* James Miller G3RUH
*

In time the program accumulated several other modules, with the mathematics described in interim O.N. articles. When issued for the BBC micro the program acquired the name "PLAN10", and later "PLAN13", and it's been transcribed for several other machines including the IBM-PC and Commodore 64 etc.

In recent years or so my mail has featured more and more queries about tracking matters than was usual. The flavour of these suggests that a new generation of satellite enthusiasts is emerging and are rightly asking the old questions anew.

So for a long while I've been intending to re-publish these routines, but in their most up-to-date embodiment. Hence this pamphlet. I don't intend to provide 100% mathematical details. To do so means essentially writing the whole program out anyway, and references are plentiful.

Old hands will recognise the coding style. Additions include the Sun's position, eclipses, visibility, Sun angle, sub-satellite point, mode switching, date algorithms and direct range-rate calculations for doppler shift.

Almost every publication to do with satellites has information about orbits and their properties. For general reading *The Satellite Experimenter's Handbook* [2] is unsurpassed, and itself documents several hundred references. *Fundamentals of Astrodynamics* [3] is a no-nonsense text that's full of insights, and is incredibly cheap.

Models for the Propagation of NORAD Element Sets [4] documents five ways to use a set of keps properly. SGP, the simplest model, runs to 4 pages of higher mathematics just to compute satellite position and velocity. All amateur implementations (i.e. lines 2000-2210 of PROCsatvec) use only the first half page of this, skipping the long term perturbatory effects. Sun's Up [5] is about modelling the Sun's position, and describes applications including illumination and eclipses. 30.6 days Hath September [6] documents date algorithms. *The Astronomical Almanac* [7] is invaluable for checking calculations; an annual, but useful for many years.

Surprisingly when OSCAR10.BAS was published there were complaints that "filling Oscar News with programs was a waste of space" etc etc. My reply to this myopic view was, and remains:

*"The program was designed to be more than a mere listing to be blindly typed in and run. It is a treatise giving all the equations for solving the elliptic orbit and related problems. To do this, as well as provide a tool suitable for immediate use the program was written in a quasi-algebraic style. Variable names were chosen to be meaningful, there are copious comments, and routines are written so that users can adapt them as they wish. However the published arrangement is not the only one. The program is a clear formulation of the general satellite orbit problem, and is sure to provide the interested amateur with several months of absorbing self-training material".* Plunder! Enjoy!

2. Davidoff M.R., *Satellite Experimenter's Handbook*, 2nd edition, ARRL 1990. ISBN 0-87259-004-6.

3. Bate R., Mueller D. & White J., *Fundamentals of Astrodynamics*, Dover 1971. ISBN 0-486-60061-0.

4. Hoots F.R. & Roerich R.L., *Models for Propagation of NORAD Element Sets*, USAF Spacetrack Report No.3, 1988 December.

5. Miller J.R., *Sun's Up* Part 1, Oscar News, 1984 Dec, No.50 p.2-6; Part 2, 1985 Feb, No.51 p.2-6; "Part 3", 1985 Apr, No.52 p.8-11.

6. Miller J.R., *30.6 days hath September*, Oscar News, 1985 Dec, No.56 p.6-9.

7. *The Astronomical Almanac 1984*, Section C, TSO and USGPO. ISBN 0-11-886919-1

PLAN13 v2.0 SATELLITE PREDICTIONS ----------------------------------- Enter start date (e.g. 1990, 12, 25) ? 1990,11,3 Enter number of days for printout ? 1 OSCAR-13 - G3RUH AMSAT DAY 4689 1990 Nov 3 [Sat] ORBIT: 1828 AP/RAAN: 239/128 ALON/ALAT: 180/0 SAZ/SEL: 210/-72 ILL: 96% UTC MA MODE RANGE EL AZ SQ RR ECL? HGT SLAT SLON ----------------------------------------------------------------------------- 0100 38 B 25929 3 89 48 2.1 vis 20635 13 80 0115 43 B 27716 8 87 43 1.9 vis 22880 17 79 0130 49 B 29345 12 86 38 1.7 vis 24917 21 78 0145 55 B 30825 16 85 34 1.6 vis 26763 24 77 0200 60 B 32160 20 84 30 1.4 vis 28432 26 75 etc

PROCinit (lines 1000-1840): All the constants needed by the routines are established here. Most relate to a physical quantity, but are then combined with others or need units changing to form a derived quantity. Some of these constants are only used transiently within "init" itself, but are included for clarity.

PROCsatvec (lines 2000-2430): Takes input variables day (DN) and fraction of day (TN) and computes satellite position, velocity and antenna direction in orbit plane coordinates. These are then output in celestial coordinates and geocentric coordinates.

PROCrangevec (lines 4000-4200): Calculates Range vector from satellite and observer vector, and derives range, azimuth, elevation, squint, range rate and sub-satellite point.

PROCsunvec (lines 3000-3260): Computes Sun's position in celestial coordinates, and then Sun angle, illumination, visibility and eclipse details. Actual visibility also depends on brightness of satellite, and darkness of sky.

FNday(Y,M,D) Function returns a general day number from year, month and day. Value is Julian Day - 1721409.5 or Amsat day + 722100

FNdate(DN) Function converts day number to a string, e.g. 1990 Nov 3 [Sat]

FNatn(Y,X) Crash-proof four quadrant arctangent. Returns value in range 0-2pi

BBC BASIC Effect OTHER BASICS --------------------------------------------------------------------- PROCname .... ENDPROC subroutine GOSUB nnnn .... RETURN DEG(X) rad -> deg X*180/PI ) where PI = RAD(X) deg -> rad X*PI/180 ) 3.141592654 ASN(X) arcsin FNatn(X,SQR(1-X*X)) ACS(X) arccos PI/2 - FNatn(X,SQR(1-X*X)) ---------------------------------------------------------------------

Origin: centre of the Earth.

X: Directed towards the First point in Aries (Right Ascension = 0)

Y: 90 degrees East around the equator (RA = 90)

Z: Directed towards the North pole

Origin: centre of the Earth.

X: Directed to intersection of Equator and Greenwich meridian (Longitude = 0)

Y: 90 degrees East around the Earth's Equator (Longitude = 90E)

Z: Directed towards the North pole

Origin: Earth's centre.

X: Directed towards perigee

Y: 90 degrees from perigee round the orbit in direction of satellite's motion

Z: Perpendicular to orbit plane

10 T$="PLAN13": REM OSCAR-13 POSITION, SUN + ECLIPSE PLANNER 20 REM 30 IS$="v2.0": REM Last modified 1990 Aug 12 by JRM 40 REM 50 REM (C)1990 J.R. Miller G3RUH 60 REM 70 REM Proceeds from the sale of this software go directly to the 80 REM Amateur Satellite Programme that helped fund AO-13. 90 REM If you take a copy PLEASE also send a small donation to: 100 REM AMSAT-UK, LONDON, E12 5EQ. 110 120 MODE 3: REM Screen 80 columns 130 PROCinit: REM Set up constants 140 150 INPUT "Enter start date (e.g. 1990, 12, 25) ";YR,MN,DY 160 INPUT "Enter number of days for printout ";ND 170 DS = FNday(YR,MN,DY): REM Start day No. 180 DF = DS + ND - 1: REM Finish day No. 190 FOR DN = DS TO DF 200 FOR HR = 0 TO 23 210 FOR MIN = 0 TO 45 STEP 15 220 TN = (HR+MIN/60)/24 230 PROCsatvec 240 PROCrangevec 250 PROCsunvec 260 IF EL > 0 THEN PROCprintdata 270 NEXT:NEXT:NEXT 280 END 1000 DEF PROCinit 1010 REM SATELLITE EPHEMERIS 1020 REM ------------------- 1030 SAT$="OSCAR-13" 1040 YE = 1990 : REM Epoch Year year 1050 TE = 191.145409: REM Epoch time days 1060 IN = 56.9975 : REM Inclination deg 1070 RA = 146.4527 : REM R.A.A.N. deg 1080 EC = 0.6986 : REM Eccentricity - 1090 WP = 231.0027 : REM Arg perigee deg 1100 MA = 43.2637 : REM Mean anomaly deg 1110 MM = 2.09695848: REM Mean motion rev/d 1120 M2 = 1E-8 : REM Decay Rate rev/d/d 1130 RV = 1585 : REM Orbit number - 1140 ALON = 180 : REM Sat attitude, deg. 180 = nominal ) See bulletins 1150 ALAT = 0 : REM Sat attitude, deg. 0 = nominal ) for latest 1160 1170 REM Observer's location + North, + East, ASL(m) 1180 LOC$="G3RUH": LA = 52.21: LO = 0.06: HT = 79: REM Cambridge, UK 1190 1200 LA = RAD(LA): LO = RAD(LO): HT = HT/1000 1210 CL = COS(LA): SL = SIN(LA): CO = COS(LO): SO = SIN(LO) 1220 RE = 6378.137: FL = 1/298.257224: REM WGS-84 Earth ellipsoid 1230 RP = RE*(1-FL): XX = RE*RE: ZZ = RP*RP 1240 D = SQR(XX*CL*CL + ZZ*SL*SL) 1250 Rx = XX/D + HT: Rz = ZZ/D + HT 1260 1270 REM Observer's unit vectors UP EAST and NORTH in GEOCENTRIC coords. 1280 Ux =CL*CO: Ex =-SO: Nx =-SL*CO 1290 Uy =CL*SO: Ey = CO: Ny =-SL*SO 1300 Uz =SL : Ez = 0: Nz = CL 1310 1320 REM Observer's XYZ coords at Earth's surface 1330 Ox = Rx*Ux: Oy = Rx*Uy: Oz = Rz*Uz 1340 1350 REM Convert angles to radians etc. 1360 RA = RAD(RA): IN = RAD(IN): WP = RAD(WP) 1370 MA = RAD(MA): MM = MM*2*PI: M2 = M2*2*PI 1380 1390 YM = 365.25: REM Mean Year, days 1400 YT = 365.2421874: REM Tropical year, days 1410 WW = 2*PI/YT: REM Earth's rotation rate, rads/whole day 1420 WE = 2*PI + WW: REM ditto radians/day 1430 W0 = WE/86400: REM ditto radians/sec 1440 1450 VOx=-Oy*W0: VOy=Ox*W0: REM Observer's velocity, GEOCENTRIC coords. (VOz=0) 1460 1470 REM Convert satellite Epoch to Day No. and Fraction of day 1480 DE = FNday(YE,1,0)+INT(TE): TE = TE-INT(TE) 1490 1500 REM Average Precession rates 1510 GM = 3.986E5: REM Earth's Gravitational constant km^3/s^2 1520 J2 = 1.08263E-3: REM 2nd Zonal coeff, Earth's Gravity Field 1530 N0 = MM/86400: REM Mean motion rad/s 1540 A0 = (GM/N0/N0)^(1/3): REM Semi major axis km 1550 B0 = A0*SQR(1-EC*EC): REM Semi minor axis km 1560 SI = SIN(IN): CI = COS(IN) 1570 PC = RE*A0/(B0*B0): PC = 1.5*J2*PC*PC*MM: REM Precession const, rad/Day 1580 QD = -PC*CI: REM Node precession rate, rad/day 1590 WD = PC*(5*CI*CI-1)/2: REM Perigee precession rate, rad/day 1600 DC = -2*M2/MM/3: REM Drag coeff. (Angular momentum rate)/(Ang mom) s^-1 1610 1615 REM *** Please see end of listing for newer values; use old ones for test. *** 1617 1620 REM Sidereal and Solar data. Rarely needs changing. Valid to year ~2015 1630 YG = 2000: G0 = 98.9821: REM GHAA, Year YG, Jan 0.0 1640 MAS0 = 356.0507: MASD = 0.98560028: REM MA Sun and rate, deg, deg/day 1650 INS = RAD(23.4393): CNS = COS(INS): SNS = SIN(INS): REM Sun's inclination 1660 EQC1=0.03342: EQC2=0.00035: REM Sun's Equation of centre terms 1670 1680 REM Bring Sun data to Satellite Epoch 1690 TEG = (DE-FNday(YG,1,0)) + TE: REM Elapsed Time: Epoch - YG 1700 GHAE = RAD(G0) + TEG*WE: REM GHA Aries, epoch 1710 MRSE = RAD(G0) + TEG*WW + PI: REM Mean RA Sun at Sat epoch 1720 MASE = RAD(MAS0 + MASD*TEG): REM Mean MA Sun .. 1730 1740 REM Antenna unit vector in orbit plane coordinates. 1750 CO=COS(RAD(ALON)): SO=SIN(RAD(ALON)) 1760 CL=COS(RAD(ALAT)): SL=SIN(RAD(ALAT)) 1770 ax = -CL*CO: ay = -CL*SO: az = -SL 1780 1790 REM Miscellaneous 1800 @%=&507: REM 5 decimals, field 7 1810 OLDRN=-99999 1820 PRINT T$;" ";IS$;" SATELLITE PREDICTIONS" 1830 PRINT STRING$(35,"-") 1840 ENDPROC 2000 DEF PROCsatvec 2010 REM Calculate Satellite Position at DN,TN 2020 T = (DN - DE) + (TN-TE):REM Elapsed T since epoch, days 2030 DT = DC*T/2: KD = 1+4*DT:KDP= 1-7*DT: REM Linear drag terms 2040 M = MA + MM*T*(1-3*DT): REM Mean anomaly at YR,TN 2050 DR = INT(M/(2*PI)): REM Strip out whole no of revs 2060 M = M - DR*2*PI: REM M now in range 0 - 2pi 2070 RN = RV + DR: REM Current Orbit number 2080 2090 REM Solve M = EA - EC*SIN(EA) for EA given M, by Newton's Method 2100 EA = M: REM Initial solution 2110 REPEAT 2120 C = COS(EA): S = SIN(EA): DNOM=1-EC*C 2130 D = (EA-EC*S-M)/DNOM: REM Change to EA for better solution 2140 EA = EA - D: REM by this amount 2150 UNTIL ABS(D) < 1E-5: REM Until converged 2160 2170 A = A0*KD: B = B0*KD: RS = A*DNOM: REM Distances 2180 2190 REM Calc satellite position & velocity in plane of ellipse 2200 Sx = A*(C-EC): Vx=-A*S/DNOM*N0 2210 Sy = B*S: Vy= B*C/DNOM*N0 2220 2230 AP = WP + WD*T*KDP: CW = COS(AP): SW = SIN(AP) 2240 RAAN = RA + QD*T*KDP: CQ = COS(RAAN): SQ = SIN(RAAN) 2250 2260 REM Plane -> celestial coordinate transformation, [C] = [RAAN]*[IN]*[AP] 2270 CXx=CW*CQ-SW*CI*SQ: CXy=-SW*CQ-CW*CI*SQ: CXz= SI*SQ 2280 CYx=CW*SQ+SW*CI*CQ: CYy=-SW*SQ+CW*CI*CQ: CYz=-SI*CQ 2290 CZx=SW*SI: CZy= CW*SI: CZz= CI 2300 2310 REM Compute SATellite's position vector, ANTenna axis unit vector 2320 REM and VELocity in CELESTIAL coordinates. (Note: Sz=0, Vz=0) 2330 SATx=Sx*CXx+Sy*CXy: ANTx=ax*CXx+ay*CXy+az*CXz: VELx=Vx*CXx+Vy*CXy 2340 SATy=Sx*CYx+Sy*CYy: ANTy=ax*CYx+ay*CYy+az*CYz: VELy=Vx*CYx+Vy*CYy 2350 SATz=Sx*CZx+Sy*CZy: ANTz=ax*CZx+ay*CZy+az*CZz: VELz=Vx*CZx+Vy*CZy 2360 2370 REM Also express SAT,ANT and VEL in GEOCENTRIC coordinates: 2380 GHAA = GHAE + WE*T: REM GHA Aries at elapsed time T 2390 C = COS(-GHAA): S = SIN(-GHAA) 2400 Sx=SATx*C - SATy*S: Ax=ANTx*C - ANTy*S: Vx=VELx*C - VELy*S 2410 Sy=SATx*S + SATy*C: Ay=ANTx*S + ANTy*C: Vy=VELx*S + VELy*C 2420 Sz=SATz: Az=ANTz: Vz=VELz 2430 ENDPROC 3000 DEF PROCsunvec 3010 MAS = MASE + RAD(MASD*T): REM MA of Sun round its orbit 3020 TAS = MRSE + WW*T + EQC1*SIN(MAS) + EQC2*SIN(2*MAS) 3030 C = COS(TAS): S=SIN(TAS): REM Sin/Cos Sun's true anomaly 3040 SUNx=C: SUNy=S*CNS: SUNz=S*SNS: REM Sun unit vector - CELESTIAL coords 3050 3060 REM Find Solar angle, illumination, and eclipse status. 3070 SSA = -(ANTx*SUNx + ANTy*SUNy + ANTz*SUNz):REM Sin of Sun angle -a.h 3080 ILL = SQR(1-SSA*SSA): REM Illumination 3090 CUA = -(SATx*SUNx+SATy*SUNy+SATz*SUNz)/RS: REM Cos of umbral angle -h.s 3100 UMD = RS*SQR(1-CUA*CUA)/RE: REM Umbral dist, Earth radii 3110 IF CUA>=0 THEN ECL$=" +" ELSE ECL$=" -": REM + for shadow side 3120 IF UMD <= 1 AND CUA>=0 THEN ECL$=" ECL": REM - for sunny side 3130 3140 REM Obtain SUN unit vector in GEOCENTRIC coordinates 3150 C = COS(-GHAA): S = SIN(-GHAA) 3160 Hx=SUNx*C - SUNy*S 3170 Hy=SUNx*S + SUNy*C: REM If Sun more than 10 deg below horizon 3180 Hz=SUNz: REM satellite possibly visible 3190 IF (Hx*Ux+Hy*Uy+Hz*Uz < -0.17) AND (ECL$ <> " ECL") THEN ECL$=" vis" 3200 3210 REM Obtain Sun unit vector in ORBIT coordinates 3220 Hx = SUNx*CXx + SUNy*CYx + SUNz*CZx 3230 Hy = SUNx*CXy + SUNy*CYy + SUNz*CZy 3240 Hz = SUNx*CXz + SUNy*CYz + SUNz*CZz 3250 SEL = ASN(Hz): SAZ= FNatn(Hy,Hx) 3260 ENDPROC 4000 DEF PROCrangevec 4010 REM Compute and manipulate range/velocity/antenna vectors 4020 Rx = Sx-Ox: Ry = Sy-Oy: Rz = Sz-Oz: REM Rangevec = Satvec - Obsvec 4030 R = SQR(Rx*Rx+Ry*Ry+Rz*Rz): REM Range magnitude 4040 Rx=Rx/R: Ry=Ry/R: Rz=Rz/R: REM Normalise Range vector 4050 U = Rx*Ux+Ry*Uy+Rz*Uz: REM UP Component of unit range 4060 E = Rx*Ex+Ry*Ey: REM EAST do (Ez=0) 4070 N = Rx*Nx+Ry*Ny+Rz*Nz: REM NORTH do 4080 AZ = DEG(FNatn(E,N)): REM Azimuth 4090 EL = DEG(ASN(U)): REM Elevation 4100 4110 REM Resolve antenna vector along unit range vector, -r.a = Cos(SQ) 4120 SQ = DEG(ACS(-(Ax*Rx + Ay*Ry + Az*Rz))): REM Hi-gain ant SQuint 4130 4140 REM Calculate sub-satellite Lat/Lon 4150 SLON = DEG(FNatn(Sy,Sx)): REM Lon, + East 4160 SLAT = DEG(ASN(Sz/RS)): REM Lat, + North 4170 4180 REM Resolve Sat-Obs velocity vector along unit range vector. (VOz=0) 4190 RR = (Vx-VOx)*Rx + (Vy-VOy)*Ry + Vz*Rz: REM Range rate, km/s 4200 ENDPROC 4220 DEF FNatn(Y,X) 4230 IF X <> 0 THEN A=ATN(Y/X) ELSE A=PI/2*SGN(Y) 4240 IF X < 0 THEN A=A+PI 4250 IF A < 0 THEN A=A+2*PI 4260 =A 4280 DEF PROCmode 4290 M=INT(M*128/PI) 4300 REM Mode switching MA/256 4310 MD$="-" 4320 IF M >= 0 THEN MD$="B" 4330 IF M >= 100 THEN MD$="L" 4340 IF M >= 130 THEN MD$="S" 4350 IF M >= 135 THEN MD$="B" 4360 IF M >= 220 THEN MD$="-" 4370 ENDPROC 5000 DEF FNdate(D) 5010 REM Convert day-number to date; valid 1900 Mar 01 - 2100 Feb 28 5020 D=D+428: DW=(D+5)MOD7 5030 Y=INT((D-122.1)/YM): D=D-INT(Y*YM) 5040 MN=INT(D/30.61): D=D-INT(MN*30.6) 5050 MN=MN-1: IF MN>12 THEN MN=MN-12: Y=Y+1 5060 D$=STR$(Y)+" "+MID$("JanFebMarAprMayJunJulAugSepOctNovDec",3*MN-2,3) 5070 =D$+" "+STR$(D)+" ["+MID$("SunMonTueWedThuFriSat",3*DW+1,3)+"]" 5080 5090 DEF FNday(Y,M,D) 5100 REM Convert date to day-number 5110 IF M<=2 THEN Y=Y-1: M=M+12 5120 =INT(Y*YM) + INT((M+1)*30.6) + D-428 6000 DEF PROCprintdata 6010 REM Construct time as a string 6020 HR$=STR$(HR): MIN$=STR$(MIN) 6030 IF LEN(HR$) < 2 THEN HR$="0"+HR$ 6040 IF LEN(MIN$) < 2 THEN MIN$="0"+MIN$ 6050 TIM$=HR$+MIN$+" " 6060 6070 PROCmode: REM Get AO-13 mode. Now round-off data 6080 R=FNrn(R): EL=FNrn(EL): AZ=FNrn(AZ): SQ=FNrn(SQ): RR=FNrn(RR*10)/10 6090 HGT=FNrn(RS-RE): SLON=FNrn(SLON): SLAT=FNrn(SLAT) 6100 IF RN <> OLDRN THEN OLDRN=RN: PROCheader 6110 PRINT TIM$;STR$(M);" ";MD$,R,EL,AZ,SQ,RR,ECL$,HGT,SLAT,SLON 6120 ENDPROC 6130 6140 DEF PROCheader 6150 RAAN=FNrn(DEG(RAAN)): AP=FNrn(DEG(AP)): SAZ=FNrn(DEG(SAZ)) 6160 SEL=FNrn(DEG(SEL)): ILL=FNrn(100*ILL) 6170 PRINT: PRINT 6180 PRINT SAT$;" - "LOC$;SPC(16);"AMSAT DAY ";STR$(DN-722100);SPC(12);FNdate(DN) 6190 PRINT "ORBIT: ";RN;" AP/RAAN: ";AP;"/";RAAN;" ALON/ALAT:";ALON;"/";ALAT; 6200 PRINT" SAZ/SEL: ";SAZ;"/";SEL;" ILL: ";ILL;"%" 6210 PRINT 6220 PRINT " UTC MA MODE RANGE EL AZ SQ RR ECL? "; 6230 PRINT "HGT SLAT SLON" 6240 PRINT STRING$(77,"-") 6250 ENDPROC 6260 6270 DEF FNrn(X) = INT(X+0.5) *** Astronomical data for 2023 onwards �1620�REM�Sidereal�and�Solar�data. �1630�YG�=�2023:�G0�=�99.4057:��REM�GHAA,�Year�YG,�Jan�0.0 �1640�MAS0�=�356.0787:�MASD�=�0.98560028:�REM�MA�Sun�and�rate,�deg,�deg/day �1650�INS�=�RAD(23.4363):�CNS�=�COS(INS):�SNS�=�SIN(INS):�REM�Sun's�inclination �1660�EQC1=0.03340:�EQC2=0.00035:���REM�Sun's�Equation�of�centre�terms ***

In earlier versions of the program, the conversion of a vector from Orbit plane coordinates to Celestial coordinates was written out stage by stage. That is, a rotation in argument of perigee, followed by a rotation through inclination, and finally through RAAN. Since there are actually several vectors (satellite position, velocity and antenna axis) to transform, this became a little clumsy, so the three rotations are now amalgamated into one. The array [C] that does this is computed at lines 2260-2290.

**Celestial -> Orbit Plane Coordinate Transformation**

The array [C] can also be used in the reverse direction, to convert a vector from Celestial to Orbit plane coordinates. This is done by "transposing the array", using C(J,I) for C(I,J) throughout. This is used to calculate the Sun's position as seen from the orbit plane (lines 3210-3240).

**Decay**

The Keplerian element DECAY is half the rate of change of mean motion. DECAY is usually very small, of order 10^-7, and its short term influence is small. For example, MA is affected by an amount DECAY*T*T over T days. Over 100 days that makes 10^-7*100*100 = 0.001 revolutions, or about 7.2 seconds for a 2 hour period satellite.

Decay is only really significant for low altitude satellites such as the space shuttle, ISS, or satellites nearing burn-up. Then the value can be as large as 10^-4. This causes in addition a noticeable second order effect on semi-major axis, argument of perigee and RAAN. The program accounts for this with terms calculated at line 2030.

For many purposes, especially if the Keplerian elements are regularly updated, drag can be ignored. This is true in particular for Oscar-13, where the unmodelled luni-solar perturbations are substantially larger than "decay".

**Doppler Shift**

The program calculates range-rate RR (line 4190) in km/s. If a satellite transmits a frequency FT, then this is received as a frequency FR:

FR = FT * (1 - RR/299792). [Note: 299792 is the speed of light in km/s]

Thus the "doppler shift" FD is given by: FD = -FT*RR/299792.

The calculation of range-rate is accurate in vacuo. However, at low elevations the radio path through the Earth's atmosphere is not perfectly straight. This refraction causes the actual observed doppler shift at AOS and LOS to be as much as 1 part in 10^7 in error, say 50-100 Hz at 435 MHz. It should also be noted that simple crystal oscillators have a similar order of stability, and satellites usually experience large temperature changes.

**Footprint**

In some programs there is a requirement to draw a circle around the sub- satellite point to indicate the field of view of the satellite. The following routine indicates how to do this. It is coded for clarity, not for speed. It would be better to store SIN(A) and COS(A) in a table rather than compute them every call. Since the footprint is left-right symmetric, only one half of the circle's points need be computed, and the other half can be inferred logically.

The sub-routine's output is a unit vector {X,Y,Z} in geocentric coordinates for the I'th point on the Earth's surface. This will then need transforming to map coordinates (mercator, spherical, linear or whatever), and then screen coordinates to suit the computer.

DEF PROCfoot(RS, slat, slon) REM Take satellite distance, sub-satellite lat/lon and compute unit vectors' REM x,y,z of N points of footprint on Earth's surface in Geocentric REM Coordinates. Also terrestrial latitude and longitude of points. : srad = ACS(RE/RS): REM Radius of footprint circle cla = COS(slat): sla = SIN(slat): REM Sin/Cos these to save time clo = COS(slon): slo = SIN(slon) sra = SIN(srad): cra = COS(srad) FOR I = 0 TO N: REM N points to the circle A = 2*PI*I/N: REM Angle around the circle : X = cra: REM Circle of points centred on Lat=0, Lon=0 Y = sra*SIN(A): REM assuming Earth's radius = 1 Z = sra*COS(A) REM [ However, use a table for SIN(.) COS(.) ] : x = X*cla - Z*sla: REM Rotate point "up" by latitude slat y = Y z = X*sla + Z*cla : X = x*clo - y*slo: REM Rotate point "around" through longitude slon Y = x*slo + y*clo Z = z : LON(I) = FNatn(Y,X): REM Convert point to Lat/Lon (or as required by map LAT(I) = ASN(Z): REM projection and display system). NEXT I ENDPROC

**Squint Angle**

Squint (or pointing) angle is the angle between the spacecraft's antenna and the observer. This is calculated at line 4120 for Oscar-13's high-gain antennas. As the low gain antennas are simple monopoles radiating normal to the **a** axis, low-gain squint is given by:

SQ_low_gain = 90 - SQ_high_gain (deg)Some satellites are stabilised so that they are continuously Earth pointing (e. g. UOSATs). For these, the antenna axis unit vector is aligned with the spacecraft's position vector, i.e.

4120 SQ = DEG(ACS( (Sx*Rx + Sy*Ry + SZ*Rz)/RS )) (deg)

The calculations of Earth rotation and Sun position are the most accurate part of this program! There is no need to update the solar constants at lines 1630-1660 for several decades.

This program was written and evolved over a period of years from 1983 to 1990. Constructive comment and suggestions are welcome.

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*Created: 1994 Dec 09 -- Last modified: 2023 Apr 12
*