// AberrationRonVondrak uses the Ron-Vondrák expression to compute corrections
// due to aberration for equatorial coordinates of an object.
func AberrationRonVondrak(α unit.RA, δ unit.Angle, jd float64) (Δα unit.HourAngle, Δδ unit.Angle) {
T := base.J2000Century(jd)
r := &rv{
T: T,
L2: 3.1761467 + 1021.3285546*T,
L3: 1.7534703 + 628.3075849*T,
L4: 6.2034809 + 334.0612431*T,
L5: 0.5995465 + 52.9690965*T,
L6: 0.8740168 + 21.3299095*T,
L7: 5.4812939 + 7.4781599*T,
L8: 5.3118863 + 3.8133036*T,
Lp: 3.8103444 + 8399.6847337*T,
D: 5.1984667 + 7771.3771486*T,
Mp: 2.3555559 + 8328.6914289*T,
F: 1.6279052 + 8433.4661601*T,
}
var Xp, Yp, Zp float64
// sum smaller terms first
for i := 35; i >= 0; i-- {
x, y, z := rvTerm[i](r)
Xp += x
Yp += y
Zp += z
}
sα, cα := α.Sincos()
sδ, cδ := δ.Sincos()
// (23.4) p. 156
Δα = unit.HourAngle((Yp*cα - Xp*sα) / (c * cδ))
Δδ = unit.Angle(-((Xp*cα+Yp*sα)*sδ - Zp*cδ) / c)
return
}
// Topocentric2 returns topocentric corrections including parallax.
//
// This function implements the "non-rigorous" method descripted in the text.
//
// Note that results are corrections, not corrected coordinates.
func Topocentric2(α unit.RA, δ unit.Angle, Δ, ρsφʹ, ρcφʹ float64, L unit.Angle, jde float64) (Δα unit.HourAngle, Δδ unit.Angle) {
π := Horizontal(Δ)
θ0 := sidereal.Apparent(jde)
H := (θ0.Angle() - L - unit.Angle(α)).Mod1()
sH, cH := H.Sincos()
sδ, cδ := δ.Sincos()
Δα = unit.HourAngle(-π.Mul(ρcφʹ * sH / cδ)) // (40.4) p. 280
Δδ = -π.Mul(ρsφʹ*cδ - ρcφʹ*cH*sδ) // (40.5) p. 280
return
}
// Nutation returns corrections due to nutation for equatorial coordinates
// of an object.
//
// Results are invalid for objects very near the celestial poles.
func Nutation(α unit.RA, δ unit.Angle, jd float64) (Δα1 unit.HourAngle, Δδ1 unit.Angle) {
ε := nutation.MeanObliquity(jd)
sε, cε := ε.Sincos()
Δψ, Δε := nutation.Nutation(jd)
sα, cα := α.Sincos()
tδ := δ.Tan()
// (23.1) p. 151
Δα1 = unit.HourAngle((cε+sε*sα*tδ)*Δψ.Rad() - cα*tδ*Δε.Rad())
Δδ1 = Δψ.Mul(sε*cα) + Δε.Mul(sα)
return
}
// Topocentric returns topocentric positions including parallax.
//
// Arguments α, δ are geocentric right ascension and declination in radians.
// Δ is distance to the observed object in AU. ρsφʹ, ρcφʹ are parallax
// constants (see package globe.) L is geographic longitude of the observer,
// jde is time of observation.
//
// Results are observed topocentric ra and dec in radians.
func Topocentric(α unit.RA, δ unit.Angle, Δ, ρsφʹ, ρcφʹ float64, L unit.Angle, jde float64) (αʹ unit.RA, δʹ unit.Angle) {
π := Horizontal(Δ)
θ0 := sidereal.Apparent(jde)
H := (θ0.Angle() - L - unit.Angle(α)).Mod1()
sπ := π.Sin()
sH, cH := H.Sincos()
sδ, cδ := δ.Sincos()
// (40.2) p. 279
Δα := unit.HourAngle(math.Atan2(-ρcφʹ*sπ*sH, cδ-ρcφʹ*sπ*cH))
αʹ = α.Add(Δα)
// (40.3) p. 279
δʹ = unit.Angle(math.Atan2((sδ-ρsφʹ*sπ)*Δα.Cos(), cδ-ρcφʹ*sπ*cH))
return
}
// ESmart computes the "equation of time" for the given JDE.
//
// Result is equation of time as an hour angle.
//
// Result is less accurate that E() but the function has the advantage
// of not requiring the V87Planet object.
func ESmart(jde float64) unit.HourAngle {
ε := nutation.MeanObliquity(jde)
t := ε.Mul(.5).Tan()
y := t * t
T := base.J2000Century(jde)
L0 := l0(T * .1)
e := solar.Eccentricity(T)
M := solar.MeanAnomaly(T)
s2L0, c2L0 := L0.Mul(2).Sincos()
sM := M.Sin()
// (28.3) p. 185, with double angle identity
return unit.HourAngle(y*s2L0 - 2*e*sM + 4*e*y*sM*c2L0 -
y*y*s2L0*c2L0 - 1.25*e*e*M.Mul(2).Sin())
}
// Topocentric3 returns topocentric hour angle and declination including parallax.
//
// This function implements the "alternative" method described in the text.
// The method should be similarly rigorous to that of Topocentric() and results
// should be virtually consistent.
func Topocentric3(α unit.RA, δ unit.Angle, Δ, ρsφʹ, ρcφʹ float64, L unit.Angle, jde float64) (Hʹ unit.HourAngle, δʹ unit.Angle) {
π := Horizontal(Δ)
θ0 := sidereal.Apparent(jde)
H := (θ0.Angle() - L - unit.Angle(α)).Mod1()
sπ := π.Sin()
sH, cH := H.Sincos()
sδ, cδ := δ.Sincos()
A := cδ * sH
B := cδ*cH - ρcφʹ*sπ
C := sδ - ρsφʹ*sπ
q := math.Sqrt(A*A + B*B + C*C)
Hʹ = unit.HourAngle(math.Atan2(A, B))
δʹ = unit.Angle(math.Asin(C / q))
return
}
// 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)
}
// Aberration returns corrections due to aberration for equatorial
// coordinates of an object.
func Aberration(α unit.RA, δ unit.Angle, jd float64) (Δα2 unit.HourAngle, Δδ2 unit.Angle) {
ε := nutation.MeanObliquity(jd)
T := base.J2000Century(jd)
s, _ := solar.True(T)
e := solar.Eccentricity(T)
π := perihelion(T)
sα, cα := α.Sincos()
sδ, cδ := δ.Sincos()
ss, cs := s.Sincos()
sπ, cπ := π.Sincos()
cε := ε.Cos()
tε := ε.Tan()
q1 := cα * cε
// (23.3) p. 152
Δα2 = unit.HourAngle(κ.Rad() * (e*(q1*cπ+sα*sπ) - (q1*cs + sα*ss)) / cδ)
q2 := cε * (tε*cδ - sα*sδ)
q3 := cα * sδ
Δδ2 = κ.Mul(e*(cπ*q2+sπ*q3) - (cs*q2 + ss*q3))
return
}
// NutationInRA returns "nutation in right ascension" or "equation of the
// equinoxes."
func NutationInRA(jde float64) unit.HourAngle {
// ch 12, p.88
Δψ, Δε := Nutation(jde)
ε0 := MeanObliquity(jde)
return unit.HourAngle(Δψ.Rad() * math.Cos((ε0 + Δε).Rad()))
}