What is Benford’s Law

Benford’s Law of First Digits and the Mass of ExoplanetsThomas W. Hair, Ph.D.Florida Gulf Coast University

What is Benford’s Law Benford’s Law refers to the frequency distribution of first digits in many natural and human-constructed sources of data. In this distribution, the number 1 occurs as the leading digit approximately 30% of the time, while larger numbers occur in that position with decreasing frequency.

What is Benford’s Law The discovery of Benford’s Law dates back to 1881, when the American astronomer Simon Newcomb noticed that in logarithm tables the earlier pages, which contained numbers that started with 1, were more heavily worn than other pages. The phenomenon was noted again in 1938 by physicist Frank Benford, who tested it on data from 20 different domains ranging from the surface areas of rivers to the populations of US cities.

Efficacy of Benford’s Law Benford’s Law is so well established that forensic accountants can reference it in their testimony during trial proceedings based upon the plausible assumption that people who make up figures tend to distribute their digits fairly uniformly. In the United States Benford’s Law has been admitted into testimony at the federal, state and local level.

How does Benford’s Law Work? Generally speaking, Benford’s Law applies very well when a data set is both exponentially distributed and spans at least several orders of magnitude. 20 24 27 210 214 217 6 of the first 18 powers of 2 have a 1 as their first digit.

Exoplanet Mass Data Set The exoplanet data set used is from the Exoplanet Orbit Database maintained at the website Exoplanets.org. As of November 2013 the data set contained 758 confirmed exoplanets and 3455 Kepler candidate exoplanets.

Kepler Mission Objective Kepler is designed to orbit the sun in the gravitationally stable LaGrange Point 2 (L2) some 1,500,000km from Earth and constantly stare at a fixed region of the galactic arm we inhabit within the Milky Way galaxy. From this vantage point it surveys the approximately 150,000 stars within its field of view for the slight dimming caused by the small fraction of planets in proper alignment.

Exoplanet Mass First Digit Exoplanet masses range from just under the size of Mars to over 27 times the size of Jupiter giving the data a spread of almost six orders of magnitude on both the Jupiter-mass and Earth-mass scales. First Digit Frequencies (%) 1 2 3 4 5 6 7 8 9 Benford 30.1 17.6 12.5 9.7 7.9 6.7 5.8 5.1 4.6 Jupiter-mass 32.9 18.4 11.2 8.0 6.7 7.3 5.8 5.5 4.2 Earth-mass 29.3 18.4 11.0 8.6 8.5 8.6 6.5 5.7 3.6 *Data courtesy of Exoplanets.org Χ2 goodness-of-fit p < 0.02

Exoplanet Mass and Benford’s Law Expected percentage according to Benford’s Law compared with the exoplanet first digit distributions in both Jupiter and Earth mass equivalent units.

Exoplanet Mass Distribution The distribution of exoplanet masses as measured from Earth, Kepler, Hubble, et. al. shows an exponential distribution of nearly six orders of magnitude. The close fit with Benford’s Law suggests that our techniques for measurement are sound.

Conclusions As of August 15, 2013, with the failure of a second reaction wheel, Kepler is no longer able to maintain its fixed position and thus cannot confirm through repeated transits many of the candidates it has detected. However, researchers associated with the Kepler exoplanet search are confident that a significant majority (>90%) of these Kepler candidates are indeed actual planets.

Conclusions The fact that these 3455 Kepler candidates, along with the confirmed 758 known exoplanets, nicely comport with Benford’s Law further suggests that the optimism expressed by the scientists involved in this great endeavor is not misplaced. It is somewhat amazing to me that an obscure and simple mathematical fact such as Benford’s Law can be used to help confirm such profound cutting edge science that spans thousands of light-years.

Benford’s Law of First Digits and the Mass of ExoplanetsThomas W. Hair, Ph.D.Florida Gulf Coast University

What is Benford’s Law