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Signal Propagation

Signal Propagation. Electro-Magnetic Signal Geometric Approximation ~ Fast Particle Approximation Speed of Light in Vacuum. 1-Way Propagation. t = t 0. Source. Linear Motion of Photon Fast Motion + Non-Relativistic. photon. t = t 1. Observer. Passive Observables. Arrival Time

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Signal Propagation

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  1. Signal Propagation • Electro-Magnetic Signal • Geometric Approximation ~ Fast Particle Approximation • Speed of Light in Vacuum

  2. 1-Way Propagation t = t0 Source • Linear Motion of Photon • Fast Motion + Non-Relativistic photon t = t1 Observer

  3. Passive Observables • Arrival Time • Incoming Direction • Received Wavelength

  4. Equation of Light Time S • within Solar System • Departure Time • Arrival Time • Light Time = Travel Time • Obtain Light Time O

  5. Derivation of Eq. of Light Time • Beginning/End of Photon Motion • Taking the norm • Assumption: Body Motions are known

  6. Velocity Expression (Newtonian) Velocity Expression (Special Relativity) Derivation (contd.)

  7. Solving Eq. of Light Time • Newton Method

  8. Approximate Solution • Initial Guess: Infinite c= Zero Solution • First Newton Corrector • Further Correction: General Relativity

  9. Light Direction • Aberration: Observer’s Velocity • Parallax: Offset of Observer’s Position • Periodic: Annual, Diurnal, Monthly, … • Correction for Light Time: within Solar System

  10. Aberration • Finiteness of Speed of Light • Bradley (1727) • Track of Raindrops on Car’s Side Window

  11. Annual Aberration • Order of Magnitude = Aberration Constant • Angle Expression S q’ q E0 vE E1

  12. Annual Aberration (contd.) • Adopting Ecliptic Coordinates • Approximate Formula • Mean Longitude of Sun: L • Aberration Ellipse

  13. Diurnal Aberration • Adopting Equatorial Coordinates • Approximate Formula • Sidereal Rotation Angle: Q • Geocentric Latitude: f

  14. Parallax • Offset of Observer’s Position • Bessel (1838): 81 Cyg • Direction Difference between L&R Eyes

  15. Annual Parallax S • Order of Magnitude = Parallax • Angle Expression q0 q Sun E

  16. Annual Parallax (contd.) • Ecliptic Coordinates • Approximate Formula • 90°Phase Shift from Aberration • Parallactic Ellipse

  17. Diurnal (Geocentric) Parallax • Very close objects only: Moon • Adopting Equatorial Coordinates • Approximate Formula • Geocentric Parallax

  18. Doppler Shift • Newtonian Approximation • Outgoing = Red shift • Incoming = Blue shift

  19. Approximate Doppler Shift • Order of Magnitude = Aberration Constant • Annual Doppler • Diurnal Doppler

  20. Propagation Delay/Diffractions • Vacuum (= Gravitational) • Wavelength independent • Non-Vacuum • Eminent in Radio wavelength • Intrergalactic, Interstellar, Solar corona • Ionospheric, Tropospheric • Atmospheric

  21. Wavelength-Dependent Delay • Cancellation by 2 waves measurements • Geodetic VLBI: S-, X-bands • GPS: L1-, L2-bands • Artificial Satellites: Up- and Down-links • Empirical Model • Solar corona, Ionospheric, Tropospheric

  22. Delay Models • Solar Corona (Muhleman and Anderson 1981) • Tropospheric (Chao 1970)

  23. Atmospheric Refraction • Variation of Zenith Distance • Saastamoinen (1972) P: Pressure in hP, PW: Water Vapor Press. T: Temperature in K z

  24. Multi-Way Propagation t0 • Variation of 1-Way Propagation • Series of Light-Time Eq. • Ex.: t3, t2, t1, t0 • Transponder Delay • Optical: 0 • Radio: Constant Source t1 Transponder 1 t2 Transponder 2 t3 Observer

  25. Round Trip Propagation • Typical Active Observation • Emission/Arrival Times • No Need of Target Motion Info • Sum of 1-Way Propagations • Cancellation of 1-st Order Effects Target t1 t0 t2 Observer

  26. Round Trip Light Time • Approximate Mid-Time • Approximate Distance at Mid-Time

  27. Simultaneous Propagation t0 Source • Almost Simultaneous Arrivals • Summed Light Time Eq. • Light Time of Mid-Point • Baseline Vector b • Mid-Direction k k t1 Observer 1 b t2 Observer 2

  28. Summed Light Time Eq. • Approximate Equation

  29. Simult. Propagation (contd.) t0 Source • Differenced Light Time Eq. • Arrival Time Delay • Baseline Vector b • Mid-Direction k k t1 Observer 1 b t2 Observer 2

  30. Eq. of Interferometric Obs. • Approximate Equation = Equation of VLBI Observation

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