| spkez |
|
Table of contents
Procedure
SPKEZ ( S/P Kernel, easy reader )
SUBROUTINE SPKEZ ( TARG, ET, REF, ABCORR, OBS, STARG, LT )
Abstract
Return the state (position and velocity) of a target body
relative to an observing body, optionally corrected for light
time (planetary aberration) and stellar aberration.
Required_Reading
SPK
NAIF_IDS
FRAMES
TIME
Keywords
EPHEMERIS
Declarations
IMPLICIT NONE
INCLUDE 'frmtyp.inc'
INCLUDE 'zzabcorr.inc'
INCLUDE 'zzctr.inc'
INTEGER TARG
DOUBLE PRECISION ET
CHARACTER*(*) REF
CHARACTER*(*) ABCORR
INTEGER OBS
DOUBLE PRECISION STARG ( 6 )
DOUBLE PRECISION LT
Brief_I/O
VARIABLE I/O DESCRIPTION
-------- --- --------------------------------------------------
TARG I Target body.
ET I Observer epoch.
REF I Reference frame of output state vector.
ABCORR I Aberration correction flag.
OBS I Observing body.
STARG O State of target.
LT O One way light time between observer and target.
Detailed_Input
TARG is the NAIF ID code for a target body. The target
and observer define a state vector whose position
component points from the observer to the target.
ET is the ephemeris time, expressed as seconds past J2000
TDB, at which the state of the target body relative to
the observer is to be computed. ET refers to time at
the observer's location.
REF is the name of the reference frame relative to which
the output state vector should be expressed. This may
be any frame supported by the SPICE system, including
built-in frames (documented in the Frames Required
Reading) and frames defined by a loaded frame kernel
(FK).
When REF designates a non-inertial frame, the
orientation of the frame is evaluated at an epoch
dependent on the selected aberration correction.
See the description of the output state vector STARG
for details.
ABCORR indicates the aberration corrections to be applied
to the state of the target body to account for one-way
light time and stellar aberration. See the discussion
in the $Particulars section for recommendations on
how to choose aberration corrections.
ABCORR may be any of the following:
'NONE' Apply no correction. Return the
geometric state of the target body
relative to the observer.
The following values of ABCORR 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 this routine 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.
Neither special nor general relativistic effects are
accounted for in the aberration corrections applied
by this routine.
Case and blanks are not significant in the string
ABCORR.
OBS is the NAIF ID code for an observing body.
Detailed_Output
STARG is a Cartesian state vector representing the position
and velocity of the target body relative to the
specified observer. STARG is corrected for the
specified aberrations, and is expressed with respect
to the reference frame specified by REF. The first
three components of STARG represent the x-, y- and
z-components of the target's position; the last three
components form the corresponding velocity vector.
The position component of STARG points from the
observer's location at ET to the aberration-corrected
location of the target. Note that the sense of the
position vector is independent of the direction of
radiation travel implied by the aberration
correction.
The velocity component of STARG is the derivative
with respect to time of the position component of
STARG.
Units are always km and km/sec.
Non-inertial frames are treated as follows: letting
LTCENT be the one-way light time between the observer
and the central body associated with the frame, the
orientation of the frame is evaluated at ET-LTCENT,
ET+LTCENT, or ET depending on whether the requested
aberration correction is, respectively, for received
radiation, transmitted radiation, or is omitted.
LTCENT is computed using the method indicated by
ABCORR.
LT is the one-way light time between the observer and
target in seconds. If the target state is corrected
for aberrations, then LT is the one-way light time
between the observer and the light time corrected
target location.
Parameters
None.
Exceptions
1) If the reference frame REF is not a recognized reference
frame, the error SPICE(UNKNOWNFRAME) is signaled.
2) If the loaded kernels provide insufficient data to compute the
requested state vector, an error is signaled by a routine in
the call tree of this routine.
3) If an error occurs while reading an SPK or other kernel file,
the error is signaled by a routine in the call tree
of this routine.
4) If any of the required attributes of the reference frame REF
cannot be determined, the error SPICE(UNKNOWNFRAME2) is
signaled.
Files
This routine computes states using SPK files that have been
loaded into the SPICE system, normally via the kernel loading
interface routine FURNSH. See the routine FURNSH and the SPK
and KERNEL Required Reading for further information on loading
(and unloading) kernels.
If the output state STARG is to be expressed relative to a
non-inertial frame, or if any of the ephemeris data used to
compute STARG are expressed relative to a non-inertial frame in
the SPK files providing those data, additional kernels may be
needed to enable the reference frame transformations required to
compute the state. Normally these additional kernels are PCK
files or frame kernels. Any such kernels must already be loaded
at the time this routine is called.
Particulars
This routine is part of the user interface to the SPICE ephemeris
system. It allows you to retrieve state information for any
ephemeris object relative to any other in a reference frame that
is convenient for further computations.
Aberration corrections
======================
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.
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 correction performed by this routine.
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 this routine 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.
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.
Below, we indicate the aberration corrections to use for some
common applications:
1) Find the apparent direction of a target 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' or 'CN')
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 '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 this routine cannot perform the relativistic
aberration corrections required to compute states
with the highest possible accuracy, it can supply the
geometric states required as inputs to these computations.
Use 'NONE', then apply relativistic aberration
corrections (not available in the SPICE Toolkit).
Below, we discuss in more detail how the aberration corrections
applied by this routine are computed.
Geometric case
==============
SPKEZ begins by computing the geometric position T(ET) of the
target body relative to the solar system barycenter (SSB).
Subtracting the geometric position of the observer O(ET) gives
the geometric position of the target body relative to the
observer. The one-way light time, LT, is given by
| T(ET) - O(ET) |
LT = -------------------
c
The geometric relationship between the observer, target, and
solar system barycenter is as shown:
SSB ---> O(ET)
| /
| /
| /
| / T(ET) - O(ET)
V V
T(ET)
The returned state consists of the position vector
T(ET) - O(ET)
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, SPKEZ 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(ET-LT) - O(ET) |
LT = ------------------------ (1)
c
The ratio
| T(ET) - O(ET) |
--------------------- (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").
Subtracting the geometric position of the observer O(ET) gives
the position of the target body relative to the observer:
T(ET-LT) - O(ET).
SSB ---> O(ET)
| \ |
| \ |
| \ | T(ET-LT) - O(ET)
| \ |
V V V
T(ET) T(ET-LT)
The position component of the light time corrected state
is the vector
T(ET-LT) - O(ET)
The velocity component of the light time corrected state
is the difference
T_vel(ET-LT)*(1-dLT/dET) - O_vel(ET)
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, SPKEZ 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(ET+LT) - O(ET) |
LT = ------------------------ (3)
c
Subtracting the geometric position of the observer, O(ET),
gives the position of the target body relative to the
observer: T(ET-LT) - O(ET).
SSB --> O(ET)
/ | *
/ | * T(ET+LT) - O(ET)
/ |*
/ *|
V V V
T(ET+LT) T(ET)
The position component of the light-time corrected state
is the vector
T(ET+LT) - O(ET)
The velocity component of the light-time corrected state
consists of the difference
T_vel(ET+LT)*(1+dLT/dET) - O_vel(ET)
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. Velocities are adjusted to account
for the rate of change of the stellar aberration correction.
Precision of light time corrections
===================================
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
(V/C)**2
----------
1 - (V/C)
which is well approximated by (V/C)**2, where V is the
velocity of the target relative to an inertial frame and C is
the speed of light.
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 (V/C)**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
(V/C)**4
----------
1 - (V/C)
which is well approximated by (V/C)**4.
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.
The magnitude of the corresponding position error, given
the above assumptions, may be as large as (V/C)**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.
Relativistic Corrections
=========================
This routine does 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 however, 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.
Examples
1) Load a planetary ephemeris SPK; then look up a series of
geometric states of the moon relative to the earth,
referenced to the J2000 frame.
IMPLICIT NONE
C
C Local constants
C
CHARACTER*(*) FRAME
PARAMETER ( FRAME = 'J2000' )
CHARACTER*(*) ABCORR
PARAMETER ( ABCORR = 'NONE' )
C
C The name of the SPK file shown here is fictitious;
C you must supply the name of an SPK file available
C on your own computer system.
C
CHARACTER*(*) SPK
PARAMETER ( SPK = 'planet.bsp' )
C
C ET0 represents the date 2000 Jan 1 12:00:00 TDB.
C
DOUBLE PRECISION ET0
PARAMETER ( ET0 = 0.0D0 )
C
C Use a time step of 1 hour; look up 100 states.
C
DOUBLE PRECISION STEP
PARAMETER ( STEP = 3600.0D0 )
INTEGER MAXITR
PARAMETER ( MAXITR = 100 )
C
C The NAIF IDs of the earth and moon are 399 and 301
C respectively.
C
INTEGER OBSRVR
PARAMETER ( OBSRVR = 399 )
INTEGER TARGET
PARAMETER ( TARGET = 301 )
C
C Local variables
C
DOUBLE PRECISION ET
DOUBLE PRECISION LT
DOUBLE PRECISION STATE ( 6 )
INTEGER I
C
C Load the SPK file.
C
CALL FURNSH ( SPK )
C
C Step through a series of epochs, looking up a
C state vector at each one.
C
DO I = 1, MAXITR
ET = ET0 + (I-1)*STEP
CALL SPKEZ ( TARGET, ET, FRAME, ABCORR, OBSRVR,
. STATE, LT )
WRITE (*,*) 'ET = ', ET
WRITE (*,*) 'J2000 x-position (km): ', STATE(1)
WRITE (*,*) 'J2000 y-position (km): ', STATE(2)
WRITE (*,*) 'J2000 z-position (km): ', STATE(3)
WRITE (*,*) 'J2000 x-velocity (km/s): ', STATE(4)
WRITE (*,*) 'J2000 y-velocity (km/s): ', STATE(5)
WRITE (*,*) 'J2000 z-velocity (km/s): ', STATE(6)
WRITE (*,*) ' '
END DO
END
Restrictions
None.
Literature_References
None.
Author_and_Institution
C.H. Acton (JPL)
N.J. Bachman (JPL)
J. Diaz del Rio (ODC Space)
J.E. McLean (JPL)
H.A. Neilan (JPL)
B.V. Semenov (JPL)
M.J. Spencer (JPL)
W.L. Taber (JPL)
I.M. Underwood (JPL)
E.D. Wright (JPL)
Version
SPICELIB Version 5.1.1, 16-AUG-2021 (JDR)
Edited the header to comply with NAIF standard.
SPICELIB Version 5.1.0, 03-JUL-2014 (NJB) (BVS)
Discussion of light time corrections was updated. Assertions
that converged light time corrections are unlikely to be
useful were removed.
Bug fix: replaced calls to ZZPRSCOR with calls to
ZZVALCOR. The latter routine rejects all aberration
corrections not supported by the SPK subsystem.
Bug fix: added a check and an exception for the FOUND flag
returned by FRINFO.
Updated to save the input frame name and POOL state counter
and to do frame name-ID conversion only if the counter has
changed.
Updated various in-line comments.
SPICELIB Version 5.0.1, 18-MAY-2010 (BVS)
Removed "C$" marker from text in the header.
SPICELIB Version 5.0.0, 27-DEC-2007 (NJB)
This routine was upgraded to more accurately compute
aberration-corrected velocity, and in particular, make it
more consistent with observer-target positions.
When light time corrections are used, the derivative of light
time with respect to time is now accounted for in the
computation of observer-target velocities. When the reference
frame associated with the output state is time-dependent, the
derivative of light time with respect to time is now accounted
for in the computation of the rate of change of orientation of
the reference frame.
When stellar aberration corrections are used, velocities
now reflect the rate of range of the stellar aberration
correction.
SPICELIB Version 4.1.0, 05-JAN-2005 (NJB)
Tests of routine FAILED() were added.
Minor header error was corrected.
SPICELIB Version 4.0.2, 20-OCT-2003 (EDW)
Added mention that LT returns in seconds.
SPICELIB Version 4.0.1, 29-JUL-2003 (NJB) (CHA)
Various minor header changes were made to improve clarity.
SPICELIB Version 4.0.0, 28-DEC-2001 (NJB)
Updated to handle aberration corrections for transmission
of radiation. Formerly, only the reception case was
supported. The header was revised and expanded to explain
the functionality of this routine in more detail.
SPICELIB Version 3.1.0, 09-JUL-1996 (WLT)
Corrected the description of LT in the Detailed Output
section of the header.
SPICELIB Version 3.0.0, 26-MAY-1995 (WLT)
The routine was upgraded to support non-inertial frames.
SPICELIB Version 2.1.1, 05-AUG-1994 (HAN) (MJS)
Added code so that routine accepts lower case, mixed case
and upper case versions of the string ABCORR.
SPICELIB Version 2.0.1, 10-MAR-1992 (WLT)
Comment section for permuted index source lines was added
following the header.
SPICELIB Version 2.0.0, 18-JUL-1991 (JEM) (NJB)
The old SPKEZ did not compute the geometric state of one body
with respect to another unless data existed for each body with
respect to the solar system barycenter.
SPICELIB Version 1.0.1, 22-MAR-1990 (HAN)
Literature references added to the header.
SPICELIB Version 1.0.0, 31-JAN-1990 (IMU)
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Fri Dec 31 18:36:52 2021