// 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
}
// Kepler1 solves Kepler's equation by iteration.
//
// The iterated formula is
//
// E1 = M + e * sin(E0)
//
// Argument e is eccentricity, M is mean anomaly,
// places is the desired number of decimal places in the result.
//
// Result E is eccentric anomaly.
//
// For some vaues of e and M it will fail to converge and the
// function will return an error.
func Kepler1(e float64, M unit.Angle, places int) (E unit.Angle, err error) {
f := func(E0 float64) float64 {
return M.Rad() + e*math.Sin(E0) // (30.5) p. 195
}
ea, err := iterate.DecimalPlaces(f, M.Rad(), places, places*5)
return unit.Angle(ea), err
}
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
}
// Kepler2 solves Kepler's equation by iteration.
//
// The iterated formula is
//
// E1 = E0 + (M + e * sin(E0) - E0) / (1 - e * cos(E0))
//
// Argument e is eccentricity, M is mean anomaly,
// places is the desired number of decimal places in the result.
//
// Result E is eccentric anomaly.
//
// The function converges over a wider range of inputs than does Kepler1
// but it also fails to converge for some values of e and M.
func Kepler2(e float64, M unit.Angle, places int) (E unit.Angle, err error) {
f := func(E0 float64) float64 {
se, ce := math.Sincos(E0)
return E0 + (M.Rad()+e*se-E0)/(1-e*ce) // (30.7) p. 199
}
ea, err := iterate.DecimalPlaces(f, M.Rad(), places, places)
return unit.Angle(ea), err
}
// 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
}
// Kepler2a solves Kepler's equation by iteration.
//
// The iterated formula is the same as in Kepler2 but a limiting function
// avoids divergence.
//
// Argument e is eccentricity, M is mean anomaly,
// places is the desired number of decimal places in the result.
//
// Result E is eccentric anomaly.
func Kepler2a(e float64, M unit.Angle, places int) (E unit.Angle, err error) {
f := func(E0 float64) float64 {
se, ce := math.Sincos(E0)
// method of Leingärtner, p. 205
return E0 + math.Asin(math.Sin((M.Rad()+e*se-E0)/(1-e*ce)))
}
ea, err := iterate.DecimalPlaces(f, M.Rad(), places, places*5)
return unit.Angle(ea), err
}
// ApparentEccentricity returns apparent eccenticity of a binary star
// given true orbital elements.
//
// e is eccentricity of the true orbit
// i is inclination relative to the line of sight
// ω is longitude of periastron
func ApparentEccentricity(e float64, i, ω unit.Angle) float64 {
ci := i.Cos()
sω, cω := ω.Sincos()
A := (1 - e*e*cω*cω) * ci * ci
B := e * e * sω * cω * ci
C := 1 - e*e*sω*sω
d := A - C
sD := math.Sqrt(d*d + 4*B*B)
return math.Sqrt(2 * sD / (A + C + sD))
}
// 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()))
}
// Times computes UT rise, transit and set times for a celestial object on
// a day of interest.
//
// The function argurments do not actually include the day, but do include
// a number of values computed from the day.
//
// p is geographic coordinates of observer.
// ΔT is delta T.
// h0 is "standard altitude" of the body.
// Th0 is apparent sidereal time at 0h UT at Greenwich.
// α3, δ3 are slices of three right ascensions and declinations.
//
// h0 unit is radians.
//
// Th0 must be the time on the day of interest, in seconds.
// See sidereal.Apparent0UT.
//
// α3, δ3 must be values at 0h dynamical time for the day before, the day of,
// and the day after the day of interest. Units are radians.
//
// Result units are seconds of day and are in the range [0,86400).
func Times(p globe.Coord, ΔT unit.Time, h0 unit.Angle, Th0 unit.Time, α3 []unit.RA, δ3 []unit.Angle) (tRise, tTransit, tSet unit.Time, err error) {
tRise, tTransit, tSet, err = ApproxTimes(p, h0, Th0, α3[1], δ3[1])
if err != nil {
return
}
αf := make([]float64, 3)
for i, α := range α3 {
αf[i] = α.Rad()
}
δf := make([]float64, 3)
for i, δ := range δ3 {
δf[i] = δ.Rad()
}
var d3α, d3δ *interp.Len3
d3α, err = interp.NewLen3(-86400, 86400, αf)
if err != nil {
return
}
d3δ, err = interp.NewLen3(-86400, 86400, δf)
if err != nil {
return
}
// adjust tTransit
{
th0 := (Th0 + tTransit.Mul(360.985647/360)).Mod1()
α := d3α.InterpolateX((tTransit + ΔT).Sec())
// local hour angle as Time
H := th0 - unit.TimeFromRad(p.Lon.Rad()+α)
tTransit -= H
}
// adjust tRise, tSet
sLat, cLat := p.Lat.Sincos()
adjustRS := func(m unit.Time) (unit.Time, error) {
th0 := (Th0 + m.Mul(360.985647/360)).Mod1()
ut := (m + ΔT).Sec()
α := d3α.InterpolateX(ut)
δ := d3δ.InterpolateX(ut)
Hrad := th0.Rad() - p.Lon.Rad() - α
sδ, cδ := math.Sincos(δ)
sH, cH := math.Sincos(Hrad)
h := math.Asin(sLat*sδ + cLat*cδ*cH)
md := (unit.TimeFromRad(h) - h0.Time()).Div(cδ * cLat * sH)
return m + md, nil
}
tRise, err = adjustRS(tRise)
if err != nil {
return
}
tSet, err = adjustRS(tSet)
return
}
func (m *moon) pa(λ, β, b unit.Angle) unit.Angle {
V := m.Ω + m.Δψ + m.σ.Div(sI)
sV, cV := V.Sincos()
sIρ, cIρ := (_I + m.ρ).Sincos()
X := sIρ * sV
Y := sIρ*cV*m.cε - cIρ*m.sε
ω := math.Atan2(X, Y)
α, _ := coord.EclToEq(λ+m.Δψ, β, m.sε, m.cε)
P := unit.Angle(math.Asin(math.Hypot(X, Y) * math.Cos(α.Rad()-ω) / b.Cos()))
if P < 0 {
P += 2 * math.Pi
}
return P
}
// Kepler2b solves Kepler's equation by iteration.
//
// The iterated formula is the same as in Kepler2 but a (different) limiting
// function avoids divergence.
//
// Argument e is eccentricity, M is mean anomaly,
// places is the desired number of decimal places in the result.
//
// Result E is eccentric anomaly.
func Kepler2b(e float64, M unit.Angle, places int) (E unit.Angle, err error) {
f := func(E0 float64) float64 {
se, ce := math.Sincos(E0)
d := (M.Rad() + e*se - E0) / (1 - e*ce)
// method of Steele, p. 205
if d > .5 {
d = .5
} else if d < -.5 {
d = -.5
}
return E0 + d
}
ea, err := iterate.DecimalPlaces(f, M.Rad(), places, places)
return unit.Angle(ea), err
}
// TopocentricEcliptical returns topocentric ecliptical coordinates including parallax.
//
// Arguments λ, β are geocentric ecliptical longitude and latitude of a body,
// s is its geocentric semidiameter. φ, h are the observer's latitude and
// and height above the ellipsoid in meters. ε is the obliquity of the
// ecliptic, θ is local sidereal time, π is equatorial horizontal parallax
// of the body (see Horizonal()).
//
// Results are observed topocentric coordinates and semidiameter.
func TopocentricEcliptical(λ, β, s, φ unit.Angle, h float64, ε unit.Angle, θ unit.Time, π unit.Angle) (λʹ, βʹ, sʹ unit.Angle) {
S, C := globe.Earth76.ParallaxConstants(φ, h)
sλ, cλ := λ.Sincos()
sβ, cβ := β.Sincos()
sε, cε := ε.Sincos()
sθ, cθ := θ.Angle().Sincos()
sπ := π.Sin()
N := cλ*cβ - C*sπ*cθ
λʹ = unit.Angle(math.Atan2(sλ*cβ-sπ*(S*sε+C*cε*sθ), N))
if λʹ < 0 {
λʹ += 2 * math.Pi
}
cλʹ := λʹ.Cos()
βʹ = unit.Angle(math.Atan(cλʹ * (sβ - sπ*(S*cε-C*sε*sθ)) / N))
sʹ = unit.Angle(math.Asin(cλʹ * βʹ.Cos() * s.Sin() / N))
return
}
// Time computes the time at which a moving body is on a straight line (great
// circle) between two fixed points, such as stars.
//
// Coordinates may be right ascensions and declinations or longitudes and
// latitudes. Fixed points are r1, d1, r2, d2. Moving body is an ephemeris
// of 5 rows, r3, d3, starting at time t1 and ending at time t5. Time scale
// is arbitrary.
//
// Result is time of alignment.
func Time(r1, d1, r2, d2 unit.Angle, r3, d3 []unit.Angle, t1, t5 float64) (float64, error) {
if len(r3) != 5 || len(d3) != 5 {
return 0, errors.New("r3, d3 must be length 5")
}
gc := make([]float64, 5)
for i, r3i := range r3 {
// (19.1) p. 121
gc[i] = d1.Tan()*(r2-r3i).Sin() +
d2.Tan()*(r3i-r1).Sin() +
d3[i].Tan()*(r1-r2).Sin()
}
l5, err := interp.NewLen5(t1, t5, gc)
if err != nil {
return 0, err
}
return l5.Zero(false)
}
// Position computes apparent position angle and angular distance of
// components of a binary star.
//
// e is eccentricity of the true orbit
// a is angular apparent semimajor axis
// i is inclination relative to the line of sight
// Ω is position angle of the ascending node
// ω is longitude of periastron
// E is eccentric anomaly, computed for example with package kepler
// and the mean anomaly as returned by function M in this package.
//
// Return value θ is the apparent position angle, ρ is the angular distance.
func Position(e float64, a, i, Ω, ω, E unit.Angle) (θ, ρ unit.Angle) {
r := a.Mul(1 - e*E.Cos())
ν := unit.Angle(2 * math.Atan(math.Sqrt((1+e)/(1-e))*E.Div(2).Tan()))
sνω, cνω := (ν + ω).Sincos()
ci := i.Cos()
num := sνω * ci
θ = (unit.Angle(math.Atan2(num, cνω)) + Ω).Mod1()
ρ = r.Mul(math.Sqrt(num*num + cνω*cνω))
return
}
// ApproxTimes computes approximate UT rise, transit and set times for
// a celestial object on a day of interest.
//
// The function argurments do not actually include the day, but do include
// values computed from the day.
//
// p is geographic coordinates of observer.
// h0 is "standard altitude" of the body.
// Th0 is apparent sidereal time at 0h UT at Greenwich.
// α, δ are right ascension and declination of the body.
//
// Th0 must be the time on the day of interest.
// See sidereal.Apparent0UT.
//
// α, δ must be values at 0h dynamical time for the day of interest.
func ApproxTimes(p globe.Coord, h0 unit.Angle, Th0 unit.Time, α unit.RA, δ unit.Angle) (tRise, tTransit, tSet unit.Time, err error) {
// approximate local hour angle
sLat, cLat := p.Lat.Sincos()
sδ1, cδ1 := δ.Sincos()
cH0 := (h0.Sin() - sLat*sδ1) / (cLat * cδ1) // (15.1) p. 102
if cH0 < -1 || cH0 > 1 {
err = ErrorCircumpolar
return
}
H0 := unit.TimeFromRad(math.Acos(cH0))
// approximate transit, rise, set times.
// (15.2) p. 102.
mt := unit.TimeFromRad(α.Rad()+p.Lon.Rad()) - Th0
tTransit = mt.Mod1()
tRise = (mt - H0).Mod1()
tSet = (mt + H0).Mod1()
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)
}
// Kepler3 solves Kepler's equation by binary search.
//
// Argument e is eccentricity, M is mean anomaly.
//
// Result E is eccentric anomaly.
func Kepler3(e float64, M unit.Angle) (E unit.Angle) {
// adapted from BASIC, p. 206
MR := M.Mod1().Rad()
f := 1
if MR > math.Pi {
f = -1
MR = 2*math.Pi - MR
}
E0 := math.Pi * .5
d := math.Pi * .25
for i := 0; i < 53; i++ {
M1 := E0 - e*math.Sin(E0)
if MR-M1 < 0 {
E0 -= d
} else {
E0 += d
}
d *= .5
}
if f < 0 {
E0 = -E0
}
return unit.Angle(E0)
}
// 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
}
// Smallest finds the smallest circle containing three points.
//
// Arguments should represent coordinates in right ascension and declination
// or longitude and latitude. Result Δ is the diameter of the circle, typeI
// is true if solution is of type I.
//
// type I Two points on circle, one interior.
// type II All three points on circle.
func Smallest(r1, d1, r2, d2, r3, d3 unit.Angle) (Δ unit.Angle, typeI bool) {
// Using haversine formula, but reimplementing SepHav here to reuse
// the computed cosines.
cd1 := d1.Cos()
cd2 := d2.Cos()
cd3 := d3.Cos()
a := 2 * math.Asin(math.Sqrt(base.Hav(d2-d1)+cd1*cd2*base.Hav(r2-r1)))
b := 2 * math.Asin(math.Sqrt(base.Hav(d3-d2)+cd2*cd3*base.Hav(r3-r2)))
c := 2 * math.Asin(math.Sqrt(base.Hav(d1-d3)+cd3*cd1*base.Hav(r1-r3)))
if b > a {
a, b = b, a
}
if c > a {
a, c = c, a
}
if a*a >= b*b+c*c {
return unit.Angle(a), true
}
// (20.1) p. 128
return unit.Angle(2 * a * b * c /
math.Sqrt((a+b+c)*(a+b-c)*(b+c-a)*(a+c-b))), false
}
// 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
}
// Illuminated returns the illuminated fraction of a body's disk.
//
// The illuminated body can be the Moon or a planet.
//
// Argument i is the phase angle.
func Illuminated(i unit.Angle) float64 {
// (41.1) p. 283, also (48.1) p. 345.
return (1 + i.Cos()) * .5
}
// 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()))
}
// Radius returns radius distance r for given eccentric anomaly E.
//
// Argument e is eccentricity, a is semimajor axis.
//
// Result unit is the unit of semimajor axis a (typically AU.)
func Radius(E unit.Angle, e, a float64) float64 {
// (30.2) p. 195
return a * (1 - e*E.Cos())
}
// True returns true anomaly ν for given eccentric anomaly E.
//
// Argument e is eccentricity. E must be in radians.
func True(E unit.Angle, e float64) unit.Angle {
// (30.1) p. 195
return unit.Angle(2 * math.Atan(math.Sqrt((1+e)/(1-e))*E.Mul(.5).Tan()))
}
// 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
}
// SepHav returns the angular separation between two celestial bodies.
//
// The algorithm uses the haversine function and is superior to the naïve
// algorithm of the Sep function.
func SepHav(r1, d1, r2, d2 unit.Angle) unit.Angle {
// using (17.5) p. 115
return unit.Angle(2 * math.Asin(math.Sqrt(base.Hav(d2-d1)+
d1.Cos()*d2.Cos()*base.Hav(r2-r1))))
}
// Hav implements the haversine trigonometric function.
//
// See https://en.wikipedia.org/wiki/Haversine_formula.
func Hav(a unit.Angle) float64 {
// (17.5) p. 115
return .5 * (1 - a.Cos())
}
// 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))
}