# -*- coding: utf-8 -*- # PyMeeus: Python module implementing astronomical algorithms. # Copyright (C) 2018 Dagoberto Salazar # # This program is free software: you can redistribute it and/or modify # it under the terms of the GNU Lesser General Public License as published by # the Free Software Foundation, either version 3 of the License, or # (at your option) any later version. # # This program is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU Lesser General Public License for more details. # # You should have received a copy of the GNU Lesser General Public License # along with this program. If not, see . from math import sin, cos, tan, acos, atan2, sqrt, radians, log10 from pymeeus.Angle import Angle from pymeeus.Epoch import Epoch, JDE2000 from pymeeus.Interpolation import Interpolation from pymeeus.Coordinates import ( geometric_vsop_pos, apparent_vsop_pos, orbital_elements, nutation_longitude, true_obliquity, ecliptical2equatorial, passage_nodes_elliptic ) from pymeeus.Earth import Earth from pymeeus.Sun import Sun """ .. module:: Saturn :synopsis: Class to model Saturn planet :license: GNU Lesser General Public License v3 (LGPLv3) .. moduleauthor:: Dagoberto Salazar """ VSOP87_L = [ # L0 [ [87401354.029, 0.00000000000, 0.00000000000], [11107659.780, 3.96205090194, 213.29909543800], [1414150.958, 4.58581515873, 7.11354700080], [398379.386, 0.52112025957, 206.18554843720], [350769.223, 3.30329903015, 426.59819087600], [206816.296, 0.24658366938, 103.09277421860], [79271.288, 3.84007078530, 220.41264243880], [23990.338, 4.66976934860, 110.20632121940], [16573.583, 0.43719123541, 419.48464387520], [14906.995, 5.76903283845, 316.39186965660], [15820.300, 0.93808953760, 632.78373931320], [14609.562, 1.56518573691, 3.93215326310], [13160.308, 4.44891180176, 14.22709400160], [15053.509, 2.71670027883, 639.89728631400], [13005.305, 5.98119067061, 11.04570026390], [10725.066, 3.12939596466, 202.25339517410], [5863.207, 0.23657028777, 529.69096509460], [5227.771, 4.20783162380, 3.18139373770], [6126.308, 1.76328499656, 277.03499374140], [5019.658, 3.17787919533, 433.71173787680], [4592.541, 0.61976424374, 199.07200143640], [4005.862, 2.24479893937, 63.73589830340], [2953.815, 0.98280385206, 95.97922721780], [3873.696, 3.22282692566, 138.51749687070], [2461.172, 2.03163631205, 735.87651353180], [3269.490, 0.77491895787, 949.17560896980], [1758.143, 3.26580514774, 522.57741809380], [1640.183, 5.50504966218, 846.08283475120], [1391.336, 4.02331978116, 323.50541665740], [1580.641, 4.37266314120, 309.27832265580], [1123.515, 2.83726793572, 415.55249061210], [1017.258, 3.71698151814, 227.52618943960], [848.643, 3.19149825839, 209.36694217490], [1087.237, 4.18343232481, 2.44768055480], [956.752, 0.50740889886, 1265.56747862640], [789.205, 5.00745123149, 0.96320784650], [686.965, 1.74714407827, 1052.26838318840], [654.470, 1.59889331515, 0.04818410980], [748.811, 2.14398149298, 853.19638175200], [633.980, 2.29889903023, 412.37109687440], [743.584, 5.25276954625, 224.34479570190], [852.677, 3.42141350697, 175.16605980020], [579.857, 3.09259007048, 74.78159856730], [624.904, 0.97046831256, 210.11770170030], [529.861, 4.44938897119, 117.31986822020], [542.643, 1.51824320514, 9.56122755560], [474.279, 5.47527185987, 742.99006053260], [448.542, 1.28990416161, 127.47179660680], [546.358, 2.12678554211, 350.33211960040], [478.054, 2.96488054338, 137.03302416240], [354.944, 3.01286483030, 838.96928775040], [451.827, 1.04436664241, 490.33408917940], [347.413, 1.53928227764, 340.77089204480], [343.475, 0.24604039134, 0.52126486180], [309.001, 3.49486734909, 216.48048917570], [322.185, 0.96137456104, 203.73786788240], [372.308, 2.27819108625, 217.23124870110], [321.543, 2.57182354537, 647.01083331480], [330.196, 0.24715617844, 1581.95934828300], [249.116, 1.47010534421, 1368.66025284500], [286.688, 2.37043745859, 351.81659230870], [220.225, 4.20422424873, 200.76892246580], [277.775, 0.40020408926, 211.81462272970], [204.500, 6.01082206600, 265.98929347750], [207.663, 0.48349820488, 1162.47470440780], [208.655, 1.34516255304, 625.67019231240], [182.454, 5.49122292426, 2.92076130680], [226.609, 4.91003163138, 12.53017297220], [207.659, 1.28302218900, 39.35687591520], [173.914, 1.86305806814, 0.75075952540], [184.690, 3.50344404958, 149.56319713460], [183.511, 0.97254952728, 4.19278569400], [146.068, 6.23102544071, 195.13984817330], [164.541, 0.44005517520, 5.41662597140], [147.526, 1.53529320509, 5.62907429250], [139.666, 4.29450260069, 21.34064100240], [131.283, 4.06828961903, 10.29494073850], [117.283, 2.67920400584, 1155.36115740700], [149.299, 5.73594349789, 52.69019803950], [122.373, 1.97588777199, 4.66586644600], [113.747, 5.59427544714, 1059.38193018920], [102.702, 1.19748124058, 1685.05212250160], [118.156, 5.34072933900, 554.06998748280], [109.275, 3.43812715686, 536.80451209540], [110.399, 0.16604024090, 1.48447270830], [124.969, 6.27737805832, 1898.35121793960], [89.949, 5.80392934702, 114.13847448250], [103.956, 2.19210363069, 88.86568021700], [112.437, 1.10502663534, 191.20769491020], [106.570, 4.01156608514, 956.28915597060], [91.430, 1.87521577510, 38.13303563780], [83.791, 5.48810655641, 0.11187458460], [83.461, 2.28972767279, 628.85158605010], [96.987, 4.53666595763, 302.16477565500], [100.631, 4.96513666539, 269.92144674060], [75.491, 2.18045274099, 728.76296653100], [96.330, 2.83319189210, 275.55052103310], [82.363, 3.05469876064, 440.82528487760], [73.888, 5.08914205084, 1375.77379984580], [71.633, 5.10940743430, 65.22037101170], [70.409, 4.86846451411, 0.21244832110], [69.760, 3.71029022489, 14.97785352700], [88.772, 3.86334563977, 278.51946644970], [68.090, 0.73415460990, 1478.86657406440], [66.501, 0.02677580336, 70.84944530420], [65.682, 2.02165559602, 142.44965013380], [75.765, 1.61410487792, 284.14854074220], [63.153, 3.49493353034, 479.28838891550], [62.539, 2.58713611532, 422.66603761290], [69.313, 3.43979731402, 515.46387109300], [79.021, 4.45154941586, 35.42472265210], [63.664, 3.31749528708, 62.25142559510], [52.939, 5.51392725227, 0.26063243090], [53.011, 3.18480701697, 8.07675484730], [54.492, 2.45674090515, 22.09140052780], [50.514, 4.26749346978, 99.16062095550], [55.170, 0.96797446150, 942.06206196900], [49.288, 2.38641424063, 1471.75302706360], [47.199, 2.02515248245, 312.19908396260], [61.080, 1.50295092063, 210.85141488320], [45.126, 0.93109376473, 2001.44399215820], [60.556, 2.68715551585, 388.46515523820], [43.452, 2.52602011714, 288.08069400530], [42.544, 3.81793980322, 330.61896365820], [39.915, 5.71378652900, 408.43894361130], [50.145, 6.03164759907, 2214.74308759620], [45.860, 0.54229721801, 212.33588759150], [54.165, 0.78154835399, 191.95845443560], [47.016, 4.59934671151, 437.64389113990], [42.362, 1.90070070955, 430.53034413910], [39.722, 1.63259419913, 1066.49547719000], [36.345, 0.84756992711, 213.34727954780], [35.468, 4.18603772925, 215.74677599280], [36.344, 3.93295730315, 213.25091132820], [38.005, 0.31313803095, 423.41679713830], [44.746, 1.12488341174, 6.15033915430], [37.902, 1.19795851115, 2.70831298570], [43.402, 1.37363944007, 563.63121503840], [43.764, 3.93043802956, 525.49817940060], [34.825, 1.01566605408, 203.00415469950], [31.755, 1.69273634405, 0.16005869440], [30.880, 6.13525703832, 417.03696332040], [36.388, 6.00586032647, 18.15924726470], [29.032, 1.19660544505, 404.50679034820], [32.812, 0.53649479713, 107.02492748170], [30.433, 0.72335287989, 222.86032299360], [32.644, 0.81204701486, 1795.25844372100], [37.769, 3.69666903716, 1272.68102562720], [27.679, 1.45663979401, 7.16173111060], [27.187, 1.89731951902, 1045.15483618760], [37.699, 4.51997049537, 24.37902238820], [34.885, 4.46095761791, 214.26230328450], [32.650, 0.66372395761, 692.58748435350], [30.324, 5.30369950147, 33.94024994380], [27.480, 6.22702216249, 1.27202438720], [26.657, 4.56713198392, 7.06536289100], [31.745, 5.49798599565, 56.62235130260], [28.050, 5.64447420566, 128.95626931510], [24.277, 3.93966553574, 414.06801790380], [32.017, 5.22260660455, 92.04707395470], [26.976, 0.06705123981, 205.22234059070], [22.974, 3.65817751770, 207.67002114550], [31.775, 5.59198119173, 6069.77675455340], [23.153, 2.10054506119, 1788.14489672020], [31.025, 0.37190053329, 703.63318461740], [29.376, 0.14742155778, 131.40394986990], [22.562, 5.24009182383, 212.77783057620], [26.185, 5.41311252822, 140.00196957900], [25.673, 4.36038885283, 32.24332891440], [20.392, 2.82413909260, 429.77958461370], [20.659, 0.67091805084, 2317.83586181480], [24.397, 3.08740396398, 145.63104387150], [23.735, 2.54365387567, 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2627.11418447060], [0.203, 0.13969067385, 1485.98012106520], [0.204, 3.38137545713, 699.70103135430], [0.212, 4.52370676085, 2310.72231481400], [0.218, 5.79277335862, 2221.85663459700], [0.213, 0.50441377637, 934.94851496820], [0.210, 5.04017633795, 2420.92863603340], [0.214, 4.64286758581, 2317.83586181480], [0.178, 0.84588580004, 128.36556848410], [0.170, 2.75006619605, 710.74673161820], [0.171, 4.32615182967, 291.26208774300], [0.172, 3.46971306920, 501.37978944330], [0.170, 1.05408992106, 526.50957135690], [0.162, 1.15683042950, 519.39602435610], [0.180, 4.96266204107, 1670.82502850000], [0.172, 1.65385549578, 916.93228005540], [0.170, 2.30821101766, 429.04587143080], [0.170, 5.98716489326, 643.82943957710], [0.173, 5.19933564968, 1354.43315884340], [0.195, 4.50165508529, 2214.74308759620], [0.156, 4.16290662749, 572.22923474750], [0.153, 1.23776248578, 2413.81508903260], [0.150, 0.63076983213, 1478.86657406440], [0.169, 4.28090123029, 305.34616939270], [0.174, 6.23077892653, 3384.33133900480], [0.149, 3.13274908516, 9573.38825989700], [0.162, 6.25601818345, 213.25091132820], [0.149, 4.81749019484, 945.99421523210], [0.162, 0.88610129190, 216.21985674480], [0.133, 2.31915371262, 156.67674413540], [0.165, 6.06456216591, 732.69511979410], [0.141, 6.14293754333, 1795.25844372100], [0.133, 0.06530337135, 218.71572140940], [0.162, 3.17058130506, 213.34727954780], [0.125, 2.07143636845, 425.11371816770], [0.146, 1.88627500632, 211.81462272970], [0.113, 2.79541965778, 235.39049596580], [0.117, 0.76464798684, 479.28838891550], [0.108, 3.95650672786, 570.74476203920], [0.106, 0.12820734703, 188.02630117250], [0.134, 3.58244908862, 849.26422848890], [0.114, 0.25990388555, 398.14400287280], [0.112, 2.39181495831, 217.49188113200], [0.091, 2.50716605179, 121.25202148330], [0.091, 1.75367948574, 213.82036029980], [0.088, 5.26121947108, 395.57870223900], [0.096, 3.98832609364, 289.56516671360], [0.091, 0.35318362186, 312.45971639350], [0.112, 1.14387590923, 1802.37199072180], [0.082, 3.73605217214, 207.67002114550], [0.082, 6.06283262812, 210.37833413120], [0.084, 3.34470673492, 67.66805156650], [0.086, 2.73917300180, 5863.59120611620], [0.083, 2.81499116485, 776.93031047640], [0.091, 1.26160093170, 212.77783057620], [0.090, 2.08722491981, 2111.65031337760], [0.080, 2.13136842916, 421.93232443000], [0.082, 4.16358350281, 9793.80090233580], [0.077, 2.96973341607, 431.26405732200], [0.079, 3.42790361067, 417.03696332040], [0.079, 3.18693585419, 320.32402291970], [0.080, 0.78975763683, 204.70107572890], [0.077, 1.89354243952, 556.51766803760], [0.073, 4.85923277221, 2118.76386037840], [0.071, 3.64551577433, 198.32124191100], ], # L3 [ [16038.734, 5.73945377424, 7.11354700080], [4249.793, 4.58539675603, 213.29909543800], [1906.524, 4.76082050205, 220.41264243880], [1465.687, 5.91326678323, 206.18554843720], [1162.041, 5.61973132428, 14.22709400160], [1066.581, 3.60816533142, 426.59819087600], [239.377, 3.86088273439, 433.71173787680], [236.975, 5.76826451465, 199.07200143640], [165.641, 5.11641150216, 3.18139373770], [131.409, 4.74327544615, 227.52618943960], [151.352, 2.73594641861, 639.89728631400], [61.630, 4.74287052463, 103.09277421860], [63.365, 0.22850089497, 419.48464387520], [40.437, 5.47298059144, 21.34064100240], [40.205, 5.96420266720, 95.97922721780], [38.746, 5.83386199529, 110.20632121940], [28.025, 3.01235311514, 647.01083331480], [25.029, 0.98808170740, 3.93215326310], [18.101, 1.02506397063, 412.37109687440], [17.879, 3.31913418974, 309.27832265580], [16.208, 3.89825272754, 440.82528487760], [15.763, 5.61667809625, 117.31986822020], [19.014, 1.91614237463, 853.19638175200], [18.262, 4.96738415934, 10.29494073850], [12.947, 1.18068953942, 88.86568021700], [17.919, 4.20376505349, 216.48048917570], [11.453, 5.57520615096, 11.04570026390], [10.548, 5.92906266269, 191.95845443560], [10.389, 3.94838736947, 209.36694217490], [8.650, 3.39335369698, 302.16477565500], [7.580, 4.87736913157, 323.50541665740], [6.697, 0.38198725552, 632.78373931320], [5.864, 1.05621157685, 210.11770170030], [5.449, 4.64268475485, 234.63973644040], [6.327, 2.25492722762, 522.57741809380], [3.602, 2.30677010956, 515.46387109300], [3.229, 2.20309400066, 860.30992875280], [3.701, 3.14159265359, 0.00000000000], [2.583, 4.93447677059, 224.34479570190], [2.543, 0.42393884183, 625.67019231240], [2.213, 3.19814958289, 202.25339517410], [2.421, 4.76621391814, 330.61896365820], [2.850, 0.58604395010, 529.69096509460], [1.965, 4.39525359412, 124.43341522100], [2.154, 1.35488209144, 405.25754987360], [2.296, 3.34809165905, 429.77958461370], [2.018, 3.06693569701, 654.12438031560], [1.979, 1.02981005658, 728.76296653100], [1.868, 3.09383546177, 422.66603761290], [1.846, 4.15225985450, 536.80451209540], [2.194, 1.18918501013, 1066.49547719000], [2.090, 4.15631351317, 223.59403617650], [1.481, 0.37916705169, 316.39186965660], [1.720, 5.82865773356, 195.13984817330], [1.460, 1.57663426355, 81.75213321620], [1.623, 6.03706764648, 742.99006053260], [1.286, 1.66154726117, 63.73589830340], [1.304, 5.02409881054, 956.28915597060], [1.446, 2.10575519127, 838.96928775040], [1.245, 3.88109752770, 269.92144674060], [1.018, 3.72599601656, 295.05122865420], [1.323, 1.38492882986, 735.87651353180], [1.318, 2.33460998999, 217.23124870110], [0.943, 2.75813531246, 284.14854074220], [0.906, 0.71155526266, 846.08283475120], [0.886, 3.83754799777, 447.93883187840], [0.943, 3.31480217015, 18.15924726470], [0.800, 4.71386673963, 56.62235130260], [0.908, 2.02119147951, 831.85574074960], [0.787, 0.80410269937, 1045.15483618760], [0.709, 4.27064410504, 437.64389113990], [0.651, 6.17565900032, 942.06206196900], [0.785, 2.40767785311, 203.00415469950], [0.702, 1.64585301418, 423.41679713830], [0.543, 2.86326941725, 184.84490743480], [0.532, 6.25762144463, 1059.38193018920], [0.521, 3.43013038466, 149.56319713460], [0.484, 4.88366060720, 1272.68102562720], [0.437, 5.40220619672, 408.43894361130], [0.388, 2.57589594168, 508.35032409220], [0.421, 4.05836524024, 543.91805909620], [0.375, 1.22747948298, 2324.94940881560], [0.347, 0.59237194522, 22.09140052780], [0.433, 1.69090148012, 1155.36115740700], [0.389, 1.46170367972, 1073.60902419080], [0.307, 1.82185086955, 628.85158605010], [0.409, 1.21858750514, 1052.26838318840], [0.309, 0.33610530663, 6076.89030155420], [0.309, 1.42279282226, 6062.66320755260], [0.340, 1.83325770310, 1141.13406340540], [0.303, 2.41584747330, 127.47179660680], [0.305, 5.34154702988, 131.40394986990], [0.298, 2.28594631393, 635.96513305090], [0.372, 1.03723911390, 313.21047591890], [0.338, 0.69100012338, 1361.54670584420], [0.325, 1.78816356937, 1148.24761040620], [0.322, 1.18628805010, 721.64941953020], [0.271, 2.45663156460, 415.55249061210], [0.251, 3.12046701975, 1382.88734684660], [0.254, 3.00353256829, 618.55664531160], [0.295, 0.35280179538, 2730.20695868920], [0.242, 1.52154324392, 70.84944530420], [0.296, 0.89576757167, 2104.53676637680], [0.264, 3.00987438634, 661.23792731640], [0.267, 0.31623829657, 1677.93857550080], [0.270, 2.56774718753, 643.07868005170], [0.261, 1.55058302472, 1457.52593306200], [0.246, 2.29214585472, 867.42347575360], [0.269, 3.18157515051, 750.10360753340], [0.272, 1.12208982319, 1788.14489672020], [0.256, 0.37673546414, 1279.79457262800], [0.206, 1.81129778306, 497.44763618020], [0.251, 0.61933213502, 2413.81508903260], [0.237, 3.35941544147, 436.89313161450], [0.247, 0.10102936687, 99.91138048090], [0.247, 0.93125798111, 52.69019803950], [0.221, 2.07880035795, 824.74219374880], [0.197, 6.16682223437, 1258.45393162560], [0.229, 5.57917534840, 2943.50605412720], [0.227, 0.43324651601, 2737.32050569000], [0.203, 4.12623986247, 337.73251065900], [0.214, 3.57607524509, 934.94851496820], [0.212, 1.25688162158, 1773.91780271860], [0.215, 0.88867647880, 1038.04128918680], [0.244, 5.51572084570, 231.45834270270], [0.181, 2.13821830481, 416.30325013750], [0.210, 4.19139167658, 2221.85663459700], [0.178, 2.91685344537, 74.78159856730], [0.201, 0.46214583002, 2854.64037391020], [0.236, 4.64388694899, 1905.46476494040], [0.199, 1.54991619669, 1471.75302706360], [0.199, 0.70725247497, 2420.92863603340], [0.162, 2.51488345020, 430.53034413910], [0.160, 1.23508694599, 1596.18644228460], [0.175, 4.14605894816, 2090.30967237520], [0.152, 0.05796022559, 32.24332891440], [0.176, 1.29002070623, 490.33408917940], [0.154, 3.60622857548, 650.94298657790], [0.185, 4.74969742128, 319.57326339430], [0.154, 1.54587199996, 1464.63948006280], [0.108, 4.25125786500, 145.63104387150], [0.106, 1.04047809351, 1162.47470440780], [0.114, 2.64055737100, 362.86229257260], [0.093, 3.36746275886, 483.22054217860], [0.091, 2.05796248692, 210.85141488320], [0.091, 4.53336314765, 241.75328344120], [0.072, 3.74361312157, 1485.98012106520], [0.076, 3.33892447677, 195.89060769870], ], # L4 [ [1661.894, 3.99826248978, 7.11354700080], [257.107, 2.98436499013, 220.41264243880], [236.344, 3.90241428075, 14.22709400160], [149.418, 2.74110824208, 213.29909543800], [109.598, 1.51515739251, 206.18554843720], [113.953, 3.14159265359, 0.00000000000], [68.390, 1.72120953337, 426.59819087600], [37.699, 1.23795458356, 199.07200143640], [40.060, 2.04644897412, 433.71173787680], [31.219, 3.01094184090, 227.52618943960], [15.111, 0.82897064529, 639.89728631400], [9.444, 3.71485300868, 21.34064100240], [5.690, 2.41995290633, 419.48464387520], [4.470, 1.45120818748, 95.97922721780], [5.608, 1.15607095740, 647.01083331480], [4.463, 2.11783225176, 440.82528487760], [3.229, 4.09278077834, 110.20632121940], [2.871, 2.77203153866, 412.37109687440], [2.796, 3.00730249564, 88.86568021700], [2.638, 0.00255721254, 853.19638175200], [2.574, 0.39246854091, 103.09277421860], [1.862, 5.07955457727, 309.27832265580], [2.225, 3.77689198137, 117.31986822020], [1.769, 5.19176876406, 302.16477565500], [1.921, 2.82884328662, 234.63973644040], [1.805, 2.23816036743, 216.48048917570], [1.211, 1.54685246534, 191.95845443560], [0.765, 3.44501766503, 323.50541665740], [0.763, 4.83197222448, 210.11770170030], [0.613, 4.19052656353, 515.46387109300], [0.648, 2.28591710303, 209.36694217490], [0.616, 4.03194472161, 522.57741809380], [0.630, 2.37952532019, 632.78373931320], [0.639, 0.29772678242, 860.30992875280], [0.559, 2.17110060530, 124.43341522100], [0.442, 2.23500083592, 447.93883187840], [0.407, 5.44515970990, 1066.49547719000], [0.469, 1.26889429317, 654.12438031560], [0.488, 3.20329778617, 405.25754987360], [0.415, 3.12435410343, 330.61896365820], [0.442, 3.38933498625, 81.75213321620], [0.332, 4.12464206608, 838.96928775040], [0.320, 3.18332026736, 529.69096509460], [0.312, 1.40962796637, 429.77958461370], [0.291, 3.18885372262, 1464.63948006280], [0.333, 2.94355912397, 728.76296653100], [0.235, 3.67049647573, 1148.24761040620], [0.286, 2.57895004576, 1045.15483618760], [0.223, 3.57980034401, 1155.36115740700], [0.261, 2.04564143519, 1677.93857550080], [0.218, 2.61967125327, 536.80451209540], [0.262, 2.48322150677, 625.67019231240], [0.191, 4.39064160974, 1574.84580128220], [0.176, 1.26161895188, 422.66603761290], [0.190, 2.32693171200, 223.59403617650], [0.185, 1.08713469614, 742.99006053260], [0.168, 0.69946458053, 824.74219374880], [0.177, 5.02663339078, 203.00415469950], [0.218, 0.40426546037, 867.42347575360], [0.178, 3.67593243311, 831.85574074960], [0.175, 5.75326979098, 1073.60902419080], [0.156, 3.02120117572, 1781.03134971940], [0.148, 2.28313808274, 295.05122865420], [0.150, 3.48436135302, 956.28915597060], [0.152, 1.91404443241, 942.06206196900], [0.146, 6.16519696640, 316.39186965660], [0.096, 2.93247663741, 224.34479570190], [0.088, 4.48383632427, 423.41679713830], ], # L5 [ [123.615, 2.25923345732, 7.11354700080], [34.190, 2.16250652689, 14.22709400160], [27.546, 1.19868150215, 220.41264243880], [5.818, 1.21584270184, 227.52618943960], [5.318, 0.23550400093, 433.71173787680], [3.677, 6.22669694355, 426.59819087600], [3.057, 2.97372046322, 199.07200143640], [2.861, 4.28710932685, 206.18554843720], [1.617, 6.25265362286, 213.29909543800], [1.279, 5.27612561266, 639.89728631400], [0.932, 5.56741549127, 647.01083331480], [0.756, 6.17716234487, 191.95845443560], [0.760, 0.69475544472, 302.16477565500], [1.038, 0.23516951637, 440.82528487760], [1.007, 3.14159265359, 0.00000000000], [0.549, 4.87733288264, 88.86568021700], [0.504, 4.77955496203, 419.48464387520], [0.346, 4.31847547394, 853.19638175200], [0.392, 5.69922389094, 654.12438031560], [0.242, 2.05052677361, 323.50541665740], [0.266, 1.11384528244, 234.63973644040], [0.199, 0.88505901097, 309.27832265580], [0.258, 5.10074489186, 95.97922721780], [0.166, 2.40063312194, 515.46387109300], [0.155, 4.70433216164, 860.30992875280], [0.089, 1.36371070380, 412.37109687440], [0.102, 0.49450039082, 117.31986822020], ], ] """This table contains Saturn's periodic terms (all of them) from the planetary theory VSOP87 for the heliocentric longitude at the equinox of date (taken from the 'D' solution). In Meeus' book a shortened version can be found in pages 435-440.""" VSOP87_B = [ # B0 [ [4330678.040, 3.60284428399, 213.29909543800], [240348.303, 2.85238489390, 426.59819087600], [84745.939, 0.00000000000, 0.00000000000], [30863.357, 3.48441504465, 220.41264243880], [34116.063, 0.57297307844, 206.18554843720], [14734.070, 2.11846597870, 639.89728631400], [9916.668, 5.79003189405, 419.48464387520], [6993.564, 4.73604689179, 7.11354700080], [4807.587, 5.43305315602, 316.39186965660], [4788.392, 4.96512927420, 110.20632121940], [3432.125, 2.73255752123, 433.71173787680], [1506.129, 6.01304536144, 103.09277421860], [1060.298, 5.63099292414, 529.69096509460], [969.071, 5.20434966103, 632.78373931320], [942.050, 1.39646678088, 853.19638175200], [707.645, 3.80302329547, 323.50541665740], [552.313, 5.13149109045, 202.25339517410], [399.675, 3.35891413961, 227.52618943960], [316.063, 1.99716764199, 647.01083331480], [319.380, 3.62571550980, 209.36694217490], [284.494, 4.88648481625, 224.34479570190], [314.225, 0.46510272410, 217.23124870110], [236.442, 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2.28640485589, 1898.35121793960], [0.168, 4.63122081226, 1891.23767093880], [0.129, 0.05327999225, 211.81462272970], [0.134, 3.49720535944, 488.84961647110], [0.117, 3.43819501459, 436.15941843160], [0.125, 5.87007326241, 963.40270297140], [0.120, 0.70795300239, 81.75213321620], [0.125, 4.50999471095, 2317.83586181480], [0.116, 6.11600926571, 558.00214074590], [0.117, 4.78666549046, 601.76425067620], [0.108, 0.45464469172, 1802.37199072180], [0.111, 1.44669239244, 2531.13495725280], [0.109, 6.14289264597, 151.04766984290], [0.113, 4.05600865495, 1286.90811962880], [0.129, 5.13936946160, 849.26422848890], [0.127, 3.88189432056, 98.89998852460], [0.133, 2.38290634070, 2111.65031337760], [0.122, 4.40757611742, 778.41478318470], [0.095, 0.07909774752, 213.41097002260], [0.095, 1.66925524906, 213.18722085340], [0.103, 1.88058957173, 99.16062095550], [0.119, 3.62785705509, 248.72381809010], [0.090, 4.63029999228, 228.27694896500], [0.092, 5.48700119144, 767.36908292080], [0.089, 4.61331934339, 431.26405732200], [0.099, 3.60670326134, 776.93031047640], [0.085, 4.93878023673, 2.44768055480], [0.089, 6.24541644164, 661.23792731640], [0.085, 0.45896349060, 1382.88734684660], [0.088, 3.81144552178, 1788.14489672020], [0.103, 3.20558404998, 312.19908396260], [0.080, 2.28889729136, 213.08664711690], [0.080, 5.74264101239, 213.51154375910], [0.082, 3.23546757052, 198.32124191100], [0.078, 6.03841191050, 835.03713448730], [0.080, 0.22601918692, 427.56139872250], [0.072, 2.05164614795, 2634.22773147140], [0.091, 5.97938003596, 556.51766803760], [0.087, 2.71469794199, 617.80588578620], ], # B2 [ [20629.977, 0.50482422817, 213.29909543800], [3719.555, 3.99833475829, 206.18554843720], [1627.158, 6.18189939500, 220.41264243880], [1346.067, 0.00000000000, 0.00000000000], [705.842, 3.03914308836, 419.48464387520], [365.042, 5.09928680706, 426.59819087600], [329.632, 5.27899210039, 433.71173787680], [219.335, 3.82841533795, 639.89728631400], [139.393, 1.04272623499, 7.11354700080], [103.980, 6.15730992966, 227.52618943960], [92.961, 1.97994412845, 316.39186965660], [71.242, 4.14754353431, 199.07200143640], [51.927, 2.88364833898, 632.78373931320], [48.961, 4.43390206741, 647.01083331480], [41.373, 3.15927770079, 853.19638175200], [28.602, 4.52978327558, 210.11770170030], [23.969, 1.11595912146, 14.22709400160], [20.511, 4.35095844197, 217.23124870110], [19.532, 5.30779711223, 440.82528487760], [18.263, 0.85391476786, 110.20632121940], [15.742, 4.25767226302, 103.09277421860], [16.840, 5.68112084135, 216.48048917570], [13.613, 2.99904334066, 412.37109687440], [11.567, 2.52679928410, 529.69096509460], [7.963, 3.31512423920, 202.25339517410], [6.599, 0.28766025146, 323.50541665740], [6.312, 1.16121321336, 117.31986822020], [5.891, 3.58260177246, 309.27832265580], [6.648, 5.55714129949, 209.36694217490], [5.590, 2.47783944511, 1066.49547719000], [6.192, 3.61231886519, 860.30992875280], [4.231, 3.02212363572, 846.08283475120], [3.612, 4.79935735435, 625.67019231240], [3.398, 3.76732731354, 423.41679713830], [3.387, 6.04222745633, 234.63973644040], [2.578, 5.63610668746, 735.87651353180], [2.831, 4.81642822334, 429.77958461370], [2.817, 4.47516563908, 654.12438031560], [2.573, 0.22467245054, 522.57741809380], [2.610, 3.29126967191, 95.97922721780], [2.419, 0.02986335489, 415.55249061210], [2.112, 4.55964179603, 422.66603761290], [2.304, 6.25081073546, 330.61896365820], [1.758, 5.53430456858, 536.80451209540], [1.814, 5.05675881426, 277.03499374140], [1.550, 5.60375604692, 223.59403617650], [1.457, 4.47767649852, 430.53034413910], [1.607, 5.53599550100, 224.34479570190], [1.172, 4.71017775994, 203.00415469950], [1.231, 0.25115931880, 3.93215326310], [1.105, 1.01595427676, 21.34064100240], [0.868, 4.84623483952, 949.17560896980], [0.939, 1.35429452093, 742.99006053260], [0.693, 6.03599130692, 124.43341522100], [0.712, 4.45550701473, 191.95845443560], [0.690, 5.44243765037, 437.64389113990], [0.810, 0.46198177342, 515.46387109300], [0.694, 5.23748122403, 447.93883187840], [0.604, 2.95749705544, 88.86568021700], [0.669, 0.08457977809, 215.74677599280], [0.579, 0.65329445948, 3.18139373770], [0.712, 6.05964117622, 11.04570026390], [0.698, 2.91371419321, 1073.60902419080], [0.526, 2.24947851818, 1059.38193018920], [0.511, 2.86838724347, 408.43894361130], [0.589, 5.79268515755, 63.73589830340], [0.519, 1.76641574095, 1279.79457262800], [0.503, 5.73762297081, 728.76296653100], [0.482, 4.68234512154, 838.96928775040], [0.494, 4.04363805503, 490.33408917940], [0.458, 1.17998315936, 210.85141488320], [0.380, 5.28045750432, 1052.26838318840], [0.404, 4.58953258519, 302.16477565500], [0.377, 5.20131800999, 74.78159856730], [0.328, 0.11893501088, 956.28915597060], [0.290, 3.99300398632, 1162.47470440780], [0.262, 2.04320741578, 1471.75302706360], [0.259, 3.76206113036, 635.96513305090], [0.254, 0.16694559092, 195.13984817330], [0.309, 5.44921175960, 543.91805909620], [0.237, 1.27761853769, 231.45834270270], [0.288, 1.32449995239, 203.73786788240], [0.229, 4.19748765966, 1265.56747862640], [0.238, 4.02925601887, 643.07868005170], [0.238, 0.49997895983, 10.29494073850], [0.257, 3.69107889837, 867.42347575360], [0.191, 0.17807919948, 628.85158605010], [0.246, 5.62469599682, 351.81659230870], [0.183, 3.38184740572, 636.71589257630], [0.172, 3.83173494030, 1581.95934828300], [0.220, 1.03443668151, 483.22054217860], [0.217, 4.65210162713, 750.10360753340], [0.143, 2.31969979791, 18.15924726470], [0.137, 5.50046852846, 1169.58825140860], [0.120, 3.70151294359, 416.30325013750], [0.136, 3.38453909352, 1155.36115740700], [0.149, 0.85459831932, 1375.77379984580], [0.150, 5.71949902293, 618.55664531160], [0.125, 4.82446274394, 436.89313161450], [0.120, 3.26968058035, 1478.86657406440], [0.131, 0.11496484259, 1898.35121793960], [0.099, 4.57241894541, 643.82943957710], [0.095, 4.92115458463, 650.94298657790], [0.090, 2.09300085806, 621.73803904930], [0.111, 0.11975665259, 831.85574074960], [0.089, 2.54351587616, 85.82729883120], [0.080, 5.09103451442, 340.77089204480], [0.078, 3.17395501851, 497.44763618020], [0.085, 0.18997660997, 1258.45393162560], [0.081, 1.16732337173, 217.96496188400], [0.072, 5.47328223678, 337.73251065900], ], # B3 [ [666.252, 1.99006340181, 213.29909543800], [632.350, 5.69778316807, 206.18554843720], [398.051, 0.00000000000, 0.00000000000], [187.838, 4.33779804809, 220.41264243880], [91.884, 4.84104208217, 419.48464387520], [42.369, 2.38073239056, 426.59819087600], [51.548, 3.42149490328, 433.71173787680], [25.661, 4.40167213109, 227.52618943960], [20.551, 5.85313509872, 199.07200143640], [18.081, 1.99321433229, 639.89728631400], [10.874, 5.37344546547, 7.11354700080], [9.590, 2.54901825866, 647.01083331480], [7.085, 3.45518372721, 316.39186965660], [6.002, 4.80055225135, 632.78373931320], [5.778, 0.01680378777, 210.11770170030], [4.881, 5.63719730884, 14.22709400160], [4.501, 1.22424419010, 853.19638175200], [5.542, 3.51756747774, 440.82528487760], [3.548, 4.71299370890, 412.37109687440], [2.851, 0.62679207578, 103.09277421860], [2.173, 3.71982274459, 216.48048917570], [1.991, 6.10867071657, 217.23124870110], [1.435, 1.69177141453, 860.30992875280], [1.217, 4.30778838827, 234.63973644040], [1.157, 5.75027789902, 309.27832265580], [0.795, 5.69026441157, 117.31986822020], [0.733, 0.59842720676, 1066.49547719000], [0.713, 0.21700311697, 625.67019231240], [0.773, 5.48361981990, 202.25339517410], [0.897, 2.65577866867, 654.12438031560], [0.509, 2.86079833766, 429.77958461370], [0.462, 4.17742567173, 529.69096509460], [0.390, 6.11288036049, 191.95845443560], [0.505, 4.51905764563, 323.50541665740], [0.379, 3.74436004151, 223.59403617650], [0.332, 5.49370890570, 21.34064100240], [0.377, 5.25624813434, 95.97922721780], [0.384, 4.48187414769, 330.61896365820], [0.367, 5.03190929680, 846.08283475120], [0.281, 1.14133888637, 735.87651353180], [0.245, 5.81618253250, 423.41679713830], [0.241, 1.70335120180, 522.57741809380], [0.258, 3.69110118716, 447.93883187840], [0.231, 4.15697626494, 110.20632121940], [0.305, 5.97746884029, 302.16477565500], [0.284, 0.66224572127, 203.00415469950], [0.204, 1.54683820621, 209.36694217490], [0.194, 4.21193801453, 124.43341522100], [0.145, 4.79689259614, 88.86568021700], [0.151, 3.82010884134, 536.80451209540], [0.100, 0.03596545368, 949.17560896980], [0.097, 0.91303450276, 1073.60902419080], [0.110, 2.21197473966, 515.46387109300], [0.084, 2.53842533109, 422.66603761290], [0.085, 5.11102520704, 3.93215326310], [0.077, 6.04074586787, 838.96928775040], [0.085, 1.18898817378, 728.76296653100], [0.084, 4.10158366806, 224.34479570190], ], # B4 [ [80.384, 1.11918414679, 206.18554843720], [31.660, 3.12218745098, 213.29909543800], [17.143, 2.48073200414, 220.41264243880], [11.844, 3.14159265359, 0.00000000000], [9.005, 0.38441424927, 419.48464387520], [6.164, 1.56186379537, 433.71173787680], [4.660, 1.28235639570, 199.07200143640], [4.775, 2.63498295487, 227.52618943960], [1.487, 1.43096671616, 426.59819087600], [1.424, 0.66988083613, 647.01083331480], [1.075, 6.18092274059, 639.89728631400], [1.145, 1.72041928134, 440.82528487760], [0.682, 3.84841098180, 14.22709400160], [0.655, 3.49486258327, 7.11354700080], [0.456, 0.47338193402, 632.78373931320], [0.509, 0.31432285584, 412.37109687440], [0.343, 5.86413875355, 853.19638175200], [0.270, 2.50125594913, 234.63973644040], [0.197, 5.39156324804, 316.39186965660], [0.236, 2.11084590211, 210.11770170030], [0.172, 6.09682874401, 860.30992875280], [0.159, 5.95049154821, 216.48048917570], [0.100, 1.98534903594, 625.67019231240], [0.112, 0.85526419268, 654.12438031560], [0.115, 5.03884718594, 117.31986822020], [0.115, 0.44589613974, 110.20632121940], ], # B5 [ [7.895, 2.81927558645, 206.18554843720], [1.014, 0.51187210270, 220.41264243880], [0.772, 2.99484124049, 199.07200143640], [0.967, 3.14159265359, 0.00000000000], [0.583, 5.96456944075, 433.71173787680], [0.588, 0.78008666397, 227.52618943960], [0.445, 2.38630799074, 419.48464387520], [0.098, 5.10622131539, 647.01083331480], [0.091, 5.81659714144, 7.11354700080], [0.088, 6.17828532308, 440.82528487760], [0.089, 0.58396864530, 213.29909543800], ], ] """This table contains Saturn's periodic terms (all of them) from the planetary theory VSOP87 for the heliocentric latitude at the equinox of date (taken from the 'D' solution). In Meeus' book a shortened version can be found in pages 440-442.""" VSOP87_R = [ # R0 [ [955758135.801, 0.00000000000, 0.00000000000], [52921382.465, 2.39226219733, 213.29909543800], [1873679.934, 5.23549605091, 206.18554843720], [1464663.959, 1.64763045468, 426.59819087600], [821891.059, 5.93520025371, 316.39186965660], [547506.899, 5.01532628454, 103.09277421860], [371684.449, 2.27114833428, 220.41264243880], [361778.433, 3.13904303264, 7.11354700080], [140617.548, 5.70406652991, 632.78373931320], [108974.737, 3.29313595577, 110.20632121940], [69007.015, 5.94099622447, 419.48464387520], [61053.350, 0.94037761156, 639.89728631400], [48913.044, 1.55733388472, 202.25339517410], [34143.794, 0.19518550682, 277.03499374140], [32401.718, 5.47084606947, 949.17560896980], [20936.573, 0.46349163993, 735.87651353180], [20839.118, 1.52102590640, 433.71173787680], [20746.678, 5.33255667599, 199.07200143640], [15298.457, 3.05943652881, 529.69096509460], [14296.479, 2.60433537909, 323.50541665740], [11993.314, 5.98051421881, 846.08283475120], [11380.261, 1.73105746566, 522.57741809380], [12884.128, 1.64892310393, 138.51749687070], [7752.769, 5.85191318903, 95.97922721780], [9796.061, 5.20475863996, 1265.56747862640], [6465.967, 0.17733160145, 1052.26838318840], [6770.621, 3.00433479284, 14.22709400160], [5850.443, 1.45519636076, 415.55249061210], [5307.481, 0.59737534050, 63.73589830340], [4695.746, 2.14919036956, 227.52618943960], [4043.988, 1.64010323863, 209.36694217490], [3688.132, 0.78016133170, 412.37109687440], [3376.457, 3.69528478828, 224.34479570190], [2885.348, 1.38764077631, 838.96928775040], [2976.033, 5.68467931117, 210.11770170030], [3419.551, 4.94549148887, 1581.95934828300], [3460.943, 1.85088802878, 175.16605980020], [3400.616, 0.55386747515, 350.33211960040], [2507.630, 3.53851863255, 742.99006053260], [2448.325, 6.18412386316, 1368.66025284500], [2406.138, 2.96559220267, 117.31986822020], [2881.181, 0.17960757891, 853.19638175200], [2173.959, 0.01508587396, 340.77089204480], [2024.483, 5.05411271271, 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5863.59120611620], [1.315, 4.73430848989, 241.75328344120], [1.516, 4.99488503706, 1162.47470440780], [1.310, 1.98714265058, 563.63121503840], [1.286, 2.12891372062, 251.43213107580], [1.271, 5.70165238307, 2.92076130680], [1.312, 1.68811514551, 2207.62954059540], [1.259, 0.35924965717, 334.55111692130], [1.252, 2.14513440216, 1055.44977692610], [1.401, 6.13250261735, 1802.37199072180], [1.343, 5.79995727295, 9793.80090233580], [1.228, 3.29059284057, 661.23792731640], [1.202, 2.88792018909, 2413.81508903260], [1.286, 5.72360160371, 298.23262239190], [1.357, 0.93175963411, 217.49188113200], [1.356, 2.28121627817, 601.76425067620], [1.190, 1.94993809928, 501.37978944330], [1.304, 0.37337923280, 3473.19701922180], [1.350, 2.87235622320, 142.44965013380], [1.349, 3.21102203937, 175.16605980020], [1.312, 3.70149813509, 2111.65031337760], [1.129, 1.08860603834, 842.15068148810], [1.237, 0.08698781252, 526.50957135690], [1.217, 3.89835349840, 209.10630974400], [1.467, 1.16228775027, 621.73803904930], [1.044, 0.30512759901, 436.15941843160], [1.140, 5.33720637097, 114.13847448250], [1.295, 4.70261675421, 9786.68735533500], [1.037, 4.07846687083, 156.67674413540], [1.391, 4.73554028436, 398.14400287280], [1.167, 5.68899703631, 479.28838891550], [1.035, 5.34279429465, 327.43756992050], [0.997, 1.19323192891, 710.74673161820], [1.193, 5.17722376816, 98.89998852460], [1.165, 4.58588490135, 732.69511979410], [1.161, 4.90854984994, 10206.17199921020], [1.144, 0.50394784140, 3906.90875709860], [1.182, 3.69482624364, 2854.64037391020], [0.970, 2.89031410383, 1987.21689815660], [1.039, 0.48694895443, 525.49817940060], [1.079, 3.61750956217, 2097.42321937600], [1.148, 3.31015591733, 5856.47765911540], [1.241, 4.31971543677, 230.70758317730], [0.910, 4.59825926062, 380.12776796000], [0.907, 1.34912454077, 685.47393735270], [1.166, 1.61085609717, 5849.36411211460], [0.882, 6.12045540405, 519.39602435610], [0.963, 4.96065454054, 699.70103135430], [1.062, 5.13323858077, 2751.54759969160], [0.865, 6.12821112133, 245.54242435240], [1.100, 2.18435744407, 1699.27921650320], [0.822, 5.55083534581, 739.05790726950], [0.926, 2.01158276144, 417.03696332040], [0.813, 5.18401872205, 214.78356814630], [1.033, 5.48677848094, 3995.77443731560], [0.872, 3.02363724703, 306.09692891810], [0.796, 0.44343664540, 486.40193591630], [0.878, 1.82164034386, 135.33610313300], [0.791, 2.14989417962, 2620.00063746980], [0.881, 2.39697554334, 289.56516671360], [0.782, 4.50471317138, 980.66817835880], [0.783, 1.14229319753, 540.73666535850], [0.831, 0.69937251013, 421.93232443000], [0.770, 2.40292326155, 576.16138801060], [0.950, 5.97621460162, 196.62432088160], [0.814, 4.19303098086, 831.10498122420], [0.969, 4.78071024754, 326.68681039510], [0.760, 0.44860533530, 425.63498302950], [0.907, 0.94781730418, 525.75881183150], [0.788, 0.14287187051, 916.93228005540], [0.801, 1.86383119100, 3039.48528134500], [0.801, 0.46947170994, 3466.08347222100], [0.747, 6.05374861925, 211.81462272970], [0.968, 3.02618272726, 2634.22773147140], [0.739, 2.27110740297, 2303.60876781320], [0.750, 5.48554383902, 173.94221952280], [1.024, 1.91564925560, 229.97386999440], [0.816, 4.98990432666, 4209.07353275360], [0.728, 1.30997967935, 511.53171782990], [0.716, 3.74192651696, 3053.71237534660], [0.727, 0.39191881243, 1493.09366806600], [0.717, 2.68899513085, 228.27694896500], [0.739, 2.12749199443, 1176.70179840940], [0.805, 0.07187193910, 556.51766803760], [0.835, 3.48287855700, 84.93352695390], [0.790, 0.48073040004, 4017.11507831800], [0.725, 1.96643065215, 220.46082654860], [0.683, 2.68825142163, 151.04766984290], [0.739, 3.33688408107, 953.10776223290], [0.745, 6.22304530635, 1269.49963188950], ], # R3 [ [20315.005, 3.02186626038, 213.29909543800], [8923.581, 3.19144205755, 220.41264243880], [6908.677, 4.35174889353, 206.18554843720], [4087.129, 4.22406927376, 7.11354700080], [3879.041, 2.01056445995, 426.59819087600], [1070.788, 4.20360341236, 199.07200143640], [907.332, 2.28344368029, 433.71173787680], [606.121, 3.17458570534, 227.52618943960], [596.639, 4.13455753351, 14.22709400160], [483.181, 1.17345973258, 639.89728631400], [393.174, 0.00000000000, 0.00000000000], [229.472, 4.69838526383, 419.48464387520], [188.250, 4.59003889007, 110.20632121940], [149.508, 3.20199444400, 103.09277421860], [121.442, 3.76831374104, 323.50541665740], [101.215, 5.81884137755, 412.37109687440], [102.146, 4.70974422803, 95.97922721780], [93.078, 1.43531270909, 647.01083331480], [72.601, 4.15395598507, 117.31986822020], [84.347, 2.63462379693, 216.48048917570], [62.198, 2.31239345505, 440.82528487760], [45.145, 4.37317047297, 191.95845443560], [49.536, 2.38854232908, 209.36694217490], [54.829, 0.30526468471, 853.19638175200], [40.498, 1.83836569765, 302.16477565500], [38.089, 5.94455115525, 88.86568021700], [32.243, 4.01146349387, 21.34064100240], [40.671, 0.68845183210, 522.57741809380], [28.209, 5.77193013961, 210.11770170030], [24.976, 3.06249709014, 234.63973644040], [20.824, 4.92570695678, 625.67019231240], [25.070, 0.73137425284, 515.46387109300], [17.485, 5.73135068691, 728.76296653100], [18.009, 1.45593152612, 309.27832265580], [16.927, 3.52771580455, 3.18139373770], [13.437, 3.36479898106, 330.61896365820], [11.090, 3.37212682914, 224.34479570190], [11.082, 3.41719974793, 956.28915597060], [9.978, 1.58791582772, 202.25339517410], [11.551, 5.99093726182, 735.87651353180], [10.500, 6.06911092266, 405.25754987360], [9.144, 2.93557421591, 124.43341522100], [8.737, 4.65432480769, 632.78373931320], [10.023, 0.58247011625, 860.30992875280], [7.482, 4.50669216436, 942.06206196900], [10.091, 0.28268774007, 838.96928775040], [9.243, 2.57034547708, 223.59403617650], [8.652, 1.75808100881, 429.77958461370], [7.564, 1.45635107202, 654.12438031560], [7.058, 5.47394786065, 1045.15483618760], [6.970, 1.51811695028, 422.66603761290], [8.067, 4.48457709292, 742.99006053260], [6.817, 4.83084424818, 316.39186965660], [7.693, 0.43769724671, 831.85574074960], [7.934, 4.20112367712, 195.13984817330], [6.119, 2.33960392135, 269.92144674060], [5.589, 1.14518720694, 284.14854074220], [5.564, 4.18123189068, 529.69096509460], [5.034, 2.12020038657, 295.05122865420], [6.556, 3.42459866876, 10.29494073850], [5.544, 2.46823271699, 536.80451209540], [6.189, 6.01433827520, 1066.49547719000], [5.649, 0.82784598388, 217.23124870110], [4.264, 3.23245736673, 1272.68102562720], [4.450, 0.92477808590, 203.00415469950], [3.268, 4.32777516976, 1258.45393162560], [3.655, 0.05832123987, 81.75213321620], [3.951, 0.11124996745, 1155.36115740700], [3.773, 6.01157059552, 1052.26838318840], [2.915, 5.64342950039, 3.93215326310], [3.019, 2.19411778004, 447.93883187840], [2.977, 1.89387342550, 149.56319713460], [3.146, 0.19215180096, 1148.24761040620], [2.763, 0.92363342001, 508.35032409220], [2.790, 4.97199778427, 1677.93857550080], [2.608, 2.99591016813, 1589.07289528380], [2.881, 5.40535671721, 1361.54670584420], [2.337, 1.30362271569, 184.84490743480], [2.536, 3.71412120849, 408.43894361130], [2.450, 3.22118361135, 319.57326339430], [2.585, 2.31346415454, 543.91805909620], [2.324, 5.87500715503, 721.64941953020], [1.990, 0.51565577383, 416.30325013750], [2.490, 4.24017800021, 1059.38193018920], [1.935, 2.41463084855, 337.73251065900], [1.886, 0.53809070779, 635.96513305090], [1.893, 5.62352727352, 11.04570026390], [2.389, 5.73399981234, 313.21047591890], [1.900, 2.41000566465, 131.54696222180], [1.743, 4.57646237847, 1994.33044515740], [1.913, 5.17436386408, 2854.64037391020], [1.946, 6.23355845623, 1471.75302706360], [1.963, 6.17814558628, 1464.63948006280], [1.838, 5.59464577559, 1038.04128918680], [1.541, 0.60765337379, 210.85141488320], [1.617, 1.75479346067, 195.89060769870], [1.577, 0.55789908488, 2324.94940881560], [1.492, 0.26624235633, 497.44763618020], [1.659, 2.57526072926, 2090.30967237520], [1.809, 1.82317819973, 436.89313161450], [1.566, 6.15328100324, 490.33408917940], [1.771, 6.11741716855, 1073.60902419080], [1.456, 0.85374460914, 415.55249061210], [1.645, 2.95335775161, 437.64389113990], [1.391, 4.12025028560, 1574.84580128220], [1.585, 5.96841377266, 1781.03134971940], [1.507, 3.84895122542, 1251.34038462480], [1.442, 5.32547705924, 2538.24850425360], [1.805, 1.50973093681, 750.10360753340], [1.462, 3.28599831588, 1884.12412393800], [1.482, 0.99340744053, 643.07868005170], [1.312, 3.79347668996, 1567.73225428140], [1.665, 0.02551523913, 423.41679713830], [1.469, 5.35285153471, 1354.43315884340], [1.352, 0.69945139243, 867.42347575360], [1.124, 1.79624810407, 618.55664531160], [1.126, 4.70052329245, 113.38771495710], [1.122, 3.95537224270, 1891.23767093880], [1.458, 1.50198846753, 430.53034413910], [1.145, 5.13093399117, 25.27279426550], [1.178, 2.97062300389, 241.75328344120], [1.274, 2.29089799814, 2420.92863603340], [1.071, 0.04888943982, 63.73589830340], [1.377, 5.58271514873, 1382.88734684660], [1.145, 3.10797488346, 2200.51599359460], [1.076, 0.79465514815, 127.47179660680], [1.046, 5.85060227045, 215.74677599280], [1.082, 3.72589445510, 131.40394986990], [1.218, 0.47504349592, 824.74219374880], [1.116, 3.78039049056, 1375.77379984580], [0.969, 5.90752273481, 265.98929347750], [1.230, 1.41325962069, 2634.22773147140], [1.070, 4.80334493874, 1987.21689815660], [0.946, 6.25968535931, 2015.67108615980], [1.030, 1.08973644893, 362.86229257260], [1.072, 5.41838042079, 1279.79457262800], [0.880, 1.92224908504, 483.22054217860], [0.878, 2.96591300878, 934.94851496820], [0.879, 2.65659265685, 145.63104387150], [0.872, 6.26261969664, 2.44768055480], [1.082, 4.48298283322, 2214.74308759620], [0.959, 0.74479087918, 16.67477455640], [1.035, 4.05664979327, 231.45834270270], [0.851, 0.09360495322, 628.85158605010], [0.888, 5.98816755324, 2524.02141025200], [0.866, 3.16259265630, 2207.62954059540], [0.843, 1.23731248821, 74.78159856730], [0.809, 2.89742868175, 2008.55753915900], [0.779, 2.28434811609, 1478.86657406440], [0.990, 5.32604038017, 2428.04218303420], [0.795, 2.38178135810, 2228.97018159780], [0.765, 4.70033674940, 1670.82502850000], [1.024, 4.23352869513, 1802.37199072180], [0.831, 5.87457134912, 1368.66025284500], [0.717, 5.92144324994, 1685.05212250160], [0.772, 1.15596098579, 3053.71237534660], [0.691, 3.13193109668, 56.62235130260], ], # R4 [ [1202.050, 1.41499446465, 220.41264243880], [707.796, 1.16153570102, 213.29909543800], [516.121, 6.23973568330, 206.18554843720], [426.664, 2.46924890293, 7.11354700080], [267.736, 0.18659206741, 426.59819087600], [170.171, 5.95926972384, 199.07200143640], [145.113, 1.44211060143, 227.52618943960], [150.339, 0.47970167140, 433.71173787680], [121.033, 2.40527320817, 14.22709400160], [47.332, 5.56857488676, 639.89728631400], [15.745, 2.90112466278, 110.20632121940], [16.668, 0.52920774279, 440.82528487760], [18.954, 5.85626429118, 647.01083331480], [14.074, 1.30343550656, 412.37109687440], [12.708, 2.09349305926, 323.50541665740], [14.724, 0.29905316786, 419.48464387520], [11.133, 2.46304825990, 117.31986822020], [11.320, 0.21785507019, 95.97922721780], [9.233, 2.28127318068, 21.34064100240], [9.246, 1.56496312830, 88.86568021700], [8.970, 0.68301278041, 216.48048917570], [7.674, 3.59367715368, 302.16477565500], [7.823, 4.48688804175, 853.19638175200], [8.360, 1.27239488455, 234.63973644040], [9.552, 3.14159265359, 0.00000000000], [4.834, 2.58836294602, 515.46387109300], [6.059, 5.16774448740, 103.09277421860], [4.410, 0.02211643085, 191.95845443560], [4.364, 1.59622746023, 330.61896365820], [3.676, 3.29899839673, 210.11770170030], [4.364, 5.97349927933, 654.12438031560], [4.447, 4.97415112184, 860.30992875280], [3.220, 2.72684237392, 522.57741809380], [4.005, 1.59858435636, 405.25754987360], [3.099, 0.75235436533, 209.36694217490], [2.464, 1.19167306488, 124.43341522100], [3.088, 1.32258934286, 728.76296653100], [2.220, 3.28087994088, 203.00415469950], [2.127, 6.14648095022, 429.77958461370], [2.110, 0.75462855247, 295.05122865420], [2.020, 3.89394929749, 1066.49547719000], [2.248, 0.49319150178, 447.93883187840], [2.180, 0.72761059998, 625.67019231240], [1.809, 0.09057839517, 942.06206196900], [1.672, 1.39635398184, 224.34479570190], [1.641, 3.02468307550, 184.84490743480], [1.772, 0.81879250825, 223.59403617650], [1.902, 2.00472814984, 831.85574074960], [1.600, 5.41185167676, 824.74219374880], [1.505, 5.95520747253, 422.66603761290], [1.133, 1.11512973946, 838.96928775040], [1.190, 1.89600567803, 956.28915597060], [1.487, 2.11906469507, 529.69096509460], [1.409, 0.72254420236, 536.80451209540], [1.125, 0.89062692183, 721.64941953020], [1.301, 1.64867038984, 17.40848773930], [1.164, 5.96957981840, 195.13984817330], [0.950, 5.36080713290, 316.39186965660], [0.985, 3.05768671768, 1574.84580128220], [1.050, 1.59202481523, 735.87651353180], [0.817, 4.92838813598, 56.62235130260], [0.780, 2.72125404102, 508.35032409220], [0.969, 1.00708261792, 1045.15483618760], [0.716, 1.11042181341, 1169.58825140860], ], # R5 [ [128.612, 5.91282565136, 220.41264243880], [32.273, 0.69256228602, 7.11354700080], [26.698, 5.91428528629, 227.52618943960], [19.923, 0.67370653385, 14.22709400160], [20.223, 4.95136801768, 433.71173787680], [13.537, 1.45669521408, 199.07200143640], [14.097, 2.67074280191, 206.18554843720], [13.364, 4.58826996370, 426.59819087600], [7.257, 4.62966127155, 213.29909543800], [4.876, 3.61448275002, 639.89728631400], [3.136, 4.65661021909, 191.95845443560], [2.917, 0.48665273315, 323.50541665740], [3.759, 4.89624165044, 440.82528487760], [3.303, 4.07190859545, 647.01083331480], [2.883, 3.18003019204, 419.48464387520], [2.338, 3.69553554327, 88.86568021700], [1.950, 5.32729247780, 302.16477565500], [2.052, 3.31663577368, 95.97922721780], [1.591, 2.67009215574, 853.19638175200], [2.028, 0.56025552769, 117.31986822020], [1.560, 0.85608042681, 515.46387109300], [1.678, 0.00000000000, 0.00000000000], [1.102, 5.98011943842, 3.18139373770], [1.285, 5.82563377753, 234.63973644040], [0.896, 5.22791858719, 216.48048917570], [1.141, 0.15741228205, 412.37109687440], [0.798, 0.37452846153, 28.45418800320], [0.837, 5.04769794123, 124.43341522100], ], ] """This table contains Saturn's periodic terms (all of them) from the planetary theory VSOP87 for the radius vector at the equinox of date (taken from the 'D' solution). In Meeus' book a shortened version can be found in pages 442-445.""" ORBITAL_ELEM = [ [50.077444, 1223.5110686, 0.00051908, -0.00000003], # L [9.554909192, -0.000002139, 0.000000004, 0.0], # a [0.05554814, -0.000346641, -0.0000006436, 0.0000000034], # e [2.488879, -0.0037362, -0.00001519, 0.000000087], # i [113.665503, 0.877088, -0.00012176, -0.000002249], # Omega [93.057237, 1.9637613, 0.00083753, 0.000004928] # pie ] """This table contains the parameters to compute Saturn's orbital elements for the mean equinox of date. Based in Table 31.A, page 213""" ORBITAL_ELEM_J2000 = [ [50.077444, 1222.1138488, 0.00021004, -0.000000046], # L [2.488879, 0.0025514, -0.00004906, 0.000000017], # i [113.665503, -0.2566722, -0.00018399, 0.00000048], # Omega [93.057237, 0.5665415, 0.0005285, 0.000004912] # pie ] """This table contains the parameters to compute Saturn's orbital elements for the standard equinox J2000.0. Based on Table 31.B, page 215""" class Saturn(object): """ Class Saturn models that planet. """ @staticmethod def geometric_heliocentric_position(epoch, tofk5=True): """This method computes the geometric heliocentric position of planet Saturn for a given epoch, using the VSOP87 theory. :param epoch: Epoch to compute Saturn position, as an Epoch object :type epoch: :py:class:`Epoch` :param tofk5: Whether or not the small correction to convert to the FK5 system will be applied or not :type tofk5: bool :returns: A tuple with the heliocentric longitude and latitude (as :py:class:`Angle` objects), and the radius vector (as a float, in astronomical units), in that order :rtype: tuple :raises: TypeError if input values are of wrong type. >>> epoch = Epoch(2018, 10, 27.0) >>> l, b, r = Saturn.geometric_heliocentric_position(epoch) >>> print(round(l.to_positive(), 4)) 279.5108 >>> print(round(b, 4)) 0.6141 >>> print(round(r, 5)) 10.06266 """ return geometric_vsop_pos(epoch, VSOP87_L, VSOP87_B, VSOP87_R, tofk5) @staticmethod def apparent_heliocentric_position(epoch): """This method computes the apparent heliocentric position of planet Saturn for a given epoch, using the VSOP87 theory. :param epoch: Epoch to compute Saturn position, as an Epoch object :type epoch: :py:class:`Epoch` :returns: A tuple with the heliocentric longitude and latitude (as :py:class:`Angle` objects), and the radius vector (as a float, in astronomical units), in that order :rtype: tuple :raises: TypeError if input values are of wrong type. """ return apparent_vsop_pos(epoch, VSOP87_L, VSOP87_B, VSOP87_R) @staticmethod def orbital_elements_mean_equinox(epoch): """This method computes the orbital elements of Saturn for the mean equinox of the date for a given epoch. :param epoch: Epoch to compute orbital elements, as an Epoch object :type epoch: :py:class:`Epoch` :returns: A tuple containing the following six orbital elements: - Mean longitude of the planet (Angle) - Semimajor axis of the orbit (float, astronomical units) - eccentricity of the orbit (float) - inclination on the plane of the ecliptic (Angle) - longitude of the ascending node (Angle) - argument of the perihelion (Angle) :rtype: tuple :raises: TypeError if input values are of wrong type. >>> epoch = Epoch(2065, 6, 24.0) >>> l, a, e, i, ome, arg = Saturn.orbital_elements_mean_equinox(epoch) >>> print(round(l, 6)) 131.196871 >>> print(round(a, 8)) 9.55490779 >>> print(round(e, 7)) 0.0553209 >>> print(round(i, 6)) 2.486426 >>> print(round(ome, 5)) 114.23974 >>> print(round(arg, 6)) -19.896331 """ return orbital_elements(epoch, ORBITAL_ELEM, ORBITAL_ELEM) @staticmethod def orbital_elements_j2000(epoch): """This method computes the orbital elements of Saturn for the standard equinox J2000.0 for a given epoch. :param epoch: Epoch to compute orbital elements, as an Epoch object :type epoch: :py:class:`Epoch` :returns: A tuple containing the following six orbital elements: - Mean longitude of the planet (Angle) - Semimajor axis of the orbit (float, astronomical units) - eccentricity of the orbit (float) - inclination on the plane of the ecliptic (Angle) - longitude of the ascending node (Angle) - argument of the perihelion (Angle) :rtype: tuple :raises: TypeError if input values are of wrong type. >>> epoch = Epoch(2065, 6, 24.0) >>> l, a, e, i, ome, arg = Saturn.orbital_elements_j2000(epoch) >>> print(round(l, 6)) 130.28188 >>> print(round(a, 8)) 9.55490779 >>> print(round(e, 7)) 0.0553209 >>> print(round(i, 6)) 2.490529 >>> print(round(ome, 5)) 113.49736 >>> print(round(arg, 6)) -20.068943 """ return orbital_elements(epoch, ORBITAL_ELEM, ORBITAL_ELEM_J2000) @staticmethod def geocentric_position(epoch): """This method computes the geocentric position of Saturn (right ascension and declination) for the given epoch, as well as the elongation angle. :param epoch: Epoch to compute geocentric position, as an Epoch object :type epoch: :py:class:`Epoch` :returns: A tuple containing the right ascension, the declination and the elongation angle as Angle objects :rtype: tuple :raises: TypeError if input value is of wrong type. >>> epoch = Epoch(1992, 12, 20.0) >>> ra, dec, elon = Saturn.geocentric_position(epoch) >>> print(ra.ra_str(n_dec=1)) 21h 11' 41.8'' >>> print(dec.dms_str(n_dec=1)) -17d 15' 40.8'' >>> print(elon.dms_str(n_dec=1)) 46d 51' 47.7'' """ # First check that input value is of correct types if not isinstance(epoch, Epoch): raise TypeError("Invalid input type") # Compute the heliocentric position of Saturn l, b, r = Saturn.geometric_heliocentric_position(epoch, tofk5=False) # Compute the heliocentric position of the Earth l0, b0, r0 = Earth.geometric_heliocentric_position(epoch, tofk5=False) # Convert to radians lr = l.rad() br = b.rad() l0r = l0.rad() b0r = b0.rad() # Compute first iteration x = r * cos(br) * cos(lr) - r0 * cos(b0r) * cos(l0r) y = r * cos(br) * sin(lr) - r0 * cos(b0r) * sin(l0r) z = r * sin(br) - r0 * sin(b0r) delta = sqrt(x * x + y * y + z * z) tau = 0.0057755183 * delta # Adjust the epoch for light-time epoch -= tau # Compute again Saturn coordinates with this correction l, b, r = Saturn.geometric_heliocentric_position(epoch, tofk5=False) # Compute second iteration lr = l.rad() br = b.rad() x = r * cos(br) * cos(lr) - r0 * cos(b0r) * cos(l0r) y = r * cos(br) * sin(lr) - r0 * cos(b0r) * sin(l0r) z = r * sin(br) - r0 * sin(b0r) # Compute longitude and latitude lamb = atan2(y, x) beta = atan2(z, sqrt(x * x + y * y)) # Now, let's compute the aberration effect t = (epoch - JDE2000) / 36525 e = 0.016708634 + t * (-0.000042037 - t * 0.0000001267) pie = 102.93735 + t * (1.71946 + t * 0.00046) pie = radians(pie) lon = l0 + 180.0 lon = lon.rad() k = 20.49552 # The constant of aberration deltal1 = k * (-cos(lon - lamb) + e * cos(pie - lamb)) / cos(beta) deltab1 = -k * sin(beta) * (sin(lon - lamb) - e * sin(pie - lamb)) deltal1 = Angle(0, 0, deltal1) deltab1 = Angle(0, 0, deltab1) # Correction to FK5 system lamb = Angle(lamb, radians=True) lamb = lamb.to_positive() beta = Angle(beta, radians=True) l_prime = lamb - t * (1.397 + t * 0.00031) deltal2 = Angle(0, 0, -0.09033) a = 0.03916 * (cos(l_prime.rad()) + sin(l_prime.rad())) a = a * tan(b.rad()) deltal2 += Angle(0, 0, a) deltab2 = 0.03916 * (cos(l_prime.rad()) - sin(l_prime.rad())) deltab2 = Angle(0, 0, deltab2) # Apply the corrections lamb = lamb + deltal1 + deltal2 beta = beta + deltab1 + deltab2 # Correction for nutation dpsi = nutation_longitude(epoch) lamb += dpsi e = true_obliquity(epoch) ra, dec = ecliptical2equatorial(lamb, beta, e) # Let's compute the elongation angle lons, lats, rs = Sun.apparent_geocentric_position(epoch) lambr = lamb.rad() lsr = lons.rad() betar = beta.rad() elon = acos(cos(betar) * cos(lambr - lsr)) elon = Angle(elon, radians=True) return ra, dec, elon @staticmethod def conjunction(epoch): """This method computes the time of the conjunction closest to the given epoch. :param epoch: Epoch close to the desired conjunction :type epoch: :py:class:`Epoch` :returns: The time when the conjunction happens, as an Epoch :rtype: :py:class:`Epoch` :raises: TypeError if input value is of wrong type. :raises: ValueError if input epoch outside the -2000/4000 range. >>> epoch = Epoch(2125, 6, 1.0) >>> conj = Saturn.conjunction(epoch) >>> y, m, d = conj.get_date() >>> print(y) 2125 >>> print(m) 8 >>> print(round(d, 4)) 26.4035 """ # First check that input value is of correct types if not isinstance(epoch, Epoch): raise TypeError("Invalid input type") # Check that the input epoch is within valid range y = epoch.year() if y < -2000.0 or y > 4000.0: raise ValueError("Epoch outside the -2000/4000 range") # Set some specific constants for Saturn's conjunction a = 2451681.124 b = 378.091904 m0 = 131.6934 m1 = 12.647487 k = round((365.2425 * y + 1721060.0 - a) / b) jde0 = a + k * b m = m0 + k * m1 m = Angle(m).to_positive() m = m.rad() t = (jde0 - 2451545.0) / 36525.0 # Compute auxiliary angles aa = 82.74 + 40.76 * t bb = 29.86 + 1181.36 * t cc = 14.13 + 590.68 * t dd = 220.02 + 1262.87 * t # Convert to radians aa = Angle(aa).rad() bb = Angle(bb).rad() cc = Angle(cc).rad() dd = Angle(dd).rad() corr = (0.0172 + t * (-0.0006 + t * 0.00023) + sin(m) * (-8.5885 + t * (0.0411 + t * 0.0002)) + cos(m) * (-1.147 + t * (0.0352 - t * 0.00011)) + sin(2.0 * m) * (0.3331 + t * (-0.0034 - t * 0.00001)) + cos(2.0 * m) * (0.1145 + t * (-0.0045 + t * 0.00002)) + sin(3.0 * m) * (-0.0169 + t * 0.0002) + cos(3.0 * m) * (-0.0109 + t * 0.0004) + sin(aa) * (0.0 + t * (-0.0337 + t * 0.00018)) + cos(aa) * (-0.851 + t * (0.0044 + t * 0.00068)) + sin(bb) * (0.0 + t * (-0.0064 + t * 0.00004)) + cos(bb) * (0.2397 + t * (-0.0012 - t * 0.00008)) + sin(cc) * (0.0 - t * 0.001) + cos(cc) * (0.1245 + t * 0.0006) + sin(dd) * (0.0 + t * (0.0024 - t * 0.00003)) + cos(dd) * (0.0477 + t * (-0.0005 - t * 0.00006))) to_return = jde0 + corr return Epoch(to_return) @staticmethod def opposition(epoch): """This method computes the time of the opposition closest to the given epoch. :param epoch: Epoch close to the desired opposition :type epoch: :py:class:`Epoch` :returns: The time when the opposition happens, as an Epoch :rtype: :py:class:`Epoch` :raises: TypeError if input value is of wrong type. :raises: ValueError if input epoch outside the -2000/4000 range. >>> epoch = Epoch(-6, 9, 1.0) >>> oppo = Saturn.opposition(epoch) >>> y, m, d = oppo.get_date() >>> print(y) -6 >>> print(m) 9 >>> print(round(d, 4)) 14.3709 """ # First check that input value is of correct types if not isinstance(epoch, Epoch): raise TypeError("Invalid input type") # Check that the input epoch is within valid range y = epoch.year() if y < -2000.0 or y > 4000.0: raise ValueError("Epoch outside the -2000/4000 range") # Set some specific constants for Saturn's opposition a = 2451870.17 b = 378.091904 m0 = 318.0172 m1 = 12.647487 k = round((365.2425 * y + 1721060.0 - a) / b) jde0 = a + k * b m = m0 + k * m1 m = Angle(m).to_positive() m = m.rad() t = (jde0 - 2451545.0) / 36525.0 # Compute an auxiliary angle aa = 82.74 + 40.76 * t bb = 29.86 + 1181.36 * t cc = 14.13 + 590.68 * t dd = 220.02 + 1262.87 * t # Convert to radians aa = Angle(aa).rad() bb = Angle(bb).rad() cc = Angle(cc).rad() dd = Angle(dd).rad() corr = (-0.0209 + t * (0.0006 + t * 0.00023) + sin(m) * (4.5795 + t * (-0.0312 - t * 0.00017)) + cos(m) * (1.1462 + t * (-0.0351 + t * 0.00011)) + sin(2.0 * m) * (0.0985 - t * 0.0015) + cos(2.0 * m) * (0.0733 + t * (-0.0031 + t * 0.00001)) + sin(3.0 * m) * (0.0025 - t * 0.0001) + cos(3.0 * m) * (0.005 - t * 0.0002) + sin(aa) * (0.0 + t * (-0.0337 + t * 0.00018)) + cos(aa) * (-0.851 + t * (0.0044 + t * 0.00068)) + sin(bb) * (0.0 + t * (-0.0064 + t * 0.00004)) + cos(bb) * (0.2397 + t * (-0.0012 - t * 0.00008)) + sin(cc) * (0.0 - t * 0.001) + cos(cc) * (0.1245 + t * 0.0006) + sin(dd) * (0.0 + t * (0.0024 - t * 0.00003)) + cos(dd) * (0.0477 + t * (-0.0005 - t * 0.00006))) to_return = jde0 + corr return Epoch(to_return) @staticmethod def station_longitude_1(epoch): """This method computes the time of the 1st station in longitude (i.e. when the planet is stationary and begins to move westward - retrograde - among the starts) closest to the given epoch. :param epoch: Epoch close to the desired opposition :type epoch: :py:class:`Epoch` :returns: Time when the 1st station in longitude happens, as an Epoch :rtype: :py:class:`Epoch` :raises: TypeError if input value is of wrong type. :raises: ValueError if input epoch outside the -2000/4000 range. >>> epoch = Epoch(2018, 11, 1.0) >>> sta1 = Saturn.station_longitude_1(epoch) >>> y, m, d = sta1.get_date() >>> print(y) 2018 >>> print(m) 4 >>> print(round(d, 4)) 17.9433 """ # First check that input value is of correct types if not isinstance(epoch, Epoch): raise TypeError("Invalid input type") # Check that the input epoch is within valid range y = epoch.year() if y < -2000.0 or y > 4000.0: raise ValueError("Epoch outside the -2000/4000 range") # Set some specific constants for Saturn's opposition a = 2451870.17 b = 378.091904 m0 = 318.0172 m1 = 12.647487 k = round((365.2425 * y + 1721060.0 - a) / b) jde0 = a + k * b m = m0 + k * m1 m = Angle(m).to_positive() m = m.rad() t = (jde0 - 2451545.0) / 36525.0 # Compute an auxiliary angle aa = 82.74 + 40.76 * t bb = 29.86 + 1181.36 * t cc = 14.13 + 590.68 * t dd = 220.02 + 1262.87 * t # Convert to radians aa = Angle(aa).rad() bb = Angle(bb).rad() cc = Angle(cc).rad() dd = Angle(dd).rad() corr = (-68.884 + t * (0.0009 + t * 0.00023) + sin(m) * (5.5452 + t * (-0.0279 - t * 0.0002)) + cos(m) * (3.0727 + t * (-0.043 + t * 0.00007)) + sin(2.0 * m) * (0.1101 + t * (-0.0006 - t * 0.00001)) + cos(2.0 * m) * (0.1654 + t * (-0.0043 + t * 0.00001)) + sin(3.0 * m) * (0.001 + t * 0.0001) + cos(3.0 * m) * (0.0095 - t * 0.0003) + sin(aa) * (0.0 + t * (-0.0337 + t * 0.00018)) + cos(aa) * (-0.851 + t * (0.0044 + t * 0.00068)) + sin(bb) * (0.0 + t * (-0.0064 + t * 0.00004)) + cos(bb) * (0.2397 + t * (-0.0012 - t * 0.00008)) + sin(cc) * (0.0 - t * 0.001) + cos(cc) * (0.1245 + t * 0.0006) + sin(dd) * (0.0 + t * (0.0024 - t * 0.00003)) + cos(dd) * (0.0477 + t * (-0.0005 - t * 0.00006))) to_return = jde0 + corr return Epoch(to_return) @staticmethod def station_longitude_2(epoch): """This method computes the time of the 2nd station in longitude (i.e. when the planet is stationary and begins to move eastward - prograde - among the starts) closest to the given epoch. :param epoch: Epoch close to the desired opposition :type epoch: :py:class:`Epoch` :returns: Time when the 2nd station in longitude happens, as an Epoch :rtype: :py:class:`Epoch` :raises: TypeError if input value is of wrong type. :raises: ValueError if input epoch outside the -2000/4000 range. >>> epoch = Epoch(2018, 11, 1.0) >>> sta2 = Saturn.station_longitude_2(epoch) >>> y, m, d = sta2.get_date() >>> print(y) 2018 >>> print(m) 9 >>> print(round(d, 4)) 6.4175 """ # First check that input value is of correct types if not isinstance(epoch, Epoch): raise TypeError("Invalid input type") # Check that the input epoch is within valid range y = epoch.year() if y < -2000.0 or y > 4000.0: raise ValueError("Epoch outside the -2000/4000 range") # Set some specific constants for Saturn's opposition a = 2451870.17 b = 378.091904 m0 = 318.0172 m1 = 12.647487 k = round((365.2425 * y + 1721060.0 - a) / b) jde0 = a + k * b m = m0 + k * m1 m = Angle(m).to_positive() m = m.rad() t = (jde0 - 2451545.0) / 36525.0 # Compute an auxiliary angle aa = 82.74 + 40.76 * t bb = 29.86 + 1181.36 * t cc = 14.13 + 590.68 * t dd = 220.02 + 1262.87 * t # Convert to radians aa = Angle(aa).rad() bb = Angle(bb).rad() cc = Angle(cc).rad() dd = Angle(dd).rad() corr = (68.872 + t * (-0.0007 + t * 0.00023) + sin(m) * (5.9399 + t * (-0.04 - t * 0.00015)) + cos(m) * (-0.7998 + t * (-0.0266 + t * 0.00014)) + sin(2.0 * m) * (0.1738 - t * 0.0032) + cos(2.0 * m) * (-0.0039 + t * (-0.0024 + t * 0.00001)) + sin(3.0 * m) * (0.0073 - t * 0.0002) + cos(3.0 * m) * (0.002 - t * 0.0002) + sin(aa) * (0.0 + t * (-0.0337 + t * 0.00018)) + cos(aa) * (-0.851 + t * (0.0044 + t * 0.00068)) + sin(bb) * (0.0 + t * (-0.0064 + t * 0.00004)) + cos(bb) * (0.2397 + t * (-0.0012 - t * 0.00008)) + sin(cc) * (0.0 - t * 0.001) + cos(cc) * (0.1245 + t * 0.0006) + sin(dd) * (0.0 + t * (0.0024 - t * 0.00003)) + cos(dd) * (0.0477 + t * (-0.0005 - t * 0.00006))) to_return = jde0 + corr return Epoch(to_return) @staticmethod def perihelion_aphelion(epoch, perihelion=True): """This method computes the time of Perihelion (or Aphelion) closer to a given epoch. :param epoch: Epoch close to the desired Perihelion (or Aphelion) :type epoch: :py:class:`Epoch` :param peihelion: If True, the epoch of the closest Perihelion is computed, if False, the epoch of the closest Aphelion is found. :type bool: :returns: The epoch of the desired Perihelion (or Aphelion) :rtype: :py:class:`Epoch` :raises: TypeError if input values are of wrong type. >>> epoch = Epoch(1944, 1, 1.0) >>> e = Saturn.perihelion_aphelion(epoch) >>> y, m, d, h, mi, s = e.get_full_date() >>> print(y) 1944 >>> print(m) 9 >>> print(d) 8 >>> print(h) 1 >>> epoch = Epoch(2047, 1, 1.0) >>> e = Saturn.perihelion_aphelion(epoch, perihelion=False) >>> y, m, d, h, mi, s = e.get_full_date() >>> print(y) 2047 >>> print(m) 7 >>> print(d) 15 >>> print(h) 0 """ if not isinstance(epoch, Epoch): raise TypeError("Invalid input value") # First approximation k = 0.03393 * (epoch.year() - 2003.52) if perihelion: k = round(k) else: k = round(k + 0.5) - 0.5 jde = 2452830.12 + k * (10764.21676 - k * 0.000827) # Compute the epochs three months before and after jde_before = jde - 90.0 jde_after = jde + 90.0 # Compute the Sun-Saturn distance for each epoch l, b, r_b = Saturn.geometric_heliocentric_position(Epoch(jde_before)) l, b, r = Saturn.geometric_heliocentric_position(Epoch(jde)) l, b, r_a = Saturn.geometric_heliocentric_position(Epoch(jde_after)) # Call an interpolation object m = Interpolation([jde_before, jde, jde_after], [r_b, r, r_a]) sol = m.minmax() return Epoch(sol) @staticmethod def passage_nodes(epoch, ascending=True): """This function computes the time of passage by the nodes (ascending or descending) of Saturn, nearest to the given epoch. :param epoch: Epoch closest to the node passage :type epoch: :py:class:`Epoch` :param ascending: Whether the time of passage by the ascending (True) or descending (False) node will be computed :type ascending: bool :returns: Tuple containing: - Time of passage through the node (:py:class:`Epoch`) - Radius vector when passing through the node (in AU, float) :rtype: tuple :raises: TypeError if input values are of wrong type. >>> epoch = Epoch(2019, 1, 1) >>> time, r = Saturn.passage_nodes(epoch) >>> year, month, day = time.get_date() >>> print(year) 2034 >>> print(month) 5 >>> print(round(day, 1)) 30.2 >>> print(round(r, 4)) 9.0546 """ if not isinstance(epoch, Epoch): raise TypeError("Invalid input types") # Get the orbital parameters l, a, e, i, ome, arg = Saturn.orbital_elements_mean_equinox(epoch) # Compute the time of passage through perihelion t = Saturn.perihelion_aphelion(epoch) # Get the time of passage through the node time, r = passage_nodes_elliptic(arg, e, a, t, ascending) return time, r @staticmethod def magnitude(sun_dist, earth_dist, delta_u, b): """This function computes the approximate magnitude of Saturn. :param sun_dist: Distance from Saturn to the Sun, in Astronomical Units :type sun_dist: float :param earth_dist: Distance from Saturn to Earth, in Astronomical Units :type earth_dist: float :param delta_u: Difference between the Saturnicentric longitudes of the Sun and the Earth, measured in the plane of the ring :type delta_u: float, :py:class:`Angle` :param b: Saturnicentric latitude of the Earth refered to the plane of the ring, positive towards the north :type b: float, :py:class:`Angle` :returns: Saturn's magnitude :rtype: float :raises: TypeError if input values are of wrong type. >>> sun_dist = 9.867882 >>> earth_dist = 10.464606 >>> delta_u = Angle(16.442) >>> b = Angle(4.198) >>> m = Saturn.magnitude(sun_dist, earth_dist, delta_u, b) >>> print(m) 1.9 """ # WARNING: According to Example 41.d in page 286 of Meeus book, the # result for the example above is 0.9 (instead of 1.9). However, after # carefully checking the formula implemented here, I'm sure that the # book has an error if not (isinstance(sun_dist, float) and isinstance(earth_dist, float) and isinstance(delta_u, (float, Angle)) and isinstance(b, (float, Angle))): raise TypeError("Invalid input types") delta_u = float(delta_u) b = Angle(b).rad() m = (-8.68 + 5.0 * log10(sun_dist * earth_dist) + 0.044 * abs(delta_u) - 2.6 * sin(abs(b)) + 1.25 * sin(b) * sin(b)) return round(m, 1) def main(): # Let's define a small helper function def print_me(msg, val): print("{}: {}".format(msg, val)) # Let's show some uses of Saturn class print("\n" + 35 * "*") print("*** Use of Saturn class") print(35 * "*" + "\n") # Let's now compute the heliocentric position for a given epoch epoch = Epoch(2018, 10, 27.0) lon, lat, r = Saturn.geometric_heliocentric_position(epoch) print_me("Geometric Heliocentric Longitude", lon.to_positive()) print_me("Geometric Heliocentric Latitude", lat) print_me("Radius vector", r) print("") # Compute the geocentric position for 1992/12/20: epoch = Epoch(1992, 12, 20.0) ra, dec, elon = Saturn.geocentric_position(epoch) print_me("Right ascension", ra.ra_str(n_dec=1)) print_me("Declination", dec.dms_str(n_dec=1)) print_me("Elongation", elon.dms_str(n_dec=1)) print("") # Print mean orbital elements for Saturn at 2065.6.24 epoch = Epoch(2065, 6, 24.0) l, a, e, i, ome, arg = Saturn.orbital_elements_mean_equinox(epoch) print_me("Mean longitude of the planet", round(l, 6)) # 131.196871 print_me("Semimajor axis of the orbit (UA)", round(a, 8)) # 9.55490779 print_me("Eccentricity of the orbit", round(e, 7)) # 0.0553209 print_me("Inclination on plane of the ecliptic", round(i, 6)) # 2.486426 print_me("Longitude of the ascending node", round(ome, 5)) # 114.23974 print_me("Argument of the perihelion", round(arg, 6)) # -19.896331 print("") # Compute the time of the conjunction close to 2125/6/1 epoch = Epoch(2125, 6, 1.0) conj = Saturn.conjunction(epoch) y, m, d = conj.get_date() d = round(d, 4) date = "{}/{}/{}".format(y, m, d) print_me("Conjunction date", date) # Compute the time of the opposition close to -6/9/1 epoch = Epoch(-6, 9, 1.0) oppo = Saturn.opposition(epoch) y, m, d = oppo.get_date() d = round(d, 4) date = "{}/{}/{}".format(y, m, d) print_me("Opposition date", date) print("") # Compute the time of the station in longitude #1 close to 2018/11/1 epoch = Epoch(2018, 11, 1.0) sta1 = Saturn.station_longitude_1(epoch) y, m, d = sta1.get_date() d = round(d, 4) date = "{}/{}/{}".format(y, m, d) print_me("Date of station in longitude #1", date) # Compute the time of the station in longitude #2 close to 2018/11/1 epoch = Epoch(2018, 11, 1.0) sta2 = Saturn.station_longitude_2(epoch) y, m, d = sta2.get_date() d = round(d, 4) date = "{}/{}/{}".format(y, m, d) print_me("Date of station in longitude #2", date) print("") # Find the epoch of the Perihelion closer to 2000/1/1 epoch = Epoch(2000, 1, 1.0) e = Saturn.perihelion_aphelion(epoch) y, m, d, h, mi, s = e.get_full_date() peri = str(y) + '/' + str(m) + '/' + str(d) + ' at ' + str(h) + ' hours' print_me("The Perihelion closest to 2000/1/1 happened on", peri) print("") # Compute the time of passage through an ascending node epoch = Epoch(2019, 1, 1) time, r = Saturn.passage_nodes(epoch) y, m, d = time.get_date() d = round(d, 1) print("Time of passage through ascending node: {}/{}/{}".format(y, m, d)) # 2034/5/30.2 print("Radius vector at ascending node: {}".format(round(r, 4))) # 9.0546 if __name__ == "__main__": main()