Symbolic Expressions of Differences between Earth Radii

doi:10.11947/j.JGGS.2020.0105

Abstract

Abstract:

A systematic and comprehensive comparison of the five commonly used earth radii in geodesy and cartography is carried out, and the differences between the most common points of the earth radii, their corresponding maximum values, and the latitudes of equal points between them are derived with the help of computer algebraic systems. The symbolic expressions are expressed as a power series of the first eccentricity. Taking the CGCS2000 ellipsoid as an example, the differences between the commonly used earth radii are clarified to numerical values. The results show that the difference between the commonly used earth radii has a maximum at 90 degrees and a minimum at 0 degree. The difference between the average radius of curvature and the rectifying sphere radius is the biggest, and the difference between the average radius of curvature and the average sphere radius is the smallest. These results can provide a theoretical basis for corresponding research in the geosciences, space science, navigation and positioning.

Key words: earth radius; difference extrema value; extrema value point; symbolic expression; CGCS200

Jingwen ZONG,Houpu LI,Shaofeng BIAN,Qinghui TANG. Symbolic Expressions of Differences between Earth Radii[J]. Journal of Geodesy and Geoinformation Science, 2020, 3(1): 45-51.

Figures/Tables 9

Tab.1

The symbolic expressions of different max value between the average radius of curvature and four common sphere radii"

ΔR/a	Value of the maximum difference point	Difference maximum symbol expression
(R_a-R_e)/a	$π 2$	$23$ e²+ $512$ e⁴+ $13$ e⁶+ $55192$ e⁸
(R_a-R_F)/a	$π 2$	$23$ e²+ $1945$ e⁴+ $253756$ e⁶+ $32453113400$ e⁸
(R_a-R_V)/a	$π 2$	$23$ e²+ $49$ e⁴+ $115324$ e⁶+ $295972$ e⁸
(R_a-R_S)/a	$π 2$	$34$ e²+ $2764$ e⁴+ $85256$ e⁶+ $465516384$ e⁸

Tab.1

Tab.2

The symbolic expressions of different min value between average radius of curvature and four common sphere radii"

ΔR/a	Value of the minimum difference point	Difference minimum symbol expression
(R_a-R_e)/a	0	- $13$ e²- $112$ e⁴- $124$ e⁶- $5192$ e⁸
(R_a-R_F)/a	0	- $13$ e²- $790$ e⁴- $611512$ e⁶- $5969226800$ e⁸
(R_a-R_V)/a	0	- $13$ e²- $118$ e⁴- $13648$ e⁶- $353888$ e⁸
(R_a-R_S)/a	0	- $14$ e²- $564$ e⁴- $11256$ e⁶- $46516384$ e⁸

Tab.2

Tab.3

The equal geodetic latitude symbolic expressions between average radius of curvature and four common sphere radii"

ΔR	Geodetic latitude B
R_a-R_e	$12$ arccos $13$ + $5224$ e²+ $24722304$ e⁴+ $231192331776$ e⁶+ $1076243221233664$ e⁸
R_a-R_F	$12$ arccos $13$ + $210$ e²+ $677212600$ e⁴+ $135132378000$ e⁶+ $9190336923492720000$ e⁸
R_a-R_V	$12$ arccos $13$ + $212$ e²+ $372864$ e⁴+ $9723456$ e⁶+ $3073121492992$ e⁸
R_a-R_S	$π 6$ + $3232$ e²+ $15723072$ e⁴+ $413212288$ e⁶+ $13633324718592$ e⁸

Tab.3

Tab.4

The symbolic expressions of different extrema between four common sphere radii"

ΔR/a	Difference symbol expression
(R_e-R_V)/a	$136$ e⁴+ $7324$ e⁶+ $25615552$ e⁸
(R_F-R_V)/a	$145$ e⁴+ $231134$ e⁶+ $5891340200$ e⁸
(R_e-R_F)/a	$1180$ e⁴+ $1756$ e⁶- $251907200$ e⁸
(R_F-R_S)/a	$112$ e²- $12880$ e⁴- $12748384$ e⁶- $479119232243200$ e⁸
(R_e-R_S)/a	$112$ e²+ $1192$ e⁴- $1768$ e⁶- $11549152$ e⁸
(R_S-R_V)/a	- $112$ e²+ $13576$ e⁴+ $47520736$ e⁶+ $771553981312$ e⁸

Tab.4

Fig.1

Tab.5

Fig.2

Tab.6

Tab.7

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B	0°	15°	30°	45°	60°	75°	90°
R_a-R_e	-14256.5	-11404.6	-3599.9	7092.3	17820.5	25697.0	28584.9
R_a-R_F	-14254.9	-11403.0	-3598.4	7093.9	17822.1	25698.6	28586.4
R_a-R_S	-10696.8	-7844.9	-4036.8	10651.9	21380.1	29256.6	32144.5
R_a-R_V	-14248.5	-11396.6	-3592.0	7100.2	17828.5	25705.0	28592.8