Table of Contents
Aberration Corrections Required Reading
Abstract
Purpose
Intended Audience
References
Introduction
Types of Corrections
Oneway 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 Noninertial Frames
Relativistic Corrections
Appendix A  Revision History
2020 MAY 26 by E. D. Wright.
2015 AUG 11 by E. D. Wright.
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Aberration Corrections Required Reading
Last revised on 2020 MAY 26 by 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 FitzRandolph, 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: oneway light time (also
called "planetary aberration") and stellar aberration.
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Types of Corrections
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Oneway Light Time
Correcting for oneway 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.
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 lighttime corrected epoch
ETLT and *arrive* at the observer's location at ET:
Correct for oneway 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.
Correct for oneway 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 targetthe position
and velocity of the target as seen by the observer.
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.
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 lighttime corrected epoch ET+LT:
"Transmission" case: correct for oneway 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.
"Transmission" case: correct for oneway 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.
"Transmission" case: converged Newtonian light time correction.
"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
remotesensing 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.
5. Use a geometric state vector as a lowaccuracy estimate of the apparent
state for an application where execution speed is critical.
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 oneway 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 etlt, where `lt' is the oneway 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 lighttime
equation
 T(tlt)  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 lighttime equation (1) yields the "oneiteration"
estimate of the oneway 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(tlt)  O(t).
SSB > O(t)
 \ 
 \ 
 \  T(tlt)  O(t)
 \ 
 \ 
 \
V V
T(t) T(tlt)
Note, in general, the vectors defined by T(t), O(t), T(tlt)  O(t), and
T(tlt) are not coplanar.
The position component of the light time corrected state is the vector
T(tlt)  O(t)
The velocity component of the light time corrected state is the
difference
d(T(tlt)  O(t)) d(lt)
 = T_vel(tlt) * (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 etlt
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.
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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 oneway light time. `lt' is the solution of the
lighttime 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 lighttime corrected state is the vector
T(t+lt)  O(t)
The velocity component of the lighttime 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)
(1x)
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 oneiteration solution for `lt'
has a potential relative error of not more than 4e8. This is a
potential light time error of approximately 2e5 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
oneiteration 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
oneiteration 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 oneiteration light time correction, applied to the
spacecrafttoMars center vector, is at the 1 cm level.
Comparison of results obtained using the oneiteration and converged
light time solutions is recommended when adequacy of the oneiteration
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 roundoff error) is better
than 4e11 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 = 2.001e4, beta**4 = 1.604e15).
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 Noninertial Frames
When applying corrections in a non inertial reference frame, the epoch
at which to evaluate frame orientation is adjusted by the oneway 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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2020 MAY 26 by E. D. Wright.
Removed block describing derivative of lighttime in the geometric case.
Not needed, as well as the derivation shown in the wrong document
section.
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2015 AUG 11 by E. D. Wright.
Created a Required Reading document from Nat Bachman's original
derivations and writeup 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.
