Ellipsoidal model of planet with WSG84 values for Earth

# Reference Ellipsoid (WSG84)

Reference ellipsoidal gravitational and rotational model, instantiated with WSG84 values for Earth.

Expected error of calculated values for gravity is less than 0.02% in the surface normal component and 0.07% in the transverse component, where percentages are given relative to the total magnitude of the computed gravity vector (assuming correct implementation).

This should be evaluated relative to the fact the value of felt gravity differs from its nominal value (9.80665 m/s^2) by extremes of +0.28% and -0.44% over Earth's surface.

Basic coordinate transformations are included for convenience, allowing for full account of non–inertial effects of the (geo)synchronous reference frame.

# Installation

## PyPi

pip3 install --user rellipsoid

## From source

Clone or download the source from github Numpy and (for testing) pytest are required.

# Usage

• class Planet defines the model.
• earth instantiates it with WSG84 values.

## Methods

get_free_air_gravity: calculate magnitude of free air gravity (including
centrifugal force) near the surface of the reference ellipsoid at
specified latitude and height (geodetic)

get_analytic_gravity: calculate gravity vector at a specified latitude
and height (geodetic); contribution of centrifugal force is optional

prep_local_cartesian: return transforms to and from a local Cartesian
surface coordinate system and geodetic coordinates.

prep_local_cartesian_inertial: return transformations between a local
Cartesian surface coordinate system and an inertial frame
coincident with it at time 0


# References

A. NIMA Technical Report TR8350.2, "Department of Defense World Geodetic System 1984, Its Definition and Relationships With Local Geodetic Systems", Third Edition, Amendment 1, 3 January 2000: link

B. NIMA Technical Report TR8350.2, "Department of Defense World Geodetic System 1984, Its Definition and Relationships With Local Geodetic Systems", Second Edition: Chapters 3 and 4

C. Bernhard Hofmann-Wellenhof, Helmut Moritz; Physical Geodesy (2006)

D. Zhu, "Conversion of Earth-centered Earth-fixed coordinates to geodetic coordinates," Aerospace and Electronic Systems, IEEE Transactions on, vol. 30, pp. 957–961, 1994.

# Discussion of model

A planet's rotation deforms its shape and mass distribution (and thus experienced gravity close to its surface). Solving for the equilibrium between gravity and centrifugal force (i.e. an equipotential surface) yields an ellipsoid, which we use to approximate the surface of Earth. This approximation is standard and is used nearly universally as a reference coordinate system (e.g. for GPS or geospacial data sets). The current standard reference ellipsoid for Earth is part of the "WGS84" standard maintained by IERS.

## Unimplemented Corrections

Empirical refinement of the model is possible with a spherical harmonic expansions of gravitational potential (the gradient of which is local gravity).

Such corrections are appropriate for highly accurate predictions of surface gravity or orbital operations in the neighborhood of Earth. At extreme distances however, the ellipsoidal model, or even a point–mass model is preferable.

Reference A. demonstrates calculations with the current IERS standard spherical harmonic coefficients for Earth: the Earth Gravitational Model 1996 (EGM96).

More refined coefficients for Earth are provided by the NASA GRACE mission (to be followed by GRACE-FO in 2019), which have allowed observation of temporal variations.

ftp://ftp.csr.utexas.edu/pub/grace/GGM05

# Values used for Earth (WSG 84)

The current ellipsoidal model for Earth is the World Geodetic System 1984 (WSG 84): parameters which completely specify instantiation of the model.

Values for WSG 84 model obtained from Ref. A.:

6378137,       # a, semi-major axis (equatorial) [m]
298.257223563, # 1/f, inverse flatness [dimensionless]
7.292115e-5,   # omega, angular velocity [rad s^-1]
3.9860009e14   # GM, Universal grav. const. * mass of Earth [m^3 s^-2]


## Notes on Values Used

### omega

Treated as average contstant angular velocity for Earth, ignoring precession. For very precise orbital calculations, such nuances may not be lightly ignored.

### GM

Different values for GM are used for different purposes

• old value 3.986005e14
• new value 3.986004418e14
• value without atmosphere 3.9860009e14 <-- We use this

In this package, we have used the value without atmosphere for near–surface gravity estimations, justified by appeal to the shell theorem (approximately).

For the sake of completeness, orbital computations should use the updated value, however GPS broadcasts assume that receivers continue to use the old value and encode appropriately (no accuracy is lost while complicated software updates to many deployed receivers are avoided).

## Evaluating Uncertainty

From the GRACE data, we note that gravitational potential can differ from the predictions of the ellipsoidal model by 0.02% near the surface of Earth, though such deviations are typically less than 0.005%. Ignoring the transverse components of gravity caused by such irregularities, this translates to errors of up to 0.02% in the magnitude of the calculated gravity vector.

Accounting for vertical deflection, we note maximum deflections at Earth's surface of up to 100 arc-seconds (4.8e-4 radians), yielding 0.00001% uncertainty in the vertical component (as a percentage of the total magnitude) and 0.048% in the transverse component. Note that we use the geodetic definition of "vertical" (normal to the reference ellipsoid).

Finally, in consideration of our choice of value for GM (which differs from the updated value by subtracting the mass of the atmosphere), we note that the mass of the atmosphere is ~0.0001% of that of the planet and consider that the force of gravity linear in the planet's mass (this is approximately true: only for pure gravitation, not the effects of centrifugal force).

We expect the ellipsoidal gravitational model, as implemented here for Earth, to be accurate to +/- 0.02% in the vertical component and +/- 0.07% in the transverse component (where percentages are given relative to the total magnitude of the calculated gravity vector) everywhere on or near Earth's surface.

All of this assumes of course that our neither our implementation nor our analysis is flawed. This package it NOT intended for mission-critical calculations or simulation. See license.txt for more information.

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