{
 "description": "Numbers used by the Orbits playground: a point-mass Earth with a Molniya constellation and a geostationary satellite, and six Gaia DR3 stars in the MWPotential2014 model of the Milky Way.",
 "built_utc": "2026-10-26T02:56:45Z",
 "earth": {
  "gm_km3_s2": 398600.4418,
  "gm_source": "IERS Conventions (2010), Petit and Luzum (eds.), IERS Technical Note 36, Table 1.1: GM_E = 3.986004418 x 10^14 m^3 s^-2; the NSSDC Earth fact sheet gives 0.39860 x 10^6 km^3 s^-2",
  "radius_km": 6378.137,
  "radius_source": "NSSDC Earth fact sheet, equatorial radius 6378.137 km (also the WGS 84 semi-major axis); the Earth is treated as a sphere of this radius",
  "rotation_rate_rad_s": 7.292115e-05,
  "rotation_source": "IERS Conventions (2010), Table 1.1, nominal mean angular velocity of the Earth 7.292115 x 10^-5 rad s^-1; the NSSDC fact sheet gives the sidereal rotation period as 23.9345 h",
  "sidereal_day_s": 86164.10063718943
 },
 "molniya": {
  "period_s": 43082.05031859472,
  "period_note": "half a sidereal day (semi-synchronous), so the ground track repeats daily; Kolyuka et al. (2009) give 718 min, the NASA catalogue 12 h",
  "eccentricity": 0.74,
  "eccentricity_source": "Kidder and Vonder Haar (1990), J. Atmos. Oceanic Technol. 7, 517, as relayed by the Molniya orbit article of Wikipedia (0.74, 'approximately 0.737'); the NASA Earth Observatory catalogue gives 0.722 for the orbit type. Operational Molniya satellites varied between these values",
  "inclination_deg": 63.4,
  "inclination_source": "NASA Earth Observatory, Catalog of Earth Satellite Orbits (63.4 degrees); Kidder and Vonder Haar (1990). At this critical inclination the Earth's oblateness does not rotate the line of apsides, which keeps the apogee over the northern hemisphere",
  "argument_of_perigee_deg": 270.0,
  "argument_of_perigee_source": "the perigee at the southernmost point of the orbit puts the apogee over the northern hemisphere (Kidder and Vonder Haar 1990); 270 degrees is the value listed for the orbit type",
  "constellation": {
   "count": 3,
   "raan_spacing_deg": 120.0,
   "mean_anomaly_spacing_deg": -240.0,
   "apogee_longitude_deg": 37.62,
   "note": "three planes spaced by 120 degrees in the longitude of the ascending node, with mean anomalies spaced by -240 degrees, so one satellite passes apogee every two thirds of a period above the same longitude; the apogee longitude is set to Moscow's, as in the coursework test. The Greenwich meridian lies along the inertial x axis at t = 0"
  },
  "semi_major_axis_km": 26561.764513686318,
  "perigee_radius_km": 6906.058773558443,
  "apogee_radius_km": 46217.470253814194,
  "perigee_altitude_km": 527.9217735584434,
  "apogee_altitude_km": 39839.33325381419,
  "perigee_speed_km_s": 10.021399534177746,
  "apogee_speed_km_s": 1.4974505051070195,
  "derived_note": "semi-major axis from Kepler's third law with the period and GM above; radii, altitudes and speeds (vis-viva) follow from it and the eccentricity. Kidder and Vonder Haar (1990) give about 26,600 km, a perigee altitude of about 600 km and an apogee altitude of 39,700 km"
 },
 "geo": {
  "period_s": 86164.10063718943,
  "longitude_deg": 0.0,
  "source": "NASA Earth Observatory, Catalog of Earth Satellite Orbits: 42,164 km from the centre of the Earth, about 36,000 km above the surface, one orbit per day; ESA, Types of orbits: altitude 35,786 km, period 23 h 56 min 4 s",
  "radius_km": 42164.17293115725,
  "altitude_km": 35786.03593115725,
  "speed_km_s": 3.0746599789388602
 },
 "stations": [
  {
   "id": "porto",
   "name": "Porto",
   "short_name": "Porto",
   "lat_deg": 41.15,
   "lon_deg": -8.61
  },
  {
   "id": "santiago",
   "name": "Santiago",
   "short_name": "Santiago",
   "lat_deg": -33.45,
   "lon_deg": -70.67
  },
  {
   "id": "north",
   "name": "Polar station",
   "short_name": "Polar",
   "lat_deg": 85.0,
   "lon_deg": 0.0,
   "adjustable_latitude": [
    60.0,
    90.0
   ],
   "note": "a movable station on the geostationary satellite's own meridian, where the satellite's elevation is highest for that latitude; it starts at 85 degrees north, above the limit of acos(R_E / r_geo) = 81.3 degrees beyond which the geostationary satellite never rises. The coursework placed it at 80 degrees north"
  }
 ],
 "potential": {
  "name": "MWPotential2014",
  "source": "Bovy (2015), ApJS 216, 29, doi:10.1088/0067-0049/216/2/29, Table 1 and Section 3.5; parameters checked against galpy's definition (galpy.potential.MWPotential2014)",
  "R0_kpc": 8.0,
  "v0_km_s": 220.0,
  "bulge": {
   "form": "power-law density with exponential cut-off, rho(r) proportional to r^-alpha exp(-(r/rc)^2)",
   "alpha": 1.8,
   "rc_kpc": 1.9,
   "force_fraction_at_R0": 0.05
  },
  "disc": {
   "form": "Miyamoto and Nagai (1975) potential",
   "a_kpc": 3.0,
   "b_kpc": 0.28,
   "force_fraction_at_R0": 0.6
  },
  "halo": {
   "form": "Navarro, Frenk and White (1997) profile",
   "a_kpc": 16.0,
   "force_fraction_at_R0": 0.35
  }
 },
 "solar": {
  "galcen_distance_kpc": 8.0,
  "z_sun_pc": 20.8,
  "v_sun_km_s": [
   11.1,
   232.24,
   7.25
  ],
  "note": "distance from the Sun to the Galactic centre 8.0 kpc and circular speed 220 km/s, the normalisation of MWPotential2014 (Bovy 2015); height of the Sun above the plane 20.8 pc (Bennett and Bovy 2019, MNRAS 482, 1417); solar peculiar motion (U, V, W) = (11.10, 12.24, 7.25) km/s (Schoenrich, Binney and Dehnen 2010, MNRAS 403, 1829), so the Sun's Galactocentric velocity is (11.10, 232.24, 7.25) km/s. Frame: astropy Galactocentric, right-handed, x from the Sun towards the centre, y along Galactic rotation, z towards the north Galactic pole; the Sun sits at x = -8 kpc",
  "distance_note": "distances are 1 / parallax; every selected star has parallax / error above 600. No parallax zero-point correction is applied: the Lindegren et al. (2021) offsets are of order -0.02 mas, 0.1 percent of the smallest parallax here"
 },
 "stars": [
  {
   "name": "61 Virginis",
   "other": "HD 115617",
   "source_id": 3507879565090229888,
   "gaia": {
    "source_id": 3507879565090229888,
    "ref_epoch": 2016.0,
    "ra": 199.596298132351,
    "dec": -18.31592210064546,
    "parallax": 117.1726460991387,
    "parallax_error": 0.14559289813041687,
    "pmra": -1070.2015735200084,
    "pmra_error": 0.15287327766418457,
    "pmdec": -1063.8486605375717,
    "pmdec_error": 0.12301944941282272,
    "radial_velocity": -7.8643388748168945,
    "radial_velocity_error": 0.1293913871049881,
    "rv_nb_transits": 16.0,
    "rv_template_teff": 5500.0,
    "grvs_mag": 3.836165428161621,
    "ruwe": 1.1792638301849365,
    "phot_g_mean_mag": 4.532528400421143,
    "bp_rp": 0.8984379768371582,
    "parallax_pmra_corr": -0.0677686482667923,
    "parallax_pmdec_corr": 0.527199387550354,
    "pmra_pmdec_corr": -0.49559152126312256
   },
   "galactocentric": {
    "distance_pc": 8.5344,
    "distance_error_pc": 0.0107,
    "xyz_kpc": [
     -7.995868,
     -0.004566,
     0.026727
    ],
    "xyz_error_kpc": [
     5e-06,
     6e-06,
     7e-06
    ],
    "vxyz_km_s": [
     -12.473,
     184.8847,
     -24.2263
    ],
    "vxyz_error_km_s": [
     0.066,
     0.0954,
     0.095
    ]
   },
   "reference": {
    "eccentricity": 0.176,
    "zmax_kpc": 0.3775,
    "rperi_kpc": 5.635,
    "rapo_kpc": 8.0417,
    "energy_km2_s2": -114124.7,
    "energy_galpy_km2_s2": -49030.6,
    "Lz_kpc_km_s": 1478.38,
    "galpy_xyz0_kpc": [
     7.995895,
     -0.004566,
     0.026727
    ],
    "galpy_vxyz0_km_s": [
     12.473,
     184.8847,
     -24.2263
    ]
   }
  },
  {
   "name": "Proxima Centauri",
   "other": "GJ 551",
   "source_id": 5853498713190525696,
   "gaia": {
    "source_id": 5853498713190525696,
    "ref_epoch": 2016.0,
    "ra": 217.39232147200883,
    "dec": -62.67607511676666,
    "parallax": 768.0665391873573,
    "parallax_error": 0.049872905015945435,
    "pmra": -3781.741008265163,
    "pmra_error": 0.03138607740402222,
    "pmdec": 769.4650146478623,
    "pmdec_error": 0.05052453279495239,
    "radial_velocity": -21.942726135253906,
    "radial_velocity_error": 0.21612651646137238,
    "rv_nb_transits": 7.0,
    "rv_template_teff": 3500.0,
    "grvs_mag": 6.990696907043457,
    "ruwe": 0.9714532494544983,
    "phot_g_mean_mag": 8.984748840332031,
    "bp_rp": 3.8045802116394043,
    "parallax_pmra_corr": -0.11339641362428665,
    "parallax_pmdec_corr": -0.09566396474838257,
    "pmra_pmdec_corr": 0.6296852827072144
   },
   "galactocentric": {
    "distance_pc": 1.302,
    "distance_error_pc": 0.0001,
    "xyz_kpc": [
     -7.99907,
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     0.020754
    ],
    "xyz_error_kpc": [
     0.0,
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     0.0
    ],
    "vxyz_km_s": [
     -18.1299,
     233.8945,
     21.0956
    ],
    "vxyz_error_km_s": [
     0.1498,
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     0.0077
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   },
   "reference": {
    "eccentricity": 0.0966,
    "zmax_kpc": 0.3511,
    "rperi_kpc": 7.8303,
    "rapo_kpc": 9.5053,
    "energy_km2_s2": -103828.4,
    "energy_galpy_km2_s2": -38734.2,
    "Lz_kpc_km_s": 1870.96,
    "galpy_xyz0_kpc": [
     7.999097,
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    ],
    "galpy_vxyz0_km_s": [
     18.1299,
     233.8945,
     21.0956
    ]
   }
  },
  {
   "name": "Barnard's star",
   "other": "GJ 699",
   "source_id": 4472832130942575872,
   "gaia": {
    "source_id": 4472832130942575872,
    "ref_epoch": 2016.0,
    "ra": 269.44850252543836,
    "dec": 4.739420051112412,
    "parallax": 546.975939730948,
    "parallax_error": 0.04011635482311249,
    "pmra": -801.5509783684709,
    "pmra_error": 0.0318208672106266,
    "pmdec": 10362.394206546573,
    "pmdec_error": 0.036070454865694046,
    "radial_velocity": -110.46822357177734,
    "radial_velocity_error": 0.13125522434711456,
    "rv_nb_transits": 12.0,
    "rv_template_teff": 3300.0,
    "grvs_mag": 6.365547180175781,
    "ruwe": 1.084850549697876,
    "phot_g_mean_mag": 8.193974494934082,
    "bp_rp": 2.8336968421936035,
    "parallax_pmra_corr": -0.05781335011124611,
    "parallax_pmdec_corr": 0.5593565106391907,
    "pmra_pmdec_corr": 0.03058500960469246
   },
   "galactocentric": {
    "distance_pc": 1.8282,
    "distance_error_pc": 0.0001,
    "xyz_kpc": [
     -7.998453,
     0.000915,
     0.021241
    ],
    "xyz_error_kpc": [
     0.0,
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     0.0
    ],
    "vxyz_km_s": [
     -130.1119,
     237.1044,
     26.0235
    ],
    "vxyz_error_km_s": [
     0.1092,
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   "reference": {
    "eccentricity": 0.4416,
    "zmax_kpc": 0.6664,
    "rperi_kpc": 5.9156,
    "rapo_kpc": 15.272,
    "energy_km2_s2": -94659.8,
    "energy_galpy_km2_s2": -29565.7,
    "Lz_kpc_km_s": 1896.36,
    "galpy_xyz0_kpc": [
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    ],
    "galpy_vxyz0_km_s": [
     130.1119,
     237.1044,
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   }
  },
  {
   "name": "Lalande 21185",
   "other": "HD 95735, GJ 411",
   "source_id": 762815470562110464,
   "gaia": {
    "source_id": 762815470562110464,
    "ref_epoch": 2016.0,
    "ra": 165.83095967577933,
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    "parallax_error": 0.032066650688648224,
    "pmra": -580.0570872139048,
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    "radial_velocity": -85.11064147949219,
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    "rv_nb_transits": 11.0,
    "rv_template_teff": 3900.0,
    "grvs_mag": 4.989918231964111,
    "ruwe": 0.9638494253158569,
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    "parallax_pmra_corr": 0.308246910572052,
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   "galactocentric": {
    "distance_pc": 2.5461,
    "distance_error_pc": 0.0002,
    "xyz_kpc": [
     -8.001021,
     -9.5e-05,
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    "vxyz_km_s": [
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   "reference": {
    "eccentricity": 0.2512,
    "zmax_kpc": 1.6918,
    "rperi_kpc": 5.3331,
    "rapo_kpc": 8.9105,
    "energy_km2_s2": -111704.8,
    "energy_galpy_km2_s2": -46610.7,
    "Lz_kpc_km_s": 1428.12,
    "galpy_xyz0_kpc": [
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  },
  {
   "name": "Kapteyn's star",
   "other": "GJ 191",
   "source_id": 4810594479418041856,
   "gaia": {
    "source_id": 4810594479418041856,
    "ref_epoch": 2016.0,
    "ra": 77.9599373502188,
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    "radial_velocity": 245.05313110351562,
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    "rv_nb_transits": 21.0,
    "rv_template_teff": 3600.0,
    "grvs_mag": 6.611197471618652,
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    "phot_g_mean_mag": 8.063551902770996,
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   "galactocentric": {
    "distance_pc": 3.9339,
    "distance_error_pc": 0.0003,
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    "vxyz_km_s": [
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   },
   "reference": {
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    "rapo_kpc": 8.1163,
    "energy_km2_s2": -128508.0,
    "energy_galpy_km2_s2": -63413.8,
    "Lz_kpc_km_s": -445.62,
    "galpy_xyz0_kpc": [
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   }
  },
  {
   "name": "HD 140283",
   "other": "the metal-poor subgiant",
   "source_id": 6268770373590148224,
   "gaia": {
    "source_id": 6268770373590148224,
    "ref_epoch": 2016.0,
    "ra": 235.75785685172764,
    "dec": -10.934848057403192,
    "parallax": 16.26724221803098,
    "parallax_error": 0.02603433094918728,
    "pmra": -1115.1411639955027,
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    "pmdec": -303.5727744644891,
    "pmdec_error": 0.020811788737773895,
    "radial_velocity": -170.3622283935547,
    "radial_velocity_error": 0.15684741735458374,
    "rv_nb_transits": 14.0,
    "rv_template_teff": 5750.0,
    "grvs_mag": 6.352089881896973,
    "ruwe": 1.0640039443969727,
    "phot_g_mean_mag": 7.035969257354736,
    "bp_rp": 0.7588825225830078,
    "parallax_pmra_corr": -0.2317069172859192,
    "parallax_pmdec_corr": -0.0050647747702896595,
    "pmra_pmdec_corr": -0.16991709172725677
   },
   "galactocentric": {
    "distance_pc": 61.4732,
    "distance_error_pc": 0.0988,
    "xyz_kpc": [
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    "xyz_error_kpc": [
     8.2e-05,
     5e-06,
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    "vxyz_km_s": [
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    "vxyz_error_km_s": [
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   },
   "reference": {
    "eccentricity": 0.9222,
    "zmax_kpc": 2.6884,
    "rperi_kpc": 0.661,
    "rapo_kpc": 16.34,
    "energy_km2_s2": -99068.9,
    "energy_galpy_km2_s2": -33974.8,
    "Lz_kpc_km_s": -309.88,
    "galpy_xyz0_kpc": [
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    ],
    "galpy_vxyz0_km_s": [
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   }
  }
 ],
 "credit": "Gaia data: ESA/Gaia/DPAC, CC BY-SA 3.0 IGO"
}