QUICK USER GUIDE Contents 1. Introduction 1 2. Overview of
Contents
1. 8 EXAMPLE INPUT FILE See the below example input file Notice that asteroid name together with asteroid number is a single String Example txt 24 Themis 22 13 55 7 825 13 28 7 853 7 26 7 600 6 89 7 586 4 95 7 478 4 75 7 452 4 36 7 439 3 95 7 417 2 94 7 366 2 80 7 351 2 40 7 317 1 98 7 295 1 71 7 278 1 57 7 279 1 39 7 250 1 17 7 239 0 74 7 207 0 57 7 197 0 44 7 172 0 34 7 146 7 79 7 626 20 79 8 110 44 Nysa 23 19 00 7 551 18 52 7 524 17 16 7 511 12 DAGMARA OSZKIEWICZ 13 81 7 437 13 20 7 426 8 27 7 304 0 98 7 052 0 63 7 014 0 17 6 911 0 36 6 972 0 75 7 033 1 23 7 080 1 62 7 105 2 02 7 126 4 95 7 235 9 78 7 341 11 59 7 385 12 94 7 425 13 27 7 427 13 58 7 433 13 89 7 434 19 40 7 545 21 47 7 599 69 Hesperia 21 16 00 7 82 12 19 7 71 11 90 7 68 11 62 7 67 9 89 7 64 5 10 7 48 4 04 7 42 3 26 7 28 2 46 7 25 1 67 7 13 1 28 7 12 0 92 7 08 0 13 7 00 0 29 6 99 1 04 7 10 2 23 7 23 5 69 7 43 6 03 7 43 QUICK USER GUIDE 13 8 59 7 53 12 72 7 64 13 00 7 69 9 EXAMPLE RESULTS For each asteroid three plots are produced H G phase function H Gi Ga phase function and H Gj phase function Example result plots are show in Fig File result which is a text file containing detailed numerical output is produced and also a summary file resultsSummary is produced 2 4 04 41 NN AAN 4 34 4 44 4 5 NN x SS h 4 6 N N gi uH ala N T 4 84 Nee ge a scenes Ve 8 4 S2. 7 G a2 a a2 a3 The coefficients a1 aa a3 are estimated from the observations using linear least squares The basis functions a 2 a 3 a are defined as e For 0 lt a lt 7 5 a 1 a a 1 20 o is defined using a cubic spline as defined in Table e For 7 5 lt a lt 30 o is defined using a cubic spline as defined in Table o is defined using a cubic spline as defined in Table li 3 q is defined using a cubic spline as defined in Table e For 30 lt a lt 150 a is defined using a cubic spline as defined in Table li o is defined using a cubic spline as defined in Table 1 3 a 0 8 G2 a deg 7 5 7 5 x107 9 25 x107 30 0 3 3486016 x10 6 2884169 x107 60 0 1 3410560 x107 3 1755495 x 10 1 90 0 5 1104756 x107 1 2716367 x 10 120 0 2 1465687 x107 2 2373903 x107 150 0 3 6396989 x107 1 6505689 x1074 TABLE 1 Knots for splines used in and The first derivatives per radian at the ends of splines are 4 S E 9 1328612 x 10 Z 2 FE 8 6573138 x 1078 85 0 0 10630097 4 0 4 DAGMARA OSZKIEWICZ a deg s 0 0 1 0 3 8 3381185 x107 1 0 5 7735424 x107 2 0 4 2144772 x107 4 0 2 3174230 x107 8 0 1 0348178 x107 12 0 6 1733473 x107 20 0 1 6107006 x 10 30 0 0 TABLE 2 Knots for splin
3. 0 098 0 52 0 52 0 30 0 32 419 Aurelia 849 849 09 095 0 057 0 057 0 14 0 13 0 54 0 52 0 399 0 367 0 16 0 16 0 69 0 66 0 318 0 326 1862 Apollo 16 249 16 249 0 38 0 38 10 354 0 354 0 097 0 097 0 12 0 12 0 051 0 051 0 100 0 105 0 15 0 15 0 052 0 053 Moon 0 126 0 126 0 36 036 10 338 0 338 0 084 0 085 0 12 0 12 0 052 0 048 0 091 0 090 0 14 0 14 0 049 0 049 TABLE 5 Absolute magnitudes two sided 99 73 errors and Comparison with Muinonen et al slope parameters with K Muinonen et al 2010 for the H G G2 phase function I re sults obtained using the Java software II results obtained using the Fortran software PhaseAngleN ReducedMagnitudeN First line contains asteroid designation and N number of data points to follow Next N lines contain phase angles and corresponding reduced magnitudes File can contain more then one object the above record style would be then repeated and next asteroid record could be added following the first record For example input files see Example txt below 10 DAGMARA OSZKIEWICZ TABLE 6 Absolute magnitudes two sided 99 73 errors tran software and slope parameters with Comparison with Muinonen et al K Muinonen et al 2010 for the H G 2 phase function I re sults obtained using Java software II results obtained using For As
4. For the H G and H G1 G phase functions the a coefficients are fitted using linear least squares and for the H G 2 phase function the Lo and gt parameters are fitted using simplex non linear regression J A Nelder and R Mead 1965 3 2 Error estimation The absolute magnitude computation requires Monte Carlo error estimation because of its nonlinearity Gaussian errors in parameters a and ag in the H G phase function or a a2 and az in the H G1 Ga phase function result in non Gaussian errors in the H G or H G1 G parameters To estimate those errors we make use of the least squares solution for a and as or ay a2 and az and its error covariance matrix We set the error covariance matrix and the least squares solution as a covariance and mean of a multi normal distribution to produce a sample a a2 Or a1 a2 a3 using a multi normal random number generator The a samples are then converted to H G or H G1 G samples using either Eqs 3 and 4 or 6 7 and 8 Next the samples are ordered in descending goodness of fit and then 68 27 equivalent to 1 o and 99 73 equivalent to 3 0 error cut offs are computed resulting in a list of subsamples The limiting maximum and minimum H G or H G1 G parameters are selected from that list and the two sided errors are computed Error estimation for the H G parameters derives from the Markov chain Monte Carlo MCMC technique From non linear least squares fitting we obta
5. 1 6 Java Runtime Environment JRE The core of the applet constitutes a multi threaded Java program composed of a number of packages the contents of which are summarized in Table The program uses the Java Scientific Library written by Flanagan The program is multi threaded and uses a thread pool pattern in which a number of threads are created one per computer core to perform absolute magnitude computations one per asteroid in parallel Tasks are organized in a queue As soon as the thread has completed a task the next task from the queue is requested until all the tasks are completed The programming details need not be known to applet users The applet is straightforward and very easy to use A user guide is also provided It is however expected that the user be familiar with the methods behind state of the art empirical phase functions as well as non Gaussian error estimation The applet primarily produces absolute magnitudes and slope parameters together with two sided uncertainties for all three phase functions at the same time for a QUICK USER GUIDE 7 Package Classes included Content absoluteMagnitude AsteroidData java Main package with Thread Pool AsteroidPhotometricSolution java creating jobs for different asteroids AbsoluteMagnitude and distributing them among cores CalculatorThreaded java Contains data object and object ThreadPool java with collected results phaseCurves Function1D java Contai
6. 25 26 27 28 Phase angle deg data HG12 HG12 68 27 error envelope HG12 99 73 error envelope FIGURE 3 H G z phase function for asteroid 2 Pallas REFERENCES Bowell et al 1989 Bowell E Hapke B Domingue D Lumme K Peltoniemi J and Harris A W Application of photometric models to asteroids In T Gehrels M T Matthews R P Binzel editors Asteroids II University of Arizona Press 1989 p 524 555 K Muinonen et al 2010 K Muinonen I N Belskaya A Cellino M Delbo A C Levasseur Regourd A Penttil E F Tedesco A three parameter magnitude phase function for as teroids Icarus 2010 209 542 555 J A Nelder and R Mead 1965 J A Nelder and R Mead A simplex method for function minimization Computer Journal 1965 7 308 313 QUICK USER GUIDE 15 Tedesco E F at al 2002a Tedesco E F Noah P V Noah M Price S D The supplemental IRAS minor planet survey Astron J 2002a 123 10561085 M T Enga et al 2011 in prep M T Enga et al Asteroid family albedos from the Spritzer asteroid catalog in preparation http www ee ucl ac uk mflanaga java Michael Thomas Flanagan s Java Scientific Li brary ftp ftp lowell edu pub elgb astorb html The Asteroid Orbital Elements Database Zeljko Ivezic et al 2001 Z Ivezic S Tabachnik R Rafikov R H Lupton T Quinn M Hammergren L Eyer J Chu J C Armstrong X Fan K Finlator T R
7. Geballe J E Gunn G S Hennessy G R Knapp S K Leggett J A Munn J R Pier C M Rockosi D P Schneider M A Strauss B Yanny J Brinkmann I Csabai R B Hindsley S Kent D Q Lamb B Margon T A McKay J A Smith P Waddel and D G York Solar System objects observed in the Sloan Digita Sky Survey Commissioning data Astron J 2001 122 2749 2784 C T Rogers et al 2006 C T Rodgers R Canterna J A Smith M J Pierce and D L Tucker Improved u g r i z to UBVRelc transformation equations for main sequence stars Astron J 2006 132 989 993 Harris et al 1989a Harris A W Young J W Bowell E Martin L J Millis R L Pouta nen M Scaltriti F Zappal V Schober H J Debehogne H and Zeigler K W 1989a Photoelectric observation of asteroids 3 24 60 261 863 Icarus 1989a 77 171 186 Harris et al 1989b Harris A W Young J W Contreiras L Dockweiler T Belkora L Salo H Harris W D Bowell E Poutanen M Binzel R P Tholen D J and Wang S Phase relations of high albedo asteroids the unusual opposition brightening of 44 Nysa and 64 Angelina Icarus 1989b 81 365 374 Harris et al 1987 Harris A W Young J W Goguen J Hammel H B Hahn G Tedesco E F Tholen D J Photoelectric lightcurves of the Asteroid 1862 Apollo Icarus 1987 70 246 256 Poutanen et al 1985 Poutanen M Bowell E Martin L J and Thompson D T Photoelec tri
8. QUICK USER GUIDE DAGMARA OSZKIEWICZ CONTENTS Introduction Overview of phase functions H G phase function H G1 s phase function 3 H G13 phase function Numerical method 1 Least Squares 2 Error estimation 3 Fitting data for individual asteroids A Fitting data for asteroid families Software description requirements and dependences Testing the code Running instructions 1 Processing single asteroid phase curve data Processing multiple asteroids at the same time Processing big amount of data Example input file Example results _ Comment Acknowledgments References N N RDO NASD DNR RR Rw HE Ori Or OU el Col Co SO Co CHl m um w aD N Co rm Heco a en aa wow wore KH ww 1 INTRODUCTION 2 OVERVIEW OF PHASE FUNCTIONS Absolute magnitude computation relies on magnitude phase curve fitting A number of different mathematical formulations for magnitude phase curves have been developed Here we make use of the H G phase function the H G G2 phase function and the H Gj2 phase function Date February 17 2011 2 DAGMARA OSZKIEWICZ The H G phase function was developed to predict the magnitude of an aster oid as a function of solar phase angle Bowell et al 1989 This function was adopted by the International Astronomical Union in 1985 to redefine the absolute magnitudes of asteroids The H G phase function is not v
9. alid for phase angles greater than 120 The H G1 G2 and H G 2 phase functions K Muinonen et al 2010 are based on cubic splines The H G1 Gs phase function is designed to fit asteroid phase curves containing substantial numbers of observations whereas the H G 12 phase function is applicable to asteroids that have sparse or low accurancy photometric data 2 1 H G phase function In the H G magnitude phase function the reduced apparent magnitudes can be obtained from 10 4V a By a aza 1047 1 G a G o 1 where a is the phase angle V is the reduced magnitude The basis functions are defined as 0 986 si 2 vfi sin a 0 119 1 341 sina 0 754 sin a 1 1 ons tan 3 0 238 si i i sin a 0 119 1 341 sina 0 754 sin a 1 1 w exp 1 862 tan sa zl exp 90 56 tan 59 The coefficients a and aa are estimated from observations using linear least squares The absolute magnitude H and slope parameter G can then be obtained from 2 w a2 ay ag 4 G QUICK USER GUIDE 3 2 2 H G1 Ga phase function The reduced magnitudes V a can be obtained from K Muinonen ef al 2010 10479 ay By a aza a33 a 10044 G18 a Go o a 1 Gy G2 63 a 5 where the absolute magnitude H and slope parameters Gi and Gs are 6 H 2 5log G a2 a3 ay 1 a a2 a3
10. c photometry of asteroid 69 Hesperia Astron Astroph Suppl Ser 1985 61 291 297 Harris et al 1984b Harris A W Young J W Scaltriti F and Zappal V The lightcurve and phase relation of the asteroids 82 Alkmene and 444 Gyptis Icarus 1984b 57 251 258 Harris et al 1984a Harris A W Carlsson M Young J W and Lagerkvist C I The lightcurve and phase relation of the asteroid 133 Cyrene Icarus 1984a 58 246 256 Harris and Young 1988 Harris A W and Young J W Two dark asteroids with very small opposition effects Lunar Planet Sci 1988 XIX 2 447 448 http ssd jpl nasa gov sbdb cgi JPL Small Body Database Browser NASA 2003 NASA Study to determine the feasibility of extending the search for near Earth objects to smaller limiting diameters NASA Office of Space Science Solar System Explo ration Division Washington DC 2003
11. e used in 3 2 3 H G 2 phase function G and Gs from the three parameter phase function are replaced by a single slope parameter gt which relates to the G slope parameter in the H G system though there is not an exact correspondence The reduced flux densities can be obtained from K Muinonen et al 2010 9 10770 Lo G1 a Gs o 1 Gi G2 3 a where a J 0 7527G12 0 06164 if Giz lt 0 2 171 0 9529Gy5 0 02162 otherwise G 0 9612G 12 0 6270 if Gi2 lt 0 2 21 0 6125G12 0 5572 otherwise 10 and Lp is the disk integrated brightness at zero phase angle The basis functions are as in the H G1 Ga magnitude phase function Coefficients Lo and G2 are estimated from observations using non linear least squares 3 NUMERICAL METHODS 3 1 Least Squares Least squares fitting is carried out in the flux density do main because it reduces the problem to a linear problem for the H G G2 and H G phase functions and the errors are symmetric about the fit The flux for the it observation is computed using L s y 12 of Lat 1 QUICK USER GUIDE 5 where ol are standard deviations of the magnitude measurements The y value to be minimized here with respect to the parameters a is N 2 2 Li Lila a 13 la Soe i 1 0 The computed disk integrated brightnesses are expressed via N basis functions a a oiii x a 14 Lilana X a ai
12. in the least squares values of Lo and Gj The errors in Lo Gig are non Gaussian so the error estimation used for the H G and H G G2 phase functions cannot be used To proceed we create one long Markov chain by sampling possible Lo Gi2 solutions The chain is started at the least squares point for Lo gt and makes use of a multivariate Gaussian proposal distribution where a covariance matrix for Lo Gio is taken from the least squares solution After obtaining 10 000 different solutions the two sided errors are computed based on the equivalents of the 1 0 and 3 0 cut offs as in the H G and H G G phase functions Error envelopes are based on the 1 0 and 3 0 Monte Carlo samples We compute reduced magnitude for all the sampled values of H G and or H G1 Ga and or Lo Gig for a set of phase angles and choose the maximum and minimum of reduced magnitude at each phase angle 6 DAGMARA OSZKIEWICZ 3 3 Fitting data for individual asteroids We start our computation by per forming least squares fits with all three phase functions assuming 0 3 mag stan dard deviation for all of the observations see section We extract magnitude residual rms value from that computation and repeat least square fitting assuming magnitude uncertainties equal to the rms value for each phase function Next we perform the Monte Carlo error computation to obtain two sided errors in the photometric parameters together with the error envelopes for the phase functio
13. inty for all the objects except for the Moon where we assume 0 023 mag uncertainty for phase angles a lt 100 and 0 2 mag uncertainty for phase angles a gt 100 Tables 5 and 6 contain the results obtained using Java software I and Fortran software II The results from and the Asteroid Phase Function Analyzer agree very well There exist some small differences in the error analysis which can be explained by the limited number of samples in the Monte Carlo simulations Asteroid Class py Nobs min max References 24 Themis C B 0 08 22 0 34 20 8 Harris et al 1989a 44 Nysa 0 54123 0 17 21 5 Harris et al 1989b 69 Hesperia 0 14 21 0 13 16 0 Poutanen et al 1985 82 Alkmene 0 21 11 2 29 27 2 133 Cyrene 0 26 11 0 20 13 2 Harris et al 1984a 0 05 7 0 62 15 4 1862 Apollo 0 26 18 0 2 89 0 The Moon 0 17 17 0 5 140 0 Bowell et al 1989 TABLE 4 Objects used to illustrate the Asteroid Phase Function Analyzer capabilities We show the V band geometric albedo py Tedesco E F at al 2002a the number of observations Nops the minimum and maximum phase angles of the observations Amin and Qmax and references to the observations 419 Aurelia Harris and Young 1988 OVUM os 6 RUNNING INSTRUCTIONS 6 1 Processing single asteroid phase curve data Go to File Input tab click Load file import data file from your file s
14. m iy 2 5 04 TH i Th TT 5 14 il Mh igs 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Phase angle deg data HG HG 68 27 error envelope HG 99 73 error envelope FIGURE 1 H G phase function for asteroid 2 Pallas 10 COMMENTS Comments bugs etc can be reported to dagmara oszkiewicz helsinki fi 11 ACKNOWLEDGMENTS Research supported by the Magnus Ehrnrooth Foundation Academy of Fin land Lowell Observatory and the Spitzer Science Center We would like to thank Michael Thomas Flanagan for developing and maintaining the Java Scientific Li brary which we have used in the Asteroid Phase Function Analyzer DO thanks Berry Holl for help with Java plotters and Saeid Zoonemat Kermani for valuable advice on Java applets We thank the Department of Physics of Northern Arizona University for CPU time on its Javelina open cluster allocated for our computing 14 DAGMARA OSZKIEWICZ 2 SERC a9 um St 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Phase angle deg data HG1G2 HG1G2 68 27 error envelope HG1G2 99 73 error envelope FIGURE 2 H Gi Ga phase function for asteroid 2 Pallas 4 14 4 2 4 34 4 44 4 54 4 64 4 73 g 5 4 84 E 5 4 94 8 8 5 01 3 5 14 5 24 5 34 5 47 5 55 5 64 Em 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
15. ns 3 4 Fitting data for asteroid families We fit family data sets using the H G G2 phase function Family membership was established by agglomerating asteroids in relative velocity phase space M T Enga et al 2011 in prep Ve locity cutoffs for each family are related to the background population velocity We devise a 20 x 20 400 nodes grid in G1 Ga phase space G4 0 1 G2 0 1 and step through that grid In each step G1 Gz parameters are fixed to the node G1 Ga values and for a number of asteroids in a given family i 1 2 N a linear least squares fit of Lo is performed Global family x is computed assuming no correlation in the noise among different asteroids From the 400 nodes we select the one with the best x and create a new denser grid in the vicinity of that node stretching 1 node distance on each side from the best x solution We repeat the least squares analysis in the nodes of the new grid This operation is repeated a number of times to obtain improved precision of the Gi Gz parameters Once the family Gr G2 have been obtained we perform Monte Carlo error analysis in the N 2 parameters phase space to obtain the two sided 1 0 and 3 0 uncertainties 4 SOFTWARE DESCRIPTION REQUIREMENTS AND DEPENDENCES The Asteroid Phase Function Analyzer is an online free interactive applet written in Java It is cross platform and it runs in a Web browser using a Java Virtual Machine JVM It requirers
16. ns the phase functions HGfunction java and methods for evaluating HG1G2function java them for different sets of parameters HG12function java regression LinearRegressor java Contains tools for performing FittedFunctionHG12 java linear and nonlinear least squares FitterHG java phase curve fitting and least squares FitterHG1G2 java error analysis FitterHG12 java error Analysis NonGaussianErrorEstimator java Contains methods for MCMCSampler java two sided error estimation RandomNumberGenerator java a multinormal random number PdfComparator java generator a Markov chain Monte ParameterComparator java Carlo sampler comparators PhaseCurveSolution java inputOutput DataFileReader java Contains methods dealing Output Writer java with input output Utils java ui PhaseCurveAnalyserApplet java Contains the applet and graphical DataEntry java user interface FileInputPane java Input TableModel java Task java Result Pane java Logger java TextInputPane java TABLE 3 Description of main packages given data file Other photometric parameters and plots are also produced The tool is available at http asteroid astro helsinki fi AstPhase 5 TESTING THE CODE We compared the results obtained from this software and those from Fortran software K Muinonen et al 2010 We have repeated calculations for all the asteroids listed in Table 4 using the same data and error estimates that is 0 03 8 DAGMARA OSZKIEWICZ mag uncerta
17. teroid H II H 1 Gio II Gio 1 24 Themis 7 121 7 121 068 0 68 0 042 0 043 0 23 0 24 0 044 0 044 40 25 0 26 44 Nysa 6 896 6 896 0 066 0 066 0 041 0 038 0 077 0 075 0 044 0 043 0 072 0 074 69 Hesperia 6 987 6 987 0 41 0 41 0 036 0 041 0 21 0 22 0 040 0 043 0 26 0 28 82 Alkmene 8 187 8 187 0 30 0 30 0 032 0 032 0 14 0 14 0 034 0 033 0 19 0 19 133 Cyrene 7 882 7 882 0 20 0 20 0 026 0 027 0 51 0 016 0 070 0 03 0 49 0 015 419 Aurelia 8 514 8 514 1 04 1 04 0 074 0 077 0 46 0 50 0 052 0 084 0 16 0 55 1862 Apollo 16 209 16 209 10 334 0 334 0 022 0 024 0 077 0 081 0 023 0 025 0 077 0 078 Moon 0 124 0 124 0 358 0 398 0 020 0 020 0 073 0 075 0 022 0 021 0 073 0 074 7 PROCESSING BIG AMOUNT OF DATA Download PhaseCurveAnalyzer jar file and run it using java jar PhaseCurveAnalyzer jar Example txt nrOfCores ie java jar PhaseCurveAnalyzer jar Output will be written to files on your disk Output files will include e Plots HGPhaseCurve png dataFile dat 2 QUICK USER GUIDE 11 HG1G2PhaseCurve png HG12PhaseCurve png e Text files resultsSummary summary of all the objects processed results detailed output for each asteroid processed The files will be located in the input file directory
18. ystem data file has to be in format specified below Once the file is imported click Compute Go to Log tab to trace the progress of computation New tab with figures will appear Go to that tab and switch between different phase function plots To save the figures right click on the figure and pick save as to save numerical results right click on the figure and pick save numerical results as 6 2 Processing multiple asteroids at the same time Everything the same as for processing single asteroid phase curve except that the data file contains many asteroids File should be in the following format AsteroidDesignation NrOfDataPoints PhaseAnglel ReducedMagnitude1 PhaseAngle2 ReducedMagnitude2 PhaseAngle3 ReducedMagnitude3 QUICK USER GUIDE Asteroid Em FO am cO Gm GD 24 Themis 7 088 7 088 0 62 0 62 0 14 0 14 0 073 0 069 0 24 0 25 0 16 0 16 0 079 0 084 0 28 0 26 0 16 0 15 44 Nysa 16 04 6904 10 0650 0 050 1067 0 67 0 070 0 068 0 259 0 275 0 15 0 17 0 079 0 079 0 269 0 299 0 14 0 14 69 Hesperia 6 927 6 927 0 36 036 1029 0 29 0 069 0 067 0 25 0 23 0 18 0 17 0 069 0 072 0 28 0 28 0 16 0 15 82 Alkmene 8 06 806 0 17 0 17 1039 0 39 0 25 0 25 0 28 0 27 0 13 0 12 0 33 033 0 46 0 43 0 11 0 103 133 Cyrene 7 831 7831 021 021 1039 0 39 0 088 0 088 0 45 0 43 0 33 0 39 0 098
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