def prenut(epoch, mjd):
"""
Form the matrix of precession and nutation (IAU1976/fk5)
Inputs:
- epoch Julian Epoch for mean coordinates
- mjd Modified Julian Date (jd-2400000.5) for true coordinates
Returns:
- pnMat the combined precession/nutation matrix, a 3x3 numpy.array.
Notes:
- The epoch and MJD are TDB (loosely ET).
- The matrix is in the sense V(true) = pnMat * V(mean)
"""
precMat = prec(epoch, epj(mjd))
# Nutation
nutMat = nut(mjd)
# Combine the matrices: pn = N x P
return numpy.dot(nutMat, precMat)
def prenut(epoch, mjd):
"""
Form the matrix of precession and nutation (IAU1976/fk5)
Inputs:
- epoch Julian Epoch for mean coordinates
- mjd Modified Julian Date (jd-2400000.5) for true coordinates
Returns:
- pnMat the combined precession/nutation matrix, a 3x3 numpy.array.
Notes:
- The epoch and MJD are TDB (loosely ET).
- The matrix is in the sense V(true) = pnMat * V(mean)
"""
precMat = prec(epoch, epj(mjd))
# Nutation
nutMat = nut(mjd)
# Combine the matrices: pn = N x P
return numpy.dot(nutMat, precMat)
def evp(tdb, deqx = 0.0):
"""
Barycentric and heliocentric velocity and position of the Earth
Inputs:
- tdb TDB date (loosely et) as a Modified Julian Date
- deqx Julian Epoch (e.g. 2000.0) of mean equator and
equinox of the vectors returned. If deqx <= 0.0,
all vectors are referred to the mean equator and
equinox (fk5) of epoch date.
Returns a tuple consisting of (all numpy.array(3)):
- Barycentric velocity of Earth (au/s)
- Barycentric position of Earth (au)
- Heliocentric velocity of Earth (au/s)
- Heliocentric position of Earth (au)
Accuracy:
The maximum deviations from the JPL DE96 ephemeris are as
follows:
barycentric velocity 0.42 m/s
barycentric position 6900 km
heliocentric velocity 0.42 m/s
heliocentric position 1600 km
This routine is adapted from the barvel and barcor
subroutines of P.Stumpff, which are described in
Astron. Astrophys. Suppl. Ser. 41, 1-8 (1980). Most of the
changes are merely cosmetic and do not affect the results at
all. However, some adjustments have been made so as to give
results that refer to the new (IAU 1976 'fk5') equinox
and precession, although the differences these changes make
relative to the results from Stumpff's original 'fk4' version
are smaller than the inherent accuracy of the algorithm. One
minor shortcoming in the original routines that has not been
corrected is that better numerical accuracy could be achieved
if the various polynomial evaluations were nested. Note also
that one of Stumpff's precession constants differs by 0.001 arcsec
from the value given in the Explanatory Supplement to the A.E.
"""
# note: in the original code, E was a shortcut for sorbel[0]
# and G was a shortcut for forbel[0]
# time arguments
t = (tdb-15019.5)/36525.0
tsq = t*t
# Values of all elements for the instant date
forbel = [0.0]*7
for k in range(8):
dlocal = fmod(DCFEL[k,0]+t*DCFEL[k,1]+tsq*DCFEL[k,2], TWOPI)
if k == 0:
dml = dlocal
else:
forbel[k-1] = dlocal
deps = fmod(DCEPS[0]+t*DCEPS[1]+tsq*DCEPS[2], TWOPI)
sorbel = [fmod(CCSEL[k,0]+t*CCSEL[k,1]+tsq*CCSEL[k,2], TWOPI)
for k in range(17)]
# Secular perturbations in longitude
sn = [sin(fmod(CCSEC[k,1]+t*CCSEC[k,2], TWOPI))
for k in range(4)]
# Periodic perturbations of the emb (Earth-Moon barycentre)
pertl = CCSEC[0,0] *sn[0] +CCSEC[1,0]*sn[1]+ \
(CCSEC[2,0]+t*CCSEC3)*sn[2] +CCSEC[3,0]*sn[3]
pertld = 0.0
pertr = 0.0
pertrd = 0.0
for k in range(15):
A = fmod(DCARGS[k,0]+t*DCARGS[k,1], TWOPI)
cosa = cos(A)
sina = sin(A)
pertl = pertl + CCAMPS[k,0]*cosa+CCAMPS[k,1]*sina
pertr = pertr + CCAMPS[k,2]*cosa+CCAMPS[k,3]*sina
if k < 11:
pertld = pertld+ \
(CCAMPS[k,1]*cosa-CCAMPS[k,0]*sina)*CCAMPS[k,4]
pertrd = pertrd+ \
(CCAMPS[k,3]*cosa-CCAMPS[k,2]*sina)*CCAMPS[k,4]
# Elliptic part of the motion of the emb
esq = sorbel[0]*sorbel[0]
dparam = 1.0-esq
param = dparam
twoe = sorbel[0]+sorbel[0]
twog = forbel[0]+forbel[0]
phi = twoe*((1.0-esq*0.125)*sin(forbel[0])+sorbel[0]*0.625*sin(twog) \
+esq*0.5416667*sin(forbel[0]+twog) )
F = forbel[0]+phi
sinf = sin(F)
cosf = cos(F)
dpsi = dparam/(1.0+(sorbel[0]*cosf))
phid = twoe*CCSGD*((1.0+esq*1.5)*cosf+sorbel[0]*(1.25-sinf*sinf*0.5))
psid = CCSGD*sorbel[0]*sinf/sqrt(param)
# Perturbed heliocentric motion of the emb
d1pdro = 1.0+pertr
drd = d1pdro*(psid+dpsi*pertrd)
drld = d1pdro*dpsi*(DCSLD+phid+pertld)
dtl = fmod(dml+phi+pertl, TWOPI)
dsinls = sin(dtl)
dcosls = cos(dtl)
dxhd = drd*dcosls-drld*dsinls
dyhd = drd*dsinls+drld*dcosls
# Influence of eccentricity, evection and variation on the
# geocentric motion of the Moon
pertl = 0.0
pertld = 0.0
pertp = 0.0
pertpd = 0.0
for k in range(3):
A = fmod(DCARGM[k,0]+t*DCARGM[k,1], TWOPI)
sina = sin(A)
cosa = cos(A)
pertl = pertl +CCAMPM[k,0]*sina
pertld = pertld+CCAMPM[k,1]*cosa
pertp = pertp +CCAMPM[k,2]*cosa
pertpd = pertpd-CCAMPM[k,3]*sina
# Heliocentric motion of the Earth
tl = forbel[1]+pertl
sinlm = sin(tl)
coslm = cos(tl)
sigma = CCKM/(1.0+pertp)
A = sigma*(CCMLD+pertld)
B = sigma*pertpd
dxhd = dxhd+(A*sinlm)+(B*coslm)
dyhd = dyhd-(A*coslm)+(B*sinlm)
dzhd = -(sigma*CCFDI*cos(forbel[2]))
# Barycentric motion of the Earth
dxbd = dxhd*DC1MME
dybd = dyhd*DC1MME
dzbd = dzhd*DC1MME
sinlp = [0.0] * 4
coslp = [0.0] * 4
for k in range(4):
plon = forbel[k+3]
pomg = sorbel[k+1]
pecc = sorbel[k+9]
tl = fmod(plon+2.0*pecc*sin(plon-pomg), TWOPI)
sinlp[k] = sin(tl)
coslp[k] = cos(tl)
dxbd = dxbd+(CCPAMV[k]*(sinlp[k]+pecc*sin(pomg)))
dybd = dybd-(CCPAMV[k]*(coslp[k]+pecc*cos(pomg)))
dzbd = dzbd-(CCPAMV[k]*sorbel[k+13]*cos(plon-sorbel[k+5]))
# Transition to mean equator of date
dcosep = cos(deps)
dsinep = sin(deps)
dyahd = dcosep*dyhd-dsinep*dzhd
dzahd = dsinep*dyhd+dcosep*dzhd
dyabd = dcosep*dybd-dsinep*dzbd
dzabd = dsinep*dybd+dcosep*dzbd
# Heliocentric coordinates of the Earth
dr = dpsi*d1pdro
flatm = CCIM*sin(forbel[2])
A = sigma*cos(flatm)
dxh = dr*dcosls-(A*coslm)
dyh = dr*dsinls-(A*sinlm)
dzh = -(sigma*sin(flatm))
# Barycentric coordinates of the Earth
dxb = dxh*DC1MME
dyb = dyh*DC1MME
dzb = dzh*DC1MME
for k in range(4):
flat = sorbel[k+13]*sin(forbel[k+3]-sorbel[k+5])
A = CCPAM[k]*(1.0-sorbel[k+9]*cos(forbel[k+3]-sorbel[k+1]))
B = A*cos(flat)
dxb = dxb-(B*coslp[k])
dyb = dyb-(B*sinlp[k])
dzb = dzb-(A*sin(flat))
# Transition to mean equator of date
dyah = dcosep*dyh-dsinep*dzh
dzah = dsinep*dyh+dcosep*dzh
dyab = dcosep*dyb-dsinep*dzb
dzab = dsinep*dyb+dcosep*dzb
# Copy result components into vectors, correcting for fk4 equinox
depj=epj(tdb)
deqcor = DS2R*(0.035+0.00085*(depj-B1950))
helVel = numpy.array((
dxhd-deqcor*dyahd,
dyahd+deqcor*dxhd,
dzahd,
))
barVel = numpy.array((
dxbd-deqcor*dyabd,
dyabd+deqcor*dxbd,
dzabd,
))
helPos = numpy.array((
dxh-deqcor*dyah,
dyah+deqcor*dxh,
dzah,
))
barPos = numpy.array((
dxb-deqcor*dyab,
dyab+deqcor*dxb,
dzab,
))
# Was precession to another equinox requested?
if deqx > 0.0:
# Yes: compute precession matrix from mjd date to Julian epoch deqx
dprema = prec(depj,deqx)
# Rotate helVel
helVel = numpy.dot(dprema, helVel)
# Rotate barVel
barVel = numpy.dot(dprema, barVel)
# Rotate helPos
helPos = numpy.dot(dprema, helPos)
# Rotate barPos
barPos = numpy.dot(dprema, barPos)
return (barVel, barPos, helVel, helPos)
def mappa(eq, tdb):
"""
Compute star-independent parameters in preparation for
conversions between mean place and geocentric apparent place.
The parameters produced by this routine are required in the
parallax, light deflection, aberration, and precession/nutation
parts of the mean/apparent transformations.
The reference frames and timescales used are post IAU 1976.
Inputs:
- eq epoch of mean equinox to be used (Julian)
- tdb TDB as a modified Julian date (JD-2400000.5)
Returned a tuple containing the following
star-independent mean-to-apparent parameters:
- time interval for proper motion (Julian years)
- barycentric position of the Earth (au)
- heliocentric direction of the Earth (unit vector)
- (grav rad Sun)*2/(Sun-Earth distance)
- bvc: barycentric velocity of Earth in units of c
- sqrt(1-v**2) where v=modulus(bvc)
- precession/nutation (3,3) matrix
References:
1984 Astronomical Almanac, pp b39-b41.
(also Lederle & Schwan, Astron. Astrophys. 134,
1-6, 1984)
Notes:
1) For tdb, the distinction between the required TDB and TT
is always negligible. Moreover, for all but the most
critical applications UTC is adequate.
2) The accuracy of the routines using the parameters amprms is
limited by the routine EVP, used here to compute the
Earth position and velocity by the methods of Stumpff.
The maximum error in the resulting aberration corrections is
about 0.3 milliarcsecond.
3) The barycentric position of the Earth and
heliocentric direction of the Earth are referred to
the mean equinox and equator of epoch eq.
"""
# Get Earth barycentric and heliocentric position and velocity
bVel, bPos, hVel, hPos = evp(tdb, eq)
# Heliocentric direction of earth (normalised) and modulus
hDir, hDist = vn(hPos)
# Aberration parameters
vbc = bVel * _CR
vbcDir, vbcMag = vn(vbc)
return (
# Time interval for proper motion correction (years)
epj(tdb) - eq,
# Barycentric position of Earth (au)
bPos,
# Heliocentric direction of earth
hDir,
# Light deflection parameter
_GR2 / hDist,
# Barycentric velocity of Earth, in units of C
vbc,
# sqrt(1-v**2) where v=modulus(bvc)
sqrt(1.0 - (vbcMag * vbcMag)),
# Precession/nutation matrix
prenut(eq, tdb),
)
def mappa (eq, tdb):
"""
Compute star-independent parameters in preparation for
conversions between mean place and geocentric apparent place.
The parameters produced by this routine are required in the
parallax, light deflection, aberration, and precession/nutation
parts of the mean/apparent transformations.
The reference frames and timescales used are post IAU 1976.
Inputs:
- eq epoch of mean equinox to be used (Julian)
- tdb TDB as a modified Julian date (JD-2400000.5)
Returned a tuple containing the following
star-independent mean-to-apparent parameters:
- time interval for proper motion (Julian years)
- barycentric position of the Earth (au)
- heliocentric direction of the Earth (unit vector)
- (grav rad Sun)*2/(Sun-Earth distance)
- bvc: barycentric velocity of Earth in units of c
- sqrt(1-v**2) where v=modulus(bvc)
- precession/nutation (3,3) matrix
References:
1984 Astronomical Almanac, pp b39-b41.
(also Lederle & Schwan, Astron. Astrophys. 134,
1-6, 1984)
Notes:
1) For tdb, the distinction between the required TDB and TT
is always negligible. Moreover, for all but the most
critical applications UTC is adequate.
2) The accuracy of the routines using the parameters amprms is
limited by the routine EVP, used here to compute the
Earth position and velocity by the methods of Stumpff.
The maximum error in the resulting aberration corrections is
about 0.3 milliarcsecond.
3) The barycentric position of the Earth and
heliocentric direction of the Earth are referred to
the mean equinox and equator of epoch eq.
"""
# Get Earth barycentric and heliocentric position and velocity
bVel, bPos, hVel, hPos = evp(tdb, eq)
# Heliocentric direction of earth (normalised) and modulus
hDir, hDist = vn(hPos)
# Aberration parameters
vbc = bVel * _CR
vbcDir, vbcMag = vn(vbc)
return (
# Time interval for proper motion correction (years)
epj(tdb)-eq,
# Barycentric position of Earth (au)
bPos,
# Heliocentric direction of earth
hDir,
# Light deflection parameter
_GR2/hDist,
# Barycentric velocity of Earth, in units of C
vbc,
# sqrt(1-v**2) where v=modulus(bvc)
sqrt(1.0 - (vbcMag * vbcMag)),
# Precession/nutation matrix
prenut(eq,tdb),
)
