Geopotential harmonics of order 15 and odd degree from analysis of resonant orbits

We have analysed the variations of inclination in 13 satellite orbits as they pass slowly, under the action of air drag, through 15th-order resonance with the geopotential, when successive equatorial crossings are 24° apart and the ground track repeats after 15 rev. The size and form of the change in inclination are determined mainly by the values of the geopotential harmonics of 15th order and odd degree, $\text{C}\text{?}{}_{\mathrm{l,15}}$ and $\text{S}\text{?}{}_{\mathrm{l,15}}$ (with l = 15, 17, 19, …) in the usual notation. Our analysis gives values of these coefficients up to l = 33 as follows:

     

2.

Evaluation of harmonics in the geopotential of order 15 and odd degree     

D.G. King-Hele  Doreen M.C. Walker  R.H. Gooding 《Planetary and Space Science》1974,22(10):1349-1373

When a satellite orbit decaying slowly under the action of air drag experiences 15th-order resonance with the Earth's gravitational field, so that the ground track repeats after 15 rev, the orbital inclination suffers appreciable changes due to the perturbations from the harmonics in the geopotential of order 15 and odd degree (15,17,19 …). In this paper the changes in inclination at resonance of 11 satellites at inclinations between 30° and 90° have been analysed to determine values of the geopotential coefficients of order 15 and degree $\text{l,}\text{C}\text{?}{}_{\mathrm{l,15}}$ and $\text{S}\text{?}{}_{\mathrm{l,15}}$ in the usual notation. The recommended solution, going up to l = 31, is:

l	109C?l,15	109S?l,15
15	?23.5 ± 0.8	?7.7 ± 0.8
17	6.3 ± 1.5	5.6 ± 1.5
19	?25.1 ± 2.5	?7.3 ± 2.3
21	27.8 ± 3.6	?0.7 ± 3.4
23	17.1 ± 4.1	13.9 ± 4.8
25	?1.1 ± 3.0	8.5 ± 4.2
27	10.0 ± 3.3	6.7 ± 2.7
29	?9.4 ± 3.5	0.1 ± 4.7
31	10.1 ± 5.4	3.8 ± 5.6
33	1.1 ± 5.7	3.1 ± 5.8

     

3.

Evaluation of 14th-order harmonics in the geopotential     

D.G. King-Hele  Doreen M.C. Walker  R.H. Gooding 《Planetary and Space Science》1979,27(1):1-18

The Earth's gravitational potential is now usually expressed in terms of a double series of tesseral harmonics with several hundred terms, up to order and degree at least 20. The harmonics of order 14 can be evaluated by analysing changes in satellite orbits which experience 14th-order resonance, when the track over the Earth repeats after 14 revolutions.In this paper we describe our first evaluation of individual 14th-order coefficients in the geopotential from analysis of the variations in inclination and eccentricity of satellite orbits passing through 14th-order resonance under the action of air drag. Using results from eleven satellites, we find the following values for normalized coefficients of harmonics of order 14 and degree l, C?_{l, 14} and S?_{l, 14}, for l=14, 154. 22:

l	109C?l,15	109S?l,15
15	?21.5 ± 0.9	?8.4 ± 0.9
17	4.4 ± 1.6	9.0 ± 1.5
19	?15.6 ± 2.6	?14.1 ± 2.7
21	10.4 ± 3.0	7.3 ± 3.5
23	22.5 ± 2.8	1.2 ± 4.4
25	?0.9 ± 4.7	?3.8 ± 5.3
27	?11.2 ±3.3	9.1 ± 3.2
29	?20.5 ± 5.4	?1.2 ± 6.1
31	17.7 ± 6.6	?1.0 ± 7.1

     

4.

Geopotential harmonics of order 15 and even degree,from changes in orbital eccentricity at resonance     

D.G. King-Hele  Doreen M.C. Walker  R.H. Gooding 《Planetary and Space Science》1975,23(2):229-246

When a satellite orbit decaying slowly under the action of air drag experiences 15th-order resonance with the Earth's gravitational field, so that the ground track repeats after 15 rev, the orbital eccentricity may suffer appreciable changes due to perturbations from the gravitational harmonics of order 15 and even degree (16, 18, 20…). In this paper the changes in eccentricity at resonance for six satellites in near-circular orbits at inclinations between 56 and 90° have been analysed to derive 11 pairs of equations linking the harmonic coefficients of order 15 and (even) degree l, $\text{C}{}_{\mathrm{l,15}}\text{and}\text{S}{}_{\mathrm{l,15}}$ in the usual notation. These equations (together with eight constraint equations) are solved to give:

$\text{l}$	109C?$\text{l,14}$	109S?l,14
-	-	-
14	?38.5 ±2.9	?7.8 ±2.2
15	4.5 ±1.1	?23.8 ±0.3
16	?22.3 ±3.6	?36.0 ±3.8
17	?15.0 ±2.6	16.8 ±1.2
18	?24.0±4.9	?3.2 ±3.7
19	?1.6 ±2.8	?7.6 ±1.0
20	8.8 ±5.8	?15.4 ±4.6
21	18.2 ±3.6	?10.6 ±1.9
22	?14.5 ±8.1	9.9 ±6.4

     

5.

The Meteoritical Bulletin,No. 98, September 2010     

Michael K. WEISBERG  Caroline SMITH  Christopher HERD  Henning HAACK  Akira YAMAGUCHI  Hasnaa CHENNAOUI AOUDJEHANE  Linda WELZENBACH  Jeffrey N. GROSSMAN 《Meteoritics & planetary science》2010,45(9):1530-1551

     

6.

A general solution of Poisson''s equation in three dimensions for disk galaxies     

Tong Yi  Zheng Xue-tang 《Chinese Astronomy and Astrophysics》1982,6(4):319-324

     

7.

Stratigraphic signatures of climatic change during the Holocene evolution of the Tigris–Euphrates delta, lower Mesopotamia     

Adnan A. M. Aqrawi   《Global and Planetary Change》2001,28(1-4)

     

8.

Darwin—an experimental astronomy mission to search for extrasolar planets     

《Experimental Astronomy》2009,23(1):435-461

As a response to ESA call for mission concepts for its Cosmic Vision 2015–2025 plan, we propose a mission called Darwin. Its primary goal is the study of terrestrial extrasolar planets and the search for life on them. In this paper, we describe different characteristics of the instrument.

$\text{l}$	109$\text{C}{}_{\mathrm{l,15}}$	109$\text{S}{}_{\mathrm{l,15}}$
16	?13.7 ± 1.3	?18.5 ± 2.7
18	?42.3 ± 1.8	?34.7 ± 3.4
20	10.5 ± 3.1	29.8 ± 5.2
22	?8.6 ± 3.8	?20.2 ± 7.4

Charles S. CockellEmail:

     

Evaluation of harmonics in the geopotential of order 15 and odd degree

When a satellite orbit decaying slowly under the action of air drag experiences 15th-order resonance with the Earth's gravitational field, so that the ground track repeats after 15 rev, the orbital inclination suffers appreciable changes due to the perturbations from the harmonics in the geopotential of order 15 and odd degree (15,17,19 …). In this paper the changes in inclination at resonance of 11 satellites at inclinations between 30° and 90° have been analysed to determine values of the geopotential coefficients of order 15 and degree $\text{l,}\text{C}\text{?}{}_{\mathrm{l,15}}$ and $\text{S}\text{?}{}_{\mathrm{l,15}}$ in the usual notation. The recommended solution, going up to l = 31, is:

Evaluation of 14th-order harmonics in the geopotential

The Earth's gravitational potential is now usually expressed in terms of a double series of tesseral harmonics with several hundred terms, up to order and degree at least 20. The harmonics of order 14 can be evaluated by analysing changes in satellite orbits which experience 14th-order resonance, when the track over the Earth repeats after 14 revolutions.In this paper we describe our first evaluation of individual 14th-order coefficients in the geopotential from analysis of the variations in inclination and eccentricity of satellite orbits passing through 14th-order resonance under the action of air drag. Using results from eleven satellites, we find the following values for normalized coefficients of harmonics of order 14 and degree l, C?_{l, 14} and S?_{l, 14}, for l=14, 154. 22:

Geopotential harmonics of order 15 and even degree,from changes in orbital eccentricity at resonance

When a satellite orbit decaying slowly under the action of air drag experiences 15th-order resonance with the Earth's gravitational field, so that the ground track repeats after 15 rev, the orbital eccentricity may suffer appreciable changes due to perturbations from the gravitational harmonics of order 15 and even degree (16, 18, 20…). In this paper the changes in eccentricity at resonance for six satellites in near-circular orbits at inclinations between 56 and 90° have been analysed to derive 11 pairs of equations linking the harmonic coefficients of order 15 and (even) degree l, $\text{C}{}_{\mathrm{l,15}}\text{and}\text{S}{}_{\mathrm{l,15}}$ in the usual notation. These equations (together with eight constraint equations) are solved to give:

Darwin—an experimental astronomy mission to search for extrasolar planets

As a response to ESA call for mission concepts for its Cosmic Vision 2015–2025 plan, we propose a mission called Darwin. Its primary goal is the study of terrestrial extrasolar planets and the search for life on them. In this paper, we describe different characteristics of the instrument.

PEGASE, an infrared interferometer to study stellar environments and low mass companions around nearby stars

PEGASE is a mission dedicated to the exploration of the environment (including habitable zone) of young and solar-type stars (particularly those in the DARWIN catalogue) and the observation of low mass companions around nearby stars. It is a space interferometer project composed of three free flying spacecraft, respectively featuring two 40 cm siderostats and a beam combiner working in the visible and near infrared. It has been proposed to ESA as an answer to the first “Cosmic Vision” call for proposals, as an M mission. The concept also enables full-scale demonstration of space nulling interferometry operation for DARWIN.

The space infrared telescope for cosmology and astrophysics: SPICA A joint mission between JAXA and ESA

The Space Infrared telescope for Cosmology and Astrophysics (SPICA) is planned to be the next space astronomy mission observing in the infrared. The mission is planned to be launched in 2017 and will feature a 3.5 m telescope cooled to <5 K through the use of mechanical coolers. These coolers will also cool the focal plane instruments thus avoiding the use of consumables and giving the mission a long lifetime. SPICA’s large, cold aperture will provide a two order of magnitude sensitivity advantage over current far infrared facilities (>30 microns wavelength). We describe the scientific advances that will be made possible by this large increase in sensitivity and give details of the mission, spacecraft and focal plane conceptual design.

Bianchi type-III non-static magnetized cosmological model for perfect fluid distribution in general relativity

We have investigated Bianchi type III non-static magnetized cosmological model for perfect fluid distribution in general relativity. We assume that F ₁₂ is the only non-vanishing component of F _ij . Maxwell’s equation

POLAR investigation of the Sun—POLARIS

The POLAR Investigation of the Sun (POLARIS) mission uses a combination of a gravity assist and solar sail propulsion to place a spacecraft in a 0.48 AU circular orbit around the Sun with an inclination of 75° with respect to solar equator. This challenging orbit is made possible by the challenging development of solar sail propulsion. This first extended view of the high-latitude regions of the Sun will enable crucial observations not possible from the ecliptic viewpoint or from Solar Orbiter. While Solar Orbiter would give the first glimpse of the high latitude magnetic field and flows to probe the solar dynamo, it does not have sufficient viewing of the polar regions to achieve POLARIS’s primary objective: determining the relation between the magnetism and dynamics of the Sun’s polar regions and the solar cycle.

The Dark UNiverse Explorer (DUNE): proposal to ESA’s cosmic vision

The Dark UNiverse Explorer (DUNE) is a wide-field space imager whose primary goal is the study of dark energy and dark matter with unprecedented precision. For this purpose, DUNE is optimised for the measurement of weak gravitational lensing but will also provide complementary measurements of baryonic accoustic oscillations, cluster counts and the Integrated Sachs Wolfe effect. Immediate auxiliary goals concern the evolution of galaxies, to be studied with unequalled statistical power, the detailed structure of the Milky Way and nearby galaxies, and the demographics of Earth-mass planets. DUNE is an Medium-class mission which makes use of readily available components, heritage from other missions, and synergy with ground based facilities to minimise cost and risks. The payload consists of a 1.2 m telescope with a combined visible/NIR field-of-view of 1 deg². DUNE will carry out an all-sky survey, ranging from 550 to 1600 nm, in one visible and three NIR bands which will form a unique legacy for astronomy. DUNE will yield major advances in a broad range of fields in astrophysics including fundamental cosmology, galaxy evolution, and extrasolar planet search. DUNE was recently selected by ESA as one of the mission concepts to be studied in its Cosmic Vision programme.

The fifth-order analytical solution of the equations of motion of a satellite in orbit around a non-spherical planet

A high-precise analytical theory of a satellite in orbit around a non-spherical planet has been developed. The Poisson's small parameter method has been used. All secular and short-periodic perturbations proportional up to and including a product of five arbitrary harmonic coefficients of the planetary potential expansion are calculated. Long-periodic perturbations are derived with the accuracy of up to the fourth-order, inclusive. The influence of the high-order perturbations on the motion of ETALON-1 satellite has been investigated. The results of comparison of the numerical and analytical integration of the equations of its motion over a five year interval are as follows:

Vacuum Cosmological Models in Einstein and Barber Theories

An attempt has been made to solve the field equations with perfect fluid in an inhomogeneous space-time governed by the metric

The Non-Linear Stability of L4 in the Restricted Three-Body Problem when the Bigger Primary is a Triaxial Rigid Body

The non-linear stability of L ₄ in the restricted three-body problem has been studied when the bigger primary is a triaxial rigid body with its equatorial plane coincident with the plane of motion. It is found that L ₄ is stable in the range of linear stability except for three mass ratios:

The fate of dwarf galaxies in clusters and the origin of intracluster stars

The main goal of this paper is to compare the relative importance of destruction by tides vs. destruction by mergers, in order to assess if tidal destruction of galaxies in clusters is a viable scenario for explaining the origin of intracluster stars. We have designed a simple algorithm for simulating the evolution of isolated clusters. The distribution of galaxies in the cluster is evolved using a direct gravitational N-body algorithm combined with a subgrid treatment of physical processes such as mergers, tidal disruption, and galaxy harassment. Using this algorithm, we have performed a total of 148 simulations. Our main results are:

Astrodynamical Space Test of Relativity Using Optical Devices I (ASTROD I)—A class-M fundamental physics mission proposal for Cosmic Vision 2015–2025

ASTROD I is a planned interplanetary space mission with multiple goals. The primary aims are: to test general relativity with an improvement in sensitivity of over three orders of magnitude, improving our understanding of gravity and aiding the development of a new quantum gravity theory; to measure key solar system parameters with increased accuracy, advancing solar physics and our knowledge of the solar system; and to measure the time rate of change of the gravitational constant with an order of magnitude improvement and the anomalous Pioneer acceleration, thereby probing dark matter and dark energy gravitationally. It is an international project, with major contributions from Europe and China and is envisaged as the first in a series of ASTROD missions. ASTROD I will consist of one spacecraft carrying a telescope, four lasers, two event timers and a clock. Two-way, two-wavelength laser pulse ranging will be used between the spacecraft in a solar orbit and deep space laser stations on Earth, to achieve the ASTROD I goals. A second mission, ASTROD (ASTROD II) is envisaged as a three-spacecraft mission which would test General Relativity to 1 ppb, enable detection of solar g-modes, measure the solar Lense–Thirring effect to 10 ppm, and probe gravitational waves at frequencies below the LISA bandwidth. In the third phase (ASTROD III or Super-ASTROD), larger orbits could be implemented to map the outer solar system and to probe primordial gravitational-waves at frequencies below the ASTROD II bandwidth.

A model of the solar atmosphere and wind

Using a combination of solar and interplanetary measurements, a topological model is developed of the overall magnetic and plasma structures.

Einstein Gravity Explorer–a medium-class fundamental physics mission

The Einstein Gravity Explorer mission (EGE) is devoted to a precise measurement of the properties of space-time using atomic clocks. It tests one of the most fundamental predictions of Einstein’s Theory of General Relativity, the gravitational redshift, and thereby searches for hints of quantum effects in gravity, exploring one of the most important and challenging frontiers in fundamental physics. The primary mission goal is the measurement of the gravitational redshift with an accuracy up to a factor 10⁴ higher than the best current result. The mission is based on a satellite carrying cold atom-based clocks. The payload includes a cesium microwave clock (PHARAO), an optical clock, a femtosecond frequency comb, as well as precise microwave time transfer systems between space and ground. The tick rates of the clocks are continuously compared with each other, and nearly continuously with clocks on earth, during the course of the 3-year mission. The highly elliptic orbit of the satellite is optimized for the scientific goals, providing a large variation in the gravitational potential between perigee and apogee. Besides the fundamental physics results, as secondary goals EGE will establish a global reference frame for the Earth’s gravitational potential and will allow a new approach to mapping Earth’s gravity field with very high spatial resolution. The mission was proposed as a class-M mission to ESA’s Cosmic Vision Program 2015–2025.