The Visualization and Measurements of Mass Functions with LEGO
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The Visualization and Measurements of Mass Functions with LEGO ∗ Kyle K. Hansotia and Stefan J. Kautsch Nova Southeastern University3301 College Ave.Fort Lauderdale, FL 33314
ABSTRACTWe intend to promote the concept of mass functions for interdisciplinary science education andcommunication. A mass function characterizes the frequency distribution of the masses of objects inthe universe. We present an experiment to demonstrate this concept to a diverse audience of sciencestudents, using popular LEGO toys. We discovered that the LEGO mass function is surprisinglysimilar compared to mass functions of astronomical objects such as stars and galaxies.
Keywords:
Astronomy data visualization (1968), Stellar mass function (1612), Initial mass function(796), Interdisciplinary astronomy (804), Pareto distribution (1897), Mass ratio (1012)INTRODUCTIONWe present an activity to visualize the concept of mass functions for the astronomy, physics, and general sciencecurriculum in higher education, and public outreach. A mass function describes the frequency distribution of themasses of many objects. Mass functions can be frequently observed at ensembles of cosmic objects such as planets,stars, galaxies, dark halos, etc. Therefore, being aware of the concept of mass functions is fundamental to understandmass distribution and structure formation in the universe. However, this concept is barely discussed in textbooks inastronomy and astrophysics for introductory courses, as well as in physics. It is even lesser known in other sciences.Only a few advanced, general astronomy textbooks (e.g., Kutner (2003), Carroll & Ostlie (2017)) cover the massfunction of stars.A simple mass function is the mathematical model f ( m ) that fits the number of objects ( n ) in different ranges ofmass ( m, m + dm ): dndm = f ( m ) = km α (1) k is a coefficient, aka the constant. α is the power index, i.e., the slope of the power function. This type of functionis also known as power law or scaling law. It shows that the number frequency of massive objects is much lower thanthe number of objects with smaller masses. Moreover, the proportion of high-to-low mass objects is constant.In astronomy, mass functions are most commonly applied to stars since Salpeter (1955). The aim of these studies isto find the initial mass function (IMF, see the reviews in Corbelli et al. (2005), and by Kroupa et al. (2013), Krumholz(2014), Hopkins (2018), and references therein) and the present-day mass function (e.g., Scalo (1986), Chabrier (2003),Bovy (2017), Sollima (2019)), i.e., the proportion of stars of various masses and within a unit of volume at the momentof their birth or at present, respectively. Moreover, mass functions can be also found for other objects like galaxies,e.g., Moffett et al. (2016), and galaxy halos, e.g., Press & Schechter (1974), Pengfei et al. (2019).Binggeli & Hascher (2007) attempted to create a unified mass function based on an idea by Zwicky (1942) to showhow cosmic objects are universally linked. They studied nearly all mass hierarchies in space, which include 36 ordersof magnitude in mass, i.e., asteroids, planets, stars and their remnants, open and globular star clusters, molecular gasand dust clouds, galaxies, galaxy groups and clusters, and even simulated cold dark matter halos. They found thatthe mass distribution of these objects roughly follows a universal mass function of form f ( m ) ∝ m − . This means Corresponding author: Stefan [email protected] ∗ Published in 2020, Research Notes of the AAS, 4, 134 a r X i v : . [ phy s i c s . pop - ph ] O c t that the proportion of high-to-low mass objects is always the same, no matter what kind of objects in the cosmos areconsidered. E.g., for stellar objects, it means that for each solar mass star, four stars with 1/2 the mass of the Sunexist, and 16 stars with 1/4 of the mass of the Sun exist.Our intention of this work is to promote the concept of mass functions to be an integrated component of modernscience education. We focus on the visualization of this concept using popular LEGO bricks. The choice of thistoy enables easy visual understanding and application for college courses (life and online), laboratory courses, andindependent studies. METHODSMany LEGO sets contain a large amount of low-mass pieces and only a few massive bricks. This makes this toyideal to show mass frequency distributions. We decided to use the set LEGO 70656 NINJAGO garmadon, Garmadon,GARMADON! , because it contains a shark model like our university’s mascot.All pieces of this LEGO set were individually massed. Those bricks were then distributed into six equal-sized massbins and counted. The number of bin intervals was naturally limited due to low numbers of massive bricks. Thesedata were fit with the power law function of Eq. 1. A nonlinear least-squares Marquardt-Levenberg fitting algorithmin gnuplot (Williams et al. 2020) was used to find the free parameters, i.e., the slope and the constant. Fig. 1 showsthe mass function as a line in a histogram with the mass of the bricks in grams in the bins on the x-axis and thenumbers of bricks on the y-axis. The best-fit result for the slope is α = − . ± .
15, and for the coefficient, it is k = 267 . ± . Figure 1.
The histogram shows the result of our measurements and fitting procedure. The LEGO bricks are sorted into sixbins of increasing mass (blue), and fitted with the mass function power law (red line). The fitting results (slope and constant)are indicated in the legend. The graphic inlay shows the proportionality of the mass distribution of the LEGO bricks from ourset.
RESULTS AND DISCUSSIONThis simple experiment enables scientists and educators to inspire the audience in understanding the concept of massfunctions. The quantitatively crucial part in understanding the mass function is the slope. This slope α describes howfast the numbers of objects increase with decreasing mass, thus, α determines the shape of the curve. Typical valuesof the slopes of stellar IMFs converge at values around − α = − α = − .12).This might be a fortunate coincidence of this particular set. However, it shows that LEGO is ideal to visualize thisconcept and should spark a classroom discussion where else this mass distribution behavior can be found in natureand how to explain it.The physical origin of individual mass functions is highly debated (for the IMF see, e.g., Chabrier (2003), Offner etal. (2016)). It is even lesser known why a universal mass function exists, which ranges from planetary bodies to galaxyclusters. Binggeli & Hascher (2007) point out that the formation scenarios of each object ensemble (planets vs. starsvs. galaxies, etc.) are totally different (bottom-up versus bottom-down). Aschwanden et al. (2016), Aschwanden etal. (2018) and references therein explain in a comprehensive review that power law-like distributions can be observedin much more than the distribution of mass, e.g., structure distribution of stellar flares, black hole objects, planetarysurface geometry, galactic structures, Pareto distributions in social science, and much more. They conclude that thesedistribution functions follow the concept of self-organizing critical processes. Applied to the mass function it couldmean the ability of complex systems to self organize on many different mass scales in the universe.Therefore, we conclude that the concept of mass functions should be an integral part of science education. It willinspire students to learn more about the mysteries in our universe and how the cosmos works. The experiential learningand teaching of mass functions using LEGO bricks can be applied in many creative ways because it is an excellenteducational tool to visualize complex concepts.ACKNOWLEDGMENTSThis project was funded by Nova Southeastern University’s President’s Faculty Research and Development Grant335510. We would like to thank Prof. Dr. B. Binggeli (University of Basel, Switzerland), Prof. Dr. D. Castano (NovaSoutheastern University, U.S.A.), and Prof. Dr. D. Veras (University of Warwick, U.K.) for assisting us in the workprocess through intellectual conversations. The graph was made with gnuplot. Adobe Photoshop was used to combinethe gnuplot histogram with the LEGO infographic. LEGO is a trademark of The LEGO Group. REFERENCES