9*" r \ NBS TECHNICAL NOTE 659 U. S. DEPARTMENT OF COMMERCE /National Bureau of Standards AN EARTH-BASED COORDINATE CLOCK NETWORK NATIONAL BUREAU OF STANDARDS The National Bureau of Standards' was established by an act of Congress March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology and facilitate their effective application for public benefit. To this end, the Bureau conducts research and provides: (I) a basis for the Nation's physical measurement system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety. The Bureau consists of the Institute for Basic Standards, the Institute for Materials Research, the Institute for Applied Technology, the Institute for Computer Sciences and Technology, and the Office for Information Programs. THE INSTITUTE FOR BASIC STANDARDS provides the central basis within the United States of a complete and consistent system of physical measurement; coordinates that system with measurement systems of other nations; and furnishes essential services leading to accurate and uniform physical measurements throughout the Nation's scientific community, industry, and commerce. The Institute consists of a Center for Radiation Research, an Office of Meas- urement Services and the following divisions: Applied Mathematics — Electricity — Mechanics — Heat — Optical Physics — Nuclear 3 ! Sciences 2 — Applied Radiation - — Quantum Electronics — Electromagnetics — Time '' and Frequency " — Laboratory Astrophysics — Cryogenics "'. 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THE INSTITUTE FOR COMPUTER SCIENCES AND TECHNOLOGY conducts research and provides technical services designed to aid Government agencies in improving cost effec- tiveness in the conduct of their programs through the selection, acquisition, and effective utilization of automatic data processing equipment; and serves as the principal focus within the executive branch for the development of Federal standards for automatic data processing equipment, techniques, and computer languages. The Institute consists of the following divisions: Computer Services — Systems and Software — Computer Systems Engineering — Informa- tion Technology. THE OFFICE FOR INFORMATION PROGRAMS promotes optimum dissemination and accessibility of scientific information generated within NBS and other agencies of the Federal Government; promotes the development of the National Standard Reference Data System and a system of information analysis centers dealing with the broader aspects of the National Measurement System; provides appropriate services to ensure that the NBS staff has optimum accessibility to the scientific information of the world. The Office consists of the following organizational units: Office of Standard Reference Data — Office of Information Activities — Office of Technical Publications — Library — Office of International Relations. 1 Headquarters and Laboratories at Gaithersburg, Maryland, unless otherwise noted; mailing address Washington, D.C. 20234. 3 Part of the Center for Radiation Research. 3 Located at Boulder, Colorado 80302. * Part of the Center for Building Technology. HAL BUREAU OF STANDARD; 1 9 i \m AN EARTH-BASED ;.. ;;: ' COORDINATE CLOCK NETWORK 15 Neil Ashby Professor of Physics and Astrophysics University of Colorado Boulder, Colorado 80302 j^(ikMA£a§ JAj^e )A0. (oS^ Work performed for the Time and Frequency Division under contract #CST-8335 i^ / U.S. DEPARTMENT OF COMMERCE, NATIONAL BUREAU OF STANDARDS. Richard W Roberts Director * ' i Issued April 1975 National Bureau of Standards Technical Note 659 Nat. Bur. Stand. (U.S.), Tech Note 659, 34 pages (April 1975) CODEN: NBTNAE For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 (Order by SD Catalog No. 013.46:659) $ .85 . Contents Page 1. Introduction 1 2 Summary of Results 3 A; A coordinate clock system measuring global coordinate time 3 B) A coordinate clock system with a discontinuity 4 C) Investigation of other effects 4 3. Synchronization in a Static Gravitational Field 5 A) Synchronization (Initialization) using light signals 6 B) Slow transport of standard clocks 8 4. Effect of Gravitational Fields of the Sun and Moon 11 5. Clock Synchronization on the Rotating Earth 15 A) Synchronization of coordinate clocks by slow transport of standard clocks 19 B) Inf initesimally slow transport on the geoid 21 C) Global coordinate clock synchronization using light signals 21 D) "Hafele-Keating" discontinuity 23 E) Use of satellites to construct a coordinate time network 24 6. Miscellaneous Effects 25 A) Mass of the atmosphere 25 B) Effect of flattening of the earth on discontinuity 26 C) Retardation 26 References and Footnotes 28 Appendix—Latitude and Polar Angle in Ellipsoidal Geometry • . An Earth-Based Coordinate Clock Network Neil Ashby This paper investigates some of the possible operational procedures for synchronizing clocks at fixed sites spread around on the earth's surface, to within a 1 nanosecond level of accuracy. Since a common synchroniza- tion procedure is by transport of standard clocks in commercial jet air- line flights, and most of the effects we shall discuss are fractional corrections to the elapsed time, as a criterion of whether an effect is significant at the 1 nanosecond level we take for comparison purposes an elapsed time T =10 hours. This is a typical time for an inter- continental airplane flight. Analysis of a number of effects which might affect clock synchronization is carried out within the framework of general relativity. These effects include the gravitational fields of the earth, sun, and moon, and orbital motion, rotation, and flattening of the earth. It is shown that the only significant effects are due to the gravitational field and rotation of the earth, and motion of the transported clocks. Operational procedures for construction of a synchronized coordinate clock network using light signals and trans- ported standard clocks are discussed and compared. Key words: Clocks; coordinate time; general relativity; time Introduction The purpose of this paper is to investigate some of the possible operational procedures for synchronizing clocks at various fixed sites spread around on the earth's surface. There are a number of effects which could conceivably play a role in the synchronization of clocks — among these are the gravitational fields of the earth, moon, and sun, and the orbi- tal motions, rotation, and flattening of the earth. Since most of the effects we shall discuss turn out to be fractional corrections to elapsed time (that is, the corrections to the time are proportional to the time) , as a criterion of whether an effect is signifi- cant at the level of 1 nanosecond (1 ns) , we shall take for comparison purposes an elapsed time T = 10 hours, which corresponds to a typical time for an intercontinental airplane flight. This "comparison flight" will, where appropriate, be assumed to occur at an altitude of h = 12,000 meters and at a speed of 450 meter /sec. The reason for selection of these parameters is that a fairly common synchronization procedure will be by transport of standard clocks in commercial jet airline flights pL] The analysis will be carried out within the framework of the general theory of rela- tivity; we shall not discuss any modification of procedure which might be necessary in to and Dicke some other gravitational theory such as that due Brans [2 J The basic assumptions and results of the general theory of relativity, which we shall repeatedly use, are as follows (we use the notation and sign conventions of Weber [3]): 1) There exists a metric tensor g (U,v = 1, 2, 3) such that the space-time interval ds defined by 2 M V -ds = g dx dx (1) is invariant with respect to arbitrary coordinate transformations. 2) Propagation of lighlight rays along the path element dx is described by the vanishing of ds: y = g dx dx . (2) 3) The proper time elapsed on a standard clock transported along the space-time path element dx is — v i i a V A (3) ds = J- 35 dx