| Aberration Corrections Required Reading |
Table of ContentsAberration 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 Aberration Corrections Required Reading
Abstract
Purpose
Intended Audience
References
Introduction
Types of CorrectionsOne-way Light Time
Stellar Aberration
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)
Common Correction Applications
Computation of Corrections
Geometric case
|| 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
|| 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 dtwhere 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. Transmission case
|| 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 dtwhere 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
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
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
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
Relativistic Corrections
Appendix A --- Revision History2015 AUG 11 by E. D. Wright.
This document now serves as the primary reference for the implementation of aberration corrections in all SPICE Toolkit distributions. Original version by N. J. Bachman
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