Accuracy of Two-Line-Element Data forGeostationary and High-Eccentricity Orbits

Abstract

T HE USSTRATCOM catalog is the only publicly available orbital element catalog of resident space objects in near Earth orbits. USSTRATCOM is one of ten unified commands of the U.S. Department of Defense. Through its Joint Space Operations Center, it operates a space surveillance network of 29 sensors [1]. The orbits, using the observations of the SurveillanceNetwork, are determined with different models [2,3]: The development of the simplified general perturbation (SGP) model for orbit determination and propagation started in the 1960s and became operational in 1970 in the Space Detection and Tracking System Center, located in Colorado Springs, Colorado. SGP is based on the two different astrodynamic solutions for the equations of motion of near Earth satellites [4–6]. A first enhancement was performed in implementing an analytical rather than an empirical drag model [7]. These model is known as SGP4. It replaced SGP as the sole model for the U.S. satellite catalog maintenance since 1979. In 1977 an extension of the model was implemented for so-called deep space modeling (SDP4) in the existing SGP4 routines [8]. In the 1980s, deficiencies in the reentry prediction of decaying objects of the SGP4/SDP4 models were mitigated, the further development led to the SGP8/SDP8 models. The SGP4/SDP4 models are, however, still used without exception for the generation of publicly available data of USSTRATCOM. The mathematical foundation of the SGP4/SDP4 model and the equations are published in Hoots et al. [2]. The catalog data of USSTRATCOM are given in the two-lineelement (TLE) format. The TLE format is a fixed format, originally developed for punch cards. For every entry a fixed number of columns is reserved, including decimal points. Among other entries, the orbital elements and the mean motion are displayed in the two lines as well as the so-called Bstar parameter. Bstar is a draglike coefficient; but as a free modeling parameter, it can take negative values. Because for every entry a fixed space is available, the accuracy of the TLE data is limited not only by the observations in the Space Surveillance Network, or the orbit determination, but also by the sheer number of decimal digits available in each field [9]. With eight decimal places the accuracy of the epoch is accurate only up to 0.0004 s. An object in a circular low Earth orbit (LEO) at an altitude of 400 kmhas a velocity of 7:6 km=s, it thereforemoves by about 3m in 0.0004 s. An object in a perfectly geostationary orbit (GEO) has a velocity of about 3:07 km=s, it movesmore than 1m in 0.0004 s. The eccentricity e is specified by sevendecimal places. This introduces an error of the order of e a e, corresponding to twometers for a GEO object with semimajor axis a. The inclination and right ascension of ascending node are only accurate to four decimal places; with a simple estimation of the semimajor axis times the inclination angle, errors of 6 m in LEO and of approximately 35 m in GEO can be estimated. Intrinsic errors of the USSTRATCOM catalog can be determined, when only the catalog itselfwithout an external source of information is available. In [10], e.g., intrinsic errors of USSTRATCOM have been determined in generating position snapshots of the USSTRATCOM catalog at equally spaced epochs over 24 h. State vectors were generated at these epochs. These 24 h state vectors were used to determine an orbit. The residuals of the orbit to the TLE state vectors, which were used for the orbit determination, are a measure for the theoretically achievable intrinsic precision of the catalog. Along-track, cross-track and radial direction distances of 0.356, 0.432, and 0.086 km were found for GEO objects, of 0.824, 1.367, and 1.056 km for objects in high-eccentricity orbits (HEO). Intrinsic errors were determined to be 0.102, 0.471, and 0.126 km in alongtrack, cross-track, and radial directions for LEOobjects. The intrinsic accuracy of the HEO TLE data is reduced compared to objects in GEO. To assess the external errors of the catalog data, an additional external source of information, independent of the catalog itself, has to be available. High-precision ephemerides are available for the GNSS satellites. Only for very few operational satellites is highaccuracy operator data available. USSTRATCOMTLE data sets, for example, have been compared to the high-precision operator data of the LEO satellites ALOS and ASTRO-F of the Japanese Space Agency [11]. The distances between the predicted TLE positions and the operator datawere of the order of 2 kmon average. Also so-called Received 3 August 2011; revision received 1 March 2012; accepted for publication 2 March 2012. Copyright © 2012 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0731-5090/12 and $10.00 in correspondence with the CCC. ∗Research Astrodynamicist, Astronomical Institute, Sidlerstrasse 5; frueh@aiub.unibe.ch Professor, Astronomical Institute, Sidlerstrasse 5; thomas.schildknecht@aiub.unibe.ch JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS Vol. 35, No. 5, September–October 2012