Constraining Recovery Observations for Trans‐Neptunian Objects with Poorly Known Orbits

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  • Constraining Recovery Observations for TransNeptunian Objects with Poorly Known OrbitsAuthor(s): JeffreyD.Goldader and CharlesAlcockSource: Publications of the Astronomical Society of the Pacific, Vol. 115, No. 813 (November2003), pp. 1330-1339Published by: The University of Chicago Press on behalf of the Astronomical Society of the PacificStable URL: .Accessed: 25/05/2014 01:31

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    Publications of the Astronomical Society of the Pacific, 115:13301339, 2003 November 2003. The Astronomical Society of the Pacific. All rights reserved. Printed in U.S.A.

    Constraining Recovery Observations for Trans-Neptunian Objectswith Poorly Known Orbits

    Jeffrey D. Goldader and Charles Alcock

    Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104;,

    Received 2003 June 9; accepted 2003 August 22

    ABSTRACT. We present a simple method for constraining the possible future positions of distant solar systemobjects observed twice over only a very short time span. The method involves taking two positions and thendetermining a large number of possible orbits compatible with the observed motion across the sky for an objectwith unknown (but constrainable) distance from Earth. A key advantage of this approach is that it assumes onlythat the object is bound and distant. Monte Carlo techniques are used to incorporate astrometric uncertainty andmap out the allowed orbital parameter space. The method allows us to compute the objects position on theselected recovery date for each potential orbit, assisting the selection of fields for recovery observations. Examplesare shown, and usage of the code is discussed.


    The problem of initial orbit determination of celestial objectsis centuries old. Marsden (1985, 1991) cites references goingback more than 200 years and reviews modern applications ofGausss method. More recent efforts include those of Bernstein& Khushalani (2000) and Virtanen et al. (2001).

    Reliable orbit determinations require several observations,because a proper orbit is described by six orbital parameters(semimajor axis, eccentricity, inclination, longitude of the as-cending node, argument of pericenter, and time of pericenterpassage). Since a simple observation of the celestial coordinatesof an object does not tell us the distance to the object, twosuch observations spaced closely in time are insufficient toyield a reliable orbit.

    Circumstances sometimes unavoidably result in very fewobservations being taken of a newly discovered object, and thisis the case for our own Southern Edgeworth-Kuiper Surveyfor Trans-Neptunian Objects (TNOs), described by Marshall etal. (2001) and Moody et al. (2003). The survey took data atMount Stromlo for about 3 years. It used the MACHO telescopesystem to search large patches of sky ( ) for slowly0.7# 0.7moving objects that were possibly TNOs.

    Fitting reliable orbits from our archive observations aloneis not possible. With the observing strategy of our survey, thearc of observations was short, and a TNO would be visible inat most three images, taken over a time span of at most1 week.1 As Bernstein & Khushalani (2000) discuss, only the

    1 Typically, we would take two images of a field a few hours apart on onenight, both on the same side of the meridian, followed within the next fewnights by another single image of the same field.

    inclination of the orbit and heliocentric distance can be con-strained with such a short arc, and even that assumes a circularorbit, with the inherent degeneracies in the remaining orbitalparameters. The longer between the discovery and recoveryobservations, the larger the area of sky that must be searched.We needed to find a way to use only our few observations topredict and constrain the position of a candidate at a laterapparition, so as to maximize the possibility of recovery whileminimizing the area of sky we would have to search to recoverthe object. Once recovered, a candidates orbit could be de-termined using one of the conventional methods (such as thoseused by the IAU Minor Planet Center or that of Bernstein &Khushalani 2000).

    In the case of poorly constrained orbits, recovery observa-tions for distant moving objects become increasingly more dif-ficult and expensive (in terms of telescope time) as one movesfarther in time from the epoch of discovery. The instantaneousobserved motion of a TNO is primarily caused by the Earthsown motion around the Sun. But a TNO in an orbit with semi-major axis of 40 AU would complete an orbit in about 250years. If the orbit were nearly circular, the TNO would moveat a nearly constant1.44 across the celestial sphere over thecourse of a single year. In reality, most orbits are reasonablynoncircular, and so there may be significant acceleration in themotion of the TNO, even in arcs as short as 1 yr (e.g., theTNO 2000 CR105, discussed in Millis et al. 2002 and Gladmanet al. 2002).

    This paper documents a method to better constrain the likelypositions of TNOs with short discovery arcs at some futureepoch, in order to enhance the probability of recovering theobjects. Our source code will be made publicly available

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    2003 PASP,115:13301339

    Fig. 1.Demonstration of the method, using Pluto observations on 2003June 1.0 and June 8.0 UT. The heliocentric equatorialX-Y plane is shown.The observed Earth-Pluto position vectors are drawn in; the dots are Plutosactual positions. At Plutos true distance from the Sun, the greatest distanceit could travel between the two observation times is shown as the circle centeredat the first position. The arrows show some possible motion vectors betweenthe two observations. Since the position vectors eventually cross, a second setof solutions at much greater heliocentric distance yields retrograde orbits. Asof this writing, no known TNO has a retrograde orbit.

    on-line,2 and we hope to have a Web-enabled version of thecode by the time this article appears.

    2. THE MODEL

    We work in a Cartesian, heliocentric equatorial coordinatesystem. The model assumes that the TNO is in a bound orbit.Two observations (the first the discovery observation, and asecond observation that we assume is taken no more than1 week later) give us the vectors along which the TNO mustbe located, but not the distance to the TNO (see Fig. 1). Thefirst example we will give here is for Pluto, observed usingpositions calculated on 2003 June 1.0 UT and 2003 June 8.0UT. (Since Plutos orbit is known, we can find its celestial andthree-dimensional Cartesian coordinates at any time, and so itprovides a good example to check the method.)

    Since Pluto is indeed a TNO, at discovery, it is located atsome great distance ( ), which we assume to be at least,dET1say, 27 AU from the Earth.3 The Earths coordinates at thetimes of the first and second observations are ( , , ) andx y zE1 E1 E1( , , ). Since the Earths orbit is well known and thex y zE2 E2 E2coordinate system is well defined, simply specifying the datesof the observations allows the determination of the Earthsthree-dimensional coordinates using routines fromslalib, apackage used extensively in this work (available on theInternet.4

    We know only the projection of the TNOs position on thecelestial sphere at the moment of the first observation, not itsthree-dimensional coordinates. Since the right ascension anddeclination are known, we can calculate the TNOs geocentricequatorial coordinates relative to the unit vector. This gives usa vector with origin at Earths position at time 1 (T1), alongwhich the TNO must lie, with the form

    i i j j k k, (1)1 1 1

    where , , and are the coefficients of the unit vectori j k1 1 1( ). (We ignore diurnal aberration, which can2 2 2i j k p 11 1 1cause offsets of0.2 in observed and geocentric coordinates;this introduces errors of magnitude similar to the astrometricuncertainty. The maximum error comes when the observationsare taken at opposite ends of the night; the error is minimizedwhen both observations are taken at the same clock time, par-

    2 The distance of 27 AU was chosen arbitrarily, but the intent was to favor

    objects exterior to Neptunes orbit. As shown in the numerical integrations byDuncan, Levison, & Budd (1995) and other authors, orbits with periheliondistances35 AU are dynamically unstable on timescales of108 yr, unlessthey are located in Neptune mean motion resonances. Holman & Wisdom(1993) found that orbits between Uranus and Neptune are unstable on time-scales of roughly106 yr. Such objects are occasionally found as Centaurs,of which (2060) Chiron is the best known, whose orbits are temporarily locatedbetween those of Jupiter and Neptune. As we show later, however, our codedoes appear to work as intended for Centaurs.


    ticularly near midnight. Our observing strategy was to obtainexposures on different nights at nearly the same clock time,reducing the error due to diurnal aberration.)

    The vector for the first observation itself, now transformedto heliocentric by adding in Earths own coordinates, is givenin parametric form as

    x p x d i ,T E1 1 11

    y p y d j ,T E1 1 11

    z p z d k , (2)T E1 1 11

    where the distance parameter at the origin of the vectord p 01(i.e., at Earth) and increases away from Earth; at the trued1geocentric distance of the TNO, .d p d1 ET1

    The explicit assumption of the distance to the TNO at agiven time (say, setting the Earth-TNO distance at the timed1of the first observation to 27 AU) allows us to identify oneunique possible location (in Cartesian geocentric equatorial co-ordinates) of the TNO along the vector.

    We can construct the geocentric Earth-TNO vector at thesecond observation time in the same way. The position vectorfor the second observation is

    i i j j k k, (3)2 2 2

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    2003 PASP,115:13301339

    where , , and are lengths of the three coefficients of thei j k2 2 2unit vector ( ).2 2 2i j k p 12 2 2

    The sticking point is setting the Earth-TNO distance at thesecond observation. This is where we use the requirement thatthe TNOs orbit be bound. Given the Sun-TNO distance at time1, (determined assuming some Earth-TNO distance ),d dST1 1the escape velocity is just . Multiplying1/2v p (2GM /d )Sun ST1escthis by the elapsed time between observations 1 and 2 givesus the maximum distance the TNO could possibly have traveledsince the initial observation, if it is in fact ( just barely) bound.

    We can visualize the situation as follows. We know the TNOmust lie on some vectorET1 during the initial observation, attime 1. We do not know the location along the vector, but wecan set it to be some interestingly large geocentric distance.We know that the object must be found at some later time (T2)within a sphere centered at the original coordi-(x , y , z )T T T1 1 1nates, the radius of the sphere being set by the product of theescape velocity and the elapsed time between the twoobservations.

    We further know that at the time of the second observation,the TNO is again along a vector extending out from Earth, butagain we do not know the distance. However, we now knowthat the TNO must lie within the sphere whose dimensionswere found above. The Earth-TNO vector at time 2 (ET2) mustintersect the sphere at two points. If we can find those twopoints, then we will have narrowed the possible range of TNOpositions down to a line segment within the sphere, joining thetwo points of intersection (see Fig. 1).

    The radius (r) of the sphere defined by the maximum distancethe TNO could travel is defined by , wherer p v DT DT pesc

    is the time difference between the two observations.T T2 1The equation of the sphere is

    2 2 2 2(x x ) (y y ) (z z ) r p 0, (4)T T T T T T2 1 2 1 2 1

    wherex, y, z are the possible coordinates of the TNO at times1 and 2.

    Substituting in the parametric equations for the TNOs lo-cation along the Earth-TNO vector at time 2,

    2 20 p (x i d x ) (y j d y )E2 2 2 T E2 2 2 T1 1

    2 2(z k d z ) r (5)E2 2 2 T1

    2 2 2p i d 2i d (x x ) (x x )2 2 2 2 E2 T E2 T1 1

    2 2 2j d 2j d (y y ) (y y )2 2 2 2 E2 T E2 T1 1

    2 2 2 2k d 2k d (z z ) (z z ) r (6)2 2 2 2 E2 T E2 T1 1

    2 2 2 2p d (i j k ) d [2i (x x )2 2 2 2 2 2 E2 T1

    22j (y y ) 2k (z z )] [(x x )2 E2 T 2 E2 T E2 T1 1 1

    2 2 2(y y ) (z z ) r ]. (7)E2 T E2 T1 1

    This is a simple quadratic equation in , whose rootsd2and can be found in the usual way.d d2,low 2,high

    We are helped by the fact that , since they2 2 2i j k p 12 2 2are the components of a unit vector. The Earths coordinatesare known, the TNOs coordinates are known from the positionvector and the assumed distance at time 1, andr depends onthe assumed distance at time 1.

    Solving for the two roots of and using the parametricd2TNO position equation (2) gives the coordinates of the inter-sections of the circle of greatest motion and the vectorET2.This defines a line segment that is the locus of possible positionsof the TNO at time 2. Any given point along that line segmentcan be connected to in order to create a seven-(x , y , z )T T T1 1 1dimensional vector . This vector can be (x , y , z , x, y, z, T )T T T 11 1 1to recover the corresponding possible orbit. Universal elementsare used byslalib for orbit computations (see Sterne 1960 ora good textbook on methods of astrodynamics). Because a realobject in a real orbit will experience acceleration, and w...


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