Пример #1
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// 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.
}
Пример #2
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func eqProperMotionToEcl(mα, mδ, epoch float64, pos *coord.Ecliptic) (mλ, mβ float64) {
	ε := nutation.MeanObliquity(base.JulianYearToJDE(epoch))
	sε, cε := math.Sincos(ε)
	α, δ := coord.EclToEq(pos.Lon, pos.Lat, sε, cε)
	sα, cα := math.Sincos(α)
	sδ, cδ := math.Sincos(δ)
	cβ := math.Cos(pos.Lat)
	mλ = (mδ*sε*cα + mα*cδ*(cε*cδ+sε*sδ*sα)) / (cβ * cβ)
	mβ = (mδ*(cε*cδ+sε*sδ*sα) - mα*sε*cα*cδ) / cβ
	return
}
Пример #3
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func eqProperMotionToEcl(mα unit.HourAngle, mδ unit.Angle, epoch float64, pos *coord.Ecliptic) (mλ, mβ unit.Angle) {
	ε := nutation.MeanObliquity(base.JulianYearToJDE(epoch))
	sε, cε := ε.Sincos()
	α, δ := coord.EclToEq(pos.Lon, pos.Lat, sε, cε)
	sα, cα := α.Sincos()
	sδ, cδ := δ.Sincos()
	cβ := pos.Lat.Cos()
	mλ = (mδ.Mul(sε*cα) + unit.Angle(mα).Mul(cδ*(cε*cδ+sε*sδ*sα))).Div(cβ * cβ)
	mβ = (mδ.Mul(cε*cδ+sε*sδ*sα) - unit.Angle(mα).Mul(sε*cα*cδ)).Div(cβ)
	return
}
Пример #4
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// 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
}
Пример #5
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func (m *moon) pa(λ, β, b unit.Angle) unit.Angle {
	V := m.Ω + m.Δψ + m.σ.Div(sI)
	sV, cV := V.Sincos()
	sIρ, cIρ := (_I + m.ρ).Sincos()
	X := sIρ * sV
	Y := sIρ*cV*m.cε - cIρ*m.sε
	ω := math.Atan2(X, Y)
	α, _ := coord.EclToEq(λ+m.Δψ, β, m.sε, m.cε)
	P := unit.Angle(math.Asin(math.Hypot(X, Y) * math.Cos(α.Rad()-ω) / b.Cos()))
	if P < 0 {
		P += 2 * math.Pi
	}
	return P
}
Пример #6
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func (m *moon) pa(λ, β, b float64) float64 {
	V := m.Ω + m.Δψ + m.σ/sI
	sV, cV := math.Sincos(V)
	sIρ, cIρ := math.Sincos(_I + m.ρ)
	X := sIρ * sV
	Y := sIρ*cV*m.cε - cIρ*m.sε
	ω := math.Atan2(X, Y)
	α, _ := coord.EclToEq(λ+m.Δψ, β, m.sε, m.cε)
	P := math.Asin(math.Hypot(X, Y) * math.Cos(α-ω) / math.Cos(b))
	if P < 0 {
		P += 2 * math.Pi
	}
	return P
}
Пример #7
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// 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)
}
Пример #8
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// 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
}
Пример #9
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// 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
}