def nutation(tee):
"""Return the longitudinal nutation at moment, tee."""
c = julian_centuries(tee)
cap_A = poly(c, [mpf(124.90), mpf(-1934.134), mpf(0.002063)])
cap_B = poly(c, [mpf(201.11), mpf(72001.5377), mpf(0.00057)])
return (mpf(-0.004778) * sin_degrees(cap_A) +
mpf(-0.0003667) * sin_degrees(cap_B))
def equation_of_time(tee):
"""Return the equation of time (as fraction of day) for moment, tee.
Adapted from "Astronomical Algorithms" by Jean Meeus,
Willmann_Bell, Inc., 1991."""
c = julian_centuries(tee)
lamb = poly(c, [mpf(280.46645), mpf(36000.76983), mpf(0.0003032)])
anomaly = poly(
c,
[mpf(357.52910),
mpf(35999.05030),
mpf(-0.0001559),
mpf(-0.00000048)])
eccentricity = poly(
c, [mpf(0.016708617),
mpf(-0.000042037),
mpf(-0.0000001236)])
varepsilon = obliquity(tee)
y = pow(tan_degrees(varepsilon / 2), 2)
equation = (
(1 / 2 / pi) *
(y * sin_degrees(2 * lamb) + -2 * eccentricity * sin_degrees(anomaly) +
(4 * eccentricity * y * sin_degrees(anomaly) * cos_degrees(2 * lamb))
+ -0.5 * y * y * sin_degrees(4 * lamb) +
-1.25 * eccentricity * eccentricity * sin_degrees(2 * anomaly)))
return signum(equation) * min(abs(equation), Clock.days_from_hours(
mpf(12)))
def ephemeris_correction(tee):
"""Return Dynamical Time minus Universal Time (in days) for
moment, tee. Adapted from "Astronomical Algorithms"
by Jean Meeus, Willmann_Bell, Inc., 1991."""
year = GregorianDate.to_year(int(math.floor(tee)))
c = GregorianDate.date_difference(
GregorianDate(1900, MonthOfYear.JANUARY, 1),
GregorianDate(year, MonthOfYear.JULY, 1)) / mpf(36525)
if 1988 <= year <= 2019:
return 1 / 86400 * (year - 1933)
elif 1900 <= year <= 1987:
return poly(c, [
mpf(-0.00002),
mpf(0.000297),
mpf(0.025184),
mpf(-0.181133),
mpf(0.553040),
mpf(-0.861938),
mpf(0.677066),
mpf(-0.212591)
])
elif 1800 <= year <= 1899:
return poly(c, [
mpf(-0.000009),
mpf(0.003844),
mpf(0.083563),
mpf(0.865736),
mpf(4.867575),
mpf(15.845535),
mpf(31.332267),
mpf(38.291999),
mpf(28.316289),
mpf(11.636204),
mpf(2.043794)
])
elif 1700 <= year <= 1799:
return 1 / 86400 * poly(
year - 1700,
[8.118780842, -0.005092142, 0.003336121, -0.0000266484])
elif 1620 <= year <= 1699:
return 1 / 86400 * poly(
year - 1600,
[mpf(196.58333), mpf(-4.0675),
mpf(0.0219167)])
else:
x = Clock.days_from_hours(mpf(12)) + GregorianDate.date_difference(
GregorianDate(1810, MonthOfYear.JANUARY, 1),
GregorianDate(year, MonthOfYear.JANUARY, 1))
return 1 / 86400 * (((x * x) / mpf(41048480)) - 15)
def obliquity(tee):
"""Return (mean) obliquity of ecliptic at moment tee."""
c = julian_centuries(tee)
return (angle(23, 26, mpf(21.448)) + poly(c, [
mpf(0),
angle(0, 0, mpf(-46.8150)),
angle(0, 0, mpf(-0.00059)),
angle(0, 0, mpf(0.001813))
]))
def sidereal_from_moment(tee):
"""Return the mean sidereal time of day from moment tee expressed
as hour angle. Adapted from "Astronomical Algorithms"
by Jean Meeus, Willmann_Bell, Inc., 1991."""
c = (tee - J2000) / mpf(36525)
return poly(c, [
mpf(280.46061837),
mpf(36525) * mpf(360.98564736629),
mpf(0.000387933),
mpf(-1) / mpf(38710000)
]) % 360
def solar_anomaly(c):
"""Return mean anomaly of sun (in degrees) at moment
given in Julian centuries c.
Adapted from eq. 47.3 in "Astronomical Algorithms" by Jean Meeus,
Willmann_Bell, Inc., 2nd ed. with corrections, 2005."""
return normalized_degrees(
poly(c, [
mpf(357.5291092),
mpf(35999.0502909),
mpf(-0.0001536),
mpf(1.0 / 24490000.0)
]))
def moon_node(c):
"""Return Moon's argument of latitude (in degrees) at moment
given in Julian centuries 'c'.
Adapted from eq. 47.5 in "Astronomical Algorithms" by Jean Meeus,
Willmann_Bell, Inc., 2nd ed. with corrections, 2005."""
return normalized_degrees(
poly(c, [
mpf(93.2720950),
mpf(483202.0175233),
mpf(-0.0036539),
mpf(-1.0 / 3526000.0),
mpf(1.0 / 863310000.0)
]))
def lunar_anomaly(c):
"""Return mean anomaly of moon (in degrees) at moment
given in Julian centuries c.
Adapted from eq. 47.4 in "Astronomical Algorithms" by Jean Meeus,
Willmann_Bell, Inc., 2nd ed. with corrections, 2005."""
return normalized_degrees(
poly(c, [
mpf(134.9633964),
mpf(477198.8675055),
mpf(0.0087414),
mpf(1.0 / 69699.0),
mpf(-1.0 / 14712000.0)
]))
def lunar_elongation(c):
"""Return elongation of moon (in degrees) at moment
given in Julian centuries c.
Adapted from eq. 47.2 in "Astronomical Algorithms" by Jean Meeus,
Willmann_Bell, Inc., 2nd ed. with corrections, 2005."""
return normalized_degrees(
poly(c, [
mpf(297.8501921),
mpf(445267.1114034),
mpf(-0.0018819),
mpf(1 / 545868),
mpf(-1 / 113065000)
]))
def lunar_longitude(tee):
"""Return longitude of moon (in degrees) at moment tee.
Adapted from "Astronomical Algorithms" by Jean Meeus,
Willmann_Bell, Inc., 2nd ed., 1998."""
c = julian_centuries(tee)
cap_L_prime = mean_lunar_longitude(c)
cap_D = lunar_elongation(c)
cap_M = solar_anomaly(c)
cap_M_prime = lunar_anomaly(c)
cap_F = moon_node(c)
# see eq. 47.6 in Meeus
cap_E = poly(c, [1, mpf(-0.002516), mpf(-0.0000074)])
args_lunar_elongation = \
[0, 2, 2, 0, 0, 0, 2, 2, 2, 2, 0, 1, 0, 2, 0, 0, 4, 0, 4, 2, 2, 1,
1, 2, 2, 4, 2, 0, 2, 2, 1, 2, 0, 0, 2, 2, 2, 4, 0, 3, 2, 4, 0, 2,
2, 2, 4, 0, 4, 1, 2, 0, 1, 3, 4, 2, 0, 1, 2]
args_solar_anomaly = \
[0, 0, 0, 0, 1, 0, 0, -1, 0, -1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1,
0, 1, -1, 0, 0, 0, 1, 0, -1, 0, -2, 1, 2, -2, 0, 0, -1, 0, 0, 1,
-1, 2, 2, 1, -1, 0, 0, -1, 0, 1, 0, 1, 0, 0, -1, 2, 1, 0]
args_lunar_anomaly = \
[1, -1, 0, 2, 0, 0, -2, -1, 1, 0, -1, 0, 1, 0, 1, 1, -1, 3, -2,
-1, 0, -1, 0, 1, 2, 0, -3, -2, -1, -2, 1, 0, 2, 0, -1, 1, 0,
-1, 2, -1, 1, -2, -1, -1, -2, 0, 1, 4, 0, -2, 0, 2, 1, -2, -3,
2, 1, -1, 3]
args_moon_node = \
[0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, -2, 2, -2, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, -2, 2, 0, 2, 0, 0, 0, 0,
0, 0, -2, 0, 0, 0, 0, -2, -2, 0, 0, 0, 0, 0, 0, 0]
sine_coefficients = \
[6288774,1274027,658314,213618,-185116,-114332,
58793,57066,53322,45758,-40923,-34720,-30383,
15327,-12528,10980,10675,10034,8548,-7888,
-6766,-5163,4987,4036,3994,3861,3665,-2689,
-2602, 2390,-2348,2236,-2120,-2069,2048,-1773,
-1595,1215,-1110,-892,-810,759,-713,-700,691,
596,549,537,520,-487,-399,-381,351,-340,330,
327,-323,299,294]
correction = ((1.0 / 1000000.0) * sigma([
sine_coefficients, args_lunar_elongation, args_solar_anomaly,
args_lunar_anomaly, args_moon_node
], lambda v, w, x, y, z: v * pow(cap_E, abs(x)) * sin_degrees(
(w * cap_D) + (x * cap_M) + (y * cap_M_prime) + (z * cap_F))))
A1 = mpf(119.75) + (c * mpf(131.849))
venus = ((3958 / 1000000) * sin_degrees(A1))
A2 = mpf(53.09) + c * mpf(479264.29)
jupiter = ((318 / 1000000) * sin_degrees(A2))
flat_earth = ((1962 / 1000000) * sin_degrees(cap_L_prime - cap_F))
return (cap_L_prime + correction + venus + jupiter + flat_earth +
nutation(tee)) % 360
def lunar_perigee(tee):
"""Return Angular distance of the perigee from the equinoctal point
at moment, tee.
Adapted from eq. 47.7 in "Astronomical Algorithms"
by Jean Meeus, Willmann_Bell, Inc., 2nd ed., 1998
with corrections June 2005."""
return normalized_degrees(
poly(julian_centuries(tee), [
mpf(83.3532465),
mpf(4069.0137287),
mpf(-0.0103200),
mpf(-1.0 / 80053.0),
mpf(1.0 / 18999000.0)
]))
def lunar_latitude(tee):
"""Return the latitude of moon (in degrees) at moment, tee.
Adapted from "Astronomical Algorithms" by Jean Meeus,
Willmann_Bell, Inc., 1998."""
c = julian_centuries(tee)
cap_L_prime = mean_lunar_longitude(c)
cap_D = lunar_elongation(c)
cap_M = solar_anomaly(c)
cap_M_prime = lunar_anomaly(c)
cap_F = moon_node(c)
cap_E = poly(c, [1, mpf(-0.002516), mpf(-0.0000074)])
args_lunar_elongation = \
[0, 0, 0, 2, 2, 2, 2, 0, 2, 0, 2, 2, 2, 2, 2, 2, 2, 0, 4, 0, 0, 0,
1, 0, 0, 0, 1, 0, 4, 4, 0, 4, 2, 2, 2, 2, 0, 2, 2, 2, 2, 4, 2, 2,
0, 2, 1, 1, 0, 2, 1, 2, 0, 4, 4, 1, 4, 1, 4, 2]
args_solar_anomaly = \
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, -1, -1, -1, 1, 0, 1,
0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 1,
0, -1, -2, 0, 1, 1, 1, 1, 1, 0, -1, 1, 0, -1, 0, 0, 0, -1, -2]
args_lunar_anomaly = \
[0, 1, 1, 0, -1, -1, 0, 2, 1, 2, 0, -2, 1, 0, -1, 0, -1, -1, -1,
0, 0, -1, 0, 1, 1, 0, 0, 3, 0, -1, 1, -2, 0, 2, 1, -2, 3, 2, -3,
-1, 0, 0, 1, 0, 1, 1, 0, 0, -2, -1, 1, -2, 2, -2, -1, 1, 1, -2,
0, 0]
args_moon_node = \
[1, 1, -1, -1, 1, -1, 1, 1, -1, -1, -1, -1, 1, -1, 1, 1, -1, -1,
-1, 1, 3, 1, 1, 1, -1, -1, -1, 1, -1, 1, -3, 1, -3, -1, -1, 1,
-1, 1, -1, 1, 1, 1, 1, -1, 3, -1, -1, 1, -1, -1, 1, -1, 1, -1,
-1, -1, -1, -1, -1, 1]
sine_coefficients = \
[5128122, 280602, 277693, 173237, 55413, 46271, 32573,
17198, 9266, 8822, 8216, 4324, 4200, -3359, 2463, 2211,
2065, -1870, 1828, -1794, -1749, -1565, -1491, -1475,
-1410, -1344, -1335, 1107, 1021, 833, 777, 671, 607,
596, 491, -451, 439, 422, 421, -366, -351, 331, 315,
302, -283, -229, 223, 223, -220, -220, -185, 181,
-177, 176, 166, -164, 132, -119, 115, 107]
beta = ((1.0 / 1000000.0) * sigma([
sine_coefficients, args_lunar_elongation, args_solar_anomaly,
args_lunar_anomaly, args_moon_node
], lambda v, w, x, y, z: (v * pow(cap_E, abs(x)) * sin_degrees(
(w * cap_D) + (x * cap_M) + (y * cap_M_prime) + (z * cap_F)))))
venus = ((175 / 1000000) *
(sin_degrees(mpf(119.75) + c * mpf(131.849) + cap_F) +
sin_degrees(mpf(119.75) + c * mpf(131.849) - cap_F)))
flat_earth = ((-2235 / 1000000) * sin_degrees(cap_L_prime) +
(127 / 1000000) * sin_degrees(cap_L_prime - cap_M_prime) +
(-115 / 1000000) * sin_degrees(cap_L_prime + cap_M_prime))
extra = ((382 / 1000000) * sin_degrees(mpf(313.45) + c * mpf(481266.484)))
return beta + venus + flat_earth + extra
def alt_lunar_node(tee):
"""Return Angular distance of the node from the equinoctal point
at fixed moment, tee.
Adapted from eq. 47.7 in "Astronomical Algorithms"
by Jean Meeus, Willmann_Bell, Inc., 2nd ed., 1998
with corrections June 2005."""
return normalized_degrees(
poly(julian_centuries(tee), [
mpf(125.0445479),
mpf(-1934.1362891),
mpf(0.0020754),
mpf(1.0 / 467441.0),
mpf(-1.0 / 60616000.0)
]))
def precession(tee):
"""Return the precession at moment tee using 0,0 as J2000 coordinates.
Adapted from "Astronomical Algorithms" by Jean Meeus,
Willmann-Bell, Inc., 1991."""
c = julian_centuries(tee)
eta = poly(
c, [0, secs(mpf(47.0029)),
secs(mpf(-0.03302)),
secs(mpf(0.000060))]) % 360
cap_P = poly(c,
[mpf(174.876384),
secs(mpf(-869.8089)),
secs(mpf(0.03536))]) % 360
p = poly(
c,
[0, secs(mpf(5029.0966)),
secs(mpf(1.11113)),
secs(mpf(0.000006))]) % 360
cap_A = cos_degrees(eta) * sin_degrees(cap_P)
cap_B = cos_degrees(cap_P)
arg = arctan_degrees(cap_A, cap_B)
return (p + cap_P - arg) % 360
def mean_lunar_longitude(c):
"""Return mean longitude of moon (in degrees) at moment
given in Julian centuries c (including the constant term of the
effect of the light-time (-0".70).
Adapted from eq. 47.1 in "Astronomical Algorithms" by Jean Meeus,
Willmann_Bell, Inc., 2nd ed. with corrections, 2005."""
return normalized_degrees(
poly(c, [
mpf(218.3164477),
mpf(481267.88123421),
mpf(-0.0015786),
mpf(1 / 538841.0),
mpf(-1.0 / 65194000.0)
]))
def precise_obliquity(tee):
"""Return precise (mean) obliquity of ecliptic at moment tee."""
u = julian_centuries(tee) / 100
#assert(abs(u) < 1,
# 'Error! This formula is valid for +/-10000 years around J2000.0')
return (poly(u, [
angle(23, 26, mpf(21.448)),
angle(0, 0, mpf(-4680.93)),
angle(0, 0, mpf(-1.55)),
angle(0, 0, mpf(+1999.25)),
angle(0, 0, mpf(-51.38)),
angle(0, 0, mpf(-249.67)),
angle(0, 0, mpf(-39.05)),
angle(0, 0, mpf(+7.12)),
angle(0, 0, mpf(+27.87)),
angle(0, 0, mpf(+5.79)),
angle(0, 0, mpf(+2.45))
]))
def lunar_distance(tee):
"""Return the distance to moon (in meters) at moment, tee.
Adapted from "Astronomical Algorithms" by Jean Meeus,
Willmann_Bell, Inc., 2nd ed."""
c = julian_centuries(tee)
cap_D = lunar_elongation(c)
cap_M = solar_anomaly(c)
cap_M_prime = lunar_anomaly(c)
cap_F = moon_node(c)
cap_E = poly(c, [1, mpf(-0.002516), mpf(-0.0000074)])
args_lunar_elongation = \
[0, 2, 2, 0, 0, 0, 2, 2, 2, 2, 0, 1, 0, 2, 0, 0, 4, 0, 4, 2, 2, 1,
1, 2, 2, 4, 2, 0, 2, 2, 1, 2, 0, 0, 2, 2, 2, 4, 0, 3, 2, 4, 0, 2,
2, 2, 4, 0, 4, 1, 2, 0, 1, 3, 4, 2, 0, 1, 2, 2,]
args_solar_anomaly = \
[0, 0, 0, 0, 1, 0, 0, -1, 0, -1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1,
0, 1, -1, 0, 0, 0, 1, 0, -1, 0, -2, 1, 2, -2, 0, 0, -1, 0, 0, 1,
-1, 2, 2, 1, -1, 0, 0, -1, 0, 1, 0, 1, 0, 0, -1, 2, 1, 0, 0]
args_lunar_anomaly = \
[1, -1, 0, 2, 0, 0, -2, -1, 1, 0, -1, 0, 1, 0, 1, 1, -1, 3, -2,
-1, 0, -1, 0, 1, 2, 0, -3, -2, -1, -2, 1, 0, 2, 0, -1, 1, 0,
-1, 2, -1, 1, -2, -1, -1, -2, 0, 1, 4, 0, -2, 0, 2, 1, -2, -3,
2, 1, -1, 3, -1]
args_moon_node = \
[0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, -2, 2, -2, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, -2, 2, 0, 2, 0, 0, 0, 0,
0, 0, -2, 0, 0, 0, 0, -2, -2, 0, 0, 0, 0, 0, 0, 0, -2]
cosine_coefficients = \
[-20905355, -3699111, -2955968, -569925, 48888, -3149,
246158, -152138, -170733, -204586, -129620, 108743,
104755, 10321, 0, 79661, -34782, -23210, -21636, 24208,
30824, -8379, -16675, -12831, -10445, -11650, 14403,
-7003, 0, 10056, 6322, -9884, 5751, 0, -4950, 4130, 0,
-3958, 0, 3258, 2616, -1897, -2117, 2354, 0, 0, -1423,
-1117, -1571, -1739, 0, -4421, 0, 0, 0, 0, 1165, 0, 0,
8752]
correction = sigma([
cosine_coefficients, args_lunar_elongation, args_solar_anomaly,
args_lunar_anomaly, args_moon_node
], lambda v, w, x, y, z: (v * pow(cap_E, abs(x)) * cos_degrees(
(w * cap_D) + (x * cap_M) + (y * cap_M_prime) + (z * cap_F))))
return 385000560 + correction
def geometric_solar_mean_longitude(tee):
"""Return the geometric mean longitude of the Sun at moment, tee,
referred to mean equinox of the date."""
c = julian_centuries(tee)
return poly(c, [mpf(280.46646), mpf(36000.76983), mpf(0.0003032)])
def nth_new_moon(n):
"""Return the moment of n-th new moon after (or before) the new moon
of January 11, 1. Adapted from "Astronomical Algorithms"
by Jean Meeus, Willmann_Bell, Inc., 2nd ed., 1998."""
n0 = 24724
k = n - n0
c = k / mpf(1236.85)
approx = (J2000 + poly(c, [
mpf(5.09766), MEAN_SYNODIC_MONTH * mpf(1236.85),
mpf(0.0001437),
mpf(-0.000000150),
mpf(0.00000000073)
]))
cap_E = poly(c, [1, mpf(-0.002516), mpf(-0.0000074)])
solar_anomaly = poly(c, [
mpf(2.5534), (mpf(1236.85) * mpf(29.10535669)),
mpf(-0.0000014),
mpf(-0.00000011)
])
lunar_anomaly = poly(c, [
mpf(201.5643), (mpf(385.81693528) * mpf(1236.85)),
mpf(0.0107582),
mpf(0.00001238),
mpf(-0.000000058)
])
moon_argument = poly(c, [
mpf(160.7108), (mpf(390.67050284) * mpf(1236.85)),
mpf(-0.0016118),
mpf(-0.00000227),
mpf(0.000000011)
])
cap_omega = poly(c, [
mpf(124.7746), (mpf(-1.56375588) * mpf(1236.85)),
mpf(0.0020672),
mpf(0.00000215)
])
E_factor = [
0, 1, 0, 0, 1, 1, 2, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
]
solar_coeff = [
0, 1, 0, 0, -1, 1, 2, 0, 0, 1, 0, 1, 1, -1, 2, 0, 3, 1, 0, 1, -1, -1,
1, 0
]
lunar_coeff = [
1, 0, 2, 0, 1, 1, 0, 1, 1, 2, 3, 0, 0, 2, 1, 2, 0, 1, 2, 1, 1, 1, 3, 4
]
moon_coeff = [
0, 0, 0, 2, 0, 0, 0, -2, 2, 0, 0, 2, -2, 0, 0, -2, 0, -2, 2, 2, 2, -2,
0, 0
]
sine_coeff = [
mpf(-0.40720),
mpf(0.17241),
mpf(0.01608),
mpf(0.01039),
mpf(0.00739),
mpf(-0.00514),
mpf(0.00208),
mpf(-0.00111),
mpf(-0.00057),
mpf(0.00056),
mpf(-0.00042),
mpf(0.00042),
mpf(0.00038),
mpf(-0.00024),
mpf(-0.00007),
mpf(0.00004),
mpf(0.00004),
mpf(0.00003),
mpf(0.00003),
mpf(-0.00003),
mpf(0.00003),
mpf(-0.00002),
mpf(-0.00002),
mpf(0.00002)
]
correction = (
(mpf(-0.00017) * sin_degrees(cap_omega)) +
sigma([sine_coeff, E_factor, solar_coeff, lunar_coeff, moon_coeff],
lambda v, w, x, y, z: (v * pow(cap_E, w) * sin_degrees(
(x * solar_anomaly) + (y * lunar_anomaly) +
(z * moon_argument)))))
add_const = [
mpf(251.88),
mpf(251.83),
mpf(349.42),
mpf(84.66),
mpf(141.74),
mpf(207.14),
mpf(154.84),
mpf(34.52),
mpf(207.19),
mpf(291.34),
mpf(161.72),
mpf(239.56),
mpf(331.55)
]
add_coeff = [
mpf(0.016321),
mpf(26.651886),
mpf(36.412478),
mpf(18.206239),
mpf(53.303771),
mpf(2.453732),
mpf(7.306860),
mpf(27.261239),
mpf(0.121824),
mpf(1.844379),
mpf(24.198154),
mpf(25.513099),
mpf(3.592518)
]
add_factor = [
mpf(0.000165),
mpf(0.000164),
mpf(0.000126),
mpf(0.000110),
mpf(0.000062),
mpf(0.000060),
mpf(0.000056),
mpf(0.000047),
mpf(0.000042),
mpf(0.000040),
mpf(0.000037),
mpf(0.000035),
mpf(0.000023)
]
extra = (mpf(0.000325) * sin_degrees(
poly(c, [mpf(299.77), mpf(132.8475848),
mpf(-0.009173)])))
additional = sigma([add_const, add_coeff, add_factor],
lambda i, j, l: l * sin_degrees(i + j * k))
return universal_from_dynamical(approx + correction + extra + additional)