func newMp(y, q float64) *mp {
m := &mp{k: snap(y, q)}
m.T = m.k * ck // (49.3) p. 350
m.E = base.Horner(m.T, 1, -.002516, -.0000074)
m.M = base.Horner(m.T, 2.5534*p, 29.1053567*p/ck,
-.0000014*p, -.00000011*p)
m.Mʹ = base.Horner(m.T, 201.5643*p, 385.81693528*p/ck,
.0107582*p, .00001238*p, -.000000058*p)
m.F = base.Horner(m.T, 160.7108*p, 390.67050284*p/ck,
-.0016118*p, -.00000227*p, .000000011*p)
m.Ω = base.Horner(m.T, 124.7746*p, -1.56375588*p/ck,
.0020672*p, .00000215*p)
m.A[0] = 299.7*p + .107408*p*m.k - .009173*m.T*m.T
m.A[1] = 251.88*p + .016321*p*m.k
m.A[2] = 251.83*p + 26.651886*p*m.k
m.A[3] = 349.42*p + 36.412478*p*m.k
m.A[4] = 84.66*p + 18.206239*p*m.k
m.A[5] = 141.74*p + 53.303771*p*m.k
m.A[6] = 207.17*p + 2.453732*p*m.k
m.A[7] = 154.84*p + 7.30686*p*m.k
m.A[8] = 34.52*p + 27.261239*p*m.k
m.A[9] = 207.19*p + .121824*p*m.k
m.A[10] = 291.34*p + 1.844379*p*m.k
m.A[11] = 161.72*p + 24.198154*p*m.k
m.A[12] = 239.56*p + 25.513099*p*m.k
m.A[13] = 331.55*p + 3.592518*p*m.k
return m
}
// Nutation returns nutation in longitude (Δψ) and nutation in obliquity (Δε)
// for a given JDE.
//
// JDE = UT + ΔT, see package deltat.
//
// Computation is by 1980 IAU theory, with terms < .0003″ neglected.
func Nutation(jde float64) (Δψ, Δε unit.Angle) {
T := base.J2000Century(jde)
D := base.Horner(T,
297.85036, 445267.11148, -0.0019142, 1./189474) * math.Pi / 180
M := base.Horner(T,
357.52772, 35999.050340, -0.0001603, -1./300000) * math.Pi / 180
N := base.Horner(T,
134.96298, 477198.867398, 0.0086972, 1./5620) * math.Pi / 180
F := base.Horner(T,
93.27191, 483202.017538, -0.0036825, 1./327270) * math.Pi / 180
Ω := base.Horner(T,
125.04452, -1934.136261, 0.0020708, 1./450000) * math.Pi / 180
// sum in reverse order to accumulate smaller terms first
var Δψs, Δεs float64
for i := len(table22A) - 1; i >= 0; i-- {
row := table22A[i]
arg := row.d*D + row.m*M + row.n*N + row.f*F + row.ω*Ω
s, c := math.Sincos(arg)
Δψs += s * (row.s0 + row.s1*T)
Δεs += c * (row.c0 + row.c1*T)
}
Δψ = unit.AngleFromSec(Δψs * .0001)
Δε = unit.AngleFromSec(Δεs * .0001)
return
}
// Extremum returns the x and y values at the extremum.
//
// Results are restricted to the range of the table given to the constructor
// NewLen5. (Meeus actually recommends restricting the range to one unit of
// the tabular interval, but that seems a little harsh.)
func (d *Len5) Extremum() (x, y float64, err error) {
// (3.9) p. 29
nCoeff := []float64{
6*(d.b+d.c) - d.h - d.j,
0,
3 * (d.h + d.k),
2 * d.k,
}
den := d.k - 12*d.f
if den == 0 {
return 0, 0, ErrorExtremumOutside
}
n0, ok := iterate(0, func(n0 float64) float64 {
return base.Horner(n0, nCoeff...) / den
})
if !ok {
return 0, 0, ErrorNoConverge
}
if n0 < -2 || n0 > 2 {
return 0, 0, ErrorExtremumOutside
}
x = .5*d.xSum + .25*d.xDiff*n0
y = base.Horner(n0, d.interpCoeff...)
return x, y, nil
}
// 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
}
// True returns true geometric longitude and anomaly of the sun referenced to the mean equinox of date.
//
// Argument T is the number of Julian centuries since J2000.
// See base.J2000Century.
//
// Results:
// s = true geometric longitude, ☉
// ν = true anomaly
func True(T float64) (s, ν unit.Angle) {
// (25.2) p. 163
L0 := unit.AngleFromDeg(base.Horner(T, 280.46646, 36000.76983, 0.0003032))
M := MeanAnomaly(T)
C := unit.AngleFromDeg(base.Horner(T, 1.914602, -0.004817, -.000014)*
M.Sin() +
(0.019993-.000101*T)*M.Mul(2).Sin() +
0.000289*M.Mul(3).Sin())
return (L0 + C).Mod1(), (M + C).Mod1()
}
func newLa(y, h float64) *la {
l := &la{k: snap(y, h)}
l.T = l.k * ck // (50.3) p. 350
l.D = base.Horner(l.T, 171.9179*p, 335.9106046*p/ck,
-.0100383*p, -.00001156*p, .000000055*p)
l.M = base.Horner(l.T, 347.3477*p, 27.1577721*p/ck,
-.000813*p, -.000001*p)
l.F = base.Horner(l.T, 316.6109*p, 364.5287911*p/ck,
-.0125053*p, -.0000148*p)
return l
}
// True returns true geometric longitude and anomaly of the sun referenced to the mean equinox of date.
//
// Argument T is the number of Julian centuries since J2000.
// See base.J2000Century.
//
// Results:
// s = true geometric longitude, ☉, in radians
// ν = true anomaly in radians
func True(T float64) (s, ν float64) {
// (25.2) p. 163
L0 := base.Horner(T, 280.46646, 36000.76983, 0.0003032) *
math.Pi / 180
M := MeanAnomaly(T)
C := (base.Horner(T, 1.914602, -0.004817, -.000014)*
math.Sin(M) +
(0.019993-.000101*T)*math.Sin(2*M) +
0.000289*math.Sin(3*M)) * math.Pi / 180
return base.PMod(L0+C, 2*math.Pi), base.PMod(M+C, 2*math.Pi)
}
func dmf(T float64) (D, M, Mʹ, F float64) {
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)
F = base.Horner(T, 93.272095*p, 483202.0175233*p,
-.0036539*p, -p/3526000, p/863310000)
return
}
// Sum computes a sum of periodic terms.
func sum(T, M float64, c [][]float64) float64 {
j := base.Horner(T, c[0]...)
mm := 0.
for i := 1; i < len(c); i++ {
mm += M
smm, cmm := math.Sincos(mm)
j += smm * base.Horner(T, c[i]...)
i++
j += cmm * base.Horner(T, c[i]...)
}
return j
}
// SumA computes the sum of periodic terms with "additional angles"
func sumA(T, M float64, c [][]float64, aa []caa) float64 {
i := len(c) - 2*len(aa)
j := sum(T, M, c[:i])
for k := 0; k < len(aa); k++ {
saa, caa := math.Sincos((aa[k].c + aa[k].f*T) * math.Pi / 180)
j += saa * base.Horner(T, c[i]...)
i++
j += caa * base.Horner(T, c[i]...)
i++
}
return j
}
func (q *qs) iapetus() (r r4) {
L := 261.1582*d + 22.57697855*d*q.t4
ϖʹ := 91.796*d + .562*d*q.t7
ψ := 4.367*d - .195*d*q.t7
θ := 146.819*d - 3.198*d*q.t7
φ := 60.47*d + 1.521*d*q.t7
Φ := 205.055*d - 2.091*d*q.t7
eʹ := .028298 + .001156*q.t11
ϖ0 := 352.91*d + 11.71*d*q.t11
μ := 76.3852*d + 4.53795125*d*q.t10
iʹ := base.Horner(q.t11, 18.4602*d, -.9518*d, -.072*d, .0054*d)
Ωʹ := base.Horner(q.t11, 143.198*d, -3.919*d, .116*d, .008*d)
l := μ - ϖ0
g := ϖ0 - Ωʹ - ψ
g1 := ϖ0 - Ωʹ - φ
ls := q.W5 - ϖʹ
gs := ϖʹ - θ
lT := L - q.W4
gT := q.W4 - Φ
u1 := 2 * (l + g - ls - gs)
u2 := l + g1 - lT - gT
u3 := l + 2*(g-ls-gs)
u4 := lT + gT - g1
u5 := 2 * (ls + gs)
sl, cl := math.Sincos(l)
su1, cu1 := math.Sincos(u1)
su2, cu2 := math.Sincos(u2)
su3, cu3 := math.Sincos(u3)
su4, cu4 := math.Sincos(u4)
slu2, clu2 := math.Sincos(l + u2)
sg1gT, cg1gT := math.Sincos(g1 - gT)
su52g, cu52g := math.Sincos(u5 - 2*g)
su5ψ, cu5ψ := math.Sincos(u5 + ψ)
su2φ, cu2φ := math.Sincos(u2 + φ)
s5, c5 := math.Sincos(l + g1 + lT + gT + φ)
a := 58.935028 + .004638*cu1 + .058222*cu2
e := eʹ - .0014097*cg1gT + .0003733*cu52g +
.000118*cu3 + .0002408*cl + .0002849*clu2 + .000619*cu4
w := .08077*d*sg1gT + .02139*d*su52g - .00676*d*su3 +
.0138*d*sl + .01632*d*slu2 + .03547*d*su4
p := ϖ0 + w/eʹ
λʹ := μ - .04299*d*su2 - .00789*d*su1 - .06312*d*math.Sin(ls) -
.00295*d*math.Sin(2*ls) - .02231*d*math.Sin(u5) + .0065*d*su5ψ
i := iʹ + .04204*d*cu5ψ + .00235*d*c5 + .0036*d*cu2φ
wʹ := .04204*d*su5ψ + .00235*d*s5 + .00358*d*su2φ
Ω := Ωʹ + wʹ/math.Sin(iʹ)
return q.subr(λʹ, p, e, a, Ω, i)
}
// PhaseAngle3 computes the phase angle of the Moon given a julian day.
//
// Less accurate than PhaseAngle functions taking coordinates.
func PhaseAngle3(jde float64) unit.Angle {
T := base.J2000Century(jde)
D := unit.AngleFromDeg(base.Horner(T, 297.8501921,
445267.1114034, -.0018819, 1/545868, -1/113065000)).Mod1().Rad()
M := unit.AngleFromDeg(base.Horner(T,
357.5291092, 35999.0502909, -.0001535, 1/24490000)).Mod1().Rad()
Mʹ := unit.AngleFromDeg(base.Horner(T, 134.9633964,
477198.8675055, .0087414, 1/69699, -1/14712000)).Mod1().Rad()
return math.Pi - unit.Angle(D) + unit.AngleFromDeg(
-6.289*math.Sin(Mʹ)+
2.1*math.Sin(M)+
-1.274*math.Sin(2*D-Mʹ)+
-.658*math.Sin(2*D)+
-.214*math.Sin(2*Mʹ)+
-.11*math.Sin(D))
}
// MeanObliquity returns mean obliquity (ε₀) following the IAU 1980
// polynomial.
//
// Accuracy is 1″ over the range 1000 to 3000 years and 10″ over the range
// 0 to 4000 years.
func MeanObliquity(jde float64) unit.Angle {
// (22.2) p. 147
return unit.AngleFromSec(base.Horner(base.J2000Century(jde),
unit.FromSexaSec(' ', 23, 26, 21.448),
-46.815,
-0.00059,
0.001813))
}
// PhaseAngle3 computes the phase angle of the Moon given a julian day.
//
// Less accurate than PhaseAngle functions taking coordinates.
//
// Result in radians.
func PhaseAngle3(jde float64) float64 {
T := base.J2000Century(jde)
const p = math.Pi / 180
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)
return math.Pi - base.PMod(D, 2*math.Pi) +
-6.289*p*math.Sin(Mʹ) +
2.1*p*math.Sin(M) +
-1.274*p*math.Sin(2*D-Mʹ) +
-.658*p*math.Sin(2*D) +
-.214*p*math.Sin(2*Mʹ) +
-.11*p*math.Sin(D)
}
// Apparent0UT returns apparent sidereal time at Greenwich at 0h UT
// on the given JD.
//
// The result is in seconds of time and is in the range [0,86400).
func Apparent0UT(jd float64) float64 {
j0, f := math.Modf(jd + .5)
cen := (j0 - .5 - base.J2000) / 36525
s := base.Horner(cen, iau82...) + f*1.00273790935*86400
n := nutation.NutationInRA(j0) // angle (radians) of RA
ns := n * 3600 * 180 / math.Pi / 15 // convert RA to time in seconds
return base.PMod(s+ns, 86400)
}
// Apparent0UT returns apparent sidereal time at Greenwich at 0h UT
// on the given JD.
//
// The result is in the range [0,86400).
func Apparent0UT(jd float64) unit.Time {
j0, f := math.Modf(jd + .5)
cen := (j0 - .5 - base.J2000) / 36525
s := unit.Time(base.Horner(cen, iau82...)) +
unit.TimeFromDay(f*1.00273790935)
n := nutation.NutationInRA(j0) // HourAngle
return (s + n.Time()).Mod1()
}
// Position returns geocentric location of the Moon.
//
// Results are referenced to mean equinox of date and do not include
// the effect of nutation.
//
// λ Geocentric longitude, in radians.
// β Geocentric latidude, in radians.
// Δ Distance between centers of the Earth and Moon, in km.
func Position(jde float64) (λ, β, Δ float64) {
T := base.J2000Century(jde)
Lʹ := base.Horner(T, 218.3164477*p, 481267.88123421*p,
-.0015786*p, p/538841, -p/65194000)
D, M, Mʹ, F := dmf(T)
A1 := 119.75*p + 131.849*p*T
A2 := 53.09*p + 479264.29*p*T
A3 := 313.45*p + 481266.484*p*T
E := base.Horner(T, 1, -.002516, -.0000074)
E2 := E * E
Σl := 3958*math.Sin(A1) + 1962*math.Sin(Lʹ-F) + 318*math.Sin(A2)
Σr := 0.
Σb := -2235*math.Sin(Lʹ) + 382*math.Sin(A3) + 175*math.Sin(A1-F) +
175*math.Sin(A1+F) + 127*math.Sin(Lʹ-Mʹ) - 115*math.Sin(Lʹ+Mʹ)
for i := range ta {
r := &ta[i]
sa, ca := math.Sincos(D*r.D + M*r.M + Mʹ*r.Mʹ + F*r.F)
switch r.M {
case 0:
Σl += r.Σl * sa
Σr += r.Σr * ca
case 1, -1:
Σl += r.Σl * sa * E
Σr += r.Σr * ca * E
case 2, -2:
Σl += r.Σl * sa * E2
Σr += r.Σr * ca * E2
}
}
for i := range tb {
r := &tb[i]
sb := math.Sin(D*r.D + M*r.M + Mʹ*r.Mʹ + F*r.F)
switch r.M {
case 0:
Σb += r.Σb * sb
case 1, -1:
Σb += r.Σb * sb * E
case 2, -2:
Σb += r.Σb * sb * E2
}
}
λ = base.PMod(Lʹ, 2*math.Pi) + Σl*1e-6*p
β = Σb * 1e-6 * p
Δ = 385000.56 + Σr*1e-3
return
}
// MeanObliquity returns mean obliquity (ε₀) following the IAU 1980
// polynomial.
//
// Accuracy is 1″ over the range 1000 to 3000 years and 10″ over the range
// 0 to 4000 years.
//
// Result unit is radians.
func MeanObliquity(jde float64) float64 {
// (22.2) p. 147
return base.Horner(base.J2000Century(jde),
base.NewAngle(false, 23, 26, 21.448).Rad(),
-46.815/3600*(math.Pi/180),
-0.00059/3600*(math.Pi/180),
0.001813/3600*(math.Pi/180))
}
// 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
}
func eq(y int, c []float64) float64 {
J0 := base.Horner(float64(y)*.001, c...)
T := base.J2000Century(J0)
W := 35999.373*math.Pi/180*T - 2.47*math.Pi/180
Δλ := 1 + .0334*math.Cos(W) + .0007*math.Cos(2*W)
S := 0.
for i := len(terms) - 1; i >= 0; i-- {
t := &terms[i]
S += t.a * math.Cos((t.b+t.c*T)*math.Pi/180)
}
return J0 + .00001*S/Δλ
}
// 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 eq2(y int, e *pp.V87Planet, q unit.Angle, c []float64) float64 {
J0 := base.Horner(float64(y)*.001, c...)
for {
λ, _, _ := solar.ApparentVSOP87(e, J0)
c := 58 * (q - λ).Sin() // (27.1) p. 180
J0 += c
if math.Abs(c) < .000005 {
break
}
}
return J0
}
// NewEclipticPrecessor constructs an EclipticPrecessor object and initializes
// it to precess coordinates from epochFrom to epochTo.
func NewEclipticPrecessor(epochFrom, epochTo float64) *EclipticPrecessor {
// (21.5) p. 136
ηCoeff := ηt
πCoeff := πt
pCoeff := pt
if epochFrom != 2000 {
T := (epochFrom - 2000) * .01
ηCoeff = []float64{
base.Horner(T, ηT...),
-0.03302*s + 0.000598*s*T,
0.000060 * s}
πCoeff = []float64{
base.Horner(T, πT...),
-869.8089*s - 0.50491*s*T,
0.03536 * s}
pCoeff = []float64{
base.Horner(T, pT...),
1.11113*s - 0.000042*s*T,
-0.000006 * s}
}
t := (epochTo - epochFrom) * .01
p := &EclipticPrecessor{
π: base.Horner(t, πCoeff...),
p: base.Horner(t, pCoeff...) * t,
}
η := base.Horner(t, ηCoeff...) * t
p.sη, p.cη = math.Sincos(η)
return p
}
// NewPrecessor constructs a Precessor object and initializes it to precess
// coordinates from epochFrom to epochTo.
func NewPrecessor(epochFrom, epochTo float64) *Precessor {
// (21.2) p. 134
ζCoeff := ζt
zCoeff := zt
θCoeff := θt
if epochFrom != 2000 {
T := (epochFrom - 2000) * .01
ζCoeff = []float64{
base.Horner(T, ζT...),
0.30188*s - 0.000344*s*T,
0.017998 * s}
zCoeff = []float64{
base.Horner(T, zT...),
1.09468*s + 0.000066*s*T,
0.018203 * s}
θCoeff = []float64{
base.Horner(T, θT...),
-0.42665*s - 0.000217*s*T,
-0.041833 * s}
}
t := (epochTo - epochFrom) * .01
p := &Precessor{
ζ: base.Horner(t, ζCoeff...) * t,
z: base.Horner(t, zCoeff...) * t,
}
θ := base.Horner(t, θCoeff...) * t
p.sθ, p.cθ = math.Sincos(θ)
return p
}
// PositionEquinox returns rectangular coordinates referenced to an arbitrary epoch.
//
// Position will be computed for given Julian day "jde" but referenced to mean
// equinox "epoch" (year).
func PositionEquinox(e *pp.V87Planet, jde, epoch float64) (xp, yp, zp float64) {
x0, y0, z0 := PositionJ2000(e, jde)
t := (epoch - 2000) * .01
ζ := base.Horner(t, ζt...) * t * math.Pi / 180 / 3600
z := base.Horner(t, zt...) * t * math.Pi / 180 / 3600
θ := base.Horner(t, θt...) * t * math.Pi / 180 / 3600
sζ, cζ := math.Sincos(ζ)
sz, cz := math.Sincos(z)
sθ, cθ := math.Sincos(θ)
xx := cζ*cz*cθ - sζ*sz
xy := sζ*cz + cζ*sz*cθ
xz := cζ * sθ
yx := -cζ*sz - sζ*cz*cθ
yy := cζ*cz - sζ*sz*cθ
yz := -sζ * sθ
zx := -cz * sθ
zy := -sz * sθ
zz := cθ
return xx*x0 + yx*y0 + zx*z0,
xy*x0 + yy*y0 + zy*z0,
xz*x0 + yz*y0 + zz*z0
}
// Len5Zero finds a zero of the quartic function represented by the table.
//
// That is, it returns an x value that yields y=0.
//
// Argument strong switches between two strategies for the estimation step.
// when iterating to converge on the zero.
//
// Strong=false specifies a quick and dirty estimate that works well
// for gentle curves, but can work poorly or fail on more dramatic curves.
//
// Strong=true specifies a more sophisticated and thus somewhat more
// expensive estimate. However, if the curve has quick changes, This estimate
// will converge more reliably and in fewer steps, making it a better choice.
//
// Results are restricted to the range of the table given to the constructor
// NewLen5.
func (d *Len5) Zero(strong bool) (x float64, err error) {
var f iterFunc
if strong {
// (3.11), p. 29
M := d.k / 24
N := (d.h + d.j) / 12
P := d.f/2 - M
Q := (d.b+d.c)/2 - N
numCoeff := []float64{d.y3, Q, P, N, M}
denCoeff := []float64{Q, 2 * P, 3 * N, 4 * M}
f = func(n0 float64) float64 {
return n0 -
base.Horner(n0, numCoeff...)/base.Horner(n0, denCoeff...)
}
} else {
// (3.10), p. 29
numCoeff := []float64{
-24 * d.y3,
0,
d.k - 12*d.f,
-2 * (d.h + d.j),
-d.k,
}
den := 12*(d.b+d.c) - 2*(d.h+d.j)
f = func(n0 float64) float64 {
return base.Horner(n0, numCoeff...) / den
}
}
n0, ok := iterate(0, f)
if !ok {
return 0, ErrorNoConverge
}
if n0 > 2 || n0 < -2 {
return 0, ErrorZeroOutside
}
x = .5*d.xSum + .25*d.xDiff*n0
return x, nil
}
// MeanObliquityLaskar returns mean obliquity (ε₀) following the Laskar
// 1986 polynomial.
//
// Accuracy over the range 1000 to 3000 years is .01″.
//
// Accuracy over the valid date range of -8000 to +12000 years is
// "a few seconds."
func MeanObliquityLaskar(jde float64) unit.Angle {
// (22.3) p. 147
return unit.AngleFromSec(base.Horner(base.J2000Century(jde)*.01,
unit.FromSexaSec(' ', 23, 26, 21.448),
-4680.93,
-1.55,
1999.25,
-51.38,
-249.67,
-39.05,
7.12,
27.87,
5.79,
2.45))
}
// MeanObliquityLaskar returns mean obliquity (ε₀) following the Laskar
// 1986 polynomial.
//
// Accuracy over the range 1000 to 3000 years is .01″.
//
// Accuracy over the valid date range of -8000 to +12000 years is
// "a few seconds."
//
// Result unit is radians.
func MeanObliquityLaskar(jde float64) float64 {
// (22.3) p. 147
return base.Horner(base.J2000Century(jde)*.01,
base.NewAngle(false, 23, 26, 21.448).Rad(),
-4680.93/3600*(math.Pi/180),
-1.55/3600*(math.Pi/180),
1999.25/3600*(math.Pi/180),
-51.38/3600*(math.Pi/180),
-249.67/3600*(math.Pi/180),
-39.05/3600*(math.Pi/180),
7.12/3600*(math.Pi/180),
2787/3600*(math.Pi/180),
5.79/3600*(math.Pi/180),
2.45/3600*(math.Pi/180))
}
// Mean returns mean orbital elements for a planet
//
// Argument p must be a planet const as defined above, argument e is
// a result parameter. A valid non-nil pointer to an Elements struct
// must be passed in.
//
// Results are referenced to mean dynamical ecliptic and equinox of date.
//
// Semimajor axis is in AU, angular elements are in radians.
func Mean(p int, jde float64, e *Elements) {
T := base.J2000Century(jde)
c := &cMean[p]
e.Lon = base.PMod(base.Horner(T, c.L...)*math.Pi/180, 2*math.Pi)
e.Axis = base.Horner(T, c.a...)
e.Ecc = base.Horner(T, c.e...)
e.Inc = base.Horner(T, c.i...) * math.Pi / 180
e.Node = base.Horner(T, c.Ω...) * math.Pi / 180
e.Peri = base.Horner(T, c.ϖ...) * math.Pi / 180
}
// Mean returns mean orbital elements for a planet
//
// Argument p must be a planet const as defined above, argument e is
// a result parameter. A valid non-nil pointer to an Elements struct
// must be passed in.
//
// Results are referenced to mean dynamical ecliptic and equinox of date.
//
// Semimajor axis is in AU, angular elements are in radians.
func Mean(p int, jde float64, e *Elements) {
T := base.J2000Century(jde)
c := &cMean[p]
e.Lon = unit.AngleFromDeg(base.Horner(T, c.L...)).Mod1()
e.Axis = base.Horner(T, c.a...)
e.Ecc = base.Horner(T, c.e...)
e.Inc = unit.AngleFromDeg(base.Horner(T, c.i...))
e.Node = unit.AngleFromDeg(base.Horner(T, c.Ω...))
e.Peri = unit.AngleFromDeg(base.Horner(T, c.ϖ...))
}