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/)