 
Aberration Corrections Required Reading
===========================================================================
 
   Last revised on 2020 MAY 26 by E. D. Wright.
 
 
Abstract
--------------------------------------------------------
 
   The SPICE Toolkit can calculate positions, velocities, and orientations
   corrected for aberrations caused by the finite speed of light, and the
   relative velocity of the target to observer.
 
 
Purpose
 
   This document is a reference guide describing the details of the
   aberration correction calculations as implemented in the SPICE system.
 
 
Intended Audience
 
   This document is for SPICE users who need specifics concerning the
   application of aberration corrections to state calculations.
 
 
References
 
       1.   Jesperson and Fitz-Randolph, From Sundials to Atomic Clocks,
            Dover Publications, New York, 1977.
 
 
Introduction
===========================================================================
 
   In space science or engineering applications one frequently wishes to
   know where to point a remote sensing instrument, such as an optical
   camera or radio antenna, in order to observe or otherwise receive
   radiation from a target. This pointing problem is complicated by the
   finite speed of light: one needs to point to where the target appears to
   be as opposed to where it actually is at the epoch of observation. We
   use the adjectives "geometric," "uncorrected," or "true" to refer to an
   actual position or state of a target at a specified epoch. When a
   geometric position or state vector is modified to reflect how it appears
   to an observer, we describe that vector by any of the terms "apparent,"
   "corrected," "aberration corrected," or "light time and stellar
   aberration corrected." The SPICE Toolkit can correct for two phenomena
   affecting the apparent location of an object: one-way light time (also
   called "planetary aberration") and stellar aberration.
 
 
Types of Corrections
--------------------------------------------------------
 
 
One-way Light Time
 
   Correcting for one-way light time is done by computing, given an
   observer and observation epoch, where a target was when the observed
   photons departed the target's location. The vector from the observer to
   this computed target location is called a "light time corrected" vector.
   The light time correction depends on the motion of the target relative
   to the solar system barycenter, but it is independent of the velocity of
   the observer relative to the solar system barycenter. Relativistic
   effects such as light bending and gravitational delay are not accounted
   for in the light time corrections.
 
 
Stellar Aberration
 
   The velocity of the observer also affects the apparent location of a
   target: photons arriving at the observer are subject to a "raindrop
   effect" whereby their velocity relative to the observer is, using a
   Newtonian approximation, the photons' velocity relative to the solar
   system barycenter minus the velocity of the observer relative to the
   solar system barycenter. This effect is called "stellar aberration."
   Stellar aberration is independent of the velocity of the target. The
   stellar aberration formula used by SPICE routines does not include (the
   much smaller) relativistic effects.
 
   Stellar aberration corrections are applied after light time corrections:
   the light time corrected target position vector is used as an input to
   the stellar aberration correction.
 
   When light time and stellar aberration corrections are both applied to a
   geometric position vector, the resulting position vector indicates where
   the target "appears to be" from the observer's location.
 
   As opposed to computing the apparent position of a target, one may wish
   to compute the pointing direction required for transmission of photons
   to the target. This also requires correction of the geometric target
   position for the effects of light time and stellar aberration, but in
   this case the corrections are computed for radiation traveling *from*
   the observer to the target. We will refer to this situation as the
   "transmission" case.
 
   The "transmission" light time correction yields the target's location as
   it will be when photons emitted from the observer's location at `et'
   arrive at the target. The transmission stellar aberration correction is
   the inverse of the traditional stellar aberration correction: it
   indicates the direction in which radiation should be emitted so that,
   using a Newtonian approximation, the sum of the velocity of the
   radiation relative to the observer and of the observer's velocity,
   relative to the solar system barycenter, yields a velocity vector that
   points in the direction of the light time corrected position of the
   target.
 
   One may object to using the term "observer" in the transmission case, in
   which radiation is emitted from the observer's location. The terminology
   was retained for consistency with earlier documentation.
 
 
SPICE Aberration Identifiers (also called Flags)
 
   SPICE uses a set of string flags to indicate the particular aberration
   corrections to apply to state evaluations.
 
       --   'NONE'
 
            Apply no correction. Return the geometric state of the target
            body relative to the observer.
 
   The following flags apply to the "reception" case in which photons
   depart from the target's location at the light-time corrected epoch
   ET-LT and *arrive* at the observer's location at ET:
 
       --   'LT'
 
            Correct for one-way light time (also called "planetary
            aberration") using a Newtonian formulation. This correction
            yields the state of the target at the moment it emitted photons
            arriving at the observer at ET.
 
   The light time correction uses an iterative solution of the light time
   equation (see Particulars for details). The solution invoked by the 'LT'
   option uses one iteration.
 
       --   'LT+S'
 
            Correct for one-way light time and stellar aberration using a
            Newtonian formulation. This option modifies the state obtained
            with the 'LT' option to account for the observer's velocity
            relative to the solar system barycenter. The result is the
            apparent state of the target---the position and velocity of the
            target as seen by the observer.
 
       --   'CN'
 
            Converged Newtonian light time correction. In solving the light
            time equation, the 'CN' correction iterates until the solution
            converges (three iterations on all supported platforms).
            Whether the 'CN+S' solution is substantially more accurate than
            the 'LT' solution depends on the geometry of the participating
            objects and on the accuracy of the input data. In all cases,
            the correction calculation will execute more slowly when a
            converged solution is computed. See the Particulars section
            below for a discussion of precision of light time corrections.
 
       --   'CN+S'
 
            Converged Newtonian light time correction and stellar
            aberration correction.
 
   The following values of ABCORR apply to the "transmission" case in which
   photons *depart* from the observer's location at ET and arrive at the
   target's location at the light-time corrected epoch ET+LT:
 
       --   'XLT'
 
            "Transmission" case: correct for one-way light time using a
            Newtonian formulation. This correction yields the state of the
            target at the moment it receives photons emitted from the
            observer's location at ET.
 
       --   'XLT+S'
 
            "Transmission" case: correct for one-way light time and stellar
            aberration using a Newtonian formulation. This option modifies
            the state obtained with the 'XLT' option to account for the
            observer's velocity relative to the solar system barycenter.
            The position component of the computed target state indicates
            the direction that photons emitted from the observer's location
            must be "aimed" to hit the target.
 
       --   'XCN'
 
            "Transmission" case: converged Newtonian light time correction.
 
       --   'XCN+S'
 
            "Transmission" case: converged Newtonian light time correction
            and stellar aberration correction.
 
 
Common Correction Applications
--------------------------------------------------------
 
   Below, we indicate the aberration corrections to use for some common
   applications:
 
       1.   Find the apparent direction of a target. This is the most
            common case for a remote-sensing observation.
 
            Use 'LT+S' or 'CN+S': apply both light time and stellar
            aberration corrections.
 
            Note that using light time corrections alone ('LT') is
            generally not a good way to obtain an approximation to an
            apparent target vector: since light time and stellar aberration
            corrections often partially cancel each other, it may be more
            accurate to use no correction at all than to use light time
            alone.
 
       2.   Find the corrected pointing direction to radiate a signal to a
            target. This computation is often applicable for implementing
            communications sessions.
 
            Use 'XLT+S' or 'XCN+S': apply both light time and stellar
            aberration corrections for transmission.
 
       3.   Compute the apparent position of a target body relative to a
            star or other distant object.
 
            Use one of 'LT', 'CN', 'LT+S', or 'CN+S' as needed to match the
            correction applied to the position of the distant object. For
            example, if a star position is obtained from a catalog, the
            position vector may not be corrected for stellar aberration. In
            this case, to find the angular separation of the star and the
            limb of a planet, the vector from the observer to the planet
            should be corrected for light time but not stellar aberration.
 
       4.   Obtain an uncorrected state vector derived directly from data
            in an SPK file.
 
            Use 'NONE'.
 
       5.   Use a geometric state vector as a low-accuracy estimate of the
            apparent state for an application where execution speed is
            critical.
 
            Use 'NONE'.
 
       6.   While the correction routines do not perform the relativistic
            aberration corrections required to compute states with the
            highest possible accuracy, they can supply the geometric states
            required as inputs to these computations.
 
            Use 'NONE', then apply relativistic aberration corrections (not
            available in the SPICE Toolkit).
 
 
Computation of Corrections
--------------------------------------------------------
 
   Below, we discuss in more detail how the aberration corrections are
   computed.
 
 
Geometric case
 
   The algorithm begins by computing the geometric position T(t) of the
   target body relative to the solar system barycenter (SSB). Subtracting
   the geometric position of the observer O(t) gives the geometric position
   of the target body relative to the observer. The one-way light time, lt,
   is given by
 
                 || T(t) - O(t) ||
            lt = -----------------
                         c
 
   The geometric relationship between the observer, target, and solar
   system barycenter is as shown:
 
            SSB ---> O(t)
             |      /
             |     /
             |    /
             |   /  T(t) - O(t)
             |  /
             | /
             |/
             V
            T(t)
 
   The returned state consists of the position vector
 
            T(t) - O(t)
 
   and a velocity obtained by taking the difference of the corresponding
   velocities. In the geometric case, the returned velocity is actually the
   time derivative of the position.
 
 
Reception case
 
   When any of the options 'LT', 'CN', 'LT+S', 'CN+S' is selected for
   `abcorr', the algorithm computes the position of the target body at
   epoch et-lt, where `lt' is the one-way light time. Let T(t) and O(t)
   represent the positions of the target and observer relative to the solar
   system barycenter at time t; then `lt' is the solution of the light-time
   equation
 
                   || T(t-lt) - O(t) ||
            lt =   --------------------                              (1)
                           c
 
   The ratio
 
             || T(t) - O(t) ||
             -----------------                                       (2)
                     c
 
   is used as a first approximation to `lt'; inserting (2) into the right
   hand side of the light-time equation (1) yields the "one-iteration"
   estimate of the one-way light time ('LT'). Repeating the process until
   the estimates of `lt' converge yields the "converged Newtonian" light
   time estimate ('CN'). This methodology performs a contraction mapping.
 
   Subtracting the geometric position of the observer O(t) gives the
   position of the target body relative to the observer: T(t-lt) - O(t).
 
            SSB ---> O(t)
             | \     |
             |  \    |
             |   \   | T(t-lt) - O(t)
             |    \  |
             |     \ |
             |      \|
             V       V
            T(t)  T(t-lt)
 
   Note, in general, the vectors defined by T(t), O(t), T(t-lt) - O(t), and
   T(t-lt) are not coplanar.
 
   The position component of the light time corrected state is the vector
 
            T(t-lt) - O(t)
 
   The velocity component of the light time corrected state is the
   difference
 
      d(T(t-lt) - O(t))                      d(lt)
      ----------------- = T_vel(t-lt) * (1 - -----) - O_vel(t)
      dt                                      dt
 
   where T_vel and O_vel are, respectively, the velocities of the target
   and observer relative to the solar system barycenter at the epochs et-lt
   and `et'.
 
   If correction for stellar aberration is requested, the target position
   is rotated toward the solar system barycenter- relative velocity vector
   of the observer. The rotation is computed as follows:
 
   Let r be the light time corrected vector from the observer to the
   object, and v be the velocity of the observer with respect to the solar
   system barycenter. Let w be the angle between them. The aberration angle
   phi is given by
 
                  sin(phi) = v sin(w)
                             --------
                             c
 
   Let h be the vector given by the cross product
 
                  h = r X v
 
   Rotate r by phi radians about h to obtain the apparent position of the
   object.
 
   When stellar aberration corrections are used, the rate of change of the
   stellar aberration correction is accounted for in the computation of the
   output velocity.
 
 
Transmission case
 
   When any of the options 'XLT', 'XCN', 'XLT+S', 'XCN+S' is selected, the
   algorithm computes the position of the target body T at epoch et+lt,
   where `lt' is the one-way light time. `lt' is the solution of the
   light-time equation
 
                 || T(t+lt) - O(t) ||
            lt = ---------------------                               (3)
                          c
 
   Subtracting the geometric position of the observer, O(t), gives the
   position of the target body relative to the observer: T(t+lt) - O(t).
 
                     O(t) <--- SSB
                        |     / |
                        |    /  |
         T(t+lt) - O(t) |   /   |
                        |  /    |
                        | /     |
                        |/      |
                        V       V
                    T(t+lt)  T(t)
 
   Note, in general, the vectors defined by T(t), O(t), T(t+lt) - O(t), and
   T(t+lt) are not coplanar.
 
   The position component of the light-time corrected state is the vector
 
            T(t+lt) - O(t)
 
   The velocity component of the light-time corrected state consists of the
   difference
 
      d(T(t+lt) - O(t))                      d(lt)
      ----------------- = T_vel(t+lt) * (1 + -----) - O_vel(t)
      dt                                     dt
 
   where T_vel and O_vel are, respectively, the velocities of the target
   and observer relative to the solar system barycenter at the epochs et+lt
   and `et'.
 
   If correction for stellar aberration is requested, the target position
   is rotated away from the solar system barycenter- relative velocity
   vector of the observer. The rotation is computed as in the reception
   case, but the sign of the rotation angle is negated.
 
 
Precision of light time corrections
--------------------------------------------------------
 
   Let:
 
                    V
             beta = -
                    C
 
   where V is the velocity of the target relative to an inertial frame and
   C is the speed of light.
 
 
Corrections using one iteration of the light time solution
 
   When the requested aberration correction is 'LT', 'LT+S', 'XLT', or
   'XLT+S', only one iteration is performed in the algorithm used to
   compute lt.
 
   The relative error in this computation
 
            || lt_actual - lt_computed ||
            ---------------------------
                   lt_actual
 
   is at most
 
                 2
             beta
            ----------
             1 - beta
 
   which is well approximated by beta**2 for beta << 1 since
 
              1               2    3    4    5      6
            ----- ~= 1 + x + x  + x  + x  + x  + O(x )               (4)
            (1-x)
 
            about x = 0.
 
            So with x = beta
 
                 2
             beta              2      3      4         5
            ----------  ~= beta + beta + beta + O( beta )
             1 - beta
 
   For nearly all objects in the solar system V is less than 60 km/sec. The
   value of C is ~300000 km/sec. Thus the one-iteration solution for `lt'
   has a potential relative error of not more than 4e-8. This is a
   potential light time error of approximately 2e-5 seconds per
   astronomical unit of distance separating the observer and target. Given
   the bound on V cited above:
 
   As long as the observer and target are separated by less than 50
   astronomical units, the error in the light time returned using the
   one-iteration light time corrections is less than 1 millisecond.
 
   The magnitude of the corresponding position error, given the above
   assumptions, may be as large as beta**2 * the distance between the
   observer and the uncorrected target position: 300 km or equivalently 6
   km/AU.
 
   In practice, the difference between positions obtained using
   one-iteration and converged light time is usually much smaller than the
   value computed above and can be insignificant. For example, for the
   spacecraft Mars Reconnaissance Orbiter and Mars Express, the position
   error for the one-iteration light time correction, applied to the
   spacecraft-to-Mars center vector, is at the 1 cm level.
 
   Comparison of results obtained using the one-iteration and converged
   light time solutions is recommended when adequacy of the one-iteration
   solution is in doubt.
 
 
Converged corrections
 
   When the requested aberration correction is 'CN', 'CN+S', 'XCN', or
   'XCN+S', as many iterations as are required for convergence are
   performed in the computation of LT. Usually the solution is found after
   three iterations.
 
   The relative error present in this case is at most
 
                 4
             beta
            ----------
             1 - beta
 
   which is well approximated by beta**4 for beta << 1 since using (4) with
   x = beta as before
 
                 4
             beta              4      5      6         7
            ----------  ~= beta + beta + beta + O( beta )
             1 - beta
 
   The precision of this computation (ignoring round-off error) is better
   than 4e-11 seconds for any pair of objects less than 50 AU apart, and
   having speed relative to the solar system barycenter less than 60 km/s (
   beta = 2.001e-4, beta**4 = 1.604e-15).
 
   The magnitude of the corresponding position error, given the above
   assumptions, may be as large as beta**4 * the distance between the
   observer and the uncorrected target position: 1.2 cm at 50 AU or
   equivalently 0.24 mm/AU.
 
   However, to very accurately model the light time between target and
   observer one must take into account effects due to general relativity.
   These may be as high as a few hundredths of a millisecond for some
   objects.
 
 
Corrections in Non-inertial Frames
 
   When applying corrections in a non inertial reference frame, the epoch
   at which to evaluate frame orientation is adjusted by the one-way light
   time, `lt', between the observer and the frame's center. The orientation
   of the frame is evaluated at the time of interest - lt, the time of
   interest + lt, or the time of interest depending on whether the
   requested aberration correction is, respectively, for received
   radiation, transmitted radiation, or is omitted. `lt' is computed using
   the method indicated by the aberration correction flag.
 
 
Relativistic Corrections
--------------------------------------------------------
 
   SPICE aberration correction routines do not attempt to perform either
   general or special relativistic corrections in computing the various
   aberration corrections. For many applications relativistic corrections
   are not worth the expense of added computation cycles. If your
   application requires these additional corrections we suggest you consult
   the astronomical almanac (page B36) for a discussion of how to carry out
   these corrections.
 
 
Appendix A --- Revision History
===========================================================================
 
 
2020 MAY 26 by E. D. Wright.
 
   Removed block describing derivative of light-time in the geometric case.
   Not needed, as well as the derivation shown in the wrong document
   section.
 
 
2015 AUG 11 by E. D. Wright.
 
   Created a Required Reading document from Nat Bachman's original
   derivations and write-up as shown in SPICELIB routine SPKEZ. Edited as
   required only for fake TeX format, corrections, and improvement of
   descriptions.
 
   This document serves as a reference for the implementation of aberration
   corrections in all SPICE Toolkit distributions.
 
