Appendix B
————————————————————–
FORTRAN Codes
This appendix is to provide the reader with FORTRAN codes, which, when
applied, generate directly the results for the test cases of the book.
All the FORTRAN codes listed in this appendix work in a double precision
mode, which means that the word length amounts to 64 bit. The SI units are
used throughout this book, Appendix D. All the FORTRAN codes listed in
this appendix can be downloaded from http://extra.springer.com.
B.1 General Equations for Planetary Flight – Three
Degree of Freedom Simulation
This is a program code for the numerical integration of the general equations
of planetary flight – three degree of freedom simulation – in the form of eq.
(6.24) where the Earth rotation is included, as well as in the form of eq. (6.26)
where the Earth rotation is not included, see Chapter 6.
The given status of the program code is the test case 1 of Section 5.6, which
is discussed in Sub-Section 5.8.3.
On unit 40 the output file for plotting with MATLAB routines is generated. This file contains in the following order:
the time T, the velocity V, the flight path angle γ (GAM), the altitude H,
the position vector of the space vehicle measured from the central point of
the planet R, the longitude angle θ (THETA), the azimuth angle χ (CHI1),
the latitude angle φ (PHI), the total length of the flight path STOT, and the
length projected onto the planet’s surface in latitude direction SLAT.
C
*************************************************************
C
PROGRAM CODE FOR COMPUTING THE SPACE FLIGHT MECHANICS OF
C
SPACE VEHICLES
C
USE OF HIGHLY ACCURATE ”IMSL” INTEGRATION PROCEDURES
C
*************************************************************
USE NUMERICAL LIBRARIES
IMPLICIT REAL*8 (A-H,O-Z)
PARAMETER (MIN=1000001)
PARAMETER (MXPARM=50,N=6)
240
Appendix B FORTRAN Codes
REAL*8 M,MUEA,TOL
COMMON ARC
DIMENSION PARAM(MXPARM),Y(N)
EXTERNAL FCN
OPEN(40,FILE=’E:\MATLAB6p5\MSF 04 2\ Kepler 2’,FORM=’FORMATTED’,
* STATUS=’UNKNOWN’)
RE = 6378000
PI = 3.141592653
IEND = 582850
ITD = 1000
IT1 = 1
DT = 0.1
STOT = 0.0
SLAT = 0.0
ARC = PI/180.
C
*************************************************************
C
INITIAL CONDITIONS
C
*************************************************************
T = 0.0
Y(1) = 10510.47
Y(2) = 0.0
Y(3) = 0.0
Y(4) = 122000
Y(5) = 90.0*ARC
Y(6) = -20.0*ARC
VM1 = Y(1)
GAMM1 = Y(2)
CHIM1 = Y(3)
RM1 = RE + Y(4)
THETAM1 = Y(5)
PHIM1 = Y(6)
TOL = 0.0005
PARAM(4) = IEND+1
PARAM(10) = 1.0
WRITE(10,15)
IDO = 1
ISTEP = 0
C
*************************************************************
C
BEGIN OF INTEGRATION PROCEDURE
C
*************************************************************
10 CONTINUE
ISTEP = ISTEP + 1
ITE = ISTEP/ITD
TEND = ISTEP*DT
CALL DIVPRK(IDO,N,FCN,T,TEND,TOL,PARAM,Y)
B.1 General Equations for Planetary Flight
IF(Y(1).LE.0.OR.Y(4).LE.0)THEN
WRITE(10,16)T,V,GAM,H/1000,R/1000,THETA,CHI1,PHI,STOT/1000,
* SLAT/1000
STOP ’V OR H LESS THAN ZERO’
END IF
V00 = Y(1)
GAM00 = Y(2)
CHI00 = Y(3)
R00 = RE + Y(4)
THETA00 = Y(5)
PHI00 = Y(6)
C
*************************************************************
C
BEGIN OF COMPUTATION OF LONGITUDINAL REGIME
C
*************************************************************
XI1 = R00 * COS(-PHI00) * COS(THETA00)
XI = RM1 * COS(-PHIM1) * COS(THETAM1)
YI1 = R00 * COS(-PHI00) * SIN(THETA00)
YI = RM1 * COS(-PHIM1) * SIN(THETAM1)
ZI1 = R00 * SIN(-PHI00)
ZI = RM1 * SIN(-PHIM1)
DXI = ABS(XI1 - XI)
IF(DXI.LT.1.E-8)DXI = 1.E-8
DYI = YI1 - YI
DZI = ZI1 - ZI
DYDXS = (DYI/DXI)**2
DZDXS = (DZI/DXI)**2
ROOTI = SQRT(1. + DYDXS + DZDXS)
DSTOT = ROOTI * DXI
STOT = STOT + DSTOT
C
*************************************************************
C
END OF COMPUTATION OF LONGITUDE REGIME
C
*************************************************************
C
*************************************************************
C
BEGIN OF COMPUTATION OF LATITUDE REGIME
C
*************************************************************
XJ1 = RE * COS(-PHI00)
XJ = RE * COS(-PHIM1)
ZJ1 = RE * SIN(-PHI00)
ZJ = RE * SIN(-PHIM1)
DXJ = XJ1 - XJ
DZJ = ZJ1 - ZJ
IF(ABS(DZJ).EQ.0)DZJ = 1.E-08
DXDZS = (DXJ/DZJ)**2
ROOTJ = SQRT(1.+DXDZS)
IF(DZJ.EQ.1.E-08)DZJ = 0.
241
242
Appendix B FORTRAN Codes
DSLAT = ROOTJ * DZJ
SLAT = SLAT + DSLAT
C
*************************************************************
C
END OF COMPUTATION OF LATITUDE REGIME
C
*************************************************************
VM1 = Y(1)
GAMM1 = Y(2)
CHIM1 = Y(3)
RM1 = RE + Y(4)
THETAM1 = Y(5)
PHIM1 = Y(6)
IF(ISTEP.LE.IEND.AND.IT1.EQ.ITE)THEN
V = Y(1)
GAM = Y(2)/ARC
CHI1 = Y(3)/ARC
H = Y(4)
R = RE + H
THETA = Y(5)/ARC
PHI = Y(6)/ARC
IT1 = IT1 + 1
WRITE(10,16)T,V,GAM,H/1000,R/1000,THETA,CHI1,PHI,STOT/1000,
* SLAT/1000
C
*************************************************************
C
OUTPUT FILE FOR PLOT WITH MATLAB ROUTINES
C
*************************************************************
WRITE(40,16)T,V,GAM,H/1000,R/1000,THETA,CHI1,PHI,STOT/1000,
* SLAT/1000
END IF
IF(ISTEP.EQ.IEND)THEN
IDO = 3
STOP ’END OF INTEGRATION’
END IF
GOTO 10
C
*************************************************************
C
END OF INTEGRATION PROCEDURE
C
*************************************************************
15 FORMAT(/,8X,’T’,6X,’V’,11X,’GAM’,11X,’H’,11X,’R’,8X,’THETA’,6X,
* ’CHI1’,8X,’PHI’,6X,’STOT’,8X,’SLAT’,/)
16 FORMAT(F10.2,F10.3,F13.6,2E14.6,3F10.4,2E13.5)
END
C
*************************************************************
C
*************************************************************
SUBROUTINE FCN(N,T,Y,YPRIME)
IMPLICIT REAL*8 (A-H,O-Z)
B.1 General Equations for Planetary Flight
243
COMMON ARC
DIMENSION Y(N),YPRIME(N)
REAL*8 M,MUEA
KROT = 0
C
*************************************************************
C
KROT = 0 ==> EQUATIONS WITH NO EARTH ROTATION IN P-FRAME
C
KROT = 1 ==> EQUATIONS WITH EARTH ROTATION IN O-FRAME
C
*************************************************************
S = 12.02
! REFERENCE AREA, [m2]
CD = 0.0
! DRAG COEFFICIENT
CL = 0.0
! LIFT COEFFICIENT
M = 50.0
! VEHICLE MASS, [kg]
RE = 6378000
! RADIUS OF EARTH, [m]
RHOS = 1.2930
! DENSITY AT SEA-LEVEL; [kg/m3]
GS = 9.8066
! GRAVITATIONAL CONSTANT AT SEA-LEVEL, [m/s2]
BETA = 0.000140845
! SCALING COEFFICIENT
OMEGA = 0.000072920
! ANGULAR VELOCITY OF EARTH, [rad/s]
MUEA = 0.0*ARC
! BANK ANGLE
RHO = RHOS * EXP(-BETA * Y(4))
RRED = RE/(RE + Y(4))
D = RHO * S * CD * 0.5 * Y(1)**2
L = RHO * S * CL * 0.5 * Y(1)**2
G = GS * RRED**2
R = RE + Y(4)
IF(KROT.EQ.1)THEN
YPRIME(1) = - D/M - G * SIN(Y(2)) + OMEGA**2 * R * (COS(Y(6)))**2*
* (SIN(Y(2)) - COS(Y(2)) * TAN(Y(6)) * SIN(Y(3)))
YPRIME(2) = (L/M * COS(MUEA) - G*COS(Y(2)) + Y(1)**2/R * COS(Y(2))
* + 2. * OMEGA * Y(1) * COS(Y(6)) * COS(Y(3)) + OMEGA**2 * R *
* (COS(Y(6)))**2 * (COS(Y(2)) + SIN(Y(2)) * TAN(Y(6)) *
* SIN(Y(3))))/Y(1)
YPRIME(3) = (L/M * SIN(MUEA)/COS(Y(2)) - Y(1)**2/R * COS(Y(2)) *
* COS(Y(3)) * TAN(Y(6)) + 2. * OMEGA * Y(1) *
* (TAN(Y(2)) * COS(Y(6)) * SIN(Y(3)) - SIN(Y(6)))
* - OMEGA**2 * R * SIN(Y(6)) * COS(Y(6)) * COS(Y(3))/COS(Y(2)))
* /Y(1)
ELSE
YPRIME(1) = - D/M - G * SIN(Y(2))
YPRIME(2) = (L/M * COS(MUEA) - G * COS(Y(2)) + Y(1)**2/R * COS(Y(2)))
* /Y(1)
YPRIME(3) = (L/M * SIN(MUEA)/COS(Y(2)) - Y(1)**2/R * COS(Y(2)) *
* COS(Y(3)) * TAN(Y(6)))/Y(1)
244
Appendix B FORTRAN Codes
END IF
YPRIME(4) = Y(1) * SIN(Y(2))
YPRIME(5) = Y(1) * COS(Y(2)) * COS(Y(3))/(R * COS(Y(6)))
YPRIME(6) = Y(1) * COS(Y(2)) * SIN(Y(3))/R
RETURN
END
C
*************************************************************
C
END OF SUBROUTINE FCN
C
*************************************************************
B.2 Orbit Determination with Orbital Elements
245
B.2 Orbit Determination with Orbital Elements
This is a program code for evaluation of the relations for the orbital elements
and for determination of the position vector R(t) and the velocity vector
V(t) for the whole orbit using the eqs. (5.53) and (5.54) of Sub-Section 5.8.1.
The given status of the program code is the test case 1 of Section 5.6 in
the form applied in Sub-Section 5.8.2.
On unit 11 the output file for a plot with MATLAB routines is generated.
It contains in the following order:
the time T, the true anomaly θ (THETA1), the x-component of the position
vector R in the planetocentric frame (xE , y E , z E ) RT1EXE, the y-component
RT1EYE, the z-component RT1EZE (see Fig. 5.14 and eq. (5.57)), and the
magnitude of the velocity vector |V| (VSCALAR1).
PROGRAM ORBITAL ELEMENTS
C
*************************************************************
C
Program calculates the orbital elements from the initial
C
conditions R(to) and V(to) and supplies for any time t the
C
position vector R(t) and the velocity vector V(t)
C
*************************************************************
IMPLICIT REAL*8 (A-H,O-Z)
PARAMETER (NEND = 5830)
DIMENSION THETA1(NEND),RT1EXE(NEND),RT1EYE(NEND),RT1EZE(NEND)
DIMENSION T(NEND)
REAL*8 INC,NX,NY,NZ,NABS
OPEN(11,FILE=’E:\MATLAB6p5\MSF 04 2 \Orbital elements\case 1’,
*
FORM=’FORMATTED’,STATUS=’UNKNOWN’)
C
*****************************************************************
C
Constants
C
*****************************************************************
GAMMA = 3.98920D14
PI = 3.141592653D0
ARC = PI/180.D0
C
*****************************************************************
C
Initial condition at time TO
C
*****************************************************************
RT0EXE = 0.0D0
RT0EYE = 6.10800D6
RT0EZE = 2.22313D6
VT0EXE = -10510.47D0
VT0EYE = 0.0D0
VT0EZE = 0.0D0
RSCALAR = 6.500D6
246
Appendix B FORTRAN Codes
VSCALAR = 10510.47D0
T0 = 0.0D0
C
*****************************************************************
C
From energy equation to semi-major axis A
C
*****************************************************************
ENERGY = 0.5D0*VSCALAR**2 - GAMMA/RSCALAR
A = - 0.5D0*GAMMA/ENERGY
C
*****************************************************************
C
The period time TT
C
*****************************************************************
C
*****************************************************************
C
The angular momentum vector F
C
*****************************************************************
TT = 2.D0*PI*SQRT(A**3/GAMMA)
FX = RT0EYE*VT0EZE - RT0EZE*VT0EYE
FY = -(RT0EXE*VT0EZE - RT0EZE*VT0EXE)
FZ = RT0EXE*VT0EYE - RT0EYE*VT0EXE
C
*****************************************************************
C
The unit vector n of the lines of nodes
C
*****************************************************************
FABS = SQRT(FX**2 + FY**2 + FZ**2)
GNX = -FY
GNY = FX
GNZ = 0.0D0
GNABS = SQRT(GNX**2 + GNY**2)
NX = GNX/GNABS
NY = GNY/GNABS
NZ = GNZ/GNABS
C
*****************************************************************
C
The eccentricity e
C
*****************************************************************
ECCX = (VT0EYE*FZ - VT0EZE*FY - GAMMA*RT0EXE/RSCALAR)/GAMMA
ECCY = (-(VT0EXE*FZ - VT0EZE*FX) - GAMMA*RT0EYE/RSCALAR)/GAMMA
ECCZ = (VT0EXE*FY - VT0EYE*FX - GAMMA*RT0EZE/RSCALAR)/GAMMA
EABS = SQRT(ECCX**2 + ECCY**2 + ECCZ**2)
C
*****************************************************************
C
Determination of the inclination angle i
C
*****************************************************************
C
*****************************************************************
C
Determination of the right ascension of the ascending node Omega
C
*****************************************************************
INC = ACOS(FZ/FABS)
OMGCOS = ACOS(NX)
OMGSIN = ASIN(NY)
OMG = OMGCOS
B.2 Orbit Determination with Orbital Elements
C
*****************************************************************
C
Determination of the unit vectors of the orbit plane
C
*****************************************************************
247
EXOXE = ECCX/EABS
EXOYE = ECCY/EABS
EXOZE = ECCZ/EABS
EZOXE = FX/FABS
EZOYE = FY/FABS
EZOZE = FZ/FABS
EYOXE = EZOYE*EXOZE - EZOZE*EXOYE
EYOYE = -(EZOXE*EXOZE - EZOZE*EXOXE)
EYOZE = EZOXE*EXOYE - EZOYE*EXOXE
C
*****************************************************************
C
Determination of the argument of perigee omega
C
*****************************************************************
OMKCOS = ACOS(NX*EXOXE + NY*EXOYE)
OMKSIN = ASIN(NX*EYOXE + NY*EYOYE)
OMK = OMKCOS
C
*****************************************************************
C
The true anomaly theta at time TO
C
*****************************************************************
ARGU = (RT0EXE*EXOXE + RT0EYE*EXOYE + RT0EZE*EXOZE)/RSCALAR
DARGU = ABS(1.D0 - ARGU)
IF(DARGU.LT.1D-6)ARGU = 1.0D0
THETACOS = ACOS(ARGU)
THETASIN = ASIN((RT0EXE*EYOXE + RT0EYE*EYOYE +
*
RT0EZE*EYOZE)/RSCALAR)
THETA = THETACOS
C
*****************************************************************
C
The time TP as function of the true anomaly theta
C
*****************************************************************
E = 2.D0*ATAN(SQRT((1D0-EABS)/(1D0+EABS))*TAN(THETA/2D0))
TP = T0 - TT/(2D0*PI)*(E - EABS*SIN(E))
C
*****************************************************************
C
End of determination of orbital elements
C
*****************************************************************
C
*****************************************************************
C
Write the output for the orbital elements on file 20
C
*****************************************************************
WRITE(20,101) A
101 FORMAT(/,’MAJOR AXIS A:’,27X,E12.5)
WRITE(20,102) EABS
102 FORMAT(’ECCENTRICITY:’,27X,E12.5)
WRITE(20,103) INC/ARC
103 FORMAT(’INCLINATION ANGLE:’,22X,E12.5)
248
Appendix B FORTRAN Codes
WRITE(20,104) OMG/ARC
104 FORMAT(’RIGHT ASCENSION OF THE ASCENDING NODE:’,2X,E12.5)
WRITE(20,105) OMK/ARC
105 FORMAT(’ARGUMENT OF PERIGEE:’,20X,E12.5)
WRITE(20,106) THETA/ARC
106 FORMAT(’TRUE ANOMALY:’,27X,E12.5)
WRITE(20,107) TP
107 FORMAT(’PERIGEE PASSAGE TP:’,21X,E12.5)
WRITE(20,108) TT
108 FORMAT(’PERIOD TIME TT:’,25X,E12.5,//)
C
*****************************************************************
C
Determination of the complete solution
C
*****************************************************************
EPS = 1.D-6
DT = 10.0D0
E0 = 1.D0
DO 10 N = 1,NEND
T(N) = (N-1)*DT
CO1 = 2.D0*PI*(T(N)-TP)/TT
20
FE = E0 - EABS*SIN(E0) - CO1
DFE = 1.D0 - EABS*COS(E0)
E1 = E0 - FE/DFE
DIFF = ABS(E1 - E0)
E0 = E1
IF(DIFF.GT.EPS)GOTO20
THETA1(N) = 2.D0*ATAN(SQRT((1.D0+EABS)/(1.D0-EABS))*TAN(E0/2.D0))
IF(THETA1(N).LT.0)THETA1(N) = 2.D0*PI + THETA1(N)
10
CONTINUE
P = A*(1.D0-EABS**2)
WRITE(20,111)
DO 30 N = 1,NEND
EXOXE1 = COS(OMG)*COS(OMK) - SIN(OMG)*SIN(OMK)*COS(INC)
EXOYE1 = SIN(OMG)*COS(OMK) + COS(OMG)*SIN(OMK)*COS(INC)
EXOZE1 = SIN(OMK)*SIN(INC)
EYOXE1 =-COS(OMG)*SIN(OMK) - SIN(OMG)*COS(OMK)*COS(INC)
EYOYE1 =-SIN(OMG)*SIN(OMK) + COS(OMG)*COS(OMK)*COS(INC)
EYOZE1 = COS(OMK)*SIN(INC)
RSCALAR1 = P/(1.D0+EABS*COS(THETA1(N)))
RT1EXE(N)=RSCALAR1*(COS(THETA1(N))*EXOXE1+SIN(THETA1(N))*EYOXE1)
RT1EYE(N)=RSCALAR1*(COS(THETA1(N))*EXOYE1+SIN(THETA1(N))*EYOYE1)
RT1EZE(N)=RSCALAR1*(COS(THETA1(N))*EXOZE1+SIN(THETA1(N))*EYOZE1)
VSCALAR1=SQRT(GAMMA/P)*SQRT(1D0+2D0*EABS*COS(THETA1(N))+EABS**2)
B.2 Orbit Determination with Orbital Elements
C
*****************************************************************
C
Write the output of R(t) and V(t) for the orbit on file 20
C
*****************************************************************
WRITE(20,110)T(N),THETA1(N)/ARC,RT1EXE(N)/1000,RT1EYE(N)/1000,
* RT1EZE(N)/1000,VSCALAR1
C
*****************************************************************
C
Output file 11 of R(t) and V(t) for plot with MATLAB routines
C
*****************************************************************
WRITE(11,110)T(N),THETA1(N)/ARC,RT1EXE(N)/1000,RT1EYE(N)/1000,
* RT1EZE(N)/1000,VSCALAR1
110 FORMAT(F10.2,F8.2,4F12.2)
111 FORMAT(5X,’TIME’,4X,’THETA’,5X,’X-COORD’, 5X,’Y-COORD’,5X,
*
’Z-COORD’,4X,’VELOCITY’,/)
30
CONTINUE
END
249
250
Appendix B FORTRAN Codes
B.3 Lagrange’s Planet Equations
This is a program code for the numerical integration of Lagrange’s planet
equations in the form of eqs. (5.63) together with eqs (5.70) and (5.71). For
more information regarding Lagrange’s planet equations see also [2] to [5].
The given status of the program code is the test case 1 of Section 5.6 in
the form applied in Sub-Section 5.8.2 and discussed in Section 5.9.2. On units
11 and 12 the output files for plots with MATLAB routines are generated.
Unit 11 contains in the following order:
the time T, the true anomaly θ (THETA1), the x-component of the position
vector R in the planetocentric frame (xE , y E , z E ) RT1EXE, the y-component
RT1EYE, the z-component RT1EZE (see Fig. 5.14 and eq. (5.57)), and the
magnitude of the velocity vector |V| (VSCALAR1).
Unit 12 contains in the following order:
the six orbital elements, viz. the semi-major axis a (Y(1)), the eccentricity
e (Y(2)), the inclination angle i (Y(3)), the right ascension of the ascending
node Ω (Y(4)), the argument of perigee ω (Y(5)), the mean anomaly M0
(Y(6)), the true anomaly θ (THETA1), the magnitude of the position vector
|R| (RLA) and the time step N.
PROGRAM ORBITAL ELEMENTS AND LAGRANGE’S PERTURBATION THEORY
C
*************************************************************
C
1)
C
The undisturbed orbital elements from the initial
C
conditions R(to) and V(to) are computed, where the equations of
C
the two-body theory are used.
C
2)
C
With the undisturbed orbital elements as initial conditions, the
C
deviations of these orbital elements due to perturbations are
C
computed by integration of the Lagrange’s planet equations and
C
subsequent determination of the position vector R(t) and the
C
velocity vector V(t).
C
C
*****************************************************************
USE NUMERICAL LIBRARIES
IMPLICIT REAL*8 (A-H,O-Z)
PARAMETER (MXPARM=50,NN=6)
PARAMETER (NEND = 582000)
COMMON GAM,TP,T0,EE0
COMMON ALA(0:NEND),EABSLA(0:NEND),INCLA(0:NEND),OMGLA(0:NEND)
COMMON OMKLA(0:NEND),CHILA(0:NEND),THETALA(0:NEND),N,KPERIOD2
DIMENSION RT1EXE(NEND),RT1EYE(NEND),RT1EZE(NEND)
DIMENSION TTLA(0:NEND),T(0:NEND),PARAM(MXPARM),Y(NN)
B.3 Lagrange’s Planet Equations
REAL*8 INC0,INCLA,NX,NY,NZ,NABS
EXTERNAL FCN
OPEN(11,FILE=’E:\MATLAB6p5\Buch Wei\MSF 04 2 \Orbital elements
*
\case 1 A’,FORM=’FORMATTED’,STATUS=’UNKNOWN’)
OPEN(12,FILE=’E:\MATLAB6p5\Buch Wei\MSF 04 2 \Orbital elements
*
\case 1 B’,FORM=’FORMATTED’,STATUS=’UNKNOWN’)
C
*****************************************************************
C
Constants
C
*****************************************************************
GAM = 3.98920D14
PI = 3.141592653D0
ARC = PI/180.D0
EE0 = 1.D-1
C
*****************************************************************
C
Initial conditions at time T0
C
*****************************************************************
RT0EXE = 0.0D6
RT0EYE = 6.10800D6
RT0EZE = 2.22313D6
VT0EXE = -10510.47D0
VT0EYE = 0.0D0
VT0EZE = 0.0D0
RSCALAR = 6.500D6
VSCALAR = 10510.47D0
T0 = 0.0D0
C
*****************************************************************
C
From energy equation to semimajor axis A
C
*****************************************************************
ENERGY = 0.5D0*VSCALAR**2 - GAM/RSCALAR
A0 = - 0.5D0*GAM/ENERGY
C
*****************************************************************
C
The period time TT
C
*****************************************************************
C
*****************************************************************
TT0 = 2.D0*PI*SQRT(A0**3/GAM)
C
The angular momentum vector F
C
*****************************************************************
FX = RT0EYE*VT0EZE - RT0EZE*VT0EYE
FY = -(RT0EXE*VT0EZE - RT0EZE*VT0EXE)
FZ = RT0EXE*VT0EYE - RT0EYE*VT0EXE
C
*****************************************************************
C
The unit vector n of the lines of nodes
C
*****************************************************************
FABS = SQRT(FX**2 + FY**2 + FZ**2)
251
252
Appendix B FORTRAN Codes
GNX = -FY
GNY = FX
GNZ = 0.0D0
GNABS = SQRT(GNX**2 + GNY**2)
NX = GNX/GNABS
NY = GNY/GNABS
NZ = GNZ/GNABS
C
*****************************************************************
C
The eccentricity e
C
*****************************************************************
ECCX = (VT0EYE*FZ - VT0EZE*FY - GAM*RT0EXE/RSCALAR)/GAM
ECCY = (-(VT0EXE*FZ - VT0EZE*FX) - GAM*RT0EYE/RSCALAR)/GAM
ECCZ = (VT0EXE*FY - VT0EYE*FX - GAM*RT0EZE/RSCALAR)/GAM
EABS0 = SQRT(ECCX**2 + ECCY**2 + ECCZ**2)
C
*****************************************************************
C
Determination of the inclination angle i
C
*****************************************************************
C
*****************************************************************
C
Determination of the right ascension of the ascending node Omega
C
*****************************************************************
INC0 = ACOS(FZ/FABS)
OMGCOS = ACOS(NX)
OMGSIN = ASIN(NY)
OMG0 = OMGCOS
C
*****************************************************************
C
Determination of the unit vectors of the orbit plane
C
*****************************************************************
EXOXE = ECCX/EABS0
EXOYE = ECCY/EABS0
EXOZE = ECCZ/EABS0
EZOXE = FX/FABS
EZOYE = FY/FABS
EZOZE = FZ/FABS
EYOXE = EZOYE*EXOZE - EZOZE*EXOYE
EYOYE = -(EZOXE*EXOZE - EZOZE*EXOXE)
EYOZE = EZOXE*EXOYE - EZOYE*EXOXE
C
*****************************************************************
C
Determination of the argument of perigee omega
C
*****************************************************************
OMKCOS = ACOS(NX*EXOXE + NY*EXOYE)
OMKSIN = ASIN(NX*EYOXE + NY*EYOYE)
OMK0 = OMKCOS
C
*****************************************************************
C
The true anomaly theta at time T0
C
*****************************************************************
B.3 Lagrange’s Planet Equations
253
ARGU = (RT0EXE*EXOXE + RT0EYE*EXOYE + RT0EZE*EXOZE)/RSCALAR
DARGU = ABS(1.D0 - ARGU)
IF(DARGU.LT.1D-5)ARGU = 1.0D0
THETACOS = ACOS(ARGU)
THETASIN = ASIN((RT0EXE*EYOXE + RT0EYE*EYOYE +
*
RT0EZE*EYOZE)/RSCALAR)
THETA0 = THETASIN
C
*****************************************************************
C
The time TP as function of the true anomaly theta
C
*****************************************************************
E = 2.D0*ATAN(SQRT((1D0-EABS0)/(1D0+EABS0))*TAN(THETA0/2D0))
TP = T0 - TT0/(2D0*PI)*(E - EABS0*SIN(E))
CHI0 = 2.D0*PI*(T0-TP)/TT0
C
*****************************************************************
C
End of determination of orbital elements
C
*****************************************************************
C
*****************************************************************
C
Write the output for the orbital elements on file 20
C
*****************************************************************
WRITE(20,101) A0
101 FORMAT(/,’MAJOR AXIS A:’,27X,E12.5)
WRITE(20,102) EABS0
102 FORMAT(’ECCENTRICITY:’,27X,E12.5)
WRITE(20,103) INC0/ARC
103 FORMAT(’INCLINATION ANGLE:’,22X,E12.5)
WRITE(20,104) OMG0/ARC
104 FORMAT(’RIGHT ASCENSION OF THE ASCENDING NODE:’,2X,E12.5)
WRITE(20,105) OMK0/ARC
105 FORMAT(’ARGUMENT OF PERIGEE:’,20X,E12.5)
WRITE(20,106) THETA0/ARC
106 FORMAT(’TRUE ANOMALY:’,27X,E12.5)
WRITE(20,107) TP
107 FORMAT(’PERIGEE PASSAGE TP:’,21X,E12.5)
WRITE(20,108) TT0
108 FORMAT(’PERIOD TIME TT:’,25X,E12.5,//)
C
*****************************************************************
C
Calculation of the disturbed orbtial elements by Lagrange’s planet
C
equations
C
*****************************************************************
EPS = 1.D-6
DT = 0.1D0
EE0 = 1.D0
NDIFF = 50
NWRITE = 1
254
Appendix B FORTRAN Codes
KPERIOD1 = 0
KPERIOD2 = 0
C
*****************************************************************
C
The initial conditions
C
*****************************************************************
ALA(0) = A0
EABSLA(0) = EABS0
INCLA(0) = INC0
OMGLA(0) = OMG0
OMKLA(0) = OMK0
CHILA(0) = CHI0
TTLA(0) = TT0
THETALA(0) = 0.0D0
Y(1) = A0
Y(2) = EABS0
Y(3) = INC0
Y(4) = OMG0
Y(5) = OMK0
Y(6) = CHI0
IDO = 1
TT = 0.0D0
TOL = 0.01D0
PARAM(4) = NEND+1
PARAM(10) = 1.0D0
WRITE(22,113)
C
*****************************************************************
C
Loop for calculation of the disturbed orbital elements as function
C
of time, along a full orbit cycle (2pi).
C
*****************************************************************
DO 10 N = 1,NEND
NSTEP = N/NDIFF
T(N) = (N)*DT
CO1 = 2.D0*PI*(T(N)-TP)/TTLA(N-1)
20 FE = EE0 - Y(2)*SIN(EE0) - CO1
DFE = 1.D0 - Y(2)*COS(EE0)
EE1 = EE0 - FE/DFE
DIFF1 = ABS(EE1 - EE0)
EE0 = EE1
IF(DIFF1.GT.EPS)GOTO20
THETALA(N) = 2.D0*ATAN(SQRT((1.D0 + Y(2))/
*
(1.D0 - Y(2)))*TAN(EE0/2.D0))
IF(THETALA(N).GE.0D0.AND.KPERIOD1.EQ.1)THEN
THETALA(N) = 2.D0*PI + THETALA(N)
END IF
IF(THETALA(N).LT.0D0)THEN
B.3 Lagrange’s Planet Equations
THETALA(N) = 2.D0*PI + THETALA(N)
KPERIOD1 = 1
END IF
C
*****************************************************************
C
IMSL function for integration of ordinary differential equations
C
*****************************************************************
CALL DIVPRK(IDO,NN,FCN,TT,T(N),TOL,PARAM,Y)
ALA(N) = Y(1)
EABSLA(N) = Y(2)
INCLA(N) = Y(3)
OMGLA(N) = Y(4)
OMKLA(N) = Y(5)
CHILA(N) = Y(6)
RLA = Y(1)*(1.D0-Y(2)**2)/(1.D0 + Y(2)*COS(THETALA(N)))
TTLA(N) = 2.D0*PI*SQRT(ALA(N)**3/GAM)
IF(NSTEP.EQ.NWRITE)THEN
C
*****************************************************************
C
Write the output of the orbital elements as function of time on
C
file 22
C
*****************************************************************
C
*****************************************************************
C
Write the output of the orbital elements as function of time for
C
plot with MATLAB routines on file 12
C
*****************************************************************
WRITE(22,112)Y(1),Y(2),Y(3),Y(4),Y(5),Y(6),THETALA(N),RLA,N
WRITE(12,112)Y(1),Y(2),Y(3),Y(4),Y(5),Y(6),THETALA(N),RLA,N
NWRITE = NWRITE + 1
END IF
10 CONTINUE
113 FORMAT(/,1X,’SEMIMAJOR A’,3X,’ECCENT. E’,2X,’ORB.
*
INC. I’,3X,’Omega’,4X,’omega’,2X,’AVERA.ANO.CHI’,2X,
** ’THETA’,6X,’R’,/)
112 FORMAT(E12.4,F12.4,4F10.4,F12.4,F12.0,I10)
WRITE(20,111)
C
*****************************************************************
C
Compute the new orbit based on the disturbed orbital elements
C
*****************************************************************
DO 30 N = 1,NEND,100
P = ALA(N)*(1.D0-EABSLA(N)**2)
EXOXE1 = COS(OMGLA(N))*COS(OMKLA(N))
*
- SIN(OMGLA(N))*SIN(OMKLA(N))*COS(INCLA(N))
*
+ COS(OMGLA(N))*SIN(OMKLA(N))*COS(INCLA(N))
EXOYE1 = SIN(OMGLA(N))*COS(OMKLA(N))
255
256
Appendix B FORTRAN Codes
EXOZE1 = SIN(OMKLA(N))*SIN(INCLA(N))
EYOXE1 =-COS(OMGLA(N))*SIN(OMKLA(N))
*
- SIN(OMGLA(N))*COS(OMKLA(N))*COS(INCLA(N))
*
+ COS(OMGLA(N))*COS(OMKLA(N))*COS(INCLA(N))
EYOYE1 =-SIN(OMGLA(N))*SIN(OMKLA(N))
EYOZE1 = COS(OMKLA(N))*SIN(INCLA(N))
RSCALAR1 = P/(1.D0 + EABSLA(N)*COS(THETALA(N)))
RT1EXE(N) = RSCALAR1*(COS(THETALA(N))*EXOXE1 +
*
SIN(THETALA(N))*EYOXE1)
*
SIN(THETALA(N))*EYOYE1)
*
SIN(THETALA(N))*EYOZE1)
*
EABSLA(N)**2)
RT1EYE(N) = RSCALAR1*(COS(THETALA(N))*EXOYE1 +
RT1EZE(N) = RSCALAR1*(COS(THETALA(N))*EXOZE1 +
VSCALAR1=SQRT(GAM/P)*SQRT(1.D0+2.D0*EABSLA(N)*COS(THETALA(N))+
C
*****************************************************************
C
Write the output of R(t) and V(t) for the disturbed orbit on
C
file 20
C
*****************************************************************
WRITE(20,110)T(N),THETALA(N)/ARC,RT1EXE(N)/1000,RT1EYE(N)/1000,
*
RT1EZE(N)/1000,VSCALAR1
C
*****************************************************************
C
Output file 11 of R(t) and V(t) for plot with MATLAB routines
C
*****************************************************************
WRITE(11,110)T(N),THETALA(N)/ARC,RT1EXE(N)/1000,RT1EYE(N)/1000,
*
RT1EZE(N)/1000,VSCALAR1
110 FORMAT(F8.2,F8.2,4F12.2)
111 FORMAT(3X,’TIME’,4X,’THETA’,5X,’X-COORD’,5X,’Y-COORD’,5X,
*
’Z-COORD’,4X,’VELOCITY’,/)
30 CONTINUE
END
SUBROUTINE FCN(NN,TT,Y,YPRIME)
C
*************************************************************
C
Subprogram provides the relations of the set of ordinary
C
differential equations for the integration procedure
C
*****************************************************************
IMPLICIT REAL*8 (A-H,O-Z)
PARAMETER (NEND = 582000)
COMMON GAM,TP,T0,EE0
COMMON ALA(0:NEND),EABSLA(0:NEND),INCLA(0:NEND),OMGLA(0:NEND)
COMMON OMKLA(0:NEND),CHILA(0:NEND),THETALA(0:NEND),N,KPERIOD2
DIMENSION Y(NN),YPRIME(NN)
REAL*8 INCLA,J2
RE = 6.378D6
B.3 Lagrange’s Planet Equations
J2 = 1082.63D-6
EPS = 1.D-6
PI = 3.141592653D0
TTLA = 2.D0*PI*SQRT(Y(1)**3/GAM)
CO1 = 2.D0*PI*(TT-TP)/TTLA
C
*****************************************************************
C
Computation of the eccentric anomaly E
C
C *****************************************************************
20 FE = EE0 - Y(2)*SIN(EE0) - CO1
DFE = 1.D0 - Y(2)*COS(EE0)
EE1 = EE0 - FE/DFE
DIFF = ABS(EE1 - EE0)
EE0 = EE1
IF(DIFF.GT.EPS)GOTO20
C
*****************************************************************
C
Determination of the true anomaly theta as function of the
C
eccentric anomaly E
C
*****************************************************************
THETA1 = 2.D0*ATAN(SQRT((1.D0+Y(2))/(1.D0-Y(2)))*TAN(EE0/2.D0))
IF(THETA1.GE.0.AND.KPERIOD2.EQ.1)THEN
THETA1 = 2.D0*PI + THETA1
END IF
IF(THETA1.LT.0)THEN
THETA1 = 2.D0*PI + THETA1
KPERIOD2 = 1
END IF
VORB = SQRT(GAM/Y(1)**3)
RLA = Y(1)*(1.D0-Y(2)**2)/(1.D0 + Y(2)*COS(THETA1))
COEF1 = GAM*J2*RE**2
COEF2 = COS(2.D0*(THETA1 + Y(5)))
COEF3 = 0.5D0 - 0.75D0*SIN(Y(3))**2 + 0.75D0*SIN(Y(3))**2*COEF2
COEF4 = 3.D0*Y(2)*SIN(THETA1)/
*
(RLA**3*(1.D0 + Y(2)*COS(THETA1)))
COEF5 = -2.D0*SIN(2.*(THETA1 + Y(5)))
COEF6 = 0.75D0*SIN(Y(3))**2/RLA**3
C
*****************************************************************
C
Numerical approximation of the derivatives of theta with respect
C
to the orbital elements
C
*****************************************************************
IF(N.GT.1)THEN
DTHE = THETALA(N) - THETALA(N-1)
DTHDA = DTHE/(ALA(N) - ALA(N-1))
DTHDE = DTHE/(EABSLA(N) - EABSLA(N-1))
DTHDI = DTHE/(INCLA(N) - INCLA(N-1))
DTHDOG = DTHE/(OMGLA(N) - OMGLA(N-1))
257
258
Appendix B FORTRAN Codes
DTHDOK = DTHE/(OMKLA(N) - OMKLA(N-1))
DTHDCHI = DTHE/(CHILA(N) - CHILA(N-1))
ELSE
DTHDA = 0.D0
DTHDE = 0.D0
DTHDI = 0.D0
DTHDOG = 0.D0
DTHDOK = 0.D0
DTHDCHI = 0.D0
END IF
C
C *****************************************************************
C
Numerical approximation of Lagrange’s planet equations
C
C *****************************************************************
D1R3DA = -3.D0/(RLA**3*Y(1)) - COEF4*DTHDA
DCOSDA = COEF5*DTHDA
DWDA =-COEF1*(COEF3*D1R3DA + COEF6*DCOSDA)
D1R3DE = 3.D0*(2.D0*Y(2)*Y(1)/RLA + COS(THETA1))/
*
(RLA**3*(1.D0 + Y(2)*COS(THETA1))) - COEF4*DTHDE
DCOSDE = COEF5*DTHDE
DWDE =-COEF1*(COEF3*D1R3DE + COEF6*DCOSDE)
D1R3DI =-COEF4*DTHDI
DCOSDI = COEF5*DTHDI
DWDI =-COEF1*(COEF3*D1R3DI + COEF6*DCOSDI +
*
0.75D0*2.D0*SIN(Y(3))*COS(Y(3))/RLA**3*(COEF2-1.D0))
D1R3DOG =-COEF4*DTHDOG
DCOSDOG = COEF5*DTHDOG
DWDOG =-COEF1*(COEF3*D1R3DOG + COEF6*DCOSDOG)
D1R3DOK =-COEF4*DTHDOK
DCOSDOK = COEF5*(DTHDOK + 1.D0)
DWDOK =-COEF1*(COEF3*D1R3DOK + COEF6*DCOSDOK)
D1R3DCHI =-COEF4*DTHDCHI
DCOSDCHI = COEF5*DTHDCHI
DWDCHI =-COEF1*(COEF3*D1R3DCHI + COEF6*DCOSDCHI)
COEF7 = (1.D0 - Y(2)**2)/(VORB*Y(1)**2*Y(2))
COEF8 = SQRT(1.D0 - Y(2)**2)/(VORB*Y(1)**2*Y(2))
COEF9 = 1.D0/(SQRT(1.D0 - Y(2)**2)*VORB*Y(1)**2)
COEF10 = 2.D0/(VORB*Y(1))
YPRIME(1) = -COEF10*DWDCHI
YPRIME(2) = -COEF7 *DWDCHI + COEF8*DWDOK
YPRIME(3) = -COEF9 *DWDOK/TAN(Y(3)) + COEF9*DWDOG/SIN(Y(3))
YPRIME(4) = -COEF9 *DWDI /SIN(Y(3))
YPRIME(5) = -COEF8 *DWDE + COEF9*DWDI/TAN(Y(3))
YPRIME(6) = COEF7 *DWDE + COEF10*DWDA
RETURN
END
References
259
References
1. Vinh, N.X., Busemann, A., Culp, R.D.: Hypersonic and Planetary Entry Flight
Mechanics. The University of Michigan Press, Ann Arbor (1980)
2. Escobal, P.R.: Methods of Orbit Determination. John Wiley & Sons, New York
(1965)
3. Steiner, W., Schagerl, M.: Raumflugmechanik. Springer, Heidelberg (2004)
4. Sterne, T.E.: An Introduction to Celestial Mechanics. Interscience Publisher,
New York (1960)
5. Chobotov, V.A. (ed.): Orbital Mechanics. 3rd edn. AIAA Education Series, Reston, VA (2002)