variations in the BC are not accounted for duringreentry prediction and therefore even if the initial state isvery accurate the prediction may not be accurate

414 10 to 180 Days before Reentry Finally the reentry pre-diction results for 10 20 30 60 90 and 180 days before reen-try using single BC estimates are shown in Figure 9 togetherwith the cumulative distribution and 90-confidence inter-val of all predictionsThepredictions at 60 days before reentryare on average most accurate The predictions at 10 and 20days before reentry on the other hand are significantly lessaccurate than the overall result It should however be noticedhere that the given reentry epochs are only accurate withinone day (as they are given at midnight) which can result in a10 reentry prediction error 10 days before reentry even if theprediction is perfect The fact that the short-term predictionsare less accurate is possibly due to the fast-changing dynamicsclose to reentry The local atmosphere changes largely andthe BC can vary quickly at lower altitudes see for exampleFigure 1 Assuming a constant value for the BCmay thereforenot be a good approximation and accurate computation of theatmospheric drag becomes difficult

10 Mathematical Problems in Engineering

0

01

02

03

04

05

06

07

08

09

1

CDF

of re

entr

y pr

edic

tion

erro

r (-)

10 20 30 400Reentry prediction error ()

BC only (CDF)BC only(confidence region)

OD - state + BC (CDF)OD - state + BC(confidence region)

(a) Prediction errors using only BC estimate and after OD

0

01

02

03

04

05

06

07

08

09

1

CDF

of re

entr

y pr

edic

tion

erro

r (-)

10 20 30 400Reentry prediction error ()

BC only (CDF)BC only(confidence region)

OD + BC reestimate (CDF)OD + BC reestimate(confidence region)

(b) Prediction errors using only BC estimate and after OD with subse-quent BC reestimation

Figure 8 Cumulative distributions and 90-confidence regions of reentry prediction error of 91 objects 30 days before reentry using onlyan estimate for BC and (a) after OD to estimate state and BC and (b) subsequently reestimate the BC

0

01

02

03

04

05

06

07

08

09

1

CDF

of re

entr

y pr

edic

tion

erro

r (-)

10 20 30 40 500Reentry prediction error ()

All predictions (CDF)All predictions(confidence region)10 days (CDF)

20 days (CDF)30 days (CDF)

(a) All prediction errors and at 10 20 and 30 days before reentry

0

01

02

03

04

05

06

07

08

09

1

CDF

of re

entr

y pr

edic

tion

erro

r (-)

10 20 30 40 500Reentry prediction error ()

All predictions (CDF)All predictions(confidence region)60 days (CDF)

90 days (CDF)180 days (CDF)

(b) All prediction errors and at 60 90 and 180 days before reentry

Figure 9 Cumulative distributions of reentry prediction error 10 20 30 60 90 and 180 days before reentry and all prediction errors togetherwith 90-confidence region using only an estimate for BC

Mathematical Problems in Engineering 11

Table 1Mean position residuals and reentry prediction errors before OD (only BC estimation) and after OD (see Section 23 for OD settings)for six objects with the lowest residuals after OD at 30 days before reentry

NORAD ID e [-] Mean position residual [km] Prediction error []Before OD After OD Before OD After OD

19332 0153 6600 99 23 147252 0070 6623 78 22 487794 0050 1055 30 63 619017 0084 5132 73 77 6425240 0087 4226 67 82 9725372 0046 3033 79 119 165

Overall with 90 confidence 62 to 72 of the predic-tions is within 10 error and 85 to 95 within 20 errorUsing a single BC estimate one can thus obtain a first-order estimate of the reentry date irrespective of TLE qualityand availability More sophisticated methods such as 6DoFpropagation and density corrections should subsequentlybe applied to accurately estimate the impact point of thereentering object

5 Conclusion

The estimation of the BC is tailored for reentry predictions bycomparing the decay of the mean semimajor axis accordingto TLE data with the decay of the average semimajor axisdue to drag according to a high-fidelity propagator con-sidering all perturbations The BC estimation results showthat the estimated BC depends strongly on the initial statebecause TLE outliers and noise in the perigee radius resultin outliers and noise in BC estimates Therefore filteringTLEs on eccentricity or perigee radius is important Becauseof the dependency on the initial state it is important touse the same initial state for BC estimation and reentryprediction as inaccuracy in the state is absorbed by a singleBC estimate such that they provide the correct decay of thesemimajor axis Taking the median of multiple BC estimatesfor predicting the reentry does not give good results becausethe median BC is not related to the initial state The accuracyof reentry predictions after state and BC estimation usingODare not significantly different from using only a single BCestimate Moreover an accurate initial state and BC do notnecessarily give accurate reentry predictions Overall usinga single BC estimate 62 to 72 of the reentry predictions iswithin 10 error (with 90 confidence) These conclusionsare based on reentry predictions using TLE data and are thussubject to their accuracy and availability that vary largely fordifferent objects

Besides using more accurate orbital data the fixed-BCapproach can be improved by using more accurate atmo-spheric density models and by applying a wind model toincrease the accuracy of density and velocity calculations dur-ing both BC estimation and reentry prediction Furthermoreif the accuracy of the orbital data is very low estimation of theeccentricity or perigee radius could improve the predictionsas they strongly affect the BC estimate and reentry prediction

However if the drag coefficient or frontal area of the objectchanges over time then the achievable accuracy using afixed BC is limited Knowledge of the objectrsquos attitude and6DoF propagation or a forecasting model for the BC couldsignificantly reduce the reentry prediction error

Appendix

Test Objects

Rocket bodies with the following NORAD catalog numberswere used for reentry prediction

625 2609 7252 7794 8479 9017 9787 9859 1098311072 11718 11719 12562 12810 13025 13087 13098 1313613294 13447 13599 13684 13940 14130 14168 14287 1433214369 14423 14787 14989 15157 15165 15679 16600 1835218923 19218 19332 19877 20042 20123 20254 20778 2092021057 21141 21654 21766 21895 21990 22118 22254 2290622928 22932 22997 23315 23416 23572 23797 23916 2431424666 24770 24799 24847 25051 25129 25154 25240 2531325372 25496 25776 26560 26576 26579 26641 27514 2771927808 28185 28239 28253 28418 28452 28623 28703 2949732764 36829 37211 37239 37257 37482 37764 37805 3794939499 40142

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was partly carried out within the EuropeanSpace Agency project ITT AO1-815515DSR titled ldquoTech-nology for Improving Re-Entry Predictions of EuropeanUpper Stages through Dedicated Observationsrdquo The authorsacknowledge Dr Hugh G Lewis of the University ofSouthampton (UoS) Dr Camilla Colombo of Politecnicodi Milano and Dr Tim Flohrer and Quirin Funke of theEuropean Space Agency for their valuable contributions Inaddition the use of the IRIDIS High Performance Com-puting Facility and associated support services at UoS inthe completion of this work are acknowledged David JGondelachwas funded by anEPSRCDoctoral TrainingGrant

12 Mathematical Problems in Engineering

awarded by the Faculty of Engineering and the Environmentof UoS Aleksander A Lidtke would like to acknowledge thefunding he received from theMinistry of Education CultureSports Science and Technology of Japan Roberto Armellinacknowledges the support received by theMarie Skłodowska-Curie Grant 627111 (HOPT Merging Lie perturbation theoryand Taylor Differential algebra to address space debris chal-lenges)

References

[1] C Pardini and L Anselmo ldquoRe-entry predictions for uncon-trolled satellites results and challengesrdquo inProceedings of the 6thIAASS Conference-Safety is Not an Option Montreal Canada2013

[2] National Research CouncilContinuing Keplerrsquos Quest AssessingAir Force Space Commandrsquos Astrodynamics Standards NationalAcademies Press Washington DC 2012

[3] J Woodburn and S Lynch ldquoA Numerical Study of Orbit Life-timerdquo in Proceedings of the AASAIAAAstrodynamics SpecialistsConference Lake Tahoe CA USA 2005

[4] B Naasz K Berry and K Schatten ldquoOrbit decay predic-tion sensitivity to solar flux variationsrdquo in Proceedings ofthe AIAAAAS Astrodynamics Specialist Conference MackinacIsland MI USA 2007

[5] P J Cefola R J Proulx A I Nazarenko and V S YurasovldquoAtmospheric density correction using two line element sets asthe observation datardquo Advances in the Astronautical Sciencesvol 116 pp 1953ndash1978 2004

[6] M F Storz B R Bowman J I Branson S J Casali and WK Tobiska ldquoHigh accuracy satellite drag model (HASDM)rdquoAdvances in Space Research vol 36 no 12 pp 2497ndash2505 2005

[7] V S Yurasov A I Nazarenko K T Alfriend and P JCefola ldquoReentry time prediction using atmospheric densitycorrectionsrdquo in Proceedings of the 4th European Conference onSpace Debris pp 325ndash330 Darmstadt Germany April 2005

[8] G Koppenwallner B Fritsche T Lips and H KlinkradldquoSCARAB - AMulti-Disciplinary Code for Destruction Analy-sis of Spacecraft during Re-Entryrdquo in Fifth European Symposiumon Aerothermodynamics for Space Vehicles vol 563 p 281 ESASpecial Publication 2005

[9] J Geul E Mooij and R Noomen ldquoGOCE statistical re-entrypredictionsrdquo in Proceedings of 7th EuropeanConference on SpaceDebris Darmstadt Germany ESACommunications April 2017

[10] D P Drob J T Emmert G Crowley et al ldquoAn empiricalmodel of the Earthrsquos horizontal wind fields HWM07rdquo Journalof Geophysical Research Space Physics vol 113 no 12 ArticleID A12304 2008

[11] F R Hoots and R L Roehrich ldquoModels for Propagation ofNORAD Element Setsrdquo Defense Technical Information Center1980

[12] D Vallado P Crawford R Hujsak and T Kelso ldquoRevisitingSpacetrack Report 3rdquo in Proceedings of the AIAAAAS Astrody-namics Specialist Conference and Exhibit Keystone ColoradoUSA 2006

[13] T Flohrer H Krag H Klinkrad B B Virgili and C FruhldquoImproving ESArsquos collision risk estimates by an assessment ofthe TLE orbit errors of the US SSN cataloguerdquo in Proceedingsof the 5th European Conference on Space Debris DarmstadtGermany April 2009

[14] D A Vallado B Bastida Virgili and T Flohrer ldquoImprovedSSA through orbit determination of two-line element setsrdquo inProceedings of the in 6th European Conference on Space DebrisESA Communications Darmstadt Germany April 2013

[15] M D Hejduk S J Casali D A Cappellucci N L Ericsonand D E Snow ldquoA catalogue-wide implementation of generalperturbations orbit determination extrapolated from higherorder orbital theory solutionsrdquo in Proceedings of the 23rdAASAIAA Space Flight Mechanics Meeting Kauai HI USA2013

[16] R K Sharma P Bandyopadhyay and V Adimurthy ldquoLifetimeestimation of upper stages re-entering from GTO by geneticalgorithmwith response surface approximationrdquo in Proceedingsof the International Astronautical Congress 2006

[17] A Saunders G G Swinerd and H G Lewis ldquoDerivingaccurate satellite ballistic coefficients from two-line elementdatardquo Journal of Spacecraft and Rockets vol 49 no 1 pp 175ndash184 2012

[18] J Sang J C Bennett and C H Smith ldquoEstimation of ballisticcoefficients of low altitude debris objects from historical twoline elementsrdquoAdvances in Space Research vol 52 no 1 pp 117ndash124 2013

[19] J C Dolado-Perez L Aivar Garcia A Agueda Mate and ILlamas de la Sierra ldquoOPERA A tool for lifetime predictionbased on orbit determination from TLE datardquo in Proceedingsof the 24th International Symposium on Space Flight DynamicsLaurel Maryland USA 2014

[20] S Gupta andA K Anilkumar ldquoIntegratedmodel for predictionof reentry time of risk objectsrdquo Journal of Spacecraft andRocketsvol 52 no 1 pp 295ndash299 2015

[21] R K Sharma and M Mutyalarao ldquoOptimal reentry timeestimation of an upper stage from geostationary transfer orbitrdquoJournal of Spacecraft and Rockets vol 47 no 4 pp 686ndash6902010

[22] M Mutyalarao and R K Sharma ldquoOn prediction of re-entrytime of an upper stage from GTOrdquo Advances in Space Researchvol 47 no 11 pp 1877ndash1884 2011

[23] J F Jeyakodi David and R K Sharma ldquoLifetime Estimation ofthe Upper Stage of GSAT-14 in Geostationary Transfer OrbitrdquoInternational Scholarly Research Notices vol 2014 pp 1ndash8 2014

[24] R Russell N Arora V Vittaldev D Gaylor and J AndersonldquoBallistic coefficient prediction for resident space objectsrdquo inProceedings of theAdvancedMauiOptical and Space SurveillanceTechnologies Conference vol 1 p 88 2012

[25] K Moe and M M Moe ldquoGas-surface interactions and satellitedrag coefficientsrdquo Planetary and Space Science vol 53 no 8 pp793ndash801 2005

[26] C Levit and W Marshall ldquoImproved orbit predictions usingtwo-line elementsrdquo Advances in Space Research vol 47 no 7pp 1107ndash1115 2011

[27] A A Lidtke D J Gondelach R Armellin et al ldquoProcessing twoline element sets to facilitate re-entry prediction of spent rocketbodies from the geostationary transfer orbitrdquo in Proceedings ofthe 6th International Conference on Astrodynamics Tools andTechniques Darmstadt Germany 2016

[28] A Morselli R Armellin P Di Lizia and F Bernelli Zazzera ldquoAhigh order method for orbital conjunctions analysis Sensitivityto initial uncertaintiesrdquo Advances in Space Research vol 53 no3 pp 490ndash508 2014

[29] E Doornbos and B Fritsche ldquoEvaluation of satellite aero-dynamic and radiation pressure acceleration models using

Mathematical Problems in Engineering 13

accelerometer datardquo in Proceedings of the 6th InternationalConference on Astrodynamics Tools and Techniques DarmstadtGermany 2016

[30] J M Picone J T Emmert and J L Lean ldquoThermosphericdensities derived from spacecraft orbits Accurate processing oftwo-line element setsrdquo Journal of Geophysical Research SpacePhysics vol 110 no 3 Article ID A03301 2005

[31] D A Vallado andWDMcClain Fundamentals of Astrodynam-ics and Applications Microcosm Press Hawthorn CA USA4th edition 2013

[32] O Montenbruck and E Gill Satellite Orbits Models Methodsand Applications Springer Berlin Germany 2000

[33] M J H Walker B Ireland and J Owens ldquoA set modifiedequinoctial orbit elementsrdquo Celestial Mechanics vol 36 no 4pp 409ndash419 1985

[34] D J Gondelach A Lidtke R Armellin et al ldquoRe-entryPrediction of Spent Rocket Bodies in GTOrdquo in Proceedings ofthe 26th AASAIAA Space Flight Mechanics Meeting Napa CAUSA 2016

[35] A Dvoretzky J Kiefer and J Wolfowitz ldquoAsymptotic minimaxcharacter of the sample distribution function and of the classicalmultinomial estimatorrdquo Annals of Mathematical Statistics vol27 pp 642ndash669 1956

[36] T Flohrer H Krag and H Klinkrad ldquoAssessment and cate-gorization of TLE orbit errors for the US SSN cataloguerdquo inProceedings of theAdvancedMauiOptical and Space SurveillanceTechnologies Conference Wailea HI USA 2008

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of