scholarly journals Constants related to the Earth and Moon

1965 ◽  
Vol 21 ◽  
pp. 67-79
Author(s):  
Harold Jeffreys
Keyword(s):  
Gravitational Field ◽  
The Moon ◽  
The Earth ◽  
Earth’S Gravitational Field ◽  
Tesseral Harmonics

The author discusses various determinations of zonal and tesseral harmonics of the Earth's gravitational field, the values of the solar parallax, and the constants related to the figure of the Moon and its motion.

The orbits of Ariel 4 and Prospero

1975 ◽  
Vol 343 (1633) ◽  
pp. 257-264 ◽  
Keyword(s):  
Gravitational Field ◽  
Orbital Parameters ◽  
Zonal Harmonics ◽  
Earth’S Gravitational Field ◽  
Tesseral Harmonics ◽  
Visual Observations

Orbital parameters for Ariel 4 and Prospero have been determined at the Royal Aircraft Establishment and made available for use with the respective telemetry analysis programs. Ariel 4 orbit determinations were based on N.A.S.A. Minitrack observations, and Prospero orbit determinations on U.S. Navy observations and a small number of visual observations. Both orbits are near polar (inclination 83° for Ariel 4 and 82° for Prospero) but not otherwise similar. The initial perigee and apogee heights were, respectively, 500 and 600 km for Ariel 4, as against 550 and 1600 km for Prospero. Hence Ariel 4 has experienced much more drag than Prospero and orbital parameters had to be determined at much closer intervals for the former than for the latter, 3 days as against 7 days. The Ariel 4 orbit is being analysed to study the effects of 15th-order tesseral harmonics in the Earth’s gravitational field, and the Prospero orbit has been used in a recent determination of odd zonal harmonics.

scholarly journals Accurate Time Calculations of Falling Bodies in the Earth's Gravitational Field and Comparisons with Newton's Laws of Vertical Motion

2021 ◽  
pp. 44-53
Author(s):  
A. Ebaid ◽  
Shorouq M. S. Al-Qahtani ◽  
Afaf A. A. Al-Jaber ◽  
Wejdan S. S. Alatwai ◽  
Wafaa T. M. Alharbi
Keyword(s):  
Gravitational Field ◽  
Real Life ◽  
Vertical Motion ◽  
Exact Form ◽  
Potential Danger ◽  
Accurate Calculation ◽  
The Earth ◽  
Earth’S Gravitational Field ◽  
Celestial Bodies ◽  
Newton’S Laws

The Earth is exposed annually to the fall of some meteorites and probably other celestial bodies which cause a potential danger to vital areas in several countries. Consequently, the accurate calculation of the falling time of such bodies is useful in order to take the necessary procedures for protecting these areas. In this paper, Newton’s law of general gravitation is applied to analyze the vertical motion in the Earth’s gravitational field. The falling time is obtained in exact form. The results are applied on several objects in real life.

scholarly journals Rotation of the Rigid Earth

1997 ◽  
Vol 165 ◽  
pp. 295-300
Author(s):  
P. Bretagnon
Keyword(s):  
Time Domain ◽  
Time Span ◽  
Euler Angles ◽  
The Sun ◽  
The Moon ◽  
The Earth ◽  
Rigid Earth ◽  
The Time Domain ◽  
Tesseral Harmonics ◽  
External Forces

AbstractWe present the results of a solution of the Earth’s rotation built with analytical solutions of the planets and of the Moon’s motion. We take into account the influence of the Moon, the Sun and all the planets on the potential of the Earth for the zonal harmonics Cj,0 for j from 2 to 5, and also for the tesseral harmonics C2,2, S2,2C3,k, S3,k for k from 1 to 3 and C4,1, S4,1. We determine three Euler angles ψ, ω, and φ by calculating the components of the torque of the external forces with respect to the geocenter in the case of the rigid Earth. The analytical solution of the precession-nutation has been compared to a numerical integration over the time span 1900–2050. The differences do not exceed 16 μas for ψ and 8 μas for ω whereas the contribution of the tesseral harmonics reaches 150 μas in the time domain.

scholarly journals Vito Volterra, 1860 - 1940

1941 ◽  
Vol 3 (10) ◽  
pp. 691-729 ◽  
Keyword(s):  
Gravitational Field ◽  
Restricted Problem ◽  
The Moon ◽  
Short Intervals ◽  
The Earth ◽  
Before And After ◽  
Vito Volterra ◽  
The Subject ◽  
The Town ◽  
Mathematics And Physics

Vito Volterra was born at Ancona on 3 May 1860, the only child of Abramo Volterra and Angelica Almagià. When he was three months old the town was besieged by the Italian army and the infant had a narrow escape from death, his cradle being actually destroyed by a bomb which fell near it. When he was barely two years old his father died, leaving the mother, now almost penniless, to the care of her brother Alfonso Almagia, an employee of the Banca Nazionale, who took his sister into his house and was like a father to her child. They lived for some time in Terni, then in Turin, and after that in Florence, where Vito passed the greater part of his youth and came to regard himself as a Florentine. At the age of eleven he began to study Bertrand’s Arithmetic and Legendre’s Geometry , and from this time on his inclination to mathematics and physics became very pronounced. At thirteen, after reading Jules Verne’s scientific novel Around the Moon , he tried to solve the problem of determining the trajectory of a projectile in the combined gravitational field of the earth and moon: this is essentially the ‘restricted Problem of Three Bodies’, and has been the subject of extensive memoirs by eminent mathematicians both before and after the youthful Volterra’s effort: his method was to partition the time into short intervals, in each of which the force could be regarded as constant, so that the trajectory was obtained as a succession of small parabolic arcs. Forty years later, in 1912, he demonstrated this solution in a course of lectures given at the Sorbonne.

Gravitational acceleration of a weakly relativistic electron in a conducting drift tube

2018 ◽  
Vol 33 (33) ◽  
pp. 1850192 ◽  
Author(s):  
V. I. Denisov ◽  
I. P. Denisova ◽  
M. G. Gapochka ◽  
A. F. Korolev ◽  
N. N. Koshelev
Keyword(s):  
Gravitational Field ◽  
Relativistic Electron ◽  
Horizontal Plane ◽  
Drift Tube ◽  
Gravitational Force ◽  
Vertical Direction ◽  
Gravitational Acceleration ◽  
The Earth ◽  
Earth’S Gravitational Field

We propose the idea of method for observing the effect of the Earth’s gravitational field on the motion of an electron. Earlier attempts to measure such an effect proved unsuccessful due to the fact that under the conductive sheath, the gravitational force acting on the non-relativistic electron is completely compensated by Barnhill–Schiff force. Therefore, experiments of this kind were unable to measure the effect of the Earth’s gravitational field on the motion of electrons. In this paper, we propose to use electrons moving with relativistic speeds in the horizontal plane, and with non-relativistic speeds in the vertical direction, in which case the gravitational force on these electrons is not fully compensated by the Barnhill–Schiff force. Calculations showed that in this case, it is possible to measure the force exerted on an electron by the gravitational field of the Earth.

scholarly journals Satellite Gravimetry: A Review of Its Realization

2021 ◽  
Author(s):  
Frank Flechtner ◽  
Christoph Reigber ◽  
Reiner Rummel ◽  
Georges Balmino
Keyword(s):  
Gravitational Field ◽  
Gravity Field ◽  
Water Cycle ◽  
Temporal Changes ◽  
Sensor Technology ◽  
Satellite Gravimetry ◽  
The Earth ◽  
Earth’S Gravitational Field ◽  
Gravity Field Determination ◽  
Space Age

AbstractSince Kepler, Newton and Huygens in the seventeenth century, geodesy has been concerned with determining the figure, orientation and gravitational field of the Earth. With the beginning of the space age in 1957, a new branch of geodesy was created, satellite geodesy. Only with satellites did geodesy become truly global. Oceans were no longer obstacles and the Earth as a whole could be observed and measured in consistent series of measurements. Of particular interest is the determination of the spatial structures and finally the temporal changes of the Earth's gravitational field. The knowledge of the gravitational field represents the natural bridge to the study of the physics of the Earth's interior, the circulation of our oceans and, more recently, the climate. Today, key findings on climate change are derived from the temporal changes in the gravitational field: on ice mass loss in Greenland and Antarctica, sea level rise and generally on changes in the global water cycle. This has only become possible with dedicated gravity satellite missions opening a method known as satellite gravimetry. In the first forty years of space age, satellite gravimetry was based on the analysis of the orbital motion of satellites. Due to the uneven distribution of observatories over the globe, the initially inaccurate measuring methods and the inadequacies of the evaluation models, the reconstruction of global models of the Earth's gravitational field was a great challenge. The transition from passive satellites for gravity field determination to satellites equipped with special sensor technology, which was initiated in the last decade of the twentieth century, brought decisive progress. In the chronological sequence of the launch of such new satellites, the history, mission objectives and measuring principles of the missions CHAMP, GRACE and GOCE flown since 2000 are outlined and essential scientific results of the individual missions are highlighted. The special features of the GRACE Follow-On Mission, which was launched in 2018, and the plans for a next generation of gravity field missions are also discussed.

The gravity field of the Earth

1980 ◽  
Vol 294 (1410) ◽  
pp. 317-328 ◽  
Keyword(s):  
Gravitational Field ◽  
Space Flight ◽  
Double Series ◽  
Earth Model ◽  
Resonant Orbits ◽  
The Earth ◽  
Goddard Space Flight Center ◽  
Improving Accuracy ◽  
Tesseral Harmonics ◽  
Optical Laser

In recent years the Earth’s gravitational field has been determined with continually improving accuracy, by using hundreds of thousands of observations of Earth satellites, chiefly optical, laser and Doppler, together with surface gravimetry and, most recently, altimeter measurements from the Geos 3 satellite. The geopotential is usually expressed as a double series of tesseral harmonics, and several hundred of the harmonic coefficients are evaluated. Progress in this work during the 1970s is briefly outlined, and some attempt is made to assess the accuracy of current geoid maps and sets of harmonic coefficients, as exemplified in the latest models derived at the Goddard Space Flight Center. The harmonic coefficients of order 14, 15 and 30 in the Goddard Earth Model 10B are compared with values obtained independently by analysis of resonant orbits: the results suggest that the values in GEM 10B are realistic for these orders, and presumably others. It appears that the accuracy of the geoid maps is now approaching 1 m.

Structure of the terrain and gravitational field of the planets: Kaula’s rule as a consequence of the probability laws by A.N. Kolmogorov and his school

2019 ◽  
Vol 485 (4) ◽  
pp. 493-496
Author(s):  
E. B. Gledzer ◽  
G. S. Golitsyn
Keyword(s):  
Random Walk ◽  
Gravitational Field ◽  
Spherical Harmonics ◽  
Empirical Data ◽  
The Moon ◽  
Linear Coefficient ◽  
Water Flows ◽  
The Earth ◽  
Order Of Magnitude ◽  
Celestial Bodies

Kaula’s empirical rule has been known for more than 50 years: the coefficients of expansion over spherical harmonics for the fluctuations of the gravitational field and terrain of the planets decrease as the number of the harmonic squared. This was found for Venus, the Moon, Mars, the asteroid Vesta, and very small celestial bodies. The inverse-square line spectra were also found for various types of the Earth’s surface on a scale of up to a hundred kilometers. From this it follows that the spectra of the terrain slope angles are constant, i.e., “white noise”. This, they are delta-correlated horizontally. These are the assumptions under which the random walk laws were derived by A.N. Kolmogorov in 1934. Using them, the equation of the horizontal probability diffusion of the terrain with the linear coefficient diffusion D is derived. Based on the empirical data, D = 1.3 ± 0.3 m for the Earth, while for Venus it is almost an order of magnitude less. The slopes resist the wind; the rock crumbles, and the water flows down the slopes as well. This consideration turns Kaula’s rule into the random walk laws (over terrain) developed by Kolmogorov in 1934.

scholarly journals Squeezing of the Super-Elastic Double-Helix Photon in the Gravitational Field (Hidden in Very-Well Known Old Formulae). Pound - Rebka - Snider Effect Studied by the Advanced LIGO Instrument, Pioneer Anomaly and CMB (Cosmic Microwave Background) (23.11.201

2019 ◽  
Vol 12 (1) ◽  
pp. 8
Author(s):  
Jiri Stavek
Keyword(s):  
Gravitational Field ◽  
Solar System ◽  
Double Helix ◽  
The Sun ◽  
The Moon ◽  
Gravitational Fields ◽  
The Earth ◽  
Pioneer Anomaly ◽  
Squeeze Rate ◽  
Squeeze Effect

In our approach we have combined knowledge of Old Masters (working in this field before the year 1905), New Masters (working in this field after the year 1905) and Dissidents under the guidance of Louis de Broglie and David Bohm. Based on the great experimental work of Robert Pound, Glen A. Rebka and J.L. Snider we have proposed a squeezing of the super-elastic double-helix photon in the gravitational field. We have newly defined the squeeze rate of that photon particle on the helical path. We have inserted this squeeze rate into the very-well old formulae of Newton, Soldner, Gerber and Einstein and might glimpse traces of the quantum gravity. The squeeze rate of photons can be studied in details using the Great instrument - the Advanced LIGO - located on the surface of the Earth (USA, Italy, Japan). The observed strains on the level 5*10-19 should be caused by the gravitational field of our Earth. The observed strains on the level 5*10-22 should be caused by the gravitational fields of the Moon and the Sun. We estimate that the experimental value of the gravitational constant G studied by the LIGO instrument can achieve the accuracy to the level of ppb (parts per billion) after the removal of those strains from the measured signal and the removal of the gravitational influences of the Earth, the Moon, the Sun, Venus and Jupiter. To study the squeeze effect on a bigger scale we propose to analyze the Pioneer anomaly where Pioneer´s photons have been flying around the planets in our Solar system causing the squeeze effect - the anomalous blueshift. Similarly, we can study cosmic microwave photons flying around the objects in our Solar system that might create “the axis of evil” - temperature fluctuations in the CMB map (Wien displacement law). Can we prepare in our Solar system “tired” light by frequent blueshift - redshift transitions? Can it be that Nature cleverly inserted the squeeze rate into our very-well known Old Formulae? We want to pass this concept into the hands of Readers of this Journal better educated in the Mathematics and Physics.

scholarly journals Interaction of Gravitational Field and Orbit in Sun-planet-moon system

2021 ◽  
Author(s):  
Yin Zhu
Keyword(s):  
Gravitational Field ◽  
Attractive Force ◽  
The Sun ◽  
The Moon ◽  
Moon System ◽  
Gravitational Fields ◽  
The Earth ◽  
Repulsive Gravity ◽  
Single Field ◽  
Gravitational Unit

Studying the two famous old problems that why the moon can move around the Sun and why the orbit of the Moon around the Earth cannot be broken off by the Sun under the condition that calculating with F=GMm/R^2, the attractive force of the Sun on the Moon is almost 2.2 times that of the Earth. We found that the planet and moon are unified as one single gravitational unit which results in that the Sun cannot have the force of F=GMm/R^2 on the moon. The moon is moved by the gravitational unit orbiting around the Sun. It could indicate that the gravitational field of the moon is limited inside the unit and the gravitational fields of both the planet and moon is unified as one single field interacting with the Sun. The findings are further clarified by reestablishing Newton’s repulsive gravity.

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