arXiv:1401.5794v2 [physics.pop-ph] 10 Sep 2014 where e hssuyadh eie h a fteanomalous the of law the derived [2]. contin- he numbers Benford and F. study 1938 this In ued favored. smaller are the but digits probability significant conclu- equal with the datasets distributed to physical not various him are of were led digits pages This significant the first last. that the sion the that than out noticed worn He more log- [1]. to books respect with arithmic observation remarkable a made comb blt ofidanme hs rtlfms ii s2, is is 5 digit is third leftmost and first 3 whose is number digit a second find to ability )=log = 5) { pcr 8,fl it fhdos[]adhl ietimes life half and [9] hadrons and of alpha Studies width for atomic full complex [7]. like [8], data exams spectra physical choice in multiple performed also to from were and [6] [5] sciences data geophysical financial to [4] statistics from ing ne h hieo nt ftedtst(cl invariance) (scale dataset the of units of choice the under ∗ † 1 [email protected] [email protected] P o h rtsgicn ii a ewitnas written be can digit significant first the For h eea infiatdgtlw[]frall for [3] law digit significant general The New- S. mathematician and astronomer the 1881 In hslwhsbe etdaantvrosdtst rang- datasets various against tested been has law This nitrsigpoet fti a sta ti invariant is it that is law this of property interesting An , 2 ( , . . . , d 1 d d , k 10 P sthe is 2 9 1+1 + (1 d , . . . , } ( k and elwt h rt eodadtidsgicn digit. significant third and w second reasonable Benfo first, law digit the with first with probabilities the well follow ga digit of of significant distances distances the third prove investigate and we been da have paper second law this financial Benford’s In from of understood. ranging features world, many the although over that all sources various log = ) β efr’ a rdcsteocrec fthe of occurrence the predicts law Benford’s k / .INTRODUCTION I. eas[0 11]. [10, decays th k d 3)=0 = 235) log = ) k etotdgt o xml,teprob- the example, For digit. leftmost 10 { ∈  0 + 1 , 10 1 . 8%. 18 , . . . ,   k 1 + 1  k , 9

} P for , X i =1 1 = k ( d 1 d ainlTcnclUiest fAthens of University Technical National k efr’ a nAstronomy in Law Benford’s , ainlTcnclUiest fAthens of University Technical National i 2 = 2 × ≥ , . . . , rohvnNtoa Laboratory National Brookhaven 10 d , is 2 k Dtd etme 1 2014) 11, September (Dated: k 2 hooo Alexopoulos Theodoros − (2) 9 ∈ 3 = i tfnsLeontsinis Stefanos ! N − , d , 1 d   3 1 (1) = ∈ n th ihi a a xmndb e ol 1]uigthe using [14] Boyle loga- Jeff the method. by of series examined explanation Fourier was the the law of in explanation rithmic approach complete Another distri- most different the law. from made mixture that he a Combin- datasets butions, are the law all [13]. that Benford’s realising Hill follow and which Theodore features in these by base ing scale written) the of of are extension dependance the numbers (the with invariance done base was to step or great feet A meters, ically. be to in chosen invariant are is units the the miles. digits lengths, that significant contains case first dataset the the the if of probability example, For [12]. h he diinlnmrclsqecscniee in considered are: sequences paper Fi- [15]. numerical law this the Benford’s additional that the three proven obey The numbers already Lucas is and It bonacci sequences. numerical ii fnmesi aaesoiiaigfrom originating datasets in numbers of digit tl,Bnodslwi o ul nesodmathemat- understood fully not is law Benford’s Still, ipeeapeo efr’ a spromdon performed is law Benford’s of example simple A I E UEIA EUNE AND SEQUENCES NUMERICAL NEW II. • • • l,adta h trdsacsarevery agree distances the that and ell, orinterval tour n enul ubr ( numbers Bernoulli and aoshlnmes( numbers Jacobsthal aoshlLcsnmes( numbers Jacobsthal-Lucas † aisadsasb oprn h first, the comparing by and laxies dspeitos ti on htthe that found is It predictions. rd’s – – – – – – – at tmcseta ti intriguing is It spectra. atomic to ta ∗ ,i ssilntflymathematically fully not still is it n, B JL JL JL J J J 0 1 n 0 0 = 1 = 0 1 n = 1 = 2 = 1 = = J n EFR’ LAW BENFORD’S JL − 1 n 2 + − 1 J 2 + n J − n JL 2 ,dfie as defined ), , B n ∀ n − JL > n ,dfie ytecon- the by defined ), 2 , n ∀ ,dfie as defined ), 1 > n 1 2

– n! z dz Bn = 2πi ez −1 zn+1 0.4 First Digit Measured 0.35 First Digit Benford H Second Digit Measured A sample of the first 1000 numbers of these sequences Second Digit Benford Third Digit Measured probability 0.3 is used to extract the probabilities of the first significant Third Digit Benford digit to take the values 1, 2, ..., 9 and the second and third 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 analysis of the Jacobsthal and Jacobsthal-Lucas numbers 0.15 and the empty circles indicate the probabilities calculated 0.1 from Benford’s formula (equation 1). It is clear that all 0.05 three sequences follow Benford’s law for the first (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 stars and galaxies and compare the probabilities of occur- (a) rence of the first, second and third significant digit with Benford’s logarithmic law. If the location of the galaxies 0.4 First Digit Measured in our universe and the stars in our are caused 0.35 First Digit Benford Second Digit Measured by uncorrelated random processes, Benford’s law might Second Digit Benford Third Digit Measured probability 0.3 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 correlate cosmological observables with 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. 0.4

0.35 First Digit Measured

A. Galaxies probability 0.3 First Digit Benford 0.25

We use the measured distances of the galaxies from ref- 0.2 erences [16, 87]. The distances considered on this dataset 0.15 are based on measurements from type II and all the units are chosen to be Mpc. The type-II supernova 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 × 1023 ergs/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) sults can be seen in figure 3(a) where with open circles we notate Benford’s law predictions and the measurements FIG. 1. Comparison of Benford’s law (empty circles) pre- with the circle. Unfortunately due to lack of statistics the dictions and the distribution of the first, second and third second and the third significant digit cannot be analyzed. 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 the third with blue circles according to Benford’s law.

B. Stars

The information for the distances of the stars is taken from the HYG database [88]. In this list 115 256 stars are be seen in figure 3(b). The first (black full circles) and included, with distances reaching up to 14 kpc. The full especially the second (red full circles) and the third (blue dataset used for the extraction of the result can be seen full circles) significant digits follow well the probabilities in figure III A. The result after analysing this dataset can predicted by Benford’s law (empty circles). 3

120 IV. SUMMARY 10000 100

8000 80 Number of Stars 6000

Number of Galaxies 60 The Benford law of significant digits was applied for

4000 the first time to astronomical measurements. It is shown 40 that the stellar distances in the HYG database follow this 20 2000 law quite well for the first, second and third significant

0 0 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 digits. Also, the probabilities of the first significant digit Distance [Gpc] Distance [kpc] of galactic distances using the Type II supernova pho- tosphere method is in good agreement with the Benford FIG. 2. Complete dataset from where the measurements for distribution; however, the errors are sufficiently large so the galaxies (left) and stars (right) is shown. that additional digits cannot be analyzed. We note, how- ever, that the plots in figure III A indicate that selection effects due to the magnitude limits of both samples may 0.4 be responsible for this behaviour and so it is not firmly 0.35 First Digit Measured established. Therefore it is necessary to repeat this study

probability 0.3 First Digit Benford using different galactic distance measures and larger cat- 0.25 alogs of both galaxies and stars to see if the Benford law 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)

0.4

First Digit Measured 0.35 First Digit Benford Second Digit Measured Second Digit Benford Third Digit Measured probability 0.3 Third Digit Benford ACKNOWLEDGMENTS 0.25

0.2 0.15 We would like to thank I.P. Karananas for the lengthy 0.1 discussions on this subject. We would like also to thank 0.05 Emeritus Professor Anastasios Filippas, the editor of 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. Comparisons of Benford’s law (empty circles) and the tional funds through the Operational Program ”Educa- distribution of the first (black), second (red) and third (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, Note on the frequency of use of the different 327 (1995) digits in natural numbers, Am. J. Math., vol 4, 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.De andU. Sen, Benford’s law detects quantum phase Phil. Soc. 78, 551-572 (1938) transitions similarly as earthquakes, Europhys. Lett., vol [3] Theodore P. Hill, The significant-digit phenomenon, The 95, 50008 (2011) American Mathematical Monthly, vol 102, No. 4, 322- [6] P. Clippe and M. Ausloos, Benford’s law and Theil trans- 4

form on financial data, arXiv:1208.5896v1 (2012) p.1292 (1992) [7] F. M. Hopee, Benford’s law and distractors in multiple [25] Schmidt-Kaler, T., The Distance to the Large Magellanic choice exams, arXiv:1311.7606 (2013) Cloud from Observations of SN1987A, Variable Stars and [8] J. C. Pain, Benford’s law and complex atomic spectra, Galaxies, in honour of M. W. Feast on his retirement, Phys. Rev. E, vol 77, 012102 (2008) ASP Conference Series, vol 30, B. Warner, Ed., p. 195 [9] L. Shao and B. Q. Ma, First digit distribution of hadron (1992) full width, Mod. Phys. Lett. A 24, 3275-3282 (2009) [26] Brian P. Schmidt et al., The unusual supernova SN1993J [10] B. Buck et al., An illustration of Benford’s first digit law in the galaxy M81, Nature 364, 600 - 602 (1993) using alpha decay half lives, Eur. J. Phys. 14, 59 (1993) [27] Brian P. Schmidt et al., Type II Supernovae, Expanding [11] NI Dong-Dong et al., Benford’s law and β-decay half- Photospheres, and the Extragalactic Distance Scale, The- lives, Commun. Theor. Phys., vol 51, No. 4, 713-716 sis (PH.D.) - Harvard University, 1993. Source: Disser- (2009) tation Abstracts International, Volume: 54-11, Section: [12] M. R. Wojcik, Notes on scale-invariance and base- B, page: 5717 (1993) invariance for Benford’s Law, arXiv:1307.3620 (2013) [28] Fernley J. et al., The absolute magnitudes of RR Lyraes [13] T. P. Hill, Base-Invariance Implies Benford’s Law, Pro- from HIPPARCOS parallaxes and proper motions., As- ceedings of the American Mathematical Society 123.3, tron. Astrophys., vol 330, 515-520 (1998) 887-895 (1995) [29] Panagia, N., New Distance Determination to the LMC, [14] J. Boyle, An application of Fourier series to the most Memorie della Societa Astronomia Italiana, vol 69, p.225 significant digit problem, The American Mathematical (1998) Monthly, vol 101, No. 9, pp. 879-886 (1994) [30] Garnavich, P. et al., Supernova 1987A in the Large Mag- [15] J. Wlodarksi, Fibonacci and Lucas numbers tend to obey ellanic Cloud, IAU Circ., 7102, 1 (1999) Benford’s law, Westhoven, Federal Republic of Germany [31] Panagia N., Distance to SN 1987 A and the LMC, New (1971) Views of the Magellanic Clouds, IAU Symposium #190, [16] Gurugubelli et al., Photometric and spectroscopic evolu- Edited by Y.-H. Chu, N. Suntzeff, J. Hesser, & D. tion of type II-P supernova SN 2004A, Bulletin of the Bohlender. ISBN: 1-58381-021-8, p.549 (1999) Astronomical Society of India, 36 (2-3), 79-97 (2008) [32] Leonard, Douglas C. et al., A Study of the Type II- [17] Bartel N., Angular diameter determinations of radio su- Plateau Supernova 1999gi and the Distance to its Host pernovae and the distance scale, Supernovae as distance Galaxy, NGC 3184, The Astronomical Journal, vol 124, indicators; Proceedings of the Workshop, Cambridge, Issue 5, pp. 2490-2505 (2002) MA, September 27, 28, 1984 (A86-38101 17-90). Berlin [33] Dessart, L. et al., Quantitative spectroscopic analysis of and New York, Springer-Verlag, 107-122 (1985) and distance to SN1999em Astronomy and Astrophysics, [18] Bartel N., Hubble’s constant determined using very-long vol 447, Issue 2, pp.691-707 (2006) baseline interferometry of a supernova, Nature 318, 25-30 [34] Takats V. J., Improved distance determination to M 51 (1985) from supernovae 2011dh and 2005cs, Astronomy & As- [19] Chugaj N. N., Supernova 1987A - Ha Profile and the Dis- trophysics, vol 540, id.A93, 7 pp. (2012) tance of the Large Magellanic Cloud, Astronomicheskii [35] Sonnenorn G. et al., The Evolution of Ultraviolet Emis- Tsirkulyar NO. 1494/MAY (1987) sion Lines from Circumstellar Material Surrounding SN [20] Hoeflich, P., Model calculations for scattering dominated 1987A, The Astrophysical Journal, vol 477, 848-864 atmospheres and the use of supernovae as distance indi- (1997) cators, Nuclear astrophysics; Proceedings of the Work- [36] Gould A. et al., Upper Limit to the Distance to the Large shop, Tegernsee, Federal Republic of Germany, Apr. Magellanic Cloud, The Astrophysical Journal, vol 494, 21-24, 1987 (A89-10202 01-90). Berlin and New York, 118-124 (1998) Springer-Verlag, 307-315 (1987) [37] Romaniello M. et al., Hubble Observa- [21] R. V. Wagoner et al., Supernova 1987A: A Test of the Im- tions of the Large Magellanic Cloud Field Around SN proved Baade Method of Distance Determination, BAAS 1987A: Distance Detetrmination with Red Clump and 08/1988; 20:985 (1988) Tup of the Red Giant Branch Stars The Astrophysical [22] Bartel N., Determinations of distances to radio sources Journal, vol 530, 738-743 (2000) with VLBI, The impact of VLBI on astrophysics and geo- [38] Hamuy M. et al., The Distance to SN 1999em from the physics; Proceedings of the 129th IAU Symposium, Cam- Expanding Photosphere Method, The Astrophysical Jour- bridge, MA, May 10-15, 1987 (A89-13726 03-90). Dor- nal, vol 558, 615-642 (2001) drecht, Kluwer Academic Publishers, 175-184 (1988) [39] Mitchell R. C. et al., Detailed Spectroscopic Analysis [23] Chilukuri, M. et al., Type-II Supernova Photospheres and of SN 1987A: The Distance to the Large Magellanic the Distance to Supernova 1987A, Atmospheric Diagnos- Cloud Using the Spectral-Fitting Expanding Atmosphere tics of Stellar Evolution. Chemical Peculiarity, Mass Loss, Method, The Astrophysical Journal, vol 574, 293-305 and Explosion. Proceedings of the 108th. Colloquium of (2002) the International Astronomical Union, held at the Uni- [40] Bartel N. et al., SN 1979C VLBI: 22 Years of Almost Free versity of Tokyo, Japan, September 1-4, 1987. Lecture Expansion, The Astrophysical Journal, vol 591, 301-315 Notes in Physics, Volume 305, Editor, K. Nomoto; Pub- (2003) lisher, Springer-Verlag, Berlin, New York, 1988. ISBN # [41] Leonard D. C. et al., The Cepheid Distance to NGC 1637: 3-540-19478-9. LC # QB806 .I18 1987, P. 295, (1987) A Direct Test of the Expanding Photosphere Method Dis- [24] Schmidt, B. P., Expanding Photospheres of Type II Su- tance to SN 1999em, The Astrophysical Journal, vol pernovae and the Extragalactic Distance Scale, American 594, 247-278 (2003) Astronomical Society, 181st AAS Meeting, #107.04D; [42] Nugent P. et al., Toward a Cosmological Hubble Diagram Bulletin of the American Astronomical Society, vol 24, for Type II-P Supernovae, The Astrophysical Journal, 5

vol 645, 841-850 (2006) ical Journal, vol 426, 334-339 (1994) [43] Baron E. et al., Reddening, Abundances and Line For- [64] Schmidt B. P. et al., The Distance to Five Type II Super- mation in SNe II, The Astrophysical Journal, vol 662, novae Using the Expanding Photosphere Method and the 1148-1155 (2007) Value of H0, The Astrophysical Journal, vol 432, 42-48 [44] Bartel N. et al., SN 1993J VLBI. IV. A Geometric Dis- (1994) tance to M81 with the Expanding Shock Front Method, [65] Sparks W. B., A Direct Way to Measure the Distances The Astrophysical Journal, vol 668, 924-940 (2007) of Galaxies, The Astrophysical Journal, vol 433, 19-28 [45] Dessart L. et al., Using Quantitative Spectroscopic Anal- (1994) ysis to Determine the Properties and Distances of Type [66] Branchini E. et al., Testing the Lease Action Principle II Plateau Supernovae: SN 2005cs and SN 2006bp, The in an Ω0 = 1 Universe, The Astrophysical Journal, vol Astrophysical Journal, vol 675, 644-669 (2008) 434, 37-45 (1994) [46] Poaznanski D. et al., Improved Standardization of Type [67] Iwamoto K. et al., Theoretical Light Curves for the Type II-P Supernovae: Application to an Expanded Sample, Ic Supernova SN 1994I, The Astrophysical Journal, vol The Astrophysical Journal, vol 694, 1067-1079 (2009) 437, 115-118 (1994) [47] Jones M. I. et al., Distance Determination to 12 Type [68] Crotts A. P. S., et al., The Circumstellar Envelope of SN II Supernovae Using the Expanding Photosphere Method, 1987A. I. The Shape of the Double-Lobed and its The Astrophysical Journal, vol 696, 1176-1194 (2009) Rings and the Distance to the Large Maggelanic Cloud, [48] Kasen D. et al., Type II Supernovae: Model Light Curves The Astrophysical Journal, vol 438, 724-734 (1995) and Standard Candle Relationships, The Astrophysical [69] Baron E., et al., Non-LTE Spectral Analysis and Model Journal, vol 703, 2205-2216 (2009) Constraints on SN 1993J, The Astrophysical Journal, [49] D’Andrea C. B. et al., Type II-P Supernovae from the vol 441, 170-181 (1995) SDSS-II Supernova Survey and the Standardized Candle [70] Clocchiatti A., et al., Spectrophotometric Study of SN Method, The Astrophysical Journal, vol 708, 661-674 1993J at First Maximum Light, The Astrophysical Jour- (2010) nal, vol 446, 167-176 (1995) [50] Olivares F. E. et al., The Standardized Candle Method for [71] Gould A., Supernova Ring Revisited. II. Distance to the type II Plateau Supernovae, The Astrophysical Journal, Large Magellanic Cloud, The Astrophysical Journal, vol vol 715, 833-853 (2010) 452, 189-1994 (1995) [51] Roy R. et al., SN 2008in - Bridging the Gap Between [72] Richmond M. W., et al., UBVRI Photometry of the Type Normal and Faint Supernovae of Type IIP, The Astro- Ic SN 1994I in M51, The Astronomical Journal, vol 111, physical Journal, vol 736, 76 (2011) Number 1 (1996) [52] Schmidt B. P. et al., The Expandind Photosphere Method [73] Eastman R. G., et. al., The Atmospheres of Type II Su- Applied to SN 1992am At cz = 14 600 km/s, The Astro- pernovae and the Expanding Photosphere Method, The nomical Journal, vol 107, 4 (1994) Astrophysical Journal, vol 466, 911-937 (1996) [53] Kirshner R. P. at al., Distances to Extragalactic Super- [74] Botticella M. T. et al., Supernova 2009kf: An Ultraviolet novae, The Astrophysical Journal, vol 193, 27-36 (1974) Bright Type IIP Supernova Discovered with Pan-Starrs [54] Branch D. et al., The Type II Supernova 1979c in M100 1 and GALEX, The Astrophysical Journal Letters, vol and the Distance to the Virgo Cluster, The Astrophysical 717, 52-56 (2010) Journal, vol 244, 780-804 (1981) [75] Leonard D. C. et al., The Distance to SN 1999em in NGC [55] Panagia N. et al., Subluminous, Radio Emitting Type I 1637 from the Expanding Photosphere Method, Publica- Supernovae, The Astrophysical Journal, vol 300, 55-58 tions of the Astronomical Society of the Pacific, vol 114, (1986) 35-64 (2002) [56] Eastman R. G. et al., Model Atmospheres for SN 1987A [76] Van Dyk S. D. et al., The Light Echo Around Supernova and the Distance to the Large Magellanic Cloud, The As- 2003gd in Messier 74, Publications of the Astronomical trophysical Journal, vol 347, 771-793 (1989) Society of the Pacific, vol 118, 351-357 (2006) [57] Schmutz W. et al., NON-LTE Model Calculations for SN [77] Walker A. R., The Distances of the Maggelanic Clouds, 1987A and the Extragalactic Distance Scale, The Astro- arXiv:astro-ph/9808336v1 (1998) physical Journal, vol 355, 255-270 (1990) [78] Hanuschik R. W. et al., Absorption Line Velocities and [58] Panagia N. et al., Properties of the SN 1987A Circumstel- the Distance to Supernova 1987A, Astronomy and Astro- lar Ring and the Distance to the Large Magellanic Cloud, physics vol 249, 36-42 (1991) The Astrophysical Journal, vol 380, 23-26 (1991) [79] Baron E. et al., Prelimianry spectral analysis of SN [59] Schmidt B. P. et al., Expanding Photospheres of Type 1994I, Mon. Not. R. Astron. Soc. vol 279, 799-803 (1996) II Supernovae and the Extragalactic Distance Scale, The [80] Hendry M. A. et al., SN 2004A: Another Type II-P Su- Astrophysical Journal, vol 395, 366-386 (1992) pernova with a Red Supergiant Progenitor, Mon. Not. R. [60] Baron E., et al., Interpretation of the Early Spectra of Astron. Soc. vol 369, 1303-1320 (2006) SN 1993J in M81, The Astrophysical Journal, vol 416, [81] Nadyozhin D. K., Explosion Energies, Nickel Masses and 21-23 (1993) Distances of Type II Plateau Supernovae, Mon. Not. R. [61] McCall M. L., The Distance to the Large Magellanic Astron. Soc. vol 346, 97-104 (2003) Cloud from SN 1987A, The Astrophysical Journal, vol [82] Vinko J et al., The First Year of SN 2004dj in NGC 2403, 417, 75-77 (1993) Mon. Not. R. Astron. Soc. vol 369, 1780-1796 (2006) [62] Gould A., The Ring Around Supernova 1987A Revisited. [83] Takats K. et al, Distance Estimate and Progenitor Char- I. Ellipticity of the Ring, The Astrophysical Journal, vol acteristics of SN 2005cs in M51, Mon. Not. R. Astron. 425, 51-56 (1994) Soc. vol 372, 1735-1740 (2006) [63] Baron E. et al., Modeling and Interpretation of the Opti- [84] Fraser M. et al., SN 2009md: Another Faint Supernova cal and HST UV Spectrum of SN 1993J, The Astrophys- from a Low-Mass Progenitor, Mon. Not. R. Astron. Soc. 6

vol 417, 1417-1433 (2011) R. Astron. Soc. vol 422, 1178-1185 (2012) [85] Takats K. et al., Measuring Expansion Velocities in Type [87] Elmhamdi A. et al., Photometry and Spectroscopy of the II-P Supernovae, Mon. Not. R. Astron. Soc. vol 419, Type IIP SN 1999em from Outburst to Dust Formation, 2783-2796 (2012) Mon. Not. R. Astron. Soc. vol 338, 939-956 (2003) [86] Inserra C. et al., Quantitative Photospheric Spectral [88] The HYG Database, http://www.astronexus.com, Analysis of the Type IIP Supernova 2007od, Mon. Not. (2011).