# -*- 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.Coordinates import ( geometric_vsop_pos, apparent_vsop_pos, orbital_elements, nutation_longitude, true_obliquity, ecliptical2equatorial ) from pymeeus.Earth import Earth from pymeeus.Sun import Sun """ .. module:: Neptune :synopsis: Class to model Neptune planet :license: GNU Lesser General Public License v3 (LGPLv3) .. moduleauthor:: Dagoberto Salazar """ VSOP87_L = [ # L0 [ [531188633.047, 0.00000000000, 0.00000000000], [1798475.509, 2.90101273050, 38.13303563780], [1019727.662, 0.48580923660, 1.48447270830], [124531.845, 4.83008090682, 36.64856292950], [42064.450, 5.41054991607, 2.96894541660], [37714.589, 6.09221834946, 35.16409022120], [33784.734, 1.24488865578, 76.26607127560], [16482.741, 0.00007729261, 491.55792945680], [9198.582, 4.93747059924, 39.61750834610], [8994.249, 0.27462142569, 175.16605980020], [4216.235, 1.98711914364, 73.29712585900], [3364.818, 1.03590121818, 33.67961751290], [2284.800, 4.20606932559, 4.45341812490], [1433.512, 2.78340432711, 74.78159856730], [900.240, 2.07606702418, 109.94568878850], [744.996, 3.19032530145, 71.81265315070], [506.206, 5.74785370252, 114.39910691340], [399.552, 0.34972342569, 1021.24889455140], [345.195, 3.46186210169, 41.10198105440], [306.338, 0.49684039897, 0.52126486180], [287.322, 4.50523446022, 0.04818410980], [323.004, 2.24815188609, 32.19514480460], [340.323, 3.30369900416, 77.75054398390], [266.605, 4.88932609483, 0.96320784650], [227.079, 1.79713054538, 453.42489381900], [244.722, 1.24693337933, 9.56122755560], [232.887, 2.50459795017, 137.03302416240], [282.170, 2.24565579693, 146.59425171800], [251.941, 5.78166597292, 388.46515523820], [150.180, 2.99706110414, 5.93789083320], [170.404, 3.32390630650, 108.46121608020], [151.401, 2.19153094280, 33.94024994380], [148.295, 0.85948986145, 111.43016149680], [118.672, 3.67706204305, 2.44768055480], [101.821, 5.70539236951, 0.11187458460], [97.873, 2.80518260528, 8.07675484730], [103.054, 4.40441222000, 70.32818044240], [103.305, 0.04078966679, 0.26063243090], [109.300, 2.41599378049, 183.24281464750], [73.938, 1.32805041516, 529.69096509460], [77.725, 4.16446516424, 4.19278569400], [86.379, 4.22834506045, 490.07345674850], [81.536, 5.19908046216, 493.04240216510], [71.503, 5.29530386579, 350.33211960040], [64.418, 3.54541016050, 168.05251279940], [62.570, 0.15028731465, 182.27960680100], [58.488, 3.50106873945, 145.10977900970], [48.276, 1.11259925628, 112.91463420510], [47.229, 4.57373229818, 46.20979048510], [39.124, 1.66569356050, 213.29909543800], [47.728, 0.12906212461, 484.44438245600], [46.858, 3.01699530327, 498.67147645760], [38.659, 2.38685706479, 2.92076130680], [47.046, 4.49844660400, 173.68158709190], [47.565, 2.58404814824, 219.89137757700], [44.714, 5.47302733614, 176.65053250850], [32.279, 3.45759151220, 30.71067209630], [28.249, 4.13282446716, 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3.01948719112, 10.93382567930], [0.264, 5.70536379124, 490.33408917940], [0.280, 3.99993658196, 674.80074410430], [0.246, 0.37698964335, 157.63995198190], [0.219, 5.67679857772, 52099.54021187280], [0.251, 1.52353965506, 6.85291456990], [0.203, 5.44328656642, 145.63104387150], [0.238, 0.96169723853, 497.18700374930], [0.219, 4.52300776062, 1615.89959822680], [0.275, 2.37619210741, 2118.76386037840], [0.258, 5.12448148780, 608.87779767700], [0.260, 3.88543008475, 513.07988101300], [0.191, 3.72574595369, 65.22037101170], [0.211, 0.06484535455, 215.43795945210], [0.236, 3.95835282821, 141.48644228730], [0.189, 5.28135043909, 377.15882254340], [0.243, 4.35559878377, 482.95990974770], [0.243, 6.06808644973, 154.01661525950], [0.249, 1.57215637373, 14.22709400160], [0.238, 1.93340192445, 500.15594916590], [0.209, 5.02893682321, 364.55921360200], [0.227, 5.72984298540, 1543.82631264520], [0.217, 2.45036922991, 187.17496791060], [0.181, 1.65699502247, 1627.20593092160], [0.214, 1.60213179145, 11.30633269480], [0.203, 0.74638490279, 14.55716240170], [0.192, 3.17719161639, 343.47920503050], [0.177, 1.50027795761, 9.44935297100], [0.177, 0.03038098292, 165.60483224460], [0.176, 4.64462444674, 315.16802937920], [0.208, 2.65835778368, 496.01134758170], [0.174, 2.76155855705, 49.17873590170], [0.196, 1.95549714182, 335.77495719870], [0.200, 4.16839394758, 285.11174858870], [0.199, 0.06168021293, 73.55775828990], [0.188, 6.17288913873, 535.32003938710], [0.215, 1.92414563346, 552.69738935910], [0.166, 5.49038139690, 10135.53500222710], [0.192, 0.96973434120, 304.23420369990], [0.209, 5.34065233845, 13.64213866500], [0.203, 5.11234865419, 324.72925693480], [0.177, 3.50680841790, 207.36120460480], [0.174, 1.95010708561, 319.31263096340], [0.187, 5.57685931698, 266.10116806210], [0.181, 1.43525075751, 279.74330672710], [0.165, 4.00537112057, 493.56366702690], [0.191, 1.68313683465, 563.37058260750], [0.173, 3.93200456456, 238.90195810360], [0.161, 5.96143146317, 36.12729806770], [0.194, 2.37664231450, 944.98282327580], [0.165, 0.97421918976, 556.51766803760], [0.189, 1.11279570541, 1127.26243007680], [0.172, 0.75085513952, 267.58564077040], [0.193, 2.12636756833, 20350.30502114640], [0.181, 2.10814562080, 113.87784205160], [0.194, 1.13504964219, 57.25549074900], [0.181, 6.23699820519, 355.96119389290], [0.198, 5.68125942959, 6280.10690457480], [0.173, 5.15083799917, 474.94684537520], [0.151, 1.66981962338, 116.53797092750], [0.150, 5.42593657173, 526.98265210890], [0.205, 4.16096717573, 711.44930703380], [0.177, 3.49360697678, 421.22974901440], [0.168, 0.52839230204, 487.10451133190], [0.160, 4.77712663799, 524.01370669230], [0.145, 2.81448128781, 1512.80682400820], [0.146, 4.99570112660, 142.66209845490], [0.188, 0.82104161550, 10210.31660079440], [0.145, 4.96888131586, 1189.30140735080], [0.181, 2.99704790590, 75.74480641380], [0.176, 0.41626373842, 222.86032299360], [0.137, 2.96534226337, 6206.80977871580], [0.138, 1.22260849471, 187.69623277240], [0.128, 2.53394068407, 276.77436131050], [0.130, 3.04810765699, 310.71461125430], [0.122, 3.01323006886, 70.84944530420], [0.111, 0.77449448649, 179.35884549420], [0.141, 0.18423889807, 131.40394986990], [0.126, 5.77648809669, 525.23754696970], [0.124, 2.93225731024, 179.61947792510], [0.111, 6.18471578216, 981.63138620530], [0.141, 2.63342951123, 381.61224066830], [0.110, 5.25053027081, 986.08480433020], [0.096, 3.86591534559, 240.12579838100], [0.120, 3.78755085035, 1057.89745748090], [0.093, 4.54014016637, 36.69674703930], [0.109, 1.53327585900, 419.74527630610], [0.094, 4.21870300178, 1024.21783996800], [0.109, 2.15905156247, 289.56516671360], [0.104, 0.20665642552, 564.85505531580], [0.081, 1.89134135215, 36.60037881970], [0.080, 4.38832594589, 10137.01947493540], [0.080, 1.73940577376, 39.50563376150], [0.084, 0.81316746605, 170.71264167530], [0.090, 0.60145818457, 36.76043751410], [0.074, 4.92511651321, 1549.45538693770], [0.072, 5.06852406179, 249.94765836750], ], # L1 [ [3837687716.731, 0.00000000000, 0.00000000000], [16604.187, 4.86319129565, 1.48447270830], [15807.148, 2.27923488532, 38.13303563780], [3334.701, 3.68199676020, 76.26607127560], [1305.840, 3.67320813491, 2.96894541660], [604.832, 1.50477747549, 35.16409022120], [178.623, 3.45318524147, 39.61750834610], [106.537, 2.45126138334, 4.45341812490], [105.747, 2.75479326550, 33.67961751290], [72.684, 5.48724732699, 36.64856292950], [57.069, 5.21649804970, 0.52126486180], [57.355, 1.85767603384, 114.39910691340], [35.368, 4.51676827545, 74.78159856730], [32.216, 5.90411489680, 77.75054398390], [29.871, 3.67043294114, 388.46515523820], [28.866, 5.16877529164, 9.56122755560], [28.742, 5.16732589024, 2.44768055480], [25.507, 5.24526281928, 168.05251279940], [24.869, 4.73193067810, 182.27960680100], [20.205, 5.78945415677, 1021.24889455140], [19.022, 1.82981144269, 484.44438245600], [18.661, 1.31606255521, 498.67147645760], [15.063, 4.95003893760, 137.03302416240], [15.094, 3.98705254940, 32.19514480460], [10.720, 2.44148149225, 4.19278569400], [11.725, 4.89255650674, 71.81265315070], [9.581, 1.23188039594, 5.93789083320], [9.606, 1.88534821556, 41.10198105440], [8.968, 0.01758559103, 8.07675484730], [9.882, 6.08165628679, 7.11354700080], [7.632, 5.51307048241, 73.29712585900], [6.992, 0.61688864282, 2.92076130680], [5.543, 2.24141557794, 46.20979048510], [4.845, 3.71055823750, 112.91463420510], [3.700, 5.25713252333, 111.43016149680], [3.233, 6.10303038418, 70.32818044240], [2.939, 4.86520586648, 98.89998852460], [2.403, 2.90637675099, 601.76425067620], [2.398, 1.04343654629, 6.59228213900], [2.784, 4.95821114677, 108.46121608020], [2.894, 4.20148844767, 381.35160823740], [2.111, 5.93089610785, 25.60286266560], [2.075, 5.20632201951, 30.71067209630], [2.126, 0.54976393136, 41.05379694460], [2.235, 2.38045158073, 453.42489381900], [1.859, 0.89409373259, 24.11838995730], [2.018, 3.42245274178, 31.01948863700], [1.700, 3.91715254287, 11.04570026390], [1.776, 3.86571077241, 395.57870223900], [1.644, 0.15855999051, 152.53214255120], [1.646, 3.34591387314, 44.72531777680], [1.876, 2.59784179105, 33.94024994380], [1.614, 0.42137145545, 175.16605980020], [1.468, 6.12983933526, 1550.93985964600], [1.408, 6.13722948564, 490.07345674850], [1.207, 0.59525736062, 312.19908396260], [1.336, 3.28611928206, 493.04240216510], [1.176, 5.87266726996, 5.41662597140], [1.517, 3.12967210501, 491.55792945680], [1.053, 4.60375516830, 79.23501669220], [1.037, 4.89007314395, 1.27202438720], [1.034, 5.93741289103, 32.71640966640], [1.038, 1.13470380744, 1014.13534755060], [1.002, 1.85850922283, 5.10780943070], [0.983, 0.05345050384, 7.42236354150], [0.998, 1.73689827444, 1028.36244155220], [1.193, 4.63176675581, 60.76695288680], [0.940, 3.09103721222, 62.25142559510], [0.994, 4.11489180313, 4.66586644600], [0.890, 0.87049255398, 31.23193695810], [0.852, 5.35508394316, 144.14657116320], [0.922, 5.12373360511, 145.10977900970], [0.789, 0.37496785039, 26.82670294300], [0.828, 4.06035194600, 115.88357962170], [0.711, 3.14189997439, 278.25883401880], [0.727, 1.39718382835, 213.29909543800], [0.781, 0.10946327923, 173.68158709190], [0.793, 6.13086312116, 567.82400073240], [0.669, 4.50554989443, 27.08733537390], [0.825, 1.35568908148, 129.91947716160], [0.738, 3.56766018960, 176.65053250850], [0.714, 6.24797992301, 106.97674337190], [0.654, 1.13177751192, 68.84370773410], [0.624, 0.01567750666, 28.57180808220], [0.608, 4.60180625368, 189.39315380180], [0.595, 0.00857468445, 42.58645376270], [0.530, 5.61201247153, 12.53017297220], [0.521, 1.02371768017, 415.29185818120], [0.639, 0.68930265745, 529.69096509460], [0.526, 3.02138731705, 5.62907429250], [0.456, 4.44331571392, 43.24084506850], [0.524, 3.43316448349, 38.65430049960], [0.436, 2.41630174435, 82.85835341460], [0.424, 1.95736011325, 477.33083545520], [0.443, 3.39350946329, 357.44566660120], [0.383, 1.90232196422, 22.63391724900], [0.479, 5.55141744216, 37.61177077600], [0.462, 3.80436154644, 343.21857259960], [0.384, 5.60377408953, 594.65070367540], [0.369, 4.45577410338, 6.90109867970], [0.358, 3.69126616347, 3.93215326310], [0.352, 3.10952926034, 135.54855145410], [0.368, 3.53577440355, 40.58071619260], [0.424, 5.27159202779, 181.75834193920], [0.361, 0.29018303419, 72.07328558160], [0.390, 5.49512204296, 350.33211960040], [0.378, 2.74122401337, 488.37653571910], [0.372, 0.39980033572, 494.73932319450], [0.353, 1.10614174053, 20.60692781950], [0.296, 0.86351261285, 149.56319713460], [0.307, 5.39420288683, 160.93896579860], [0.395, 1.93577214824, 10137.01947493540], [0.288, 2.28755739359, 47.69426319340], [0.295, 2.48737537240, 19.12245511120], [0.290, 0.18636083306, 143.62530630140], [0.266, 3.09977370364, 69.36497259590], [0.266, 1.21002824826, 505.78502345840], [0.252, 3.12745026026, 460.53844081980], [0.328, 0.50849285663, 6206.80977871580], [0.257, 3.64119914774, 446.31134681820], [0.239, 5.54080102299, 911.04257333200], [0.265, 0.62702473701, 253.57099508990], [0.287, 2.44403568436, 16.67477455640], [0.231, 2.47026250085, 454.90936652730], [0.230, 3.24571542922, 1066.49547719000], [0.282, 1.48595620175, 983.11585891360], [0.212, 5.41931177641, 64.95973858080], [0.213, 1.64175339637, 1089.12939443900], [0.238, 2.69801319489, 882.94384600180], [0.210, 4.53976756699, 1093.32218013300], [0.220, 2.30038816175, 1052.26838318840], [0.256, 0.42073598460, 23.90594163620], [0.216, 5.44225918870, 39.09624348430], [0.201, 2.58746514605, 119.50691634410], [0.224, 4.43751392203, 639.89728631400], [0.186, 2.50651218075, 487.36514376280], [0.189, 4.05785534221, 120.99138905240], [0.184, 2.24245977278, 815.06334611420], [0.202, 3.43517732411, 45.24658263860], [0.175, 4.49165234532, 171.23390653710], [0.171, 5.50633466316, 179.09821306330], [0.200, 6.12663205401, 14.22709400160], [0.173, 2.61090344107, 389.94962794650], [0.167, 3.94754384833, 77.22927912210], [0.166, 3.41009128748, 81.37388070630], [0.163, 3.88198848446, 556.51766803760], [0.164, 1.49614763046, 63.73589830340], [0.176, 3.86129425367, 148.33935685720], [0.161, 2.22215642318, 574.93754773320], [0.171, 0.66899426684, 179.31066138440], [0.161, 1.21480182441, 1024.43028828910], [0.155, 3.25842414799, 10251.41858184880], [0.183, 5.45168150656, 218.40690486870], [0.152, 3.35145509017, 285.37238101960], [0.152, 0.42398786475, 274.06604832480], [0.146, 5.70714579127, 419.48464387520], [0.156, 0.64321524870, 1029.84691426050], [0.147, 4.30958930740, 157.63995198190], [0.147, 1.80689177510, 377.41945497430], [0.140, 1.49826604627, 386.98068252990], [0.137, 2.14480243915, 563.63121503840], [0.127, 3.98726599710, 84.34282612290], [0.134, 4.16039455079, 169.53698550770], [0.121, 0.29300927469, 206.18554843720], [0.129, 2.67625057010, 180.79513409270], [0.134, 3.18868986487, 166.56804009110], [0.135, 5.07517561780, 426.59819087600], [0.136, 1.81672451740, 151.04766984290], [0.129, 3.64795525602, 183.76407950930], [0.116, 6.06435563172, 220.41264243880], [0.123, 4.46641157829, 1022.73336725970], [0.112, 4.34485256988, 138.51749687070], [0.116, 5.58946529961, 35.68535508300], [0.108, 1.03796693383, 488.58898404020], [0.108, 2.10378485880, 494.52687487340], [0.106, 0.87068583107, 1059.38193018920], [0.097, 0.74486741478, 485.92885516430], [0.095, 5.54259914856, 497.18700374930], [0.085, 3.16062141266, 522.57741809380], [0.097, 6.05634803604, 482.95990974770], [0.095, 0.23111852730, 500.15594916590], [0.084, 2.64687252518, 536.80451209540], [0.074, 3.90678924318, 1019.76442184310], ], # L2 [ [53892.649, 0.00000000000, 0.00000000000], [281.251, 1.19084538887, 38.13303563780], [295.693, 1.85520292248, 1.48447270830], [270.190, 5.72143228148, 76.26607127560], [23.023, 1.21035596452, 2.96894541660], [7.333, 0.54033306830, 2.44768055480], [9.057, 4.42544992035, 35.16409022120], [5.223, 0.67427930044, 168.05251279940], [5.201, 3.02338671812, 182.27960680100], [4.288, 3.84351844003, 114.39910691340], [3.925, 3.53214557374, 484.44438245600], [3.741, 5.90238217874, 498.67147645760], [2.966, 0.31002477611, 4.45341812490], [3.415, 0.55971639038, 74.78159856730], [3.255, 1.84921884906, 175.16605980020], [2.157, 1.89135758747, 388.46515523820], [2.211, 4.37997092240, 7.11354700080], [1.847, 3.48574435762, 9.56122755560], [2.451, 4.68586840176, 491.55792945680], [1.844, 5.12281562096, 33.67961751290], [2.204, 1.69321574906, 77.75054398390], [1.652, 2.55859494053, 36.64856292950], [1.309, 4.52400192922, 1021.24889455140], [1.124, 0.38710602242, 137.03302416240], [0.664, 0.88101734307, 4.19278569400], [0.497, 2.24615784762, 395.57870223900], [0.512, 6.22609200672, 381.35160823740], [0.582, 5.25716719826, 31.01948863700], [0.446, 0.36647221351, 98.89998852460], [0.383, 5.48585528762, 5.93789083320], [0.375, 4.61250246774, 8.07675484730], [0.354, 1.30783918287, 601.76425067620], [0.259, 5.66033623678, 112.91463420510], [0.247, 2.89695614593, 189.39315380180], [0.245, 4.26572913391, 220.41264243880], [0.200, 0.52604535784, 64.95973858080], [0.191, 4.88786653062, 39.61750834610], [0.233, 3.16423779113, 41.10198105440], [0.248, 5.85877831382, 1059.38193018920], [0.194, 2.37949641473, 73.29712585900], [0.227, 0.20028518978, 60.76695288680], [0.184, 3.01962045713, 1014.13534755060], [0.190, 5.57500985081, 343.21857259960], [0.172, 3.66036463613, 477.33083545520], [0.172, 0.59550457102, 46.20979048510], [0.182, 1.92429384025, 183.76407950930], [0.171, 1.61368476689, 357.44566660120], [0.173, 6.23717119485, 493.04240216510], [0.217, 1.46218158211, 71.81265315070], [0.178, 0.34928799031, 1028.36244155220], [0.169, 4.91086673212, 166.56804009110], [0.157, 5.89200571154, 169.53698550770], [0.182, 2.33457064554, 152.53214255120], [0.151, 3.81621340568, 146.59425171800], [0.136, 2.75150881988, 144.14657116320], [0.104, 6.03262825314, 529.69096509460], [0.076, 0.20932812381, 453.42489381900], ], # L3 [ [31.254, 0.00000000000, 0.00000000000], [12.461, 6.04431418812, 1.48447270830], [14.541, 1.35337075856, 76.26607127560], [11.547, 6.11257808366, 38.13303563780], [1.351, 4.93951495175, 2.96894541660], [0.741, 2.35936954597, 168.05251279940], [0.715, 1.27409542804, 182.27960680100], [0.537, 5.23632185196, 484.44438245600], [0.523, 4.16769839601, 498.67147645760], [0.664, 0.55871435877, 31.01948863700], [0.301, 2.69253200796, 7.11354700080], [0.194, 2.05904114139, 137.03302416240], [0.206, 2.51012178002, 74.78159856730], [0.160, 5.63111039032, 114.39910691340], [0.149, 3.09327713923, 35.16409022120], ], # L4 [ [113.998, 3.14159265359, 0.00000000000], [0.605, 3.18211885677, 76.26607127560] ], # L5 [ [0.874, 3.14159265359, 0.00000000000] ], ] """This table contains Neptune'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 452-453.""" VSOP87_B = [ # B0 [ [3088622.933, 1.44104372626, 38.13303563780], [27780.087, 5.91271882843, 76.26607127560], [27623.609, 0.00000000000, 0.00000000000], [15355.490, 2.52123799481, 36.64856292950], [15448.133, 3.50877080888, 39.61750834610], [1999.919, 1.50998669505, 74.78159856730], [1967.540, 4.37778195768, 1.48447270830], [1015.137, 3.21561035875, 35.16409022120], [605.767, 2.80246601405, 73.29712585900], [594.878, 2.12892708114, 41.10198105440], [588.805, 3.18655882497, 2.96894541660], [401.830, 4.16883287237, 114.39910691340], [254.333, 3.27120499438, 453.42489381900], [261.647, 3.76722704749, 213.29909543800], [279.964, 1.68165309699, 77.75054398390], [205.590, 4.25652348864, 529.69096509460], [140.455, 3.52969556376, 137.03302416240], [98.530, 4.16774829927, 33.67961751290], [51.257, 1.95121181203, 4.45341812490], [67.971, 4.66970781659, 71.81265315070], [41.931, 5.41783694467, 111.43016149680], [41.822, 5.94832001477, 112.91463420510], [30.637, 0.93620571932, 42.58645376270], [11.084, 5.88898793049, 108.46121608020], [9.620, 0.03944255108, 70.32818044240], [9.664, 0.22455797403, 79.23501669220], [9.728, 5.30069593532, 32.19514480460], [7.386, 3.00684933642, 426.59819087600], [7.087, 0.12535040656, 109.94568878850], [6.021, 6.20514068152, 115.88357962170], [6.169, 3.62098109648, 983.11585891360], [4.777, 0.75210194972, 5.93789083320], [6.391, 5.84646101060, 148.07872442630], [6.251, 2.41678769385, 152.53214255120], [4.539, 5.58182098700, 175.16605980020], [5.006, 4.60815664851, 1059.38193018920], [4.289, 4.19647392821, 47.69426319340], [5.795, 5.07516716087, 415.29185818120], [4.749, 2.51605725604, 37.61177077600], [4.119, 1.72779509865, 28.57180808220], [4.076, 6.00252170354, 145.10977900970], [4.429, 5.65995321659, 98.89998852460], [3.950, 2.74104636753, 350.33211960040], [4.091, 1.61787956945, 39.09624348430], [4.131, 4.40682554313, 37.16982779130], [4.710, 3.50929350767, 38.65430049960], [4.440, 4.78977105547, 38.08485152800], [4.433, 1.23386935925, 38.18121974760], [3.762, 4.83940791709, 491.55792945680], [2.606, 1.20956732792, 451.94042111070], [2.537, 2.18628045751, 454.90936652730], [2.328, 5.19779918719, 72.07328558160], [2.502, 0.85987904350, 106.97674337190], [2.342, 0.81387240947, 4.19278569400], [1.981, 0.46617960831, 184.72728735580], [1.963, 6.01909114576, 44.07092647100], [2.180, 0.70099749844, 206.18554843720], [1.811, 0.40456996647, 40.58071619260], [1.814, 3.64699555185, 220.41264243880], [1.705, 6.13551142362, 181.75834193920], [1.855, 5.61635630213, 35.68535508300], [1.595, 2.97147156093, 37.87240320690], [1.785, 2.42154818096, 388.46515523820], [1.595, 3.05266110075, 38.39366806870], [1.437, 1.48678704605, 135.54855145410], [1.387, 2.46149266117, 138.51749687070], [1.366, 1.52026779665, 68.84370773410], [1.575, 3.58964541604, 38.02116105320], [1.297, 5.06156596196, 33.94024994380], [1.487, 0.20211121607, 30.05628079050], [1.504, 5.80298577327, 46.20979048510], [1.192, 0.87275514483, 42.32582133180], [1.569, 2.43405967107, 38.24491022240], [1.207, 1.84658687853, 251.43213107580], [1.015, 0.53439848924, 129.91947716160], [0.999, 2.47463873948, 312.19908396260], [0.990, 3.41514319052, 144.14657116320], [0.963, 4.31733242907, 151.04766984290], [1.020, 0.98226686775, 143.62530630140], [0.941, 1.02993053785, 221.37585028530], [0.938, 2.43648356625, 567.82400073240], [1.111, 0.65175024456, 146.59425171800], [0.777, 0.00175975222, 218.40690486870], [0.895, 0.25123869620, 30.71067209630], [0.795, 5.80519741659, 149.56319713460], [0.737, 3.40060492866, 446.31134681820], [0.719, 1.43795191278, 8.07675484730], [0.720, 0.00651007550, 460.53844081980], [0.766, 4.03399506246, 522.57741809380], [0.666, 1.39457824982, 84.34282612290], [0.584, 1.01405548136, 536.80451209540], [0.596, 0.62390100715, 35.21227433100], [0.598, 5.39946724188, 41.05379694460], [0.475, 5.80072248338, 7.42236354150], [0.510, 1.34478579740, 258.02441321480], [0.458, 5.25325523118, 80.71948940050], [0.421, 3.24496387889, 416.77633088950], [0.446, 1.19167306357, 180.27386923090], [0.471, 0.92632922375, 44.72531777680], [0.387, 1.68488418788, 183.24281464750], [0.375, 0.15223869165, 255.05546779820], [0.354, 4.21526988674, 0.96320784650], [0.379, 2.16947487177, 105.49227066360], [0.341, 4.79194051680, 110.20632121940], [0.427, 5.15774894584, 31.54075349880], [0.302, 3.45706306280, 100.38446123290], [0.298, 2.26790695187, 639.89728631400], [0.279, 0.25689162963, 39.50563376150], [0.320, 3.58085653166, 45.24658263860], [0.269, 5.72024180826, 36.76043751410], [0.247, 0.61040148804, 186.21176006410], [0.245, 0.64173616273, 419.48464387520], [0.235, 0.73189197665, 10213.28554621100], [0.232, 0.37399822852, 490.07345674850], [0.230, 5.76570492457, 12.53017297220], [0.240, 4.13447692727, 0.52126486180], [0.279, 1.62614865256, 294.67297614430], [0.238, 2.18528916550, 219.89137757700], [0.262, 3.08384135298, 6.59228213900], [0.217, 2.93214905312, 27.08733537390], [0.217, 4.69210602828, 406.10313764110], [0.219, 1.35212712560, 216.92243216040], [0.200, 2.35215465744, 605.95703637020], [0.232, 3.92583619589, 1512.80682400820], [0.223, 5.52392277606, 187.69623277240], [0.190, 0.29169556516, 291.70403072770], [0.236, 3.12464145036, 563.63121503840], [0.193, 0.53675942386, 60.76695288680], [0.215, 3.78391259001, 103.09277421860], [0.172, 5.63262770743, 7.11354700080], [0.164, 4.14700645532, 77.22927912210], [0.162, 0.72021213236, 11.04570026390], [0.160, 4.23490438166, 487.36514376280], [0.191, 0.37651439206, 31.01948863700], [0.157, 1.02419759383, 6283.07584999140], [0.157, 4.42530429545, 6206.80977871580], [0.178, 6.24797160202, 316.39186965660], [0.161, 5.65988283675, 343.21857259960], [0.153, 5.58405022784, 252.08652238160], [0.189, 4.80791039970, 641.12112659140], [0.166, 5.50438043692, 662.53120356300], [0.146, 5.08949604858, 286.59622129700], [0.145, 2.13015521881, 2042.49778910280], [0.156, 2.19452173251, 274.06604832480], [0.148, 4.85696640135, 442.75170057060], [0.187, 4.96121139073, 1589.07289528380], [0.155, 2.28260574227, 142.14083359310], [0.134, 1.29277093566, 456.39383923560], [0.126, 5.59769497652, 179.35884549420], [0.146, 2.53359213478, 256.53994050650], [0.140, 1.57962199954, 75.74480641380], [0.123, 0.05442220184, 944.98282327580], [0.122, 1.90676379802, 418.26080359780], [0.154, 1.86865302773, 331.32153907380], [0.144, 5.52229258454, 14.01464568050], [0.138, 2.80728175526, 82.85835341460], [0.107, 0.66995358132, 190.66517818900], [0.114, 1.48894980280, 253.57099508990], [0.110, 5.32587573069, 240.12579838100], [0.105, 0.65548440578, 173.68158709190], [0.102, 2.58735617801, 450.45594840240], [0.098, 0.44044795266, 328.35259365720], [0.101, 4.71267656829, 117.36805233000], [0.094, 0.54938580474, 293.18850343600], [0.095, 2.17636214523, 101.86893394120], [0.093, 0.63687810471, 377.15882254340], [0.091, 5.84828809934, 10137.01947493540], [0.089, 1.02830167997, 1021.24889455140], [0.094, 1.79320597168, 493.04240216510], [0.080, 1.58140274465, 69.15252427480], [0.075, 0.23453373368, 63.73589830340], [0.071, 1.51961989690, 488.58898404020], ], # B1 [ [227279.214, 3.80793089870, 38.13303563780], [1803.120, 1.97576485377, 76.26607127560], [1385.733, 4.82555548018, 36.64856292950], [1433.300, 3.14159265359, 0.00000000000], [1073.298, 6.08054240712, 39.61750834610], [147.903, 3.85766231348, 74.78159856730], [136.448, 0.47764957338, 1.48447270830], [70.285, 6.18782052139, 35.16409022120], [51.899, 5.05221791891, 73.29712585900], [37.273, 4.89476629246, 41.10198105440], [42.568, 0.30721737205, 114.39910691340], [37.104, 5.75999349109, 2.96894541660], [26.399, 5.21566335936, 213.29909543800], [16.949, 4.26463671859, 77.75054398390], [18.747, 0.90426522185, 453.42489381900], [12.951, 6.17709713139, 529.69096509460], [10.502, 1.20336443465, 137.03302416240], [4.416, 1.25478204684, 111.43016149680], [4.383, 6.14147099615, 71.81265315070], [3.694, 0.94837702528, 33.67961751290], [2.957, 4.77532871210, 4.45341812490], [2.698, 1.92435531119, 112.91463420510], [1.989, 3.96637567224, 42.58645376270], [1.150, 4.30568700024, 37.61177077600], [0.871, 4.81775882249, 152.53214255120], [0.944, 2.21777772050, 109.94568878850], [0.936, 1.17054983940, 148.07872442630], [0.925, 2.40329074000, 206.18554843720], [0.690, 1.57381082857, 38.65430049960], [0.624, 2.79466003645, 79.23501669220], [0.726, 4.13829519132, 28.57180808220], [0.640, 2.46161252327, 115.88357962170], [0.531, 2.96991530500, 98.89998852460], [0.537, 1.95986772922, 220.41264243880], [0.539, 2.06690307827, 40.58071619260], [0.716, 0.55781847010, 350.33211960040], [0.563, 1.84072805158, 983.11585891360], [0.533, 1.34787677940, 47.69426319340], [0.566, 1.80111775954, 175.16605980020], [0.449, 1.62191691011, 144.14657116320], [0.371, 2.74239666472, 415.29185818120], [0.381, 6.11910193382, 426.59819087600], [0.366, 2.39752585360, 129.91947716160], [0.456, 3.19611413854, 108.46121608020], [0.327, 3.62341506247, 38.18121974760], [0.328, 0.89613145346, 38.08485152800], [0.341, 3.87265469070, 35.68535508300], [0.331, 4.48858774501, 460.53844081980], [0.414, 1.03543720726, 70.32818044240], [0.310, 0.51297445145, 37.16982779130], [0.287, 2.18351651800, 491.55792945680], [0.274, 6.11504724934, 522.57741809380], [0.281, 3.81657117512, 5.93789083320], [0.298, 4.00532631258, 39.09624348430], [0.265, 5.26569823181, 446.31134681820], [0.319, 1.34097217817, 184.72728735580], [0.203, 6.02944475303, 149.56319713460], [0.205, 5.53935732020, 536.80451209540], [0.226, 6.17710997862, 454.90936652730], [0.186, 3.24302117645, 4.19278569400], [0.179, 4.91458426239, 451.94042111070], [0.198, 2.30775852880, 146.59425171800], [0.166, 1.16793600058, 72.07328558160], [0.147, 2.10574339673, 44.07092647100], [0.123, 1.98250467171, 46.20979048510], [0.159, 3.46955908364, 145.10977900970], [0.116, 5.88971113590, 38.02116105320], [0.115, 4.73412534395, 38.24491022240], [0.125, 3.42713474801, 251.43213107580], [0.128, 1.51108932026, 221.37585028530], [0.127, 0.17176461812, 138.51749687070], [0.124, 5.85160407534, 1059.38193018920], [0.091, 2.38273591235, 30.05628079050], [0.118, 5.27114846878, 37.87240320690], [0.117, 5.35267669439, 38.39366806870], [0.099, 5.19920708255, 135.54855145410], [0.114, 4.37452353441, 388.46515523820], [0.093, 4.64183693718, 106.97674337190], [0.084, 1.35269684746, 33.94024994380], [0.111, 3.56226463770, 181.75834193920], [0.082, 3.18401661435, 42.32582133180], [0.084, 5.51669920239, 8.07675484730], ], # B2 [ [9690.766, 5.57123750291, 38.13303563780], [78.815, 3.62705474219, 76.26607127560], [71.523, 0.45476688580, 36.64856292950], [58.646, 3.14159265359, 0.00000000000], [29.915, 1.60671721861, 39.61750834610], [6.472, 5.60736756575, 74.78159856730], [5.800, 2.25341847151, 1.48447270830], [4.309, 1.68126737666, 35.16409022120], [3.502, 2.39142672984, 114.39910691340], [2.649, 0.65061457644, 73.29712585900], [1.518, 0.37600329684, 213.29909543800], [1.223, 1.23116043030, 2.96894541660], [0.766, 5.45279753249, 453.42489381900], [0.779, 2.07081431472, 529.69096509460], [0.496, 0.26552533921, 41.10198105440], [0.469, 5.87866293959, 77.75054398390], [0.482, 5.63056237954, 137.03302416240], [0.345, 1.80085651594, 71.81265315070], [0.274, 2.86650141006, 33.67961751290], [0.158, 4.63868656467, 206.18554843720], [0.166, 1.24877330835, 220.41264243880], [0.153, 2.87376446497, 111.43016149680], [0.116, 3.63838544843, 112.91463420510], [0.085, 0.43712705655, 4.45341812490], [0.104, 6.12597614674, 144.14657116320], ], # B3 [ [273.423, 1.01688979072, 38.13303563780], [2.274, 2.36805657126, 36.64856292950], [2.029, 5.33364321342, 76.26607127560], [2.393, 0.00000000000, 0.00000000000], [0.538, 3.21934211365, 39.61750834610], [0.242, 4.52650721578, 114.39910691340], [0.185, 1.04913770083, 74.78159856730], [0.155, 3.62376309338, 35.16409022120], [0.157, 3.94195369610, 1.48447270830], ], # B4 [[5.728, 2.66872693322, 38.13303563780]], # B5 [[0.113, 4.70646877989, 38.13303563780]], ] """This table contains Neptune'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 page 453.""" VSOP87_R = [ # R0 [ [3007013206.102, 0.00000000000, 0.00000000000], [27062259.490, 1.32999458930, 38.13303563780], [1691764.281, 3.25186138896, 36.64856292950], [807830.737, 5.18592836167, 1.48447270830], [537760.613, 4.52113902845, 35.16409022120], [495725.642, 1.57105654815, 491.55792945680], [274571.970, 1.84552256801, 175.16605980020], [135134.095, 3.37220607384, 39.61750834610], [121801.825, 5.79754444303, 76.26607127560], [100895.397, 0.37702748681, 73.29712585900], [69791.722, 3.79617226928, 2.96894541660], [46687.838, 5.74937810094, 33.67961751290], [24593.778, 0.50801728204, 109.94568878850], [16939.242, 1.59422166991, 71.81265315070], [14229.686, 1.07786112902, 74.78159856730], [12011.825, 1.92062131635, 1021.24889455140], [8394.731, 0.67816895547, 146.59425171800], [7571.800, 1.07149263431, 388.46515523820], [5720.852, 2.59059512267, 4.45341812490], [4839.672, 1.90685991070, 41.10198105440], [4483.492, 2.90573457534, 529.69096509460], [4270.202, 3.41343865825, 453.42489381900], [4353.790, 0.67985662370, 32.19514480460], [4420.804, 1.74993796503, 108.46121608020], [2881.063, 1.98600105123, 137.03302416240], [2635.535, 3.09755943422, 213.29909543800], [3380.930, 0.84810683275, 183.24281464750], [2878.942, 3.67415901855, 350.33211960040], [2306.293, 2.80962935724, 70.32818044240], [2530.149, 5.79839567009, 490.07345674850], [2523.132, 0.48630800015, 493.04240216510], [2087.303, 0.61858378281, 33.94024994380], [1976.522, 5.11703044560, 168.05251279940], [1905.254, 1.72186472126, 182.27960680100], [1654.039, 1.92782545887, 145.10977900970], [1435.072, 1.70005157785, 484.44438245600], [1403.029, 4.58914203187, 498.67147645760], [1499.193, 1.01623299513, 219.89137757700], [1398.860, 0.76220317620, 176.65053250850], [1403.377, 6.07659416908, 173.68158709190], [1128.560, 5.96661179805, 9.56122755560], [1228.304, 1.59881465324, 77.75054398390], [835.414, 3.97066884218, 114.39910691340], [811.186, 3.00258880870, 46.20979048510], [731.925, 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4.79301628209, 143.62530630140], [5.346, 3.13071964722, 386.98068252990], [5.245, 6.06245070403, 171.23390653710], [5.129, 0.79394555531, 179.09821306330], [5.168, 4.73765992885, 522.57741809380], [6.422, 0.64684316894, 350.33211960040], [5.006, 2.37645082899, 77.22927912210], [5.005, 4.70632786971, 460.53844081980], [5.167, 5.20246616570, 446.31134681820], [5.119, 2.17338058771, 494.73932319450], [5.025, 4.21265519856, 536.80451209540], [4.722, 6.22814313946, 63.73589830340], [5.125, 5.38138329172, 179.31066138440], [4.918, 4.09031782903, 488.37653571910], [4.652, 5.10765073368, 274.06604832480], [4.711, 5.56542374115, 42.32582133180], [4.459, 1.30784829830, 69.36497259590], [5.485, 3.88088464259, 218.40690486870], [4.416, 3.05353893868, 27.08733537390], [4.559, 4.92224120952, 285.37238101960], [4.393, 4.18047835584, 5.41662597140], [4.687, 2.21401153210, 1029.84691426050], [4.644, 1.87902594973, 1433.61999142580], [5.639, 3.05596737234, 983.11585891360], [6.045, 5.68817982786, 351.81659230870], [4.430, 3.37768805833, 377.41945497430], [4.683, 2.14346624864, 97.41551581630], [5.845, 4.62301099402, 1024.21783996800], [4.536, 2.45860473853, 496.01134758170], [4.398, 5.65312496227, 3.93215326310], [4.287, 0.66340266603, 1012.65087484230], [4.086, 0.14551174994, 385.28376150050], [4.029, 5.98399329775, 178.34745353790], [4.276, 3.68205082970, 348.84764689210], [5.257, 3.75263242432, 379.86713552910], [4.012, 0.42559540783, 104313.47953065898], [4.025, 2.40645188238, 84.34282612290], [3.957, 0.86846121055, 171.98466606250], [3.961, 3.04953080906, 1017.31674128830], [5.559, 0.77714806229, 1447.84708542740], [5.071, 2.61075526868, 1536.71276564440], [4.052, 5.00014006312, 391.64654897590], [5.182, 4.73444634983, 382.83608094570], [3.763, 4.29449373755, 313.68355667090], [4.038, 2.82857942788, 1661.14618086540], [4.067, 5.73169928960, 169.53698550770], [3.841, 1.62580928420, 0.96320784650], [3.901, 2.70874386576, 14.01464568050], [3.721, 1.20062375429, 1026.87796884390], [3.911, 3.01809123569, 100.38446123290], [3.489, 4.28865448963, 1025.18104781450], [3.714, 5.05021268365, 292.48592802040], [3.816, 3.93084933114, 39.09624348430], [3.988, 2.82832650224, 134.11226285560], [3.745, 4.24728135115, 180.79513409270], [3.836, 1.02685786071, 1018.27994913480], [3.941, 5.21895739331, 183.76407950930], [4.669, 4.38080962573, 1066.49547719000], [3.780, 6.03723468132, 1022.73336725970], [3.647, 3.98130320367, 608.87779767700], [3.456, 5.54052355058, 846.08283475120], [4.047, 3.71041480907, 1018.06750081370], [3.865, 4.76002199091, 166.56804009110], [3.629, 3.29053233846, 447.79581952650], [3.564, 4.36703678321, 397.06317494730], [3.304, 1.49289552229, 1505.69327700740], [3.976, 2.42476188945, 106.01353552540], [4.217, 4.21677652639, 1052.26838318840], [3.294, 0.42088065654, 22.63391724900], [3.615, 3.68096122231, 494.52687487340], [3.230, 5.10786091356, 69.15252427480], [3.280, 3.62226152032, 531.17543780290], [3.337, 2.72502876320, 481.47543703940], [3.187, 0.08677634706, 399.51085550210], [3.389, 1.79454271219, 1519.92037100900], [3.179, 3.40418030121, 423.62924545940], [3.154, 3.69356460843, 470.21728845440], [3.706, 2.79048710497, 462.02291352810], [3.136, 4.38015969606, 385.49620982160], [3.122, 0.48346644637, 79.18683258240], [3.392, 0.48037804731, 521.09294538550], [3.465, 0.93152295589, 2183.72359895920], [3.735, 0.98809808606, 487.41332787260], [3.998, 3.38773325131, 6283.07584999140], [2.998, 2.61728063127, 487.62577619370], [3.295, 2.53821501556, 4.66586644600], [2.964, 3.66274645375, 495.49008271990], [3.901, 1.65463523144, 210.33015002140], [2.950, 1.99904237956, 872.90953769420], [2.948, 2.90769224206, 391.43410065480], [2.971, 0.31626092637, 5.10780943070], [3.085, 0.95725590904, 109.94568878850], [2.995, 3.34433305798, 394.09422953070], [3.126, 5.89472116854, 105.49227066360], [3.904, 3.01022809543, 556.51766803760], [3.388, 6.24936444215, 535.32003938710], [2.930, 6.15005257333, 164.12035953630], [3.267, 4.19718045293, 518.38463239980], [3.946, 2.88842759670, 151.26011816400], [3.076, 6.04134449219, 142.14083359310], [2.823, 0.60712626756, 214.78356814630], [2.917, 2.74502617182, 138.51749687070], [3.347, 6.09373507569, 6246.42728706190], [3.659, 5.12211619716, 79.23501669220], [3.010, 0.24656411754, 91.78644152380], [2.861, 6.17465663902, 422.40540518200], [2.989, 2.31620917965, 485.92885516430], [3.088, 2.29186342974, 110.20632121940], [3.030, 3.69866149100, 532.61172640140], [3.020, 2.36422658177, 290.21955801940], [3.170, 1.23078934548, 10176.63698328150], [2.652, 3.35836234807, 148.07872442630], [2.673, 6.03366372927, 196.50670080260], [2.630, 0.46957619348, 1970.42450352120], [2.599, 4.86022081674, 439.19779981740], [2.878, 2.61946597178, 488.58898404020], [2.720, 1.71836225398, 364.55921360200], [3.333, 3.25126857354, 30.05628079050], [3.053, 2.49346960035, 6243.45834164530], [3.062, 6.23776299963, 419.48464387520], [2.786, 0.83078219939, 497.18700374930], [2.834, 3.52926079424, 457.87831194390], [2.932, 1.80245810977, 500.15594916590], [3.030, 5.10152500393, 367.97010200330], [2.956, 5.76230870725, 986.08480433020], [3.116, 2.20042242739, 495.70253104100], [2.554, 0.65945973992, 67.35923502580], [2.901, 3.91891656185, 10173.66803786490], [2.840, 1.34453183591, 482.95990974770], [2.458, 1.20012815574, 489.11024890200], [2.556, 3.86921927085, 487.10451133190], [2.614, 1.51881085312, 463.50738623640], [2.386, 4.58400538443, 615.99134467780], [2.438, 5.19827220476, 501.11915701240], [2.537, 1.64802783144, 519.60847267720], [2.444, 3.87859489652, 185.24855221760], [2.795, 4.04265752580, 255.05546779820], [2.895, 3.26202698812, 1646.91908686380], [2.225, 5.75197574692, 605.95703637020], [2.324, 3.99503920129, 481.26298871830], [2.962, 1.74151265966, 2080.63082474060], [2.621, 1.74442251671, 35.21227433100], ], # R2 [ [4247.412, 5.89910679117, 38.13303563780], [217.570, 0.34581829080, 1.48447270830], [163.025, 2.23872947130, 168.05251279940], [156.285, 4.59414467342, 182.27960680100], [117.940, 5.10295026024, 484.44438245600], [112.429, 1.19000583596, 498.67147645760], [127.141, 2.84786298079, 35.16409022120], [99.467, 3.41578558739, 175.16605980020], [64.814, 3.46214064840, 388.46515523820], [77.286, 0.01659281785, 491.55792945680], [49.509, 4.06995509133, 76.26607127560], [39.330, 6.09521855958, 1021.24889455140], [36.450, 5.17130059988, 137.03302416240], [37.080, 5.97288967681, 2.96894541660], [30.484, 3.58259801313, 33.67961751290], [21.099, 0.76843555176, 36.64856292950], [13.886, 3.59248623971, 395.57870223900], [13.117, 5.09263515697, 98.89998852460], [11.379, 1.18060018898, 381.35160823740], [9.132, 2.34787658568, 601.76425067620], [8.527, 5.25134685897, 2.44768055480], [8.136, 4.96270726986, 4.45341812490], [7.417, 4.46775409796, 189.39315380180], [7.225, 1.92287508629, 9.56122755560], [7.289, 1.65519525780, 1028.36244155220], [8.076, 5.84268048311, 220.41264243880], [9.654, 0.00000000000, 0.00000000000], [6.554, 0.69397520733, 144.14657116320], [7.782, 1.14341656235, 1059.38193018920], [5.665, 6.25378258571, 74.78159856730], [5.628, 5.23383764266, 46.20979048510], [5.523, 4.59041448911, 1014.13534755060], [5.177, 5.23116646157, 477.33083545520], [5.503, 3.49522319102, 183.76407950930], [4.878, 3.52934357721, 39.61750834610], [4.787, 2.08260524745, 41.10198105440], [5.055, 0.19949888617, 166.56804009110], [4.751, 1.18054948270, 169.53698550770], [4.747, 1.50608965076, 73.29712585900], [6.113, 6.18326155595, 71.81265315070], [4.606, 3.91970908886, 587.53715667460], [5.756, 2.23667359233, 176.65053250850], [4.536, 2.84337336954, 7.11354700080], [4.338, 0.51553847388, 446.31134681820], [3.891, 0.26338839265, 1550.93985964600], [4.465, 3.01487041298, 129.91947716160], [3.727, 2.37977930658, 160.93896579860], [3.840, 3.79290381880, 111.43016149680], [4.142, 1.70293820961, 983.11585891360], [3.296, 1.07748822909, 505.78502345840], [4.008, 0.30663868827, 494.73932319450], [3.974, 5.97351783840, 488.37653571910], [3.925, 4.85736421123, 60.76695288680], [2.966, 2.01608546009, 822.17689311500], [3.972, 1.07780371834, 374.23806123660], [3.843, 5.23002047199, 350.33211960040], [2.848, 6.17799253802, 704.85702489480], [3.527, 0.79317138165, 274.06604832480], [2.828, 1.32275775835, 386.98068252990], [2.773, 5.37132330836, 251.43213107580], [3.113, 5.12622288690, 426.59819087600], [3.344, 5.61433537548, 1124.34166877000], [2.597, 0.67759426519, 312.19908396260], [2.581, 3.55847612121, 567.82400073240], [2.578, 1.45603792456, 1035.47598855300], [2.541, 5.19427579702, 1227.43444298860], [2.510, 4.12148891512, 171.23390653710], [2.511, 2.71606957319, 179.09821306330], [2.342, 0.96469916587, 1019.76442184310], [2.500, 0.70282276030, 707.77778620160], [2.480, 4.59623030219, 693.55069220000], [2.253, 0.74334306011, 976.00231191280], ], # R3 [ [166.297, 4.55243893489, 38.13303563780], [22.380, 3.94830879358, 168.05251279940], [21.348, 2.86296778794, 182.27960680100], [16.233, 0.54226725872, 484.44438245600], [15.623, 5.75702251906, 498.67147645760], [11.867, 4.40280192710, 1.48447270830], [6.448, 5.19003066847, 31.01948863700], [3.655, 5.91335292846, 1007.02180054980], [3.681, 1.62865545676, 388.46515523820], [3.198, 0.70197118575, 1558.05340664680], [3.243, 1.88035665980, 522.57741809380], [3.269, 2.94301808574, 76.26607127560], [2.688, 1.87062743473, 402.69224923980], [3.246, 0.79381356193, 536.80451209540], [2.650, 5.76858449026, 343.21857259960], [2.644, 4.64542905401, 500.15594916590], [2.541, 4.79217120822, 482.95990974770], [2.523, 1.72869889780, 395.57870223900], [2.690, 2.21096415618, 446.31134681820], [2.355, 5.77381398401, 485.92885516430], [2.874, 6.19643340540, 815.06334611420], [2.278, 3.66579603119, 497.18700374930], ], # R4 [ [4.227, 2.40375758563, 477.33083545520], [4.333, 0.10459484545, 395.57870223900], [3.545, 4.78431259422, 1028.36244155220], [3.154, 3.88192942366, 505.78502345840], [3.016, 1.03609346831, 189.39315380180], [2.294, 1.10879658603, 182.27960680100], [2.295, 5.67776133184, 168.05251279940], ], ] """This table contains Neptune'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 page 454.""" ORBITAL_ELEM = [ [304.348665, 219.8833092, 0.00030882, 0.000000018], # L [30.110386869, -0.0000001663, 0.00000000069, 0.0], # a [0.00945575, 0.000006033, 0.0, -0.00000000005], # e [1.769953, -0.0093082, -0.00000708, 0.000000027], # i [131.748057, 1.1022039, 0.00025952, -0.000000637], # Omega [48.120276, 1.4262957, 0.00038434, 0.00000002] # pie ] """This table contains the parameters to compute Neptune's orbital elements for the mean equinox of date. Based in Table 31.A, page 213""" ORBITAL_ELEM_J2000 = [ [304.348665, 218.4862002, 0.00000059, -0.000000002], # L [1.769953, 0.0002256, 0.00000023, 0.0], # i [131.748057, -0.0061651, -0.00000219, -0.000000078], # Omega [48.120276, 0.0291866, 0.0000761, 0.0] # pie ] """This table contains the parameters to compute Neptune's orbital elements for the standard equinox J2000.0. Based on Table 31.B, page 215""" class Neptune(object): """ Class Neptune models that planet. """ @staticmethod def geometric_heliocentric_position(epoch, tofk5=True): """This method computes the geometric heliocentric position of planet Neptune for a given epoch, using the VSOP87 theory. :param epoch: Epoch to compute Neptune 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 = Neptune.geometric_heliocentric_position(epoch) >>> print(round(l.to_positive(), 4)) 345.3776 >>> print(round(b, 4)) -0.9735 >>> print(round(r, 5)) 29.93966 """ 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 Neptune for a given epoch, using the VSOP87 theory. :param epoch: Epoch to compute Neptune 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 Neptune 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 = Neptune.orbital_elements_mean_equinox(epoch) >>> print(round(l, 6)) 88.321947 >>> print(round(a, 8)) 30.11038676 >>> print(round(e, 7)) 0.0094597 >>> print(round(i, 6)) 1.763855 >>> print(round(ome, 5)) 132.46986 >>> print(round(arg, 6)) -83.415521 """ return orbital_elements(epoch, ORBITAL_ELEM, ORBITAL_ELEM) @staticmethod def orbital_elements_j2000(epoch): """This method computes the orbital elements of Neptune 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 = Neptune.orbital_elements_j2000(epoch) >>> print(round(l, 6)) 87.407029 >>> print(round(a, 8)) 30.11038676 >>> print(round(e, 7)) 0.0094597 >>> print(round(i, 6)) 1.770101 >>> print(round(ome, 5)) 131.74402 >>> print(round(arg, 6)) -83.6046 """ return orbital_elements(epoch, ORBITAL_ELEM, ORBITAL_ELEM_J2000) @staticmethod def geocentric_position(epoch): """This method computes the geocentric position of Neptune (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 = Neptune.geocentric_position(epoch) >>> print(ra.ra_str(n_dec=1)) 19h 17' 14.5'' >>> print(dec.dms_str(n_dec=1)) -21d 34' 15.1'' >>> print(elon.dms_str(n_dec=1)) 19d 44' 59.6'' """ # First check that input value is of correct types if not isinstance(epoch, Epoch): raise TypeError("Invalid input type") # Compute the heliocentric position of Neptune l, b, r = Neptune.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 Neptune coordinates with this correction l, b, r = Neptune.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(1993, 10, 1.0) >>> conj = Neptune.conjunction(epoch) >>> y, m, d = conj.get_date() >>> print(y) 1994 >>> print(m) 1 >>> print(round(d, 4)) 11.3057 """ # 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 Neptune's conjunction a = 2451569.379 b = 367.486703 m0 = 21.5569 m1 = 2.194998 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 a couple auxiliary angles ee = 207.83 + 8.51 * t gg = 276.74 + 209.98 * t # Convert to radians ee = Angle(ee).rad() gg = Angle(gg).rad() corr = (0.0168 + sin(m) * (-2.5606 + t * (0.0088 + t * 0.00002)) + cos(m) * (-0.8611 + t * (-0.0037 + t * 0.00002)) + sin(2.0 * m) * (0.0118 + t * (-0.0004 + t * 0.00001)) + cos(2.0 * m) * (0.0307 - t * 0.0003) + cos(ee) * (-0.5964) + cos(gg) * (0.0728)) 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(1846, 8, 1) >>> oppo = Neptune.opposition(epoch) >>> y, m, d = oppo.get_date() >>> print(y) 1846 >>> print(m) 8 >>> print(round(d, 4)) 20.1623 """ # 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 Neptune's opposition a = 2451753.122 b = 367.486703 m0 = 202.6544 m1 = 2.194998 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 a couple auxiliary angles ee = 207.83 + 8.51 * t gg = 276.74 + 209.98 * t # Convert to radians ee = Angle(ee).rad() gg = Angle(gg).rad() corr = (-0.014 + t * t * 0.00001 + sin(m) * (-1.3486 + t * (0.001 + t * 0.00001)) + cos(m) * (0.8597 + t * 0.0037) + sin(2.0 * m) * (-0.0082 + t * (-0.0002 + t * 0.00001)) + cos(2.0 * m) * (0.0037 - t * 0.0003) + cos(ee) * (-0.5964) + cos(gg) * (0.0728)) to_return = jde0 + corr return Epoch(to_return) @staticmethod def magnitude(sun_dist, earth_dist): """This function computes the approximate magnitude of Neptune. :param sun_dist: Distance from Neptune to Sun, in Astronomical Units :type sun_dist: float :param earth_dist: Distance Neptune to Earth, in Astronomical Units :type earth_dist: float :returns: Neptune's magnitude :rtype: float :raises: TypeError if input values are of wrong type. """ if not (isinstance(sun_dist, float) and isinstance(earth_dist, float)): raise TypeError("Invalid input types") m = -7.05 + 5.0 * log10(sun_dist * earth_dist) 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 Neptune class print("\n" + 35 * "*") print("*** Use of Neptune class") print(35 * "*" + "\n") # Let's now compute the heliocentric position for a given epoch epoch = Epoch(2018, 10, 27.0) lon, lat, r = Neptune.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 = Neptune.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 Neptune at 2065.6.24 epoch = Epoch(2065, 6, 24.0) l, a, e, i, ome, arg = Neptune.orbital_elements_mean_equinox(epoch) print_me("Mean longitude of the planet", round(l, 6)) # 88.321947 print_me("Semimajor axis of the orbit (UA)", round(a, 8)) # 30.11038676 print_me("Eccentricity of the orbit", round(e, 7)) # 0.0094597 print_me("Inclination on plane of the ecliptic", round(i, 6)) # 1.763855 print_me("Longitude of the ascending node", round(ome, 5)) # 132.46986 print_me("Argument of the perihelion", round(arg, 6)) # -83.415521 print("") # Compute the time of the conjunction close to 1993/10/1 epoch = Epoch(1993, 10, 1.0) conj = Neptune.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 1846/8/1 epoch = Epoch(1846, 8, 1) oppo = Neptune.opposition(epoch) y, m, d = oppo.get_date() d = round(d, 4) date = "{}/{}/{}".format(y, m, d) print_me("Opposition date", date) if __name__ == "__main__": main()