Pencil-Beam Surveys for Trans-Neptunian Objects: Novel Methods for Optimization and Characterization

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  • Pencil-Beam Surveys for Trans-Neptunian Objects: Novel Methods for Optimization andCharacterizationAuthor(s): Alex H. Parker and J. J. KavelaarsSource: Publications of the Astronomical Society of the Pacific, Vol. 122, No. 891 (May 2010),pp. 549-559Published by: The University of Chicago Press on behalf of the Astronomical Society of the PacificStable URL: .Accessed: 25/05/2014 23:41

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  • Pencil-Beam Surveys for Trans-Neptunian Objects: Novel Methodsfor Optimization and Characterization

    ALEX H. PARKERDepartment of Astronomy, University of Victoria, Victoria, BC V8P 5C2 Canada;


    J. J. KAVELAARSHerzberg Institute of Astrophysics, National Research Council of Canada, Victoria,

    BC V9E 2E7, Canada

    Received 2010 January 7; accepted 2010 March 1; published 2010 April 14

    ABSTRACT. Digital co-addition of astronomical images is a common technique for increasing signal-to-noiseratio (S/N) and image depth. A modification of this simple technique has been applied to the detection of minorbodies in the solar system: first stationary objects are removed through the subtraction of a high-S/N template image,then the sky motion of the solar system bodies of interest is predicted and compensated for by shifting pixels insoftware prior to the co-addition step. This shift-and-stack approach has been applied with great success in di-rected surveys for minor solar system bodies. In these surveys, the shifts have been parameterized in a variety ofways. However, these parameterizations have not been optimized and in most cases cannot be effectively applied todata sets with long observation arcs due to objects real trajectories diverging from linear tracks on the sky. Thispaper presents two novel probabilistic approaches for determining a near-optimum set of shift vectors to apply toany image set given a desired region of orbital space to search. The first method is designed for short observationalarcs, and the second for observational arcs long enough to require nonlinear shift vectors. Using these techniquesand other optimizations, we derive optimized grids for previous surveys that have used shift-and-stack approachesto illustrate the improvements that can be made with our method, and at the same time derive new limits on the rangeof orbital parameters these surveys searched. We conclude with a simulation of a future application for this approachwith LSST, and show that combining multiple nights of data from such next-generation facilities is within the realmof computational feasibility.


    Detecting faint objects in the solar system is a technicallychallenging problem. The approaches taken in the past canbe broken into two basic categories: interframe source detectionand linking, and the shift-and-stack approach. Interframe de-tection, where transient sources are detected in individual expo-sures and motion is linked between multiple exposures, iswidely employed and favored in planned proposed surveys likePan-STARRS (Denneau et al. 2007) and Large Synoptic SurveyTelescope (LSST; Axelrod et al. 2009) due to its simplicity androbustness. This method, however, does not exploit imagingdata to the fullest extent possible, because detections are subjectto the completeness limit of individual frames.

    Since solar system objects are moving across the sky, theirimages trail and the noise contribution from the sky backgroundincreases. Trailing losses create an upper limit to individualframe exposure times for efficient detection of solar system ob-jects. It is advantageous to combine multiple exposures, eachshort enough that trailing effects are negligible. Co-addition

    is routinely done to improve the S/N of stationary sourcesand to remove cosmic ray events, but in order to apply it tothe detection of moving objects, several other steps must be per-formed. First, contamination from stationary sources can be re-moved by subtracting a high-S/N template image from eachimage in the set. Next, the sky motion of the objects of interestmust be predicted and this motion compensated for by shiftingeach images pixels in software. When the images are then com-bined, only the flux from sources that moved at the predictedrates will be constructively added.

    This method first appeared in literature in Tyson et al. (1992),though details of the application were limited. An early detaileddescription of this technique was presented by Cochran et al.(1995). A set of images from the Hubble Space Telescope(HST) Wide Field Planetary Camera 2 (WFPC2) were com-bined to allow statistical detection of very faint trans-Neptunianobjects (TNOs). The sky motion of the sources was parameter-ized as angular rates ( _) and angles on the sky (). Gladmanet al. (1997) used the same technique in order to combine a se-ries of images from the Palomar 5 m telescope in the hope of


    PUBLICATIONS OF THE ASTRONOMICAL SOCIETY OF THE PACIFIC, 122:549559, 2010 May 2010. The Astronomical Society of the Pacific. All rights reserved. Printed in U.S.A.

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  • constraining the size distribution of Kuiper Belt Objects (KBOs)by discovering objects with smaller radii than had been detectedin most previous wide-area surveys that employed interframedetection. This survey was followed by further searches fromKeck (Luu & Jewitt 1998; Chaing & Brown 1999), CTIO (Allenet al. 2001 and 2002; Fraser et al. 2008), Palomar (Gladmanet al. 1998), VLT and CFHT (Gladman et al. 2001), which em-ployed the same technique to varying degrees of success. Re-cently, Fraser & Kavelaars (2009) and Fuentes et al. (2009) bothused similar techniques to search Subaru data for KBOs as faintas R 27.

    All of these surveys had observational arcs less than 2 days inlength. Over periods this short, the motion of outer solar systemsources can be well approximated (with respect to ground-basedseeing) by the linear trajectories that the method these surveysimplemented implicitly assumes. For longer arcs or higher-resolution data, however, the approximation of a linear trajec-tory is no longer adequate. Bernstein et al. (2004) employed amore advanced approach, which they called digital tracking,to extremely high-resolution HST Advanced Camera for Sur-veys (ACS) data. Even in arcs on the order of a day in length,nonlinear components of motion were nonnegligible relative tothe ACS psf. In order to stack frames such that distant movingsources add constructively in this regime, it becomes necessaryto generate nonlinear shift vectors via full ephemerides fromorbits of interest. They stacked arcs up to 24 hr in length usingshift vectors parameterized as a grid over three parameters: twolinear components of motion and heliocentric distance d. Bysampling densely over this grid, they were able to recover orbitswith 25 AU d and i < 45, and detect objects as faint asR 28:5. However, this dense sampling, somewhat uncon-strained discovery volume, and extremely small psf required7 105 shift vectors, which translated into 1014 pixels to besearched. Bernstein et al. found that stacking more than 24 hr oftheir data became computationally prohibitive.

    In the age of ground-based gigapixel CCD mosaic imagers,how can data obtained over long arcs be efficiently exploited tosearch for faint sources? In the interest of exploring massivedata sets with long observation arcs to as deep a limit as pos-sible, we have created a simple probabilistic method for gener-ating an optimized set of shift vectors for any imaging surveyand any interesting volume of orbital element space. By gener-ating shift vectors in a probabilistic way, exploring an explicitlydemarcated region of orbital parameter space becomes straight-forward. The targeted region of orbital parameter space issearched with uniform sensitivity, simplifying the accurate char-acterization of any detected sample of objects. This method alsoavoids searching nonphysical or uninteresting parameter space,thereby maximizing scientific return for minimum computa-tional cost.

    In 2.2, we define the maximum tracking error, the basicparameter that defines the density of a grid. Section 2 describesprevious analytical methods for search-grid parameterization,

    our probabilistic approach, and a grid-tree structure for optimiz-ing multiple-night arcs. Section 3 compares the efficiency of ourmethod to those used in previous surveys, and in 4 we show anexample of application of this method to the kind of data thatmay be acquired by the LSST.


    2.1. Estimates of On-Sky Motion

    Previous on-ecliptic surveys (e.g., Fraser & Kavelaars 2009)have used simple analytical estimates for the apparent on-skyrates of motion for distant solar system objects in order to gen-erate shift rates. For an on-ecliptic observation of a field de-grees from opposition, the range of on-sky angular rates ofmotion _ and angle from the ecliptic for a distant object atheliocentric distance d, and geocentric distance with inclina-tion i and eccentricity e, can be approximated by the vector ad-dition of the reflex motion of the object due to Earths motion(inversely proportional to ) and the intrinsic on-sky orbitalmotion of the object by Keplers law (proportional to d3=2)

    _ 148fcos1 vdd32 cosi2 vdd32 sini2g12 hr1 (1)

    arcsin148vd _1d32 sini; (2)where vd

    ffiffiffiffiffiffiffiffiffiffiffi1 ep , representing the fraction of the mean an-

    gular velocity of an orbit at pericenter (e) or apocenter (e),compared to a circular orbit at heliocentric distance d.

    This motion can also be parameterized as two components ofangular rates of motion, one parallel and one perpendicular tothe ecliptic. Rearranging equation (1) into parallel and perpen-dicular components, we find

    _ 148cos1 vdd32 cosi hr1 (3)

    _ 148vdd32 sini hr1: (4)The _ and parameterization is convenient for visualizationpurposes, but in 2.2 we show that the _ and _ parameter-ization is more appropriate for generating a search grid.

    The angular rate of motion is lowest (for a given d and d) for i 0 orbits at pericenter, since vd is at its highestand the parallax and orbital motion vectors are antialigned.1 Thehighest angular rate of motion at opposition is reached in twocases, depending on the maximum inclination and eccentricity

    1 This assumes that the objects distance and eccentricity are not such thatvd=d

    32 vin other words, that the object is not overtaking Earth at opposition.

    If this were the case, _ is lowest at i 180.


    2010 PASP, 122:549559

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  • considered (imax and emax):

    _max wheni 0; apocenter; if imax < i0i imax; pericenter; if imax > i0;


    i0 arccos

    v2q v2Q2




    and vq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 emax

    p, vQ

    ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 emax


    Given this approximation, the highest any orbit can achieveis a strong function of d. If we consider an orbit with e 1 atpericenter with i 90, and approximate d, then by equa-tion (2) it can be shown that

    max arcsin ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi


    cos2d 2

    s : (5)

    2.2. Grid Geometry and Maximum Tracking Error max

    In using digital tracking, the motions of real sources are ap-proximated by a grid or some distribution of shift- or motion-vectors. In order to quantize the effectiveness of this strategy, wedefine a maximum tracking error to be the largest possibleerror tolerated between any real objects motion and the nearestpredicted motion vector. To estimate its effect, we compare thefinal signal-to-noise ratio S=N 0 for a faint circular source withflux f and area A0 before and after a linear tracking error isadded in an image with exposure time t. The linear trackingerror adds an additional rectangular region to the area ofthe source, with width 2R (where R is the aperture radius ap-plied to the initial circular source) and length . This rectanglehas area 2R, and the total area of the blurred source is now

    A0 R2 2R A01 2


    The final signal-to-noise S=N 0 is given by


    N 0 f


    pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif skyA01 2R

    q SN0

    1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2R

    q for f f sky:(6)

    So, for a search grid with maximum tracking error max, theworst possible S/N degradation is a factor of F 1 2R


    Given a Gaussian psf with a full-width at half-maximum of ,the radius at which the S/N of a sky-dominated source is max-imized is R 0:68. If we define this as our initial apertureradius, then the maximum allowable tracking error given a de-sired limit on F becomes

    max 2 0:68F2 1: (7)

    Previous surveys have used widely varying values for max withrespect to the typical seeing of that survey, ranging from0:13. Surveys that searched data by eye were limited inthe number of shift vectors they could apply, and leveragedthe eyes robust noise discrimination to compensate for the lossin S/N due to using large max (e.g., Fraser & Kavelaars 2009).Surveys using automated detection pipelines have decreasedmax to limit their S/N loss, and leveraged massive computa-tional resources to compensate for the increased number of re-quired shift vectors (e.g., Fuentes et al. 2009).

    In order to evaluate the maximum tracking error of anysearch grid of shift vectors applied to a given data set, we de-termine the distribution of shifts from the initial image to thefinal image, where the shift for orbit k in image i is

    vk;i dk;i; dk;i cosk;0k;i k;0; k;i k;0:(8)

    The maxi...


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