Benford’s Law in Astronomy Theodoros Alexopoulos∗ National Technical University of Athens Stefanos Leontsinis† National Technical University of Athens Brookhaven National Laboratory (Dated: September11, 2014) Benford’s law predicts the occurrence of the nth digit of numbers in datasets originating from 4 1 various sources all over the world, ranging from financial data to atomic spectra. It is intriguing 0 that although many features of Benford’s law havebeen proven,it is still not fully mathematically 2 understood. Inthis paperweinvestigate thedistances of galaxies andstars bycomparing thefirst, second and third significant digit probabilities with Benford’s predictions. It is found that the p distancesofgalaxiesfollowthefirstdigitlawreasonablewell,andthatthestardistancesagreevery e well with the first,second and third significant digit. S 0 1 I. INTRODUCTION [12]. For example, if the dataset contains lengths, the probability of the first significant digits is invariant in ] h In 1881 the astronomer and mathematician S. New- the case that the units are chosen to be meters, feet or p combmadearemarkableobservationwithrespecttolog- miles. - arithmic books [1]. He noticed that the first pages were Still, Benford’s law is not fully understoodmathemat- p morewornoutthanthe last. Thisledhimtotheconclu- ically. A great step was done with the extension of scale o p sionthatthesignificantdigitsofvariousphysicaldatasets to base invariance (the dependance of the base in which . arenotdistributedwithequalprobabilitybutthesmaller numbers are written) by Theodore Hill [13]. Combin- s c significant digits are favored. In 1938 F. Benford contin- ingthese features andrealisingthatallthe datasetsthat i ued this study and he derived the law of the anomalous follow Benford’s law are a mixture from different distri- s y numbers [2]. butions, he made the most complete explanation of the h The generalsignificant digit law [3] for all k ∈N, d1 ∈ law. Another approach in the explanation of the loga- p {1,2,...,9} and dk ∈{0,1,...,9}, for k ≥2 is rithmic law was examined by Jeff Boyle [14] using the [ Fourier series method. 2 k −1 v 4 P(d1,d2,...,dk)=log10 1+ di×10k−i II. NEW NUMERICAL SEQUENCES AND 9  i=1 !  BENFORD’S LAW X 7  (1) 5 wheredk isthekth leftmostdigit. Forexample,theprob- A simple example of Benford’s law is performed on . 1 ability to find a number whose first leftmost digit is 2, numerical sequences. It is already proven that the Fi- 0 second digit is 3 and third is 5 is P(d1 =2,d2 =3,d3 = bonacci and Lucas numbers obey the Benford’s law [15]. 4 5)=log10(1+1/235)=0.18%. The three additional numerical sequences considered in 1 For the first significant digit can be written as this paper are: : v i • Jacobsthal numbers (Jn), defined as X 1 P(k)=log 1+ , k =1,2,...,9 (2) r 10 k – J0 =0 a (cid:18) (cid:19) Thislawhasbeentestedagainstvariousdatasetsrang- – J1 =1 ingfromstatistics[4]togeophysicalsciences[5]andfrom – Jn =Jn−1+2Jn−2, ∀ n>1 financial data [6] to multiple choice exams [7]. Studies werealsoperformedinphysicaldatalikecomplexatomic • Jacobsthal-Lucas numbers (JLn), defined as spectra [8], full width of hadrons [9] and half life times – JL0 =2 for alpha and β decays [10, 11]. Aninterestingpropertyofthislawisthatitisinvariant – JL1 =1 underthechoiceofunitsofthedataset(scaleinvariance) – JLn =JLn−1+2JLn−2, ∀ n>1 • and Bernoulli numbers (Bn), defined by the con- tour interval ∗ [email protected] † [email protected] – B0 =1 2 A sam–pleBonf=th2enπ!fiirHstez1z−0100zndn+zu1mbers of these sequences bability0.03.54 FFSiierrcssott nDDdiigg Diitti gMBieet naMfsoeurardseudred is used to extract the probabilities of the first significant pro 0.3 STheicrodn Ddi gDiitg Mite Baesnufroerdd Third Digit Benford digittotakethevalues1,2,...,9andthesecondandthird 0.25 significant digits to be 0,1,...,9. The results can be seen 0.2 in figure 1. Full circles represent the result from the analysisoftheJacobsthalandJacobsthal-Lucasnumbers 0.15 andtheemptycirclesindicatetheprobabilitiescalculated 0.1 from Benford’s formula (equation 1). It is clear that all 0.05 threesequencesfollowBenford’slawforthefirst(black), 0 second (red) and third (blue) significant digit. 0 1 2 3 4 5 6 7 8 9 In the following sections we examine the distances of significant digit starsandgalaxiesandcomparetheprobabilitiesofoccur- (a) rence of the first, second and third significant digit with Benford’s logarithmic law. If the locationof the galaxies y 0.4 in our universe and the stars in our galaxy are caused abilit0.35 FFiirrsstt DDiiggiitt MBeenafsourrded by uncorrelated random processes, Benford’s law might prob 0.3 SSTheeiccroodnn Dddi gDDiiitgg Miitte MBaeesnaufsroeurdrded not be followed because each digit would be equiproba- Third Digit Benford ble to appear. To our knowledge this is the first paper 0.25 that attempts to correlatecosmologicalobservableswith 0.2 Benford’s law. 0.15 0.1 III. APPLICATIONS TO ASTRONOMY 0.05 0 0 1 2 3 4 5 6 7 8 9 Cosmological data with accurate measurements of ce- significant digit lestial objects are available since the 1970s. We examine if the frequencies of occurrence of the first digits of the (b) distances of galaxies and stars follow Benford’s law. y 0.4 babilit0.35 First Digit Measured o A. Galaxies pr 0.3 First Digit Benford 0.25 Weusethemeasureddistancesofthegalaxiesfromref- 0.2 erences[16,87]. Thedistancesconsideredonthisdataset 0.15 are based on measurements from type II Supernova and alltheunitsarechosentobeMpc. Thetype-IIsupernova 0.1 (SNII) radio standard candle is based on the maximum 0.05 absolute radio magnitude reached by these explosions, 0 which is 5.5 ×1023ergs/s/Hz. 0 1 2 3 4 5 6 7 8 9 The total number of galaxies selected is 702 with dis- significant digit tances reaching 1660Mpc (see figure IIIA left). The re- (c) sultscanbeseeninfigure3(a)wherewithopencircleswe notate Benford’s law predictions and the measurements FIG. 1. Comparison of Benford’s law (empty circles) pre- withthecircle. Unfortunatelyduetolackofstatisticsthe dictions and the distribution of the first, second and third secondandthethirdsignificantdigitcannotbeanalyzed. significant digit of the (a) Jacobsthal, (b) Jacobsthal-Lucas The trend of the distribution tends to follow Benford’s and (c) Bernoulli sequences (full circles). The probabilities prediction reasonably well. for the first digit is plotted with black, the second with red and thethird with blue circles according to Benford’s law. B. Stars The information for the distances of the stars is taken fromtheHYGdatabase[88]. Inthislist115256starsare be seen in figure 3(b). The first (black full circles) and included, with distances reaching up to 14kpc. The full especiallythesecond(redfullcircles)andthethird(blue dataset used for the extraction of the result can be seen full circles) significant digits follow well the probabilities infigureIIIA.Theresultafteranalysingthisdatasetcan predicted by Benford’s law (empty circles). 3 axies120 Stars10000 IV. SUMMARY mber of Gal10800 Number of 8000 u 6000 The Benford law of significant digits was applied for N 60 the firsttime toastronomicalmeasurements. Itis shown 40 4000 thatthestellardistancesintheHYGdatabasefollowthis 20 2000 law quite well for the first, second and third significant digits. Also,the probabilities ofthe firstsignificantdigit 00 0.5 1 1.5 2 2.5 3 00 2 4 6 8 10 12 14 of galactic distances using the Type II supernova pho- Distance [Gpc] Distance [kpc] tosphere method is in good agreement with the Benford distribution; however, the errors are sufficiently large so FIG. 2. Complete dataset from where the measurements for thegalaxies (left) and stars (right) is shown. thatadditionaldigitscannotbeanalyzed. Wenote,how- ever,that the plots in figure IIIA indicate that selection effects due to the magnitude limits of both samples may bability0.03.54 First Digit Measured besetarbeslipsohnedsi.bTlehfeorreftohriesibteishanveicoeussraarnydtosoreipteiastnthoitsfisrtumdlyy pro 0.3 First Digit Benford usingdifferentgalacticdistancemeasuresandlargercat- alogsof both galaxiesandstars to see if the Benfordlaw 0.25 is still followed when larger distances are probed. Such 0.2 larger samples of galaxies would also allow the examina- 0.15 tion of second and perhaps third significant digits. 0.1 0.05 0 0 1 2 3 4 5 6 7 8 9 Significant digit of galaxy distances (a) y 0.4 abilit0.35 FFiirrsstt DDiiggiitt MBeenafsourrded b Second Digit Measured pro 0.3 STheicrodn Ddi gDiitg Mite Baesnufroerdd ACKNOWLEDGMENTS Third Digit Benford 0.25 0.2 0.15 We wouldliketo thank I.P.Karananasforthe lengthy 0.1 discussions on this subject. We would like also to thank Emeritus Professor Anastasios Filippas, the editor of 0.05 JOAA and the reviewer for the valuable comments and 0 0 1 2 3 4 5 6 7 8 9 suggestions. Sigificant digit of star distances (b) The present work was co-funded by the European Union (European Social Fund ESF) and Greek na- FIG.3. ComparisonsofBenford’slaw(emptycircles)andthe tional funds through the Operational Program ”Educa- distributionofthefirst(black),second(red)andthird(blue) tion and Lifelong Learning” of the National Strategic significant digit of the distances of the (a) galaxies and (b) Reference Framework (NSRF) 2007-1013. ARISTEIA- stars (full circles). 1893-ATLAS MICROMEGAS. [1] S.Newcomb,Noteonthefrequencyofuseofthedifferent 327 (1995) digits in natural numbers, Am.J. Math., vol4, No.1/4, [4] L. Shao and B. Q. Ma, First-digit law in nonextensive 39-40 (1881) statistics, Phys. Rev.E, vol 82, 041110 (2010) [2] F. Benford, The law of anomalous numbers, Proc. Am. [5] A.S.DeandU.Sen,Benford’slawdetectsquantumphase Phil. 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