On the Mathematical Formulation of Empirical Laws. Andrés Escala DAS, Universidad de Chile

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1 On the Mathematical Formulation of Empirical Laws Andrés Escala DAS, Universidad de Chile

2 Examples of Scaling (Kennicutt 1998) Relations Kennicutt-Schmidt Law: Σ SF Σ SF0 = Σ gas Σ (Kleiber 1932) Kleiber s Law: B MR B 0 = M M

3 However as datasets improves, differences appear Kennicutt-Schmidt Law: Σ SF Σ SF0 = Σ gas Σ (Daddi et al 2010) Kleiber s Law: B MR B 0 = M M

4 Evidence for secondary parameters or something beyond that? In the Kennicutt-Schmidt Law: Σ SF Σ = gas 1.4 Σ Σ Σ SF = f (Σ gas, Ω, f H2, M, αα CO,Σ star, P turb,etc) 0 SF0 Σ SF Σ SF0 = Σ gas 1.4 Ω Σ 0 Ω (Principal Component Analysis) However, mathematically speaking a relation that requires as many constants as variables CANNOT BE a meaningful LAW of nature (Bridgman 1922), since laws must be independent of the units employed to measure the variables (Fourier 1822, Théorie de la Chaleur). Only homogeneous equations in their various units of measurement fulfil this requirement, for example: Σ SF = ε 0 Σ gas Ω

5 Free Fall Acceleration Experiments h Principal Component Analysis - Dimensionally Homogeneous Equation

6 Educated Guess: ππ theorem F (A 1, A 2,..., A n ) = 0 f (ππ 1, ππ 2,..., ππ n-k ) = 0; For k=3 (mass, length and time): In the FREE FALL example, if n=5 (a, G, M, R, h) a= GM R 2 In STAR FORMATION LAW: if n=4 ( Σ SF, gas, v, L) f (h/r) Σ SF = ε o gas vl -1 = ε o gas Ω (Silk 1997; Elmegreen 1997) if n=4 ( Σ SF, gas, G, L) Σ SF = ε o G L gas 1.5 (Corrected K-S; Escala 2015) if n=5 (+ Ω) Σ SF = ε (Ω / G gas /L) G L gas 3/2.Etc t ff -1

7 Star Formation Laws

8 Numerical Experiments Galactic Disk Simulations using ENZO (Adaptive Mesh Refinement), for both Spiral and Starburst Disks. To isolate the the effects of galactic rotation, we run the SAME gas configuration, only varying Ω thru the galactic potential.

9 Kennicutt-Schmidt & Silk-Elmegreen Relations: Utreras, Becerra & Escala (2016)

10 Dimensionally Corrected Kennicutt-Schmidt Law Escala (2015) Utreras, Becerra & Escala (2016) L=length integration in the Line of Sight Krumholz et al. (2012

11 Ω in the Star Formation Relations

12 Kennicutt-Schmidt vs this work law

13 Metabolic Rate Relation

14 Kleiber s Law: The Fire of Life (Kleiber 1932) B MR B 0 = M M 0 3/4 Basal metabolic rate (energy consumption under resting conditions) as a function of the animal s mass. It s scales as ¾, instead of the 2/3 expected for surface energy losses in isometric bodies: 2 B MR SS L 2 3 M M 2/3 ρρ o

15 It is possible to fulfil dimensional homogeneity in Biology? Brainbridge (1958) von Kármán (1957) Allometry: scaling of a physiological variable with size ( M )

16 Metabolic Rate in Running vs Weibel, Bacigalupe et al (2004) Resting Mammals Weibel & Hoppeler (2005) Different allometric exponents but VV O2 f H M 1.0

17 ππ Theorem F (A 1,A 2,...,A n ) = 0 f (ππ 1, ππ 2,..., ππ n-k ) = 0; For k=3 (kg, mlo 2 & sec): In the METABOLIC RATE: if n=4 ( V O2, f H, ηη O2, W) V O2 = ε o ηη O2 f H W if n=6 (+ T, T a ) V O2 = ε T/T a ηη O2 f H W...

18 O2 in Birds(green) and Mammals(blue) Escala (2018) (ηη O2 = 1 ) Slope ~ 1 [0.98, 1.01] V O2 = ε o ηη O2 f H W

19 V O2 in Running (yellow) vs Resting Escala (2018) (red) Mammals (ηη O2 = 1 ) ~ x5 Slope ~ 1 [0.94, 1.06] V O2 = ε o ηη O2 f H W

20 Unique homogeneous equation for the metabolic rates V O2 ηη O2 max ~ 5 ηη O2 rest Escala (2018) Red: flying Birds Cyan: resting & running Penguins

21 Number of Constants with Dimensions ALLOMETRY (sub sub area of Biology) PHYSICS B MR B 0 = M M G, c, ħ m p, t p, l p B MAX B MAX0 = M M ff MMMMMM ff MMMMMM0 = M M εε 0 ηη O2 = mlo 2 g 1 ff BB ff BB0 = M M Etc.

22 Physical Theories G 1/c ħ

23 otal Metabolic Energy per life-span & Mass Atanasov (2007) B MR T ls = A ls+ M 1.05 T ls ~ N f H -1 (Cook et al. 2006) A ls+ / N = mlo 2 g 1 = ε 0 η O2

24 Summary Well defined empirical laws must satisfy dimensional homogeneity (Rayleigh s similitude principle). We reformulated the star formation law to fulfil this principle and in addition, we use homogeneity to constrain the role of the orbital frequency in the SF efficiency. We reformulated the metabolic rate relation, unifying the relation for different classes of animals and aerobic conditions into a single formula.

25 THANKS!

26 Free Fall Acceleration Experiments h -

27 Confounding Variables Problem

28 Secondary physical parameters on the Kennicutt-Schmidt law

29 SFR = ε gas 3/2 L -1/2 L=length integration in LOS= ηr The SF Law as a Single Function Escala (2015)

30 More sophisticated (n>4) Law: Simulations Variations on the efficiency are up to a factor of 10 & parameters going on the efficiency are harder to measure -> cannot be currently constrained by observations. A better approach is to study variations on the efficiency by numerical experiments.

31 Caveat: multivariable function Star formation should depend on multiple parameters (not only Σ gas ) and a law described as function should have (at least) correct units. For example, free fall terminal velocity: V t = (2mg/ρ A Cd)