{
    "metadata": {
        "calculation_date": "2025-09-25T17:12:25.088360",
        "author": "Yoshiyuki Matsuyama",
        "article": "#005 Polarization Prediction",
        "version": "1.0"
    },
    "physical_constants": {
        "G": 6.6743e-11,
        "c": 299792458,
        "M_sun": 1.98847e+30,
        "unit_G": "m^3 kg^-1 s^-2",
        "unit_c": "m\/s",
        "unit_M_sun": "kg"
    },
    "system_parameters": {
        "black_hole_mass": 6500000000,
        "schwarzschild_radius": 19196705983434.707,
        "major_radius": 191967059834347.06,
        "minor_radius": 67188470942021.48,
        "R_over_r": 2.8571428571428568,
        "W_wavelength": 38393411966869.414,
        "W_amplitude": 19196705983434.707
    },
    "results": {
        "magnetic_field": {
            "reference_strength": 100,
            "configurations": [
                {
                    "w_phase": 0,
                    "B_toroidal": 53.45224838248488,
                    "B_poloidal": 16.035674514745462,
                    "B_w": 0,
                    "B_total": 55.80578567035603
                },
                {
                    "w_phase": 0.41887902047863906,
                    "B_toroidal": 53.45224838248488,
                    "B_poloidal": 16.035674514745462,
                    "B_w": 4.0673664307580015,
                    "B_total": 55.95381295289689
                },
                {
                    "w_phase": 0.8377580409572781,
                    "B_toroidal": 53.45224838248488,
                    "B_poloidal": 16.035674514745462,
                    "B_w": 7.431448254773941,
                    "B_total": 56.29842038147338
                },
                {
                    "w_phase": 1.2566370614359172,
                    "B_toroidal": 53.45224838248488,
                    "B_poloidal": 16.035674514745462,
                    "B_w": 9.510565162951535,
                    "B_total": 56.61039272081109
                },
                {
                    "w_phase": 1.6755160819145563,
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                    "B_poloidal": 16.035674514745462,
                    "B_w": 9.945218953682733,
                    "B_total": 56.6850341300277
                },
                {
                    "w_phase": 2.0943951023931953,
                    "B_toroidal": 53.45224838248488,
                    "B_poloidal": 16.035674514745462,
                    "B_w": 8.660254037844387,
                    "B_total": 56.473761290405605
                },
                {
                    "w_phase": 2.5132741228718345,
                    "B_toroidal": 53.45224838248488,
                    "B_poloidal": 16.035674514745462,
                    "B_w": 5.877852522924733,
                    "B_total": 56.11447999016802
                },
                {
                    "w_phase": 2.9321531433504733,
                    "B_toroidal": 53.45224838248488,
                    "B_poloidal": 16.035674514745462,
                    "B_w": 2.079116908177597,
                    "B_total": 55.844502338221126
                },
                {
                    "w_phase": 3.3510321638291125,
                    "B_toroidal": 53.45224838248488,
                    "B_poloidal": 16.035674514745462,
                    "B_w": -2.079116908177591,
                    "B_total": 55.844502338221126
                },
                {
                    "w_phase": 3.7699111843077517,
                    "B_toroidal": 53.45224838248488,
                    "B_poloidal": 16.035674514745462,
                    "B_w": -5.87785252292473,
                    "B_total": 56.11447999016802
                },
                {
                    "w_phase": 4.1887902047863905,
                    "B_toroidal": 53.45224838248488,
                    "B_poloidal": 16.035674514745462,
                    "B_w": -8.660254037844386,
                    "B_total": 56.473761290405605
                },
                {
                    "w_phase": 4.607669225265029,
                    "B_toroidal": 53.45224838248488,
                    "B_poloidal": 16.035674514745462,
                    "B_w": -9.945218953682733,
                    "B_total": 56.6850341300277
                },
                {
                    "w_phase": 5.026548245743669,
                    "B_toroidal": 53.45224838248488,
                    "B_poloidal": 16.035674514745462,
                    "B_w": -9.510565162951536,
                    "B_total": 56.6103927208111
                },
                {
                    "w_phase": 5.445427266222308,
                    "B_toroidal": 53.45224838248488,
                    "B_poloidal": 16.035674514745462,
                    "B_w": -7.4314482547739455,
                    "B_total": 56.29842038147338
                },
                {
                    "w_phase": 5.864306286700947,
                    "B_toroidal": 53.45224838248488,
                    "B_poloidal": 16.035674514745462,
                    "B_w": -4.0673664307580095,
                    "B_total": 55.95381295289689
                },
                {
                    "w_phase": 6.283185307179586,
                    "B_toroidal": 53.45224838248488,
                    "B_poloidal": 16.035674514745462,
                    "B_w": -2.4492935982947065e-15,
                    "B_total": 55.80578567035603
                }
            ],
            "helical_pitch": 0.3366748193867272,
            "helical_period": 1408166.2235638094
        },
        "polarization_pattern": {
            "description": "Helical polarization pattern due to W-axis",
            "rotation_measure": "Variable with W-phase",
            "period_hours": 391.1572843232804,
            "data": [
                {
                    "w_phase": 0,
                    "polarization_angle_mean": 1.5707963267948966,
                    "polarization_angle_std": 1.7927953850684968,
                    "polarization_degree_mean": 0.20156249999999998,
                    "polarization_degree_max": 0.30000000000000004,
                    "polarization_degree_min": 0.10012430787810778,
                    "stokes_Q_mean": -0.006090806547049322,
                    "stokes_U_mean": 4.336808689942018e-19
                },
                {
                    "w_phase": 0.41887902047863906,
                    "polarization_angle_mean": 1.5221814824672122,
                    "polarization_angle_std": 1.784424772232135,
                    "polarization_degree_mean": 0.2014274147775666,
                    "polarization_degree_max": 0.2999801072770524,
                    "polarization_degree_min": 0.10004475677164967,
                    "stokes_Q_mean": -0.002454378285451431,
                    "stokes_U_mean": -6.093505652478878e-5
                },
                {
                    "w_phase": 0.8377580409572781,
                    "polarization_angle_mean": 1.5220535714746797,
                    "polarization_angle_std": 1.782890395355215,
                    "polarization_degree_mean": 0.20104551657243575,
                    "polarization_degree_max": 0.29992043702261795,
                    "polarization_degree_min": 0.1000049733044057,
                    "stokes_Q_mean": -0.0004758165900344939,
                    "stokes_U_mean": 4.1504974574631e-6
                },
                {
                    "w_phase": 1.2566370614359172,
                    "polarization_angle_mean": 1.570877032722985,
                    "polarization_angle_std": 1.7829655247487488,
                    "polarization_degree_mean": 0.20048283905371087,
                    "polarization_degree_max": 0.29992043702261795,
                    "polarization_degree_min": 0.1000049733044057,
                    "stokes_Q_mean": 9.157569422888964e-5,
                    "stokes_U_mean": 0.0001145457803344047
                },
                {
                    "w_phase": 1.6755160819145563,
                    "polarization_angle_mean": 1.5708053876569958,
                    "polarization_angle_std": 1.7826151025703636,
                    "polarization_degree_mean": 0.19983667427614432,
                    "polarization_degree_max": 0.2999801072770524,
                    "polarization_degree_min": 0.10004475677164967,
                    "stokes_Q_mean": -0.0001029261628702656,
                    "stokes_U_mean": 0.00015686028441035007
                },
                {
                    "w_phase": 2.0943951023931953,
                    "polarization_angle_mean": 1.5710055728759147,
                    "polarization_angle_std": 1.7836860703970974,
                    "polarization_degree_mean": 0.19921875,
                    "polarization_degree_max": 0.30000000000000004,
                    "polarization_degree_min": 0.10012430787810778,
                    "stokes_Q_mean": -0.0006949262235746451,
                    "stokes_U_mean": 3.6245482859284745e-5
                },
                {
                    "w_phase": 2.5132741228718345,
                    "polarization_angle_mean": 1.5221460602631705,
                    "polarization_angle_std": 1.7832912779611825,
                    "polarization_degree_mean": 0.19873591094628915,
                    "polarization_degree_max": 0.2999801072770524,
                    "polarization_degree_min": 0.10004475677164967,
                    "stokes_Q_mean": -0.0012884107456001808,
                    "stokes_U_mean": -8.518016488462074e-5
                },
                {
                    "w_phase": 2.9321531433504733,
                    "polarization_angle_mean": 1.522159553797765,
                    "polarization_angle_std": 1.7867115157262725,
                    "polarization_degree_mean": 0.19847164437385345,
                    "polarization_degree_max": 0.29992043702261795,
                    "polarization_degree_min": 0.1000049733044057,
                    "stokes_Q_mean": -0.0022038422210205015,
                    "stokes_U_mean": -4.8303142234728744e-5
                },
                {
                    "w_phase": 3.3510321638291125,
                    "polarization_angle_mean": 1.6194330997920283,
                    "polarization_angle_std": 1.7867115157262725,
                    "polarization_degree_mean": 0.19847164437385342,
                    "polarization_degree_max": 0.29992043702261795,
                    "polarization_degree_min": 0.1000049733044057,
                    "stokes_Q_mean": -0.0022038422210204998,
                    "stokes_U_mean": 4.830314223472679e-5
                },
                {
                    "w_phase": 3.7699111843077517,
                    "polarization_angle_mean": 1.6194465933266229,
                    "polarization_angle_std": 1.7832912779611827,
                    "polarization_degree_mean": 0.19873591094628917,
                    "polarization_degree_max": 0.2999801072770524,
                    "polarization_degree_min": 0.10004475677164967,
                    "stokes_Q_mean": -0.001288410745600186,
                    "stokes_U_mean": 8.518016488461987e-5
                },
                {
                    "w_phase": 4.1887902047863905,
                    "polarization_angle_mean": 1.6196744659262188,
                    "polarization_angle_std": 1.7830162559628053,
                    "polarization_degree_mean": 0.19921875,
                    "polarization_degree_max": 0.30000000000000004,
                    "polarization_degree_min": 0.10012430787810778,
                    "stokes_Q_mean": -0.000694926223574659,
                    "stokes_U_mean": -3.6245482859279324e-5
                },
                {
                    "w_phase": 4.607669225265029,
                    "polarization_angle_mean": 1.5707872659327973,
                    "polarization_angle_std": 1.7826151025703636,
                    "polarization_degree_mean": 0.19983667427614427,
                    "polarization_degree_max": 0.2999801072770524,
                    "polarization_degree_min": 0.10004475677164967,
                    "stokes_Q_mean": -0.00010292616287027774,
                    "stokes_U_mean": -0.00015686028441035143
                },
                {
                    "w_phase": 5.026548245743669,
                    "polarization_angle_mean": 1.570715620866808,
                    "polarization_angle_std": 1.7829655247487488,
                    "polarization_degree_mean": 0.20048283905371084,
                    "polarization_degree_max": 0.29992043702261795,
                    "polarization_degree_min": 0.1000049733044057,
                    "stokes_Q_mean": 9.157569422889571e-5,
                    "stokes_U_mean": -0.00011454578033440681
                },
                {
                    "w_phase": 5.445427266222308,
                    "polarization_angle_mean": 1.6195390821151134,
                    "polarization_angle_std": 1.782890395355215,
                    "polarization_degree_mean": 0.20104551657243575,
                    "polarization_degree_max": 0.29992043702261795,
                    "polarization_degree_min": 0.1000049733044057,
                    "stokes_Q_mean": -0.000475816590034487,
                    "stokes_U_mean": -4.150497457462341e-6
                },
                {
                    "w_phase": 5.864306286700947,
                    "polarization_angle_mean": 1.619411171122581,
                    "polarization_angle_std": 1.784424772232135,
                    "polarization_degree_mean": 0.2014274147775666,
                    "polarization_degree_max": 0.2999801072770524,
                    "polarization_degree_min": 0.10004475677164966,
                    "stokes_Q_mean": -0.002454378285451423,
                    "stokes_U_mean": 6.09350565247903e-5
                },
                {
                    "w_phase": 6.283185307179586,
                    "polarization_angle_mean": 1.6198837120072371,
                    "polarization_angle_std": 1.7921232439025825,
                    "polarization_degree_mean": 0.2015625,
                    "polarization_degree_max": 0.30000000000000004,
                    "polarization_degree_min": 0.10012430787810778,
                    "stokes_Q_mean": -0.0060908065470493145,
                    "stokes_U_mean": 6.071532165918825e-18
                }
            ]
        },
        "observational_signatures": {
            "W_axis_period": {
                "seconds": 1408166.2235638094,
                "hours": 391.1572843232804,
                "days": 16.29822018013668
            },
            "polarization_variation": {
                "amplitude_degrees": 30,
                "period_hours": 391.1572843232804,
                "pattern": "Sinusoidal with W-phase"
            },
            "frequency_dependence": {
                "frequencies_GHz": [
                    86000000000,
                    230000000000,
                    345000000000
                ],
                "faraday_rotation_rad": [
                    0.012151908852580011,
                    0.0016989700921300897,
                    0.0007550978187244845
                ],
                "rotation_measure": 1000
            },
            "detectability": {
                "ALMA": "Detectable with polarimetry mode",
                "EHT": "Requires multiple epochs",
                "time_resolution_needed": "< 1 hour"
            }
        },
        "helical_structure": {
            "pitch_angle": {
                "radians": 0.3366748193867272,
                "degrees": 19.290046219188735
            },
            "helical_wavelength": {
                "meters": 1277908493025936,
                "Rs_units": 66.56915483982895
            },
            "twist_rate": {
                "per_meter": 4.91677247742657e-15,
                "per_Rs": 0.09438583563660173
            },
            "number_of_turns": 17.951958020513104,
            "handedness": "Right-handed (positive helicity)",
            "stability": {
                "kink_mode": "Stable for n_turns < 2π",
                "current_status": "Marginally stable"
            }
        },
        "visualization": {
            "file": "polarization_pattern_005.html",
            "type": "3D polarization pattern"
        }
    }
}