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Gravitational and Related Constants for Accurate Space Navigation

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Abstract

Gravitational and Related Constants for Accurate Space Navigation. The paper first compares the Gaussian gravitational constant, k s , and the astronomical unit with the cgs value of G and the meter, showing that the latter two are valueless for heliocentric orbits. For geocentric orbits a quasi-Gaussian geocentric gravitational constant, k e , is determined from the equatorial acceleration of gravity, g e , the equatorial radius, a e , the flattening, f, and the related coefficients, J and K, of the second and fourth harmonics in the potential of the terrestrial ellipsoid, after improved values of these data are ascertained and adopted:

$$\begin{array}{*{20}{l}} {{g_e} = 9.780,368(1 + g' + \frac{1}{3}f')meters/{{\sec }^2}}&{g' = 0\pm 3 \times {{10}^{ - 6}}}\\ {{a_e} = 6,378,270(1 + a' + \frac{4}{3}f')meters}&{a' = 0\pm 10 \times {{10}^{ - 6}}}\\ {f = + 0.003,367,00 + f'}&{f' = 0\pm 4 \times {{10}^{ - 6}}}\\ {J = + 0.001,638,08 + f'}&{K = + 9.04 \times {{10}^{ - 6}}} \end{array}$$

The value of k e is ascertained for three units of distance, the megameter = 106 meters, the “q-radius” =the equatorial radius, and the “g-radius” =the “gravitational radius”, so defined by \(a_e = 1 + \frac{3}{1}a' - \frac{1}{3}g' + \frac{3}{2}f'\) that the uncertainty of k e will be zero:

$$\begin{array}{*{20}{l}} {{k_e} = 1.197,918,5(1 + a' + \frac{1}{2}g' + f')megameter{s^{3/2}}/\min }&{\frac{{\Delta {k_e}}}{{{k_e}}} = \pm 11 \times {{10}^{ - 6}}}\\ {{k_e} = 0.074,365,74(1 - \frac{1}{2}a' + \frac{1}{2}g' - f')q - radi{i^{3/2}}/\min }&{\frac{{\Delta {k_e}}}{{{k_e}}} = \pm 6\frac{1}{2} \times {{10}^{ - 6}}}\\ {{k_e} = 0.074,365,74\,g - radii{\,^{3/2}}/\min }&{\frac{{\Delta {k_e}}}{{{k_e}}} = 0} \end{array}$$

Basic expressions due to de Sitter, Lambert, and others, that are alternative to those used in the foregoing determinations are compared with them in the appendix.

The moon’s parallax, as an alternate source of k e , is shown to be inferior to g e . A new value of the dynamical parallax is ascertained to be \({\pi _\varepsilon } = 3422.650(1\pm 4\frac{1}{2} \times {10^{ - 6}})\). Other possible sources are considered; the effects of the constants on ICBM and satellite trajectories are developed; extensive tables of conversion factors are supplied for mass, length, time, velocity, angular velocity, acceleration, and period.

Keywords

  • Equatorial Radius
  • Astronomical Unit
  • Monthly Notice
  • Related Constant
  • Gravitational Radius

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Zusammenfassung

Gravitationskonstante und verwandte Konstanten für exakte Navigation im Raum. Die vorliegende Arbeit vergleicht die Gausssche Gravitationskonstante k s und die Astronomische Einheit mit den CGS-Werten von G und dem Meter. Dabei wird gezeigt, daß die beiden letztgenannten für heliozentrische Bahnen ohne Wert sind. Für geozentrische Bahnen wird eine quasi-Gausssche geozentrische Gravitations-konstante k e aus der äquatorialen Schwerebeschleunigung g e , dem äquatorialen Radius a e , der Abplattung f und den entsprechenden Koeffizienten J und K der zweiten und vierten Harmonie im Potential des Erdellipsoids bestimmt. Hierauf werden verbesserte Werte dieser Daten ermittelt und angenommen:

$$\begin{array}{*{20}{l}} {{g_e} = 9.780,368(1 + g' + \frac{1}{3}f')m/{{\sec }^2}}&{g' = 0\pm 3 \times {{10}^{ - 6}}}\\ {{a_e} = 6,378,270(1 + a' + \frac{4}{3}f')m}&{a' = 0\pm 10 \times {{10}^{ - 6}}}\\ {f = + 0.003,367,00 + f'}&{f' = 0\pm 4 \times {{10}^{ - 6}}}\\ {J = + 0.001,638,08 + f'}&{K = + 9.04 \times {{10}^{ - 6}}} \end{array}$$

Der Wert von k e wird für drei Entfernungseinheiten ermittelt: für das Megameter = 106 Meter, den „q-Radius“ = Äquatorradius und den „g-Radius“ = „Gravitationsradius“, durch \(a_e = 1 + \frac{3}{1}a' - \frac{1}{3}g' + \frac{3}{2}f'\) so definiert, daß die Unsicherheit von k e Null wird:

$$\begin{array}{*{20}{l}} {{k_e} = 1.197,918,5(1 + a' + \frac{1}{2}g' + f')Megameter{s^{3/2}}/\min }&{\frac{{\Delta {k_e}}}{{{k_e}}} = \pm 11 \times {{10}^{ - 6}}}\\ {{k_e} = 0.074,365,74(1 - \frac{1}{2}a' + \frac{1}{2}g' - f')q - radi{i^{3/2}}/\min }&{\frac{{\Delta {k_e}}}{{{k_e}}} = \pm 6\frac{1}{2} \times {{10}^{ - 6}}}\\ {{k_e} = 0.074,365,74\,g - Radien{\,^{3/2}}/\min }&{\frac{{\Delta {k_e}}}{{{k_e}}} = 0} \end{array}$$

Grundgleichungen nach de Sitter, Lambert und anderen Autoren, die neben den in den vorstehenden Bestimmungen verwendeten möglich sind, werden mit diesen im Anhang verglichen.

Eine andere Berechnungsquelle für k e , die Mondparallaxe, erwies sich als dem g e unterlegen. Als ein neuer Wert der dynamischen Parallaxe wird: \(\pi _\varepsilon = 3422.650\left( {1 \pm 4\frac{1}{2} \times 10^{ - 6} } \right)\) ermittelt. Noch andere Berechnungsquellen werden betrachtet; die Auswirkungen der Konstanten auf interkontinentale ballistische Geschosse und Satellitenbahnen werden entwickelt; umfangreiche Tabellen der Umrechnungsfaktoren für Masse, Länge, Zeit, Geschwindigkeit, Winkelgeschwindigkeit, Beschleunigung und Periode werden aufgestellt.

Keywords

  • Equatorial Radius
  • Astronomical Unit
  • Monthly Notice
  • Related Constant
  • Gravitational Radius

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Résumé

Constantes de gravitation et constantes associées pour une navigation spatiale de précision. L’article compare d’abord la constante de gravitation k s de Gauss et l’unité astronomique avec l’unité cgs de G et le mètre, montrant que ces deux dernières sont sans signification pour les orbites héliocentriques. Une constante de gravitation quasi-gaussienne k e est déterminée pour les orbites géocentriques à partir de l’accélération équatoriale de la pesanteur g e , du rayon équatorial a e , de l’applatissement f et des coefficients J etK relatifs au second et au quatrième harmonique du potentiel du géoide. Les valeurs améliorées et finalement adoptées pour ces grandeurs sont les suivantes:

$$\begin{array}{*{20}{l}} {{g_e} = 9.780,368(1 + g' + \frac{1}{3}f')m/{{\sec }^2}}&{g' = 0\pm 3 \times {{10}^{ - 6}}}\\ {{a_e} = 6,378,270(1 + a' + \frac{4}{3}f')m}&{a' = 0\pm 10 \times {{10}^{ - 6}}}\\ {f = + 0.003,367,00 + f'}&{f' = 0\pm 4 \times {{10}^{ - 6}}}\\ {J = + 0.001,638,08 + f'}&{K = + 9.04 \times {{10}^{ - 6}}} \end{array}$$

La valeur de k e est obtenue pour trois unités de distance: le mégamètre (106 mètres) le “rayon q” ou rayon équatorial et le “rayon g” ou rayon gravitationnel défini par la formule \(a_e = 1 + \frac{3}{1}a' - \frac{1}{3}g' + \frac{3}{2}f'\) telle que l’inexactitude sur k e soit nulle:

$$\begin{array}{*{20}{l}} {{k_e} = 1.197,918,5(1 + a' + \frac{1}{2}g' + f')Megameter{s^{3/2}}/\min }&{\frac{{\Delta {k_e}}}{{{k_e}}} = \pm 11 \times {{10}^{ - 6}}}\\ {{k_e} = 0.074,365,74(1 - \frac{1}{2}a' + \frac{1}{2}g' - f')q - radi{i^{3/2}}/\min }&{\frac{{\Delta {k_e}}}{{{k_e}}} = \pm 6\frac{1}{2} \times {{10}^{ - 6}}}\\ {{k_e} = 0.074,365,74\,g - Radien{\,^{3/2}}/\min }&{\frac{{\Delta {k_e}}}{{{k_e}}} = 0} \end{array}$$

.

Les expressions fondamentales de de Sitter, Lambert et autres, qui peuvent remplacer celles utilisees dans la détermination précédente, leurs sont comparées en appendice.

La parallaxe de la lune, autre source possible de détermination de k e , est montrée être inférieure à g e . Une nouvelle valeur de la parallaxe dynamique est établie: \(\pi _\varepsilon = 3422.650\left( {1 \pm 4\frac{1}{2} \times 10^{ - 6} } \right)\). D’autres sources possibles sont envisagées; l’influence de ces constantes sur les trajectoires des engins balistiques intercontinentaux et des satellites artificiels est analysée. Enfin des tables de conversion étendues sont fournies pour la masse, la longueur, le temps, la vitesse, la vitesse angulaire, l’accélération et la période.

Keywords

  • Equatorial Radius
  • Astronomical Unit
  • Monthly Notice
  • Related Constant
  • Gravitational Radius

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Revision of a January 1957 paper circulated under the title “Units and Constants for Geocentric Orbits” and supported in part by the International Geophysical Year Satellite program, through the Smithsonian Astrophysical Observatory.

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© 1958 Springer-Verlag Wien

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Herrick, S., Baker, R.M.L., Hilton, C.G. (1958). Gravitational and Related Constants for Accurate Space Navigation. In: Hecht, F. (eds) VIIIth International Astronautical Congress Barcelona 1957 / VIII. Internationaler Astronautischer Kongress / VIIIe Congrès International D’Astronautique. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-39990-3_17

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  • DOI: https://doi.org/10.1007/978-3-662-39990-3_17

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