| eqncpv_c |
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Table of contents
Procedure
eqncpv_c (Equinoctial Elements to position and velocity)
void eqncpv_c ( SpiceDouble et,
SpiceDouble epoch,
ConstSpiceDouble eqel[9],
SpiceDouble rapol,
SpiceDouble decpol,
SpiceDouble state[6] )
AbstractCompute the state (position and velocity) of an object whose trajectory is described via equinoctial elements relative to some fixed plane (usually the equatorial plane of some planet). Required_ReadingSPK KeywordsEPHEMERIS Brief_I/OVARIABLE I/O DESCRIPTION -------- --- -------------------------------------------------- et I Epoch in seconds past J2000 to find state epoch I Epoch of elements in seconds past J2000 eqel I Array of equinoctial elements rapol I Right Ascension of the pole of the reference plane decpol I Declination of the pole of the reference plane state O State of the object described by `eqel'. Detailed_Input
et is the epoch (ephemeris time) at which the state
of the target body is to be computed. `et' is measured
in seconds past the J2000 epoch.
epoch is the epoch of the equinoctial elements in seconds
past the J2000 epoch.
eqel is an array of 9 double precision numbers that are the
equinoctial elements for some orbit expressed relative to
the equatorial frame of the central body defined as
- The Z-axis of the equatorial frame is the direction
of the pole of the central body relative to some
inertial frame;
- The X-axis is given by the cross product of the Z-axis
of the inertial frame with the direction of the pole
of the central body; and
- The Y-axis completes a right handed frame.
If the X-axis of the equatorial frame is aligned with the
X-axis of the inertial frame, then the X-axis of the
equatorial frame will be located at 90 degrees + rapol in
the inertial frame.
The specific arrangement of the elements is spelled out
below:
eqel[0] is the semi-major axis (A) of the orbit in
km.
eqel[1] is the value of H at the specified epoch.
( E*sin(argp+node) ).
eqel[2] is the value of K at the specified epoch
( E*cos(argp+node) ).
eqel[3] is the mean longitude (mean0+argp+node) at
the epoch of the elements measured in
radians.
eqel[4] is the value of P (tan(inc/2)*sin(node))at
the specified epoch.
eqel[5] is the value of Q (tan(inc/2)*cos(node))at
the specified epoch.
eqel[6] is the rate of the longitude of periapse
(dargp/dt + dnode/dt ) at the epoch of
the elements. This rate is assumed to hold
for all time. The rate is measured in
radians per second.
eqel[7] is the derivative of the mean longitude
( dm/dt + dargp/dt + dnode/dt ). This
rate is assumed to be constant and is
measured in radians/second.
eqel[8] is the rate of the longitude of the
ascending node ( dnode/dt). This rate is
measured in radians per second.
where
inc is the inclination of the orbit,
argp is the argument of periapse,
node is longitude of the ascending node, and
E is eccentricity of the orbit.
rapol is the Right Ascension of the pole of the reference plane
with respect to some inertial frame (measured in
radians).
decpol is the Declination of the pole of the reference plane
with respect to some inertial frame (measured in
radians).
Detailed_Output
state is the state of the object described by `eqel' relative to
the inertial frame used to define `rapol' and `decpol'. Units
are in km and km/sec.
ParametersNone. Exceptions
1) If the eccentricity corresponding to the input elements is
greater than 0.9, the error SPICE(ECCOUTOFRANGE) is signaled
by a routine in the call tree of this routine.
2) If the semi-major axis of the elements is non-positive, the
error SPICE(BADSEMIAXIS) is signaled by a routine in the call
tree of this routine.
FilesNone. ParticularsThis routine evaluates the input equinoctial elements for the specified epoch and return the corresponding state. This routine was adapted from a routine provided by Bob Jacobson of the Planetary Dynamics Group of the Navigation and Flight Mechanics Section at JPL. Examples
The numerical results shown for this example may differ across
platforms. The results depend on the SPICE kernels used as
input, the compiler and supporting libraries, and the machine
specific arithmetic implementation.
1) Compute a state vector from a set of equinoctial elements.
Suppose you have classical elements and rates of change of the
ascending node and argument of periapse for some satellite of
the Earth.
By transforming the classical elements this routine computes the
state of the object at an arbitrary epoch. The code below
illustrates how to do this.
The table below illustrates the meanings of the various
variables used in the discussion below.
Variable Meaning
-------- ----------------------------------
a Semi-major axis in km.
ecc Eccentricity of orbit.
inc Inclination of orbit.
node Longitude of the ascending node at epoch.
omega Argument of periapse at epoch.
m Mean anomaly at epoch.
dmdt Mean anomaly rate in radians/second.
dnode Rate of change of longitude of ascending node
in radians/second.
domega Rate of change of argument of periapse in
radians/second.
epoch is the epoch of the elements in seconds past
the J2000 epoch.
Example code begins here.
/.
Program eqncpv_ex1
./
#include <stdio.h>
#include <math.h>
#include "SpiceUsr.h"
int main()
{
/.
Local variables.
./
SpiceInt i;
SpiceDouble a;
SpiceDouble argp;
SpiceDouble decpol;
SpiceDouble ecc;
SpiceDouble eqel [9];
SpiceDouble et;
SpiceDouble gm;
SpiceDouble inc;
SpiceDouble m0;
SpiceDouble n;
SpiceDouble node;
SpiceDouble p;
SpiceDouble rapol;
SpiceDouble t0;
SpiceDouble state [6];
p = 1.0e4;
gm = 398600.436;
ecc = 0.1;
a = p/( 1. - ecc );
n = sqrt ( gm / a ) / a;
argp = 30. * rpd_c();
node = 15. * rpd_c();
inc = 10. * rpd_c();
m0 = 45. * rpd_c();
t0 = -100000000.;
/.
Define the input equinoctial elements.
eqel[0] = a
eqel[1] = ecc * sin( omega + node )
eqel[2] = ecc * cos( omega + node )
eqel[3] = m + omega + node
eqel[4] = tan(inc/2.0) * sin(node)
eqel[5] = tan(inc/2.0) * cos(node)
eqel[6] = domega
eqel[7] = domega + dmdt + dnode
eqel[8] = dnode
In this case, the rates of node and argument of
periapse are zero and the pole of the central
frame is aligned with the pole of an inertial frame.
./
eqel[0] = a;
eqel[1] = ecc*sin(argp+node);
eqel[2] = ecc*cos(argp+node);
eqel[3] = m0 + argp + node;
eqel[4] = tan(inc/2.)*sin(node);
eqel[5] = tan(inc/2.)*cos(node);
eqel[6] = 0.;
eqel[7] = n;
eqel[8] = 0.;
rapol = -halfpi_c();
decpol = halfpi_c();
et = t0 - 10000.0;
for ( i = 0; i < 10; i++)
{
et = et + 250.;
eqncpv_c ( et, t0, eqel, rapol, decpol, state );
printf ("\nPos = %16.6f %16.6f %16.6f \n",
state[0], state[1], state[2] );
printf ( "Vel = %16.6f %16.6f %16.6f \n",
state[3], state[4], state[5] );
}
return ( 0 );
}
When this program was executed on a Mac/Intel/cc/64-bit
platform, the output was:
Pos = -10732.167433 3902.505791 1154.451615
Vel = -2.540767 -5.152269 -0.761576
Pos = -11278.382863 2586.179875 955.184099
Vel = -1.827156 -5.362916 -0.830020
Pos = -11645.295454 1228.612448 740.709574
Vel = -1.108096 -5.482811 -0.883256
Pos = -11832.799901 -147.990984 514.805250
Vel = -0.393421 -5.515905 -0.921508
Pos = -11843.089312 -1522.469846 281.175257
Vel = 0.308288 -5.466565 -0.945128
Pos = -11680.364607 -2874.784755 43.424394
Vel = 0.989520 -5.339364 -0.954552
Pos = -11350.589872 -4186.049765 -194.958526
Vel = 1.643649 -5.138938 -0.950269
Pos = -10861.293274 -5438.536175 -430.610411
Vel = 2.264759 -4.869899 -0.932792
Pos = -10221.410986 -6615.660644 -660.298988
Vel = 2.847476 -4.536794 -0.902651
Pos = -9441.170335 -7701.967890 -880.925189
Vel = 3.386822 -4.144103 -0.860382
Restrictions
1) The equinoctial elements used by this routine are taken
from "Tangent" formulation of equinoctial elements
P = tan(inclination/2) * sin(R.A. of ascending node)
Q = tan(inclination/2) * cos(R.A. of ascending node)
Other formulations use Sine instead of Tangent. We shall
call these the "Sine" formulations.
P = sin(inclination/2) * sin(R.A. of ascending node)
Q = sin(inclination/2) * cos(R.A. of ascending node)
If you have equinoctial elements from this alternative
formulation you should replace P and Q by the
expressions below.
P = P / sqrt( 1.0 - P*P - Q*Q )
Q = Q / sqrt( 1.0 - P*P - Q*Q )
This will convert the Sine formulation to the Tangent
formulation.
Literature_References
[1] W. Owen and R. Vaughan, "Optical Navigation Program
Mathematical Models," JPL Engineering Memorandum 314-513,
August 9, 1991.
Author_and_InstitutionJ. Diaz del Rio (ODC Space) E.D. Wright (JPL) Version
-CSPICE Version 1.0.1, 02-AUG-2021 (JDR)
Edited the header to comply with NAIF standard.
Added example's problem statement. Removed unused variable from code
example. Added SPK required reading and -Exceptions section.
Removed unnecessary comments from the code.
-CSPICE Version 1.0.0, 20-MAR-2012 (EDW)
Index_EntriesCompute a state from equinoctial elements |
Fri Dec 31 18:41:06 2021