Constants¶

Common constant values used in AHRS and Geodesy. The constants are defined in SI Units (second, metre, kilogram) unless otherwise noted, or when constants are unitless.

Basic Trigonometry

Name	Description	Value
`M_PI`	Pi	`3.141592653589793`
`DEG2RAD`	Degrees to Radians	`0.017453292519943`
`RAD2DEG`	Radians to Degrees	`57.29577951308232`

Geodesy

The following constants are set as defined in the latest report of the World Geodetic System 1984 [WGS84] and can be compared against [Chambat]. The CODATA constants are differentiated with a suffix of their origin indicating their epoch as of [CODATA2014] or [CODATA2018].

Name	Description	Value
`DYNAMIC_ELLIPTICITY`	Dynamic Ellipticity	`3.2737949e-3`
`EARTH_ATMOSPHERE_MASS`	Mass of Atmosphere including water vapor	`5.148e18`
`EARTH_AUTHALIC_RADIUS`	Earth’s Radius of equal area sphere	`6371007.1810`
`EARTH_AXIS_RATIO`	Earth’s Axis ratio	`9.96647189335e-1`
`EARTH_C20_DYN`	Earth’s Dynamic 2nd Degree Zonal Harmonic	`-4.84165143790815e-4`
`EARTH_C22_DYN`	Earth’s Dynamic 2nd Degree Sectorial Harmonic	`2.43938357328313e-6`
`EARTH_C20_GEO`	Earth’s Geographic 2nd Degree Zonal Harmonic	`-4.84166774985e-4`
`EARTH_EQUATOR_RADIUS`	Earth’s Semi-major axis (Equatorial Radius)	`6378137.0`
`EARTH_EQUIVOLUMETRIC_RADIUS`	Earth’s Radius of equal volume sphere	`6371000.79`
`EARTH_FIRST_ECCENTRICITY`	Earth’s First Eccentricity	`8.1819190842622e-2`
`EARTH_FIRST_ECCENTRICITY_2`	Earth’s First Eccentricity Squared	`6.6943799901414e-3`
`EARTH_FLATTENING_INV`	Earth’s Flattening Factor	`298.257223563`
`EARTH_FLATTENING`	Earth’s Flattening Factor (reduced)	`1/298.257223563`
`EARTH_GM`	Earth’s Gravitational Constant (GM)	`3.986004418e14`
`EARTH_GM_1`	Earth’s GM without Atmosphere	`3.986000982e14`
`EARTH_GM_2`	Earth Atmosphere’s Gravitational Constant	`3.4359e8`
`EARTH_GM_GPSNAV`	Earth’s GM for GPS Navigation	`3.9860050e14`
`EARTH_J2`	Earth’s Dynamic Form Factor	`1.08263e-3`
`EARTH_LINEAR_ECCENTRICITY`	Earth’s Linear Eccentricity	`5.2185400842339e5`
`EARTH_MEAN_RADIUS`	Earth’s Arithmetic Mean radius	`6371200.0`
`EARTH_MEAN_AXIAL_RADIUS`	Earth’s Mean Radius of the three semi-axes	`6371008.7714`
`EARTH_MASS`	Earth’s Mass (Atmosphere inclulded)	`5.9721864e24`
`EARTH_POLAR_CURVATURE_RADIUS`	Earth’s Polar Radius of Curvature	`6_399_593.6258`
`EARTH_POLAR_RADIUS`	Earth’s Semi-minor axis (Polar Radius)	`6356752.3142`
`EARTH_ROTATION`	Earth’s Rotation rate	`7.292115e-5`
`EARTH_SECOND_ECCENTRICITY`	Earth’s Second Eccentricity	`8.2094437949696e-2`
`EARTH_SECOND_ECCENTRICITY_2`	Earth’s Second Eccentricity Squared	`6.739496742276486e-3`
`EARTH_SIDEREAL_DAY`	Earth’s duration of sidereal day	`86164.09053083288`
`EQUATORIAL_NORMAL_GRAVITY`	Earth’s Normal Gravity at the Equator	`9.7803253359`
`LIGHT_SPEED`	Velocity of light in vacuum	`2.99792458e8`
`MEAN_NORMAL_GRAVITY`	Earth’s Mean Normal Gravity	`9.7976432223`
`NORMAL_GRAVITY_FORMULA`	Constant for Normal Gravity Formula	`3.449786506841e-3`
`NORMAL_GRAVITY_POTENTIAL`	Earth’s Normal Gravity Potential	`6.26368517146`
`POLAR_NORMAL_GRAVITY`	Earth’s Normal Gravity at the Pole	`9.8321849379`
`SOMIGLIANA_GRAVITY`	Constant for Somigliana’s Formula	`1.931852652458e-3`
`UNIVERSAL_GRAVITATION_CODATA2014`	Universal Gravitation defined in CODATA2014	`6.67408e-11`
`UNIVERSAL_GRAVITATION_CODATA2018`	Universal Gravitation defined in CODATA2018	`6.67430e-11`
`UNIVERSAL_GRAVITATION_WGS84`	Universal Gravitation defined in WGS84	`6.67428e-11`

The values above are set as defined in [WGS84], although most of these values can be also obtained with the class WGS of this package, which builds the World’s Geodetic System independently.

The elemental defining parameters (equatorial radius, flattening, gravitational constant and rotational velocity) are set, by default, to that of Earth’s, and all other parameters are derived from these. Just to compare:

>>> import ahrs
>>> ahrs.EARTH_EQUATOR_RADIUS
6378137.0
>>> ahrs.EARTH_POLAR_RADIUS
6356752.3142
>>> ahrs.EARTH_FIRST_ECCENTRICITY_2
0.0066943799901414
>>> wgs = ahrs.utils.WGS()  # Default model is Earth's
>>> wgs.a
6378137.0
>>> wgs.b
6356752.314245179
>>> wgs.first_eccentricity_squared
0.0066943799901413165

These values are, so far, determined for the Earth. However, other celestial bodies have been measured and their values are, as defined in [Archinal], [Park] and [Williams], set as follows:

Name	Value
`MOON_EQUATOR_RADIUS`	`1738100.0`
`MOON_POLAR_RADIUS`	`1736000.0`
`MOON_MASS`	`7.346e22`
`MOON_ROTATION`	`1.109027709148159e-7`
`MOON_J2`	`2.027e-4`
`MERCURY_EQUATOR_RADIUS`	`2440530.0`
`MERCURY_POLAR_RADIUS`	`2438260.0`
`MERCURY_ROTATION`	`1.2399326882596827e-6`
`MERCURY_MASS`	`3.30114e23`
`MERCURY_J2`	`5.03e-5`
`VENUS_EQUATOR_RADIUS`	`6051800.0`
`VENUS_POLAR_RADIUS`	`6051800.0`
`VENUS_ROTATION`	`-2.9923691869737844e-7`
`VENUS_MASS`	`4.86747e24`
`VENUS_J2`	`4.458e-6`
`MARS_EQUATOR_RADIUS`	`3396190.0`
`MARS_POLAR_RADIUS`	`3376200.0`
`MARS_ROTATION`	`7.088235959185674e-5`
`MARS_MASS`	`6.41712e23`
`MARS_J2`	`1.96045e-3`
`JUPITER_EQUATOR_RADIUS`	`71492000.0`
`JUPITER_POLAR_RADIUS`	`66854000.0`
`JUPITER_ROTATION`	`1.758518138029551e-4`
`JUPITER_MASS`	`1.898187e27`
`JUPITER_J2`	`1.4736e-2`
`SATURN_EQUATOR_RADIUS`	`60268000.0`
`SATURN_POLAR_RADIUS`	`54364000.0`
`SATURN_ROTATION`	`1.637884057802486e-4`
`SATURN_MASS`	`5.683174e26`
`SATURN_J2`	`1.6298e-2`
`URANUS_EQUATOR_RADIUS`	`25559000.0`
`URANUS_POLAR_RADIUS`	`24973000.0`
`URANUS_ROTATION`	`-1.012376653716682e-4`
`URANUS_MASS`	`8.68127e25`
`URANUS_J2`	`3.343430e-3`
`NEPTUNE_EQUATOR_RADIUS`	`24764000.0`
`NEPTUNE_POLAR_RADIUS`	`24341000.0`
`NEPTUNE_ROTATION`	`1.083382527619075e-4`
`NEPTUNE_MASS`	`1.024126e26`
`NEPTUNE_J2`	`3.411e-3`
`PLUTO_EQUATOR_RADIUS`	`1188300.0`
`PLUTO_POLAR_RADIUS`	`1188300.0`
`PLUTO_ROTATION`	`-1.138559183467410e-05`
`PLUTO_MASS`	`1.303e22`

References

[WGS84]

(1, 2) World Geodetic System 1984. Its Definition and Relationships with Local Geodetic Systems. National Geospatial-Intelligence Agency (NGA) Standarization Document. 2014. (ftp://ftp.nga.mil/pub2/gandg/website/wgs84/NGA.STND.0036_1.0.0_WGS84.pdf)

[Chambat]

F. Chambat. Mean radius, mass, and inertia for reference Earth models. Physics of the Earth and Planetary Interiors Vol 124 (2001) p237–253.

[Archinal]

Archinal, B.A. et al. 2018. “Report of the IAU/IAG Working Group on cartographic coordinates and rotational elements: 2015” Celestial Mech. Dyn. Astr. 130:22. (https://astropedia.astrogeology.usgs.gov/download/Docs/WGCCRE/WGCCRE2015reprint.pdf)

[CODATA2018]

2018 CODATA Recommended Values of the Fundamental Constants of Physics and Chemistry. NIST. June 2019. (https://physics.nist.gov/cuu/pdf/wallet_2018.pdf)

[CODATA2014]

2014 CODATA Recommended Values of the Fundamental Constants of Physics and Chemistry. NIST. August 2015. (https://physics.nist.gov/cuu/pdf/wallet_2014.pdf)

[Park]

Ryan S. Park. Planets and Pluto: Physical Characteristics. NASA Jet Propulsion Laboratory. California Institute of Technology. 29th May 2020. (https://ssd.jpl.nasa.gov/?planet_phys_par)

[Williams]

David R. Williams. Planetary Fact Sheet - Metric. NASA Goddard Space Flight Center. 21st October 2019. (https://nssdc.gsfc.nasa.gov/planetary/factsheet/)