""" Eccentricity functions (squared) for various truncations of e at tidal order-l = 7
"""
from typing import TYPE_CHECKING
from ...utilities.performance.numba import njit
if TYPE_CHECKING:
from . import EccenOutput
from ...utilities.types import FloatArray
[docs]
@njit(cacheable=True)
def eccentricity_funcs_trunc2(eccentricity: 'FloatArray') -> 'EccenOutput':
""" Calculates the eccentricity functions (by mode) truncated to e^2 for order-l = 7
Parameters
----------
eccentricity : FloatArray
Orbital Eccentricity
Returns
-------
eccentricity_results_bymode : EccenOutput
"""
# Eccentricity functions calculated at truncation level 2.
# and order-l = 7.
# Performance and readability improvements
e = eccentricity
e2 = e * e
# Unique results.
eccentricity_results_bymode = {
0: {
-1: 9.0 * e2,
0 : 1.0 - 70.0 * e2,
1 : 121.0 * e2,
},
1: {
-1: e2,
0 : 1.0 - 22.0 * e2,
1 : 81.0 * e2,
},
2: {
-1: e2,
0 : 10.0 * e2 + 1.0,
1 : 49.0 * e2,
},
3: {
-1: -0.140625 * e2 * (5.0 * e**4 + 20.0 * e2 + 8.0)**2 / (e2 - 1.0)**13,
0 : 26.0 * e2 + 1.0,
1 : 25.0 * e2,
},
4: {
-1: 25.0 * e2,
},
5: {
-1: 49.0 * e2,
},
6: {
-1: 81.0 * e2,
},
7: {
-1: 121.0 * e2,
}
}
# Duplicate results are stored as dictionary lookups to previous calculations
# Generally leads to a 30--50% speed-up when working with large arrays.
eccentricity_results_bymode[4][0] = eccentricity_results_bymode[3][0]
eccentricity_results_bymode[4][1] = eccentricity_results_bymode[3][-1]
eccentricity_results_bymode[5][0] = eccentricity_results_bymode[2][0]
eccentricity_results_bymode[5][1] = eccentricity_results_bymode[2][-1]
eccentricity_results_bymode[6][0] = eccentricity_results_bymode[1][0]
eccentricity_results_bymode[6][1] = eccentricity_results_bymode[2][-1]
eccentricity_results_bymode[7][0] = eccentricity_results_bymode[0][0]
eccentricity_results_bymode[7][1] = eccentricity_results_bymode[0][-1]
return eccentricity_results_bymode
[docs]
@njit(cacheable=True)
def eccentricity_funcs_trunc4(eccentricity: 'FloatArray') -> 'EccenOutput':
""" Calculates the eccentricity functions (by mode) truncated to e^4 for order-l = 7
Parameters
----------
eccentricity : FloatArray
Orbital Eccentricity
Returns
-------
eccentricity_results_bymode : EccenOutput
"""
# Eccentricity functions calculated at truncation level 4.
# and order-l = 7.
# Performance and readability improvements
e = eccentricity
e2 = e * e
e4 = e * e * e * e
# Unique results.
eccentricity_results_bymode = {
0: {
-2: 9.765625 * e4,
-1: -238.5 * e4 + 9.0 * e2,
0 : 1785.21875 * e4 - 70.0 * e2 + 1.0,
1 : -5016.0 * e4 + 121.0 * e2,
2 : 4607.015625 * e4,
},
1: {
-2: 0.140625 * e4,
-1: -9.0 * e4 + e2,
0 : 183.96875 * e4 - 22.0 * e2 + 1.0,
1 : -1201.5 * e4 + 81.0 * e2,
2 : 2173.890625 * e4,
},
2: {
-3: -0.09765625 * e**6 * (3.0 * e2 + 8.0)**2 / (e2 - 1.0)**13,
-2: 2.640625 * e4,
-1: 16.5 * e4 + e2,
0 : 76.46875 * e4 + 10.0 * e2 + 1.0,
1 : 189.0 * e4 + 49.0 * e2,
2 : 862.890625 * e4,
},
3: {
-2: 47.265625 * e4,
-1: -0.140625 * e2 * (5.0 * e4 + 20.0 * e2 + 8.0)**2 / (e2 - 1.0)**13,
0 : 310.71875 * e4 + 26.0 * e2 + 1.0,
1 : 367.5 * e4 + 25.0 * e2,
2 : 260.015625 * e4,
},
4: {
-2: 260.015625 * e4,
},
5: {
-2: 862.890625 * e4,
},
6: {
-2: 2173.890625 * e4,
},
7: {
-2: 4607.015625 * e4,
}
}
# Duplicate results are stored as dictionary lookups to previous calculations
# Generally leads to a 30--50% speed-up when working with large arrays.
eccentricity_results_bymode[4][-1] = eccentricity_results_bymode[3][1]
eccentricity_results_bymode[4][0] = eccentricity_results_bymode[3][0]
eccentricity_results_bymode[4][1] = eccentricity_results_bymode[3][-1]
eccentricity_results_bymode[4][2] = eccentricity_results_bymode[3][-2]
eccentricity_results_bymode[5][-1] = eccentricity_results_bymode[2][1]
eccentricity_results_bymode[5][0] = eccentricity_results_bymode[2][0]
eccentricity_results_bymode[5][1] = eccentricity_results_bymode[2][-1]
eccentricity_results_bymode[5][2] = eccentricity_results_bymode[2][-2]
eccentricity_results_bymode[5][3] = eccentricity_results_bymode[2][-3]
eccentricity_results_bymode[6][-1] = eccentricity_results_bymode[1][1]
eccentricity_results_bymode[6][0] = eccentricity_results_bymode[1][0]
eccentricity_results_bymode[6][1] = eccentricity_results_bymode[1][-1]
eccentricity_results_bymode[6][2] = eccentricity_results_bymode[1][-2]
eccentricity_results_bymode[7][-1] = eccentricity_results_bymode[0][1]
eccentricity_results_bymode[7][0] = eccentricity_results_bymode[0][0]
eccentricity_results_bymode[7][1] = eccentricity_results_bymode[0][-1]
eccentricity_results_bymode[7][2] = eccentricity_results_bymode[0][-2]
return eccentricity_results_bymode
[docs]
@njit(cacheable=True)
def eccentricity_funcs_trunc6(eccentricity: 'FloatArray') -> 'EccenOutput':
""" Calculates the eccentricity functions (by mode) truncated to e^6 for order-l = 7
Parameters
----------
eccentricity : FloatArray
Orbital Eccentricity
Returns
-------
eccentricity_results_bymode : EccenOutput
"""
# Eccentricity functions calculated at truncation level 6.
# and order-l = 7.
# Performance and readability improvements
e = eccentricity
e2 = e * e
e4 = e * e * e * e
e6 = e * e * e * e * e * e
# Unique results.
eccentricity_results_bymode = {
0: {
-3: 1.77777777777778 * e6,
-2: -120.442708333333 * e6 + 9.765625 * e4,
-1: 2496.9375 * e6 - 238.5 * e4 + 9.0 * e2,
0 : -21364.4618055556 * e6 + 1785.21875 * e4 - 70.0 * e2 + 1.0,
1 : 83775.8333333333 * e6 - 5016.0 * e4 + 121.0 * e2,
2 : -148518.984375 * e6 + 4607.015625 * e4,
3 : 95841.8402777778 * e6,
},
1: {
-3: 0.00694444444444444 * e6,
-2: 0.140625 * e6 + 0.140625 * e4,
-1: 24.5833333333333 * e6 - 9.0 * e4 + e2,
0 : -741.545138888889 * e6 + 183.96875 * e4 - 22.0 * e2 + 1.0,
1 : 7388.4375 * e6 - 1201.5 * e4 + 81.0 * e2,
2 : -27743.8177083333 * e6 + 2173.890625 * e4,
3 : 33184.6944444444 * e6,
},
2: {
-3: -0.09765625 * e6 * (3.0 * e2 + 8.0)**2 / (e2 - 1.0)**13,
-2: 38.9322916666667 * e6 + 2.640625 * e4,
-1: 139.270833333333 * e6 + 16.5 * e4 + e2,
0 : 424.84375 * e6 + 76.46875 * e4 + 10.0 * e2 + 1.0,
1 : 1080.58333333333 * e6 + 189.0 * e4 + 49.0 * e2,
2 : 1254.55729166667 * e6 + 862.890625 * e4,
3 : 9168.0625 * e6,
},
3: {
-3: 193.673611111111 * e6,
-2: 688.932291666667 * e6 + 47.265625 * e4,
-1: -0.140625 * e2 * (5.0 * e4 + 20.0 * e2 + 8.0)**2 / (e2 - 1.0)**13,
0 : 2353.37152777778 * e6 + 310.71875 * e4 + 26.0 * e2 + 1.0,
1 : 2885.77083333333 * e6 + 367.5 * e4 + 25.0 * e2,
2 : 2727.140625 * e6 + 260.015625 * e4,
3 : 1792.11111111111 * e6,
},
4: {
-3: 1792.11111111111 * e6,
},
5: {
-3: 9168.0625 * e6,
},
6: {
-3: 33184.6944444444 * e6,
},
7: {
-3: 95841.8402777778 * e6,
}
}
# Duplicate results are stored as dictionary lookups to previous calculations
# Generally leads to a 30--50% speed-up when working with large arrays.
eccentricity_results_bymode[4][-2] = eccentricity_results_bymode[3][2]
eccentricity_results_bymode[4][-1] = eccentricity_results_bymode[3][1]
eccentricity_results_bymode[4][0] = eccentricity_results_bymode[3][0]
eccentricity_results_bymode[4][1] = eccentricity_results_bymode[3][-1]
eccentricity_results_bymode[4][2] = eccentricity_results_bymode[3][-2]
eccentricity_results_bymode[4][3] = eccentricity_results_bymode[3][-3]
eccentricity_results_bymode[5][-2] = eccentricity_results_bymode[2][2]
eccentricity_results_bymode[5][-1] = eccentricity_results_bymode[2][1]
eccentricity_results_bymode[5][0] = eccentricity_results_bymode[2][0]
eccentricity_results_bymode[5][1] = eccentricity_results_bymode[2][-1]
eccentricity_results_bymode[5][2] = eccentricity_results_bymode[2][-2]
eccentricity_results_bymode[5][3] = eccentricity_results_bymode[2][-3]
eccentricity_results_bymode[6][-2] = eccentricity_results_bymode[1][2]
eccentricity_results_bymode[6][-1] = eccentricity_results_bymode[1][1]
eccentricity_results_bymode[6][0] = eccentricity_results_bymode[1][0]
eccentricity_results_bymode[6][1] = eccentricity_results_bymode[1][-1]
eccentricity_results_bymode[6][2] = eccentricity_results_bymode[1][-2]
eccentricity_results_bymode[6][3] = eccentricity_results_bymode[1][-3]
eccentricity_results_bymode[7][-2] = eccentricity_results_bymode[0][2]
eccentricity_results_bymode[7][-1] = eccentricity_results_bymode[0][1]
eccentricity_results_bymode[7][0] = eccentricity_results_bymode[0][0]
eccentricity_results_bymode[7][1] = eccentricity_results_bymode[0][-1]
eccentricity_results_bymode[7][2] = eccentricity_results_bymode[0][-2]
eccentricity_results_bymode[7][3] = eccentricity_results_bymode[0][-3]
return eccentricity_results_bymode
[docs]
@njit(cacheable=True)
def eccentricity_funcs_trunc8(eccentricity: 'FloatArray') -> 'EccenOutput':
""" Calculates the eccentricity functions (by mode) truncated to e^8 for order-l = 7
Parameters
----------
eccentricity : FloatArray
Orbital Eccentricity
Returns
-------
eccentricity_results_bymode : EccenOutput
"""
# Eccentricity functions calculated at truncation level 8.
# and order-l = 7.
# Performance and readability improvements
e = eccentricity
e2 = e * e
e4 = e * e * e * e
e6 = e * e * e * e * e * e
e8 = e**8
# Unique results.
eccentricity_results_bymode = {
0: {
-4: 0.04449462890625 * e8,
-3: -9.77777777777778 * e8 + 1.77777777777778 * e6,
-2: 591.854519314236 * e8 - 120.442708333333 * e6 + 9.765625 * e4,
-1: -13602.09375 * e8 + 2496.9375 * e6 - 238.5 * e4 + 9.0 * e2,
0 : 142589.589938694 * e8 - 21364.4618055556 * e6 + 1785.21875 * e4 - 70.0 * e2 + 1.0,
1 : -748898.888888889 * e8 + 83775.8333333333 * e6 - 5016.0 * e4 + 121.0 * e2,
2 : 2024000.86157227 * e8 - 148518.984375 * e6 + 4607.015625 * e4,
3 : -2669833.76736111 * e8 + 95841.8402777778 * e6,
4 : 1352599.28662788 * e8,
},
1: {
-5: -0.03515625 * e**10 / (e2 - 1.0)**13,
-4: 0.0190497504340278 * e8,
-3: 0.118055555555556 * e8 + 0.00694444444444444 * e6,
-2: 1.742431640625 * e8 + 0.140625 * e6 + 0.140625 * e4,
-1: -32.8888888888889 * e8 + 24.5833333333333 * e6 - 9.0 * e4 + e2,
0 : 1576.83668348524 * e8 - 741.545138888889 * e6 + 183.96875 * e4 - 22.0 * e2 + 1.0,
1 : -24466.5 * e8 + 7388.4375 * e6 - 1201.5 * e4 + 81.0 * e2,
2 : 152617.184597439 * e8 - 27743.8177083333 * e6 + 2173.890625 * e4,
3 : -397130.923611111 * e8 + 33184.6944444444 * e6,
4 : 354871.521057129 * e8,
},
2: {
-4: 14.396491156684 * e8,
-3: -0.09765625 * e6 * (3.0 * e2 + 8.0)**2 / (e2 - 1.0)**13,
-2: 303.058295355903 * e8 + 38.9322916666667 * e6 + 2.640625 * e4,
-1: 811.631944444444 * e8 + 139.270833333333 * e6 + 16.5 * e4 + e2,
0 : 1945.65222167969 * e8 + 424.84375 * e6 + 76.46875 * e4 + 10.0 * e2 + 1.0,
1 : 4115.31944444444 * e8 + 1080.58333333333 * e6 + 189.0 * e4 + 49.0 * e2,
2 : 8894.1413031684 * e8 + 1254.55729166667 * e6 + 862.890625 * e4,
3 : -95.75 * e8 + 9168.0625 * e6,
4 : 71362.7221747504 * e8,
},
3: {
-4: 685.376037597656 * e8,
-3: 2413.96180555556 * e8 + 193.673611111111 * e6,
-2: 5301.66408962674 * e8 + 688.932291666667 * e6 + 47.265625 * e4,
-1: -0.140625 * e2 * (5.0 * e4 + 20.0 * e2 + 8.0)**2 / (e2 - 1.0)**13,
0 : 13103.2553032769 * e8 + 2353.37152777778 * e6 + 310.71875 * e4 + 26.0 * e2 + 1.0,
1 : 16012.0798611111 * e8 + 2885.77083333333 * e6 + 367.5 * e4 + 25.0 * e2,
2 : 16688.9113769531 * e8 + 2727.140625 * e6 + 260.015625 * e4,
3 : 14474.4722222222 * e8 + 1792.11111111111 * e6,
4 : 9661.76367865668 * e8,
},
4: {
-4: 9661.76367865668 * e8,
},
5: {
-4: 71362.7221747504 * e8,
},
6: {
-4: 354871.521057129 * e8,
},
7: {
-4: 1352599.28662788 * e8,
}
}
# Duplicate results are stored as dictionary lookups to previous calculations
# Generally leads to a 30--50% speed-up when working with large arrays.
eccentricity_results_bymode[4][-3] = eccentricity_results_bymode[3][3]
eccentricity_results_bymode[4][-2] = eccentricity_results_bymode[3][2]
eccentricity_results_bymode[4][-1] = eccentricity_results_bymode[3][1]
eccentricity_results_bymode[4][0] = eccentricity_results_bymode[3][0]
eccentricity_results_bymode[4][1] = eccentricity_results_bymode[3][-1]
eccentricity_results_bymode[4][2] = eccentricity_results_bymode[3][-2]
eccentricity_results_bymode[4][3] = eccentricity_results_bymode[3][-3]
eccentricity_results_bymode[4][4] = eccentricity_results_bymode[3][-4]
eccentricity_results_bymode[5][-3] = eccentricity_results_bymode[2][3]
eccentricity_results_bymode[5][-2] = eccentricity_results_bymode[2][2]
eccentricity_results_bymode[5][-1] = eccentricity_results_bymode[2][1]
eccentricity_results_bymode[5][0] = eccentricity_results_bymode[2][0]
eccentricity_results_bymode[5][1] = eccentricity_results_bymode[2][-1]
eccentricity_results_bymode[5][2] = eccentricity_results_bymode[2][-2]
eccentricity_results_bymode[5][3] = eccentricity_results_bymode[2][-3]
eccentricity_results_bymode[5][4] = eccentricity_results_bymode[2][-4]
eccentricity_results_bymode[6][-3] = eccentricity_results_bymode[1][3]
eccentricity_results_bymode[6][-2] = eccentricity_results_bymode[1][2]
eccentricity_results_bymode[6][-1] = eccentricity_results_bymode[1][1]
eccentricity_results_bymode[6][0] = eccentricity_results_bymode[1][0]
eccentricity_results_bymode[6][1] = eccentricity_results_bymode[1][-1]
eccentricity_results_bymode[6][2] = eccentricity_results_bymode[1][-2]
eccentricity_results_bymode[6][3] = eccentricity_results_bymode[1][-3]
eccentricity_results_bymode[6][4] = eccentricity_results_bymode[1][-4]
eccentricity_results_bymode[6][5] = eccentricity_results_bymode[1][-5]
eccentricity_results_bymode[7][-3] = eccentricity_results_bymode[0][3]
eccentricity_results_bymode[7][-2] = eccentricity_results_bymode[0][2]
eccentricity_results_bymode[7][-1] = eccentricity_results_bymode[0][1]
eccentricity_results_bymode[7][0] = eccentricity_results_bymode[0][0]
eccentricity_results_bymode[7][1] = eccentricity_results_bymode[0][-1]
eccentricity_results_bymode[7][2] = eccentricity_results_bymode[0][-2]
eccentricity_results_bymode[7][3] = eccentricity_results_bymode[0][-3]
eccentricity_results_bymode[7][4] = eccentricity_results_bymode[0][-4]
return eccentricity_results_bymode
[docs]
@njit(cacheable=True)
def eccentricity_funcs_trunc10(eccentricity: 'FloatArray') -> 'EccenOutput':
""" Calculates the eccentricity functions (by mode) truncated to e^10 for order-l = 7
Parameters
----------
eccentricity : FloatArray
Orbital Eccentricity
Returns
-------
eccentricity_results_bymode : EccenOutput
"""
# Eccentricity functions calculated at truncation level 10.
# and order-l = 7.
# Performance and readability improvements
e = eccentricity
e2 = e * e
e4 = e * e * e * e
e6 = e * e * e * e * e * e
e8 = e**8
e10 = e**10
# Unique results.
eccentricity_results_bymode = {
0: {
-5: 6.94444444444444e-5 * e10,
-4: -0.07119140625 * e10 + 0.04449462890625 * e8,
-3: 20.1555555555556 * e10 - 9.77777777777778 * e8 + 1.77777777777778 * e6,
-2: -1516.85926649306 * e10 + 591.854519314236 * e8 - 120.442708333333 * e6 + 9.765625 * e4,
-1: 43713.97734375 * e10 - 13602.09375 * e8 + 2496.9375 * e6 - 238.5 * e4 + 9.0 * e2,
0 : -586640.268715278 * e10 + 142589.589938694 * e8 - 21364.4618055556 * e6 + 1785.21875 * e4 - 70.0 * e2 + 1.0,
1 : 4085875.57048611 * e10 - 748898.888888889 * e8 + 83775.8333333333 * e6 - 5016.0 * e4 + 121.0 * e2,
2 : -15548508.0931641 * e10 + 2024000.86157227 * e8 - 148518.984375 * e6 + 4607.015625 * e4,
3 : 32361560.6011285 * e10 - 2669833.76736111 * e8 + 95841.8402777778 * e6,
4 : -34373096.9709961 * e10 + 1352599.28662788 * e8,
5 : 14501244.8025 * e10,
},
1: {
-5: -0.03515625 * e10 / (e2 - 1.0)**13,
-4: 0.257063802083333 * e10 + 0.0190497504340278 * e8,
-3: 0.971701388888889 * e10 + 0.118055555555556 * e8 + 0.00694444444444444 * e6,
-2: 5.8693359375 * e10 + 1.742431640625 * e8 + 0.140625 * e6 + 0.140625 * e4,
-1: 38.0814236111111 * e10 - 32.8888888888889 * e8 + 24.5833333333333 * e6 - 9.0 * e4 + e2,
0 : -2021.14040798611 * e10 + 1576.83668348524 * e8 - 741.545138888889 * e6 + 183.96875 * e4 - 22.0 * e2 + 1.0,
1 : 48573.09140625 * e10 - 24466.5 * e8 + 7388.4375 * e6 - 1201.5 * e4 + 81.0 * e2,
2 : -475122.020203993 * e10 + 152617.184597439 * e8 - 27743.8177083333 * e6 + 2173.890625 * e4,
3 : 2097863.73862847 * e10 - 397130.923611111 * e8 + 33184.6944444444 * e6,
4 : -4146304.49912109 * e10 + 354871.521057129 * e8,
5 : 2959618.45876736 * e10,
},
2: {
-5: 32.4425173611111 * e10,
-4: 185.11298828125 * e10 + 14.396491156684 * e8,
-3: -0.09765625 * e6 * (3.0 * e2 + 8.0)**2 / (e2 - 1.0)**13,
-2: 1659.23604600694 * e10 + 303.058295355903 * e8 + 38.9322916666667 * e6 + 2.640625 * e4,
-1: 3694.66293402778 * e10 + 811.631944444444 * e8 + 139.270833333333 * e6 + 16.5 * e4 + e2,
0 : 7565.199296875 * e10 + 1945.65222167969 * e8 + 424.84375 * e6 + 76.46875 * e4 + 10.0 * e2 + 1.0,
1 : 14360.1105902778 * e10 + 4115.31944444444 * e8 + 1080.58333333333 * e6 + 189.0 * e4 + 49.0 * e2,
2 : 24376.8419053819 * e10 + 8894.1413031684 * e8 + 1254.55729166667 * e6 + 862.890625 * e4,
3 : 59750.0453125 * e10 - 95.75 * e8 + 9168.0625 * e6,
4 : -78294.8146809896 * e10 + 71362.7221747504 * e8,
5 : 448693.440434028 * e10,
},
3: {
-5: 2198.82840277778 * e10,
-4: 7485.75439453125 * e10 + 685.376037597656 * e8,
-3: 16387.2709201389 * e10 + 2413.96180555556 * e8 + 193.673611111111 * e6,
-2: 28704.1544053819 * e10 + 5301.66408962674 * e8 + 688.932291666667 * e6 + 47.265625 * e4,
-1: -0.140625 * e2 * (5.0 * e4 + 20.0 * e2 + 8.0)**2 / (e2 - 1.0)**13,
0 : 58347.2753993056 * e10 + 13103.2553032769 * e8 + 2353.37152777778 * e6 + 310.71875 * e4 + 26.0 * e2 + 1.0,
1 : 70311.6384548611 * e10 + 16012.0798611111 * e8 + 2885.77083333333 * e6 + 367.5 * e4 + 25.0 * e2,
2 : 76178.1462890625 * e10 + 16688.9113769531 * e8 + 2727.140625 * e6 + 260.015625 * e4,
3 : 73701.4461805556 * e10 + 14474.4722222222 * e8 + 1792.11111111111 * e6,
4 : 61808.6661783854 * e10 + 9661.76367865668 * e8,
5 : 44163.0225 * e10,
},
4: {
-5: 44163.0225 * e10,
},
5: {
-5: 448693.440434028 * e10,
},
6: {
-5: 2959618.45876736 * e10,
},
7: {
-5: 14501244.8025 * e10,
}
}
# Duplicate results are stored as dictionary lookups to previous calculations
# Generally leads to a 30--50% speed-up when working with large arrays.
eccentricity_results_bymode[4][-4] = eccentricity_results_bymode[3][4]
eccentricity_results_bymode[4][-3] = eccentricity_results_bymode[3][3]
eccentricity_results_bymode[4][-2] = eccentricity_results_bymode[3][2]
eccentricity_results_bymode[4][-1] = eccentricity_results_bymode[3][1]
eccentricity_results_bymode[4][0] = eccentricity_results_bymode[3][0]
eccentricity_results_bymode[4][1] = eccentricity_results_bymode[3][-1]
eccentricity_results_bymode[4][2] = eccentricity_results_bymode[3][-2]
eccentricity_results_bymode[4][3] = eccentricity_results_bymode[3][-3]
eccentricity_results_bymode[4][4] = eccentricity_results_bymode[3][-4]
eccentricity_results_bymode[4][5] = eccentricity_results_bymode[3][-5]
eccentricity_results_bymode[5][-4] = eccentricity_results_bymode[2][4]
eccentricity_results_bymode[5][-3] = eccentricity_results_bymode[2][3]
eccentricity_results_bymode[5][-2] = eccentricity_results_bymode[2][2]
eccentricity_results_bymode[5][-1] = eccentricity_results_bymode[2][1]
eccentricity_results_bymode[5][0] = eccentricity_results_bymode[2][0]
eccentricity_results_bymode[5][1] = eccentricity_results_bymode[2][-1]
eccentricity_results_bymode[5][2] = eccentricity_results_bymode[2][-2]
eccentricity_results_bymode[5][3] = eccentricity_results_bymode[2][-3]
eccentricity_results_bymode[5][4] = eccentricity_results_bymode[2][-4]
eccentricity_results_bymode[5][5] = eccentricity_results_bymode[2][-5]
eccentricity_results_bymode[6][-4] = eccentricity_results_bymode[1][4]
eccentricity_results_bymode[6][-3] = eccentricity_results_bymode[1][3]
eccentricity_results_bymode[6][-2] = eccentricity_results_bymode[1][2]
eccentricity_results_bymode[6][-1] = eccentricity_results_bymode[1][1]
eccentricity_results_bymode[6][0] = eccentricity_results_bymode[1][0]
eccentricity_results_bymode[6][1] = eccentricity_results_bymode[1][-1]
eccentricity_results_bymode[6][2] = eccentricity_results_bymode[1][-2]
eccentricity_results_bymode[6][3] = eccentricity_results_bymode[1][-3]
eccentricity_results_bymode[6][4] = eccentricity_results_bymode[1][-4]
eccentricity_results_bymode[6][5] = eccentricity_results_bymode[1][-5]
eccentricity_results_bymode[7][-4] = eccentricity_results_bymode[0][4]
eccentricity_results_bymode[7][-3] = eccentricity_results_bymode[0][3]
eccentricity_results_bymode[7][-2] = eccentricity_results_bymode[0][2]
eccentricity_results_bymode[7][-1] = eccentricity_results_bymode[0][1]
eccentricity_results_bymode[7][0] = eccentricity_results_bymode[0][0]
eccentricity_results_bymode[7][1] = eccentricity_results_bymode[0][-1]
eccentricity_results_bymode[7][2] = eccentricity_results_bymode[0][-2]
eccentricity_results_bymode[7][3] = eccentricity_results_bymode[0][-3]
eccentricity_results_bymode[7][4] = eccentricity_results_bymode[0][-4]
eccentricity_results_bymode[7][5] = eccentricity_results_bymode[0][-5]
return eccentricity_results_bymode
[docs]
@njit(cacheable=True)
def eccentricity_funcs_trunc12(eccentricity: 'FloatArray') -> 'EccenOutput':
""" Calculates the eccentricity functions (by mode) truncated to e^12 for order-l = 7
Parameters
----------
eccentricity : FloatArray
Orbital Eccentricity
Returns
-------
eccentricity_results_bymode : EccenOutput
"""
# Eccentricity functions calculated at truncation level 12.
# and order-l = 7.
# Performance and readability improvements
e = eccentricity
e2 = e * e
e4 = e * e * e * e
e6 = e * e * e * e * e * e
e8 = e**8
e10 = e**10
e12 = e**12
# Unique results.
eccentricity_results_bymode = {
0: {
-6: 4.7095027970679e-10 * e12,
-5: 5.78703703703704e-5 * e12 + 6.94444444444444e-5 * e10,
-4: 0.0320917510986328 * e12 - 0.07119140625 * e10 + 0.04449462890625 * e8,
-3: -20.1716049382716 * e12 + 20.1555555555556 * e10 - 9.77777777777778 * e8 + 1.77777777777778 * e6,
-2: 2269.74851555294 * e12 - 1516.85926649306 * e10 + 591.854519314236 * e8 - 120.442708333333 * e6 + 9.765625 * e4,
-1: -89131.943671875 * e12 + 43713.97734375 * e10 - 13602.09375 * e8 + 2496.9375 * e6 - 238.5 * e4 + 9.0 * e2,
0 : 1586997.38291086 * e12 - 586640.268715278 * e10 + 142589.589938694 * e8 - 21364.4618055556 * e6 + 1785.21875 * e4 - 70.0 * e2 + 1.0,
1 : -14695176.0341667 * e12 + 4085875.57048611 * e10 - 748898.888888889 * e8 + 83775.8333333333 * e6 - 5016.0 * e4 + 121.0 * e2,
2 : 76176310.2957281 * e12 - 15548508.0931641 * e10 + 2024000.86157227 * e8 - 148518.984375 * e6 + 4607.015625 * e4,
3 : -227752424.952377 * e12 + 32361560.6011285 * e10 - 2669833.76736111 * e8 + 95841.8402777778 * e6,
4 : 387648466.517466 * e12 - 34373096.9709961 * e10 + 1352599.28662788 * e8,
5 : -347139933.975 * e12 + 14501244.8025 * e10,
6 : 126441284.408644 * e12,
},
1: {
-6: 0.0649431247475706 * e12,
-5: -0.03515625 * e10 / (e2 - 1.0)**13,
-4: 1.8597327875208 * e12 + 0.257063802083333 * e10 + 0.0190497504340278 * e8,
-3: 5.39066358024691 * e12 + 0.971701388888889 * e10 + 0.118055555555556 * e8 + 0.00694444444444444 * e6,
-2: 20.3473989486694 * e12 + 5.8693359375 * e10 + 1.742431640625 * e8 + 0.140625 * e6 + 0.140625 * e4,
-1: 30.2398958333333 * e12 + 38.0814236111111 * e10 - 32.8888888888889 * e8 + 24.5833333333333 * e6 - 9.0 * e4 + e2,
0 : 1725.84582929258 * e12 - 2021.14040798611 * e10 + 1576.83668348524 * e8 - 741.545138888889 * e6 + 183.96875 * e4 - 22.0 * e2 + 1.0,
1 : -62900.057109375 * e12 + 48573.09140625 * e10 - 24466.5 * e8 + 7388.4375 * e6 - 1201.5 * e4 + 81.0 * e2,
2 : 938547.352011341 * e12 - 475122.020203993 * e10 + 152617.184597439 * e8 - 27743.8177083333 * e6 + 2173.890625 * e4,
3 : -6464235.02122878 * e12 + 2097863.73862847 * e10 - 397130.923611111 * e8 + 33184.6944444444 * e6,
4 : 21747975.9750501 * e12 - 4146304.49912109 * e10 + 354871.521057129 * e8,
5 : -34500980.0248119 * e12 + 2959618.45876736 * e10,
6 : 20518448.8791875 * e12,
},
2: {
-6: 71.8156937026978 * e12,
-5: 390.619458912037 * e12 + 32.4425173611111 * e10,
-4: 1283.45398924368 * e12 + 185.11298828125 * e10 + 14.396491156684 * e8,
-3: -0.09765625 * e6 * (3.0 * e2 + 8.0)**2 / (e2 - 1.0)**13,
-2: 7178.43844695621 * e12 + 1659.23604600694 * e10 + 303.058295355903 * e8 + 38.9322916666667 * e6 + 2.640625 * e4,
-1: 14041.6768142361 * e12 + 3694.66293402778 * e10 + 811.631944444444 * e8 + 139.270833333333 * e6 + 16.5 * e4 + e2,
0 : 25731.1676902771 * e12 + 7565.199296875 * e10 + 1945.65222167969 * e8 + 424.84375 * e6 + 76.46875 * e4 + 10.0 * e2 + 1.0,
1 : 44644.5701736111 * e12 + 14360.1105902778 * e10 + 4115.31944444444 * e8 + 1080.58333333333 * e6 + 189.0 * e4 + 49.0 * e2,
2 : 73906.7868524128 * e12 + 24376.8419053819 * e10 + 8894.1413031684 * e8 + 1254.55729166667 * e6 + 862.890625 * e4,
3 : 97963.396875 * e12 + 59750.0453125 * e10 - 95.75 * e8 + 9168.0625 * e6,
4 : 392310.672480862 * e12 - 78294.8146809896 * e10 + 71362.7221747504 * e8,
5 : -894602.368952546 * e12 + 448693.440434028 * e10,
6 : 2412439.34078983 * e12,
},
3: {
-6: 6572.37797547564 * e12,
-5: 21292.203287037 * e12 + 2198.82840277778 * e10,
-4: 45732.4679340363 * e12 + 7485.75439453125 * e10 + 685.376037597656 * e8,
-3: 80064.9051456404 * e12 + 16387.2709201389 * e10 + 2413.96180555556 * e8 + 193.673611111111 * e6,
-2: 122862.972033925 * e12 + 28704.1544053819 * e10 + 5301.66408962674 * e8 + 688.932291666667 * e6 + 47.265625 * e4,
-1: -0.140625 * e2 * (5.0 * e4 + 20.0 * e2 + 8.0)**2 / (e2 - 1.0)**13,
0 : 219108.287543295 * e12 + 58347.2753993056 * e10 + 13103.2553032769 * e8 + 2353.37152777778 * e6 + 310.71875 * e4 + 26.0 * e2 + 1.0,
1 : 259842.755598958 * e12 + 70311.6384548611 * e10 + 16012.0798611111 * e8 + 2885.77083333333 * e6 + 367.5 * e4 + 25.0 * e2,
2 : 285788.216771507 * e12 + 76178.1462890625 * e10 + 16688.9113769531 * e8 + 2727.140625 * e6 + 260.015625 * e4,
3 : 290946.807098765 * e12 + 73701.4461805556 * e10 + 14474.4722222222 * e8 + 1792.11111111111 * e6,
4 : 271561.904476236 * e12 + 61808.6661783854 * e10 + 9661.76367865668 * e8,
5 : 224786.9475 * e12 + 44163.0225 * e10,
6 : 179171.885103873 * e12,
},
4: {
-6: 179171.885103873 * e12,
},
5: {
-6: 2412439.34078983 * e12,
},
6: {
-6: 20518448.8791875 * e12,
},
7: {
-6: 126441284.408644 * e12,
}
}
# Duplicate results are stored as dictionary lookups to previous calculations
# Generally leads to a 30--50% speed-up when working with large arrays.
eccentricity_results_bymode[4][-5] = eccentricity_results_bymode[3][5]
eccentricity_results_bymode[4][-4] = eccentricity_results_bymode[3][4]
eccentricity_results_bymode[4][-3] = eccentricity_results_bymode[3][3]
eccentricity_results_bymode[4][-2] = eccentricity_results_bymode[3][2]
eccentricity_results_bymode[4][-1] = eccentricity_results_bymode[3][1]
eccentricity_results_bymode[4][0] = eccentricity_results_bymode[3][0]
eccentricity_results_bymode[4][1] = eccentricity_results_bymode[3][-1]
eccentricity_results_bymode[4][2] = eccentricity_results_bymode[3][-2]
eccentricity_results_bymode[4][3] = eccentricity_results_bymode[3][-3]
eccentricity_results_bymode[4][4] = eccentricity_results_bymode[3][-4]
eccentricity_results_bymode[4][5] = eccentricity_results_bymode[3][-5]
eccentricity_results_bymode[4][6] = eccentricity_results_bymode[3][-6]
eccentricity_results_bymode[5][-5] = eccentricity_results_bymode[2][5]
eccentricity_results_bymode[5][-4] = eccentricity_results_bymode[2][4]
eccentricity_results_bymode[5][-3] = eccentricity_results_bymode[2][3]
eccentricity_results_bymode[5][-2] = eccentricity_results_bymode[2][2]
eccentricity_results_bymode[5][-1] = eccentricity_results_bymode[2][1]
eccentricity_results_bymode[5][0] = eccentricity_results_bymode[2][0]
eccentricity_results_bymode[5][1] = eccentricity_results_bymode[2][-1]
eccentricity_results_bymode[5][2] = eccentricity_results_bymode[2][-2]
eccentricity_results_bymode[5][3] = eccentricity_results_bymode[2][-3]
eccentricity_results_bymode[5][4] = eccentricity_results_bymode[2][-4]
eccentricity_results_bymode[5][5] = eccentricity_results_bymode[2][-5]
eccentricity_results_bymode[5][6] = eccentricity_results_bymode[2][-6]
eccentricity_results_bymode[6][-5] = eccentricity_results_bymode[1][5]
eccentricity_results_bymode[6][-4] = eccentricity_results_bymode[1][4]
eccentricity_results_bymode[6][-3] = eccentricity_results_bymode[1][3]
eccentricity_results_bymode[6][-2] = eccentricity_results_bymode[1][2]
eccentricity_results_bymode[6][-1] = eccentricity_results_bymode[1][1]
eccentricity_results_bymode[6][0] = eccentricity_results_bymode[1][0]
eccentricity_results_bymode[6][1] = eccentricity_results_bymode[1][-1]
eccentricity_results_bymode[6][2] = eccentricity_results_bymode[1][-2]
eccentricity_results_bymode[6][3] = eccentricity_results_bymode[1][-3]
eccentricity_results_bymode[6][4] = eccentricity_results_bymode[1][-4]
eccentricity_results_bymode[6][5] = eccentricity_results_bymode[1][-5]
eccentricity_results_bymode[6][6] = eccentricity_results_bymode[1][-6]
eccentricity_results_bymode[7][-5] = eccentricity_results_bymode[0][5]
eccentricity_results_bymode[7][-4] = eccentricity_results_bymode[0][4]
eccentricity_results_bymode[7][-3] = eccentricity_results_bymode[0][3]
eccentricity_results_bymode[7][-2] = eccentricity_results_bymode[0][2]
eccentricity_results_bymode[7][-1] = eccentricity_results_bymode[0][1]
eccentricity_results_bymode[7][0] = eccentricity_results_bymode[0][0]
eccentricity_results_bymode[7][1] = eccentricity_results_bymode[0][-1]
eccentricity_results_bymode[7][2] = eccentricity_results_bymode[0][-2]
eccentricity_results_bymode[7][3] = eccentricity_results_bymode[0][-3]
eccentricity_results_bymode[7][4] = eccentricity_results_bymode[0][-4]
eccentricity_results_bymode[7][5] = eccentricity_results_bymode[0][-5]
eccentricity_results_bymode[7][6] = eccentricity_results_bymode[0][-6]
return eccentricity_results_bymode
[docs]
@njit(cacheable=True)
def eccentricity_funcs_trunc14(eccentricity: 'FloatArray') -> 'EccenOutput':
""" Calculates the eccentricity functions (by mode) truncated to e^14 for order-l = 7
Parameters
----------
eccentricity : FloatArray
Orbital Eccentricity
Returns
-------
eccentricity_results_bymode : EccenOutput
"""
# Eccentricity functions calculated at truncation level 14.
# and order-l = 7.
# Performance and readability improvements
e = eccentricity
e2 = e * e
e4 = e * e * e * e
e6 = e * e * e * e * e * e
e8 = e**8
e10 = e**10
e12 = e**12
e14 = e**14
# Unique results.
eccentricity_results_bymode = {
0: {
-6: 1.14373639357363e-9 * e14 + 4.7095027970679e-10 * e12,
-5: 6.28995811287478e-5 * e14 + 5.78703703703704e-5 * e12 + 6.94444444444444e-5 * e10,
-4: -0.0112746211460658 * e14 + 0.0320917510986328 * e12 - 0.07119140625 * e10 + 0.04449462890625 * e8,
-3: 11.5056128747795 * e14 - 20.1716049382716 * e12 + 20.1555555555556 * e10 - 9.77777777777778 * e8 + 1.77777777777778 * e6,
-2: -2132.90065789769 * e14 + 2269.74851555294 * e12 - 1516.85926649306 * e10 + 591.854519314236 * e8 - 120.442708333333 * e6 + 9.765625 * e4,
-1: 121302.196799665 * e14 - 89131.943671875 * e12 + 43713.97734375 * e10 - 13602.09375 * e8 + 2496.9375 * e6 - 238.5 * e4 + 9.0 * e2,
0 : -2955172.8970563 * e14 + 1586997.38291086 * e12 - 586640.268715278 * e10 + 142589.589938694 * e8 - 21364.4618055556 * e6 + 1785.21875 * e4 - 70.0 * e2 + 1.0,
1 : 36681269.0568395 * e14 - 14695176.0341667 * e12 + 4085875.57048611 * e10 - 748898.888888889 * e8 + 83775.8333333333 * e6 - 5016.0 * e4 + 121.0 * e2,
2 : -255474118.187811 * e14 + 76176310.2957281 * e12 - 15548508.0931641 * e10 + 2024000.86157227 * e8 - 148518.984375 * e6 + 4607.015625 * e4,
3 : 1050342781.37476 * e14 - 227752424.952377 * e12 + 32361560.6011285 * e10 - 2669833.76736111 * e8 + 95841.8402777778 * e6,
4 : -2589179614.74784 * e14 + 387648466.517466 * e12 - 34373096.9709961 * e10 + 1352599.28662788 * e8,
5 : 3741770950.29402 * e14 - 347139933.975 * e12 + 14501244.8025 * e10,
6 : -2910686193.59795 * e14 + 126441284.408644 * e12,
7 : 937784248.676216 * e14,
},
1: {
-7: 0.119978389648684 * e14,
-6: 0.811787874265533 * e14 + 0.0649431247475706 * e12,
-5: -0.03515625 * e10 / (e2 - 1.0)**13,
-4: 9.57492866166264 * e14 + 1.8597327875208 * e12 + 0.257063802083333 * e10 + 0.0190497504340278 * e8,
-3: 23.2613752652392 * e14 + 5.39066358024691 * e12 + 0.971701388888889 * e10 + 0.118055555555556 * e8 + 0.00694444444444444 * e6,
-2: 64.667712789263 * e14 + 20.3473989486694 * e12 + 5.8693359375 * e10 + 1.742431640625 * e8 + 0.140625 * e6 + 0.140625 * e4,
-1: 139.310944113757 * e14 + 30.2398958333333 * e12 + 38.0814236111111 * e10 - 32.8888888888889 * e8 + 24.5833333333333 * e6 - 9.0 * e4 + e2,
0 : -752.495595552844 * e14 + 1725.84582929258 * e12 - 2021.14040798611 * e10 + 1576.83668348524 * e8 - 741.545138888889 * e6 + 183.96875 * e4 - 22.0 * e2 + 1.0,
1 : 56545.2157073103 * e14 - 62900.057109375 * e12 + 48573.09140625 * e10 - 24466.5 * e8 + 7388.4375 * e6 - 1201.5 * e4 + 81.0 * e2,
2 : -1266309.79616931 * e14 + 938547.352011341 * e12 - 475122.020203993 * e10 + 152617.184597439 * e8 - 27743.8177083333 * e6 + 2173.890625 * e4,
3 : 13062709.1406972 * e14 - 6464235.02122878 * e12 + 2097863.73862847 * e10 - 397130.923611111 * e8 + 33184.6944444444 * e6,
4 : -67933871.4747141 * e14 + 21747975.9750501 * e12 - 4146304.49912109 * e10 + 354871.521057129 * e8,
5 : 182888425.081913 * e14 - 34500980.0248119 * e12 + 2959618.45876736 * e10,
6 : -241659566.56067 * e14 + 20518448.8791875 * e12,
7 : 123255910.704605 * e14,
},
2: {
-7: 156.64211077255 * e14,
-6: 809.854400596619 * e14 + 71.8156937026978 * e12,
-5: 2563.02166482377 * e14 + 390.619458912037 * e12 + 32.4425173611111 * e10,
-4: 6363.25240061133 * e14 + 1283.45398924368 * e12 + 185.11298828125 * e10 + 14.396491156684 * e8,
-3: -0.09765625 * e6 * (3.0 * e2 + 8.0)**2 / (e2 - 1.0)**13,
-2: 26135.9927566851 * e14 + 7178.43844695621 * e12 + 1659.23604600694 * e10 + 303.058295355903 * e8 + 38.9322916666667 * e6 + 2.640625 * e4,
-1: 46429.6170135272 * e14 + 14041.6768142361 * e12 + 3694.66293402778 * e10 + 811.631944444444 * e8 + 139.270833333333 * e6 + 16.5 * e4 + e2,
0 : 78327.7874431159 * e14 + 25731.1676902771 * e12 + 7565.199296875 * e10 + 1945.65222167969 * e8 + 424.84375 * e6 + 76.46875 * e4 + 10.0 * e2 + 1.0,
1 : 126743.549716435 * e14 + 44644.5701736111 * e12 + 14360.1105902778 * e10 + 4115.31944444444 * e8 + 1080.58333333333 * e6 + 189.0 * e4 + 49.0 * e2,
2 : 197042.647860248 * e14 + 73906.7868524128 * e12 + 24376.8419053819 * e10 + 8894.1413031684 * e8 + 1254.55729166667 * e6 + 862.890625 * e4,
3 : 306304.572200056 * e14 + 97963.396875 * e12 + 59750.0453125 * e10 - 95.75 * e8 + 9168.0625 * e6,
4 : 196218.790338876 * e14 + 392310.672480862 * e12 - 78294.8146809896 * e10 + 71362.7221747504 * e8,
5 : 2608659.94096175 * e14 - 894602.368952546 * e12 + 448693.440434028 * e10,
6 : -6709222.00565983 * e14 + 2412439.34078983 * e12,
7 : 11503209.3055423 * e14,
},
3: {
-7: 18617.627869898 * e14,
-6: 56701.7701593513 * e14 + 6572.37797547564 * e12,
-5: 118393.583558477 * e14 + 21292.203287037 * e12 + 2198.82840277778 * e10,
-4: 204879.526902172 * e14 + 45732.4679340363 * e12 + 7485.75439453125 * e10 + 685.376037597656 * e8,
-3: 314588.587124842 * e14 + 80064.9051456404 * e12 + 16387.2709201389 * e10 + 2413.96180555556 * e8 + 193.673611111111 * e6,
-2: 442814.440752841 * e14 + 122862.972033925 * e12 + 28704.1544053819 * e10 + 5301.66408962674 * e8 + 688.932291666667 * e6 + 47.265625 * e4,
-1: -0.140625 * e2 * (5.0 * e4 + 20.0 * e2 + 8.0)**2 / (e2 - 1.0)**13,
0 : 719186.088857198 * e14 + 219108.287543295 * e12 + 58347.2753993056 * e10 + 13103.2553032769 * e8 + 2353.37152777778 * e6 + 310.71875 * e4 + 26.0 * e2 + 1.0,
1 : 839864.510332445 * e14 + 259842.755598958 * e12 + 70311.6384548611 * e10 + 16012.0798611111 * e8 + 2885.77083333333 * e6 + 367.5 * e4 + 25.0 * e2,
2 : 927769.057548736 * e14 + 285788.216771507 * e12 + 76178.1462890625 * e10 + 16688.9113769531 * e8 + 2727.140625 * e6 + 260.015625 * e4,
3 : 969087.477329971 * e14 + 290946.807098765 * e12 + 73701.4461805556 * e10 + 14474.4722222222 * e8 + 1792.11111111111 * e6,
4 : 953426.381187765 * e14 + 271561.904476236 * e12 + 61808.6661783854 * e10 + 9661.76367865668 * e8,
5 : 876003.884196429 * e14 + 224786.9475 * e12 + 44163.0225 * e10,
6 : 716483.787616746 * e14 + 179171.885103873 * e12,
7 : 664011.24140688 * e14,
},
4: {
-7: 664011.24140688 * e14,
},
5: {
-7: 11503209.3055423 * e14,
},
6: {
-7: 123255910.704605 * e14,
},
7: {
-7: 937784248.676216 * e14,
}
}
# Duplicate results are stored as dictionary lookups to previous calculations
# Generally leads to a 30--50% speed-up when working with large arrays.
eccentricity_results_bymode[4][-6] = eccentricity_results_bymode[3][6]
eccentricity_results_bymode[4][-5] = eccentricity_results_bymode[3][5]
eccentricity_results_bymode[4][-4] = eccentricity_results_bymode[3][4]
eccentricity_results_bymode[4][-3] = eccentricity_results_bymode[3][3]
eccentricity_results_bymode[4][-2] = eccentricity_results_bymode[3][2]
eccentricity_results_bymode[4][-1] = eccentricity_results_bymode[3][1]
eccentricity_results_bymode[4][0] = eccentricity_results_bymode[3][0]
eccentricity_results_bymode[4][1] = eccentricity_results_bymode[3][-1]
eccentricity_results_bymode[4][2] = eccentricity_results_bymode[3][-2]
eccentricity_results_bymode[4][3] = eccentricity_results_bymode[3][-3]
eccentricity_results_bymode[4][4] = eccentricity_results_bymode[3][-4]
eccentricity_results_bymode[4][5] = eccentricity_results_bymode[3][-5]
eccentricity_results_bymode[4][6] = eccentricity_results_bymode[3][-6]
eccentricity_results_bymode[4][7] = eccentricity_results_bymode[3][-7]
eccentricity_results_bymode[5][-6] = eccentricity_results_bymode[2][6]
eccentricity_results_bymode[5][-5] = eccentricity_results_bymode[2][5]
eccentricity_results_bymode[5][-4] = eccentricity_results_bymode[2][4]
eccentricity_results_bymode[5][-3] = eccentricity_results_bymode[2][3]
eccentricity_results_bymode[5][-2] = eccentricity_results_bymode[2][2]
eccentricity_results_bymode[5][-1] = eccentricity_results_bymode[2][1]
eccentricity_results_bymode[5][0] = eccentricity_results_bymode[2][0]
eccentricity_results_bymode[5][1] = eccentricity_results_bymode[2][-1]
eccentricity_results_bymode[5][2] = eccentricity_results_bymode[2][-2]
eccentricity_results_bymode[5][3] = eccentricity_results_bymode[2][-3]
eccentricity_results_bymode[5][4] = eccentricity_results_bymode[2][-4]
eccentricity_results_bymode[5][5] = eccentricity_results_bymode[2][-5]
eccentricity_results_bymode[5][6] = eccentricity_results_bymode[2][-6]
eccentricity_results_bymode[5][7] = eccentricity_results_bymode[2][-7]
eccentricity_results_bymode[6][-6] = eccentricity_results_bymode[1][6]
eccentricity_results_bymode[6][-5] = eccentricity_results_bymode[1][5]
eccentricity_results_bymode[6][-4] = eccentricity_results_bymode[1][4]
eccentricity_results_bymode[6][-3] = eccentricity_results_bymode[1][3]
eccentricity_results_bymode[6][-2] = eccentricity_results_bymode[1][2]
eccentricity_results_bymode[6][-1] = eccentricity_results_bymode[1][1]
eccentricity_results_bymode[6][0] = eccentricity_results_bymode[1][0]
eccentricity_results_bymode[6][1] = eccentricity_results_bymode[1][-1]
eccentricity_results_bymode[6][2] = eccentricity_results_bymode[1][-2]
eccentricity_results_bymode[6][3] = eccentricity_results_bymode[1][-3]
eccentricity_results_bymode[6][4] = eccentricity_results_bymode[1][-4]
eccentricity_results_bymode[6][5] = eccentricity_results_bymode[1][-5]
eccentricity_results_bymode[6][6] = eccentricity_results_bymode[1][-6]
eccentricity_results_bymode[6][7] = eccentricity_results_bymode[1][-7]
eccentricity_results_bymode[7][-6] = eccentricity_results_bymode[0][6]
eccentricity_results_bymode[7][-5] = eccentricity_results_bymode[0][5]
eccentricity_results_bymode[7][-4] = eccentricity_results_bymode[0][4]
eccentricity_results_bymode[7][-3] = eccentricity_results_bymode[0][3]
eccentricity_results_bymode[7][-2] = eccentricity_results_bymode[0][2]
eccentricity_results_bymode[7][-1] = eccentricity_results_bymode[0][1]
eccentricity_results_bymode[7][0] = eccentricity_results_bymode[0][0]
eccentricity_results_bymode[7][1] = eccentricity_results_bymode[0][-1]
eccentricity_results_bymode[7][2] = eccentricity_results_bymode[0][-2]
eccentricity_results_bymode[7][3] = eccentricity_results_bymode[0][-3]
eccentricity_results_bymode[7][4] = eccentricity_results_bymode[0][-4]
eccentricity_results_bymode[7][5] = eccentricity_results_bymode[0][-5]
eccentricity_results_bymode[7][6] = eccentricity_results_bymode[0][-6]
return eccentricity_results_bymode
[docs]
@njit(cacheable=True)
def eccentricity_funcs_trunc16(eccentricity: 'FloatArray') -> 'EccenOutput':
""" Calculates the eccentricity functions (by mode) truncated to e^16 for order-l = 7
Parameters
----------
eccentricity : FloatArray
Orbital Eccentricity
Returns
-------
eccentricity_results_bymode : EccenOutput
"""
# Eccentricity functions calculated at truncation level 16.
# and order-l = 7.
# Performance and readability improvements
e = eccentricity
e2 = e * e
e4 = e * e * e * e
e6 = e * e * e * e * e * e
e8 = e**8
e10 = e**10
e12 = e**12
e14 = e**14
e16 = e**16
# Unique results.
eccentricity_results_bymode = {
0: {
-8: 9.38596699032984e-15 * e16,
-6: 1.87967129335142e-9 * e16 + 1.14373639357363e-9 * e14 + 4.7095027970679e-10 * e12,
-5: 6.23484347442681e-5 * e16 + 6.28995811287478e-5 * e14 + 5.78703703703704e-5 * e12 + 6.94444444444444e-5 * e10,
-4: 0.00108947594250952 * e16 - 0.0112746211460658 * e14 + 0.0320917510986328 * e12 - 0.07119140625 * e10 + 0.04449462890625 * e8,
-3: -4.40869268077601 * e16 + 11.5056128747795 * e14 - 20.1716049382716 * e12 + 20.1555555555556 * e10 - 9.77777777777778 * e8 + 1.77777777777778 * e6,
-2: 1348.87160996971 * e16 - 2132.90065789769 * e14 + 2269.74851555294 * e12 - 1516.85926649306 * e10 + 591.854519314236 * e8 - 120.442708333333 * e6 + 9.765625 * e4,
-1: -115035.60841179 * e16 + 121302.196799665 * e14 - 89131.943671875 * e12 + 43713.97734375 * e10 - 13602.09375 * e8 + 2496.9375 * e6 - 238.5 * e4 + 9.0 * e2,
0 : 3928011.79991695 * e16 - 2955172.8970563 * e14 + 1586997.38291086 * e12 - 586640.268715278 * e10 + 142589.589938694 * e8 - 21364.4618055556 * e6 + 1785.21875 * e4 - 70.0 * e2 + 1.0,
1 : -66006759.5382433 * e16 + 36681269.0568395 * e14 - 14695176.0341667 * e12 + 4085875.57048611 * e10 - 748898.888888889 * e8 + 83775.8333333333 * e6 - 5016.0 * e4 + 121.0 * e2,
2 : 614791138.403345 * e16 - 255474118.187811 * e14 + 76176310.2957281 * e12 - 15548508.0931641 * e10 + 2024000.86157227 * e8 - 148518.984375 * e6 + 4607.015625 * e4,
3 : -3399179717.5205 * e16 + 1050342781.37476 * e14 - 227752424.952377 * e12 + 32361560.6011285 * e10 - 2669833.76736111 * e8 + 95841.8402777778 * e6,
4 : 11548436061.762 * e16 - 2589179614.74784 * e14 + 387648466.517466 * e12 - 34373096.9709961 * e10 + 1352599.28662788 * e8,
5 : -24236865800.0458 * e16 + 3741770950.29402 * e14 - 347139933.975 * e12 + 14501244.8025 * e10,
6 : 30511860315.4628 * e16 - 2910686193.59795 * e14 + 126441284.408644 * e12,
7 : -21052727705.1045 * e16 + 937784248.676216 * e14,
8 : 6104436340.31089 * e16,
},
1: {
-8: 0.221719882537668 * e16,
-7: 1.43968913129488 * e16 + 0.119978389648684 * e14,
-6: 5.49174032711125 * e16 + 0.811787874265533 * e14 + 0.0649431247475706 * e12,
-5: -0.03515625 * e10 / (e2 - 1.0)**13,
-4: 39.3153428329901 * e16 + 9.57492866166264 * e14 + 1.8597327875208 * e12 + 0.257063802083333 * e10 + 0.0190497504340278 * e8,
-3: 83.9811453855269 * e16 + 23.2613752652392 * e14 + 5.39066358024691 * e12 + 0.971701388888889 * e10 + 0.118055555555556 * e8 + 0.00694444444444444 * e6,
-2: 191.185387344744 * e16 + 64.667712789263 * e14 + 20.3473989486694 * e12 + 5.8693359375 * e10 + 1.742431640625 * e8 + 0.140625 * e6 + 0.140625 * e4,
-1: 373.925813039336 * e16 + 139.310944113757 * e14 + 30.2398958333333 * e12 + 38.0814236111111 * e10 - 32.8888888888889 * e8 + 24.5833333333333 * e6 - 9.0 * e4 + e2,
0 : 1001.54276161753 * e16 - 752.495595552844 * e14 + 1725.84582929258 * e12 - 2021.14040798611 * e10 + 1576.83668348524 * e8 - 741.545138888889 * e6 + 183.96875 * e4 - 22.0 * e2 + 1.0,
1 : -35946.0589629504 * e16 + 56545.2157073103 * e14 - 62900.057109375 * e12 + 48573.09140625 * e10 - 24466.5 * e8 + 7388.4375 * e6 - 1201.5 * e4 + 81.0 * e2,
2 : 1231228.06259342 * e16 - 1266309.79616931 * e14 + 938547.352011341 * e12 - 475122.020203993 * e10 + 152617.184597439 * e8 - 27743.8177083333 * e6 + 2173.890625 * e4,
3 : -18573722.9224831 * e16 + 13062709.1406972 * e14 - 6464235.02122878 * e12 + 2097863.73862847 * e10 - 397130.923611111 * e8 + 33184.6944444444 * e6,
4 : 142257563.766493 * e16 - 67933871.4747141 * e14 + 21747975.9750501 * e12 - 4146304.49912109 * e10 + 354871.521057129 * e8,
5 : -586239684.517029 * e16 + 182888425.081913 * e14 - 34500980.0248119 * e12 + 2959618.45876736 * e10,
6 : 1307624112.7669 * e16 - 241659566.56067 * e14 + 20518448.8791875 * e12,
7 : -1477892171.3568 * e16 + 123255910.704605 * e14,
8 : 660185101.545129 * e16,
},
2: {
-8: 337.448907736202 * e16,
-7: 1653.23773598415 * e16 + 156.64211077255 * e14,
-6: 5032.83500185247 * e16 + 809.854400596619 * e14 + 71.8156937026978 * e12,
-5: 12123.5319398803 * e16 + 2563.02166482377 * e14 + 390.619458912037 * e12 + 32.4425173611111 * e10,
-4: 25261.633121265 * e16 + 6363.25240061133 * e14 + 1283.45398924368 * e12 + 185.11298828125 * e10 + 14.396491156684 * e8,
-3: -0.09765625 * e6 * (3.0 * e2 + 8.0)**2 / (e2 - 1.0)**13,
-2: 83279.1471429178 * e16 + 26135.9927566851 * e14 + 7178.43844695621 * e12 + 1659.23604600694 * e10 + 303.058295355903 * e8 + 38.9322916666667 * e6 + 2.640625 * e4,
-1: 137331.061940094 * e16 + 46429.6170135272 * e14 + 14041.6768142361 * e12 + 3694.66293402778 * e10 + 811.631944444444 * e8 + 139.270833333333 * e6 + 16.5 * e4 + e2,
0 : 217273.143247915 * e16 + 78327.7874431159 * e14 + 25731.1676902771 * e12 + 7565.199296875 * e10 + 1945.65222167969 * e8 + 424.84375 * e6 + 76.46875 * e4 + 10.0 * e2 + 1.0,
1 : 332514.430101101 * e16 + 126743.549716435 * e14 + 44644.5701736111 * e12 + 14360.1105902778 * e10 + 4115.31944444444 * e8 + 1080.58333333333 * e6 + 189.0 * e4 + 49.0 * e2,
2 : 493808.611203993 * e16 + 197042.647860248 * e14 + 73906.7868524128 * e12 + 24376.8419053819 * e10 + 8894.1413031684 * e8 + 1254.55729166667 * e6 + 862.890625 * e4,
3 : 706998.515643834 * e16 + 306304.572200056 * e14 + 97963.396875 * e12 + 59750.0453125 * e10 - 95.75 * e8 + 9168.0625 * e6,
4 : 1172588.42289078 * e16 + 196218.790338876 * e14 + 392310.672480862 * e12 - 78294.8146809896 * e10 + 71362.7221747504 * e8,
5 : -1015685.91263426 * e16 + 2608659.94096175 * e14 - 894602.368952546 * e12 + 448693.440434028 * e10,
6 : 16725535.5538483 * e16 - 6709222.00565983 * e14 + 2412439.34078983 * e12,
7 : -40255987.811503 * e16 + 11503209.3055423 * e14,
8 : 49868524.3845965 * e16,
},
3: {
-8: 50553.5356753209 * e16,
-7: 143177.836475606 * e16 + 18617.627869898 * e14,
-6: 288970.153736016 * e16 + 56701.7701593513 * e14 + 6572.37797547564 * e12,
-5: 491118.834556809 * e16 + 118393.583558477 * e14 + 21292.203287037 * e12 + 2198.82840277778 * e10,
-4: 748464.032461628 * e16 + 204879.526902172 * e14 + 45732.4679340363 * e12 + 7485.75439453125 * e10 + 685.376037597656 * e8,
-3: 1054606.98825424 * e16 + 314588.587124842 * e14 + 80064.9051456404 * e12 + 16387.2709201389 * e10 + 2413.96180555556 * e8 + 193.673611111111 * e6,
-2: 1397522.06146574 * e16 + 442814.440752841 * e14 + 122862.972033925 * e12 + 28704.1544053819 * e10 + 5301.66408962674 * e8 + 688.932291666667 * e6 + 47.265625 * e4,
-1: -0.140625 * e2 * (5.0 * e4 + 20.0 * e2 + 8.0)**2 / (e2 - 1.0)**13,
0 : 2116390.96404087 * e16 + 719186.088857198 * e14 + 219108.287543295 * e12 + 58347.2753993056 * e10 + 13103.2553032769 * e8 + 2353.37152777778 * e6 + 310.71875 * e4 + 26.0 * e2 + 1.0,
1 : 2437240.94540722 * e16 + 839864.510332445 * e14 + 259842.755598958 * e12 + 70311.6384548611 * e10 + 16012.0798611111 * e8 + 2885.77083333333 * e6 + 367.5 * e4 + 25.0 * e2,
2 : 2690460.52736734 * e16 + 927769.057548736 * e14 + 285788.216771507 * e12 + 76178.1462890625 * e10 + 16688.9113769531 * e8 + 2727.140625 * e6 + 260.015625 * e4,
3 : 2847310.73636333 * e16 + 969087.477329971 * e14 + 290946.807098765 * e12 + 73701.4461805556 * e10 + 14474.4722222222 * e8 + 1792.11111111111 * e6,
4 : 2884135.51696895 * e16 + 953426.381187765 * e14 + 271561.904476236 * e12 + 61808.6661783854 * e10 + 9661.76367865668 * e8,
5 : 2784035.39506975 * e16 + 876003.884196429 * e14 + 224786.9475 * e12 + 44163.0225 * e10,
6 : 2551140.62734588 * e16 + 716483.787616746 * e14 + 179171.885103873 * e12,
7 : 2022000.82930082 * e16 + 664011.24140688 * e14,
8 : 2291716.29166404 * e16,
},
4: {
-8: 2291716.29166404 * e16,
},
5: {
-8: 49868524.3845965 * e16,
},
6: {
-8: 660185101.545129 * e16,
},
7: {
-8: 6104436340.31089 * e16,
}
}
# Duplicate results are stored as dictionary lookups to previous calculations
# Generally leads to a 30--50% speed-up when working with large arrays.
eccentricity_results_bymode[4][-7] = eccentricity_results_bymode[3][7]
eccentricity_results_bymode[4][-6] = eccentricity_results_bymode[3][6]
eccentricity_results_bymode[4][-5] = eccentricity_results_bymode[3][5]
eccentricity_results_bymode[4][-4] = eccentricity_results_bymode[3][4]
eccentricity_results_bymode[4][-3] = eccentricity_results_bymode[3][3]
eccentricity_results_bymode[4][-2] = eccentricity_results_bymode[3][2]
eccentricity_results_bymode[4][-1] = eccentricity_results_bymode[3][1]
eccentricity_results_bymode[4][0] = eccentricity_results_bymode[3][0]
eccentricity_results_bymode[4][1] = eccentricity_results_bymode[3][-1]
eccentricity_results_bymode[4][2] = eccentricity_results_bymode[3][-2]
eccentricity_results_bymode[4][3] = eccentricity_results_bymode[3][-3]
eccentricity_results_bymode[4][4] = eccentricity_results_bymode[3][-4]
eccentricity_results_bymode[4][5] = eccentricity_results_bymode[3][-5]
eccentricity_results_bymode[4][6] = eccentricity_results_bymode[3][-6]
eccentricity_results_bymode[4][7] = eccentricity_results_bymode[3][-7]
eccentricity_results_bymode[4][8] = eccentricity_results_bymode[3][-8]
eccentricity_results_bymode[5][-7] = eccentricity_results_bymode[2][7]
eccentricity_results_bymode[5][-6] = eccentricity_results_bymode[2][6]
eccentricity_results_bymode[5][-5] = eccentricity_results_bymode[2][5]
eccentricity_results_bymode[5][-4] = eccentricity_results_bymode[2][4]
eccentricity_results_bymode[5][-3] = eccentricity_results_bymode[2][3]
eccentricity_results_bymode[5][-2] = eccentricity_results_bymode[2][2]
eccentricity_results_bymode[5][-1] = eccentricity_results_bymode[2][1]
eccentricity_results_bymode[5][0] = eccentricity_results_bymode[2][0]
eccentricity_results_bymode[5][1] = eccentricity_results_bymode[2][-1]
eccentricity_results_bymode[5][2] = eccentricity_results_bymode[2][-2]
eccentricity_results_bymode[5][3] = eccentricity_results_bymode[2][-3]
eccentricity_results_bymode[5][4] = eccentricity_results_bymode[2][-4]
eccentricity_results_bymode[5][5] = eccentricity_results_bymode[2][-5]
eccentricity_results_bymode[5][6] = eccentricity_results_bymode[2][-6]
eccentricity_results_bymode[5][7] = eccentricity_results_bymode[2][-7]
eccentricity_results_bymode[5][8] = eccentricity_results_bymode[2][-8]
eccentricity_results_bymode[6][-7] = eccentricity_results_bymode[1][7]
eccentricity_results_bymode[6][-6] = eccentricity_results_bymode[1][6]
eccentricity_results_bymode[6][-5] = eccentricity_results_bymode[1][5]
eccentricity_results_bymode[6][-4] = eccentricity_results_bymode[1][4]
eccentricity_results_bymode[6][-3] = eccentricity_results_bymode[1][3]
eccentricity_results_bymode[6][-2] = eccentricity_results_bymode[1][2]
eccentricity_results_bymode[6][-1] = eccentricity_results_bymode[1][1]
eccentricity_results_bymode[6][0] = eccentricity_results_bymode[1][0]
eccentricity_results_bymode[6][1] = eccentricity_results_bymode[1][-1]
eccentricity_results_bymode[6][2] = eccentricity_results_bymode[1][-2]
eccentricity_results_bymode[6][3] = eccentricity_results_bymode[1][-3]
eccentricity_results_bymode[6][4] = eccentricity_results_bymode[1][-4]
eccentricity_results_bymode[6][5] = eccentricity_results_bymode[1][-5]
eccentricity_results_bymode[6][6] = eccentricity_results_bymode[1][-6]
eccentricity_results_bymode[6][7] = eccentricity_results_bymode[1][-7]
eccentricity_results_bymode[6][8] = eccentricity_results_bymode[1][-8]
eccentricity_results_bymode[7][-7] = eccentricity_results_bymode[0][7]
eccentricity_results_bymode[7][-6] = eccentricity_results_bymode[0][6]
eccentricity_results_bymode[7][-5] = eccentricity_results_bymode[0][5]
eccentricity_results_bymode[7][-4] = eccentricity_results_bymode[0][4]
eccentricity_results_bymode[7][-3] = eccentricity_results_bymode[0][3]
eccentricity_results_bymode[7][-2] = eccentricity_results_bymode[0][2]
eccentricity_results_bymode[7][-1] = eccentricity_results_bymode[0][1]
eccentricity_results_bymode[7][0] = eccentricity_results_bymode[0][0]
eccentricity_results_bymode[7][1] = eccentricity_results_bymode[0][-1]
eccentricity_results_bymode[7][2] = eccentricity_results_bymode[0][-2]
eccentricity_results_bymode[7][3] = eccentricity_results_bymode[0][-3]
eccentricity_results_bymode[7][4] = eccentricity_results_bymode[0][-4]
eccentricity_results_bymode[7][5] = eccentricity_results_bymode[0][-5]
eccentricity_results_bymode[7][6] = eccentricity_results_bymode[0][-6]
eccentricity_results_bymode[7][8] = eccentricity_results_bymode[0][-8]
return eccentricity_results_bymode
[docs]
@njit(cacheable=True)
def eccentricity_funcs_trunc18(eccentricity: 'FloatArray') -> 'EccenOutput':
""" Calculates the eccentricity functions (by mode) truncated to e^18 for order-l = 7
Parameters
----------
eccentricity : FloatArray
Orbital Eccentricity
Returns
-------
eccentricity_results_bymode : EccenOutput
"""
# Eccentricity functions calculated at truncation level 18.
# and order-l = 7.
# Performance and readability improvements
e = eccentricity
e2 = e * e
e4 = e * e * e * e
e6 = e * e * e * e * e * e
e8 = e**8
e10 = e**10
e12 = e**12
e14 = e**14
e16 = e**16
e18 = e**18
# Unique results.
eccentricity_results_bymode = {
0: {
-9: 7.59405842812662e-12 * e18,
-8: 3.96296384036149e-14 * e18 + 9.38596699032984e-15 * e16,
-6: 2.59607225768029e-9 * e18 + 1.87967129335142e-9 * e16 + 1.14373639357363e-9 * e14 + 4.7095027970679e-10 * e12,
-5: 6.09139095131365e-5 * e18 + 6.23484347442681e-5 * e16 + 6.28995811287478e-5 * e14 + 5.78703703703704e-5 * e12 + 6.94444444444444e-5 * e10,
-4: -0.000713012771947043 * e18 + 0.00108947594250952 * e16 - 0.0112746211460658 * e14 + 0.0320917510986328 * e12 - 0.07119140625 * e10 + 0.04449462890625 * e8,
-3: 1.15933129613299 * e18 - 4.40869268077601 * e16 + 11.5056128747795 * e14 - 20.1716049382716 * e12 + 20.1555555555556 * e10 - 9.77777777777778 * e8 + 1.77777777777778 * e6,
-2: -612.292652626918 * e18 + 1348.87160996971 * e16 - 2132.90065789769 * e14 + 2269.74851555294 * e12 - 1516.85926649306 * e10 + 591.854519314236 * e8 - 120.442708333333 * e6 + 9.765625 * e4,
-1: 79249.6311015351 * e18 - 115035.60841179 * e16 + 121302.196799665 * e14 - 89131.943671875 * e12 + 43713.97734375 * e10 - 13602.09375 * e8 + 2496.9375 * e6 - 238.5 * e4 + 9.0 * e2,
0 : -3848911.55308733 * e18 + 3928011.79991695 * e16 - 2955172.8970563 * e14 + 1586997.38291086 * e12 - 586640.268715278 * e10 + 142589.589938694 * e8 - 21364.4618055556 * e6 + 1785.21875 * e4 - 70.0 * e2 + 1.0,
1 : 88302609.8818484 * e18 - 66006759.5382433 * e16 + 36681269.0568395 * e14 - 14695176.0341667 * e12 + 4085875.57048611 * e10 - 748898.888888889 * e8 + 83775.8333333333 * e6 - 5016.0 * e4 + 121.0 * e2,
2 : -1099191510.20098 * e18 + 614791138.403345 * e16 - 255474118.187811 * e14 + 76176310.2957281 * e12 - 15548508.0931641 * e10 + 2024000.86157227 * e8 - 148518.984375 * e6 + 4607.015625 * e4,
3 : 8076809089.5857 * e18 - 3399179717.5205 * e16 + 1050342781.37476 * e14 - 227752424.952377 * e12 + 32361560.6011285 * e10 - 2669833.76736111 * e8 + 95841.8402777778 * e6,
4 : -36797348913.1132 * e18 + 11548436061.762 * e16 - 2589179614.74784 * e14 + 387648466.517466 * e12 - 34373096.9709961 * e10 + 1352599.28662788 * e8,
5 : 106327614538.294 * e18 - 24236865800.0458 * e16 + 3741770950.29402 * e14 - 347139933.975 * e12 + 14501244.8025 * e10,
6 : -194388536778.511 * e18 + 30511860315.4628 * e16 - 2910686193.59795 * e14 + 126441284.408644 * e12,
7 : 217180524610.773 * e18 - 21052727705.1045 * e16 + 937784248.676216 * e14,
8 : -134977425865.553 * e18 + 6104436340.31089 * e16,
9 : 35680459926.5605 * e18,
},
1: {
-9: 0.409936224172744 * e18,
-8: 2.5493296938572 * e18 + 0.221719882537668 * e16,
-7: 9.40277032701544 * e18 + 1.43968913129488 * e16 + 0.119978389648684 * e14,
-6: 26.6462243315436 * e18 + 5.49174032711125 * e16 + 0.811787874265533 * e14 + 0.0649431247475706 * e12,
-5: -0.03515625 * e10 / (e2 - 1.0)**13,
-4: 136.855297948203 * e18 + 39.3153428329901 * e16 + 9.57492866166264 * e14 + 1.8597327875208 * e12 + 0.257063802083333 * e10 + 0.0190497504340278 * e8,
-3: 264.879793242202 * e18 + 83.9811453855269 * e16 + 23.2613752652392 * e14 + 5.39066358024691 * e12 + 0.971701388888889 * e10 + 0.118055555555556 * e8 + 0.00694444444444444 * e6,
-2: 525.84349136254 * e18 + 191.185387344744 * e16 + 64.667712789263 * e14 + 20.3473989486694 * e12 + 5.8693359375 * e10 + 1.742431640625 * e8 + 0.140625 * e6 + 0.140625 * e4,
-1: 962.771132986083 * e18 + 373.925813039336 * e16 + 139.310944113757 * e14 + 30.2398958333333 * e12 + 38.0814236111111 * e10 - 32.8888888888889 * e8 + 24.5833333333333 * e6 - 9.0 * e4 + e2,
0 : 1415.4631693483 * e18 + 1001.54276161753 * e16 - 752.495595552844 * e14 + 1725.84582929258 * e12 - 2021.14040798611 * e10 + 1576.83668348524 * e8 - 741.545138888889 * e6 + 183.96875 * e4 - 22.0 * e2 + 1.0,
1 : 20335.218430836 * e18 - 35946.0589629504 * e16 + 56545.2157073103 * e14 - 62900.057109375 * e12 + 48573.09140625 * e10 - 24466.5 * e8 + 7388.4375 * e6 - 1201.5 * e4 + 81.0 * e2,
2 : -895869.621245169 * e18 + 1231228.06259342 * e16 - 1266309.79616931 * e14 + 938547.352011341 * e12 - 475122.020203993 * e10 + 152617.184597439 * e8 - 27743.8177083333 * e6 + 2173.890625 * e4,
3 : 19527906.1971512 * e18 - 18573722.9224831 * e16 + 13062709.1406972 * e14 - 6464235.02122878 * e12 + 2097863.73862847 * e10 - 397130.923611111 * e8 + 33184.6944444444 * e6,
4 : -213970512.55123 * e18 + 142257563.766493 * e16 - 67933871.4747141 * e14 + 21747975.9750501 * e12 - 4146304.49912109 * e10 + 354871.521057129 * e8,
5 : 1279859819.6393 * e18 - 586239684.517029 * e16 + 182888425.081913 * e14 - 34500980.0248119 * e12 + 2959618.45876736 * e10,
6 : -4329029031.54511 * e18 + 1307624112.7669 * e16 - 241659566.56067 * e14 + 20518448.8791875 * e12,
7 : 8210215164.34436 * e18 - 1477892171.3568 * e16 + 123255910.704605 * e14,
8 : -8097179451.88272 * e18 + 660185101.545129 * e16,
9 : 3218885666.05741 * e18,
},
2: {
-9: 719.31120267907 * e18,
-8: 3328.30927572957 * e18 + 337.448907736202 * e16,
-7: 9733.68739966595 * e18 + 1653.23773598415 * e16 + 156.64211077255 * e14,
-6: 22735.9555951404 * e18 + 5032.83500185247 * e16 + 809.854400596619 * e14 + 71.8156937026978 * e12,
-5: 46212.5354749881 * e18 + 12123.5319398803 * e16 + 2563.02166482377 * e14 + 390.619458912037 * e12 + 32.4425173611111 * e10,
-4: 85307.8419407169 * e18 + 25261.633121265 * e16 + 6363.25240061133 * e14 + 1283.45398924368 * e12 + 185.11298828125 * e10 + 14.396491156684 * e8,
-3: -0.09765625 * e6 * (3.0 * e2 + 8.0)**2 / (e2 - 1.0)**13,
-2: 238460.665898411 * e18 + 83279.1471429178 * e16 + 26135.9927566851 * e14 + 7178.43844695621 * e12 + 1659.23604600694 * e10 + 303.058295355903 * e8 + 38.9322916666667 * e6 + 2.640625 * e4,
-1: 370640.856538529 * e18 + 137331.061940094 * e16 + 46429.6170135272 * e14 + 14041.6768142361 * e12 + 3694.66293402778 * e10 + 811.631944444444 * e8 + 139.270833333333 * e6 + 16.5 * e4 + e2,
0 : 557030.77991627 * e18 + 217273.143247915 * e16 + 78327.7874431159 * e14 + 25731.1676902771 * e12 + 7565.199296875 * e10 + 1945.65222167969 * e8 + 424.84375 * e6 + 76.46875 * e4 + 10.0 * e2 + 1.0,
1 : 814711.469512273 * e18 + 332514.430101101 * e16 + 126743.549716435 * e14 + 44644.5701736111 * e12 + 14360.1105902778 * e10 + 4115.31944444444 * e8 + 1080.58333333333 * e6 + 189.0 * e4 + 49.0 * e2,
2 : 1163119.00350269 * e18 + 493808.611203993 * e16 + 197042.647860248 * e14 + 73906.7868524128 * e12 + 24376.8419053819 * e10 + 8894.1413031684 * e8 + 1254.55729166667 * e6 + 862.890625 * e4,
3 : 1625116.54120954 * e18 + 706998.515643834 * e16 + 306304.572200056 * e14 + 97963.396875 * e12 + 59750.0453125 * e10 - 95.75 * e8 + 9168.0625 * e6,
4 : 2116130.95690105 * e18 + 1172588.42289078 * e16 + 196218.790338876 * e14 + 392310.672480862 * e12 - 78294.8146809896 * e10 + 71362.7221747504 * e8,
5 : 5013078.31114085 * e18 - 1015685.91263426 * e16 + 2608659.94096175 * e14 - 894602.368952546 * e12 + 448693.440434028 * e10,
6 : -15929962.0572314 * e18 + 16725535.5538483 * e16 - 6709222.00565983 * e14 + 2412439.34078983 * e12,
7 : 99121020.0636772 * e18 - 40255987.811503 * e16 + 11503209.3055423 * e14,
8 : -208038708.070791 * e18 + 49868524.3845965 * e16,
9 : 200072127.641327 * e18,
},
3: {
-9: 132646.745553428 * e18,
-8: 345601.988924924 * e18 + 50553.5356753209 * e16,
-7: 671947.52842675 * e18 + 143177.836475606 * e16 + 18617.627869898 * e14,
-6: 1117258.14188041 * e18 + 288970.153736016 * e16 + 56701.7701593513 * e14 + 6572.37797547564 * e12,
-5: 1681953.98847843 * e18 + 491118.834556809 * e16 + 118393.583558477 * e14 + 21292.203287037 * e12 + 2198.82840277778 * e10,
-4: 2357941.69033208 * e18 + 748464.032461628 * e16 + 204879.526902172 * e14 + 45732.4679340363 * e12 + 7485.75439453125 * e10 + 685.376037597656 * e8,
-3: 3128076.05868162 * e18 + 1054606.98825424 * e16 + 314588.587124842 * e14 + 80064.9051456404 * e12 + 16387.2709201389 * e10 + 2413.96180555556 * e8 + 193.673611111111 * e6,
-2: 3965741.56804964 * e18 + 1397522.06146574 * e16 + 442814.440752841 * e14 + 122862.972033925 * e12 + 28704.1544053819 * e10 + 5301.66408962674 * e8 + 688.932291666667 * e6 + 47.265625 * e4,
-1: -0.140625 * e2 * (5.0 * e4 + 20.0 * e2 + 8.0)**2 / (e2 - 1.0)**13,
0 : 5689186.47353937 * e18 + 2116390.96404087 * e16 + 719186.088857198 * e14 + 219108.287543295 * e12 + 58347.2753993056 * e10 + 13103.2553032769 * e8 + 2353.37152777778 * e6 + 310.71875 * e4 + 26.0 * e2 + 1.0,
1 : 6470933.30682901 * e18 + 2437240.94540722 * e16 + 839864.510332445 * e14 + 259842.755598958 * e12 + 70311.6384548611 * e10 + 16012.0798611111 * e8 + 2885.77083333333 * e6 + 367.5 * e4 + 25.0 * e2,
2 : 7121009.55283911 * e18 + 2690460.52736734 * e16 + 927769.057548736 * e14 + 285788.216771507 * e12 + 76178.1462890625 * e10 + 16688.9113769531 * e8 + 2727.140625 * e6 + 260.015625 * e4,
3 : 7583987.33297915 * e18 + 2847310.73636333 * e16 + 969087.477329971 * e14 + 290946.807098765 * e12 + 73701.4461805556 * e10 + 14474.4722222222 * e8 + 1792.11111111111 * e6,
4 : 7811597.05086234 * e18 + 2884135.51696895 * e16 + 953426.381187765 * e14 + 271561.904476236 * e12 + 61808.6661783854 * e10 + 9661.76367865668 * e8,
5 : 7766016.72010553 * e18 + 2784035.39506975 * e16 + 876003.884196429 * e14 + 224786.9475 * e12 + 44163.0225 * e10,
6 : 7416990.69968876 * e18 + 2551140.62734588 * e16 + 716483.787616746 * e14 + 179171.885103873 * e12,
7 : 6868819.03742272 * e18 + 2022000.82930082 * e16 + 664011.24140688 * e14,
8 : 5006023.72197177 * e18 + 2291716.29166404 * e16,
9 : 7467007.1178002 * e18,
},
4: {
-9: 7467007.1178002 * e18,
},
5: {
-9: 200072127.641327 * e18,
},
6: {
-9: 3218885666.05741 * e18,
},
7: {
-9: 35680459926.5605 * e18,
}
}
# Duplicate results are stored as dictionary lookups to previous calculations
# Generally leads to a 30--50% speed-up when working with large arrays.
eccentricity_results_bymode[4][-8] = eccentricity_results_bymode[3][8]
eccentricity_results_bymode[4][-7] = eccentricity_results_bymode[3][7]
eccentricity_results_bymode[4][-6] = eccentricity_results_bymode[3][6]
eccentricity_results_bymode[4][-5] = eccentricity_results_bymode[3][5]
eccentricity_results_bymode[4][-4] = eccentricity_results_bymode[3][4]
eccentricity_results_bymode[4][-3] = eccentricity_results_bymode[3][3]
eccentricity_results_bymode[4][-2] = eccentricity_results_bymode[3][2]
eccentricity_results_bymode[4][-1] = eccentricity_results_bymode[3][1]
eccentricity_results_bymode[4][0] = eccentricity_results_bymode[3][0]
eccentricity_results_bymode[4][1] = eccentricity_results_bymode[3][-1]
eccentricity_results_bymode[4][2] = eccentricity_results_bymode[3][-2]
eccentricity_results_bymode[4][3] = eccentricity_results_bymode[3][-3]
eccentricity_results_bymode[4][4] = eccentricity_results_bymode[3][-4]
eccentricity_results_bymode[4][5] = eccentricity_results_bymode[3][-5]
eccentricity_results_bymode[4][6] = eccentricity_results_bymode[3][-6]
eccentricity_results_bymode[4][7] = eccentricity_results_bymode[3][-7]
eccentricity_results_bymode[4][8] = eccentricity_results_bymode[3][-8]
eccentricity_results_bymode[4][9] = eccentricity_results_bymode[3][-9]
eccentricity_results_bymode[5][-8] = eccentricity_results_bymode[2][8]
eccentricity_results_bymode[5][-7] = eccentricity_results_bymode[2][7]
eccentricity_results_bymode[5][-6] = eccentricity_results_bymode[2][6]
eccentricity_results_bymode[5][-5] = eccentricity_results_bymode[2][5]
eccentricity_results_bymode[5][-4] = eccentricity_results_bymode[2][4]
eccentricity_results_bymode[5][-3] = eccentricity_results_bymode[2][3]
eccentricity_results_bymode[5][-2] = eccentricity_results_bymode[2][2]
eccentricity_results_bymode[5][-1] = eccentricity_results_bymode[2][1]
eccentricity_results_bymode[5][0] = eccentricity_results_bymode[2][0]
eccentricity_results_bymode[5][1] = eccentricity_results_bymode[2][-1]
eccentricity_results_bymode[5][2] = eccentricity_results_bymode[2][-2]
eccentricity_results_bymode[5][3] = eccentricity_results_bymode[2][-3]
eccentricity_results_bymode[5][4] = eccentricity_results_bymode[2][-4]
eccentricity_results_bymode[5][5] = eccentricity_results_bymode[2][-5]
eccentricity_results_bymode[5][6] = eccentricity_results_bymode[2][-6]
eccentricity_results_bymode[5][7] = eccentricity_results_bymode[2][-7]
eccentricity_results_bymode[5][8] = eccentricity_results_bymode[2][-8]
eccentricity_results_bymode[5][9] = eccentricity_results_bymode[2][-9]
eccentricity_results_bymode[6][-8] = eccentricity_results_bymode[1][8]
eccentricity_results_bymode[6][-7] = eccentricity_results_bymode[1][7]
eccentricity_results_bymode[6][-6] = eccentricity_results_bymode[1][6]
eccentricity_results_bymode[6][-5] = eccentricity_results_bymode[1][5]
eccentricity_results_bymode[6][-4] = eccentricity_results_bymode[1][4]
eccentricity_results_bymode[6][-3] = eccentricity_results_bymode[1][3]
eccentricity_results_bymode[6][-2] = eccentricity_results_bymode[1][2]
eccentricity_results_bymode[6][-1] = eccentricity_results_bymode[1][1]
eccentricity_results_bymode[6][0] = eccentricity_results_bymode[1][0]
eccentricity_results_bymode[6][1] = eccentricity_results_bymode[1][-1]
eccentricity_results_bymode[6][2] = eccentricity_results_bymode[1][-2]
eccentricity_results_bymode[6][3] = eccentricity_results_bymode[1][-3]
eccentricity_results_bymode[6][4] = eccentricity_results_bymode[1][-4]
eccentricity_results_bymode[6][5] = eccentricity_results_bymode[1][-5]
eccentricity_results_bymode[6][6] = eccentricity_results_bymode[1][-6]
eccentricity_results_bymode[6][7] = eccentricity_results_bymode[1][-7]
eccentricity_results_bymode[6][8] = eccentricity_results_bymode[1][-8]
eccentricity_results_bymode[6][9] = eccentricity_results_bymode[1][-9]
eccentricity_results_bymode[7][-8] = eccentricity_results_bymode[0][8]
eccentricity_results_bymode[7][-7] = eccentricity_results_bymode[0][7]
eccentricity_results_bymode[7][-6] = eccentricity_results_bymode[0][6]
eccentricity_results_bymode[7][-5] = eccentricity_results_bymode[0][5]
eccentricity_results_bymode[7][-4] = eccentricity_results_bymode[0][4]
eccentricity_results_bymode[7][-3] = eccentricity_results_bymode[0][3]
eccentricity_results_bymode[7][-2] = eccentricity_results_bymode[0][2]
eccentricity_results_bymode[7][-1] = eccentricity_results_bymode[0][1]
eccentricity_results_bymode[7][0] = eccentricity_results_bymode[0][0]
eccentricity_results_bymode[7][1] = eccentricity_results_bymode[0][-1]
eccentricity_results_bymode[7][2] = eccentricity_results_bymode[0][-2]
eccentricity_results_bymode[7][3] = eccentricity_results_bymode[0][-3]
eccentricity_results_bymode[7][4] = eccentricity_results_bymode[0][-4]
eccentricity_results_bymode[7][5] = eccentricity_results_bymode[0][-5]
eccentricity_results_bymode[7][6] = eccentricity_results_bymode[0][-6]
eccentricity_results_bymode[7][8] = eccentricity_results_bymode[0][-8]
eccentricity_results_bymode[7][9] = eccentricity_results_bymode[0][-9]
return eccentricity_results_bymode
[docs]
@njit(cacheable=True)
def eccentricity_funcs_trunc20(eccentricity: 'FloatArray') -> 'EccenOutput':
""" Calculates the eccentricity functions (by mode) truncated to e^20 for order-l = 7
Parameters
----------
eccentricity : FloatArray
Orbital Eccentricity
Returns
-------
eccentricity_results_bymode : EccenOutput
"""
# Eccentricity functions calculated at truncation level 20.
# and order-l = 7.
# Performance and readability improvements
e = eccentricity
e2 = e * e
e4 = e * e * e * e
e6 = e * e * e * e * e * e
e8 = e**8
e10 = e**10
e12 = e**12
e14 = e**14
e16 = e**16
e18 = e**18
e20 = e**20
# Unique results.
eccentricity_results_bymode = {
0: {
-10: 2.52521938967461e-10 * e20,
-9 : 3.56920746121951e-11 * e20 + 7.59405842812662e-12 * e18,
-8 : 9.77755590635387e-14 * e20 + 3.96296384036149e-14 * e18 + 9.38596699032984e-15 * e16,
-6 : 3.25078870235199e-9 * e20 + 2.59607225768029e-9 * e18 + 1.87967129335142e-9 * e16 + 1.14373639357363e-9 * e14 + 4.7095027970679e-10 * e12,
-5 : 5.88175762149424e-5 * e20 + 6.09139095131365e-5 * e18 + 6.23484347442681e-5 * e16 + 6.28995811287478e-5 * e14 + 5.78703703703704e-5 * e12 + 6.94444444444444e-5 * e10,
-4 : -0.000148162983566979 * e20 - 0.000713012771947043 * e18 + 0.00108947594250952 * e16 - 0.0112746211460658 * e14 + 0.0320917510986328 * e12 - 0.07119140625 * e10 + 0.04449462890625 * e8,
-3 : -0.263712912514114 * e20 + 1.15933129613299 * e18 - 4.40869268077601 * e16 + 11.5056128747795 * e14 - 20.1716049382716 * e12 + 20.1555555555556 * e10 - 9.77777777777778 * e8 + 1.77777777777778 * e6,
-2 : 208.787078755598 * e20 - 612.292652626918 * e18 + 1348.87160996971 * e16 - 2132.90065789769 * e14 + 2269.74851555294 * e12 - 1516.85926649306 * e10 + 591.854519314236 * e8 - 120.442708333333 * e6 + 9.765625 * e4,
-1 : -41223.227052915 * e20 + 79249.6311015351 * e18 - 115035.60841179 * e16 + 121302.196799665 * e14 - 89131.943671875 * e12 + 43713.97734375 * e10 - 13602.09375 * e8 + 2496.9375 * e6 - 238.5 * e4 + 9.0 * e2,
0 : 2865580.41903142 * e20 - 3848911.55308733 * e18 + 3928011.79991695 * e16 - 2955172.8970563 * e14 + 1586997.38291086 * e12 - 586640.268715278 * e10 + 142589.589938694 * e8 - 21364.4618055556 * e6 + 1785.21875 * e4 - 70.0 * e2 + 1.0,
1 : -90219706.0333024 * e20 + 88302609.8818484 * e18 - 66006759.5382433 * e16 + 36681269.0568395 * e14 - 14695176.0341667 * e12 + 4085875.57048611 * e10 - 748898.888888889 * e8 + 83775.8333333333 * e6 - 5016.0 * e4 + 121.0 * e2,
2 : 1501519128.92349 * e20 - 1099191510.20098 * e18 + 614791138.403345 * e16 - 255474118.187811 * e14 + 76176310.2957281 * e12 - 15548508.0931641 * e10 + 2024000.86157227 * e8 - 148518.984375 * e6 + 4607.015625 * e4,
3 : -14565745106.477 * e20 + 8076809089.5857 * e18 - 3399179717.5205 * e16 + 1050342781.37476 * e14 - 227752424.952377 * e12 + 32361560.6011285 * e10 - 2669833.76736111 * e8 + 95841.8402777778 * e6,
4 : 87553341774.5094 * e20 - 36797348913.1132 * e18 + 11548436061.762 * e16 - 2589179614.74784 * e14 + 387648466.517466 * e12 - 34373096.9709961 * e10 + 1352599.28662788 * e8,
5 : -337793075412.305 * e20 + 106327614538.294 * e18 - 24236865800.0458 * e16 + 3741770950.29402 * e14 - 347139933.975 * e12 + 14501244.8025 * e10,
6 : 848099460456.673 * e20 - 194388536778.511 * e18 + 30511860315.4628 * e16 - 2910686193.59795 * e14 + 126441284.408644 * e12,
7 : -1374063443413.64 * e20 + 217180524610.773 * e18 - 21052727705.1045 * e16 + 937784248.676216 * e14,
8 : 1381790431565.48 * e20 - 134977425865.553 * e18 + 6104436340.31089 * e16,
9 : -782705671616.145 * e20 + 35680459926.5605 * e18,
10 : 190515839767.516 * e20,
},
1: {
-10: 0.758360891213849 * e20,
-9 : 4.50713038083351 * e20 + 0.409936224172744 * e18,
-8 : 16.05631286433 * e20 + 2.5493296938572 * e18 + 0.221719882537668 * e16,
-7 : 44.2439282915792 * e20 + 9.40277032701544 * e18 + 1.43968913129488 * e16 + 0.119978389648684 * e14,
-6 : 103.782001058244 * e20 + 26.6462243315436 * e18 + 5.49174032711125 * e16 + 0.811787874265533 * e14 + 0.0649431247475706 * e12,
-5 : -0.03515625 * e10 / (e2 - 1.0)**13,
-4 : 419.394715198794 * e20 + 136.855297948203 * e18 + 39.3153428329901 * e16 + 9.57492866166264 * e14 + 1.8597327875208 * e12 + 0.257063802083333 * e10 + 0.0190497504340278 * e8,
-3 : 750.71365065208 * e20 + 264.879793242202 * e18 + 83.9811453855269 * e16 + 23.2613752652392 * e14 + 5.39066358024691 * e12 + 0.971701388888889 * e10 + 0.118055555555556 * e8 + 0.00694444444444444 * e6,
-2 : 1350.80357341325 * e20 + 525.84349136254 * e18 + 191.185387344744 * e16 + 64.667712789263 * e14 + 20.3473989486694 * e12 + 5.8693359375 * e10 + 1.742431640625 * e8 + 0.140625 * e6 + 0.140625 * e4,
-1 : 2317.51811908101 * e20 + 962.771132986083 * e18 + 373.925813039336 * e16 + 139.310944113757 * e14 + 30.2398958333333 * e12 + 38.0814236111111 * e10 - 32.8888888888889 * e8 + 24.5833333333333 * e6 - 9.0 * e4 + e2,
0 : 3677.29777259859 * e20 + 1415.4631693483 * e18 + 1001.54276161753 * e16 - 752.495595552844 * e14 + 1725.84582929258 * e12 - 2021.14040798611 * e10 + 1576.83668348524 * e8 - 741.545138888889 * e6 + 183.96875 * e4 - 22.0 * e2 + 1.0,
1 : -1872.0923695604 * e20 + 20335.218430836 * e18 - 35946.0589629504 * e16 + 56545.2157073103 * e14 - 62900.057109375 * e12 + 48573.09140625 * e10 - 24466.5 * e8 + 7388.4375 * e6 - 1201.5 * e4 + 81.0 * e2,
2 : 517111.348636908 * e20 - 895869.621245169 * e18 + 1231228.06259342 * e16 - 1266309.79616931 * e14 + 938547.352011341 * e12 - 475122.020203993 * e10 + 152617.184597439 * e8 - 27743.8177083333 * e6 + 2173.890625 * e4,
3 : -15755869.6938751 * e20 + 19527906.1971512 * e18 - 18573722.9224831 * e16 + 13062709.1406972 * e14 - 6464235.02122878 * e12 + 2097863.73862847 * e10 - 397130.923611111 * e8 + 33184.6944444444 * e6,
4 : 242364596.820666 * e20 - 213970512.55123 * e18 + 142257563.766493 * e16 - 67933871.4747141 * e14 + 21747975.9750501 * e12 - 4146304.49912109 * e10 + 354871.521057129 * e8,
5 : -2037372091.39537 * e20 + 1279859819.6393 * e18 - 586239684.517029 * e16 + 182888425.081913 * e14 - 34500980.0248119 * e12 + 2959618.45876736 * e10,
6 : 9878875019.08264 * e20 - 4329029031.54511 * e18 + 1307624112.7669 * e16 - 241659566.56067 * e14 + 20518448.8791875 * e12,
7 : -28167144687.0669 * e20 + 8210215164.34436 * e18 - 1477892171.3568 * e16 + 123255910.704605 * e14,
8 : 46337824554.8877 * e20 - 8097179451.88272 * e18 + 660185101.545129 * e16,
9 : -40503407456.5363 * e20 + 3218885666.05741 * e18,
10 : 14509353796.2492 * e20,
},
2: {
-10: 1519.3827456556 * e20,
-9 : 6615.42497342978 * e20 + 719.31120267907 * e18,
-8 : 18563.9913698274 * e20 + 3328.30927572957 * e18 + 337.448907736202 * e16,
-7 : 42024.5806378428 * e20 + 9733.68739966595 * e18 + 1653.23773598415 * e16 + 156.64211077255 * e14,
-6 : 83302.2629102853 * e20 + 22735.9555951404 * e18 + 5032.83500185247 * e16 + 809.854400596619 * e14 + 71.8156937026978 * e12,
-5 : 150611.873140433 * e20 + 46212.5354749881 * e18 + 12123.5319398803 * e16 + 2563.02166482377 * e14 + 390.619458912037 * e12 + 32.4425173611111 * e10,
-4 : 254351.158324509 * e20 + 85307.8419407169 * e18 + 25261.633121265 * e16 + 6363.25240061133 * e14 + 1283.45398924368 * e12 + 185.11298828125 * e10 + 14.396491156684 * e8,
-3 : -0.09765625 * e6 * (3.0 * e2 + 8.0)**2 / (e2 - 1.0)**13,
-2 : 625323.727861521 * e20 + 238460.665898411 * e18 + 83279.1471429178 * e16 + 26135.9927566851 * e14 + 7178.43844695621 * e12 + 1659.23604600694 * e10 + 303.058295355903 * e8 + 38.9322916666667 * e6 + 2.640625 * e4,
-1 : 926316.964591722 * e20 + 370640.856538529 * e18 + 137331.061940094 * e16 + 46429.6170135272 * e14 + 14041.6768142361 * e12 + 3694.66293402778 * e10 + 811.631944444444 * e8 + 139.270833333333 * e6 + 16.5 * e4 + e2,
0 : 1334801.20394474 * e20 + 557030.77991627 * e18 + 217273.143247915 * e16 + 78327.7874431159 * e14 + 25731.1676902771 * e12 + 7565.199296875 * e10 + 1945.65222167969 * e8 + 424.84375 * e6 + 76.46875 * e4 + 10.0 * e2 + 1.0,
1 : 1880427.39612577 * e20 + 814711.469512273 * e18 + 332514.430101101 * e16 + 126743.549716435 * e14 + 44644.5701736111 * e12 + 14360.1105902778 * e10 + 4115.31944444444 * e8 + 1080.58333333333 * e6 + 189.0 * e4 + 49.0 * e2,
2 : 2596902.5440326 * e20 + 1163119.00350269 * e18 + 493808.611203993 * e16 + 197042.647860248 * e14 + 73906.7868524128 * e12 + 24376.8419053819 * e10 + 8894.1413031684 * e8 + 1254.55729166667 * e6 + 862.890625 * e4,
3 : 3519993.0781772 * e20 + 1625116.54120954 * e18 + 706998.515643834 * e16 + 306304.572200056 * e14 + 97963.396875 * e12 + 59750.0453125 * e10 - 95.75 * e8 + 9168.0625 * e6,
4 : 4734364.69296058 * e20 + 2116130.95690105 * e18 + 1172588.42289078 * e16 + 196218.790338876 * e14 + 392310.672480862 * e12 - 78294.8146809896 * e10 + 71362.7221747504 * e8,
5 : 4777840.16433186 * e20 + 5013078.31114085 * e18 - 1015685.91263426 * e16 + 2608659.94096175 * e14 - 894602.368952546 * e12 + 448693.440434028 * e10,
6 : 27201219.1571369 * e20 - 15929962.0572314 * e18 + 16725535.5538483 * e16 - 6709222.00565983 * e14 + 2412439.34078983 * e12,
7 : -129204759.791638 * e20 + 99121020.0636772 * e18 - 40255987.811503 * e16 + 11503209.3055423 * e14,
8 : 535838559.442134 * e20 - 208038708.070791 * e18 + 49868524.3845965 * e16,
9 : -962610691.5785 * e20 + 200072127.641327 * e18,
10 : 752677336.129396 * e20,
},
3: {
-10: 338314.585109017 * e20,
-9 : 801490.636272239 * e20 + 132646.745553428 * e18,
-8 : 1499194.70421724 * e20 + 345601.988924924 * e18 + 50553.5356753209 * e16,
-7 : 2433125.2841376 * e20 + 671947.52842675 * e18 + 143177.836475606 * e16 + 18617.627869898 * e14,
-6 : 3607947.58406798 * e20 + 1117258.14188041 * e18 + 288970.153736016 * e16 + 56701.7701593513 * e14 + 6572.37797547564 * e12,
-5 : 5014171.91781823 * e20 + 1681953.98847843 * e18 + 491118.834556809 * e16 + 118393.583558477 * e14 + 21292.203287037 * e12 + 2198.82840277778 * e10,
-4 : 6628280.06662149 * e20 + 2357941.69033208 * e18 + 748464.032461628 * e16 + 204879.526902172 * e14 + 45732.4679340363 * e12 + 7485.75439453125 * e10 + 685.376037597656 * e8,
-3 : 8411973.47785275 * e20 + 3128076.05868162 * e18 + 1054606.98825424 * e16 + 314588.587124842 * e14 + 80064.9051456404 * e12 + 16387.2709201389 * e10 + 2413.96180555556 * e8 + 193.673611111111 * e6,
-2 : 10311792.6455446 * e20 + 3965741.56804964 * e18 + 1397522.06146574 * e16 + 442814.440752841 * e14 + 122862.972033925 * e12 + 28704.1544053819 * e10 + 5301.66408962674 * e8 + 688.932291666667 * e6 + 47.265625 * e4,
-1 : -0.140625 * e2 * (5.0 * e4 + 20.0 * e2 + 8.0)**2 / (e2 - 1.0)**13,
0 : 14170276.7205269 * e20 + 5689186.47353937 * e18 + 2116390.96404087 * e16 + 719186.088857198 * e14 + 219108.287543295 * e12 + 58347.2753993056 * e10 + 13103.2553032769 * e8 + 2353.37152777778 * e6 + 310.71875 * e4 + 26.0 * e2 + 1.0,
1 : 15942250.9578873 * e20 + 6470933.30682901 * e18 + 2437240.94540722 * e16 + 839864.510332445 * e14 + 259842.755598958 * e12 + 70311.6384548611 * e10 + 16012.0798611111 * e8 + 2885.77083333333 * e6 + 367.5 * e4 + 25.0 * e2,
2 : 17470034.1455324 * e20 + 7121009.55283911 * e18 + 2690460.52736734 * e16 + 927769.057548736 * e14 + 285788.216771507 * e12 + 76178.1462890625 * e10 + 16688.9113769531 * e8 + 2727.140625 * e6 + 260.015625 * e4,
3 : 18652914.8224162 * e20 + 7583987.33297915 * e18 + 2847310.73636333 * e16 + 969087.477329971 * e14 + 290946.807098765 * e12 + 73701.4461805556 * e10 + 14474.4722222222 * e8 + 1792.11111111111 * e6,
4 : 19399856.8715594 * e20 + 7811597.05086234 * e18 + 2884135.51696895 * e16 + 953426.381187765 * e14 + 271561.904476236 * e12 + 61808.6661783854 * e10 + 9661.76367865668 * e8,
5 : 19634240.6082711 * e20 + 7766016.72010553 * e18 + 2784035.39506975 * e16 + 876003.884196429 * e14 + 224786.9475 * e12 + 44163.0225 * e10,
6 : 19300147.1263215 * e20 + 7416990.69968876 * e18 + 2551140.62734588 * e16 + 716483.787616746 * e14 + 179171.885103873 * e12,
7 : 18300587.1822257 * e20 + 6868819.03742272 * e18 + 2022000.82930082 * e16 + 664011.24140688 * e14,
8 : 17509571.5605009 * e20 + 5006023.72197177 * e18 + 2291716.29166404 * e16,
9 : 10377351.9010732 * e20 + 7467007.1178002 * e18,
10 : 23199404.2410762 * e20,
},
4: {
-10: 23199404.2410762 * e20,
},
5: {
-10: 752677336.129396 * e20,
},
6: {
-10: 14509353796.2492 * e20,
},
7: {
-10: 190515839767.516 * e20,
}
}
# Duplicate results are stored as dictionary lookups to previous calculations
# Generally leads to a 30--50% speed-up when working with large arrays.
eccentricity_results_bymode[4][-9] = eccentricity_results_bymode[3][9]
eccentricity_results_bymode[4][-8] = eccentricity_results_bymode[3][8]
eccentricity_results_bymode[4][-7] = eccentricity_results_bymode[3][7]
eccentricity_results_bymode[4][-6] = eccentricity_results_bymode[3][6]
eccentricity_results_bymode[4][-5] = eccentricity_results_bymode[3][5]
eccentricity_results_bymode[4][-4] = eccentricity_results_bymode[3][4]
eccentricity_results_bymode[4][-3] = eccentricity_results_bymode[3][3]
eccentricity_results_bymode[4][-2] = eccentricity_results_bymode[3][2]
eccentricity_results_bymode[4][-1] = eccentricity_results_bymode[3][1]
eccentricity_results_bymode[4][0] = eccentricity_results_bymode[3][0]
eccentricity_results_bymode[4][1] = eccentricity_results_bymode[3][-1]
eccentricity_results_bymode[4][2] = eccentricity_results_bymode[3][-2]
eccentricity_results_bymode[4][3] = eccentricity_results_bymode[3][-3]
eccentricity_results_bymode[4][4] = eccentricity_results_bymode[3][-4]
eccentricity_results_bymode[4][5] = eccentricity_results_bymode[3][-5]
eccentricity_results_bymode[4][6] = eccentricity_results_bymode[3][-6]
eccentricity_results_bymode[4][7] = eccentricity_results_bymode[3][-7]
eccentricity_results_bymode[4][8] = eccentricity_results_bymode[3][-8]
eccentricity_results_bymode[4][9] = eccentricity_results_bymode[3][-9]
eccentricity_results_bymode[4][10] = eccentricity_results_bymode[3][-10]
eccentricity_results_bymode[5][-9] = eccentricity_results_bymode[2][9]
eccentricity_results_bymode[5][-8] = eccentricity_results_bymode[2][8]
eccentricity_results_bymode[5][-7] = eccentricity_results_bymode[2][7]
eccentricity_results_bymode[5][-6] = eccentricity_results_bymode[2][6]
eccentricity_results_bymode[5][-5] = eccentricity_results_bymode[2][5]
eccentricity_results_bymode[5][-4] = eccentricity_results_bymode[2][4]
eccentricity_results_bymode[5][-3] = eccentricity_results_bymode[2][3]
eccentricity_results_bymode[5][-2] = eccentricity_results_bymode[2][2]
eccentricity_results_bymode[5][-1] = eccentricity_results_bymode[2][1]
eccentricity_results_bymode[5][0] = eccentricity_results_bymode[2][0]
eccentricity_results_bymode[5][1] = eccentricity_results_bymode[2][-1]
eccentricity_results_bymode[5][2] = eccentricity_results_bymode[2][-2]
eccentricity_results_bymode[5][3] = eccentricity_results_bymode[2][-3]
eccentricity_results_bymode[5][4] = eccentricity_results_bymode[2][-4]
eccentricity_results_bymode[5][5] = eccentricity_results_bymode[2][-5]
eccentricity_results_bymode[5][6] = eccentricity_results_bymode[2][-6]
eccentricity_results_bymode[5][7] = eccentricity_results_bymode[2][-7]
eccentricity_results_bymode[5][8] = eccentricity_results_bymode[2][-8]
eccentricity_results_bymode[5][9] = eccentricity_results_bymode[2][-9]
eccentricity_results_bymode[5][10] = eccentricity_results_bymode[2][-10]
eccentricity_results_bymode[6][-9] = eccentricity_results_bymode[1][9]
eccentricity_results_bymode[6][-8] = eccentricity_results_bymode[1][8]
eccentricity_results_bymode[6][-7] = eccentricity_results_bymode[1][7]
eccentricity_results_bymode[6][-6] = eccentricity_results_bymode[1][6]
eccentricity_results_bymode[6][-5] = eccentricity_results_bymode[1][5]
eccentricity_results_bymode[6][-4] = eccentricity_results_bymode[1][4]
eccentricity_results_bymode[6][-3] = eccentricity_results_bymode[1][3]
eccentricity_results_bymode[6][-2] = eccentricity_results_bymode[1][2]
eccentricity_results_bymode[6][-1] = eccentricity_results_bymode[1][1]
eccentricity_results_bymode[6][0] = eccentricity_results_bymode[1][0]
eccentricity_results_bymode[6][1] = eccentricity_results_bymode[1][-1]
eccentricity_results_bymode[6][2] = eccentricity_results_bymode[1][-2]
eccentricity_results_bymode[6][3] = eccentricity_results_bymode[1][-3]
eccentricity_results_bymode[6][4] = eccentricity_results_bymode[1][-4]
eccentricity_results_bymode[6][5] = eccentricity_results_bymode[1][-5]
eccentricity_results_bymode[6][6] = eccentricity_results_bymode[1][-6]
eccentricity_results_bymode[6][7] = eccentricity_results_bymode[1][-7]
eccentricity_results_bymode[6][8] = eccentricity_results_bymode[1][-8]
eccentricity_results_bymode[6][9] = eccentricity_results_bymode[1][-9]
eccentricity_results_bymode[6][10] = eccentricity_results_bymode[1][-10]
eccentricity_results_bymode[7][-9] = eccentricity_results_bymode[0][9]
eccentricity_results_bymode[7][-8] = eccentricity_results_bymode[0][8]
eccentricity_results_bymode[7][-7] = eccentricity_results_bymode[0][7]
eccentricity_results_bymode[7][-6] = eccentricity_results_bymode[0][6]
eccentricity_results_bymode[7][-5] = eccentricity_results_bymode[0][5]
eccentricity_results_bymode[7][-4] = eccentricity_results_bymode[0][4]
eccentricity_results_bymode[7][-3] = eccentricity_results_bymode[0][3]
eccentricity_results_bymode[7][-2] = eccentricity_results_bymode[0][2]
eccentricity_results_bymode[7][-1] = eccentricity_results_bymode[0][1]
eccentricity_results_bymode[7][0] = eccentricity_results_bymode[0][0]
eccentricity_results_bymode[7][1] = eccentricity_results_bymode[0][-1]
eccentricity_results_bymode[7][2] = eccentricity_results_bymode[0][-2]
eccentricity_results_bymode[7][3] = eccentricity_results_bymode[0][-3]
eccentricity_results_bymode[7][4] = eccentricity_results_bymode[0][-4]
eccentricity_results_bymode[7][5] = eccentricity_results_bymode[0][-5]
eccentricity_results_bymode[7][6] = eccentricity_results_bymode[0][-6]
eccentricity_results_bymode[7][8] = eccentricity_results_bymode[0][-8]
eccentricity_results_bymode[7][9] = eccentricity_results_bymode[0][-9]
eccentricity_results_bymode[7][10] = eccentricity_results_bymode[0][-10]
return eccentricity_results_bymode