Пример #1
0
// Position returns rectangular coordinates referenced to the mean equinox of date.
func Position(e *pp.V87Planet, jde float64) (x, y, z float64) {
	// (26.1) p. 171
	s, β, R := solar.TrueVSOP87(e, jde)
	sε, cε := math.Sincos(nutation.MeanObliquity(jde))
	ss, cs := math.Sincos(s)
	sβ := math.Sin(β)
	x = R * cs
	y = R * (ss*cε - sβ*sε)
	z = R * (ss*sε + sβ*cε)
	return
}
Пример #2
0
// 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 #3
0
// 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 #4
0
// 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
}
Пример #5
0
// 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
}
Пример #6
0
// 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
}
Пример #7
0
// 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
}