Two-line elements

Orbital parameter for the SGP4 model (or one of the SGP8, SDP4, SDP8 models) are determined for many thousands of space objects by NORAD and are freely distributed on the Internet in the form of TLEs by Celestrak (http://celestrak.com/). Here the use of these TLE is described in detail

Description of a TLE

The following is an example of a TLE (for the International Space Station)

ISS (ZARYA) 1 25544U 98067A 08264.51782528 -.00002182 00000-0 -11606-4 0 2927 2 25544 51.6416 247.4627 0006703 130.5360 325.0288 15.72125391563537

The meaning of this data is as follows:

LINE 1: FIELD COLS CONTENT EXAMPLE 1 01-01 Line number 1 2 03-07 Satellite number 25544 3 08-08 Classification (U=Unclassified) U 4 10-11 International Designator (Last two digits of launch year) 98 5 12-14 International Designator (Launch number of the year) 067 6 15-17 International Designator (Piece of the launch) A 7 19-20 Epoch Year (Last two digits of year) 08 8 21-32 Epoch (Day of the year and fractional portion of the day) 264.51782528 9 34-43 First Time Derivative of the Mean Motion -.00002182 10 45-52 Second Time Derivative of Mean Motion (decimal point assumed) 00000-0 11 54-61 BSTAR drag term (decimal point assumed) -11606-4 12 63-63 The number 0 (Originally this should have been "Ephemeris type") 0 13 65-68 Element number 292 14 69-69 Checksum (Modulo 10) 7

LINE 2: FIELD COLS CONTENT EXAMPLE 1 01-01 Line number 2 2 03-07 Satellite number 25544 3 09-16 Inclination [Degrees] 51.6416 4 18-25 Right Ascension of the Ascending Node [Degrees] 247.4627 5 27-33 Eccentricity (decimal point assumed) 0006703 6 35-42 Argument of Perigee [Degrees] 130.5360 7 44-51 Mean Anomaly [Degrees] 325.0288 8 53-63 Mean Motion [Revs per day] 15.72125391 9 64-68 Revolution number at epoch [Revs] 56353 10 69-69 Checksum (Modulo 10) 7

How a TLE should be used

The "First Time Derivative of the Mean Motion" and the "Second Time Derivative of Mean Motion" are not used for the SGP4 model.

The parameters "Inclination", "Right Ascension of the Ascending Node", "Eccentricity", "Argument of Perigee" and "Mean Anomaly" are not osculating Kepler elements but a special case of "mean elements" to be used with the SGP4 model as given in the http://celestrak.com/NORAD/documentation/spacetrk.pdf.

The original FORTRAN code for the SGP4 algorithm is as follows:

SUBROUTINE SGP4(IFLAG,TSINCE) COMMON/E1/XMO,XNODEO,OMEGAO,EO,XINCL,XNO, 1 BSTAR,X,Y,Z,XDOT,YDOT,ZDOT COMMON/C1/CK2,CK4,E6A,QOMS2T,S,TOTHRD,XJ3,XKE,XKMPER,AE C C THE FOLLOWING SAVE STATEMENTS ADDED TO THE ORIGINAL CODE C SOME FORTRAN SYSTEM DO NOT KEEP THE VALUES OTHERWISE! C SAVE COSIO,X3THM1,XNODP,AODP,ISIMP,ETA,C1,SINIO,X1MTH2,C4,C5 SAVE XMDOT,OMGDOT,XNODOT,OMGCOF,XMCOF,XNODCF,T2COF,XLCOF,AYCOF SAVE DELMO,SINMO,X7THM1,D2,D3,D4,T3COF,T4COF,T5COF IF (IFLAG .EQ. 0) GO TO 100 C RECOVER ORIGINAL MEAN MOTION (XNODP) AND SEMIMAJOR AXIS (AODP) C FROM INPUT ELEMENTS A1=(XKE/XNO)**TOTHRD COSIO=COS(XINCL) THETA2=COSIO*COSIO X3THM1=3.*THETA2-1. EOSQ=EO*EO BETAO2=1.-EOSQ BETAO=SQRT(BETAO2) DEL1=1.5*CK2*X3THM1/(A1*A1*BETAO*BETAO2) AO=A1*(1.-DEL1*(.5*TOTHRD+DEL1*(1.+134./81.*DEL1))) DELO=1.5*CK2*X3THM1/(AO*AO*BETAO*BETAO2) XNODP=XNO/(1.+DELO) AODP=AO/(1.-DELO) C INITIALIZATION C FOR PERIGEE LESS THAN 220 KILOMETERS, THE ISIMP FLAG IS SET AND C THE EQUATIONS ARE TRUNCATED TO LINEAR VARIATION IN SQRT A AND C QUADRATIC VARIATION IN MEAN ANOMALY. ALSO, THE C3 TERM, THE C DELTA OMEGA TERM, AND THE DELTA M TERM ARE DROPPED. ISIMP=0 IF((AODP*(1.-EO)/AE) .LT. (220./XKMPER+AE)) ISIMP=1 C FOR PERIGEE BELOW 156 KM, THE VALUES OF C S AND QOMS2T ARE ALTERED S4=S QOMS24=QOMS2T PERIGE=(AODP*(1.-EO)-AE)*XKMPER IF(PERIGE .GE. 156.) GO TO 10 S4=PERIGE-78. IF(PERIGE .GT. 98.) GO TO 9 S4=20. 9 QOMS24=((120.-S4)*AE/XKMPER)**4 S4=S4/XKMPER+AE 10 PINVSQ=1./(AODP*AODP*BETAO2*BETAO2) TSI=1./(AODP-S4) ETA=AODP*EO*TSI ETASQ=ETA*ETA EETA=EO*ETA PSISQ=ABS(1.-ETASQ) COEF=QOMS24*TSI**4 COEF1=COEF/PSISQ**3.5 C2=COEF1*XNODP*(AODP*(1.+1.5*ETASQ+EETA*(4.+ETASQ))+.75* 1 CK2*TSI/PSISQ*X3THM1*(8.+3.*ETASQ*(8.+ETASQ))) C1=BSTAR*C2 SINIO=SIN(XINCL) A3OVK2=-XJ3/CK2*AE**3 C3=COEF*TSI*A3OVK2*XNODP*AE*SINIO/EO X1MTH2=1.-THETA2 C4=2.*XNODP*COEF1*AODP*BETAO2*(ETA* 1 (2.+.5*ETASQ)+EO*(.5+2.*ETASQ)-2.*CK2*TSI/ 2 (AODP*PSISQ)*(-3.*X3THM1*(1.-2.*EETA+ETASQ* 3 (1.5-.5*EETA))+.75*X1MTH2*(2.*ETASQ-EETA* 4 (1.+ETASQ))*COS(2.*OMEGAO))) C5=2.*COEF1*AODP*BETAO2*(1.+2.75*(ETASQ+EETA)+EETA*ETASQ) THETA4=THETA2*THETA2 TEMP1=3.*CK2*PINVSQ*XNODP TEMP2=TEMP1*CK2*PINVSQ TEMP3=1.25*CK4*PINVSQ*PINVSQ*XNODP XMDOT=XNODP+.5*TEMP1*BETAO*X3THM1+.0625*TEMP2*BETAO* 1 (13.-78.*THETA2+137.*THETA4) X1M5TH=1.-5.*THETA2 OMGDOT=-.5*TEMP1*X1M5TH+.0625*TEMP2*(7.-114.*THETA2+ 1 395.*THETA4)+TEMP3*(3.-36.*THETA2+49.*THETA4) XHDOT1=-TEMP1*COSIO XNODOT=XHDOT1+(.5*TEMP2*(4.-19.*THETA2)+2.*TEMP3*(3.- 1 7.*THETA2))*COSIO OMGCOF=BSTAR*C3*COS(OMEGAO) XMCOF=-TOTHRD*COEF*BSTAR*AE/EETA XNODCF=3.5*BETAO2*XHDOT1*C1 T2COF=1.5*C1 XLCOF=.125*A3OVK2*SINIO*(3.+5.*COSIO)/(1.+COSIO) AYCOF=.25*A3OVK2*SINIO DELMO=(1.+ETA*COS(XMO))**3 SINMO=SIN(XMO) X7THM1=7.*THETA2-1. IF(ISIMP .EQ. 1) GO TO 90 C1SQ=C1*C1 D2=4.*AODP*TSI*C1SQ TEMP=D2*TSI*C1/3. D3=(17.*AODP+S4)*TEMP D4=.5*TEMP*AODP*TSI*(221.*AODP+31.*S4)*C1 T3COF=D2+2.*C1SQ T4COF=.25*(3.*D3+C1*(12.*D2+10.*C1SQ)) T5COF=.2*(3.*D4+12.*C1*D3+6.*D2*D2+15.*C1SQ*( 1 2.*D2+C1SQ)) 90 I

Using this original code for the SGP4 algorithm the variables in COMMON/E1/ shall be given the following values:

BSTAR shall be given the value read from line 1 cols 54-61 (example: -11606-4 means -0.11606E-4)

XINCL=FINC*6.283185/360. where FINC is the value from line 2 cols 9-16. XINCL has the dimension radians and FINC the dimension degrees.

XNODEO=GOM*6.283185/360. where GOM is the value from line 2 cols 18-25. XNODEO has the dimensionradians and GOM the dimension degrees.

EO is the value from line 2 cols 27-33 (example: 0006703 means 0.0006703).

OMEGAO=SOM*6.283185/360. where SOM is the value from line 2 cols 35-42. OMEGAO has the dimension radians and SOM the dimension degrees.

XMO=FMA*6.283185/360. where FMA is the value from line 2 cols 44-51. XMO has the dimension radians and FMA the dimension degrees.

XNO=FMM*6.283185/1440. where FMM is the value from line 2 cols 53-63. XNO has the dimension radians/minute and FMM the dimension revolutions/day.

COMMON/C1/ shall be given fixed constants that do not depend on the TLE:

CK2 = 5.413080E-4 CK4 = 0.62098875E-6 E6A = 1E-6 QOMS2T = 1.88027916E-9 S = 1.01222928 TOTHRD = 2./3. XJ3 = -0.253881E-5 XKE = 0.743669161E-1 XKMPER = 6378.135 AE = 1.

The position and velocity in km and km/s (relative the "True of Date" coordinate system) at a certain time are obtained from the parameters X,Y,Z,XDOT,YDOT,ZDOT of COMMON/E1/ as XKMPER*X XKMPER*Y XKMPER*Z XKMPER*XDOT/60. XKMPER*YDOT/60. XKMPER*ZDOT/60.

after having made a call

CALL SGP4(IFLAG,TSINCE)

where TSINCE is a REAL*4 variable giving the time in minutes relative the time specified on line 1 cols 19-32.

The INTEGER*4 variable IFLAG shall have another value then 0 for the first call of SGP4 and will be 0 after the call. For subsequent calls with unchanged values for XMO,XNODEO,OMEGAO,EO,XINCL,XNO,BSTAR (i.e no new TLE having been read) the variable IFLAG can be kept 0 resulting in a faster execution.

The accuracy that can be obtained with the SGP4 model

For a spacecraft in a typical Low Earth orbit the accuracy that can be obtaine with the SGP4 orbit model is in the order of 1 km.