func TestNode(t *testing.T) {
j := julian.CalendarGregorianToJD(2065, 6, 24)
var e pe.Elements
pe.Mean(pe.Mercury, j, &e)
if Ω := pe.Node(pe.Mercury, j); Ω != e.Node {
t.Fatal(Ω, "!=", e.Node)
}
}
// 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
}