### SLA_PLANEL

Planet Position from Elements

ACTION:
Heliocentric position and velocity of a planet, asteroid or comet, starting from orbital elements.
CALL:
CALL sla_PLANEL (DATE, JFORM, EPOCH, ORBINC, ANODE, PERIH, AORQ, E, AORL, DM, PV, JSTAT) 
##### GIVEN:
 DATE D TT MJD of observation (JD$-$2400000.5, Note 1) JFORM I choice of element set (1-3, Note 3) EPOCH D epoch of elements (${t}_{0}$ or $T$, TT MJD, Note 4) ORBINC D inclination ($i$, radians) ANODE D longitude of the ascending node ($\Omega$, radians) PERIH D longitude or argument of perihelion ($\varpi$ or $\omega$, radians) AORQ D mean distance or perihelion distance ($a$ or $q$, AU) E D eccentricity ($e$) AORL D mean anomaly or longitude ($M$ or $L$, radians, JFORM=1,2 only) DM D daily motion ($n$, radians, JFORM=1 only)

##### RETURNED:
 PV D(6) heliocentric $\left[\phantom{\rule{0.3em}{0ex}}x,y,z,ẋ,ẏ,ż\phantom{\rule{0.3em}{0ex}}\right]$, equatorial, J2000 (AU, AU/s) JSTAT I status: 0 = OK $-$1 = illegal JFORM $-$2 = illegal E $-$3 = illegal AORQ $-$4 = illegal DM $-$5 = numerical error

NOTES:
(1)
DATE is the instant for which the prediction is required. It is in the TT time scale (formerly Ephemeris Time, ET) and is a Modified Julian Date (JD$-$2400000.5).
(2)
The elements are with respect to the J2000 ecliptic and equinox.
(3)
A choice of three different element-format options is available, as follows.

JFORM=1, suitable for the major planets:

 EPOCH = epoch of elements ${t}_{0}$ (TT MJD) ORBINC = inclination $i$ (radians) ANODE = longitude of the ascending node $\Omega$ (radians) PERIH = longitude of perihelion $\varpi$ (radians) AORQ = mean distance $a$ (AU) E = eccentricity $e$ AORL = mean longitude $L$ (radians) DM = daily motion $n$ (radians)

JFORM=2, suitable for minor planets:

 EPOCH = epoch of elements ${t}_{0}$ (TT MJD) ORBINC = inclination $i$ (radians) ANODE = longitude of the ascending node $\Omega$ (radians) PERIH = argument of perihelion $\omega$ (radians) AORQ = mean distance $a$ (AU) E = eccentricity $e$ AORL = mean anomaly $M$ (radians)

JFORM=3, suitable for comets:

 EPOCH = epoch of perihelion $T$ (TT MJD) ORBINC = inclination $i$ (radians) ANODE = longitude of the ascending node $\Omega$ (radians) PERIH = argument of perihelion $\omega$ (radians) AORQ = perihelion distance $q$ (AU) E = eccentricity $e$

Unused elements (DM for JFORM=2, AORL and DM for JFORM=3) are not accessed.

(4)
Each of the three element sets defines an unperturbed heliocentric orbit. For a given epoch of observation, the position of the body in its orbit can be predicted from these elements, which are called osculating elements, using standard two-body analytical solutions. However, due to planetary perturbations, a given set of osculating elements remains usable for only as long as the unperturbed orbit that it describes is an adequate approximation to reality. Attached to such a set of elements is a date called the osculating epoch, at which the elements are, momentarily, a perfect representation of the instantaneous position and velocity of the body.

Therefore, for any given problem there are up to three different epochs in play, and it is vital to distinguish clearly between them:

• The epoch of observation: the moment in time for which the position of the body is to be predicted.
• The epoch defining the position of the body: the moment in time at which, in the absence of purturbations, the specified position—mean longitude, mean anomaly, or perihelion—is reached.
• The osculating epoch: the moment in time at which the given elements are correct.

For the major-planet and minor-planet cases it is usual to make the epoch that defines the position of the body the same as the epoch of osculation. Thus, only two different epochs are involved: the epoch of the elements and the epoch of observation. For comets, the epoch of perihelion fixes the position in the orbit and in general a different epoch of osculation will be chosen. Thus, all three types of epoch are involved.

For the present routine:

• The epoch of observation is the argument DATE.
• The epoch defining the position of the body is the argument EPOCH.
• The osculating epoch is not used and is assumed to be close enough to the epoch of observation to deliver adequate accuracy. If not, a preliminary call to sla_PERTEL may be used to update the element-set (and its associated osculating epoch) by applying planetary perturbations.
(5)
The reference frame for the result is equatorial and is with respect to the mean equinox and ecliptic of epoch J2000.
(6)
The algorithm was originally adapted from the EPHSLA program of D. H. P. Jones (private communication, 1996). The method is based on Stumpff’s Universal Variables.
REFERENCE:
Everhart, E. & Pitkin, E.T., Am. J. Phys. 51, 712, 1983.