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

2. 1-Way Propagation  Linear Motion of Photon  Fast Motion + Non-Relativistic Source Observer t = t 0 t = t 1 photon

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

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

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

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

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 E0E0 q’q’ E1E1 q vEvE

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  Order of Magnitude = Parallax  Angle Expression Sun E S q0q0 q

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, P W : Water Vapor Press. T: Temperature in K z

24. Multi-Way Propagation  Variation of 1-Way Propagation  Series of Light-Time Eq.  Ex.: t 3, t 2, t 1, t 0  Transponder Delay – Optical: 0 – Radio: Constant Source Observer Transponder 1 Transponder 2 t0t0 t1t1 t2t2 t3t3

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 Observer Target t2t2 t0t0 t1t1

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

27. Simultaneous Propagation t2t2  Almost Simultaneous Arrivals  Summed Light Time Eq.  Light Time of Mid-Point  Baseline Vector b  Mid-Direction k t1t1 t0t0 Observer 1 Observer 2 Source b k

28. Summed Light Time Eq.  Approximate Equation

29. Simult. Propagation (contd.) t2t2  Differenced Light Time Eq.  Arrival Time Delay  Baseline Vector b  Mid-Direction k t1t1 t0t0 Observer 1 Observer 2 Source b k

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