## Info

Equatorial angular momentum function xi (top) and x2 (bottom) from GPS analysis (solid line) and from meteorological data (NCEP reanalysis) two series is perhaps better illustrated by Figures 2.50, containing the same type of information as in Figure 2.49 but only for the year 2001. It is interesting to note that x2 shows more structure than xi. Interpreting Xi and X2 as the two components of one vector (what they actually are) one Fig. 2.50. Equatorial angular momentum function xi...

## Atid Grrid 5 G mQ 298

I I r - ri I3 r3 J J r - rQ 3 r3 J This acceleration may be written as the gradient of a potential function where the tide-generating potential may be written in the form (compare eqn. (I-3.28)) 1 gm< I n r 'r(I , Gme n r r0 (n i Vtid 1-r - Grrid o--b ,-r - G m o . (2.100) I r - r2 r3 r - rQ rAQ The tidal potential (2.100) is the sum of two terms, the tidal potential Vtid(j due to the Moon and that due to the Sun Vtid . Let us consider only the term due to the Moon (the tidal potential due to...

## Albedo of the Earth

For LEOs the radiation pressure due to the sunlight reflected or re-emitted from the Earth's surface must be taken into account. This particular kind of radiation pressure usually is referred to as albedo radiation pressure. Depending heavily on the actual distance of the satellite from the Earth's surface, the effect is considerably smaller than the direct radiation pressure. For LEOs it may be as much as 25 . Accurate modelling of albedo radiation pressure is rather difficult. It has to be...

## Introduction

Program ORBDET may be used to determine the orbits of minor bodies in the planetary system and the orbits of artificial Earth satellites and of space debris. ORBDET is the only interactive program of the program package. Its primary menu is reproduced in Panel ORBDET 1 (Figure 8.1). The program user first has to decide whether to determine the orbit of an object in the planetary system or in the Earth-near space. The algorithms used for the two purposes are identical. They are explained in...

## Atmospheric Drag

Above a height of about 50 km the density of the neutral atmosphere is sufficiently low to assume laminar air currents. Assuming furthermore that the atmosphere is co-rotating with the Earth (an assumption in essence ignoring winds) and neglecting the thermal motion of the molecules, it is relatively easy to calculate the transfer of linear momentum from the atmosphere to the satellite During a short time interval At the velocity r' of the satellite relative to the particles may be assumed...

## Radiation Pressure as a Dissipative Force

When speaking of non-gravitational forces one usually also means dissipative forces, i.e., forces leading to a loss of energy and angular momentum of the satellite. As the energy is in essence represented by the satellite's semi-major axis a, a dissipative force is expected to reduce the satellite's semi-major axis. Atmospheric drag is a typical example of a dissipative form. Apparently direct radiation pressure, as represented by eqn. (3.142) or its strap-down version (3.150), does not give...

## The Use of Program Numint to Generate Hill Surfaces

With each orbit integrated (in the case of the probleme restreint) the corresponding value of the reduced Jacobi constant is provided in the general output file. Also, files containing the intersections of the Hill surface with the coordinate planes (or with parallel planes) may be generated (see Panel NUMINT 3 in Figure 6.3 and the discussion associated with it). Program NUMINT allows it also, however, to generate the same files without actually performing an integration. The program option is...

## The Use of Program Numint for Numerical Integration

Panel 1 in Figure 6.1 shows that four types of algorithms suitable for numerically solving initial value problems associated with non-linear ordinary differential equations systems are implemented in program NUMINT, namely Runge-Kutta methods (see section I-7.4.4), extrapolation methods (see section I-7.4.5), collocation methods (see sections I-7.4.1 and I-7.5), multistep methods (see sections I-7.4.2 and I-7.5.6). The solution method and the problem type may be selected in Panel NUMINT 1 in...

## X6f 1 76c 06o X6f2 76 06o 6f2 1 X6f2 1 76c 06o X6f 76 06o 6f

The right-hand sides are set to zero in the first iteration step. This approxi mation separates the first two from the second two of eqns. (2.183). In order to emphasize the structure of the solution, the amplitudes of the prograde motions are characterized by the symbol +, those of the retrograde terms with . In this approximation the solution of the first of eqns. (2.183) is f2 (t) P+ cos ((1 ot + a+) P+ cos(*+t + a+) (t) P+ sin ((1 e)-176 *6ot + +) P+ sin( +t + a+) (2.184) In this...

## Program SATORB

Program SATORB may be used to generate satellite orbits or to determine the orbits of satellites using (a) tabular satellite positions as pseudoobservations or (b) astrometric positions as real observations. More precisely the following problems may be solved 1. Generation of satellite ephemerides using a wide variety of models. 2. Orbit determination using tabular positions or position differences as pseudo-observations. Two concrete problems may be addressed a) Orbit determination for GPS or...

## N

Ni+i, +i cos(iutk) cos(lutk) , i 0,1, ,m , l 0,1, ,m Ni+m+i,i+m+i sin(iwtfc)sin(lwtfc) , i 1, 2, ,m , l 1, 2, ,m Ni+i,i+m+i cos(iwtk) sin(lwtk) , i 0,1, ,m , l 1, 2, ,m . tk (k- )h, k 1,2, ,2m + l fk f (t k ) cos iujtk) cos(zwtfc) j + cos ((i + )wifc) + cos (( l)utk) sin( u tfc) sin( wifc) cos ((i + )wifc) + cos (( l)utk) (11.27) cos iujtk) sin lujtk) + sin (( + )wifc) sin (( l)utk)

## Readlfn0015010end1010stnameobjcharxmjdrade 5010 Format1xa161xa35f179f139f138

Where STNAME and OBJCHAR, the name of the observing station and the characterization of the object, are character variables. XMJD, the Modified Julian Date (MJD), RA (right ascension) and DE (declination) of the object are REAL*8 variables. The unit for the modified Julian date MJD is days. The right ascension is specified in the form hh.mmssxxxxx, where hh stands for hours, mm for minutes, ss for seconds, and xx for fractions of seconds. The declination has to be provided in the form...

## E ao eo io Qo uo wo

Introduced in Chapter I- 5 to characterize the partial derivatives of the orbit w.r.t. its initial osculating elements referring to to. So far, we either used the time To of perihelion passage or the mean anomaly ao at to as the sixth element. As indicated above, the argument of latitude wo f uo + vo is used here, instead. This particular element has the advantage to avoid quasi-singularities associated with orbits of small ecentricities. It is (obviously) a function of the classical elements....

## Density of the Upper Atmosphere

Only with the advent of the space era it was possible to gather reliable information about the density and composition of the Earth's upper atmosphere above 50 km. The information stems from analyzing the orbits of artificial satellites and from satellite missions carrying mass spectrometers and scat-terometers. First reliable information was made available in the US Standard Atmosphere (1976). In parallel, several versions of the CIRA (COSPAR International Reference Atmosphere) were developed...

## Determination of GPS and Glonass Orbits

The program SATORB may be used as an orbit determination program using tabular satellite positions as pseudo-observations. It is in particular possible to analyze one or several precise orbit files of GPS (and or GLONASS) satellites produced by one of the IGS Analysis Centers or by the IGS Analysis Coordinator in the SP3-format. One IGS precise orbit file contains (in general) 96 tabular positions per day at 15-minutes intervals for each active GPS satellite. The tabular positions of one or...

## Dynamic and Reduced Dynamics LEO Orbits Using Program SATORB

The content of the three output files *.TAB, *.SP3, and *.PPD, as generated by program LEOKIN, may be used as pseudo-observations in the program SATORB (where only one file type may be used at the time). The program will generate the best possible particular solution of the equations of motion using the parametrization already introduced in section 7.1 (more specifically in the subsection 7.1.3). It is thus only necessary to address the peculiarities of program SATORB when using LEO-data. Panel...

## Library

Eckart T. Encrenaz J. Lequeux A. Maeder A. Burkert M. A. Dopita M. Harwit R. Kippenhahn V. Trimble The Stars By E. L. Schatzman and F. Praderie Modern Astrometry 2nd Edition By J. Kovalevsky The Physics and Dynamics of Planetary Nebulae By G.A. Gurzadyan Galaxies and Cosmology By F. Combes, P. Boisse, A. Mazure and A. Blanchard Observational Astrophysics 2nd Edition By P. Lena, F. Lebrun and F. Mignard Physics of Planetary Rings Celestial Mechanics of Continuous...