Empirical mantissa distributions of pulsars

https://doi.org/10.1016/j.astropartphys.2010.02.003Get rights and content

Abstract

The occurrence of digits one through nine as the leftmost nonzero digit of numbers from real world sources is often not uniformly distributed, but instead, is distributed according to a logarithmic law, known as Benford’s law. Here, we investigate systematically the mantissa distributions of some pulsar quantities, and find that for most quantities their first digits conform to this law. However, the barycentric period shows significant deviation from the usual distribution, but satisfies a generalized Benford’s law roughly. Therefore pulsars can serve as an ideal assemblage to study the first digit distributions of real world data, and the observations can be used to constrain theoretical models of pulsar behavior.

Introduction

Pulsars, celestial lighthouses in the sky, are excellent natural laboratories for the research of fundamental properties of matter under the circumstances of strong gravity, strong magnetic field, high density, and extremely relativistic condition. Pulsar physics has been a forefront field of both astronomy and physics for more than 40 years since its discovery. Though exciting progresses in this domain have significantly enlarged our knowledge of astronomical environment and physical processes, there still remain outstanding problems [1], [2]. Nowadays, due to the efforts made by modern observational instruments covering over various wavelength scopes, e.g., ROSAT, BeppoSAX, Chandra, XMM-Newton, HST, and VLT, people accumulated a sea of pulsar data, and the data are accreting significantly all these days. Hence data analysis and statistical synthesis become a vital step to characterize pulsar properties and reveal their inner regularities [3], [4].

In this paper, we perform a systematic investigation of the mantissa distributions of pulsars for the first time. The mantissa m(-1,-0.1][0.1,1) is the significant part of a floating-point number x, defined as x=m×10n, where n is an integer. In this paper, if not noted explicitly, we always postulate that the numbers are positive for succinct statement.

One might presume that the mantissas of any randomly chosen data set are approximately uniformly distributed, but that is not the case in real world. Instead, as stated by Newcomb [5], “the law of probability of the occurrence of numbers is such that all mantissae of their logarithms are equally likely”. Subsequently, it leads to the conclusion that the first significant digit, i.e., 1, 2, …, 9, of mantissa is logarithmically distributed, where the number 1 appears almost seven times more often than that of the number 9. The probability of the occurrence of the first digit can be expressed in an analytical formula, called Benford’s law [6] after the name of its second discoverer,P(k)=log101+1k,k=1,2,,9where P(k) is the probability of a number having the first digit k.

Empirically, the areas of lakes, the lengths of rivers, arabic numbers on the front page of a newspaper [6], physical constants and distributions [7], [8], the stock market indices [9], file sizes in a personal computer [10], survival distributions [11], widths of hadrons [12], even dynamical systems [13], [14], [15], conform to the peculiar law well. Nevertheless, there also exist other types of data, e.g., lottery and telephone numbers, which do not obey the law. Unfortunately, there is no a priori criteria yet to judge which type a data set belongs to. In practice, the law is already applicable in distinguishing and ascertaining fraud in taxing and accounting [16], [17], [18], [19], and speeding up calculation and minimizing expected storage space in computer science [20], [21], [22].

Since its second discovery in 1938, many attempts have been tried to explain the underlying reason for Benford’s law. For theoretical reviews, see papers written by Raimi [23], [24], [25] and Hill [26], [27], [28], [29], [30]. Nowadays, many breakthrough points have been achieved in this domain, though, there still lacks a universally accepted final answer. In mathematics, Benford’s law is the only digit law that is scale-invariant [31], which means that the law does not depend on any particular choice of units, discovered by Pinkham [32]. Also Benford’s law is base-invariant [26], [27], [28], which means that it is independent of the base d you use. In the binary system (d=2), octal system (d=8), or other base system, the data, as well as in the decimal system (d=10), all fit the general first digit law, P(k)=logd(1+1/k),k{1,2,,d-1}. Theoretically, Hill proved that “scale-invariance implies base-invariance” [26] and “base-invariance implies Benford’s law” [27] mathematically in the framework of probability theory. He also proved that random entries, picked from random distributions, form a sequence whose significant digit distribution approaches to Benford’s law [28].

Intending to uncover some regularities of pulsar data, and also to explore new domains of the digit law, we investigate the mantissa distributions of pulsar quantities systematically. We find that the first nonzero digit of mantissas displays unevenness according to the logarithmic law. The exceptions are the barycentric period and rotation frequency, which show significant deviations from Benford’s law. Further, we also discuss various properties of the digit law, and perform the generalized Benford’s law to data sets of barycentric period and rotation frequency of pulsars as well. Therefore the data of pulsars provide an ideal assemblage for further studies on the first digit law of the nature.

Section snippets

First digit distributions

We investigate the famous Australia Telescope National Facility (ATNF) pulsar catalogue1, which is a widely used database, maintained by Manchester et al. [33]. Thanks to their exhaustive search of pulsar literatures, at least back to 1993, data from all papers announcing the discoveries of pulsars or giving improved parameters are entered into the catalogue database. As ATNF pulsar catalogue is an updating database, to avoid ambiguity, in this

Generalized Benford’s law

Pietronero et al. [36] provided a new insight, suggesting that a process or an object m(t) with its time evolution governed by multiplicative fluctuations generates Benford’s law naturally, and they used stockmarket as a convictive example. The main idea is that m(t+δt)=r(t)×m(t), where r(t) is a random variable. After treating logr(t) as a new random variable, it is a Brownian process logm(t+δt)=logr(t)+logm(t) in the logarithmic space. Utilizing the central limit theorem in a large sample, log

Summary

In this paper, we present systematic analysis on the first digit distributions of mantissas of most fundamental quantities of pulsars, including barycentric period and rotation frequency, together with their time derivatives, and many more. The results reveal obvious departures from the uniform distribution, and small digits are more prevalent than large ones according to a logarithmic formula, called Benford’s law. However, not all data sets conform to it. Artificial and restricted data sets

Acknowledgments

This work is partially supported by National Natural Science Foundation of China (Nos. 10721063, 10975003) and National Fund for Fostering Talents of Basic Science (Nos. J0630311, J0730316). It is also supported by Hui-Chun Chin and Tsung-Dao Lee Chinese Undergraduate Research Endowment (Chun–Tsung Endowment) at Peking University.

References (39)

  • L. Shao, B.-Q. Ma, The significant digit law in statistical physics, submitted for...
  • E. Ley

    On the peculiar distribution of the US stock indexes’ digits

    Am. Stat.

    (1996)
  • J. Torres et al.

    A. Sola, how do numbers begin? (The first digit law)

    Eur. J. Phys.

    (2007)
  • L.M. Leemis et al.

    Survival distributions satisfying Benford’s law

    Am. Stat.

    (2000)
  • L. Shao et al.

    First digit distribution of hadron full width

    Mod. Phys. Lett. A

    (2009)
  • C.R. Tolle et al.

    Do dynamical systems follow Benford’s law?

    Chaos

    (2000)
  • A. Berger et al.

    One-dimensional dynamical systems and Benford’s law

    T. Am. Math. Soc.

    (2005)
  • A. Berger

    Multi-dimensional dynamical systems and Benford’s law

    Discrete Cont. Dyn. S.

    (2005)
  • M.J. Nigrini

    A taxpayer compliance application of Benford’s law

    J. Am. Tax. Assoc.

    (1996)
  • Cited by (28)

    • Do pulsar and Fast Radio Burst dispersion measures obey Benford's law?

      2023, Astroparticle Physics
      Citation Excerpt :

      FRBs are short-duration radio bursts located at extragalactic distances [25]. Previously, Benford’s law has also shown to be true for a whole slew of other pulsar properties such as the time derivative of the barycentric period, first and second time derivative of rotation frequency, period derivative, spin down age, proper motions, spin-down luminosity, fluxes, transverse velocity [26]. However, this same work also showed that Benford’s law does not hold true for barycentric period and barycentric rotation frequency.

    • A concise proof of Benford's law

      2023, Fundamental Research
    • First digit law from Laplace transform

      2019, Physics Letters, Section A: General, Atomic and Solid State Physics
      Citation Excerpt :

      Empirically, the areas of lakes, the lengths of rivers, the Arabic numbers on the front page of a newspaper [2], physical constants [3], the stock market indices [4], file sizes in a personal computer [5], survival distributions [6], etc., all conform to this peculiar law well. Due to the powerful data analyzing tools provided by computer science, Benford's law has been verified for a vast number of examples in various domains, such as economics [7,8], social science [6], environmental science [9], biology [10], geology [11], astronomy [12], statistical physics [13,14], nuclear physics [15–17], particle physics [18], and some dynamical systems [19,20]. There have been also many explorations on the applications of the law in various fields, e.g., in upgrading the description in precipitation regime shift [21].

    • Benford's law and continuous dependent random variables

      2018, Annals of Physics
      Citation Excerpt :

      The digit bias investigated by these scientists is now known as Benford’s Law. Benford’s Law arises in applied mathematics [3], auditing [4–9], biology [10,11], computer science [12], dynamical systems [13–17], economics [18,19], geology [20], number theory [21–23], physics and astrophysics [24–31], signal processing [32], statistics [33,34] and voting fraud detection [35], to name just a few. See [36,37] for extensive bibliographies and [38–49] for general surveys and explanations of the Law’s prevalence, as well as the book edited by Miller [50], which develops much of the theory and discusses at great length applications in many fields.

    • Revisiting the Benford law: When the Benford-like distribution of leading digits in sets of numerical data is expectable?

      2016, Physica A: Statistical Mechanics and its Applications
      Citation Excerpt :

      Newcomb noted “that the ten digits do not occur with equal frequency must be evident to any making use of logarithmic tables, and noticing how much faster first pages wear out than the last ones” [1]. Since that, Benford’s law was applied for the analysis of the statistical data related to a broad variety of statistical data, including atomic spectra [3], population dynamic [4], magnitude and depth of earthquakes [5], genomic data [6,7], mantissa distributions of pulsars [8], infrared spectra of polymers [9], statistical analysis of the turbulent eddy motion [10] and economic data [11,12]. Alternative to Benford distribution formulae have been proposed [13].

    View all citing articles on Scopus
    View full text