Index Page Aberration Corrections Required Reading

## Table of Contents

       Aberration Corrections Required Reading
          Abstract
             Purpose
             Intended Audience
             References
       Introduction
          Types of Corrections
             One-way Light Time
             Stellar Aberration
             SPICE Aberration Identifiers (also called Flags)
          Common Correction Applications
          Computation of Corrections
             Geometric case
             Reception case
             Transmission case
          Precision of light time corrections
             Corrections using one iteration of the light time solution
             Converged corrections
             Corrections in Non-inertial Frames
          Relativistic Corrections
       Appendix A --- Revision History
             2015 AUG 11 by E. D. Wright.
             Original version by N. J. Bachman

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# Aberration Corrections Required Reading

Last revised on 2015 AUG 11 by N. J. Bachman and E. D. Wright.

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## 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.

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### Purpose

This document is a reference guide describing the details of the aberration correction calculations as implemented in the SPICE system.

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### Intended Audience

This document is for SPICE users who need specifics concerning the application of aberration corrections to state calculations.

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### References

`1.` Jesperson and Fitz-Randolph, From Sundials to Atomic Clocks, Dover Publications, New York, 1977.

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# 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.

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## Types of Corrections

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### 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.

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### 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.

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### 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.

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## 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).

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## Computation of Corrections

Below, we discuss in more detail how the aberration corrections are computed.

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### 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.

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### 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:

Note, the term

             d(lt)
             -----
             dt

does not equal zero, nor approximate zero.

Given

                  || r2(t) - r1(t) ||
             lt = -------------------
                          c

with r2(t) the position of the target at some time, and r1(t) the position of the observer at some time.

Let

             r(t)  = r2(t) - r1(t)

             range = || r2(t) - r1(t) ||
                   = || r(t) ||

The derivative of light time (lt) with respect to time equals the instantaneous range rate between the two bodies divided by light speed.

                d(lt)    d(range)   1
                ----- =  -------- * -
                 dt        dt       c

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.

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### 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.

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## 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.

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### 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.

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### 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.

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### 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.

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## 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.

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# Appendix A --- Revision History

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### 2015 AUG 11 by E. D. Wright.

Created a Required Reading document from Nat Bachman's original derivations and write-up. Edited as required only for fake TeX format, corrections, and improvement of descriptions.

This document now serves as the primary reference for the implementation of aberration corrections in all SPICE Toolkit distributions.

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### Original version by N. J. Bachman