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        "author": "Reilly Raab",
        "author_email": "raabrp@gmail.com",
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            "License :: OSI Approved :: MIT License",
            "Operating System :: OS Independent",
            "Programming Language :: Python :: 3"
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        "description": "# Reference Ellipsoid (WSG84)\n\nReference ellipsoidal gravitational and rotational model, instantiated with \nWSG84 values for Earth.\n\nExpected error of calculated values for gravity is less than 0.02% in the\nsurface normal component and 0.07% in the transverse component, where\npercentages are given relative to the total magnitude of the computed gravity\nvector (assuming correct implementation).\n\nThis should be evaluated relative to the fact the value of felt gravity differs\nfrom its nominal value (9.80665 m/s^2) by extremes of +0.28% and -0.44% over\nEarth's surface.\n\nBasic coordinate transformations are included for convenience, allowing for full\naccount of non\u2013inertial effects of the (geo)synchronous reference frame.\n\n# Installation\n\n## PyPi\n\n`pip3 install --user rellipsoid`\n\n## From source\n\nClone or download the source from [github](https://github.com/raabrp/rellipsoid)\nNumpy and (for testing) pytest are required.\n\n# Usage\n\n* class `Planet` defines the model.\n* `earth` instantiates it with WSG84 values.\n\n## Methods\n\n    get_free_air_gravity: calculate magnitude of free air gravity (including\n        centrifugal force) near the surface of the reference ellipsoid at\n        specified latitude and height (geodetic)\n\n    get_analytic_gravity: calculate gravity vector at a specified latitude\n        and height (geodetic); contribution of centrifugal force is optional\n\n    prep_local_cartesian: return transforms to and from a local Cartesian\n        surface coordinate system and geodetic coordinates.\n\n    prep_local_cartesian_inertial: return transformations between a local\n        Cartesian surface coordinate system and an inertial frame\n        coincident with it at time 0\n\n# References\n\nA. NIMA Technical Report TR8350.2, \"Department of Defense World Geodetic System \n1984, Its Definition and Relationships With Local Geodetic Systems\", Third \nEdition, Amendment 1, 3 January 2000: [link](http://earth-info.nga.mil/GandG/publications/tr8350.2/wgs84fin.pdf)\n\nB. NIMA Technical Report TR8350.2, \"Department of Defense World Geodetic System \n1984, Its Definition and Relationships With Local Geodetic Systems\", Second\nEdition: Chapters [3](http://earth-info.nga.mil/GandG/publications/tr8350.2/tr8350.2-a/Chapter%203.pdf) and\n[4](http://earth-info.nga.mil/GandG/publications/tr8350.2/tr8350.2-a/Chapter%204.pdf)\n\nC. Bernhard Hofmann-Wellenhof, Helmut Moritz; Physical Geodesy (2006)\n\nD. Zhu, \"Conversion of Earth-centered Earth-fixed coordinates to geodetic\n   coordinates,\" Aerospace and Electronic Systems, IEEE Transactions on, vol. 30,\n   pp. 957\u2013961, 1994.\n\n# Discussion of model\n\nA planet's rotation deforms its shape and mass distribution (and thus \nexperienced gravity close to its surface). Solving for the equilibrium between\ngravity and centrifugal force (i.e. an equipotential surface) yields an\nellipsoid, which we use to approximate the surface of Earth. This approximation\nis standard and is used nearly universally as a reference coordinate system \n(e.g. for GPS or geospacial data sets). The current standard reference ellipsoid\nfor Earth is part of the \"WGS84\" standard maintained by IERS.\n\n## Unimplemented Corrections \n\nEmpirical refinement of the model is possible with a spherical harmonic\nexpansions of gravitational potential (the gradient of which is local gravity). \n\nSuch corrections are appropriate for highly accurate predictions of surface\ngravity or orbital operations in the neighborhood of Earth. At extreme\ndistances however, the ellipsoidal model, or even a point\u2013mass model is \npreferable.\n\nReference A. demonstrates calculations with the current IERS standard\nspherical harmonic coefficients for Earth: the Earth Gravitational Model\n1996 (EGM96).\n\nMore refined coefficients for Earth are provided by the NASA GRACE mission\n(to be followed by GRACE-FO in 2019), which have allowed observation of temporal\nvariations.\n\nftp://ftp.csr.utexas.edu/pub/grace/GGM05\n\n# Values used for Earth (WSG 84)\n\nThe current ellipsoidal model for Earth is the World Geodetic System 1984\n(WSG 84): parameters which completely specify instantiation of the model.\n\nValues for WSG 84 model obtained from Ref. A.:\n\n```\n6378137,       # a, semi-major axis (equatorial) [m]\n298.257223563, # 1/f, inverse flatness [dimensionless]\n7.292115e-5,   # omega, angular velocity [rad s^-1]\n3.9860009e14   # GM, Universal grav. const. * mass of Earth [m^3 s^-2]\n```\n\n## Notes on Values Used\n\n### omega\n\nTreated as average contstant angular velocity for Earth, ignoring precession.\nFor very precise orbital calculations, such nuances may not be lightly ignored.\n\n### GM\n\nDifferent values for GM are used for different purposes\n\n* old value 3.986005e14\n* new value 3.986004418e14\n* value without atmosphere 3.9860009e14  <-- We use this\n\nIn this package, we have used the value without atmosphere for near\u2013surface\ngravity estimations, justified by appeal to the shell theorem (approximately).\n\nFor the sake of completeness, orbital computations should use the updated\nvalue, however GPS broadcasts assume that receivers continue to use the old\nvalue and encode appropriately (no accuracy is lost while complicated\nsoftware updates to many deployed receivers are avoided).\n\n## Evaluating Uncertainty\n\nFrom the GRACE data, we note that gravitational potential can differ from\nthe predictions of the ellipsoidal model by 0.02% near the surface of\nEarth, though such deviations are typically less than 0.005%. Ignoring the\ntransverse components of gravity caused by such irregularities, this\ntranslates to errors of up to 0.02% in the magnitude of the calculated gravity\nvector.\n\nAccounting for \n[vertical deflection](https://en.wikipedia.org/wiki/Vertical_deflection), we \nnote maximum deflections at Earth's surface of up to 100 arc-seconds (4.8e-4 \nradians), yielding 0.00001% uncertainty in the vertical component (as a \npercentage of the total magnitude) and 0.048% in the transverse component. Note\nthat we use the geodetic definition of \"vertical\" (normal to the reference\nellipsoid).\n\nFinally, in consideration of our choice of value for GM (which differs from the\nupdated value by subtracting the mass of the atmosphere), we note that the mass\nof the atmosphere is ~0.0001% of that of the planet and consider that the force\nof gravity linear in the planet's mass (this is approximately true: only for\npure gravitation, not the effects of centrifugal force).\n\nWe expect the ellipsoidal gravitational model, as implemented here for Earth,\nto be accurate to +/- 0.02% in the vertical component and +/- 0.07% in the\ntransverse component (where percentages are given relative to the total\nmagnitude of the calculated gravity vector) everywhere on or near Earth's\nsurface.\n\nAll of this assumes of course that our neither our implementation nor our\nanalysis is flawed. This package it NOT intended for mission-critical\ncalculations or simulation. See license.txt for more information.\n\n\n",
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