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REBOUND を使うと Python で軌道積分を行うことができるので、お手軽に小惑星の軌道積分ができるよう に Python のスクリプトを書いてみたでござる。
#!/usr/pkg/bin/python3.10 # # Time-stamp: <2023/07/20 09:53:19 (CST) daisuke> # # importing argparse module import argparse # importing ssl module import ssl # importing pathlib module import pathlib # importing sys module import sys # importing datetime module import datetime # importing re module import re # importing numpy module import numpy # importing rebound module import rebound # date/time datetime_now_utc = datetime.datetime.now (tz=datetime.timezone.utc) datetime_now_utc_str = f'{datetime_now_utc.year:04d}' \ + f'-{datetime_now_utc.month:02d}' \ + f'-{datetime_now_utc.day:02d}' \ + f'_{datetime_now_utc.hour:02d}' \ + f':{datetime_now_utc.minute:02d}' \ + f':{datetime_now_utc.second:02d}' # setting for SSL ssl._create_default_https_context = ssl._create_unverified_context # constants pi = numpy.pi year = 2.0 * pi # 1 yr = 2 pi for G=1 # pattern for regular expression pattern_space = re.compile ('\s+') underscore = '_' # choices of integrator choices_integrator = [ 'ias15', 'whfast', 'BS', 'mercurius', 'saba', 'janus', 'eos', 'sei', 'tes', 'none', 'whfast512' ] # initialising a parser desc = 'backward orbital integration of asteroids' parser = argparse.ArgumentParser (description=desc) # adding arguments parser.add_argument ('-s', '--start', default='2000-01-01 12:00', \ help='start date/time (default: 2000-01-01 12:00)') parser.add_argument ('-i', '--integrator', choices=choices_integrator, \ default='mercurius', \ help='choice of integrator (default: mercurius)') parser.add_argument ('-t', '--timestep', type=float, \ default=-0.01, \ help='step size of integration in yr (default: -0.01)') parser.add_argument ('-d', '--duration', type=float, \ default=10**6, \ help='duration of integration in yr (default: 10**6)') parser.add_argument ('-n', '--noutputs', type=int, \ default=10**8, \ help='number of outputs (default: 10**8)') parser.add_argument ('-o', '--output', default='rebount.out', \ help='output file name (default: rebound.out)') parser.add_argument ('-v', '--verbose', action='store_true', \ help='verbose mode (default: off)') parser.add_argument ('asteroids', \ nargs='+', \ default='Ceres', \ help='asteroid names (default: Ceres)') # parsing arguments args = parser.parse_args () # input parameters datetime_start = args.start integrator = args.integrator timestep = args.timestep duration_yr = args.duration n_outputs = args.noutputs file_output = args.output verbose = args.verbose list_asteroids = args.asteroids # making pathlib object path_output = pathlib.Path (file_output) # existence check if (path_output.exists ()): # printing error message print (f'ERROR: file "{file_output} exists."') # exit sys.exit () # making a simulation object sim = rebound.Simulation () # adding the Sun and planets to simulation sim.add ('Sun', date=datetime_start) sim.add ('Mercury', date=datetime_start) sim.add ('Venus', date=datetime_start) sim.add ('Earth', date=datetime_start) sim.add ('Mars', date=datetime_start) sim.add ('Jupiter', date=datetime_start) sim.add ('Saturn', date=datetime_start) sim.add ('Uranus', date=datetime_start) sim.add ('Neptune', date=datetime_start) # adding asteroids to simulation for asteroid in list_asteroids: sim.add (asteroid, date=datetime_start) # time step of orbital integration # -0.01 ==> backward integration with time step of 0.01 yr sim.dt = timestep # choice of integrator sim.integrator = integrator # moving to centre of momentum frame sim.move_to_com () # pointer to particles ps = sim.particles # list for data output times times = numpy.linspace (0.0, duration_yr * year, n_outputs) # opening file for writing with open (file_output, 'w') as fh_output: # writing header to output file fh_output.write (f'#\n') fh_output.write (f'# Result of orbital integration by REBOUND\n') fh_output.write (f'#\n') fh_output.write (f'# start of simulation: {datetime_now_utc_str}\n') fh_output.write (f'#\n') fh_output.write (f'# input parameters\n') fh_output.write (f'# start date/time = {datetime_start}\n') fh_output.write (f'# integrator = {integrator}\n') fh_output.write (f'# timestep = {timestep} yr\n') fh_output.write (f'# duration_yr = {duration_yr} yr\n') fh_output.write (f'# n_outputs = {n_outputs}\n') fh_output.write (f'# file_output = {file_output}\n') fh_output.write (f'# verbose = {verbose}\n') fh_output.write (f'#\n') fh_output.write (f'# asteroids for orbital integration\n') for asteroid in list_asteroids: fh_output.write (f'# {asteroid}\n') fh_output.write (f'#\n') fh_output.write (f'# format of output data\n') fh_output.write (f'# time (in yr) from start date/time, object name,' \ + f' x, y, z, vx, vy, vz, a, e, i, Omega, omega, M\n') fh_output.write (f'#\n') # integration for i, time in enumerate (times): # integration by one timestep sim.integrate (time) # state vector and orbital elements of each asteroid for j in range (len (list_asteroids)): # time from start date/time in yr if (timestep < 0.0): time_from_start_yr = time / year * (-1.0) else: time_from_start_yr = time / year # asteroid name name = re.sub (pattern_space, underscore, list_asteroids[j]) # state vector ast_x = ps[j+9].x ast_y = ps[j+9].y ast_z = ps[j+9].z ast_vx = ps[j+9].vx ast_vy = ps[j+9].vy ast_vz = ps[j+9].vz # orbital elements ast_a = ps[j+9].a ast_e = ps[j+9].e ast_i = ps[j+9].inc ast_Omega = ps[j+9].Omega ast_omega = ps[j+9].omega ast_M = ps[j+9].M # asteroid name, time, state vector, and orbital elements result = f'{time_from_start_yr:15.6f}' \ + f' {name:16s}' \ + f' {ast_x:15.10f} {ast_y:15.10f} {ast_z:15.10f}' \ + f' {ast_vx:15.10f} {ast_vy:15.10f} {ast_vz:15.10f}' \ + f' {ast_a:15.10f} {ast_e:15.10f} {ast_i:15.10f}' \ + f' {ast_Omega:15.10f} {ast_omega:15.10f} {ast_M:15.10f}\n' # writing result into file fh_output.write (result) # printing progress if (verbose): if (i % 10000 == 0): progress_percent = i / len (times) * 100.0 print (f'progress of integration: {progress_percent:7.3f}%') # date/time datetime_now_utc = datetime.datetime.now (tz=datetime.timezone.utc) datetime_now_utc_str = f'{datetime_now_utc.year:04d}' \ + f'-{datetime_now_utc.month:02d}' \ + f'-{datetime_now_utc.day:02d}' \ + f'_{datetime_now_utc.hour:02d}' \ + f':{datetime_now_utc.minute:02d}' \ + f':{datetime_now_utc.second:02d}' fh_output.write (f'#\n') fh_output.write (f'# end of simulation : {datetime_now_utc_str}\n') fh_output.write (f'#\n')
使い方は以下の通りでござる。
% ./backint_asteroids.py -h usage: backint_asteroids.py [-h] [-s START] [-i {ias15,whfast,BS,mercurius,saba,janus,eos,sei,tes,none,whfast512}] [-t TIMESTEP] [-d DURATION] [-n NOUTPUTS] [-o OUTPUT] [-v] asteroids [asteroids ...] backward orbital integration of asteroids positional arguments: asteroids asteroid names (default: Ceres) options: -h, --help show this help message and exit -s START, --start START start date/time (default: 2000-01-01 12:00) -i {ias15,whfast,BS,mercurius,saba,janus,eos,sei,tes,none,whfast512}, --integrator {ias15,whfast,BS,mercurius,saba,janus,eos,sei,tes,none,whfast512} choice of integrator (default: mercurius) -t TIMESTEP, --timestep TIMESTEP step size of integration in yr (default: -0.01) -d DURATION, --duration DURATION duration of integration in yr (default: 10**6) -n NOUTPUTS, --noutputs NOUTPUTS number of outputs (default: 10**8) -o OUTPUT, --output OUTPUT output file name (default: rebound.out) -v, --verbose verbose mode (default: off)
小惑星 (1) Ceres を 2000 年 01 月 01 日 12:00 (UT) から過去に向かって 1000 年間だけ軌道積分するには、以下のようにすればよいでござる。
% ./backint_asteroids.py -v -d 1000 -n 100000 -o ceres.data Ceres Searching NASA Horizons for 'Sun'... Found: Sun (10) Searching NASA Horizons for 'Mercury'... Found: Mercury Barycenter (199) (chosen from query 'Mercury') Searching NASA Horizons for 'Venus'... Found: Venus Barycenter (299) (chosen from query 'Venus') Searching NASA Horizons for 'Earth'... Found: Earth-Moon Barycenter (3) (chosen from query 'Earth') Searching NASA Horizons for 'Mars'... Found: Mars Barycenter (4) (chosen from query 'Mars') Searching NASA Horizons for 'Jupiter'... Found: Jupiter Barycenter (5) (chosen from query 'Jupiter') Searching NASA Horizons for 'Saturn'... Found: Saturn Barycenter (6) (chosen from query 'Saturn') Searching NASA Horizons for 'Uranus'... Found: Uranus Barycenter (7) (chosen from query 'Uranus') Searching NASA Horizons for 'Neptune'... Found: Neptune Barycenter (8) (chosen from query 'Neptune') Searching NASA Horizons for 'Ceres'... Found: 1 Ceres (A801 AA) /usr/pkg/lib/python3.10/site-packages/rebound/horizons.py:172: RuntimeWarning: Warning: Mass cannot be retrieved from NASA HORIZONS. Set to 0. warnings.warn("Warning: Mass cannot be retrieved from NASA HORIZONS. Set to 0.", RuntimeWarning) progress of integration: 0.000% progress of integration: 10.000% progress of integration: 20.000% progress of integration: 30.000% progress of integration: 40.000% progress of integration: 50.000% progress of integration: 60.000% progress of integration: 70.000% progress of integration: 80.000% progress of integration: 90.000%
書き出される ceres.data というファイルの中身は以下のようになっているで ござる。
% ls -lF ceres.data -rw-r--r-- 1 daisuke taiwan 22500558 Jul 20 10:02 ceres.data % head -30 ceres.data # # Result of orbital integration by REBOUND # # start of simulation: 2023-07-20_02:01:14 # # input parameters # start date/time = 2000-01-01 12:00 # integrator = mercurius # timestep = -0.01 yr # duration_yr = 1000.0 yr # n_outputs = 100000 # file_output = ceres.data # verbose = True # # asteroids for orbital integration # Ceres # # format of output data # time (in yr) from start date/time, object name, x, y, z, vx, vy, vz, a, e, i, Omega, omega, M # -0.000000 Ceres -2.3864642362 0.7926901101 0.4632117513 -0.2080472740 -0.6158176149 0.0194022417 2.7789456954 0.0805478853 0.1843415718 1.4057721476 1.2775270197 0.1189638665 -0.010000 Ceres -2.3992529786 0.7539037527 0.4643757542 -0.1990195966 -0.6187534690 0.0176481546 2.7791166492 0.0805971545 0.1843407539 1.4057416086 1.2783202933 0.1318743264 -0.020000 Ceres -2.4114731284 0.7149375861 0.4654294364 -0.1899490437 -0.6215409477 0.0158907432 2.7792844192 0.0806444874 0.1843400288 1.4057111272 1.2791274691 0.1447707623 -0.030000 Ceres -2.4231220767 0.6758009541 0.4663726051 -0.1808383160 -0.6241793894 0.0141305255 2.7794489685 0.0806898550 0.1843393957 1.4056807135 1.2799477704 0.1576539347 -0.040000 Ceres -2.4341973841 0.6365032404 0.4672051004 -0.1716901249 -0.6266681940 0.0123680198 2.7796102624 0.0807332307 0.1843388542 1.4056503771 1.2807804166 0.1705246080 -0.050001 Ceres -2.4446967822 0.5970538642 0.4679267948 -0.1625071908 -0.6290068231 0.0106037440 2.7797682683 0.0807745899 0.1843384035 1.4056201277 1.2816246236 0.1833835506 -0.060001 Ceres -2.4546181737 0.5574622767 0.4685375933 -0.1532922421 -0.6311947999 0.0088382157 2.7799229560 0.0808139104 0.1843380431 1.4055899747 1.2824796049 0.1962315338 -0.070001 Ceres -2.4639596329 0.5177379572 0.4690374334 -0.1440480125 -0.6332317097 0.0070719514 2.7800742975 0.0808511722 0.1843377721 1.4055599275 1.2833445720 0.2090693309 -0.080001 Ceres -2.4727194060 0.4778904089 0.4694262852 -0.1347772401 -0.6351171994 0.0053054665 2.7802222673 0.0808863576 0.1843375900 1.4055299951 1.2842187348 0.2218977170 -0.090001 Ceres -2.4808959113 0.4379291554 0.4697041509 -0.1254826657 -0.6368509776 0.0035392753 2.7803668422 0.0809194512 0.1843374957 1.4055001866 1.2851013026 0.2347174678
軌道積分する小惑星は、小惑星の確定番号でも指定することができるでござる。 但し、番号の若い小惑星の場合、番号の後にセミコロンをつけないと JPL Horizons が惑星として取り扱ってしまうでござる。また、シェルに勘違いさ れないように、シングルクォート或いはダブルクォートでくくっておく必要が あるでござる。
% ./backint_asteroids.py -v -d 1000 -n 100000 -o ceres.data '1;' Searching NASA Horizons for 'Sun'... Found: Sun (10) Searching NASA Horizons for 'Mercury'... Found: Mercury Barycenter (199) (chosen from query 'Mercury') Searching NASA Horizons for 'Venus'... Found: Venus Barycenter (299) (chosen from query 'Venus') Searching NASA Horizons for 'Earth'... Found: Earth-Moon Barycenter (3) (chosen from query 'Earth') Searching NASA Horizons for 'Mars'... Found: Mars Barycenter (4) (chosen from query 'Mars') Searching NASA Horizons for 'Jupiter'... Found: Jupiter Barycenter (5) (chosen from query 'Jupiter') Searching NASA Horizons for 'Saturn'... Found: Saturn Barycenter (6) (chosen from query 'Saturn') Searching NASA Horizons for 'Uranus'... Found: Uranus Barycenter (7) (chosen from query 'Uranus') Searching NASA Horizons for 'Neptune'... Found: Neptune Barycenter (8) (chosen from query 'Neptune') Searching NASA Horizons for '1;'... Found: 1 Ceres (A801 AA) /usr/pkg/lib/python3.10/site-packages/rebound/horizons.py:172: RuntimeWarning: Warning: Mass cannot be retrieved from NASA HORIZONS. Set to 0. warnings.warn("Warning: Mass cannot be retrieved from NASA HORIZONS. Set to 0.", RuntimeWarning) progress of integration: 0.000% progress of integration: 10.000% progress of integration: 20.000% progress of integration: 30.000% progress of integration: 40.000% progress of integration: 50.000% progress of integration: 60.000% progress of integration: 70.000% progress of integration: 80.000% progress of integration: 90.000%
以下は、 1 とだけ指定した場合に、水星 (Mercury) だと解釈されてしまった 例でござる。
% ./backint_asteroids.py -v -d 1000 -n 100000 -o ceres.data 1 Searching NASA Horizons for 'Sun'... Found: Sun (10) Searching NASA Horizons for 'Mercury'... Found: Mercury Barycenter (199) (chosen from query 'Mercury') Searching NASA Horizons for 'Venus'... Found: Venus Barycenter (299) (chosen from query 'Venus') Searching NASA Horizons for 'Earth'... Found: Earth-Moon Barycenter (3) (chosen from query 'Earth') Searching NASA Horizons for 'Mars'... Found: Mars Barycenter (4) (chosen from query 'Mars') Searching NASA Horizons for 'Jupiter'... Found: Jupiter Barycenter (5) (chosen from query 'Jupiter') Searching NASA Horizons for 'Saturn'... Found: Saturn Barycenter (6) (chosen from query 'Saturn') Searching NASA Horizons for 'Uranus'... Found: Uranus Barycenter (7) (chosen from query 'Uranus') Searching NASA Horizons for 'Neptune'... Found: Neptune Barycenter (8) (chosen from query 'Neptune') Searching NASA Horizons for '1'... Found: Mercury Barycenter (199) progress of integration: 0.000% progress of integration: 10.000% progress of integration: 20.000% progress of integration: 30.000% progress of integration: 40.000% progress of integration: 50.000% progress of integration: 60.000% progress of integration: 70.000% progress of integration: 80.000% progress of integration: 90.000%
以下は、 1; と指定した場合に、やはり水星 (Mercury) だと解釈されてしまっ た例でござる。
% ./backint_asteroids.py -v -d 1000 -n 100000 -o ceres.data 1; Searching NASA Horizons for 'Sun'... Found: Sun (10) Searching NASA Horizons for 'Mercury'... Found: Mercury Barycenter (199) (chosen from query 'Mercury') Searching NASA Horizons for 'Venus'... Found: Venus Barycenter (299) (chosen from query 'Venus') Searching NASA Horizons for 'Earth'... Found: Earth-Moon Barycenter (3) (chosen from query 'Earth') Searching NASA Horizons for 'Mars'... Found: Mars Barycenter (4) (chosen from query 'Mars') Searching NASA Horizons for 'Jupiter'... Found: Jupiter Barycenter (5) (chosen from query 'Jupiter') Searching NASA Horizons for 'Saturn'... Found: Saturn Barycenter (6) (chosen from query 'Saturn') Searching NASA Horizons for 'Uranus'... Found: Uranus Barycenter (7) (chosen from query 'Uranus') Searching NASA Horizons for 'Neptune'... Found: Neptune Barycenter (8) (chosen from query 'Neptune') Searching NASA Horizons for '1'... Found: Mercury Barycenter (199) progress of integration: 0.000% progress of integration: 10.000% progress of integration: 20.000% progress of integration: 30.000% progress of integration: 40.000% progress of integration: 50.000% progress of integration: 60.000% progress of integration: 70.000% progress of integration: 80.000% progress of integration: 90.000%
複数の小惑星を指定し、軌道積分を実行した例は以下の通りでござる。
% ./backint_asteroids.py -v -d 1000 -n 100000 -o multiple.data Einstein Euler Hipparchus Curie 'Isaac Newton' Searching NASA Horizons for 'Sun'... Found: Sun (10) Searching NASA Horizons for 'Mercury'... Found: Mercury Barycenter (199) (chosen from query 'Mercury') Searching NASA Horizons for 'Venus'... Found: Venus Barycenter (299) (chosen from query 'Venus') Searching NASA Horizons for 'Earth'... Found: Earth-Moon Barycenter (3) (chosen from query 'Earth') Searching NASA Horizons for 'Mars'... Found: Mars Barycenter (4) (chosen from query 'Mars') Searching NASA Horizons for 'Jupiter'... Found: Jupiter Barycenter (5) (chosen from query 'Jupiter') Searching NASA Horizons for 'Saturn'... Found: Saturn Barycenter (6) (chosen from query 'Saturn') Searching NASA Horizons for 'Uranus'... Found: Uranus Barycenter (7) (chosen from query 'Uranus') Searching NASA Horizons for 'Neptune'... Found: Neptune Barycenter (8) (chosen from query 'Neptune') Searching NASA Horizons for 'Einstein'... Found: 2001 Einstein (1973 EB) /usr/pkg/lib/python3.10/site-packages/rebound/horizons.py:172: RuntimeWarning: Warning: Mass cannot be retrieved from NASA HORIZONS. Set to 0. warnings.warn("Warning: Mass cannot be retrieved from NASA HORIZONS. Set to 0.", RuntimeWarning) Searching NASA Horizons for 'Euler'... Found: 2002 Euler (1973 QQ1) Searching NASA Horizons for 'Hipparchus'... Found: 4000 Hipparchus (1989 AV) Searching NASA Horizons for 'Curie'... Found: 7000 Curie (1939 VD) Searching NASA Horizons for 'Isaac Newton'... Found: 8000 Isaac Newton (1986 RL5) progress of integration: 0.000% progress of integration: 10.000% progress of integration: 20.000% progress of integration: 30.000% progress of integration: 40.000% progress of integration: 50.000% progress of integration: 60.000% progress of integration: 70.000% progress of integration: 80.000% progress of integration: 90.000%
出力されたデータは以下の通りでござる。
% head -50 multiple.data # # Result of orbital integration by REBOUND # # start of simulation: 2023-07-20_02:16:29 # # input parameters # start date/time = 2000-01-01 12:00 # integrator = mercurius # timestep = -0.01 yr # duration_yr = 1000.0 yr # n_outputs = 100000 # file_output = multiple.data # verbose = True # # asteroids for orbital integration # Einstein # Euler # Hipparchus # Curie # Isaac Newton # # format of output data # time (in yr) from start date/time, object name, x, y, z, vx, vy, vz, a, e, i, Omega, omega, M # -0.000000 Einstein -1.5845757520 0.8842923825 0.3383303171 -0.3309353520 -0.6234363949 -0.2668746964 1.9423652033 0.0976168573 0.3960436571 -0.0468347395 3.7698320826 5.3302157144 -0.000000 Euler 2.4129809188 -0.5542653935 0.0757031944 0.1785643250 0.5927596998 -0.0893219899 2.3996225653 0.0695693306 0.1487003555 3.1213627916 0.8204927397 1.9909406310 -0.000000 Hipparchus 2.6066298991 -1.1388365225 0.0410338382 0.1966437477 0.5251734594 0.0248792650 2.5749772010 0.1172708514 0.0474139820 -0.7208120625 3.0356952589 3.6594875204 -0.000000 Curie -2.6790456061 -1.4171865681 0.2848583035 0.2800660823 -0.4112558669 -0.0776134637 2.4768078694 0.2587712374 0.1722941365 1.0571260482 5.3315162882 3.7458748795 -0.000000 Isaac_Newton -2.7814237458 1.7926105195 0.3903084558 -0.2781137547 -0.4393518905 -0.0655711917 3.0681030314 0.0889294327 0.1703055328 0.1833200456 5.2847597154 3.4212522565 -0.010000 Einstein -1.6048669241 0.8448432760 0.3214564989 -0.3148942479 -0.6321984606 -0.2702055355 1.9427216273 0.0977850404 0.3960275133 -0.0468128316 3.7688743765 5.3545689226 -0.010000 Euler 2.4238884982 -0.5169512929 0.0700813849 0.1686287217 0.5949465682 -0.0896199911 2.3995563739 0.0696777167 0.1487012414 3.1213615803 0.8207549021 2.0073816322 -0.010000 Hipparchus 2.6187622405 -1.1057425010 0.0425935332 0.1895290141 0.5282174313 0.0247661307 2.5748786716 0.1172769170 0.0474159537 -0.7207984883 3.0363059890 3.6739434986 -0.010000 Curie -2.6612601291 -1.4429263230 0.2799616759 0.2860548151 -0.4080460583 -0.0782480532 2.4768161707 0.2587911909 0.1722964472 1.0571173674 5.3313926755 3.7622331295 -0.010000 Isaac_Newton -2.7987492934 1.7649092363 0.3861676200 -0.2733645390 -0.4423910137 -0.0662332892 3.0681993760 0.0889007241 0.1703046139 0.1833251794 5.2844944795 3.4332439548 -0.020000 Einstein -1.6241404735 0.8048544204 0.3043781006 -0.2985450758 -0.6406151372 -0.2733858806 1.9430755367 0.0979580404 0.3960109636 -0.0467917576 3.7680024532 5.3788383893 -0.020000 Euler 2.4341713499 -0.4795047892 0.0644416017 0.1586792272 0.5969742529 -0.0898941114 2.3994931161 0.0697854526 0.1487021358 3.1213604565 0.8210592250 2.0237802446 -0.020000 Hipparchus 2.6304460826 -1.0724598652 0.0441459796 0.1823678264 0.5311770605 0.0246485342 2.5747805861 0.1172815134 0.0474179150 -0.7207844171 3.0369189689 3.6883966492 -0.020000 Curie -2.6430988895 -1.4684621830 0.2750254445 0.2920267832 -0.4047656888 -0.0788740631 2.4768243734 0.2588118210 0.1722987618 1.0571088684 5.3312700005 3.7785902948 -0.020000 Isaac_Newton -2.8157754032 1.7370184159 0.3819853907 -0.2685825071 -0.4453849364 -0.0668887642 3.0682954412 0.0888724456 0.1703036751 0.1833303217 5.2842240375 3.4452406933 -0.030000 Einstein -1.6423773006 0.7643478619 0.2871047166 -0.2818961002 -0.6486754757 -0.2764113301 1.9434265674 0.0981354558 0.3959940137 -0.0467715631 3.7672191101 5.4030213618 -0.030000 Euler 2.4438286775 -0.4419358701 0.0587853433 0.1487182764 0.5988432088 -0.0901444256 2.3994327598 0.0698924669 0.1487030378 3.1213594221 0.8214050736 2.0401370938 -0.030000 Hipparchus 2.6416785279 -1.0389939365 0.0456908968 0.1751608666 0.5340516494 0.0245264718 2.5746829462 0.1172846295 0.0474198655 -0.7207698518 3.0375340005 3.7028471669 -0.030000 Curie -2.6245629544 -1.4937897019 0.2700501515 0.2979815529 -0.4014143537 -0.0794913928 2.4768324710 0.2588331284 0.1723010801 1.0571005509 5.3311483070 3.7949463361 -0.030000 Isaac_Newton -2.8325000238 1.7089409079 0.3777621860 -0.2637679913 -0.4483333524 -0.0675375545 3.0683912284 0.0888446040 0.1703027163 0.1833354707 5.2839484661 3.4572423975 -0.040000 Einstein -1.6595588411 0.7233463313 0.2696462163 -0.2649561073 -0.6563686336 -0.2792775375 1.9437743464 0.0983168653 0.3959766699 -0.0467522933 3.7665269145 5.4271153162 -0.040000 Euler 2.4528598370 -0.4042544940 0.0531141028 0.1387482771 0.6005539195 -0.0903710131 2.3993752729 0.0699986901 0.1487039466 3.1213584791 0.8217917982 2.0564528204 -0.040000 Hipparchus 2.6524567221 -1.0053500803 0.0472280042 0.1679088251 0.5368404969 0.0243999399 2.5745857538 0.1172862548 0.0474218050 -0.7207547955 3.0381508840 3.7172952484 -0.040000 Curie -2.6056534183 -1.5189044078 0.2650363456 0.3039186770 -0.3979916366 -0.0800999388 2.4768404567 0.2588551138 0.1723034016 1.0570924147 5.3310276400 3.8113012135 -0.040000 Isaac_Newton -2.8489211247 1.6806795813 0.3734984278 -0.2589213259 -0.4512359558 -0.0681795980 3.0684867391 0.0888172057 0.1703017373 0.1833406245 5.2836678431 3.4692489919 -0.050001 Einstein -1.6756670996 0.6818732368 0.2520127402 -0.2477344101 -0.6636838982 -0.2819802212 1.9441184921 0.0985018289 0.3959589393 -0.0467339936 3.7659281904 5.4511179692