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Icarus 178 (2005) 284–288www.elsevier.com/locate/icaru

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Timing asteroid occultations by photometry

Yiwei Ma a,1, Feng Yana,∗,2, Jin Zhub, Wen Koub

a High School Affiliated to Renmin University of China, No. 37 Zhongguancun Road, Beijing 100080, Chinab Beijing Planetarium, No. 138 Xizhimenwai Street, Beijing 100044, China

Received 7 May 2005; revised 20 July 2005

Available online 13 September 2005

Abstract

Observations of stellar occultation by asteroids provide opportunities to directly measure dimensions and study physical parameters of asteroi. Existingasteroid occultation timing methods either record the occultation time directly or compute the occultation time from the measurement of length. Inhis work,we present a timing method to derive the occultation time from the measurement of magnitude, termed Integration Photometric Timing (IPT), whlyutilizes CCD photometry with relatively long exposures. Applications to two events, (3) Juno’s and (980) Anacostia’s, are given to demonstrate thse andfeasibility of the IPT. The proposed method has the capability of timing asteroid occultations of faint stars because of its long exposure time. Tmoreobservable events are expected, with professional large telescopes to be involved in asteroid occultation observations, especially for the case oKuiper beltobjects (KBOs) and other distant asteroids occultation. 2005 Elsevier Inc. All rights reserved.

Keywords: Asteroid occultation; Photometry

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1. Introduction

Measurement of asteroids’ sizes is one of the most challenging tasastronomers, for angular diameters of asteroids are too small to besured directly by classical methods. Various techniques were devisaddress this issue in the past decades. Among all the ground-based mof determining the dimensions of asteroids, observation of stellar octation by asteroids proved to be a powerful and direct technique(Millisand Dunham, 1989). Giving the duration of the occultation observed somwhere, lower limits of the asteroids’ diameters can be easily calculfrom the observed chord across the asteroid; cross sections of the ascan be revealed by the immersion and emersion time recorded at mple sites(Smart, 1977; Wasserman et al., 1979); presuming an ellipsoidmodel, the asteroid’s 3D shape can even be calculated from asterocultation observations(Drummond and Cocke, 1989; Dunham et al., 19Sato et al., 2000). Asteroids’ albedos are derived from its relation to asoids’ diameters and absolute magnitudes(Harris and Lagerros, 2002). Mostnotably, results of occultation observations were also used to modify

* Corresponding author. Present address: Department of Electricalneering, Tsinghua University, Beijing 100084, China.

E-mail address: frankyanfeng@126.com(F. Yan).1 Present address: Department of Precision Instruments and Mech

ogy, Tsinghua University, Beijing 100084, China.2 Address for correspondence: Room 605, Building 6, No. 33 Bawa

Haidian District, Beijing 100089, China.

0019-1035/$ – see front matter 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.icarus.2005.07.015

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rameters in other diameter derivation models, such as the widely apstandard thermal model(Lebofsky et al., 1986).

Basically, when observing an occultation event, one usually tries toserve the change of the combined brightness of the star and the asat the moment of immersion and emersion in order to time disappance/reappearance time or duration of the occultation event. Currentlyeral observational techniques have been commonly adopted by bothteur and professional campaigns, such as photoelectric timing and drifttiming. We summarized these existing observational methods intocategories: Direct Timing, Instant Lightcurve Timing, and Trail Timing. Dtailed description is given below.

• Direct Timing: The easiest way of timing an occultation event isvisual observation, which we refer to as Direct Timing. This methis widely used by amateur astronomers, since many of them arwell equipped. Immersion and emersion time is usually markedartificial signals so that a sound recorder can record them, or, sirecorded by a stopper. This method is the least reliable for two rea(1) Timing cannot be precisely done because of observer’s rea“lag.” Usually, 0.5 s is subtracted in order to correct the timing whis called the standard personal equation, while other equations cabe specified by the observer. (2) When the change of the brightnethe occulted star is not significant enough or the star is too faint,can hardly figure out if the star is being occulted(Millis and Elliot,1979).

• Instant Lightcurve Timing: This is the most preferred methodder many circumstances. High time-resolved equipments, incluvideo camera, photoelectric photometer, and frame transfer CCD

Note / Icarus 178 (2005) 284–288 285

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Fig. 1. The lightcurve model of the Integration Photometric Timing method. When the occultation occurs, the flux sharply drops to the astero

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be used to record the real-time lightcurve. Popularity of timingcultation by video camera is growing, especially among amateutronomers. The primary reason is that the equipment is easy to acand gives promising timing precision by frame-by-frame playbackBy using photoelectric photometer or frame transfer CCD(Buie etal., 1993; Dunham et al., 1985), one cannot only time the occultation event, but also various features, such as dust band, doubleor potential asteroid satellites (e.g.,Arlot et al., 1985; Bus et al., 1996Elliot et al., 1995). The sensitivity of these equipments is also surior to that of Direct Timing and video camera timing. But the dadvantages of these equipments, which are evident, are the coscomplexity(Millis and Dunham, 1989).

• Trail Timing: Due to their slow image read-out speed (deadtimrapid-varying event like occultation cannot be recorded consecutby most astronomical CCDs. However, if the telescope’s drive is turoff (drift-scan mode), the Earth’s rotation makes the star light leatrail on the CCD image at the sidereal rate. Hence, when an asterocultation happens, part of the trail would be obscured because of bing of the star light by the asteroid. The name, Trail Timing, is obvifor the method, for by measuring the starting and ending positionthe obscured part on a pixel-to-pixel basis, immersion/emersionand occultation duration can be easily calculated(Fors et al., 2001).3

Popularity of astronomical CCDs among amateur astronomers icreasing in recent years, so the use of Trail Timing method is becobroader. Furthermore, this method is most ideal for CCD observemain-belt and near-Earth asteroid occultation cases.

Under the most favorable conditions, magnitude up to 14 of the occustar can be observed with Instant Lightcurve Timing and Trail Length Ting methods using an amateur telescope with moderate aperture. In pralimit magnitude might be much lower than this estimation due to varidisturbances, such as light pollution, clouds and poor seeing, etc.

In this paper, a new timing method, termedIntegration PhotometricTiming, is introduced. The idea behind the method is rather intuitive:IPT takes a single exposure covering the time span of the whole ocction while the telescope is set in stare mode. The magnitude measuredthe total flux contributed by the asteroid and the occulted star in this fr(we refer to this frame asoccultation frame hereinafter) will drop comparedwith the magnitude prior to the occultation. Then occultation durationbe calculated from the difference of the magnitudes. For occultation evwith relatively long duration, the immersion and emersion times can be

3 For details of this technique on asteroid occultations, please refthe webpage by John Broughton,http://www.users.bigpond.com/reedycrdristscantiming.htm.

,

culated by two exposures covering the beginning and ending time ooccultation, respectively.

We shall first describe the proposed timing method in detail, andthe error analysis. Applications to two occultation events, (3) Juno ocing HIP 92918 and (980) Anacostia occulting 2UCAC 33040335, are gin the following section. Finally, prospects of the proposed method arecussed.

2. The Integration Photometric Timing method

2.1. Calculating the duration

In order to calculate the duration in the occultation frame, we mmake three basic assumptions. The first is that the occulted star is sand non-variable and it can be treated as a point emitter of light. Theond assumption is that the occulting asteroid is single, airless, and weconsider its magnitude as constant during the course of exposing the otation frame. When the asteroid’s magnitude varies slowly, according tfirst assumption, the combined magnitude of asteroid and star can besidered to be constant shortly before and after the occultation, say seminutes. Thus, the magnitude of the asteroid can be measured in aobservation instead of deriving from the lightcurve. The last assumptiothat the occulting asteroid is significantly larger than the Fresnel scathe topocentric distance∆, i.e., a � (λ∆/2)1/2, wherea is the radius ofthe asteroids,λ is the wavelength of the observations. So effect of Fresdiffraction is negligible under this assumption.

The lightcurve of the occulted star on an occultation event can be meled as follow according to the assumptions presented above: at the mof immersion, the star light is fully occulted by the asteroid promptly, athe flux of star plus asteroid is replaced by the asteroid’s flux until the octation is finished. At the moment of emersion, the occulted star reappepromptly so that the flux immediately turns back to the flux of star plusteroid. Apparently, this model yields a right concave lightcurve with randerrors (Fig. 1).

We denote the star and asteroid’s magnitudes and fluxes by(ms,Fs)

and (ma,Fa), respectively, and their combined magnitude and flux(mas,Fas), obviouslyFas= Fs + Fa. When the asteroid and star are sficiently close,mas can be directly measured in a single aperture. Finacombined magnitude and flux of the star and the asteroid measuredoccultation frame are referred to as(mo,Fo). According to the lightcurvemodel, we have

(1)Fo = [Fas(T − d) + Fad

]/T ,

whereT andd are the total exposure time and the duration of occultatiothe occultation frame, respectively. Then the duration of occultation in

286 Note / Icarus 178 (2005) 284–288

Fig. 2. This figure illustrates the lightcurve of an occultation event with relatively long duration. By two exposures covering the beginning and ending phasesof the occultation, immersion and emersion time can be calculated.

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occultation frame can be derived as:

(2)d = (Fo − Fas)T /(Fa − Fas) = (Fas− Fo)T /Fs.

Brightness and magnitude are related by the well-known Pogson’smula: m = −2.5 logF + C, whereC is a constant value. Inserting thformula into(2), we express Eq.(2) in magnitude form:

(3)d = T (100.4(ms−mas) − 100.4(ms−mo)).

When the magnitudemas, as measured from the total flux of the asterand the star, does not provide promising accuracy or whenma becomesavailable from directly measurement,ma can be used instead ofmas, whichresults a minor modification of Eq.(3):

(4)d = T(100.4(ms−ma) − 100.4(ms−mo) + 1

).

Derivations of Eqs.(3) and (4)established the basic calculating fomulae for the Integration Photometric Timing method. From the obtaduration of the occultation event, a lower limit of the asteroid’s diamei.e., the recorded chord across the asteroid, can be easily derived byapproximation of its motion.

2.2. Timing immersion and emersion

When the duration of occultation is long enough, disappearancereappearance times can also be calculated by two occultation framesing beginning and ending phases of the occultation. As illustrated inFig. 2,exposure time and calculated duration in the frame taken at beginningare denoted byTb anddb. If the exposure starts attb then disappearanctime is tb + Tb − db. Reappearance time can be obtained byte + de like-wise, wherete is the starting time of the exposure at the ending phase ooccultation, andde is the duration calculated from this frame. In orderobtain disappearance and reappearance time, the prerequisite is bothning and ending occultation frames should not cover the whole timeof the occultation. So, it is better if the duration of the occultation is loand the prediction is accurate. Another problem in applying this schemthat the read-out time of most astronomical CCDs is relatively long cpared with the duration of the occultation event, therefore either the staor ending occultation frame may not be recorded. One of the possibletions is to utilize fast CCDs (e.g., multichannel or frame transfer CCDorder to increase read-out speed. Another solution is to use two telesat the same site. The advantage of operating two telescopes is that thoccultation frames can be taken independently so that it is possible tolap the exposure processes in case of the duration is too short and thecannot be recorded in consecutive frames.

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2.3. Errors

Next, we calculate the error of the duration through error propagaanalysis.σ( ) is used to denote the variances of observational values.

The calculated durationd is a function of four variables. But error iexposure timeT can usually be neglected by rigorous timing of the shu(Massey, 1997). So the error sources in(3) arems, mo, andmas. Taking thepartial derivatives, the propagated variance of the duration is:

(5)σ2(d) = (∂d/∂ms)

2σ2(ms) + (∂d/∂mo)2σ2(mo) + (∂d/∂mas)2σ2(mas).

We denote 100.4(ms−mas) = 2.51(ms−mas) by η and 100.4(ms−mo) =2.51(ms−mo) by ξ , respectively. Multiple factor 0.4 ln 10= 0.92 is denotedby A. Substituting the partial derivatives by their explicit form in(5), wehave the calculating equation of the error:

(6)σ2(d) = A2T 2[(η − ξ)2σ2(ms) + ξ2σ2(mo) + η2σ2(mas)

].

Since the IPT calculates durations through direct photometry of sobtaining small uncertainty in the calculated duration is equivalent toducing photometric errors giving certain exposure time, as seen in Eq(6).So, good photometric quality of the atmosphere, high quantum efficiCCDs with low read-out noise and large aperture telescopes are prefe

3. Applications of the proposed method

3.1. The (3) Juno event

Stellar occultation by (3) Juno is one of the earliest observed astoccultations. Occultation result reported byMillis et al. (1981) revealedJuno’s effective diameter of 267± 5 km. On July 20, 2004, Juno occultea bright V = 7.46 mag star, HIP 92918= SAO 142809= HD 175518.Predicted maximal duration is 17.3 s on the central line(Dunham, 2004).4

Observation was conduct in the Wuyuan area, Inner Mongoliatonomous Province, China. A Nikon D70 digital camera attached o80-mm portable telescope was used as the CCD. In order to compensauncertainties in mid-time and duration, a relatively long exposure timthe occultation frame, 60 s, was chosen. Aperture photometry packagPHOT in the IRAF software was used to perform photometry for this e

4 The update is given by Steve Preston athttp://www.asteroidoccultationcom.

Note / Icarus 178 (2005) 284–288 287

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as well as the Anacostia’s event. The comparison star is a nearby sta93006 withV = 8.4 mag.

mas is determined by averaging the combined magnitude of JunoHIP 92918 measured in 5 frames which were continuously taken inperiod of 10 min before the beginning of the occultation. One aperis used to measure the combined magnitude, for they are close orsolved at the time. On the date of observation, we did not have muchto wait for the asteroid and the star to be well separated in order to othe star magnitude. So in November 2004, HIP 92918 was imaged ausing the same instruments, and its magnitude was determined by aving magnitudes in 5 well-exposed frames. The final data reduction fothatmas, ms, andmo relative to the comparison star are−0.86± 0.0124,−0.79± 0.0160, and−0.52± 0.0186 mag, respectively. Calculation yielda duration of 17.0± 1.1 s, which is consistent with the result reported bvisual observer who claimed a duration of 17.6 s. We should point outthe calculated error of the duration is underestimated, because only pstatistics are considered.

3.2. The (980) Anacostia event

On February 12, 2005, (980) Anacostia occulted a 12.4 mag (UCmagnitude5) star tabulated in the 2UCAC catalogue as 2UCAC 330403Prediction provided by Steve Preston showed that the shadow track waBeijing area. The width of projected shadow track was about 104 km.cepting (980) Anacostia’s diameter of 86.2 km, occultation would last 5on the central line. The approximate occultation mid-time is 13:56:22for our observation site which is about 15 km south of the predicted celine. To avoid uncertainty in the predicted mid-time as well as insufficS/N for the comparison star, 14 s was chosen to be the exposure timeoccultation frame, and the exposure of that frame started at 13:56:15 U

A total 13 frames including the occultation frame were taken that nusing the 60/90-cmf/3 Schmidt telescope located at Xinglong StationNational Astronomical Observatories, Chinese Academy of Sciencestelescope is equipped with a Ford 2048× 2048 CCD with an image scalof 1.67′′/pixel attached on the main focal plane. An Intermediate-bandter centered at 6600 Å with FWHM of 480 Å (designated asi in the BATCfilter system) was used in the observation(Fan et al., 1996). Both the aster-oid and the star are rather bright for our facilities. Predicted magnitud(980) Anacostia isV = 12.5 mag, but it appeared to be brighter (appro0.35 mag) than the occulted star through our filter. The data are redin a standard manner including bias subtraction and flat-field calibrawith an automatic data reduction system named PIPELINE-I develofor the BATC multicolor sky survey program. The comparison star iV = 10.89 mag star 5′ away from the occulted star, which is tabulatedTYC 229-2175-1 in the Tycho2 star catalogue. The occulted star’s matudems is determined by averaging magnitudes of the occulted star froframes exposed after the occultation. In the frames exposed shortly band after the occultation, combined magnitude of the star and the asmas is measured. The adoptedmas in the calculation is determined by averaging the combined magnitudes in the previous and successive framthe occultation frame. Finally,ms, mas, andmo relative to the comparisonstar are found to be 1.817± 0.011, 0.867± 0.006, and 0.947± 0.025 mag,respectively. Applying Eqs.(3) and (6)to these data, the calculated durtion is 2.4± 0.7 s, corresponding to a lower limit of the asteroid’s diameof 37± 11 km. The result is inconsistent with the prediction. Probableplanation is that the shadow track had a shift because of the uncertainthe prediction.

4. Discussions

The major difference between the method outlined here and the eximethods is that other methods more or less depend on the time resolutthe detector while the Trail Timing method shifts the time resolution to s

5 See README file in the UCAC2 cdroms for detail.

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tial resolution on the CCD. In contrast, the IPT method times occultaby variation of the occulted star’s luminosity. Due to the relatively longposure time of the IPT, more occultation events involving fainter starsbe observed. Furthermore, as the frequency of observable events incropportunities for a fixed telescope to observe occultations by a specasteroid also increase. As long as equipped with a CCD, the use of phoetry allows any telescope to time occultation without any modificationcurrent facilities. As a result, ground-based large telescopes could bein asteroid occultation observations, especially the KBO occultations.

Within the last decade, interest in the Kuiper belt objects continugrew because they have undertaken the least thermal process andthe most primitive information on the young Solar System. In additthey are probably progenitors of short-period comets and Centaurs(Jewitt,1999). Several occultation detection programs have been proposed into investigate the population of the KBOs(Chen et al., 2003; Cooray, 200Roques and Moncuquet, 2000). Inasmuch as KBOs have distant orbits raning from 30 to 50 AU, thermal measurement of their diameters and albare only available for several bright ones(Altenhoff et al., 2001, 2004Margot et al., 2002; Jewitt et al., 2001). Besides the difficulty in thermameasurements because of KBOs’ low radiation at long wavelength,measurements, as well as the size of (50000) Quaoar measured byble Space Telescope(Brown and Trujillo, 2004), are model-dependent. Thstellar occultation technique provides an option to directly estimate tdistant objects’ sizes. However, due to the slow motion of KBOs on thelestial sphere, occultation events of bright stars by these objects arerarer than their counterparts, i.e., main belt asteroids and near-Earthoids. The advantage of the IPT allowing timing occultations of faint smakes it possible to time more occultation events by KBOs, involving fstellar sources.

Timing KBO occultation by the Integration Photometric Timing methmay produce either duration or immersion/emersion times. In order toplify the calculation, the Earth’s and the KBO’s orbits are assumebe circular, occultation durationd is approximated byd ≈ 2a/(v cosω −vR−1/2), wherev is the Earth’s velocity,ω is the angle from oppositionto the object’s current position,R is the radius of the object’s orbit in AU(Brown and Webster, 1997). For a canonical KBO with a 600-km diameter, the longest (on the central line) occultation duration range from∼20 sat opposition to several hundreds of seconds near quadrature which aus to time immersion and emersion time at different sites. If two or mchords are recorded, circular solutions to the cross sections can be obfor short duration events. The diameter can also be estimated from agle chord with accurate measurement of the impact parameter as desin Olkin et al. (1996). Stellar occultations by distant objects Pluto, CharTriton, and Chiron have been successfully observed in turn(Elliot and Kern,2003), but there is no report on successful KBO occultation observationfar. KBO occultation observation is limited by the precision of predictcurrently. Due to their faintness, accurate astrometry of these distanjects requires the assistance of large telescopes in order to generate pephemerides. Path refinement by last-minute astrometry is also impofor precise predictions.

Acknowledgments

We deeply thank Jian Gao and Huan Meng for their valuable discussand suggestions, Xiaoyun Ma for his visual observation in the Wuyuanservation, Xiang Zhan for his help with the instruments, Mike Dotsonhis English language advice, Keke Fan and Bin Luo for their supportguidance. Sincere thanks also to David J. Asher and an anonymous revfor their suggestions to improve the manuscript. This work is supportethe National Natural Science Foundation of China (Grant No. 1037300

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