Satellite Orbits: Models, Methods and Applications / Edition 1

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This is a modern textbook that guides the reader through the theory id practice of satellite orbit prediction and determination. Starting from basic principles of orbital mechanics, it covers elaborate force models as well as precise methods of satellite tracking. Emphasis is on numerical treatment and a multitude of algorithms adopted in modern satellite trajectory computation are described in detail.

Numerous exercises and applications are provided and supplemented by a unique collection of computer programs with associated C++ source codes included on the accompanying CD-ROM. These programs are built around a powerful spaceflight dynamics library well suited to the development of individual applications.

An extensive collection of Internet resources is provided through W W W hyperlinks to detailed and frequently updated online information on spaceflight dynamics.

The book addresses students and scientists working in the field of navigation, geodesy and spaceflight technology, as well as satellite engineers and operators focusing on spaceflight dynamics.

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Editorial Reviews

From the Publisher

From the reviews:

"Not many books on the topic of satellite orbits over the past decades have been informative, comprehensive and practical. I am happy to say that this publication does fall into that category. [...] This book should certainly be in the library of students and scientists working in the fields of navigation, geodesy, and spaceflight technology, as well as satellite engineers and operators focusing on spaceflight dynamics." (The Observatory, 2001)

"Satellite Orbits: Models, Methods, and Application would be a valuable addition to the library of any engineer or scientist interested in the practical aspects of orbit prediction and determination. [...] The comprehensive reference list along with the CD supplied codes make this book unique in this area." (Applied Mechanics Reviews, 2002)

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Product Details

  • ISBN-13: 9783540672807
  • Publisher: Springer Berlin Heidelberg
  • Publication date: 3/15/2013
  • Edition description: 1st ed. 2000. Corr. 9th printing 2013
  • Edition number: 1
  • Pages: 369
  • Sales rank: 487,563
  • Product dimensions: 6.20 (w) x 9.40 (h) x 0.80 (d)

Read an Excerpt

1. Around the World in a Hundred Minutes

1.2 Navigating in Space

Irrespective of the level of autonomy that may be achieved with present-day satellites, any spacecraft would rapidly become useless if one were unable to locate it and communicate with it. Furthermore, many of the spacecraft described earlier necessitate an active control of their orbit in accordance with specific mission requirements. Navigation is therefore an essential part of spacecraft operations. It comprises the planning, determination, prediction, and correction of a satellite's trajectory in line with the established mission goals.

1.2.1 Tracking Systems

A variety of tracking systems may be used to obtain measurements related to the instantaneous position of a satellite or its rate of change. Most of these systems are based on radio signals transmitted to or from a ground antenna (Fig. 1.8). Common radio tracking systems are able to perform angle measurements by locating the direction of a radio signal transmitted by a satellite. The resolution of these measurements depends on the angular diameter of the antenna cone, which is determined by the ratio of the carrier wavelength to the antenna diameter. Given a frequency of 2 GHz as applied in common antenna systems, a diameter of 15 m is required to achieve a beam width of 0.5°. Distance and velocity information can be obtained by measuring the turn-around delay or Doppler-shift of a radio signal sent to the spacecraft and returned via a transponder. Representative ranging systems achieve an accuracy between 2 m and 20 m, depending on the frequency band used and the type of ranging signal applied. Doppler measurements can provide therange rate of an Earth-orbiting satellite with an accuracy of typically 1 mm/s. In the absence of an active transmitter or transponder onboard the spacecraft, sufficiently powerful radar may also be applied for spacecraft tracking. Its use, however, is mainly restricted to emergency cases or space surveillance tasks (Pensa & Sridharan 1997).

For low-Earth satellites, a purely ground-based tracking suffers from the limited station contacts that constrain the available tracking measurements to a small fraction of the orbit. To overcome this restriction, geostationary satellites like the United Sates' Tracking and Data Relay Satellite (TDRS) can be used to track a user satellite via a relay transponder. Going even further, GPS ranging signals offer the opportunity to obtain position measurements onboard a satellite completely independently of a ground station.

Aside from radiometric tracking, optical sensors may likewise be used to locate a satellite, as illustrated both by the early days' Baker-Nunn cameras (Henize 1957) and today's high-precision satellite laser ranging systems (Fig. 1.9). Imaging telescopes are well suited for detecting unknown spacecraft and space debris up to geostationary distances, which makes them a vital part of the United States' space surveillance network. Instead of photographic films employed in former Baker-Nunn cameras, the Ground-Based Electro-Optical Deep Space Surveillance (GEODSS) telescopes are equipped with electronic sensors that allow online image processing and removal of background stars. Other applications of optical telescopes include the monitoring of colocated geostationary satellites, which are not controlled in a coordinated way by a single control center. Besides being completely passive, telescopic images can provide the plane-of-sky position of geostationary satellites to much better accuracies (typically 1" N 200 m) than angle measurements of common tracking antennas.

Satellite laser ranging (SLR) systems provide highly accurate distance measurements by determining the turn-around light time of laser pulses transmitted to a satellite and returned by a retro-reflector. Depending on the distance and the resulting strength of the returned signal, accuracies of several centimeters may be achieved. Satellite laser ranging is mainly used for scientific and geodetic missions that require an ultimate precision. In combination with dedicated satellites like Starlet and Lageos (Rubincam 1981, Smith & Dunn 1980), satellite laser ranging has contributed significantly to the study of the Earth's gravitational field. Other applications of SLR include independent calibrations of radar tracking systems like GPS or PRARE (Zhu et al. 1997).

1.2.2 A Matter of Effort

A discussion on spacecraft navigation sooner or later ends up with a question on the achieved accuracy. As illustrated in Fig. 1.10, widely varying levels of accuracy apply for the knowledge of a satellite's orbit, depending on the particular goals of a space project. In accord with these requirements, widely varying tracking systems are employed in present space projects.

An upper threshold to the permissible position uncertainty is generally given by the need for safe communication with the spacecraft from the ground. Considering, for example, an orbital altitude of 800 km and the 0.3° (0.005 rad) half-beam width of a 15 m S-band antenna, the spacecraft trajectory must be predicted to within an accuracy of 4 km to permit accurate antenna pointing throughout an entire station pass. A similar level of accuracy is required for many scheduling functions. Spacecraft-specific events like shadows, station contacts, or payload activation are commonly considered in the operations timeline with a one-second resolution. Considering an orbital velocity of 3-7 km/s, the spacecraft position must be known to within several kilometers in order to predict an orbit-related event with the desired accuracy. An angle tracking system locating the direction of the downlink signal is generally sufficient to meet these types of basic operational requirements. Aside from a transmitter, which is employed anyway for ground communication, no specific onboard equipment is required for this type of tracking.

Quite a different accuracy can be achieved by ground-based or space-based range and Doppler measurements. Their use is typically considered for missions requiring active orbital control. Colocated geostationary satellites, for example, may experience intentional proximities down to the level of several kilometers. Accordingly, the position knowledge and the associated tracking accuracy must at least be one order of magnitude better. Similar considerations hold for remote sensing satellites. In order to enable a reliable geocoding of images with a resolution of up to 10 m, a consistent orbit determination accuracy is mandatory. Considering the visibility restrictions of common ground stations for low-Earth orbits, space-based tracking systems like TDRSS, GPS, or DORIS are often preferred to achieve the specific mission requirements. While ground-based tracking requires a conventional transponder, the use of the other systems necessitates specialized onboard equipment like steerable antennas (TDRSS) or a Doppler measurement unit (DORIS). Utilization of GPS, in contrast, offers position accuracies of 100 m (navigation solution) to 25 m (with dynamical filtering) even for simple C/A code receivers. GPS tracking is therefore considered to be the sole source of orbit information for more and more spacecraft.

Leaving the field of traditional spacecraft operations, one enters the domain of scientific satellite missions with even more stringent accuracy requirements. Among these, geodetic satellite missions like Starlet and Lageos have long been the most challenging. Using satellite laser ranging systems, their orbits have been tracked with an accuracy in the centimeter to decimeter region, thus allowing a consistent improvement in trajectory models and Earth orientation parameters. For other Earth exploration missions like TOPER (Bath et al. 1989, 1998), ERS, or JERS, the use of satellite altimeters has been a driving factor for the refinement of orbital models and tracking techniques. Besides selected laser ranging campaigns, these missions are mainly supported by space-based radio tracking systems like TDRSS, GPS, DORIS, and PRARE. Their use has enabled the achievement of orbital accuracies in the decimeter region, with focus on the exact restitution of the radial component. In the case of GPS usage, the differential processing of space based and concurrent ground-based pseudorange and carrier phase measurements provides for the required increase in precision over the Standard Positioning Service. The GPS satellite orbits themselves are determined with position accuracies of several centimeters, using GPS measurements collected by a global network of geodetic reference stations (Springer et al. 1999).

Looking at the future, a new era will be opened by the upcoming GRACE mission (Davis et al. 1999). Making use of a K/Ka-band intersatellite link that provides dual one-way range measurements, changes in the distance of the two spacecraft can be established with an accuracy of about 0.01 mm. In combination with supplementary onboard accelerometers, this will for the first time allow the detection of short-term variations in the cumulative gravity field of the solid Earth, the oceans and the atmosphere...

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Table of Contents

1 Around the World in a Hundred Minutes 1
1.1 A Portfolio of Satellite Orbits 1
1.2 Navigating in Space 8
2 Introductory Astrodynamics 15
2.1 General Properties of the Two-Body Problem 16
2.2 Prediction of Unperturbed Satellite Orbits 22
2.3 Ground-Based Satellite Observations 32
2.4 Preliminary Orbit Determination 39
3 Force Model 53
3.1 Introduction 53
3.2 Geopotential 56
3.3 Sun and Moon 69
3.4 Solar Radiation Pressure 77
3.5 Atmospheric Drag 83
3.6 Thrust Forces 104
3.7 Precision Modeling 107
4 Numerical Integration 117
4.1 Runge - Kutta Methods 118
4.2 Multistep Methods 132
4.3 Extrapolation Methods 147
4.4 Comparison 151
5 Time and Reference Systems 157
5.1 Time 157
5.2 Celestial and Terrestrial Reference Systems 169
5.3 Precession and Nutation 172
5.4 Earth Rotation and Polar Motion 181
5.5 Geodetic Datums 185
6 Satellite Tracking and Observation Models 193
6.1 Tracking Systems 193
6.2 Tracking Data Models 208
6.3 Media Corrections 219
7 Linearization 233
7.1 Two-Body State Transition Matrix 235
7.2 Variational Equations 240
7.3 Partial Derivatives of the Acceleration 244
7.4 Partials of the Measurements with Respect to the State Vector 250
7.5 Partials with Respect to Measurement Model Parameters 252
7.6 Difference Quotient Approximations 253
8 Orbit Determination and Parameter Estimation 257
8.1 Weighted Least-Squares Estimation 258
8.2 Numerical Solution of Least-Squares Problems 268
8.3 Kalman Filtering 276
8.4 Comparison of Batch and Sequential Estimation 286
9 Applications 293
9.1 Orbit Determination Error Analysis 293
9.2 Real-Time Orbit Determination 303
9.3 Relay Satellite Orbit Determination 312
Appendix A 319
A.1 Calendrical Calculations 319
A.2 GPS Orbit Models 324
Appendix B 329
B.1 Internet Resources 329
B.2 The Enclosed CD-ROM 330
List of Symbols 339
References 347
Index 361
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  • Posted January 18, 2009

    Very practical and detailed.

    This book is oriented towards the practice of satellite orbit determination. The title is accurate: models, methods and applications are treated in detail sufficient for the task. From derivation of Kepler's Laws from first principles to treatment of atmospheric drag, solution of differential equations using Runge-Kutta methods, a superb discussion of time and reference systems, orbit determination and a variety of related issues, the text is intellectually rigourous, while maintaining a distinct practical focus.<BR/><BR/>The quality of the physical book is extremely high. Binding, paper quality, heft and feel are all exquisite. <BR/><BR/>The information within is priceless. A tremendous amount of effort and attention to detail is evident, and the result is outstanding. <BR/><BR/>This book is one of a very few I would consider magnificent, in both content and execution. My highest regards to the authors.

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