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Advances in Space Research 43 (2009) 1065–1069

Propagation errors analysis of TLE data

R. Wang a,*, J. Liu a, Q.M. Zhang b

a Center for Space Science and Applied Research, Chinese Academy of Science, Beijing 100080, Chinab State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, China

Received 31 October 2006; received in revised form 20 November 2008; accepted 21 November 2008

Abstract

Orbit position uncertainty is an important factor for collision avoidance issues. For a single object with high frequency historical data,we can attain its position uncertainty easily. But sometimes data is not enough for errors analysis, orbits need to be classified. In thispaper error analysis is made from two-line element sets data (TLEs). The Simplified General Perturbations-4 (SGP4) propagator wasused. Statistical errors of debris and R/B are given for lower-altitude orbits which are classified by perigee altitude and eccentricity.The errors results and analysis for SSO (the Sun synchronous orbit) typical orbits are obtained. At last atmospheric drag as a main causeof downrange errors in lower-altitude orbit is analyzed. BSTAR in TLEs is modified to improve prediction precision.� 2008 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: Space debris; Errors analysis; SGP4/SDP4; Lower-altitude orbit; Atmospheric drag; BSTAR

1. Introduction

Debris produced with human activities pollutes theouter space, which pose great hazards to the satellite inorbit. Now three in-orbit collision events have happenedand been verified. In order to avoid the catastrophic events,orbit forecasting is necessary. Positions of space objectswere forecasted according to the historical orbit data, theirintersection relationships were obtained in order to avoidthe possible collision events. Due to data errors and fore-cast errors, the intersection relationships errors inevitablyexist. So errors analysis need to be done.

In this paper, TLEs and the corresponding SGP4/SDP4model are used. And the statistical errors are obtained fororbits of different perigee altitude, eccentricity, inclinationin different prediction duration. Classified orbit errors anal-ysis is made first for the orbits below 1000 km perigee alti-tude, then is for the objects correlated with the typicalorbits of Sun synchronous orbit. Atmospheric drag as amain cause of downrange (velocity) errors in low Earth

0273-1177/$36.00 � 2008 COSPAR. Published by Elsevier Ltd. All rights rese

doi:10.1016/j.asr.2008.11.017

* Corresponding author.E-mail address: wrl@earth.sepc.ac.cn (R. Wang).

orbits is analyzed. BSTAR is modified in TLEs to improveprediction precision.

2. Data and means

TLEs are used for errors analysis which are ‘‘mean” val-ues obtained by removing periodic variations in a particu-lar way. In order to obtain good predictions, these periodicvariations must be reconstructed by the prediction modelin exactly the same way they were removed by SGP4/SDP4 model suitable for TLEs.

Three months TLE data are picked-up includingDecember 2005, January and February 2006. In order toassure the chosen orbit is not for an object using station-keeping, only debris and R/B are used.SGP4 is used fornear-Earth satellites which is an analytical method consid-ering the gravitation and atmospheric drag. SDP4 is usedfor deep-space satellites considering the gravitationaleffects of the Moon and the Sun as well as certain sectoraland tesseral Earth harmonics which are of particularimportance for half-day and one-day period orbits.

We assumed that the state vector given is accurate andused it as a reference. We propagated the preceding

rved.

Table 2Proportion relationship among downrange, normal and conormal errors.

Prediction period (day) e0.00–0.02 e0.02–0.2 e0.2–0.7 e > 0.7

1 19:01:01 10:01:01 8:02:01 4:01:012 32:01:01 13:01:01 12:02:01 5:01:013 43:01:01 17:01:01 16:02:01 6:01:014 53:01:01 20:01:01 18:02:01 7:01:015 67:01:01 24:01:01 25:02:01 7:01:016 83:01:01 29:01:01 32:02:01 8:02:017 101:02:01 31:01:01 37:02:01 9:02:018 117:02:01 37:01:01 43:02:01 10:02:019 128:02:01 41:01:01 49:02:01 10:02:01

10 126:02:01 44:01:01 58:03:01 11:02:01

1066 R. Wang et al. / Advances in Space Research 43 (2009) 1065–1069

element set to the epoch of the reference TLE data, andcompared the position and its trivariate components:downrange, normal and conormal. We can get trivariatecomponents errors.

3. Classified orbit errors analysis

In this paper, the objects were classified by the perigeealtitude and eccentricity. There are four eccentricity classes.The first class is for orbits with eccentricities less than 0.02.The second class is for orbits with eccentricities between0.02 and 0.2. The third class is for orbit with eccentricitiesbetween 0.2 and 0.7. The last is for orbits with eccentricitiesgreater than 0.7. Considering the prediction precision, pre-diction duration is less than 10 days. Downrange/normal/conormal errors are given for different perigee altituderanges less than 1000 km.

Here we only give position errors for one-day predictionduration for orbits with perigee altitude below 1000 km.The unit of position errors is kilometer (see Table 1).

For objects in lower-altitude orbits (below about2000 km), there is an intrinsic uncertainty in the futureatmospheric drag, for instance due to random variationsin the solar activity. Current methods use the drag pertur-bation from the recent past to predict drag into the nearfuture (Matney et al., 2004). So errors can not be negligible.

From Table 1, we can see that for the obits whose peri-gee altitude is below 600 km, errors decrease as perigee alti-tude increases which means that the effects of atmosphericdrag are larger than other perturbation factors for orbitsbelow 600 km. For the orbits with perigee altitude below600 km and eccentricity less than 0.2, errors with eccentric-ity less than 0.02 are larger than the orbits with eccentricitybetween 0.02 and 0.2 which is because atmospheric dragaffects the orbits with eccentricity less than 0.02 more thanthe orbits with eccentricity between 0.02 and 0.2.

Uncertainty in position prediction is usually describedby a trivariate normal distribution. Here position errorsare still divided into three components: downrange, normaland conormal errors. Here, statistical analysis results aregiven only for the orbits with 300–400 km perigee altitude.Table 2 gives the proportion relationship among down-range, normal and conormal errors in different predictionperiod. To simplify the computation, the prediction periodshave been selected to be multiples of a full day.

Table 1Position errors for one-day prediction duration whose perigees altitude arebelow 1000 km.

Perigee altitude (km) e0.00–0.02 e0.02–0.2 e0.2–0.7 e > 0.7

200–300 26.97 5.38 6.01 8.35300–400 3.50 2.79 4.82 3.67400–500 2.34 1.15 4.28 3.89500–600 1.04 0.58 3.60 3.08600–700 0.37 0.42 3.84 3.43700–800 0.31 0.51 5.81 5.09800–900 0.34 0.34 5.80 4.02900–1000 0.20 0.58 6.34 3.58

From Table 2, we can verify that for the orbits with 300–400 km perigee altitude, the downrange errors are muchlarger than other errors. Fig. 1 shows the downrange errorsof different prediction time for orbits whose perigee altitudeis 300–400 km.

From Fig. 1, we can get that errors for orbits whoseperigee altitude is 300–400 km increase with the predictionduration and the downrange errors are almost a quadraticgrowth with prediction time. Orbits with different eccen-tricity have different downrange errors, which is mainlybecause of the atmosphere drag effects to the whole orbits.For example, for the orbits with the eccentricity less than0.02, the whole orbits are affected by the atmosphere drag;for the orbits with the eccentric orbits, the atmosphere dragaffects the orbits perigee more. The downrange errorshould be (to first order) approximately proportional tothe angular velocity, and the error will be large near perigeeand smaller near apogee, causing the uncertainty ellipse to‘‘breathe” as the satellite executes its orbit.

4. Typical orbit errors analysis

There are some typical lower-altitude orbits that areoften used. For example, the Sun synchronous orbit(SSO) is a typical orbit of the Earth observation missions.

Fig. 1. Downrange errors of different prediction time for orbits with 300–400 km perigee altitude.

R. Wang et al. / Advances in Space Research 43 (2009) 1065–1069 1067

Many special-use satellites use this kind of orbit. It is a nearcircular orbit with near 90� inclination and 400–1500 kmaltitude range.

Fig. 2 shows the inclination and perigee altitude distri-bution, based on satellite situation report on 15 May,2006. From Fig. 2, perigee altitude and inclination focuson two districts: one is the orbit with 400–1200 km perigeealtitude and 97–101� inclination; the other is the orbit with1200–1500 km perigee altitude and 100–103� inclination.

Table 3 gives position errors for the orbits with 400–1200 km perigee altitude, 97–101� inclination. N is thesample number, which is the number of TLE sets forall objects in that class. Blank entries mean no enoughsamples in our database. From Table 3, we can knowthat most of objects have eccentricities smaller than0.2. For orbits with eccentricity less than 0.02, errorsare the largest for the orbits with 400–500 km perigeealtitude which reach 3 km with 1-day prediction durationand decrease to 1 km for orbits with 500–600 km perigeealtitude. The propagation errors are small for orbits withgreater than 600 km perigee altitude which show no moredecrease with increasing altitude. That means that atmo-spheric drag effects are much smaller for orbits above

Fig. 2. Perigee altitude and inclination relationship based on SatelliteSituation Report on 15 May, 2006.

Table 3Position errors in the orbits with 400–1200 km perigee altitude, 97–101�inclination.

Perigee altitude(km)

N e0.00–0.02

N e0.02–0.2

N e0.2–0.7

400–500 2750 3.06 118 3.09 46 0.26500–600 7264 1.16 200 0.55 109 0.67600–700 11,979 0.43 1717 0.36700–800 12,428 0.31 1548 0.44 5 1.59800–900 9615 0.42 1363 0.59 49 8.33900–1000 3649 0.48 830 1.151000–1100 8797 0.45 4056 1.061100–1200 324 4.08400–1200 56,480 0.52 10,153 0.76 208 1.04

600 km altitude. For orbits with 0.02–0.2 eccentricity,errors are not large for the orbits with 500–900 km peri-gee altitude, but they are larger a bit for the orbits withabove 900 km perigee altitude, which should be causedby solar/lunar perturbations.

Table 4 gives position errors for the orbits with1200–1500 km perigee altitude, 100–103� inclination. N

is sample number. From Table 4, these orbits havesmall eccentricities, most of which are near circularorbits. Errors are decreased by the perigee altitudeincrease.

5. Atmospheric drag effect to propagation errors and BSTAR

modification

From the above analysis, we know that downrangeerrors are mostly due to atmospheric drag effects inlower-altitude orbits. Atmospheric drag is a kind of non-gravitational effect which is proportional to the area-to-mass ratio of a space object, so satellite attitude has to beconsidered. Usually, perturbation quantity of atmosphericdrag is no more than 10�6 for the satellites whose area-to-mass ratio is not large compared to the Earth gravitation.

To determine atmospheric drag, SGP4/SDP4 uses a sta-tic method to approximate the effect of atmospheric dragon satellite orbits. It models the density of the Earth’supper atmosphere using the fourth power of the orbitalaltitude. The drag coefficient (XNDT20) provided in TLEdata is determined empirically as a continually updatedfit to the changes in revolution per day observed over thelong term. Thus, the ‘‘XNDT20” drag is the first timederivative of the mean motion (which has the dimensionrevolutions per day).

For SGP4 and SDP4 users, the mean motion is firstrecovered from its altered form and the drag effect isobtained from the SGP4 drag term. BSTAR is directlyrelated to the drag term. Since the SGP4/SDP4 modelassumes a static atmospheric model and a definition ofperigee height that does not depend on the actual argumentof perigee, whenever the real drag term changes, theBSTAR term changes by the same factor.

Atmospheric drag is an important factor of positionerrors for objects in lower-altitude orbit .The downrangeerrors are considered here which atmosphere drag affectsmost. BSTAR is a pivotal parameter which representsatmospheric drag in TLE data. Fig. 3 shows the down-range errors and BSTAR changes of 28872 by epoch.

Table 4Position errors in the orbits with 1200–1500 km perigee altitude, 100–103�inclination.

Perigee altitude (km) N e0.00–0.02 N e0.02–0.2

1200–1300 1755 1.321155 784 3.3897231300–1400 2900 0.401876 825 3.5970791400–1500 8171 0.075154 2034 0.155771200–1500 12,825 0.465894 3643 2.230986

Fig. 3. Comparison with downrange errors and BSTAR of object 28872.

1068 R. Wang et al. / Advances in Space Research 43 (2009) 1065–1069

From Fig. 3, the trends of BSTAR and downrangeerrors are coincident which means that BSTAR plays animportant role in the downrange errors.

BSTAR in TLEs can be obtained by fitting the orbits inorbit determination over many observations. But it isinvariant in prediction process. Errors are small for theshort prediction duration, while large for the long predic-tion duration. So modification of BSTAR is necessary forlong-term prediction.

Fig. 4. Computed and published value co

Sometimes, the BSTAR term is missed in TLE data. Wecan estimate it with published TLE data based on the meanmotion XN0 and the first time derivative of the meanmotion XNDT20 by

XNDT20 ¼ DXNO

DT; ð1Þ

BSTAR ¼ XNDT20=XNO=C2=1:5; ð2Þ

mparison of XNDT20 and BSTAR.

Table 5TLE data of object 28872.

Epoch Two-line element

Epoch1

1 28872U 05037B 05278.91558276 + .00105847 + 57326-5 + 32753-3 0 002372 28872 096.3531 106.3601 0013042 171.3686 188.793215.91513139002036

Epoch2

1 28872U 05037B 05282.24715354 .00165557 11748-4 48963-3 02462 28872 096.3471 109.4901 0012136 155.6360 204.549815.92455058 2561

R. Wang et al. / Advances in Space Research 43 (2009) 1065–1069 1069

where XN0 and XNDT20 have been converted to units ofradians and minutes, as in SGP4 and C2 is the value com-puted during SGP4 initialization.

To verify the formula, historical TLE data of 28872 areused. Fig. 4 is computed and published value comparisonof XNDT20 and BSTAR. From Fig. 4, computed and pub-lished value of drag term XNDT20 and BSTAR have goodcoherence. So the above formulas are believable. The com-puted BSTAR is obtained by computing XNDT20.TheBSTAR trend can be known from historical BSTAR datain the prediction duration.

We may get the mean BSTAR in the prediction period by themeans of the feed-forward back propagation neural networks(BP neural network) using the historical TLE data. By inputtingthe historical the Bstar value and the corresponding epoch whichwere included in the TLE data, the output is the Bstar value atthe given prediction epoch. If the TLE data does not includetheBSTARterm,wecanestimate itbyEq. (2),usingtheBSTARmean as the input to propagate the orbit. Object 28872 is ana-lyzed as an example. Table 5 is the chosen data. Data in twoepochs are chosen, one epoch (epoch 1) is 05278.91558276 and

the other (epoch 2) is 0582.24715354. Prediction will be givenfrom epoch 1 to epoch 2.

From Table 5, BSTAR1 is 0.32753e�3 in epoch 1 andBSTAR2 is 0.48963e�3 in epoch 2. BSTAR1 in fact canonly represent the drag term before epoch 1.To predict theorbit of object 28872 in epoch 2, we need to find the meanBSTAR from epoch 1 to epoch 2. With the neural networkmeans, we fit the BSTAR before epoch 1 according to thehistorical data, and then predict the BSTAR from epoch 1to epoch 2 to get the mean BSTAR 0.40753e�3. The resultsare that the downrange error is �110.49268 km with the ori-ginal BSTAR and 13.36340 km with the modified meanBSTAR. So after BSTAR modification, orbit predictionprecision becomes better.

6. Conclusion

Errors analysis is important for collision avoidanceproblem. Errors for a single object can be obtained directlyif data is enough. But in some circumstances space objectdata are not enough to get statistical error results. In thepaper, errors analysis of classified orbits and typical orbitcorrelated with SSO is obtained by using SGP4/SDP4model and TLE data. Such results can be directly used forthe intersection relationship uncertainty analysis of orbitalobjects. In this paper we also discuss BSTAR in TLE dataand try to modify it. Results show that after BSTAR mod-ification, orbit prediction precision becomes better.

Reference

Matney, M.J., Anz-Meador, P., Foster, J.L. Covariance correlation incollision avoidance probability calculations. Adv. Space Res. 34, 1109–1114, 2004.