// Vertical computes data for a vertical sundial.
//
// Argument φ is geographic latitude at which the sundial will be located.
// D is gnomonic declination, the azimuth of the perpendicular to the plane
// of the sundial, measured from the southern meridian towards the west.
// Argument a is the length of a straight stylus perpendicular to the plane
// of the sundial.
//
// Results consist of a set of lines, a center point, and u, the length of a
// polar stylus. They are in units of a, the stylus length.
func Vertical(φ, D unit.Angle, a float64) (lines []Line, center Point, u float64) {
sφ, cφ := φ.Sincos()
tφ := sφ / cφ
sD, cD := D.Sincos()
for i := 0; i < 24; i++ {
l := Line{Hour: i}
H := float64(i-12) * 15 * math.Pi / 180
aH := math.Abs(H)
sH, cH := math.Sincos(H)
for _, d := range m {
tδ := math.Tan(d * math.Pi / 180)
H0 := math.Acos(-tφ * tδ)
if aH > H0 {
continue // sun below horizon
}
Q := sD*sH + sφ*cD*cH - cφ*cD*tδ
if Q < 0 {
continue // sun below plane of sundial
}
x := a * (cD*sH - sφ*sD*cH + cφ*sD*tδ) / Q
y := -a * (cφ*cH + sφ*tδ) / Q
l.Points = append(l.Points, Point{x, y})
}
if len(l.Points) > 0 {
lines = append(lines, l)
}
}
center.X = -a * sD / cD
center.Y = a * tφ / cD
u = a / math.Abs(cφ*cD)
return
}
func (m *moon) physical(A, bʹ unit.Angle) (lʺ, bʺ unit.Angle) {
// (53.2) p. 373
sA, cA := A.Sincos()
lʺ = -m.τ + (m.ρ.Mul(cA) + m.σ.Mul(sA)).Mul(bʹ.Tan())
bʺ = m.σ.Mul(cA) - m.ρ.Mul(sA)
return
}
// Angle returns the angle between great circles defined by three points.
//
// Coordinates may be right ascensions and declinations or longitudes and
// latitudes. If r1, d1, r2, d2 defines one line and r2, d2, r3, d3 defines
// another, the result is the angle between the two lines.
//
// Algorithm by Meeus.
func Angle(r1, d1, r2, d2, r3, d3 unit.Angle) unit.Angle {
sd2, cd2 := d2.Sincos()
sr21, cr21 := (r2 - r1).Sincos()
sr32, cr32 := (r3 - r2).Sincos()
C1 := math.Atan2(sr21, cd2*d1.Tan()-sd2*cr21)
C2 := math.Atan2(sr32, cd2*d3.Tan()-sd2*cr32)
return unit.Angle(C1 + C2)
}
// SepPauwels returns the angular separation between two celestial bodies.
//
// The algorithm is a numerically stable form of that used in Sep.
func SepPauwels(r1, d1, r2, d2 unit.Angle) unit.Angle {
sd1, cd1 := d1.Sincos()
sd2, cd2 := d2.Sincos()
cdr := (r2 - r1).Cos()
x := cd1*sd2 - sd1*cd2*cdr
y := cd2 * (r2 - r1).Sin()
z := sd1*sd2 + cd1*cd2*cdr
return unit.Angle(math.Atan2(math.Hypot(x, y), z))
}
// GalToEq converts galactic coordinates to equatorial coordinates.
//
// Resulting equatorial coordinates will be referred to the standard equinox of
// B1950.0. For subsequent conversion to other epochs, see package precess and
// utility functions in package meeus.
func GalToEq(l, b unit.Angle) (α unit.RA, δ unit.Angle) {
sdLon, cdLon := (l - galacticLon0).Sincos()
sgδ, cgδ := galacticNorth.Dec.Sincos()
sb, cb := b.Sincos()
y := math.Atan2(sdLon, cdLon*sgδ-(sb/cb)*cgδ)
α = unit.RAFromRad(y + galacticNorth.RA.Rad())
δ = unit.Angle(math.Asin(sb*sgδ + cb*cgδ*cdLon))
return
}
// HzToEq transforms horizontal coordinates to equatorial coordinates.
//
// A: azimuth
// h: elevation
// φ: latitude of observer on Earth
// ψ: longitude of observer on Earth
// st: sidereal time at Greenwich at time of observation.
//
// Sidereal time must be consistent with the equatorial coordinates
// in the sense that tf coordinates are apparent, sidereal time must be
// apparent as well.
//
// Results:
//
// α: right ascension
// δ: declination
func HzToEq(A, h, φ, ψ unit.Angle, st unit.Time) (α unit.RA, δ unit.Angle) {
sA, cA := A.Sincos()
sh, ch := h.Sincos()
sφ, cφ := φ.Sincos()
H := math.Atan2(sA, cA*sφ+sh/ch*cφ)
α = unit.RAFromRad(st.Rad() - ψ.Rad() - H)
δ = unit.Angle(math.Asin(sφ*sh - cφ*ch*cA))
return
}
// Sep returns the angular separation between two celestial bodies.
//
// The algorithm is numerically naïve, and while patched up a bit for
// small separations, remains unstable for separations near π.
func Sep(r1, d1, r2, d2 unit.Angle) unit.Angle {
sd1, cd1 := d1.Sincos()
sd2, cd2 := d2.Sincos()
cd := sd1*sd2 + cd1*cd2*(r1-r2).Cos() // (17.1) p. 109
if cd < base.CosSmallAngle {
return unit.Angle(math.Acos(cd))
}
// (17.2) p. 109
return unit.Angle(math.Hypot((r2-r1).Rad()*cd1, (d2 - d1).Rad()))
}
// General computes data for the general case of a planar sundial.
//
// Argument φ is geographic latitude at which the sundial will be located.
// D is gnomonic declination, the azimuth of the perpendicular to the plane
// of the sundial, measured from the southern meridian towards the west.
// Argument a is the length of a straight stylus perpendicular to the plane
// of the sundial, z is zenithal distance of the direction defined by the
// stylus. Units of stylus length a are arbitrary.
//
// Results consist of a set of lines, a center point, u, the length of a
// polar stylus, and ψ, the angle which the polar stylus makes with the plane
// of the sundial. The center point, the points defining the hour lines, and
// u are in units of a, the stylus length.
func General(φ, D unit.Angle, a float64, z unit.Angle) (lines []Line, center Point, u float64, ψ unit.Angle) {
sφ, cφ := φ.Sincos()
tφ := sφ / cφ
sD, cD := D.Sincos()
sz, cz := z.Sincos()
P := sφ*cz - cφ*sz*cD
for i := 0; i < 24; i++ {
l := Line{Hour: i}
H := float64(i-12) * 15 * math.Pi / 180
aH := math.Abs(H)
sH, cH := math.Sincos(H)
for _, d := range m {
tδ := math.Tan(d * math.Pi / 180)
H0 := math.Acos(-tφ * tδ)
if aH > H0 {
continue // sun below horizon
}
Q := sD*sz*sH + (cφ*cz+sφ*sz*cD)*cH + P*tδ
if Q < 0 {
continue // sun below plane of sundial
}
Nx := cD*sH - sD*(sφ*cH-cφ*tδ)
Ny := cz*sD*sH - (cφ*sz-sφ*cz*cD)*cH - (sφ*sz+cφ*cz*cD)*tδ
l.Points = append(l.Points, Point{a * Nx / Q, a * Ny / Q})
}
if len(l.Points) > 0 {
lines = append(lines, l)
}
}
center.X = a / P * cφ * sD
center.Y = -a / P * (sφ*sz + cφ*cz*cD)
aP := math.Abs(P)
u = a / aP
ψ = unit.Angle(math.Asin(aP))
return
}
// Error returns an error angle of three nearly co-linear points.
//
// For the line defined by r1, d1, r2, d2, the result is the anglular distance
// between that line and r0, d0.
//
// Algorithm by Meeus.
func Error(r1, d1, r2, d2, r0, d0 unit.Angle) unit.Angle {
sr1, cr1 := r1.Sincos()
sd1, cd1 := d1.Sincos()
sr2, cr2 := r2.Sincos()
sd2, cd2 := d2.Sincos()
X1 := cd1 * cr1
X2 := cd2 * cr2
Y1 := cd1 * sr1
Y2 := cd2 * sr2
Z1 := sd1
Z2 := sd2
A := Y1*Z2 - Z1*Y2
B := Z1*X2 - X1*Z2
C := X1*Y2 - Y1*X2
m := r0.Tan()
n := d0.Tan() / r0.Cos()
return unit.Angle(math.Asin((A + B*m + C*n) /
(math.Sqrt(A*A+B*B+C*C) * math.Sqrt(1+m*m+n*n))))
}
// 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
}
// Kepler4 returns an approximate solution to Kepler's equation.
//
// It is valid only for small values of e.
//
// Argument e is eccentricity, M is mean anomaly.
//
// Result E is eccentric anomaly.
func Kepler4(e float64, M unit.Angle) (E unit.Angle) {
sm, cm := M.Sincos()
return unit.Angle(math.Atan2(sm, cm-e)) // (30.8) p. 206
}
// AngleError returns both an angle as in the function Angle, and an error
// as in the function Error.
//
// The algorithm is by B. Pessens.
func AngleError(r1, d1, r2, d2, r3, d3 unit.Angle) (ψ, ω unit.Angle) {
sr1, cr1 := r1.Sincos()
sd1, cd1 := d1.Sincos()
sr2, cr2 := r2.Sincos()
sd2, cd2 := d2.Sincos()
sr3, cr3 := r3.Sincos()
sd3, cd3 := d3.Sincos()
a1 := cd1 * cr1
a2 := cd2 * cr2
a3 := cd3 * cr3
b1 := cd1 * sr1
b2 := cd2 * sr2
b3 := cd3 * sr3
c1 := sd1
c2 := sd2
c3 := sd3
l1 := b1*c2 - b2*c1
l2 := b2*c3 - b3*c2
l3 := b1*c3 - b3*c1
m1 := c1*a2 - c2*a1
m2 := c2*a3 - c3*a2
m3 := c1*a3 - c3*a1
n1 := a1*b2 - a2*b1
n2 := a2*b3 - a3*b2
n3 := a1*b3 - a3*b1
ψ = unit.Angle(math.Acos((l1*l2 + m1*m2 + n1*n2) /
(math.Sqrt(l1*l1+m1*m1+n1*n1) * math.Sqrt(l2*l2+m2*m2+n2*n2))))
ω = unit.Angle(math.Asin((a2*l3 + b2*m3 + c2*n3) /
(math.Sqrt(a2*a2+b2*b2+c2*c2) * math.Sqrt(l3*l3+m3*m3+n3*n3))))
return
}
// RelativePosition returns the position angle of one body with respect to
// another.
//
// The position angle result is measured counter-clockwise from North.
func RelativePosition(r1, d1, r2, d2 unit.Angle) unit.Angle {
sΔr, cΔr := (r2 - r1).Sincos()
sd2, cd2 := d2.Sincos()
return unit.Angle(math.Atan2(sΔr, cd2*d1.Tan()-sd2*cΔr))
}
// SunAltitude returns altitude of the Sun above the lunar horizon.
//
// Arguments η, θ are selenographic longitude and latitude of a site on the
// Moon, l0, b0 are selenographic coordinates of the Sun, as returned by
// Physical(), for example.
func SunAltitude(η, θ, l0, b0 unit.Angle) unit.Angle {
c0 := math.Pi/2 - l0
sb0, cb0 := b0.Sincos()
sθ, cθ := θ.Sincos()
return unit.Angle(math.Asin(sb0*sθ + cb0*cθ*(c0+η).Sin()))
}
// 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 sincos2(x unit.Angle) (s2, c2 float64) {
s, c := x.Sincos()
return s * s, c * c
}
// PhaseAngle3 computes the phase angle of a planet.
//
// Arguments L, B are heliocentric ecliptical longitude and latitude of the
// planet. x, y, z are cartesian coordinates of the planet, Δ is distance
// from Earth to the planet. All distances in AU.
func PhaseAngle3(L, B unit.Angle, x, y, z, Δ float64) unit.Angle {
// (41.4) p. 283
sL, cL := L.Sincos()
sB, cB := B.Sincos()
return unit.Angle(math.Acos((x*cB*cL + y*cB*sL + z*sB) / Δ))
}