// ApparentVSOP87 returns the apparent position of the sun as ecliptic coordinates.
//
// Result computed by VSOP87, at equator and equinox of date in the FK5 frame,
// and includes effects of nutation and aberration.
//
// λ: ecliptic longitude in radians
// β: ecliptic latitude in radians
// R: range in AU
func ApparentVSOP87(e *pp.V87Planet, jde float64) (λ, β, R float64) {
// note: see duplicated code in ApparentEquatorialVSOP87.
s, β, R := TrueVSOP87(e, jde)
Δψ, _ := nutation.Nutation(jde)
a := aberration(R)
return s + Δψ + a, β, R
}
// 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.
}
// Nutation returns corrections due to nutation for equatorial coordinates
// of an object.
//
// Results are invalid for objects very near the celestial poles.
func Nutation(α, δ, jd float64) (Δα1, Δδ1 float64) {
ε := nutation.MeanObliquity(jd)
sε, cε := math.Sincos(ε)
Δψ, Δε := nutation.Nutation(jd)
sα, cα := math.Sincos(α)
tδ := math.Tan(δ)
// (23.1) p. 151
Δα1 = (cε+sε*sα*tδ)*Δψ - cα*tδ*Δε
Δδ1 = sε*cα*Δψ + sα*Δε
return
}
// Nutation returns corrections due to nutation for equatorial coordinates
// of an object.
//
// Results are invalid for objects very near the celestial poles.
func Nutation(α unit.RA, δ unit.Angle, jd float64) (Δα1 unit.HourAngle, Δδ1 unit.Angle) {
ε := nutation.MeanObliquity(jd)
sε, cε := ε.Sincos()
Δψ, Δε := nutation.Nutation(jd)
sα, cα := α.Sincos()
tδ := δ.Tan()
// (23.1) p. 151
Δα1 = unit.HourAngle((cε+sε*sα*tδ)*Δψ.Rad() - cα*tδ*Δε.Rad())
Δδ1 = Δψ.Mul(sε*cα) + Δε.Mul(sα)
return
}
// ApparentEquatorialVSOP87 returns the apparent position of the sun as equatorial coordinates.
//
// Result computed by VSOP87, at equator and equinox of date in the FK5 frame,
// and includes effects of nutation and aberration.
//
// α: right ascension in radians
// δ: declination in radians
// R: range in AU
func ApparentEquatorialVSOP87(e *pp.V87Planet, jde float64) (α, δ, R float64) {
// note: duplicate code from ApparentVSOP87 so we can keep Δε.
// see also duplicate code in time.E().
s, β, R := TrueVSOP87(e, jde)
Δψ, Δε := nutation.Nutation(jde)
a := aberration(R)
λ := s + Δψ + a
ε := nutation.MeanObliquity(jde) + Δε
sε, cε := math.Sincos(ε)
α, δ = coord.EclToEq(λ, β, sε, cε)
return
}
func ExampleNutation() {
// Example 22.a, p. 148.
jd := julian.CalendarGregorianToJD(1987, 4, 10)
Δψ, Δε := nutation.Nutation(jd)
ε0 := nutation.MeanObliquity(jd)
ε := ε0 + Δε
fmt.Printf("%+.3d\n", sexa.FmtAngle(Δψ))
fmt.Printf("%+.3d\n", sexa.FmtAngle(Δε))
fmt.Printf("%.3d\n", sexa.FmtAngle(ε0))
fmt.Printf("%.3d\n", sexa.FmtAngle(ε))
// Output:
// -3″.788
// +9″.443
// 23°26′27″.407
// 23°26′36″.850
}
// E computes the "equation of time" for the given JDE.
//
// Parameter e must be a planetposition.V87Planet object for Earth obtained
// with planetposition.LoadPlanet.
//
// Result is equation of time as an hour angle.
func E(jde float64, e *pp.V87Planet) unit.HourAngle {
τ := base.J2000Century(jde) * .1
L0 := l0(τ)
// code duplicated from solar.ApparentEquatorialVSOP87 so that
// we can keep Δψ and cε
s, β, R := solar.TrueVSOP87(e, jde)
Δψ, Δε := nutation.Nutation(jde)
a := unit.AngleFromSec(-20.4898).Div(R)
λ := s + Δψ + a
ε := nutation.MeanObliquity(jde) + Δε
sε, cε := ε.Sincos()
α, _ := coord.EclToEq(λ, β, sε, cε)
// (28.1) p. 183
E := L0 - unit.AngleFromDeg(.0057183) - unit.Angle(α) + Δψ.Mul(cε)
return unit.HourAngle((E + math.Pi).Mod1() - math.Pi)
}
// E computes the "equation of time" for the given JDE.
//
// Parameter e must be a planetposition.V87Planet object for Earth obtained
// with planetposition.LoadPlanet.
//
// Result is equation of time as an hour angle in radians.
func E(jde float64, e *pp.V87Planet) float64 {
τ := base.J2000Century(jde) * .1
L0 := l0(τ)
// code duplicated from solar.ApparentEquatorialVSOP87 so that
// we can keep Δψ and cε
s, β, R := solar.TrueVSOP87(e, jde)
Δψ, Δε := nutation.Nutation(jde)
a := -20.4898 / 3600 * math.Pi / 180 / R
λ := s + Δψ + a
ε := nutation.MeanObliquity(jde) + Δε
sε, cε := math.Sincos(ε)
α, _ := coord.EclToEq(λ, β, sε, cε)
// (28.1) p. 183
E := L0 - .0057183*math.Pi/180 - α + Δψ*cε
return base.PMod(E+math.Pi, 2*math.Pi) - math.Pi
}
// Ephemeris returns the apparent orientation of the sun at the given jd.
//
// Results:
// P: Position angle of the solar north pole.
// B0: Heliographic latitude of the center of the solar disk.
// L0: Heliographic longitude of the center of the solar disk.
//
// All results in radians.
func Ephemeris(jd float64, e *pp.V87Planet) (P, B0, L0 float64) {
θ := (jd - 2398220) * 2 * math.Pi / 25.38
I := 7.25 * math.Pi / 180
K := 73.6667*math.Pi/180 +
1.3958333*math.Pi/180*(jd-2396758)/base.JulianCentury
L, _, R := solar.TrueVSOP87(e, jd)
Δψ, Δε := nutation.Nutation(jd)
ε0 := nutation.MeanObliquity(jd)
ε := ε0 + Δε
λ := L - 20.4898/3600*math.Pi/180/R
λp := λ + Δψ
sλK, cλK := math.Sincos(λ - K)
sI, cI := math.Sincos(I)
tx := -math.Cos(λp) * math.Tan(ε)
ty := -cλK * math.Tan(I)
P = math.Atan(tx) + math.Atan(ty)
B0 = math.Asin(sλK * sI)
η := math.Atan2(-sλK*cI, -cλK)
L0 = base.PMod(η-θ, 2*math.Pi)
return
}
// Ephemeris returns the apparent orientation of the sun at the given jd.
//
// Results:
// P: Position angle of the solar north pole.
// B0: Heliographic latitude of the center of the solar disk.
// L0: Heliographic longitude of the center of the solar disk.
func Ephemeris(jd float64, e *pp.V87Planet) (P, B0, L0 unit.Angle) {
θ := unit.Angle((jd - 2398220) * 2 * math.Pi / 25.38)
I := unit.AngleFromDeg(7.25)
K := unit.AngleFromDeg(73.6667) +
unit.AngleFromDeg(1.3958333).Mul((jd-2396758)/base.JulianCentury)
L, _, R := solar.TrueVSOP87(e, jd)
Δψ, Δε := nutation.Nutation(jd)
ε0 := nutation.MeanObliquity(jd)
ε := ε0 + Δε
λ := L - unit.AngleFromSec(20.4898).Div(R)
λp := λ + Δψ
sλK, cλK := (λ - K).Sincos()
sI, cI := I.Sincos()
tx := -(λp.Cos() * ε.Tan())
ty := -(cλK * I.Tan())
P = unit.Angle(math.Atan(tx) + math.Atan(ty))
B0 = unit.Angle(math.Asin(sλK * sI))
η := unit.Angle(math.Atan2(-sλK*cI, -cλK))
L0 = (η - θ).Mod1()
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
}
func newMoon(jde float64) *moon {
m := &moon{jde: jde}
// Δψ, F, Ω, p. 372.
var Δε unit.Angle
m.Δψ, Δε = nutation.Nutation(jde)
T := base.J2000Century(jde)
m.F = unit.AngleFromDeg(base.Horner(T,
93.272095, 483202.0175233, -.0036539, -1/3526000, 1/863310000))
F := m.F.Rad()
m.Ω = unit.AngleFromDeg(base.Horner(T,
125.0445479, -1934.1362891, .0020754, 1/467441, -1/60616000))
// true ecliptic
m.sε, m.cε = math.Sincos((nutation.MeanObliquity(jde) + Δε).Rad())
// ρ, σ, τ, p. 372,373
D := unit.AngleFromDeg(base.Horner(T,
297.8501921, 445267.1114034, -.0018819, 1/545868, -1/113065000)).Rad()
M := unit.AngleFromDeg(base.Horner(T,
357.5291092, 35999.0502909, -.0001535, 1/24490000)).Rad()
Mʹ := unit.AngleFromDeg(base.Horner(T,
134.9633964, 477198.8675055, .0087414, 1/69699, -1/14712000)).Rad()
E := base.Horner(T, 1, -.002516, -.0000074)
K1 := unit.AngleFromDeg(119.75 + 131.849*T).Rad()
K2 := unit.AngleFromDeg(72.56 + 20.186*T).Rad()
m.ρ = unit.AngleFromDeg(-.02752*math.Cos(Mʹ) +
-.02245*math.Sin(F) +
.00684*math.Cos(Mʹ-2*F) +
-.00293*math.Cos(2*F) +
-.00085*math.Cos(2*(F-D)) +
-.00054*math.Cos(Mʹ-2*D) +
-.0002*math.Sin(Mʹ+F) +
-.0002*math.Cos(Mʹ+2*F) +
-.0002*math.Cos(Mʹ-F) +
.00014*math.Cos(Mʹ+2*(F-D)))
m.σ = unit.AngleFromDeg(-.02816*math.Sin(Mʹ) +
.02244*math.Cos(F) +
-.00682*math.Sin(Mʹ-2*F) +
-.00279*math.Sin(2*F) +
-.00083*math.Sin(2*(F-D)) +
.00069*math.Sin(Mʹ-2*D) +
.0004*math.Cos(Mʹ+F) +
-.00025*math.Sin(2*Mʹ) +
-.00023*math.Sin(Mʹ+2*F) +
.0002*math.Cos(Mʹ-F) +
.00019*math.Sin(Mʹ-F) +
.00013*math.Sin(Mʹ+2*(F-D)) +
-.0001*math.Cos(Mʹ-3*F))
m.τ = unit.AngleFromDeg(.0252*math.Sin(M)*E +
.00473*math.Sin(2*(Mʹ-F)) +
-.00467*math.Sin(Mʹ) +
.00396*math.Sin(K1) +
.00276*math.Sin(2*(Mʹ-D)) +
.00196*math.Sin(m.Ω.Rad()) +
-.00183*math.Cos(Mʹ-F) +
.00115*math.Sin(Mʹ-2*D) +
-.00096*math.Sin(Mʹ-D) +
.00046*math.Sin(2*(F-D)) +
-.00039*math.Sin(Mʹ-F) +
-.00032*math.Sin(Mʹ-M-D) +
.00027*math.Sin(2*(Mʹ-D)-M) +
.00023*math.Sin(K2) +
-.00014*math.Sin(2*D) +
.00014*math.Cos(2*(Mʹ-F)) +
-.00012*math.Sin(Mʹ-2*F) +
-.00012*math.Sin(2*Mʹ) +
.00011*math.Sin(2*(Mʹ-M-D)))
return m
}
// 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
}
func newMoon(jde float64) *moon {
m := &moon{jde: jde}
// Δψ, F, Ω, p. 372.
var Δε float64
m.Δψ, Δε = nutation.Nutation(jde)
T := base.J2000Century(jde)
F := base.Horner(T, 93.272095*p, 483202.0175233*p,
-.0036539*p, -p/3526000, p/863310000)
m.F = F
m.Ω = base.Horner(T, 125.0445479*p, -1934.1362891*p, .0020754*p,
p/467441, -p/60616000)
// true ecliptic
m.sε, m.cε = math.Sincos(nutation.MeanObliquity(jde) + Δε)
// ρ, σ, τ, p. 372,373
D := base.Horner(T, 297.8501921*p, 445267.1114034*p,
-.0018819*p, p/545868, -p/113065000)
M := base.Horner(T, 357.5291092*p, 35999.0502909*p,
-.0001535*p, p/24490000)
Mʹ := base.Horner(T, 134.9633964*p, 477198.8675055*p,
.0087414*p, p/69699, -p/14712000)
E := base.Horner(T, 1, -.002516, -.0000074)
K1 := 119.75*p + 131.849*p*T
K2 := 72.56*p + 20.186*p*T
m.ρ = -.02752*p*math.Cos(Mʹ) +
-.02245*p*math.Sin(F) +
.00684*p*math.Cos(Mʹ-2*F) +
-.00293*p*math.Cos(2*F) +
-.00085*p*math.Cos(2*(F-D)) +
-.00054*p*math.Cos(Mʹ-2*D) +
-.0002*p*math.Sin(Mʹ+F) +
-.0002*p*math.Cos(Mʹ+2*F) +
-.0002*p*math.Cos(Mʹ-F) +
.00014*p*math.Cos(Mʹ+2*(F-D))
m.σ = -.02816*p*math.Sin(Mʹ) +
.02244*p*math.Cos(F) +
-.00682*p*math.Sin(Mʹ-2*F) +
-.00279*p*math.Sin(2*F) +
-.00083*p*math.Sin(2*(F-D)) +
.00069*p*math.Sin(Mʹ-2*D) +
.0004*p*math.Cos(Mʹ+F) +
-.00025*p*math.Sin(2*Mʹ) +
-.00023*p*math.Sin(Mʹ+2*F) +
.0002*p*math.Cos(Mʹ-F) +
.00019*p*math.Sin(Mʹ-F) +
.00013*p*math.Sin(Mʹ+2*(F-D)) +
-.0001*p*math.Cos(Mʹ-3*F)
m.τ = .0252*p*math.Sin(M)*E +
.00473*p*math.Sin(2*(Mʹ-F)) +
-.00467*p*math.Sin(Mʹ) +
.00396*p*math.Sin(K1) +
.00276*p*math.Sin(2*(Mʹ-D)) +
.00196*p*math.Sin(m.Ω) +
-.00183*p*math.Cos(Mʹ-F) +
.00115*p*math.Sin(Mʹ-2*D) +
-.00096*p*math.Sin(Mʹ-D) +
.00046*p*math.Sin(2*(F-D)) +
-.00039*p*math.Sin(Mʹ-F) +
-.00032*p*math.Sin(Mʹ-M-D) +
.00027*p*math.Sin(2*(Mʹ-D)-M) +
.00023*p*math.Sin(K2) +
-.00014*p*math.Sin(2*D) +
.00014*p*math.Cos(2*(Mʹ-F)) +
-.00012*p*math.Sin(Mʹ-2*F) +
-.00012*p*math.Sin(2*Mʹ) +
.00011*p*math.Sin(2*(Mʹ-M-D))
return m
}