Special Relativistic Effects in Orbiting Lidar Wind Measurements

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Special Relativistic Effects in Orbiting Lidar
Wind Measurements
Neil Ashby Dept. of Physics, University of
Colorado Boulder, CO 80309-0390 NIST
Affiliate Email ashby_at_boulder.nist.gov
Michael J. Kavaya Langley Research Center, MC
468 Hampton, VA 23641-2199 Email
Michael.J.Kavaya_at_nasa.gov
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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.) c299792458 m/s
(defined)
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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
trains midpoint finds the event at front occurs
first.
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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
before it gets into the right position to be
marked at t 0. To the moving observer, the
wavelength is
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Reference Systems
w
Instantaneous Lidar Rest Frame
V
ERF Ground-Fixed
ECI Earth-Centered Inertial
V
V
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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.
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Doppler Shift Measured by Monostatic Lidar
cDt
wDt
cDt
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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.)
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Fundamental Expression for w
Introduce the fractional frequency shift
Then solve for w
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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
where VL is the velocity of the lidar apparatus
relative to the ground.
Relativistic composition of velocities
We want
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Transformation of Propagation Distance
d
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Transformation of Propagation Direction
In summary
Unit vector
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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.
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Configuration for Simulation
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Error From Neglect of Denominator
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Using ECI Frame for Light Path
The relativistic corrections on the right-hand
side are
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Aberration Corrections
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Latitude-Dependent Contributions-Polar Orbit
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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.
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Frequency Shift for Accelerated Spacecraft
It can be shown, neglecting second-order Doppler
shifts, that
Assuming that Ns-Nt, and that wwVE,gives
This increases with increasing altitude even
though the acceleration of gravity decreases.
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Correction to Wind Measurement Arising from
SpacecraftVelocity Change
(This is independent of scan angle relative to
nadir.)
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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
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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.