Photometric study for near-Earth asteroid (155140) 2005 UD
PPhotometric study for near-Earth asteroid(155140) 2005 UD
Huang, J.-N. a,b,c , Muinonen, K. d,e , Chen, T. f , Wang, X.-B. a,b,c, ∗ a Yunnan Observatories, Chinese Academy of Sciences, Kunming, 650216, China b School of Astronomy and Space Sciences, University of Chinese Academy of Sciences,Beijing, 100049, China c Key Laboratory for Structure and Evolution of Celestial Objects, CAS, Kunming,650216, China d Department of Physics, University of Helsinki, Gustaf H¨allstr¨omin katu 2, P.O. Box 64,FI-00014 U. Helsinki, Finland e Finnish Geospatial Research Institute, Geodeetinrinne 2, FI-02430 Masala, Finland f Corona borealis Observatiories, Ali, China
Abstract
The Apollo-type near-Earth asteroid (155140) 2005 UD is thought to be amember of the Phaethon-Geminid meteor stream Complex (PGC). Its basicphysical parameters are important for unveiling its origin and its relationshipto the other PGC members as well as to the Geminid stream. Adopting theLommel-Seeliger ellipsoid method and
H, G , G phase function, we carryout spin, shape, and phase curve inversion using the photometric data of2005 UD. The data consists of 11 new lightcurves, 3 lightcurves downloadedfrom the Minor Planet Center, and 166 sparse data points downloaded fromthe Zwicky Transient Facility database. As a result, we derive the pole so-lution of (285 ◦ . +1 . − . , − ◦ . +5 . − . ) in the ecliptic frame of J2000.0 with the ∗ Corresponding author: Xiaobin [email protected], [email protected]
Preprint submitted to Planetary and Space Science August 27, 2020 a r X i v : . [ a s t r o - ph . E P ] A ug otational period of 5 . a > b > c ) is estimated as b/a = 0 . +0 . − . and c/a = 0 . +0 . − . . Usingthe calibrated photometric data of 2005 UD, the H, G , G parameters areestimated as 17 . +0 . − . mag, 0 . +0 . − . , and 0 . +0 . − . , respectively. Cor-respondingly, the phase integral q , photometric phase coefficient k , and theenhancement factor ζ are 0.2447, -1.9011, and 0.7344. From the values of G and G , 2005 UD is likely to be a C-type asteroid. We estimate the equiva-lent diameter of 2005 UD from the new H -value: it is 1.30 km using the newgeometric albedo of 0.14. Keywords:
Near-Earth-Asteroid, Photometry, Shape inversion, Phasefunction
1. Introduction
Near-Earth asteroids (NEAs) derive from the main belt of asteroids andare composed of planetesimal material remaining from the early stages of theSolar System. They contain important information about the Solar System’sformation and evolution. Thus, the physical study for NEAs can allow usto know more about the history of the Solar System, including that of theasteroids. Asteroid 2005 UD has been discovered by the Catalina Sky Surveyin 22 October 2005, and, according to its orbit, it has been classified as anApollo-type NEA. Together with (3200) Phaethon and 1996 YC, 2005 UDbelongs to the Phaethon-Geminid stream Complex, briefly PGC.Ohtsukaet al. (2006) investigated the orbital evolution of 2005 UD and Phaethon,and suggested the object may be a split nucleus of Phaethon. As a member2f PGC, the physical properties of 2005 UD should provide information aboutits origin and relationship to the other PGC members and to the Geminidmeteors.Several groups have carried out physical studies of 2005 UD. Ohtsukaet al. (2006) inferred that 2005 UD is a km-size object. Jewitt and Hsieh(2006)’s data of 2005 UD showed a periodic brightness variation of 5.2492 hwith an amplitude of 0.4 mag. They also estimated the diameter of 2005 UDto be 1 . ± . .
11 (the geometricalbedo of Phaethon). Kinoshita et al. (2007) determined the rotation pe-riod for 2005 UD, and found variation of color indices with rotational phase.They derived a rotation period of 5.23 h with a lightcurve amplitude of0 . ± .
02 mag. Their multi-band photometric observations suggested that2005 UD is of F or B type. Later, Kasuga and Jewitt (2008) suggested that2005 UD is a C-type asteroid, based on the color index. Kasuga and Jewitt(2008) determined the absolute magnitude of 2005 UD to be 17 . ± .
03 mag,and derived a diameter of 1 . ± . . ± .
02. Je-witt (2013) re-determined the surface color indices of 2005 UD and estimatedits absolute magnitude to be 17.08 mag, assuming the
H, G phase functionslope of G = 0 .
15. Based on the WISE observations, Masiero et al. (2019)determined the geometric albedo of 2005 UD to be 0 . ± .
09, and deriveda diameter of 1 . ± . − . Asfor the spin and shape parameters, there is no further information preceding3ur study.For understanding the physical properties of 2005 UD, we have carriedout 11 nights of photometric observations in 2018. In addition, we downloadthe photometric data of 2005 UD from the Zwicky Transient Facility (ZTF)and Minor Planet Center (MPC) databases. ZTF is a robotic time-domainastronomical sky survey. The small Solar System bodies are important tar-gets of the survey.In total, 166 data points of 2005 UD are downloaded fromthe ZTF database.In the article, we present our new photometric observations of 2005 UDand our results of the photometric analyses of 2005 UD with the inversemethods based on Lommel-Seeliger ellipsoids (Muinonen et al. (2015)) and H, G , G phase curves (Muinonen et al. (2010)). Therefore, Sect. 2 showsthe observations and data reductions for the object, Sect. 3 introduces themethods of lightcurve inversion and phase curve inversion, together with theresults and discussion. In the last section, a summary is presented.
2. Observations and data reductions
We obtained 11 nights of photometric observations of 2005 UD in October2018 using the 30-cm telescope and a 3326 × (cid:48) . × (cid:48) . (cid:48)(cid:48) . ∼ (cid:48)(cid:48) . /min . Consequently, the telescopepointing was shifted once or twice in some nights to keep the object in theFOV.The photometric images were reduced according to the standard proce-dures with the IRAF software. The effects of bias, flat field, and dark currentwere corrected first. The cosmic rays in the images were removed properly.The magnitudes of the celestial objects in the scientific frames were mea-sured using the Apphot task of IRAF with an optimum aperture. We tried3-5 apertures ranging 2.0-2.3 times the full width at half maximum (FWHM)to find the optimum aperture giving the minimum dispersion of the lightcurvepoints.Some systematic errors in the photometric data, related to atmosphericextinction and temporal or positional changes of stars in the CCD, weresimulated with the aid of reference stars in the images using the coarse de-correlation method (Collier Cameron et al. (2006),Wang et al. (2013)) andthe SYSREM method (Tamuz et al. (2005)). The former performs a coarseinitial de-correlation by referencing each star’s magnitude to its own mean,finding small night-to-night and frame-to-frame differences in the zero point.After the coarse de-correlation, the reduced magnitudes of celestial objectscan be derived (see Eq. 1d below) by removing the biases due to the objects’mean magnitude in each night and zero-point in each frame. The lattermethod simulates the low-level systematic errors in the reduced magnitudes5y those chosen reference stars. Then those simulated low-level errors, or,say, patterns in the reduced magnitudes are removed from the reduced magni-tudes of the 2005 UD. The ratio of signal to noise of the lightcurves (reducedmagnitudes in one night) is therefore enhanced. The time stamp of eachobservation of the asteroid is corrected for the light travel time. The dis-tance effects on the asteroid’s magnitude are also corrected by the formula − r ∆).In total, 2206 data points were obtained in 11 nights. For theaim of shape inversion, the relative intensities of 2005 UD are used which arederived by normalizing the mean intensity of each lightcurve to unity.For the phase curve analysis of 2005 UD, photometric data obtained indifferent nights and/or with different filters need to be converted into thesame photometric system, e.g., the standard V-band magnitudes. In thiswork, we firstly transformed the instrumental magnitudes of celestial objectsinto the r (cid:48) band of the Carlsberg Meridian Catalogue 15 (CMC15). Therelationship between the instrumental magnitude and the r (cid:48) -band of CMC15(Eq. 2) is derived by using stable reference stars in the images. The referencestars are chosen based on a threshold in the intrinsic variance, e.g., 0.01 magin the case of 2005 UD’s observations. The intrinsic variance of the stars σ s ( i ) in a night and the variance of the observations σ t ( j ) (the variance of referencestars’ magnitudes in a image after removing their means), are output by thecoarse de-correlation method (for details, see Collier Cameron et al. (2006)).Briefly, the two variances above are estimated iteratively by minimizing thevalue of χ (Eq. 1a below), providing zero points of the stars’ magnitudes ˆ M i in a night and the zero point of each frame ˆ Z j (calculated with Eqs. 1b and6 igure 1: Ten lightcurves of 2005 UD folded with the period of 5.2340 h. Red lines arebest-fit model values. able 1: Information on the phototometric observations of near-Earth asteroid 2005 UD. Date r ∆ α Mag-V Filter N Data source (UT ) (au) (au) (deg) .
219 0 .
245 21 .
301 15 . C Ourdata .
232 0 .
252 17 .
588 15 . C Ourdata .
246 0 .
261 14 .
198 15 . C Ourdata .
286 0291 5 .
671 15 . R Ourdata .
299 0 .
302 3 .
152 15 . V, R
Ourdata .
312 0 .
315 1 .
790 15 . C R, Stephens .
312 0 .
315 0 .
964 15 . V Ourdata .
325 0 .
327 1 .
726 15 . V Ourdata .
338 0 .
341 3 .
685 15 . V Ourdata .
350 0 .
354 4 .
642 16 . C R, Stephens .
350 0 .
354 5 .
414 16 . R Ourdata .
363 0 .
369 6 .
455 16 . C R, Stephens .
363 0 .
369 7 .
130 16 . R Ourdata .
375 0 .
383 8 .
827 16 . R Ourdata
ZT F
Facility
Note that r and ∆ are heliocentric and topocentric distances of the asteroid, α is thesolar phase angle, Mag-V is the mean of the observed V-band magnitudes in a night, Nis the number of data points, and ZT F
F acility refers to IRSA, Spitzer, WISE, Herschel,Planck, SOFIA, IRTF, IRAS, and MSX. M i and ˆ Z j are re-computed when we have new valuesfor σ s ( i ) and σ t ( j ) . The iterative procedure is ended when the four quantitiesabove no longer change significantly. For a given reference star i , the errorsin the reduced magnitudes r i,j in a night, j denoting the index of a frame, arefinally random, essentially realizations of white noise. The following equation8ives a mathematical description of the procedures outlined above: χ = (cid:80) ( m ij − ˆ M i − ˆ Z j ) σ ij + σ s ( j ) + σ t ( i ) , ( a )ˆ M i = (cid:80) i ( m ij − ˆ Z j ) ∗ w ij w ij , w ij = σ ij + σ t ( j ) , ( b )ˆ Z j = (cid:80) i ( m ij − ˆ M i ) ∗ u ij u ij , u ij = σ ij + σ s ( i ) , ( c ) r ij = m ij − ˆ M j − ˆ Z i . ( d ) (1)Here m ij represents the observed magnitude of star i in frame j , and σ ij isthe corresponding observational uncertainty.The magnitude zero points of selected reference stars in a night ˆ M i areapplied to fit the relationship of Eq. 2:ˆ M i ( C/R/V ) = M ( C/R/V )0 + M ( r (cid:48) ) i + k ( C/R/V ) × ( J i − Ks i ) (2)In detail, the parameters M ( C/R/V )0 and the color index coefficient k ( C/R/V ) in Eq. 2 are fitted by comparing the ˆ M i values of selected stars to their values M ( r (cid:48) ) i in CMC15. The color indices ( J i − Ks i ) of reference stars come fromthe 2MASS catalogue. The r (cid:48) -band of CMC15 is the same as that in theSloan Digital Sky Survey. In Eq. 2, i denotes the reference star and C, R,and V denote different filters. Using the derived parameters M ( C/R/V )0 and k ( C/R/V ) , the mean magnitude of the asteroid in a given night is transformedinto the r (cid:48) -band of the CMC15 photometric system, denoted by M ( r (cid:48) ) ast . Forno measurement value for 2005 UD’s color index ( J − Ks ) ast , we temporallyuse that of Phaethon’s at 0.275 ∗ . Then the r (cid:48) -band magnitudes of the ∗ https://sbnapps.psi.edu/ferret/SimpleSearch/results.action V ast = 0 . × ( J − Ks ) ast + 0 . × M ( r (cid:48) ) ast . (3)The calibrated mean magnitude V ast of each lightcurve of 2005 UD areused in the phase curve inversion: the results are shown in Fig. 2. Theindividual data points of the asteroid in a night are calibrated by adding thecalibrated mean magnitude V ast into the reduced magnitudes in that night.In addition, we have downloaded, from the MPC Asteroid LightcurvePhotometry Database † , three dense lightcurves of 2005 UD observed on 12,15, and 16 October 2018. These data have been observed through a clearfilter and have been converted into the V-band. We corrected the distanceeffects on the magnitude and light travel time in each recorded time stamp.Finally, we have downloaded sparse photometric data of 2005 UD fromthe ZTF Data release ‡ . The ZTF is a robotic time-domain sky survey usingthe Samuel Oschin 48-inch Schmidt telescope with a new Mosaic CCD of a 47square-degree FOV. This observational system is equipped with three filters:ZTF-g, ZTF-r, and ZTF-i. The limiting magnitude in the ZTF-r band is20.7 mag with a 5 − σ detection threshold in a 30-s exposure. It can scanmore than 3750 deg per hour. The ZTF data reduction follows the dataprocessing system of the Palomar Transient Factory (PTF) survey. They usea fixed aperture of 8 pixels to obtain instrumental magnitudes and carry out † http://alcdef.org/PHP/alcdef GenerateALCDEFPage.php ‡ https://irsa.ipac.caltech.edu/applications/ztf/ Figure 2: Phase curve of 2005 UD. . Photometric analysis An asteroid’s brightness results from the scattering of sunlight by itssurface. How bright the asteroid truly is depends on its size, shape, orien-tation, and surface scattering properties. The surface scattering propertiesdetermine the asteroid’s geometric albedo. The brightness is standardizedby locating, fictitiously, the asteroid at a 1-au distance from the Sun as wellas from the observer. As for the effects on magnitude due to the varyingdistance of asteroid from the Sun and the observer, we account for them bythe formula − r ∆). The brightness of the asteroid varies with the solarphase angle, the angle between the Sun and the observer as seen from theobject. Such brightness variation is called the phase curve. Most asteroidshave an irregular shape. For a spinning nonspherical asteroid, the bright-ness varies as a function of rotational phase due to the varying illuminatedand visible parts of surface. The resulting brightness curve is called thelightcurve. The shape of the lightcurve of an asteroid is usually dominatedby the shape of the asteroid. The lightcurve shape varies from one apparitionto another due to the changes in the so-called aspect angles, the angles be-tween the spin pole direction and the line of sight or the direction of sunlight.The lightcurve of an asteroid contains information on the asteroid’s physicalproperties, that is, size, albedo, spin status, and shape, and it depends onthe observation geometry. We may infer the asteroid’s properties from thelightcurves if the lightcurves span a sufficiently large selection of observa-12ional geometries. The procedure is called lightcurve inversion, in which abrightness model is established to invert the asteroid’s properties assuminga shape model and a surface scattering law. Normally, scattering laws suchas the Lommel-Seeliger law, Lambert law, Hapke model (Hapke (2012)) andLumme-Bowell model (Bowell and Lumme (1979)) are used. As for the shapemodel, a triaxial ellipsoid model or a more complex convex shape model canbe introduced, depending on the characteristics of the lightcurves.The regular shape of lightcurves of 2005 UD (Fig. 1) with two peaksimplies a regular shape of the asteroid. That is the main reason why we use, inthe lightcurve inversion, the triaxial ellipsoid model with the Lommel-Seeligerscattering law (LS ellipsoid; Muinonen et al. (2015)). For an elementary facet dA on the asteroid’s surface, the brightness of the facet with the Lommel-Seeliger law can be written as dL = 14 F µ (cid:36) P ( α ) 1 µ + µ µdA, (4)where πF is the incident solar flux density on the facet, (cid:36) is the single-scattering albedo, α is the solar phase angle, and P ( α ) is the single-scatteringphase function.The brightness integrated over an ellipsoid surface with the Lommel-Seeliger scattering law is given as L ( α, e (cid:12) , e ⊕ ) = 18 πF abc(cid:36) P ( α ) S (cid:12) S ⊕ S { cos( λ (cid:48) − α (cid:48) ) + cos λ (cid:48) + sin λ (cid:48) sin( λ (cid:48) − α (cid:48) ) × ln[cot 12 λ (cid:48) cot 12 ( α (cid:48) − λ (cid:48) )] } . (5)Here, e (cid:12) and e ⊕ denote the unit vectors of solar and viewer directions, a , b ,13nd c are the three semimajor axes of the ellipsoid, and the auxiliary quan-tities S (cid:12) , S ⊕ , α (cid:48) , λ (cid:48) are functions of the illumination and viewing geometry,pole orientation, and shape of the asteroid. For details, the reader is referedto Eqs. 11 and 12 in Muinonen et al. (2015).Using the relationship18 (cid:36) P ( α ) = p φ HG G ( α ) φ LS ( α ) , Eq. 5 can be re-written as L ( α, e (cid:12) , e ⊕ ) = πF abcp φ HG G ( α ) φ LS ( α ) S (cid:12) S ⊕ S { cos( λ (cid:48) − α (cid:48) ) + cos λ (cid:48) + sin λ (cid:48) sin( λ (cid:48) − α (cid:48) ) × ln[cot 12 λ (cid:48) cot 12 ( α (cid:48) − λ (cid:48) )] } , (6)where φ LS ( α ) = 1 − sin 12 α tan 12 α ln(cot 14 α ) .φ LS ( α ) is the Lommel-Seeliger disk-integrated phase function for a sphericalasteroid and p is the geometric albedo. In the model above, the H, G , G phase function (Muinonen et al. (2010)) is incorporated into the LS-ellipsoidbrightness model.In short, altogether 12 unknown parameters are involved in the LS-ellipsoid model. They are the rotational period P er , pole orientation ( λ, β )in the ecliptic frame of J2000.0, rotational phase ϕ at J D , three semimajoraxes ( a, b, c ), geometric albedo p , phase function parameters H, G , G , andthe equivalent diameter of asteroid D . To derive the solution of those un-known parameters, the flexible Nelder-Mead downhill method and a Markov-chain Monte Carlo method (MCMC) are applied in our photometric analysis14rocedure. In practice, the analysis procedure of 2005 UD consists of twoparts: the shape inversion of 2005 UD with the LS-ellipsoid model and thephase curve inversion. In the first part, we estimate rotation period, polelongitude and latitude, and three semimajor axes using 14 dense lightcurves.In the second part, using calibrated photometric data of 2005 UD, the phasecurve parameters H, G , G are retrieved. Altogether 14 dense lightcurves of 2005 UD are used to invert the spinand shape parameters by the Nelder-Mead downhill simplex method. Fur-thermore, the uncertainties of the parameters are assessed by the MCMCmethod.Using the downhill simplex least-squares method, the following χ ( P ar )is minimized: χ ( P ar ) = N (cid:88) i =1 N i (cid:88) j =1 σ ij [ L obs,ij − L ij ( P ar )] . (7)In practical shape inversion, only 7 parameters ( P er , λ , β , ϕ , a , b , c ) areestimated and the rest of the parameters are kept fixed in Eq. 7. Here N is the number of lightcurves used, N i is the number of data points in the i th lightcurve, L obs,ij is the j th data point in the i th lightcurve, and σ ij isits corresponding uncertainty. L ij ( P ar ) is the modeled brightness calculatedwith the LS-ellipsoid model.To find the most probable rotation period, a wide range of periods within2 . − . P er / T (here P er is the assumed15eriod and T is the time span of all involved data). Fig. 3 shows the χ ofthe lightcurve fitting versus the trial period, the most likely value of periodis located at 5.2338 h. The result is close to Kinoshita’s result of 5.2310 h(Kinoshita et al. (2007)). First, at each step of period scanning, hundreds ofdifferent initial poles distributed uniformly over the unit sphere are tested.Second, we have scanned the entire unit sphere with a step of 1 ◦ in longi-tude and latitude directions to find the most probably pole of 2005 UD usingthe period of 5.2338 h as the initial value. The contours of χ versus thetrial poles are shown in Fig. 4. The areas in blue color in Fig. 4 are cor-responding to relatively small χ . Two candidate poles of (73 ◦ , − ◦ ) and(285 ◦ , − ◦ ) are found with almost equal values of χ . Third, taking thescanned spin parameters as initial values, unknown parameters are resolvedwith the Nelder-Mead downhill simplex method. Finally, we arrive at thefollowing pair of pole solutions: Pole 1 at (72 ◦ . , − ◦ .
6) with axial ratiosof b/a = 0 . c/a = 0 .
40 and Pole 2 at (285 ◦ . , − ◦ .
8) with axial ratios of b/a = 0 .
76 and c/a = 0 .
40. The periods corresponding to the poles are closeto 5.2338 h.In order to derive the uncertainties for the spin and shape parametersof 2005 UD,an MCMC simulation is run based on the photometric data of2005 UD and the LS-ellipsoid model. The a posteriori probability densityfor the parameters is characterized by a large number of sample solutionsobtained by Metropolis-Hastings sampling. The proposal densities for theparameters are constructed via a collection of virtual least-squares solutionsderived from virtual photometric data. The virtual photometric data are16 igure 3: The χ -values for the trial periods. generated by adding Gaussian noise into the observations. At least 10000samples are obtained with the MCMC simulation. The joint distributions ofthe spin parameters are shown in Fig. 5 for Pole 1 and Fig. 6 for Pole 2. Thedotted lines in Figs. 5 and 6 are the best-fit values of the parameters. Theintervals between the best-fit value and the 1 − σ limits for each distributionare used to estimate the uncertainties of the parameters. Based on those jointdistributions, the best-fit values of the parameters with their uncertainties areas follows: Pole 1 at (72 ◦ . +4 . − . , − ◦ . +6 . − . ) with axial ratios b/a = 0 . +0 . − . and c/a = 0 . +0 . − . and Pole 2 at (285 ◦ . +1 . − . , − ◦ . +5 . − . ) with axial ratios b/a = 0 . +0 . − . and c/a = 0 . +0 . − . ,. The periods corresponding to the17 igure 4: The χ -values for the trial pole orientations. poles are 5 . +0 . − . and 5 . +0 . − . , respectively. Comparing thedistributions of the poles, the solution of Pole 2 is prefered. Indeed, whichone of the pole solutions is the true solution requires more observations, evenobservations with other techniques (e.g., imaging, occultation, or radar).In order to understand the inversion results intuitively, the model bright-nesses from the solution with Pole 2 solution are shown in Fig. 1 togetherwith the observations. Most observed brightnesses are fitted well by themodeled brightness, but some data (e.g., data obtained on on 10 Oct. 2018)are not. This minor caveat may be due to the fast sky-plane motion of theasteroid, resulting in a low quality of photometric data due to the elongated18mage of the asteroid. Figure 5: Joint distributions of spin and shape parameters for Pole 1.
The photometric phase curve of an asteroid shows the observed bright-ness variation as a function of the solar phase angle. There are two phase19unctions, the
H, G and
H, G , G phase functions, to be used to describethe photometric phase curves of asteroids. The H, G system ( H and G being the absolute magnitude and slope parameter of the asteroid, respec-tively) has been adopted by the International Astronomical Union (IAU) in1985 as the standard photometric system for asteroids, developed from theLumme-Bowell model (Bowell et al. (1989)). The slope parameter G char-acterizes an asteroid’s surface, whereas the absolute magnitude H is relatedto its size and geometric albedo. The H, G , G phase function is used inthe standard photometric system of asteroids adopted by the IAU in 2012.The new three-parameter phase function improves the fits of phase curves ofboth high-albedo and low-albedo asteroids. Here, we use the H, G , G phasefunction to fit the calibrated photometric data of 2005 UD. The basic modelfor linear least-squares fitting is10 − . V ( α ) = a φ ( α ) + a φ ( α ) + a φ ( α )= 10 − . H [ G φ ( α ) + G φ ( α ) + (1 − G − G ) φ ( α )) . (8)The source code for H, G , G phase curve inversion is openly available § . Inpractice, the phase curve inversion is carried out based on the disk-integratedbrightnesses of the asteroid (see Eq. 8). The linear parameters a , a , and a are derived first, then the parameters H, G , and G are calculated usingEq. 19 in Muinonen et al. (2010). The functions φ , , ( α ) are basis func-tions expressed in cubic splines with the interpolation grid of phase angles of § http://h152.it.helsinki.fi/HG1G2/ ◦ , . ◦ , ◦ , ◦ , ◦ , ◦ , ◦ , ◦ , ◦ , ◦ , ◦ , and 150 ◦ ). The detailed valuesare available in Tables 3 and 4 in Muinonen et al. (2010)).The mean values of the dense lightcurves and ZTF data are used to fitthe H, G , G function with the MCMC method. In the MCMC simulationprocedure, the Metropolis-Hastings algorithm is applied to sample using theGaussian proposal probability densities of the parameters. Fig. 7 shows thejoint distributions of the three parameters. The best values of H, G , and G are 17.19 mag, 0.573, and 0.004. Using 1 − σ limits of the distributions,we obtain H = 17 . +0 . − . mag, G = 0 . +0 . − . , G = 0 . +0 . − . . The best-fit model to the photometric data is displayed in Fig. 2. From the best-fitvalues of G and G , 2005 UD is likely to be a C-type asteroid according tothe suggestion in Shevchenko et al. (2016).
4. Summary
In order to study the near-Earth asteroid 2005 UD, we carried out 11nights of photometric observations with a 30-cm telescope at the Corona bo-realis Observatory in Ali, Tibet, China. Combining our 11 lightcurves withthe 3 lightcurves from the MPC ALCDEP database, the spin and shapeparameters of 2005 UD have been analysed with the Lommel-Seeliger ellip-soid method. Two pole solutions, Pole 1 at (72 ◦ . +4 . − . , − ◦ . +6 . − . ) and Pole2 at (285 ◦ . +1 . − . , − ◦ . +5 . − . ), have been derived. Comparing the distribu-tions of spin parameters, we prefer Pole 2, since it gives more concentrateddistributions. The axial ratios of the ellipsoid corresponding to Pole 2 are b/a = 0 . +0 . − . , c/a = 0 . +0 . − . . The spin period is 5 . +0 . − . h.21he distribution of the pole latitude (see Figs. 5 and 6) is wider thanthat of the pole longitude. This is due to a small span of the aspect anglesof the photometric data. So more photometric observations are necessary forimproving the pole orientation, especial in latitude.Our group also focuses on the physical studies of another PGC member,that is, asteroid (3200) Phaethon. We have found that Pole 2 of 2005 UD isclose to that of Phaethon. Investigating published pole solutions of Phaethon,we have noted that they show slight differences: for example, (97 o ± o , − o ± o ) and (276 o ± o , − o ± o ) from Krugly et al. (2002), (85 o , − o ) fromAnsdell et al. (2014), (319 o , − o ) from Hanuˇs et al. (2016), and (308 o ± o , − o ± o ) and (322 o ± o , − o ± o ) fromKim et al. (2018). Ourgroup gives a pair of poles of (95 o . , − o . o . , − o .
6) (a pa-per is being prepared). If considering the second pole solution of Phaethon,above pole solutions are around three directions: ( a )(276 o ± o , − o ± o ),( b )(311 o . , − o .
6) and ( c )(319 o , − o ). The differences among the three ori-entations occur in ecliptic latitude: − o , − o , and − o . In the case of (b)and (c), the longitudes are very close to each other, whereas in the case of (a),the longitude diverges from that of (b) and (c). Considering the uncertaintyof the pole solutions, Phaethon’s pole appears to align with that of 2005 UD.If it is true, we think this evidence implies 2005 UD probably originated viaa collision or rotational fission from Phaethon’s parent body, resembling thecase of the Koronis family (Slivan (2002)).Combining the mean magnitudes of the dense lightcurves and the ZTFdata, we have fitted the photometric phase curve of 2005 UD with the three22arameter H, G , G phase function. The parameters H, G and G are asfollows: 17 . +0 . − . mag, 0 . +0 . − . , . +0 . − . , respectively. Given thesevalues of the parameters, the phase integral parameter q , normalized slope ofthe phase-curve k (within α (7 o .
5) and the amplitude of the opposition effect ζ − ζ is the enhancement factor) are estimated to be 0.2441, -1.9076, and0.7309 as computed from the following relationships(Muinonen et al. (2010)): q = 0 . . G + 0 . G ,k = − G π + G π G + G = − π G + 9 G G + G ,ζ − − G − G G + G . Based on the derived H value and the relationship of the diameter andalbedo ( D = √ p v − . H ) (Bowell et al., 1989), the equivalent diameter D of2005 UD is estimated to be 1.3 km using its new derived albedo of 0 . ± . . ± .
5. Acknowledgements
The research has been funded by the National Natural Science Foun-dation of China (Grant Nos. 11073051 and 11673063) and the Academy ofFinland (Grant No. 325805). The research has made use of the NASA/IPACInfrared Science Archive, which is funded by the National Aeronautics andSpace Administration and operated by the California Institute of Technol-ogy. The work includes data from the Asteroid Terrestrial-impact Last Alert23ystem (ATLAS) project. ATLAS is primarily funded to search for near-Earth asteroids through NASA grants NN12AR55G, 80NSSC18K0284, and80NSSC18K1575; byproducts of the near-Earth-object search include imagesand catalogs from the survey area. The ATLAS science products have beenmade possible through the contributions of the University of Hawaii Institutefor Astronomy, the Queen’s University Belfast, the Space Telescope ScienceInstitute, and the South African Astronomical Observatory.
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H ( m a g )
Number G - 0 . 0 2 0 . 0 0 0 . 0 202 0 04 0 06 0 08 0 01 0 0 01 2 0 01 4 0 0 Number G Figure 7: Marginal distributions of
H, G , G for 2005 UD.for 2005 UD.