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Signal Propagation

Explore the journey of light within the solar system, from signal propagation to observer effects. Learn about the impact of geometric approximation, fast particle approximation, and the speed of light on arrival times and wavelengths. Delve into the derivation of light time equations and correction methods for factors like aberration, parallax, and Doppler shift. Understand how various atmospheric conditions and delays influence the propagation of light signals for VLBI, GPS, and satellite communications.

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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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