// cl splits the work into two closures.
func cl(jde float64, earth, saturn *pp.V87Planet) (f1 func() (ΔU, B float64),
f2 func() (Bʹ, P, aEdge, bEdge float64)) {
const p = math.Pi / 180
var i, Ω float64
var l0, b0, R float64
Δ := 9.
var λ, β float64
var si, ci, sβ, cβ, sB float64
var sbʹ, cbʹ, slʹΩ, clʹΩ float64
f1 = func() (ΔU, B float64) {
// (45.1), p. 318
T := base.J2000Century(jde)
i = base.Horner(T, 28.075216*p, -.012998*p, .000004*p)
Ω = base.Horner(T, 169.50847*p, 1.394681*p, .000412*p)
// Step 2.
l0, b0, R = earth.Position(jde)
l0, b0 = pp.ToFK5(l0, b0, jde)
sl0, cl0 := math.Sincos(l0)
sb0 := math.Sin(b0)
// Steps 3, 4.
var l, b, r, x, y, z float64
f := func() {
τ := base.LightTime(Δ)
l, b, r = saturn.Position(jde - τ)
l, b = pp.ToFK5(l, b, jde)
sl, cl := math.Sincos(l)
sb, cb := math.Sincos(b)
x = r*cb*cl - R*cl0
y = r*cb*sl - R*sl0
z = r*sb - R*sb0
Δ = math.Sqrt(x*x + y*y + z*z)
}
f()
f()
// Step 5.
λ = math.Atan2(y, x)
β = math.Atan(z / math.Hypot(x, y))
// First part of step 6.
si, ci = math.Sincos(i)
sβ, cβ = math.Sincos(β)
sB = si*cβ*math.Sin(λ-Ω) - ci*sβ
B = math.Asin(sB) // return value
// Step 7.
N := 113.6655*p + .8771*p*T
lʹ := l - .01759*p/r
bʹ := b - .000764*p*math.Cos(l-N)/r
// Setup for steps 8, 9.
sbʹ, cbʹ = math.Sincos(bʹ)
slʹΩ, clʹΩ = math.Sincos(lʹ - Ω)
// Step 9.
sλΩ, cλΩ := math.Sincos(λ - Ω)
U1 := math.Atan2(si*sbʹ+ci*cbʹ*slʹΩ, cbʹ*clʹΩ)
U2 := math.Atan2(si*sβ+ci*cβ*sλΩ, cβ*cλΩ)
ΔU = math.Abs(U1 - U2) // return value
return
}
// TrueVSOP87 returns the true geometric position of the sun as ecliptic coordinates.
//
// Result computed by full VSOP87 theory. Result is at equator and equinox
// of date in the FK5 frame. It does not include nutation or aberration.
//
// s: ecliptic longitude in radians
// β: ecliptic latitude in radians
// R: range in AU
func TrueVSOP87(e *pp.V87Planet, jde float64) (s, β, R float64) {
l, b, r := e.Position(jde)
s = l + math.Pi
// FK5 correction.
λp := base.Horner(base.J2000Century(jde),
s, -1.397*math.Pi/180, -.00031*math.Pi/180)
sλp, cλp := math.Sincos(λp)
Δβ := .03916 / 3600 * math.Pi / 180 * (cλp - sλp)
// (25.9) p. 166
return base.PMod(s-.09033/3600*math.Pi/180, 2*math.Pi), Δβ - b, r
}
// TrueVSOP87 returns the true geometric position of the sun as ecliptic coordinates.
//
// Result computed by full VSOP87 theory. Result is at equator and equinox
// of date in the FK5 frame. It does not include nutation or aberration.
//
// s: ecliptic longitude
// β: ecliptic latitude
// R: range in AU
func TrueVSOP87(e *pp.V87Planet, jde float64) (s, β unit.Angle, R float64) {
l, b, r := e.Position(jde)
s = l + math.Pi
// FK5 correction.
λp := base.Horner(base.J2000Century(jde),
s.Rad(), -1.397*math.Pi/180, -.00031*math.Pi/180)
sλp, cλp := math.Sincos(λp)
Δβ := unit.AngleFromSec(.03916).Mul(cλp - sλp)
// (25.9) p. 166
s -= unit.AngleFromSec(.09033)
return s.Mod1(), Δβ - b, r
}
func xyz(e *pp.V87Planet, jde float64) (x, y, z float64) {
l, b, r := e.Position2000(jde)
s := l + math.Pi
β := -b
ss, cs := math.Sincos(s)
sβ, cβ := math.Sincos(β)
// (26.2) p. 172
x = r * cβ * cs
y = r * cβ * ss
z = r * sβ
return
}
// Position returns observed equatorial coordinates of a planet at a given time.
//
// Argument p must be a valid V87Planet object for the observed planet.
// Argument earth must be a valid V87Planet object for Earth.
//
// Results are right ascension and declination, α and δ in radians.
func Position(p, earth *pp.V87Planet, jde float64) (α, δ float64) {
L0, B0, R0 := earth.Position(jde)
L, B, R := p.Position(jde)
sB0, cB0 := math.Sincos(B0)
sL0, cL0 := math.Sincos(L0)
sB, cB := math.Sincos(B)
sL, cL := math.Sincos(L)
x := R*cB*cL - R0*cB0*cL0
y := R*cB*sL - R0*cB0*sL0
z := R*sB - R0*sB0
{
Δ := math.Sqrt(x*x + y*y + z*z) // (33.4) p. 224
τ := base.LightTime(Δ)
// repeating with jde-τ
L, B, R = p.Position(jde - τ)
sB, cB = math.Sincos(B)
sL, cL = math.Sincos(L)
x = R*cB*cL - R0*cB0*cL0
y = R*cB*sL - R0*cB0*sL0
z = R*sB - R0*sB0
}
λ := math.Atan2(y, x) // (33.1) p. 223
β := math.Atan2(z, math.Hypot(x, y)) // (33.2) p. 223
Δλ, Δβ := apparent.EclipticAberration(λ, β, jde)
λ, β = pp.ToFK5(λ+Δλ, β+Δβ, jde)
Δψ, Δε := nutation.Nutation(jde)
λ += Δψ
sε, cε := math.Sincos(nutation.MeanObliquity(jde) + Δε)
return coord.EclToEq(λ, β, sε, cε)
// Meeus gives a formula for elongation but doesn't spell out how to
// obtaion term λ0 and doesn't give an example solution.
}
func ap2a(j1, d float64, a bool, v *pp.V87Planet) (jde, r float64) {
// Meeus doesn't give a complete algorithm for finding accurate answers.
// The algorithm here starts with the approximate result algorithm
// then crawls along the curve at resultion d until three points
// are found containing the desired extremum. It's slow if the starting
// point is far away (the case of Neptune) or if high accuracy is
// demanded. 1 day accuracy is generally quick.
j0 := j1 - d
j2 := j1 + d
rr := make([]float64, 3)
_, _, rr[0] = v.Position2000(j0)
_, _, rr[1] = v.Position2000(j1)
_, _, rr[2] = v.Position2000(j2)
for {
if a {
if rr[1] > rr[0] && rr[1] > rr[2] {
break
}
} else {
if rr[1] < rr[0] && rr[1] < rr[2] {
break
}
}
if rr[0] < rr[2] == a {
j0 = j1
j1 = j2
j2 += d
rr[0] = rr[1]
rr[1] = rr[2]
_, _, rr[2] = v.Position2000(j2)
} else {
j2 = j1
j1 = j0
j0 -= d
rr[2] = rr[1]
rr[1] = rr[0]
_, _, rr[0] = v.Position2000(j0)
}
}
l, err := interp.NewLen3(j0, j2, rr)
if err != nil {
panic(err) // unexpected.
}
jde, r, err = l.Extremum()
if err != nil {
panic(err) // unexpected.
}
return
}
func ap2a(j1, d float64, a bool, v *pp.V87Planet) (jde, r float64) {
j0 := j1 - d
j2 := j1 + d
rr := make([]float64, 3)
_, _, rr[0] = v.Position2000(j0)
_, _, rr[1] = v.Position2000(j1)
_, _, rr[2] = v.Position2000(j2)
for {
if a {
if rr[1] > rr[0] && rr[1] > rr[2] {
break
}
} else {
if rr[1] < rr[0] && rr[1] < rr[2] {
break
}
}
if rr[0] < rr[2] == a {
j0 = j1
j1 = j2
j2 += d
rr[0] = rr[1]
rr[1] = rr[2]
_, _, rr[2] = v.Position2000(j2)
} else {
j2 = j1
j1 = j0
j0 -= d
rr[2] = rr[1]
rr[1] = rr[0]
_, _, rr[0] = v.Position2000(j0)
}
}
l, err := interp.NewLen3(j0, j2, rr)
if err != nil {
panic(err) // unexpected.
}
jde, r, err = l.Extremum()
if err != nil {
panic(err) // unexpected.
}
return
}
// Positions returns positions of the eight major moons of Saturn.
//
// Results returned in argument pos, which must not be nil.
//
// Result units are Saturn radii.
func Positions(jde float64, earth, saturn *pp.V87Planet, pos *[8]XY) {
s, β, R := solar.TrueVSOP87(earth, jde)
ss, cs := s.Sincos()
sβ := β.Sin()
Δ := 9.
var x, y, z float64
var JDE float64
f := func() {
τ := base.LightTime(Δ)
JDE = jde - τ
l, b, r := saturn.Position(JDE)
l, b = pp.ToFK5(l, b, JDE)
sl, cl := l.Sincos()
sb, cb := b.Sincos()
x = r*cb*cl + R*cs
y = r*cb*sl + R*ss
z = r*sb + R*sβ
Δ = math.Sqrt(x*x + y*y + z*z)
}
f()
f()
λ0 := unit.Angle(math.Atan2(y, x))
β0 := unit.Angle(math.Atan(z / math.Hypot(x, y)))
ecl := &coord.Ecliptic{λ0, β0}
precess.EclipticPosition(ecl, ecl,
base.JDEToJulianYear(jde), base.JDEToJulianYear(base.B1950), 0, 0)
λ0, β0 = ecl.Lon, ecl.Lat
q := newQs(JDE)
s4 := [9]r4{{}, // 0 unused
q.mimas(),
q.enceladus(),
q.tethys(),
q.dione(),
q.rhea(),
q.titan(),
q.hyperion(),
q.iapetus(),
}
var X, Y, Z [9]float64
for j := 1; j <= 8; j++ {
u := s4[j].λ - s4[j].Ω
w := s4[j].Ω - 168.8112*d
su, cu := math.Sincos(u)
sw, cw := math.Sincos(w)
sγ, cγ := math.Sincos(s4[j].γ)
r := s4[j].r
X[j] = r * (cu*cw - su*cγ*sw)
Y[j] = r * (su*cw*cγ + cu*sw)
Z[j] = r * su * sγ
}
Z[0] = 1
sλ0, cλ0 := λ0.Sincos()
sβ0, cβ0 := β0.Sincos()
var A, B, C [9]float64
for j := range X {
a := X[j]
b := q.c1*Y[j] - q.s1*Z[j]
c := q.s1*Y[j] + q.c1*Z[j]
a, b =
q.c2*a-q.s2*b,
q.s2*a+q.c2*b
A[j], b =
a*sλ0-b*cλ0,
a*cλ0+b*sλ0
B[j], C[j] =
b*cβ0+c*sβ0,
c*cβ0-b*sβ0
}
D := math.Atan2(A[0], C[0])
sD, cD := math.Sincos(D)
for j := 1; j <= 8; j++ {
X[j] = A[j]*cD - C[j]*sD
Y[j] = A[j]*sD + C[j]*cD
Z[j] = B[j]
d := X[j] / s4[j].r
X[j] += math.Abs(Z[j]) / k[j] * math.Sqrt(1-d*d)
W := Δ / (Δ + Z[j]/2475)
pos[j-1].X = X[j] * W
pos[j-1].Y = Y[j] * W
}
return
}
// Physical computes quantities for physical observations of Jupiter.
//
// Results:
// DS Planetocentric declination of the Sun.
// DE Planetocentric declination of the Earth.
// ω1 Longitude of the System I central meridian of the illuminated disk,
// as seen from Earth.
// ω2 Longitude of the System II central meridian of the illuminated disk,
// as seen from Earth.
// P Geocentric position angle of Jupiter's northern rotation pole.
func Physical(jde float64, earth, jupiter *pp.V87Planet) (DS, DE, ω1, ω2, P unit.Angle) {
// Step 1.
d := jde - 2433282.5
T1 := d / base.JulianCentury
const p = math.Pi / 180
α0 := 268*p + .1061*p*T1
δ0 := 64.5*p - .0164*p*T1
// Step 2.
W1 := 17.71*p + 877.90003539*p*d
W2 := 16.838*p + 870.27003539*p*d
// Step 3.
l0, b0, R := earth.Position(jde)
l0, b0 = pp.ToFK5(l0, b0, jde)
// Steps 4-7.
sl0, cl0 := l0.Sincos()
sb0 := b0.Sin()
Δ := 4. // surely better than 0.
var l, b unit.Angle
var r, x, y, z float64
f := func() {
τ := base.LightTime(Δ)
l, b, r = jupiter.Position(jde - τ)
l, b = pp.ToFK5(l, b, jde)
sb, cb := b.Sincos()
sl, cl := l.Sincos()
// (42.2) p. 289
x = r*cb*cl - R*cl0
y = r*cb*sl - R*sl0
z = r*sb - R*sb0
// (42.3) p. 289
Δ = math.Sqrt(x*x + y*y + z*z)
}
f()
f()
// Step 8.
ε0 := nutation.MeanObliquity(jde)
// Step 9.
sε0, cε0 := ε0.Sincos()
sl, cl := l.Sincos()
sb, cb := b.Sincos()
αs := math.Atan2(cε0*sl-sε0*sb/cb, cl)
δs := math.Asin(cε0*sb + sε0*cb*sl)
// Step 10.
sδs, cδs := math.Sincos(δs)
sδ0, cδ0 := math.Sincos(δ0)
DS = unit.Angle(math.Asin(-sδ0*sδs - cδ0*cδs*math.Cos(α0-αs)))
// Step 11.
u := y*cε0 - z*sε0
v := y*sε0 + z*cε0
α := math.Atan2(u, x)
δ := math.Atan(v / math.Hypot(x, u))
sδ, cδ := math.Sincos(δ)
sα0α, cα0α := math.Sincos(α0 - α)
ζ := math.Atan2(sδ0*cδ*cα0α-sδ*cδ0, cδ*sα0α)
// Step 12.
DE = unit.Angle(math.Asin(-sδ0*sδ - cδ0*cδ*math.Cos(α0-α)))
// Step 13.
ω1 = unit.Angle(W1 - ζ - 5.07033*p*Δ)
ω2 = unit.Angle(W2 - ζ - 5.02626*p*Δ)
// Step 14.
C := unit.Angle((2*r*Δ + R*R - r*r - Δ*Δ) / (4 * r * Δ))
if (l - l0).Sin() < 0 {
C = -C
}
ω1 = (ω1 + C).Mod1()
ω2 = (ω2 + C).Mod1()
// Step 15.
Δψ, Δε := nutation.Nutation(jde)
ε := ε0 + Δε
// Step 16.
sε, cε := ε.Sincos()
sα, cα := math.Sincos(α)
α += .005693 * p * (cα*cl0*cε + sα*sl0) / cδ
δ += .005693 * p * (cl0*cε*(sε/cε*cδ-sα*sδ) + cα*sδ*sl0)
// Step 17.
tδ := sδ / cδ
Δα := (cε+sε*sα*tδ)*Δψ.Rad() - cα*tδ*Δε.Rad()
Δδ := sε*cα*Δψ.Rad() + sα*Δε.Rad()
αʹ := α + Δα
δʹ := δ + Δδ
sα0, cα0 := math.Sincos(α0)
tδ0 := sδ0 / cδ0
Δα0 := (cε+sε*sα0*tδ0)*Δψ.Rad() - cα0*tδ0*Δε.Rad()
Δδ0 := sε*cα0*Δψ.Rad() + sα0*Δε.Rad()
α0ʹ := α0 + Δα0
δ0ʹ := δ0 + Δδ0
// Step 18.
sδʹ, cδʹ := math.Sincos(δʹ)
sδ0ʹ, cδ0ʹ := math.Sincos(δ0ʹ)
sα0ʹαʹ, cα0ʹαʹ := math.Sincos(α0ʹ - αʹ)
// (42.4) p. 290
P = unit.Angle(math.Atan2(cδ0ʹ*sα0ʹαʹ, sδ0ʹ*cδʹ-cδ0ʹ*sδʹ*cα0ʹαʹ))
if P < 0 {
P += 2 * math.Pi
}
return
}
// Physical computes quantities for physical observations of Mars.
//
// Results:
// DE planetocentric declination of the Earth.
// DS planetocentric declination of the Sun.
// ω Areographic longitude of the central meridian, as seen from Earth.
// P Geocentric position angle of Mars' northern rotation pole.
// Q Position angle of greatest defect of illumination.
// d Apparent diameter of Mars.
// q Greatest defect of illumination.
// k Illuminated fraction of the disk.
func Physical(jde float64, earth, mars *pp.V87Planet) (DE, DS, ω, P, Q, d, q unit.Angle, k float64) {
// Step 1.
T := base.J2000Century(jde)
const p = math.Pi / 180
// (42.1) p. 288
λ0 := 352.9065*p + 1.1733*p*T
β0 := 63.2818*p - .00394*p*T
// Step 2.
l0, b0, R := earth.Position(jde)
l0, b0 = pp.ToFK5(l0, b0, jde)
// Steps 3, 4.
sl0, cl0 := l0.Sincos()
sb0 := b0.Sin()
Δ := .5 // surely better than 0.
τ := base.LightTime(Δ)
var l, b unit.Angle
var r, x, y, z float64
f := func() {
l, b, r = mars.Position(jde - τ)
l, b = pp.ToFK5(l, b, jde)
sb, cb := b.Sincos()
sl, cl := l.Sincos()
// (42.2) p. 289
x = r*cb*cl - R*cl0
y = r*cb*sl - R*sl0
z = r*sb - R*sb0
// (42.3) p. 289
Δ = math.Sqrt(x*x + y*y + z*z)
τ = base.LightTime(Δ)
}
f()
f()
// Step 5.
λ := math.Atan2(y, x)
β := math.Atan(z / math.Hypot(x, y))
// Step 6.
sβ0, cβ0 := math.Sincos(β0)
sβ, cβ := math.Sincos(β)
DE = unit.Angle(math.Asin(-sβ0*sβ - cβ0*cβ*math.Cos(λ0-λ)))
// Step 7.
N := 49.5581*p + .7721*p*T
lʹ := l.Rad() - .00697*p/r
bʹ := b.Rad() - .000225*p*math.Cos(l.Rad()-N)/r
// Step 8.
sbʹ, cbʹ := math.Sincos(bʹ)
DS = unit.Angle(math.Asin(-sβ0*sbʹ - cβ0*cbʹ*math.Cos(λ0-lʹ)))
// Step 9.
W := 11.504*p + 350.89200025*p*(jde-τ-2433282.5)
// Step 10.
ε0 := nutation.MeanObliquity(jde)
sε0, cε0 := ε0.Sincos()
α0, δ0 := coord.EclToEq(unit.Angle(λ0), unit.Angle(β0), sε0, cε0)
// Step 11.
u := y*cε0 - z*sε0
v := y*sε0 + z*cε0
α := math.Atan2(u, x)
δ := math.Atan(v / math.Hypot(x, u))
sδ, cδ := math.Sincos(δ)
sδ0, cδ0 := δ0.Sincos()
sα0α, cα0α := math.Sincos(α0.Rad() - α)
ζ := math.Atan2(sδ0*cδ*cα0α-sδ*cδ0, cδ*sα0α)
// Step 12.
ω = unit.Angle(W - ζ).Mod1()
// Step 13.
Δψ, Δε := nutation.Nutation(jde)
// Step 14.
sl0λ, cl0λ := math.Sincos(l0.Rad() - λ)
λ += .005693 * p * cl0λ / cβ
β += .005693 * p * sl0λ * sβ
// Step 15.
λ0 += Δψ.Rad()
λ += Δψ.Rad()
ε := ε0 + Δε
// Step 16.
sε, cε := ε.Sincos()
α0ʹ, δ0ʹ := coord.EclToEq(unit.Angle(λ0), unit.Angle(β0), sε, cε)
αʹ, δʹ := coord.EclToEq(unit.Angle(λ), unit.Angle(β), sε, cε)
// Step 17.
sδ0ʹ, cδ0ʹ := δ0ʹ.Sincos()
sδʹ, cδʹ := δʹ.Sincos()
sα0ʹαʹ, cα0ʹαʹ := (α0ʹ - αʹ).Sincos()
// (42.4) p. 290
P = unit.Angle(math.Atan2(cδ0ʹ*sα0ʹαʹ, sδ0ʹ*cδʹ-cδ0ʹ*sδʹ*cα0ʹαʹ))
if P < 0 {
P += 2 * math.Pi
}
// Step 18.
s := l0 + math.Pi
ss, cs := s.Sincos()
αs := math.Atan2(cε*ss, cs)
δs := math.Asin(sε * ss)
sδs, cδs := math.Sincos(δs)
sαsα, cαsα := math.Sincos(αs - α)
χ := math.Atan2(cδs*sαsα, sδs*cδ-cδs*sδ*cαsα)
Q = unit.Angle(χ) + math.Pi
// Step 19.
d = unit.AngleFromSec(9.36) / unit.Angle(Δ)
k = illum.Fraction(r, Δ, R)
q = d.Mul(1 - k)
return
}
// Positions computes positions of moons of Jupiter.
//
// High accuracy method based on theory "E5." Results returned in
// argument pos, which must not be nil. Returned coordinates in units
// of Jupiter radii.
func E5(jde float64, earth, jupiter *pp.V87Planet, pos *[4]XY) {
// I'll interject that I don't trust the results of this function.
// There is obviously a great chance of typographic errors.
// My Y results for the test case of the example don't agree with
// Meeus's well at all, but do agree with the results from the less
// accurate method. This would seem to indicate a typo in Meeus's
// computer implementation. On the other hand, while my X results
// agree reasonably well with his, our X results for satellite III
// don't agree well with the result from the less accurate method,
// perhaps indicating a typo in the presented algorithm.
// variables assigned in following block
var λ0, β0, t float64
Δ := 5.
{
s, β, R := solar.TrueVSOP87(earth, jde)
ss, cs := math.Sincos(s.Rad())
sβ := math.Sin(β.Rad())
τ := base.LightTime(Δ)
var x, y, z float64
f := func() {
l, b, r := jupiter.Position(jde - τ)
sl, cl := math.Sincos(l.Rad())
sb, cb := math.Sincos(b.Rad())
x = r*cb*cl + R*cs
y = r*cb*sl + R*ss
z = r*sb + R*sβ
Δ = math.Sqrt(x*x + y*y + z*z)
τ = base.LightTime(Δ)
}
f()
f()
λ0 = math.Atan2(y, x)
β0 = math.Atan(z / math.Hypot(x, y))
t = jde - 2443000.5 - τ
}
const p = math.Pi / 180
l1 := 106.07719*p + 203.48895579*p*t
l2 := 175.73161*p + 101.374724735*p*t
l3 := 120.55883*p + 50.317609207*p*t
l4 := 84.44459*p + 21.571071177*p*t
π1 := 97.0881*p + .16138586*p*t
π2 := 154.8663*p + .04726307*p*t
π3 := 188.184*p + .00712734*p*t
π4 := 335.2868*p + .00184*p*t
ω1 := 312.3346*p - .13279386*p*t
ω2 := 100.4411*p - .03263064*p*t
ω3 := 119.1942*p - .00717703*p*t
ω4 := 322.6186*p - .00175934*p*t
Γ := .33033*p*math.Sin(163.679*p+.0010512*p*t) +
.03439*p*math.Sin(34.486*p-.0161731*p*t)
Φλ := 199.6766*p + .1737919*p*t
ψ := 316.5182*p - .00000208*p*t
G := 30.23756*p + .0830925701*p*t + Γ
Gʹ := 31.97853*p + .0334597339*p*t
const Π = 13.469942 * p
Σ1 := .47259*p*math.Sin(2*(l1-l2)) +
-.03478*p*math.Sin(π3-π4) +
.01081*p*math.Sin(l2-2*l3+π3) +
.00738*p*math.Sin(Φλ) +
.00713*p*math.Sin(l2-2*l3+π2) +
-.00674*p*math.Sin(π1+π3-2*Π-2*G) +
.00666*p*math.Sin(l2-2*l3+π4) +
.00445*p*math.Sin(l1-π3) +
-.00354*p*math.Sin(l1-l2) +
-.00317*p*math.Sin(2*ψ-2*Π) +
.00265*p*math.Sin(l1-π4) +
-.00186*p*math.Sin(G) +
.00162*p*math.Sin(π2-π3) +
.00158*p*math.Sin(4*(l1-l2)) +
-.00155*p*math.Sin(l1-l3) +
-.00138*p*math.Sin(ψ+ω3-2*Π-2*G) +
-.00115*p*math.Sin(2*(l1-2*l2+ω2)) +
.00089*p*math.Sin(π2-π4) +
.00085*p*math.Sin(l1+π3-2*Π-2*G) +
.00083*p*math.Sin(ω2-ω3) +
.00053*p*math.Sin(ψ-ω2)
Σ2 := 1.06476*p*math.Sin(2*(l2-l3)) +
.04256*p*math.Sin(l1-2*l2+π3) +
.03581*p*math.Sin(l2-π3) +
.02395*p*math.Sin(l1-2*l2+π4) +
.01984*p*math.Sin(l2-π4) +
-.01778*p*math.Sin(Φλ) +
.01654*p*math.Sin(l2-π2) +
.01334*p*math.Sin(l2-2*l3+π2) +
.01294*p*math.Sin(π3-π4) +
-.01142*p*math.Sin(l2-l3) +
-.01057*p*math.Sin(G) +
-.00775*p*math.Sin(2*(ψ-Π)) +
.00524*p*math.Sin(2*(l1-l2)) +
-.0046*p*math.Sin(l1-l3) +
.00316*p*math.Sin(ψ-2*G+ω3-2*Π) +
-.00203*p*math.Sin(π1+π3-2*Π-2*G) +
.00146*p*math.Sin(ψ-ω3) +
-.00145*p*math.Sin(2*G) +
.00125*p*math.Sin(ψ-ω4) +
-.00115*p*math.Sin(l1-2*l3+π3) +
-.00094*p*math.Sin(2*(l2-ω2)) +
.00086*p*math.Sin(2*(l1-2*l2+ω2)) +
-.00086*p*math.Sin(5*Gʹ-2*G+52.225*p) +
-.00078*p*math.Sin(l2-l4) +
-.00064*p*math.Sin(3*l3-7*l4+4*π4) +
.00064*p*math.Sin(π1-π4) +
-.00063*p*math.Sin(l1-2*l3+π4) +
.00058*p*math.Sin(ω3-ω4) +
.00056*p*math.Sin(2*(ψ-Π-G)) +
.00056*p*math.Sin(2*(l2-l4)) +
.00055*p*math.Sin(2*(l1-l3)) +
.00052*p*math.Sin(3*l3-7*l4+π3+3*π4) +
-.00043*p*math.Sin(l1-π3) +
.00041*p*math.Sin(5*(l2-l3)) +
.00041*p*math.Sin(π4-Π) +
.00032*p*math.Sin(ω2-ω3) +
.00032*p*math.Sin(2*(l3-G-Π))
Σ3 := .1649*p*math.Sin(l3-π3) +
.09081*p*math.Sin(l3-π4) +
-.06907*p*math.Sin(l2-l3) +
.03784*p*math.Sin(π3-π4) +
.01846*p*math.Sin(2*(l3-l4)) +
-.0134*p*math.Sin(G) +
-.01014*p*math.Sin(2*(ψ-Π)) +
.00704*p*math.Sin(l2-2*l3+π3) +
-.0062*p*math.Sin(l2-2*l3+π2) +
-.00541*p*math.Sin(l3-l4) +
.00381*p*math.Sin(l2-2*l3+π4) +
.00235*p*math.Sin(ψ-ω3) +
.00198*p*math.Sin(ψ-ω4) +
.00176*p*math.Sin(Φλ) +
.0013*p*math.Sin(3*(l3-l4)) +
.00125*p*math.Sin(l1-l3) +
-.00119*p*math.Sin(5*Gʹ-2*G+52.225*p) +
.00109*p*math.Sin(l1-l2) +
-.001*p*math.Sin(3*l3-7*l4+4*π4) +
.00091*p*math.Sin(ω3-ω4) +
.0008*p*math.Sin(3*l3-7*l4+π3+3*π4) +
-.00075*p*math.Sin(2*l2-3*l3+π3) +
.00072*p*math.Sin(π1+π3-2*Π-2*G) +
.00069*p*math.Sin(π4-Π) +
-.00058*p*math.Sin(2*l3-3*l4+π4) +
-.00057*p*math.Sin(l3-2*l4+π4) +
.00056*p*math.Sin(l3+π3-2*Π-2*G) +
-.00052*p*math.Sin(l2-2*l3+π1) +
-.00050*p*math.Sin(π2-π3) +
.00048*p*math.Sin(l3-2*l4+π3) +
-.00045*p*math.Sin(2*l2-3*l3+π4) +
-.00041*p*math.Sin(π2-π4) +
-.00038*p*math.Sin(2*G) +
-.00037*p*math.Sin(π3-π4+ω3-ω4) +
-.00032*p*math.Sin(3*l3-7*l4+2*π3+2*π4) +
.0003*p*math.Sin(4*(l3-l4)) +
.00029*p*math.Sin(l3+π4-2*Π-2*G) +
-.00028*p*math.Sin(ω3+ψ-2*Π-2*G) +
.00026*p*math.Sin(l3-Π-G) +
.00024*p*math.Sin(l2-3*l3+2*l4) +
.00021*p*math.Sin(2*(l3-Π-G)) +
-.00021*p*math.Sin(l3-π2) +
.00017*p*math.Sin(2*(l3-π3))
Σ4 := .84287*p*math.Sin(l4-π4) +
.03431*p*math.Sin(π4-π3) +
-.03305*p*math.Sin(2*(ψ-Π)) +
-.03211*p*math.Sin(G) +
-.01862*p*math.Sin(l4-π3) +
.01186*p*math.Sin(ψ-ω4) +
.00623*p*math.Sin(l4+π4-2*G-2*Π) +
.00387*p*math.Sin(2*(l4-π4)) +
-.00284*p*math.Sin(5*Gʹ-2*G+52.225*p) +
-.00234*p*math.Sin(2*(ψ-π4)) +
-.00223*p*math.Sin(l3-l4) +
-.00208*p*math.Sin(l4-Π) +
.00178*p*math.Sin(ψ+ω4-2*π4) +
.00134*p*math.Sin(π4-Π) +
.00125*p*math.Sin(2*(l4-G-Π)) +
-.00117*p*math.Sin(2*G) +
-.00112*p*math.Sin(2*(l3-l4)) +
.00107*p*math.Sin(3*l3-7*l4+4*π4) +
.00102*p*math.Sin(l4-G-Π) +
.00096*p*math.Sin(2*l4-ψ-ω4) +
.00087*p*math.Sin(2*(ψ-ω4)) +
-.00085*p*math.Sin(3*l3-7*l4+π3+3*π4) +
.00085*p*math.Sin(l3-2*l4+π4) +
-.00081*p*math.Sin(2*(l4-ψ)) +
.00071*p*math.Sin(l4+π4-2*Π-3*G) +
.00061*p*math.Sin(l1-l4) +
-.00056*p*math.Sin(ψ-ω3) +
-.00054*p*math.Sin(l3-2*l4+π3) +
.00051*p*math.Sin(l2-l4) +
.00042*p*math.Sin(2*(ψ-G-Π)) +
.00039*p*math.Sin(2*(π4-ω4)) +
.00036*p*math.Sin(ψ+Π-π4-ω4) +
.00035*p*math.Sin(2*Gʹ-G+188.37*p) +
-.00035*p*math.Sin(l4-π4+2*Π-2*ψ) +
-.00032*p*math.Sin(l4+π4-2*Π-G) +
.0003*p*math.Sin(2*Gʹ-2*G+149.15*p) +
.00029*p*math.Sin(3*l3-7*l4+2*π3+2*π4) +
.00028*p*math.Sin(l4-π4+2*ψ-2*Π) +
-.00028*p*math.Sin(2*(l4-ω4)) +
-.00027*p*math.Sin(π3-π4+ω3-ω4) +
-.00026*p*math.Sin(5*Gʹ-3*G+188.37*p) +
.00025*p*math.Sin(ω4-ω3) +
-.00025*p*math.Sin(l2-3*l3+2*l4) +
-.00023*p*math.Sin(3*(l3-l4)) +
.00021*p*math.Sin(2*l4-2*Π-3*G) +
-.00021*p*math.Sin(2*l3-3*l4+π4) +
.00019*p*math.Sin(l4-π4-G) +
-.00019*p*math.Sin(2*l4-π3-π4) +
-.00018*p*math.Sin(l4-π4+G) +
-.00016*p*math.Sin(l4+π3-2*Π-2*G)
L1 := l1 + Σ1
L2 := l2 + Σ2
L3 := l3 + Σ3
L4 := l4 + Σ4
// variables assigned in following block
var I float64
X := make([]float64, 5)
Y := make([]float64, 5)
Z := make([]float64, 5)
var R [4]float64
{
L := [...]float64{L1, L2, L3, L4}
B := [...]float64{
math.Atan(.0006393*p*math.Sin(L1-ω1) +
.0001825*p*math.Sin(L1-ω2) +
.0000329*p*math.Sin(L1-ω3) +
-.0000311*p*math.Sin(L1-ψ) +
.0000093*p*math.Sin(L1-ω4) +
.0000075*p*math.Sin(3*L1-4*l2-1.9927*Σ1+ω2) +
.0000046*p*math.Sin(L1+ψ-2*Π-2*G)),
math.Atan(.0081004*p*math.Sin(L2-ω2) +
.0004512*p*math.Sin(L2-ω3) +
-.0003284*p*math.Sin(L2-ψ) +
.0001160*p*math.Sin(L2-ω4) +
.0000272*p*math.Sin(l1-2*l3+1.0146*Σ2+ω2) +
-.0000144*p*math.Sin(L2-ω1) +
.0000143*p*math.Sin(L2+ψ-2*Π-2*G) +
.0000035*p*math.Sin(L2-ψ+G) +
-.0000028*p*math.Sin(l1-2*l3+1.0146*Σ2+ω3)),
math.Atan(.0032402*p*math.Sin(L3-ω3) +
-.0016911*p*math.Sin(L3-ψ) +
.0006847*p*math.Sin(L3-ω4) +
-.0002797*p*math.Sin(L3-ω2) +
.0000321*p*math.Sin(L3+ψ-2*Π-2*G) +
.0000051*p*math.Sin(L3-ψ+G) +
-.0000045*p*math.Sin(L3-ψ-G) +
-.0000045*p*math.Sin(L3+ψ-2*Π) +
.0000037*p*math.Sin(L3+ψ-2*Π-3*G) +
.000003*p*math.Sin(2*l2-3*L3+4.03*Σ3+ω2) +
-.0000021*p*math.Sin(2*l2-3*L3+4.03*Σ3+ω3)),
math.Atan(-.0076579*p*math.Sin(L4-ψ) +
.0044134*p*math.Sin(L4-ω4) +
-.0005112*p*math.Sin(L4-ω3) +
.0000773*p*math.Sin(L4+ψ-2*Π-2*G) +
.0000104*p*math.Sin(L4-ψ+G) +
-.0000102*p*math.Sin(L4-ψ-G) +
.0000088*p*math.Sin(L4+ψ-2*Π-3*G) +
-.0000038*p*math.Sin(L4+ψ-2*Π-G)),
}
R = [...]float64{
5.90569 * (1 +
-.0041339*math.Cos(2*(l1-l2)) +
-.0000387*math.Cos(l1-π3) +
-.0000214*math.Cos(l1-π4) +
.000017*math.Cos(l1-l2) +
-.0000131*math.Cos(4*(l1-l2)) +
.0000106*math.Cos(l1-l3) +
-.0000066*math.Cos(l1+π3-2*Π-2*G)),
9.39657 * (1 +
.0093848*math.Cos(l1-l2) +
-.0003116*math.Cos(l2-π3) +
-.0001744*math.Cos(l2-π4) +
-.0001442*math.Cos(l2-π2) +
.0000553*math.Cos(l2-l3) +
.0000523*math.Cos(l1-l3) +
-.0000290*math.Cos(2*(l1-l2)) +
.0000164*math.Cos(2*(l2-ω2)) +
.0000107*math.Cos(l1-2*l3+π3) +
-.0000102*math.Cos(l2-π1) +
-.0000091*math.Cos(2*(l1-l3))),
14.98832 * (1 +
-.0014388*math.Cos(l3-π3) +
-.0007917*math.Cos(l3-π4) +
.0006342*math.Cos(l2-l3) +
-.0001761*math.Cos(2*(l3-l4)) +
.0000294*math.Cos(l3-l4) +
-.0000156*math.Cos(3*(l3-l4)) +
.0000156*math.Cos(l1-l3) +
-.0000153*math.Cos(l1-l2) +
.000007*math.Cos(2*l2-3*l3+π3) +
-.0000051*math.Cos(l3+π3-2*Π-2*G)),
26.36273 * (1 +
-.0073546*math.Cos(l4-π4) +
.0001621*math.Cos(l4-π3) +
.0000974*math.Cos(l3-l4) +
-.0000543*math.Cos(l4+π4-2*Π-2*G) +
-.0000271*math.Cos(2*(l4-π4)) +
.0000182*math.Cos(l4-Π) +
.0000177*math.Cos(2*(l3-l4)) +
-.0000167*math.Cos(2*l4-ψ-ω4) +
.0000167*math.Cos(ψ-ω4) +
-.0000155*math.Cos(2*(l4-Π-G)) +
.0000142*math.Cos(2*(l4-ψ)) +
.0000105*math.Cos(l1-l4) +
.0000092*math.Cos(l2-l4) +
-.0000089*math.Cos(l4-Π-G) +
-.0000062*math.Cos(l4+π4-2*Π-3*G) +
.0000048*math.Cos(2*(l4-ω4))),
}
// p. 311
T0 := (jde - 2433282.423) / base.JulianCentury
P := (1.3966626*p + .0003088*p*T0) * T0
for i := range L {
L[i] += P
}
ψ += P
T := (jde - base.J1900) / base.JulianCentury
I = 3.120262*p + .0006*p*T
for i := range L {
sLψ, cLψ := math.Sincos(L[i] - ψ)
sB, cB := math.Sincos(B[i])
X[i] = R[i] * cLψ * cB
Y[i] = R[i] * sLψ * cB
Z[i] = R[i] * sB
}
}
Z[4] = 1
// p. 312
A := make([]float64, 5)
B := make([]float64, 5)
C := make([]float64, 5)
sI, cI := math.Sincos(I)
Ω := pe.Node(pe.Jupiter, jde)
sΩ, cΩ := Ω.Sincos()
sΦ, cΦ := math.Sincos(ψ - Ω.Rad())
si, ci := pe.Inc(pe.Jupiter, jde).Sincos()
sλ0, cλ0 := math.Sincos(λ0)
sβ0, cβ0 := math.Sincos(β0)
for i := range A {
// step 1
a := X[i]
b := Y[i]*cI - Z[i]*sI
c := Y[i]*sI + Z[i]*cI
// step 2
a, b =
a*cΦ-b*sΦ,
a*sΦ+b*cΦ
// step 3
b, c =
b*ci-c*si,
b*si+c*ci
// step 4
a, b =
a*cΩ-b*sΩ,
a*sΩ+b*cΩ
// step 5
a, b =
a*sλ0-b*cλ0,
a*cλ0+b*sλ0
// step 6
A[i] = a
B[i] = c*sβ0 + b*cβ0
C[i] = c*cβ0 - b*sβ0
}
sD, cD := math.Sincos(math.Atan2(A[4], C[4]))
// p. 313
for i := 0; i < 4; i++ {
x := A[i]*cD - C[i]*sD
y := A[i]*sD + C[i]*cD
z := B[i]
// differential light time
d := x / R[i]
x += math.Abs(z) / k[i] * math.Sqrt(1-d*d)
// perspective effect
W := Δ / (Δ + z/2095)
pos[i].X = x * W
pos[i].Y = y * W
}
return
}
// LongitudeJ2000 returns geometric longitude referenced to equinox J2000.
func LongitudeJ2000(e *pp.V87Planet, jde float64) (l float64) {
l, _, _ = e.Position2000(jde)
return base.PMod(l+math.Pi-.09033/3600*math.Pi/180, 2*math.Pi)
}
// LongitudeJ2000 returns geometric longitude referenced to equinox J2000.
func LongitudeJ2000(e *pp.V87Planet, jde float64) (l unit.Angle) {
l, _, _ = e.Position2000(jde)
return (l + math.Pi - unit.AngleFromSec(.09033)).Mod1()
}