1. Special Relativistic Effects in Orbiting Lidar Wind Measurements Neil Ashby Dept. of Physics, University of Colorado Boulder, CO 80309-0390 NIST Affiliate Email: ashby@boulder.nist.gov Michael J. Kavaya Langley Research Center, MC 468 Hampton, VA 23641-2199 Email: Michael.J.Kavaya@nasa.gov

2. Constancy of c, speed of light Postulate II of Special Relativity: In any inertial frame the speed of light, c, is a constant independent of the motion of the source (or of the observer.) c=299792458 m/s (defined)

3. Relativity of Simultaneity To an observer on the ground, let two lightning strokes at the front and back of the train be simultaneous. The “moving” observer at the train’s midpoint finds the event at front occurs first.

4. Relation Between Doppler Effect and Relativity of Simultaneity Wavefronts are marked simultaneously by the non-moving observer: Moving observer says the wavefronts are marked too soon: c For example, the wavefront atneeds to move an additional distance before it gets into the right position to be marked at t’ = 0. To the moving observer, the wavelength is:

5. Reference Systems ECI: Earth-Centered Inertial ERF: Ground-Fixed Instantaneous Lidar Rest Frame V V’ V’’ 

6. Notation Unprimed quantities: measured in ECI (Earth-Centered Inertial) frame Primes on a quantity indicates it is measured in the rest frame of the ground point: e.g. w’, the desired wind velocity. Doubly primed quantities are measured in the rest frame of the lidar apparatus; transmitter and detector are assumed to coincide.

7. Doppler Shift Measured by Monostatic Lidar w”  t” c  t”

8. Frequency Shift k ”/ k ” is a unit vector in the direction of propagation of the incident wave, in the lidar rest frame. (See Gudimetla and Kavaya, Appl. Opt. 38(30), 6374-6382 (1999) for a special case.)

9. Fundamental Expression for w” Introduce the fractional frequency shift: Then solve for w ”:

10. Problem: transform w” to ECEF frame First approximation: Lorentz contraction and time dilation will be neglected. Relativity of simultaneity must be accounted for. The transformations are: Relativistic composition of velocities: where V L ’ is the velocity of the lidar apparatus relative to the ground. We want:

11. Transformation of Propagation Distance w” L d”

12. Transformation of Propagation Direction In summary: Unit vector:

13. Leading Corrections What happens if relativistic correction in denominator is neglected? Simulation: Assume some configuration of spacecraft, earth, wind, Calculate fractional frequency shift and treat as “truth;” Estimate error in measurement of w’ if denominator is neglected.

14. Configuration for Simulation Uniform wind Velocity field d” 30 o A 

15. Error From Neglect of Denominator

16. Using ECI Frame for Light Path The relativistic corrections on the right-hand side are:

17. Aberration Corrections

18. Latitude-Dependent Contributions- Polar Orbit

19. Spacecraft Velocity Changes During propagation of the lidar pulse, the spacecraft falls in the gravitational field of the earth. This is analyzed most easily in the ECI frame, but then the directions of the incident and scattered ray are different.

20. Frequency Shift for Accelerated Spacecraft It can be shown, neglecting second-order Doppler shifts, that: Assuming that N s =-N t, and that w=w’+V E, gives This increases with increasing altitude even though the acceleration of gravity decreases.

21. Correction to Wind Measurement Arising from Spacecraft Velocity Change (This is independent of scan angle relative to nadir.)

22. Higher-Order Relativistic Corrections In general, higher-order relativistic corrections involve factors of order and are negligibly small. For example, such terms arise in using the relativistic law for composition of velocities between the lidar rest frame and the ECI frame:

23. Summary Change of direction of light path (aberration) contributes relativistic corrections which can be as much as 0.18 m/s. Such corrections depend in a complicated way on spacecraft, surface, and wind velocities and on scanning direction. Principal corrections arise from apparent change in direction of the light path when tranforming from one reference frame to another. Spacecraft velocity changes arising from free fall can contribute as much as 0.04 m/s to errors in wind velocity measurements. Neglect of higher-order terms in the fractional frequency shift can contribute errors of up to 0.04 m/s. High-order relativistic corrections have been worked out in detail but are probably negligible.