==================================================================================================== HOW TO LOSE A PLANET BY CHOOSING THE WRONG INTEGRATION METHOD Energy and angular-momentum drift of forward Euler, velocity Verlet and RK4 on a two-body Kepler orbit, checked against the exact solution Science Journaling Club, Volume 3 Issue 1, Fall 2026 ==================================================================================================== numpy version : 2.4.2 python version : 3.12.3 master seed : 20260901 units : GM = 1.0, a = 1.0 -> period T = 2*pi = 6.2831853072 default eccentricity : 0.6 (used for the single-orbit figures) ==================================================================================================== VALIDATION A: is Newton's solver for Kepler's equation actually solved? ==================================================================================================== Checking M = E - e*sin(E) is satisfied to the requested tolerance over a grid of mean anomalies and eccentricities that spans everything used below. cases checked : 740 worst residual |M-E+e sinE| : 8.882e-16 (at e=0.350, M=4.363) max Newton iterations needed : 9 club value: worst residual 8.882e-16 vs accepted value: 0 (exact equation) difference: 8.882e-16, i.e. machine precision, not a real disagreement ==================================================================================================== VALIDATION B: does the analytic Kepler solution actually conserve E and L? ==================================================================================================== The whole convergence study below is only as good as the closed-form orbit it is checked against. Sampling that closed form densely and independently recomputing energy and angular momentum from position and velocity at each sample must reproduce the two theoretical constants of motion. eccentricity used : 0.6 theoretical E0 = -GM/2a : -0.500000000000000 theoretical L0 = sqrt(GMa(1-e^2)) : 0.800000000000000 samples across 7 orbits : 4000 club value: worst |dE/E0| along analytic curve : 4.441e-15 club value: worst |dL/L0| along analytic curve : 5.551e-16 accepted value (exact Kepler orbit): 0 for both difference: 4.441e-15 and 5.551e-16, both at the double-precision floor [elapsed so far: 0.03 s] ==================================================================================================== SECTION 1: the headline run -- one orbit, three methods, dt = T/100 ==================================================================================================== eccentricity e : 0.6 steps per orbit : 100 (dt = 0.062832) orbits integrated : 300 (30000 steps) This single configuration is reused as the default in the interactive model and as the headline numbers quoted in the article's opening. initial energy E0 : -0.5000000000 (theory -GM/2a = -0.5000000000) initial ang. mom. L0 : 0.8000000000 (theory sqrt(GMa(1-e^2)) = 0.8000000000) -- forward Euler -- wall time : 0.07 s orbits actually reached: 300.00 of 300 requested final relative energy error dE/E0 : +1.165578e+00 max |relative energy error| over run : 1.188178e+00 final relative ang.mom. error dL/L0 : +3.534326e-01 max |relative ang.mom. error| over run: 3.534326e-01 -- velocity Verlet -- wall time : 0.08 s orbits actually reached: 300.00 of 300 requested final relative energy error dE/E0 : +2.893916e-02 max |relative energy error| over run : 2.989145e-02 final relative ang.mom. error dL/L0 : -1.887379e-14 max |relative ang.mom. error| over run: 2.386980e-14 -- RK4 -- wall time : 0.19 s orbits actually reached: 300.00 of 300 requested final relative energy error dE/E0 : -6.448211e-02 max |relative energy error| over run : 6.448212e-02 final relative ang.mom. error dL/L0 : -7.303539e-03 max |relative ang.mom. error| over run: 7.303539e-03 Orbit at which |dE/E0| first crosses a threshold, and whether the trace later falls back under half that threshold (transient) or stays above it for the rest of the recorded run (permanent, i.e. the planet is really lost): method threshold first-cross orbit character forward Euler 1% 0.00 permanent forward Euler 5% 0.00 permanent forward Euler 10% 0.00 permanent velocity Verlet 1% 0.02 transient (recovers) velocity Verlet 5% -- not reached velocity Verlet 10% -- not reached RK4 1% 47.64 permanent RK4 5% 234.19 permanent RK4 10% -- not reached Secular-trend check: ordinary-least-squares slope of the relative energy error against orbit number. A true secular drift has a slope that does not shrink as the fitting window lengthens; a bounded oscillation's slope over the whole run should be consistent with zero relative to its own scatter. forward Euler slope = +1.484941e-06 per orbit, SE(slope) = 5.82e-07, slope/SE = +2.6 velocity Verlet slope = -1.256753e-08 per orbit, SE(slope) = 3.36e-07, slope/SE = -0.0 RK4 slope = -2.146028e-04 per orbit, SE(slope) = 9.79e-09, slope/SE = -21914.6 SVG_DATA energy-vs-orbit (one point per orbit: orbit_number,relative_energy_error), for the article's Figure 2: EVO euler 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EVO verlet 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5;284.0000,0.029677;285.0000,0.029239;286.0000,0.029025;287.0000,0.028924;288.0000,0.028873;289.0000,0.028845;290.0000,0.028829;291.0000,0.028820;292.0000,0.028815;293.0000,0.028813;294.0000,0.028813;295.0000,0.028816;296.0000,0.028822;297.0000,0.028832;298.0000,0.028849;299.0000,0.028881;300.0000,0.028939 EVO rk4 0.0000,0.000000;1.0000,-0.000210;2.0000,-0.000421;3.0000,-0.000634;4.0000,-0.000849;5.0000,-0.001067;6.0000,-0.001287;7.0000,-0.001511;8.0000,-0.001737;9.0000,-0.001966;10.0000,-0.002195;11.0000,-0.002424;12.0000,-0.002652;13.0000,-0.002876;14.0000,-0.003097;15.0000,-0.003314;16.0000,-0.003529;17.0000,-0.003742;18.0000,-0.003954;19.0000,-0.004164;20.0000,-0.004374;21.0000,-0.004583;22.0000,-0.004792;23.0000,-0.005001;24.0000,-0.005209;25.0000,-0.005418;26.0000,-0.005626;27.0000,-0.005834;28.0000,-0.006042;29.0000,-0.006250;30.0000,-0.006458;31.0000,-0.006666;32.0000,-0.006873;33.0000,-0.007081;34.0000,-0.007289;35.0000,-0.007497;36.0000,-0.007704;37.0000,-0.007912;38.0000,-0.008120;39.0000,-0.008327;40.0000,-0.008535;41.0000,-0.008742;42.0000,-0.008950;43.0000,-0.009157;44.0000,-0.009365;45.0000,-0.009572;46.0000,-0.009780;47.0000,-0.009987;48.0000,-0.010194;49.0000,-0.010402;50.0000,-0.010609;51.0000,-0.010816;52.0000,-0.011023;53.0000,-0.011231;54.0000,-0.011438;55.0000,-0.011645;56.0000,-0.011852;57.0000,-0.012059;58.0000,-0.012266;59.0000,-0.012473;60.0000,-0.012680;61.0000,-0.012887;62.0000,-0.013094;63.0000,-0.013301;64.0000,-0.013508;65.0000,-0.013715;66.0000,-0.013922;67.0000,-0.014129;68.0000,-0.014336;69.0000,-0.014543;70.0000,-0.014750;71.0000,-0.014956;72.0000,-0.015163;73.0000,-0.015370;74.0000,-0.015576;75.0000,-0.015782;76.0000,-0.015988;77.0000,-0.016193;78.0000,-0.016396;79.0000,-0.016594;80.0000,-0.016944;81.0000,-0.017215;82.0000,-0.017432;83.0000,-0.017641;84.0000,-0.017848;85.0000,-0.018055;86.0000,-0.018261;87.0000,-0.018468;88.0000,-0.018674;89.0000,-0.018880;90.0000,-0.019087;91.0000,-0.019293;92.0000,-0.019499;93.0000,-0.019705;94.0000,-0.019911;95.0000,-0.020118;96.0000,-0.020324;97.0000,-0.020530;98.0000,-0.020736;99.0000,-0.020942;100.0000,-0.021148;101.0000,-0.021354;102.0000,-0.021560;103.0000,-0.021766;104.0000,-0.021972;105.0000,-0.022178;106.0000,-0.022384;107.0000,-0.022589;108.0000,-0.022795;109.0000,-0.023000;110.0000,-0.023204;111.0000,-0.023407;112.0000,-0.023621;113.0000,-0.024003;114.0000,-0.024231;115.0000,-0.024440;116.0000,-0.024646;117.0000,-0.024852;118.0000,-0.025058;119.0000,-0.025263;120.0000,-0.025469;121.0000,-0.025674;122.0000,-0.025880;123.0000,-0.026085;124.0000,-0.026291;125.0000,-0.026496;126.0000,-0.026701;127.0000,-0.026907;128.0000,-0.027112;129.0000,-0.027317;130.0000,-0.027523;131.0000,-0.027728;132.0000,-0.027933;133.0000,-0.028138;134.0000,-0.028343;135.0000,-0.028547;136.0000,-0.028750;137.0000,-0.028947;138.0000,-0.029348;139.0000,-0.029571;140.0000,-0.029778;141.0000,-0.029984;142.0000,-0.030189;143.0000,-0.030394;144.0000,-0.030599;145.0000,-0.030803;146.0000,-0.031008;147.0000,-0.031213;148.0000,-0.031418;149.0000,-0.031623;150.0000,-0.031827;151.0000,-0.032032;152.0000,-0.032237;153.0000,-0.032442;154.0000,-0.032646;155.0000,-0.032851;156.0000,-0.033055;157.0000,-0.033257;158.0000,-0.033444;159.0000,-0.033853;160.0000,-0.034076;161.0000,-0.034282;162.0000,-0.034487;163.0000,-0.034691;164.0000,-0.034896;165.0000,-0.035100;166.0000,-0.035305;167.0000,-0.035509;168.0000,-0.035713;169.0000,-0.035917;170.0000,-0.036122;171.0000,-0.036326;172.0000,-0.036530;173.0000,-0.036734;174.0000,-0.036938;175.0000,-0.037141;176.0000,-0.037342;177.0000,-0.037634;178.0000,-0.037954;179.0000,-0.038162;180.0000,-0.038367;181.0000,-0.038571;182.0000,-0.038775;183.0000,-0.038979;184.0000,-0.039182;185.0000,-0.039386;186.0000,-0.039590;187.0000,-0.039794;188.0000,-0.039998;189.0000,-0.040202;190.0000,-0.040405;191.0000,-0.040609;192.0000,-0.040811;193.0000,-0.041009;194.0000,-0.041382;195.0000,-0.041625;196.0000,-0.041831;197.0000,-0.042034;198.0000,-0.042238;199.0000,-0.042441;200.0000,-0.042645;201.0000,-0.042848;202.0000,-0.043052;203.0000,-0.043255;204.0000,-0.043459;205.0000,-0.043662;206.0000,-0.043865;207.0000,-0.044068;208.0000,-0.044269;209.0000,-0.044514;210.0000,-0.044878;211.0000,-0.045084;212.0000,-0.045288;213.0000,-0.045491;214.0000,-0.045694;215.0000,-0.045897;216.0000,-0.046100;217.0000,-0.046303;218.0000,-0.046506;219.0000,-0.046709;220.0000,-0.046912;221.0000,-0.047115;222.0000,-0.047317;223.0000,-0.047501;224.0000,-0.047919;225.0000,-0.048129;226.0000,-0.048333;227.0000,-0.048535;228.0000,-0.048738;229.0000,-0.048941;230.0000,-0.049144;231.0000,-0.049346;232.0000,-0.049549;233.0000,-0.049752;234.0000,-0.049954;235.0000,-0.050156;236.0000,-0.050355;237.0000,-0.050721;238.0000,-0.050966;239.0000,-0.051169;240.0000,-0.051372;241.0000,-0.051574;242.0000,-0.051777;243.0000,-0.051979;244.0000,-0.052182;245.0000,-0.052384;246.0000,-0.052586;247.0000,-0.052788;248.0000,-0.052990;249.0000,-0.053179;250.0000,-0.053589;251.0000,-0.053799;252.0000,-0.054002;253.0000,-0.054204;254.0000,-0.054406;255.0000,-0.054608;256.0000,-0.054810;257.0000,-0.055012;258.0000,-0.055214;259.0000,-0.055416;260.0000,-0.055617;261.0000,-0.055809;262.0000,-0.056216;263.0000,-0.056426;264.0000,-0.056628;265.0000,-0.056830;266.0000,-0.057031;267.0000,-0.057233;268.0000,-0.057435;269.0000,-0.057637;270.0000,-0.057838;271.0000,-0.058040;272.0000,-0.058239;273.0000,-0.058552;274.0000,-0.058845;275.0000,-0.059048;276.0000,-0.059250;277.0000,-0.059451;278.0000,-0.059653;279.0000,-0.059854;280.0000,-0.060056;281.0000,-0.060257;282.0000,-0.060458;283.0000,-0.060658;284.0000,-0.060972;285.0000,-0.061263;286.0000,-0.061465;287.0000,-0.061667;288.0000,-0.061868;289.0000,-0.062069;290.0000,-0.062270;291.0000,-0.062472;292.0000,-0.062673;293.0000,-0.062873;294.0000,-0.063069;295.0000,-0.063469;296.0000,-0.063678;297.0000,-0.063879;298.0000,-0.064080;299.0000,-0.064281;300.0000,-0.064482 SVG_DATA energy-vs-orbit min/max per orbit (orbit_number,min,max), resolves the within-orbit oscillation instead of aliasing it (the single-sample-per-orbit series above can land on any phase of that oscillation and is kept only for the plain-arithmetic numbers quoted in the text): EVOMM euler 1,0.000000,1.188178;2,1.165898,1.166583;3,1.165741,1.165898;4,1.165679,1.165741;5,1.165648,1.165679;6,1.165629,1.165648;7,1.165617,1.165629;8,1.165609,1.165617;9,1.165603,1.165609;10,1.165599,1.165603;11,1.165596,1.165599;12,1.165593,1.165596;13,1.165591,1.165593;14,1.165590,1.165591;15,1.165588,1.165590;16,1.165587,1.165588;17,1.165586,1.165587;18,1.165585,1.165586;19,1.165585,1.165585;20,1.165584,1.165585;21,1.165583,1.165584;22,1.165583,1.165583;23,1.165583,1.165583;24,1.165582,1.165583;25,1.165582,1.165582;26,1.165582,1.165582;27,1.165581,1.165582;28,1.165581,1.165581;29,1.165581,1.165581;30,1.165581,1.165581;31,1.165581,1.165581;32,1.165580,1.165581;33,1.165580,1.165580;34,1.165580,1.165580;35,1.165580,1.165580;36,1.165580,1.165580;37,1.165580,1.165580;38,1.165580,1.165580;39,1.165580,1.165580;40,1.165579,1.165580;41,1.165579,1.165579;42,1.165579,1.165579;43,1.165579,1.165579;44,1.165579,1.165579;45,1.165579,1.165579;46,1.165579,1.165579;47,1.165579,1.165579;48,1.165579,1.165579;49,1.165579,1.165579;50,1.165579,1.165579;51,1.165579,1.165579;52,1.165579,1.165579;53,1.165579,1.165579;54,1.165579,1.165579;55,1.165579,1.165579;56,1.165579,1.165579;57,1.165579,1.165579;58,1.165578,1.165579;59,1.165578,1.165578;60,1.165578,1.165578;61,1.165578,1.165578;62,1.165578,1.165578;63,1.165578,1.165578;64,1.165578,1.165578;65,1.165578,1.165578;66,1.165578,1.165578;67,1.165578,1.165578;68,1.165578,1.165578;69,1.165578,1.165578;70,1.165578,1.165578;71,1.165578,1.165578;72,1.165578,1.165578;73,1.165578,1.165578;74,1.165578,1.165578;75,1.165578,1.165578;76,1.165578,1.165578;77,1.165578,1.165578;78,1.165578,1.165578;79,1.165578,1.165578;80,1.165578,1.165578;81,1.165578,1.165578;82,1.165578,1.165578;83,1.165578,1.165578;84,1.165578,1.165578;85,1.165578,1.165578;86,1.165578,1.165578;87,1.165578,1.165578;88,1.165578,1.165578;89,1.165578,1.165578;90,1.165578,1.165578;91,1.165578,1.165578;92,1.165578,1.165578;93,1.165578,1.165578;94,1.165578,1.165578;95,1.165578,1.165578;96,1.165578,1.165578;97,1.165578,1.165578;98,1.165578,1.165578;99,1.165578,1.165578;100,1.165578,1.165578;101,1.165578,1.165578;102,1.165578,1.165578;103,1.165578,1.165578;104,1.165578,1.165578;105,1.165578,1.165578;106,1.165578,1.165578;107,1.165578,1.165578;108,1.165578,1.165578;109,1.165578,1.165578;110,1.165578,1.165578;111,1.165578,1.165578;112,1.165578,1.165578;113,1.165578,1.165578;114,1.165578,1.165578;115,1.165578,1.165578;116,1.165578,1.165578;117,1.165578,1.165578;118,1.165578,1.165578;119,1.165578,1.165578;120,1.165578,1.165578;121,1.165578,1.165578;122,1.165578,1.165578;123,1.165578,1.165578;124,1.165578,1.165578;125,1.165578,1.165578;126,1.165578,1.165578;127,1.165578,1.165578;128,1.165578,1.165578;129,1.165578,1.165578;130,1.165578,1.165578;131,1.165578,1.165578;132,1.165578,1.165578;133,1.165578,1.165578;134,1.165578,1.165578;135,1.165578,1.165578;136,1.165578,1.165578;137,1.165578,1.165578;138,1.165578,1.165578;139,1.165578,1.165578;140,1.165578,1.165578;141,1.165578,1.165578;142,1.165578,1.165578;143,1.165578,1.165578;144,1.165578,1.165578;145,1.165578,1.165578;146,1.165578,1.165578;147,1.165578,1.165578;148,1.165578,1.165578;149,1.165578,1.165578;150,1.165578,1.165578;151,1.165578,1.165578;152,1.165578,1.165578;153,1.165578,1.165578;154,1.165578,1.165578;155,1.165578,1.165578;156,1.165578,1.165578;157,1.165578,1.165578;158,1.165578,1.165578;159,1.165578,1.165578;160,1.165578,1.165578;161,1.165578,1.165578;162,1.165578,1.165578;163,1.165578,1.165578;164,1.165578,1.165578;165,1.165578,1.165578;166,1.165578,1.165578;167,1.165578,1.165578;168,1.165578,1.165578;169,1.165578,1.165578;170,1.165578,1.165578;171,1.165578,1.165578;172,1.165578,1.165578;173,1.165578,1.165578;174,1.165578,1.165578;175,1.165578,1.165578;176,1.165578,1.165578;177,1.165578,1.165578;178,1.165578,1.165578;179,1.165578,1.165578;180,1.165578,1.165578;181,1.165578,1.165578;182,1.165578,1.165578;183,1.165578,1.165578;184,1.165578,1.165578;185,1.165578,1.165578;186,1.165578,1.165578;187,1.165578,1.165578;188,1.165578,1.165578;189,1.165578,1.165578;190,1.165578,1.165578;191,1.165578,1.165578;192,1.165578,1.165578;193,1.165578,1.165578;194,1.165578,1.165578;195,1.165578,1.165578;196,1.165578,1.165578;197,1.165578,1.165578;198,1.165578,1.165578;199,1.165578,1.165578;200,1.165578,1.165578;201,1.165578,1.165578;202,1.165578,1.165578;203,1.165578,1.165578;204,1.165578,1.165578;205,1.165578,1.165578;206,1.165578,1.165578;207,1.165578,1.165578;208,1.165578,1.165578;209,1.165578,1.165578;210,1.165578,1.165578;211,1.165578,1.165578;212,1.165578,1.165578;213,1.165578,1.165578;214,1.165578,1.165578;215,1.165578,1.165578;216,1.165578,1.165578;217,1.165578,1.165578;218,1.165578,1.165578;219,1.165578,1.165578;220,1.165578,1.165578;221,1.165578,1.165578;222,1.165578,1.165578;223,1.165578,1.165578;224,1.165578,1.165578;225,1.165578,1.165578;226,1.165578,1.165578;227,1.165578,1.165578;228,1.165578,1.165578;229,1.165578,1.165578;230,1.165578,1.165578;231,1.165578,1.165578;232,1.165578,1.165578;233,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EVOMM verlet 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[elapsed so far: 0.40 s] ==================================================================================================== VALIDATION C: does velocity Verlet conserve angular momentum to machine precision? ==================================================================================================== Theory: for a kick-drift-kick leapfrog step applied to a purely central force, both the kick (v changes by a vector parallel to x, so x cross dv = 0) and the drift (x changes by a vector parallel to v, so dx cross v = 0) leave L = x*vy - y*vx exactly unchanged in exact arithmetic. The only source of drift in L should therefore be floating-point roundoff, not truncation error, unlike energy, whose drift is truncation error and does depend on the order. forward Euler max |dL/L0| over 300 orbits : 3.534e-01 (1591719023413770.0 machine epsilons) velocity Verlet max |dL/L0| over 300 orbits : 2.387e-14 ( 107.5 machine epsilons) RK4 max |dL/L0| over 300 orbits : 7.304e-03 (32892216617908.1 machine epsilons) club value: velocity Verlet max |dL/L0| = 2.387e-14 accepted value (theory): exactly 0, up to floating-point roundoff (~2.2e-16 per operation, accumulating as a random walk over 30000 steps) roundoff random-walk estimate ~ eps*sqrt(n_steps) = 3.846e-14 difference: club value is 0.62x the naive roundoff random-walk estimate -- consistent, not a real disagreement for contrast, forward Euler's L drifts 14,806,688,589,896x further and RK4's 305,974,108,074x further over the same run, both from truncation error, not roundoff [elapsed so far: 0.40 s] ==================================================================================================== SECTION 2: orbit trajectories for the figure -- what each method actually draws ==================================================================================================== Same e=0.6, dt=T/100 as the headline run, 2 orbits, every step recorded. forward Euler x range [-9.451, +0.400] y range [+0.000, +2.061] points=201 velocity Verlet x range [-1.663, +0.400] y range [-0.804, +0.837] points=201 RK4 x range [-1.600, +0.400] y range [-0.800, +0.800] points=201 analytic ellipse x range [-1.600, +0.400] y range [-0.800, +0.800] points=400 SVG_DATA trajectory (5 significant figures, for the article's Figure 1): TRAJ euler 0.40000,0.00000;0.40000,0.12566;0.37533,0.25133;0.32923,0.37026;0.26705,0.47843;0.19419,0.57458;0.11492,0.65924;0.03221,0.73374;-0.05201,0.79956;-0.13656,0.85806;-0.22070,0.91042;-0.30402,0.95763;-0.38629,1.00046;-0.46736,1.03956;-0.54721,1.07546;-0.62580,1.10859;-0.70317,1.13930;-0.77934,1.16789;-0.85435,1.19460;-0.92825,1.21965;-1.00109,1.24322;-1.07291,1.26544;-1.14375,1.28646;-1.21367,1.30638;-1.28271,1.32531;-1.35089,1.34333;-1.41828,1.36051;-1.48489,1.37693;-1.55076,1.39264;-1.61593,1.40769;-1.68042,1.42214;-1.74426,1.43603;-1.80748,1.44938;-1.87010,1.46225;-1.93215,1.47465;-1.99365,1.48663;-2.05462,1.49820;-2.11507,1.50938;-2.17504,1.52021;-2.23452,1.53070;-2.29355,1.54086;-2.35213,1.55073;-2.41028,1.56030;-2.46802,1.56960;-2.52536,1.57864;-2.58230,1.58743;-2.63887,1.59599;-2.69507,1.60432;-2.75092,1.61244;-2.80643,1.62035;-2.86159,1.62806;-2.91644,1.63559;-2.97096,1.64294;-3.02518,1.65011;-3.07910,1.65712;-3.13272,1.66397;-3.18607,1.67066;-3.23913,1.67721;-3.29193,1.68362;-3.34446,1.68989;-3.39673,1.69603;-3.44875,1.70204;-3.50053,1.70793;-3.55207,1.71371;-3.60338,1.71936;-3.65445,1.72491;-3.70531,1.73035;-3.75594,1.73569;-3.80636,1.74093;-3.85657,1.74607;-3.90658,1.75112;-3.95638,1.75607;-4.00599,1.76094;-4.05541,1.76573;-4.10464,1.77043;-4.15368,1.77505;-4.20254,1.77959;-4.25122,1.78405;-4.29973,1.78844;-4.34807,1.79276;-4.39624,1.79701;-4.44424,1.80119;-4.49208,1.80531;-4.53977,1.80936;-4.58729,1.81335;-4.63467,1.81728;-4.68189,1.82114;-4.72896,1.82495;-4.77589,1.82870;-4.82268,1.83240;-4.86932,1.83604;-4.91583,1.83963;-4.96220,1.84317;-5.00843,1.84666;-5.05453,1.85010;-5.10051,1.85349;-5.14635,1.85684;-5.19207,1.86014;-5.23766,1.86339;-5.28314,1.86660;-5.32849,1.86977;-5.37372,1.87290;-5.41884,1.87598;-5.46384,1.87902;-5.50873,1.88203;-5.55350,1.88500;-5.59817,1.88793;-5.64273,1.89082;-5.68718,1.89368;-5.73152,1.89650;-5.77576,1.89928;-5.81990,1.90203;-5.86394,1.90475;-5.90787,1.90744;-5.95171,1.91009;-5.99545,1.91272;-6.03909,1.91531;-6.08264,1.91787;-6.12610,1.92040;-6.16946,1.92290;-6.21273,1.92538;-6.25591,1.92782;-6.29900,1.93024;-6.34200,1.93263;-6.38492,1.93499;-6.42775,1.93733;-6.47050,1.93965;-6.51316,1.94193;-6.55574,1.94419;-6.59823,1.94643;-6.64065,1.94865;-6.68299,1.95084;-6.72524,1.95300;-6.76742,1.95515;-6.80952,1.95727;-6.85155,1.95937;-6.89350,1.96145;-6.93537,1.96350;-6.97717,1.96554;-7.01890,1.96755;-7.06056,1.96955;-7.10214,1.97152;-7.14365,1.97348;-7.18510,1.97541;-7.22647,1.97733;-7.26777,1.97922;-7.30901,1.98110;-7.35018,1.98296;-7.39129,1.98481;-7.43232,1.98663;-7.47330,1.98844;-7.51421,1.99023;-7.55505,1.99200;-7.59583,1.99376;-7.63655,1.99550;-7.67721,1.99722;-7.71780,1.99893;-7.75834,2.00062;-7.79881,2.00230;-7.83923,2.00396;-7.87958,2.00560;-7.91988,2.00723;-7.96012,2.00885;-8.00031,2.01045;-8.04043,2.01204;-8.08050,2.01361;-8.12051,2.01517;-8.16047,2.01671;-8.20038,2.01824;-8.24023,2.01976;-8.28002,2.02127;-8.31977,2.02276;-8.35946,2.02424;-8.39909,2.02570;-8.43868,2.02716;-8.47821,2.02860;-8.51770,2.03003;-8.55713,2.03145;-8.59651,2.03285;-8.63584,2.03424;-8.67513,2.03563;-8.71436,2.03700;-8.75355,2.03835;-8.79269,2.03970;-8.83178,2.04104;-8.87082,2.04236;-8.90982,2.04368;-8.94877,2.04498;-8.98768,2.04628;-9.02653,2.04756;-9.06535,2.04883;-9.10412,2.05009;-9.14284,2.05135;-9.18152,2.05259;-9.22016,2.05382;-9.25875,2.05505;-9.29730,2.05626;-9.33580,2.05746;-9.37427,2.05866;-9.41269,2.05984;-9.45107,2.06102 TRAJ verlet 0.40000,0.00000;0.38766,0.12566;0.35271,0.24400;0.30011,0.35012;0.23543,0.44215;0.16335,0.52029;0.08730,0.58576;0.00958,0.64011;-0.06827,0.68482;-0.14530,0.72123;-0.22089,0.75050;-0.29466,0.77358;-0.36638,0.79127;-0.43592,0.80425;-0.50321,0.81309;-0.56822,0.81825;-0.63097,0.82015;-0.69147,0.81912;-0.74975,0.81547;-0.80586,0.80945;-0.85983,0.80129;-0.91171,0.79118;-0.96154,0.77929;-1.00937,0.76578;-1.05524,0.75078;-1.09920,0.73442;-1.14128,0.71681;-1.18151,0.69803;-1.21994,0.67819;-1.25660,0.65737;-1.29152,0.63564;-1.32473,0.61306;-1.35626,0.58971;-1.38613,0.56564;-1.41438,0.54090;-1.44101,0.51555;-1.46606,0.48963;-1.48954,0.46318;-1.51147,0.43626;-1.53187,0.40889;-1.55075,0.38111;-1.56813,0.35297;-1.58402,0.32449;-1.59842,0.29571;-1.61136,0.26666;-1.62284,0.23736;-1.63287,0.20786;-1.64145,0.17817;-1.64859,0.14832;-1.65430,0.11834;-1.65857,0.08826;-1.66142,0.05811;-1.66284,0.02790;-1.66283,-0.00233;-1.66139,-0.03255;-1.65853,-0.06275;-1.65423,-0.09290;-1.64849,-0.12296;-1.64132,-0.15292;-1.63270,-0.18274;-1.62262,-0.21240;-1.61108,-0.24187;-1.59808,-0.27111;-1.58359,-0.30011;-1.56761,-0.32882;-1.55012,-0.35722;-1.53111,-0.38526;-1.51056,-0.41292;-1.48847,-0.44016;-1.46480,-0.46693;-1.43954,-0.49319;-1.41267,-0.51891;-1.38416,-0.54402;-1.35399,-0.56847;-1.32213,-0.59222;-1.28855,-0.61520;-1.25323,-0.63735;-1.21613,-0.65859;-1.17721,-0.67884;-1.13644,-0.69803;-1.09378,-0.71606;-1.04919,-0.73282;-1.00262,-0.74820;-0.95403,-0.76208;-0.90337,-0.77430;-0.85059,-0.78470;-0.79565,-0.79311;-0.73849,-0.79931;-0.67907,-0.80306;-0.61734,-0.80408;-0.55327,-0.80206;-0.48685,-0.79662;-0.41806,-0.78731;-0.34694,-0.77361;-0.27357,-0.75490;-0.19812,-0.73043;-0.12087,-0.69931;-0.04227,-0.66047;0.03689,-0.61263;0.11543,-0.55434;0.19146,-0.48398;0.26212,-0.40008;0.32333,-0.30174;0.36979,-0.18963;0.39590,-0.06709;0.39787,0.05955;0.37571,0.18257;0.33321,0.29570;0.27583,0.39563;0.20874,0.48164;0.13596,0.55450;0.06029,0.61560;-0.01639,0.66643;-0.09284,0.70838;-0.16830,0.74266;-0.24224,0.77030;-0.31437,0.79216;-0.38450,0.80897;-0.45251,0.82134;-0.51836,0.82978;-0.58202,0.83472;-0.64350,0.83653;-0.70282,0.83553;-0.76001,0.83199;-0.81510,0.82617;-0.86813,0.81825;-0.91915,0.80843;-0.96819,0.79688;-1.01528,0.78372;-1.06048,0.76910;-1.10382,0.75313;-1.14533,0.73592;-1.18504,0.71755;-1.22300,0.69812;-1.25923,0.67770;-1.29376,0.65636;-1.32661,0.63418;-1.35782,0.61121;-1.38741,0.58751;-1.41539,0.56313;-1.44179,0.53812;-1.46663,0.51253;-1.48993,0.48639;-1.51170,0.45976;-1.53196,0.43267;-1.55071,0.40516;-1.56798,0.37726;-1.58378,0.34900;-1.59811,0.32042;-1.61098,0.29155;-1.62240,0.26241;-1.63238,0.23304;-1.64092,0.20347;-1.64803,0.17372;-1.65371,0.14382;-1.65796,0.11379;-1.66079,0.08367;-1.66219,0.05347;-1.66216,0.02323;-1.66071,-0.00703;-1.65782,-0.03728;-1.65350,-0.06751;-1.64773,-0.09767;-1.64053,-0.12775;-1.63186,-0.15772;-1.62174,-0.18754;-1.61014,-0.21719;-1.59706,-0.24665;-1.58249,-0.27587;-1.56641,-0.30483;-1.54881,-0.33350;-1.52967,-0.36183;-1.50898,-0.38979;-1.48671,-0.41735;-1.46285,-0.44446;-1.43738,-0.47108;-1.41026,-0.49717;-1.38148,-0.52266;-1.35100,-0.54752;-1.31881,-0.57168;-1.28486,-0.59508;-1.24912,-0.61765;-1.21156,-0.63932;-1.17214,-0.66000;-1.13082,-0.67962;-1.08756,-0.69807;-1.04231,-0.71524;-0.99502,-0.73102;-0.94564,-0.74526;-0.89412,-0.75781;-0.84041,-0.76851;-0.78446,-0.77715;-0.72620,-0.78352;-0.66560,-0.78734;-0.60259,-0.78833;-0.53715,-0.78614;-0.46926,-0.78035;-0.39891,-0.77048;-0.32615,-0.75595;-0.25108,-0.73607;-0.17390,-0.71002 TRAJ rk4 0.40000,0.00000;0.38784,0.12439;0.35326,0.24172;0.30094,0.34713;0.23630,0.43859;0.16401,0.51613;0.08755,0.58086;0.00932,0.63428;-0.06911,0.67788;-0.14670,0.71300;-0.22281,0.74081;-0.29703,0.76229;-0.36912,0.77827;-0.43893,0.78942;-0.50640,0.79634;-0.57149,0.79952;-0.63421,0.79936;-0.69457,0.79623;-0.75260,0.79043;-0.80835,0.78223;-0.86186,0.77185;-0.91318,0.75951;-0.96234,0.74537;-1.00939,0.72960;-1.05438,0.71234;-1.09735,0.69371;-1.13834,0.67384;-1.17739,0.65281;-1.21453,0.63072;-1.24981,0.60766;-1.28324,0.58371;-1.31487,0.55893;-1.34471,0.53340;-1.37280,0.50717;-1.39917,0.48030;-1.42382,0.45285;-1.44679,0.42486;-1.46810,0.39638;-1.48776,0.36746;-1.50578,0.33813;-1.52219,0.30844;-1.53699,0.27842;-1.55020,0.24812;-1.56183,0.21756;-1.57189,0.18678;-1.58038,0.15582;-1.58731,0.12470;-1.59270,0.09347;-1.59653,0.06214;-1.59882,0.03075;-1.59956,-0.00067;-1.59876,-0.03209;-1.59642,-0.06348;-1.59253,-0.09481;-1.58709,-0.12604;-1.58010,-0.15715;-1.57155,-0.18811;-1.56144,-0.21888;-1.54976,-0.24942;-1.53649,-0.27972;-1.52163,-0.30972;-1.50517,-0.33940;-1.48708,-0.36871;-1.46737,-0.39761;-1.44601,-0.42607;-1.42298,-0.45404;-1.39826,-0.48147;-1.37184,-0.50832;-1.34369,-0.53452;-1.31378,-0.56002;-1.28209,-0.58476;-1.24859,-0.60868;-1.21326,-0.63170;-1.17605,-0.65375;-1.13694,-0.67473;-1.09588,-0.69457;-1.05284,-0.71314;-1.00779,-0.73035;-0.96066,-0.74606;-0.91143,-0.76013;-0.86005,-0.77240;-0.80647,-0.78270;-0.75065,-0.79082;-0.69254,-0.79653;-0.63210,-0.79956;-0.56931,-0.79960;-0.50415,-0.79630;-0.43661,-0.78924;-0.36673,-0.77793;-0.29457,-0.76178;-0.22029,-0.74010;-0.14413,-0.71206;-0.06651,-0.67667;0.01191,-0.63277;0.09011,-0.57900;0.16646,-0.51387;0.23856,-0.43589;0.30287,-0.34398;0.35469,-0.23814;0.38860,-0.12050;0.39999,0.00401;0.38705,0.12827;0.35181,0.24527;0.29901,0.35025;0.23404,0.44125;0.16155,0.51835;0.08499,0.58268;0.00671,0.63575;-0.07171,0.67904;-0.14927,0.71389;-0.22534,0.74147;-0.29950,0.76274;-0.37153,0.77852;-0.44127,0.78951;-0.50867,0.79628;-0.57368,0.79932;-0.63632,0.79904;-0.69661,0.79579;-0.75457,0.78989;-0.81024,0.78159;-0.86368,0.77112;-0.91491,0.75870;-0.96400,0.74448;-1.01098,0.72864;-1.05589,0.71131;-1.09879,0.69262;-1.13971,0.67268;-1.17868,0.65160;-1.21575,0.62946;-1.25095,0.60635;-1.28431,0.58235;-1.31587,0.55753;-1.34564,0.53196;-1.37366,0.50569;-1.39995,0.47878;-1.42453,0.45129;-1.44743,0.42327;-1.46866,0.39476;-1.48825,0.36581;-1.50620,0.33645;-1.52253,0.30673;-1.53726,0.27669;-1.55040,0.24637;-1.56195,0.21579;-1.57193,0.18499;-1.58035,0.15401;-1.58720,0.12288;-1.59251,0.09163;-1.59626,0.06029;-1.59847,0.02889;-1.59914,-0.00254;-1.59826,-0.03397;-1.59583,-0.06536;-1.59186,-0.09669;-1.58634,-0.12792;-1.57927,-0.15903;-1.57064,-0.18999;-1.56044,-0.22075;-1.54866,-0.25129;-1.53531,-0.28157;-1.52036,-0.31156;-1.50380,-0.34122;-1.48563,-0.37052;-1.46582,-0.39941;-1.44436,-0.42784;-1.42123,-0.45578;-1.39642,-0.48319;-1.36989,-0.51000;-1.34164,-0.53616;-1.31162,-0.56162;-1.27983,-0.58632;-1.24622,-0.61019;-1.21078,-0.63316;-1.17345,-0.65514;-1.13422,-0.67606;-1.09305,-0.69582;-1.04989,-0.71432;-1.00470,-0.73143;-0.95745,-0.74705;-0.90809,-0.76101;-0.85657,-0.77316;-0.80286,-0.78333;-0.74689,-0.79130;-0.68864,-0.79685;-0.62806,-0.79970;-0.56513,-0.79954;-0.49982,-0.79602;-0.43213,-0.78870;-0.36211,-0.77709;-0.28982,-0.76061;-0.21542,-0.73855;-0.13916,-0.71007;-0.06147,-0.67418;0.01696,-0.62969;0.09508,-0.57524;0.17124,-0.50934;0.24295,-0.43050;0.30661,-0.33769;0.35744,-0.23102;0.39004,-0.11277;0.39989,0.01194 TRAJ analytic 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[elapsed so far: 0.41 s] ==================================================================================================== SECTION 3: convergence order -- does each method's error scale with dt as theory says? ==================================================================================================== Global position error against the exact Kepler solution after exactly one orbit, as a function of timestep, fitted on log-log axes. Each method uses its own range of step counts because the three methods need very different resolutions before the leading-order truncation term actually dominates: Euler's asymptotic dt^1 regime only sets in once the per-orbit error is already small, which for e=0.6 needs several thousand steps per orbit; Verlet and RK4 reach their asymptotic regimes far sooner. This is itself a finding, not a convenience: it says forward Euler's nominal 'first order' convergence rate is a poor guide to its behaviour at any resolution a real simulation would actually use. eccentricity = 0.6, duration = 1 orbit method theory order fitted slope difference forward Euler 1 0.952 -0.048 (SE of slope: 0.016, 3.0 SE from theory) N=2000:err=1.054e+00 N=4000:err=6.101e-01 N=8000:err=3.264e-01 N=16000:err=1.667e-01 N=32000:err=8.374e-02 N=64000:err=4.190e-02 N=128000:err=2.095e-02 velocity Verlet 2 1.998 -0.002 (SE of slope: 0.003, 0.8 SE from theory) N=100:err=5.270e-01 N=200:err=1.361e-01 N=400:err=3.382e-02 N=800:err=8.433e-03 N=1600:err=2.107e-03 N=3200:err=5.266e-04 N=6400:err=1.316e-04 RK4 4 4.166 +0.166 (SE of slope: 0.033, 5.0 SE from theory) N=100:err=4.006e-03 N=200:err=1.824e-04 N=400:err=9.340e-06 N=800:err=5.207e-07 N=1600:err=3.060e-08 N=3200:err=1.852e-09 N=6400:err=1.137e-10 SVG_DATA convergence (log10 dt, log10 err) pairs, for the article's convergence figure: CONV euler -2.50285,0.02273;-2.80388,-0.21462;-3.10491,-0.48623;-3.40594,-0.77810;-3.70697,-1.07705;-4.00800,-1.37779;-4.30903,-1.67887 CONV verlet -1.20182,-0.27819;-1.50285,-0.86612;-1.80388,-1.47083;-2.10491,-2.07402;-2.40594,-2.67639;-2.70697,-3.27853;-3.00800,-3.88061 CONV rk4 -1.20182,-2.39733;-1.50285,-3.73888;-1.80388,-5.02965;-2.10491,-6.28339;-2.40594,-7.51425;-2.70697,-8.73229;-3.00800,-9.94430 [elapsed so far: 0.68 s] ==================================================================================================== SECTION 4 (MONTE CARLO): is the convergence order stable across random orbit shapes? ==================================================================================================== The single-orbit fit above used one eccentricity. To check it is not a coincidence of that particular orbit, the same fit is repeated on independently drawn eccentricities, uniform on [0.10, 0.60], seeded from the master seed via numpy's SeedSequence. n trials = 40 eccentricities (first 8 of 40): [0.5876, 0.3123, 0.2361, 0.3174, 0.4699, 0.2163, 0.5465, 0.569] Monte Carlo loop wall time: 49.60 s method theory MC mean MC SD MC SE z=(mean-theory)/SE forward Euler 1 0.9866 0.0134 0.0021 -6.31 velocity Verlet 2 2.0015 0.0012 0.0002 +7.72 RK4 4 4.1183 0.0324 0.0051 +23.09 club value vs accepted (theoretical) order, restated plainly: forward Euler club value 0.9866 +/- 0.0021 (SE, n=40) vs accepted value 1 -> difference -0.0134, 6.31 SE velocity Verlet club value 2.0015 +/- 0.0002 (SE, n=40) vs accepted value 2 -> difference +0.0015, 7.72 SE RK4 club value 4.1183 +/- 0.0051 (SE, n=40) vs accepted value 4 -> difference +0.1183, 23.09 SE SVG_DATA running-mean-vs-trial-count, for the Monte Carlo convergence figure: MCRUN euler 1,0.95801;2,0.97676;3,0.98303;4,0.98614;5,0.98667;6,0.98815;7,0.98610;8,0.98359;9,0.98493;10,0.98376;11,0.98483;12,0.98552;13,0.98628;14,0.98689;15,0.98576;16,0.98635;17,0.98653;18,0.98668;19,0.98715;20,0.98757;21,0.98795;22,0.98827;23,0.98767;24,0.98638;25,0.98675;26,0.98703;27,0.98732;28,0.98761;29,0.98789;30,0.98806;31,0.98707;32,0.98732;33,0.98748;34,0.98681;35,0.98638;36,0.98663;37,0.98685;38,0.98708;39,0.98730;40,0.98665 MCRUN verlet 1,2.00010;2,2.00066;3,2.00067;4,2.00082;5,2.00133;6,2.00120;7,2.00155;8,2.00165;9,2.00155;10,2.00175;11,2.00169;12,2.00176;13,2.00165;14,2.00154;15,2.00165;16,2.00157;17,2.00167;18,2.00176;19,2.00170;20,2.00165;21,2.00159;22,2.00159;23,2.00169;24,2.00160;25,2.00157;26,2.00159;27,2.00160;28,2.00156;29,2.00155;30,2.00158;31,2.00152;32,2.00148;33,2.00151;34,2.00153;35,2.00158;36,2.00157;37,2.00153;38,2.00151;39,2.00150;40,2.00149 MCRUN rk4 1,4.16104;2,4.12808;3,4.12592;4,4.12662;5,4.12687;6,4.12542;7,4.12858;8,4.13189;9,4.12562;10,4.12790;11,4.12545;12,4.12514;13,4.11991;14,4.11380;15,4.11631;16,4.11515;17,4.11603;18,4.11687;19,4.11842;20,4.11637;21,4.11353;22,4.11364;23,4.11512;24,4.11705;25,4.11656;26,4.11664;27,4.11612;28,4.11565;29,4.11552;30,4.11576;31,4.11723;32,4.11644;33,4.11683;34,4.11797;35,4.11890;36,4.11939;37,4.11775;38,4.11746;39,4.11724;40,4.11827 [elapsed so far: 50.31 s] ==================================================================================================== SECTION 5: matched computational cost -- who actually lasts longer per force evaluation? ==================================================================================================== Cost is charged in force evaluations: 1 per step for Euler and Verlet, 4 per step for RK4. RK4 is taken as the reference at a deliberately coarse 60 steps per orbit; Euler and Verlet are given 4x as many steps (240 per orbit) for the same total cost per orbit (240 force evaluations either way). A second table repeats the same three methods at a single shared timestep (60 steps/orbit for all three) to show the comparison the club would have drawn by the more obvious, but unfair to RK4, 'same dt' convention. -- MATCHED COST: RK4 at 60 steps/orbit (cost 240/orbit) vs Euler and Verlet at 240 steps/orbit (cost 240/orbit) -- e method steps/orb cost/orb orbit@1% 1% char orbit@5% 5% char 0.60 forward Euler 240 240 0.001 permanent 0.003 permanent 0.60 velocity Verlet 240 240 >1500 not reached >1500 not reached 0.60 RK4 60 240 3.950 permanent 18.334 permanent 0.20 forward Euler 240 240 0.011 permanent 0.060 permanent 0.20 velocity Verlet 240 240 >1500 not reached >1500 not reached 0.20 RK4 60 240 1480.181 permanent >1500 not reached 0.10 forward Euler 240 240 0.019 permanent 0.100 permanent 0.10 velocity Verlet 240 240 >1500 not reached >1500 not reached 0.10 RK4 60 240 >1500 not reached >1500 not reached -- MATCHED TIMESTEP (for contrast): all three methods at 60 steps/orbit, e=0.6 -- method steps/orb cost/orb orbit@1% 1% char orbit@5% 5% char forward Euler 60 60 0.0002 permanent 0.0008 permanent velocity Verlet 60 60 0.0056 transient 0.0255 transient RK4 60 240 3.9503 permanent 18.3337 permanent Plain reading: giving Euler four times as many steps as RK4, for the same total compute, does not rescue it at e=0.6 or e=0.2 -- it still loses 1% of the orbital energy within a small fraction of a single orbit, because the failure is a numerical instability at this eccentricity and resolution, not merely a slow accumulation that more steps can outrun. Velocity Verlet, at the same matched cost, never crosses 1% in any of the three cases. [elapsed so far: 67.42 s] ==================================================================================================== SUMMARY: headline numbers for the article and the interactive model ==================================================================================================== Default configuration reproduced everywhere in the article and the interactive model: eccentricity e = 0.6, steps per orbit = 100, dt = 0.062832 = T/100 forward Euler : unbound (relative energy error > 100%) within the very first orbit velocity Verlet : bounded oscillation, max |dE/E0| = 0.0299 over 300 orbits, no secular trend RK4 : crosses 1% loss at orbit 47.64, 5% loss at orbit 234.19 convergence order (measured / theory): Euler 0.952 / 1, Verlet 1.998 / 2, RK4 4.166 / 4 angular momentum, velocity Verlet : max |dL/L0| = 2.39e-14 (machine precision) ==================================================================================================== TOTAL RUNTIME: 67.43 seconds ====================================================================================================