SLA_PERTUE - Perturbed Universal Elements

From d54fe7c1f704a63824c5bfa0ece65245572e9b27 Mon Sep 17 00:00:00 2001 From: Joseph Hunkeler Date: Wed, 4 Mar 2015 21:21:30 -0500 Subject: Initial commit --- src/slalib/sun67.htx/node145.html | 279 ++++++++++++++++++++++++++++++++++++++ 1 file changed, 279 insertions(+) create mode 100644 src/slalib/sun67.htx/node145.html (limited to 'src/slalib/sun67.htx/node145.html') diff --git a/src/slalib/sun67.htx/node145.html b/src/slalib/sun67.htx/node145.html new file mode 100644 index 0000000..8fbb853 --- /dev/null +++ b/src/slalib/sun67.htx/node145.html @@ -0,0 +1,279 @@ + + + + +SLA_PERTUE - Perturbed Universal Elements + + + + + + + + + + + + +

+ +

+
+ Next: SLA_PLANEL - Planet Position from Elements +
+Up: SUBPROGRAM SPECIFICATIONS +
+ Previous: SLA_PERTEL - Perturbed Orbital Elements +

SLA_PERTUE - Perturbed Universal Elements + +

ACTION: +: Update the universal elements of an asteroid or comet by +applying planetary perturbations. +
CALL: +: CALL sla_PERTUE (DATE, U, JSTAT) +

GIVEN: +

+
+ + + + + +

DATE1	D	final epoch (TT MJD) for the updated elements

GIVEN and RETURNED: +

+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

U	D(13)	universal elements (updated in place)
(1)		combined mass (M+m)
(2)		total energy of the orbit ( $\alpha$ )
(3)		reference (osculating) epoch (t₀)
(4-6)		position at reference epoch ( ${\rm \bf r}_0$ )
(7-9)		velocity at reference epoch ( ${\rm \bf v}_0$ )
(10)		heliocentric distance at reference epoch
(11)		${\rm \bf r}_0.{\rm \bf v}_0$
(12)		date (t)
(13)		universal eccentric anomaly ( $\psi$ ) of date, approx

RETURNED: +

+
+ + + + + + + + + + + + + + + + + + + + + + + + + +

JSTAT	I	status:
		+102 = warning, distant epoch
		+101 = warning, large timespan +(>100 years)
		+1 to +8 = coincident with major planet +(Note 5)
		0 = OK
		-1 = numerical error

NOTES: +

1. +

The ``universal'' elements are those which define the orbit for the +purposes of the method of universal variables (see reference 2). +They consist of the combined mass of the two bodies, an epoch, + and the position and velocity vectors (arbitrary reference frame) + at that epoch. The parameter set used here includes also various + quantities that can, in fact, be derived from the other + information. This approach is taken to avoiding unnecessary + computation and loss of accuracy. The supplementary quantities + are (i) $\alpha$ , which is proportional to the total energy of the + orbit, (ii) the heliocentric distance at epoch, + (iii) the outwards component of the velocity at the given epoch, + (iv) an estimate of $\psi$ , the ``universal eccentric anomaly'' at a + given date and (v) that date. +

2. +

The universal elements are with respect to the J2000 equator and + equinox. +

3. +

The epochs DATE, U(3) and U(12) are all Modified Julian Dates + (JD-2400000.5). +

4. +

The algorithm is a simplified form of Encke's method. It takes as + a basis the unperturbed motion of the body, and numerically + integrates the perturbing accelerations from the major planets. + The expression used is essentially Sterne's 6.7-2 (reference 1). + Everhart and Pitkin (reference 2) suggest rectifying the orbit at + each integration step by propagating the new perturbed position + and velocity as the new universal variables. In the present + routine the orbit is rectified less frequently than this, in order + to gain a slight speed advantage. However, the rectification is + done directly in terms of position and velocity, as suggested by + Everhart and Pitkin, bypassing the use of conventional orbital + elements. +

+The f(q) part of the full Encke method is not used. The purpose + of this part is to avoid subtracting two nearly equal quantities + when calculating the ``indirect member'', which takes account of the + small change in the Sun's attraction due to the slightly displaced + position of the perturbed body. A simpler, direct calculation in + double precision proves to be faster and not significantly less + accurate. +

+Apart from employing a variable timestep, and occasionally + ``rectifying the orbit'' to keep the indirect member small, the + integration is done in a fairly straightforward way. The + acceleration estimated for the middle of the timestep is assumed + to apply throughout that timestep; it is also used in the + extrapolation of the perturbations to the middle of the next + timestep, to predict the new disturbed position. There is no + iteration within a timestep. +

+Measures are taken to reach a compromise between execution time + and accuracy. The starting-point is the goal of achieving + arcsecond accuracy for ordinary minor planets over a ten-year + timespan. This goal dictates how large the timesteps can be, + which in turn dictates how frequently the unperturbed motion has + to be recalculated from the osculating elements. +

+Within predetermined limits, the timestep for the numerical + integration is varied in length in inverse proportion to the + magnitude of the net acceleration on the body from the major + planets. +

+The numerical integration requires estimates of the major-planet + motions. Approximate positions for the major planets (Pluto + alone is omitted) are obtained from the routine sla_PLANET. Two + levels of interpolation are used, to enhance speed without + significantly degrading accuracy. At a low frequency, the routine + sla_PLANET is called to generate updated position+velocity ``state + vectors''. The only task remaining to be carried out at the full + frequency (i.e. at each integration step) is to use the state + vectors to extrapolate the planetary positions. In place of a + strictly linear extrapolation, some allowance is made for the + curvature of the orbit by scaling back the radius vector as the + linear extrapolation goes off at a tangent. +

+Various other approximations are made. For example, perturbations + by Pluto and the minor planets are neglected, relativistic effects + are not taken into account and the Earth-Moon system is treated as + a single body. +

+In the interests of simplicity, the background calculations for + the major planets are carried out en masse. + The mean elements and + state vectors for all the planets are refreshed at the same time, + without regard for orbit curvature, mass or proximity. +

5. +

This routine is not intended to be used for major planets. + However, if major-planet elements are supplied, sensible results + will, in fact, be produced. This happens because the routine + checks the separation between the body and each of the planets and + interprets a suspiciously small value (0.001 AU) as an attempt to + apply the routine to the planet concerned. If this condition + is detected, the + contribution from that planet is ignored, and the status is set to + the planet number (Mercury=1,...,Neptune=8) as a warning. +

REFERENCES: +

1. +: Sterne, Theodore E., An Introduction to Celestial Mechanics, +Interscience Publishers, 1960. Section 6.7, p199. +
2. +: Everhart, E. & Pitkin, E.T., Am. J. Phys. 51, 712, 1983. +

+ +

+
+ Next: SLA_PLANEL - Planet Position from Elements +
+Up: SUBPROGRAM SPECIFICATIONS +
+ Previous: SLA_PERTEL - Perturbed Orbital Elements +

+SLALIB --- Positional Astronomy Library
Starlink User Note 67
P. T. Wallace
12 October 1999
E-mail:ptw@star.rl.ac.uk +

+ + -- cgit