def from_eccentric(cls, eccentric_anomaly, eccentricity):
# This would be easy to understand from a simple diagram.
x = cos(eccentric_anomaly) - eccentricity
# == radius * cos(θ)
y = sin(eccentric_anomaly) * sqrt(1 - eccentricity ** 2)
# == radius * sin(θ)
self = cls.__new__(cls)
self.__init__(x, y)
return self
def position_from_plane_of_orbit_to_ecliptic(r, θ, Ω, i, ω):
# TODO: break this down to easily understood operations
# using angle sum formulas etc.
# _ _ _ _ _
# cos(-x) = cos(x) 1 ┆╳ ╲ ╱┆╳ ╲ ╱┆ cos
# sin(-x) = -sin(x) 0 ╱┄╲┄╲┄╱┄╱┄╲┄╲┄╱┄╱ sin
# sin(x) = cos(x - 90) -1 ┆ ╲_╳_╱┆ ╲_╳_╱┆
# sin(x - 90) = -cos(x) -2pi 0 2pi
#
# a - b = a + (-b)
# cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
# cos(a - b) = cos(a)cos(b) + sin(a)sin(b)
# sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
# sin(a - b) = sin(a)cos(b) - cos(a)sin(b)
(xeclip, yeclip, zeclip) = (
r * (
cos(Ω) * cos(θ + ω)
- sin(Ω) * sin(θ + ω) * cos(i)
),
r * (
sin(Ω) * cos(θ + ω)
+ cos(Ω) * sin(θ + ω) * cos(i)
),
r * sin(θ + ω) * sin(i),
)
return (xeclip, yeclip, zeclip)
def eccentric_anomaly_first_approximation(mean_anomaly, eccentricity):
"""
This truncated Taylor series is claimed accurate enough for a small
eccentricity such as that of the Sun-Earth orbit (0.017).
Properly the eccentric anomaly is the solution E
of Kepler's equation in mean anomaly M and eccentricity e:
M = E - e sin(E).
Thanks to Paul Schlyter himself for clarifying this in a
private email which, frankly, I have not yet fully analyzed.
"""
return rev(
mean_anomaly
+ (180 / pi) * eccentricity * sin(mean_anomaly)
* (1 + eccentricity * cos(mean_anomaly)))
def moon_2_distance_perturbation_terms(Ls, Lm, Ms, Mm, D, F):
# all & only the terms larger than 0.1 Earth radii
return (
-0.58 * cos(Mm - 2 * D),
-0.46 * cos(2 * D),
)
def iterate_once_for_eccentric_anomaly(M, e, E0):
return E0 - (E0 - (180 / pi) * e * sin(E0) - M) / (1 - e * cos(E0))
def from_polar(cls, p):
x = p.r * cos(p.θ)
y = p.r * sin(p.θ)
self = cls.__new__(cls)
self.__init__(x, y)
return self
def cartesian2d_in_plane_of_orbit(eccentric_anomaly, eccentricity, distance):
(x, y) = (
distance * (cos(eccentric_anomaly) - eccentricity),
distance * sqrt(1 - eccentricity ** 2) * sin(eccentric_anomaly),
)
return (x, y)