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Astronomical Reach of Fundamental Physics

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Astronomical Reach of Fundamental Physics PERSPECTIVE PERSPECTIVE Astronomical reach of fundamental physics Adam S. Burrows1 and Jeremiah P. Ostriker Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544 Edited by Neta A. Bahcall, Princeton University, Princeton, NJ, and approved January 7, 2014 (received for review September 23, 2013) Using basic physical arguments, we derive by dimensional and physical analysis the characteristic masses and sizes of important objects in the universe in terms of just a few fundamental constants. This exercise illustrates the unifying power of physics and the profound connections between the small and the large in the cosmos we inhabit. We focus on the minimum and maximum masses of normal stars, the corresponding quantities for neutron stars, the maximum mass of a rocky planet, the maximum mass of a white dwarf, and the mass of a typical galaxy. To zeroth order, we show that all these masses can be expressed in terms of either the Planck mass or the Chandrasekar mass, in combination with various dimensionless quantities. With these examples, we expose the deep interrelationships imposed by nature between disparate realms of the universe and the amazing consequences of the unifying character of physical law. 3 1=2 −33 Fundamental Physical Constants from connections between the small and the ðRpl = ðGZ=c Þ ∼ 10 cmÞ,andthe 1=2 − Which to Build the Universe large in this universe we jointly inhabit are Planck time ðGZ=c5Þ ð∼ 10 43 sÞ.The One of the profound insights of modern invited to contemplate the examples we remaining quantities of relevance would science in general, and of physics in partic- assembled here. be dimensionless ratios derivable in this ular, is that not only is everything connected, Reducing,eveninapproximatefashion, fundamental theory. For instance, these but that everything is connected quantita- the properties of the objects of the universe to ratios could be tively. Another is that there are physical their fundamental dependencies requires first η = = ð∼ 19Þ constants of nature that by their units, con- and foremost a choice of irreducible funda- p mpl mp 10 η = = ð∼ 22Þ [2] stancy in space and time, and magnitude mental constants. Various combinations of e mpl me 10 η = = ð∼ 20Þ; encapsulate nature’s laws. The constant speed those constants of nature can also be useful, π mpl mπ 10 “ ” of light delimits and defines the fundamental so the word irreducible is used here with character of kinematics and dynamics. great liberty. We could choose among the and α, and, in principle, these ratios could be Planck’s constant tells us that there is some- following: derived in the hypothetical fundamental the- thing special about angular momentum and Z; c; e; G; m ; mπ; m ; [1] ory. We show in this paper that we can write the products of length and momentum and p e astronomical masses in terms of mpl or mp, of energy and time. What is more, in com- with mass ratios and dimensionless com- where Z is the reduced Planck’s constant bination, the constants of nature set the scales binations of fundamental constants setting (lengths, times, and masses) for all objects ðh=2πÞ, c is the speed of light, e is the elemen- the corresponding relative scale factors. We and phenomena in the universe, because tary electron charge, G is Newton’s gravita- thereby reduce all masses to combinations of scales are dimensioned entities and the only tional constant, mp is the proton mass, mπ α α only five quantities: mp, me, mπ, , and g or building blocks from which to construct is the pion mass, and me is the electron mass. m , η , η , ηπ, and α. For radii, Z and c are them are the fundamental constants around It is in principle possible that these masses pl p e explicitly needed. Note that α = 1=η2 and which all nature rotates. Although in part can be reduced to one fiducial mass, with g p that α=α ∼ 1036. The latter is a rather large merely dimensional analysis, such construc- mass ratios derived from some overarching g number, a fact with significant consequences. tions encapsulate profound insights into theory, but this is currently beyond the state diverse physical phenomena. We assume in this simplified treatment of the art in particle physics. However, di- Hence, the masses of nuclei, atoms, stars, that the proton and neutron masses are the mensionless combinations of fundamental and galaxies are set by a restricted collection same and equal to the atomic mass unit, itself constants emerge naturally in the variety of ’ of basic constants that embody the finite the reciprocal of Avogadro sconstant(NA). number of core natural laws. In this paper, contexts in which they are germane. Exam- The pion mass can be the mass of any of the α = 2=Z we demonstrate this reduction to funda- Àples areÁ e c, the fine structure constant threepions,setsthelengthscaleforthenu- ∼ 1 α = 2=Z ð∼ −38Þ mentals by deriving the characteristic masses 137 ; g Gmp c 10 , the corre- cleus, and helps set the energy scale of nu- of important astronomical objects in terms sponding gravitational coupling constant; clear binding energies (6). However, for 1=2 of just a few fundamental constants. In doing and mpl = ðZc=GÞ ,thePlanckmassspecificity and for sanity’s sake, we will as- so, we are less interested in precision than ð∼ 2 × 10−5 gÞ. sume that the number of spatial dimensions illumination and focus on the orders of As noted, if our fundamental theory was is three. Then, with only five constants we magnitude. Most of our arguments are not complete, we would be able to express – original (see refs. 1 5 and references therein), allphysicalquantitiesintermsofonly Author contributions: A.S.B. and J.P.O. performed research and although some individual arguments are. three dimensioned quantities, one each wrote the paper. This exercise will be valuable to the extent for mass, length, and time. Many would The authors declare no conflict of interest. that it provides a unified discussion of the associate this fundamental theory with This article is a PNAS Direct Submission. physical scales found in the astronomical the Planck scale, so everything could be 1To whom correspondence should be addressed. E-mail: burrows@ world. Those interested in the fundamental written in terms of mpl,thePlancklength astro.princeton.edu. www.pnas.org/cgi/doi/10.1073/pnas.1318003111 PNAS | February 18, 2014 | vol. 111 | no. 7 | 2409–2416 Downloaded by guest on October 1, 2021 2 3 1=3 can proceed to explain, in broad outline, Maximum Mass of a White Dwarf: The give PF = ð3π Z neÞ . Because ne = NAρYe, objects in the universe. In this essay, we Chandrasekhar Mass where Ye is the number of electrons per emphasize stars (and planets) and galaxies, Stars are objects in hydrostatic equilibrium baryon (∼0.5), it is easy to show that the = but in principle, life and the universe are for which inward forces of gravity are bal- pressure is proportional to ρ5 3, that the as- amenable to similar analyses (3, 7, 8). anced by outward forces due to pressure sociated γ = 5=3, and that the star is, there- Before we proceed, we make an aside on gradients. The equation of hydrostatic equi- fore, a polytrope. By the arguments above, nuclear scales and why we have introduced librium is such stars are stable. η the pion mass and π. Nucleons (protons and However, as the mass increases, the central neutrons) are comprised of three quarks, and ð Þ dP = − ρ GM r ; [3] density increases, thereby increasing the pions are comprised of quark-antiquark 2 2 dr r fermi momentum. Above cPF ∼ mec ,the pairs. Quantum chromodynamics (QCD) (9) electrons become relativistic. The formula for is the fundamental theory of the quark and where P is the pressure, r is the radius, ρ is the the fermi momentum is unchanged, but the gluon interactions and determines the mass density, and MðrÞ is the interior mass fermi energy is now linear (not quadratic) in properties, such as masses, of composite ρ hɛi × ∼ (the volume integral of ). Dimensional anal- PF.Thisfactmeansthat ne is pro- particles. The proton mass ( 938 MeV/ 4=3 2 ysis thus yields for the average pressure or the portional to ρ and that the white dwarf c ) is approximately equal to ∼ 3 × ΛQCD, central pressure (P ) an approximate relation, becomes unstable (formally neutrally stable, where ΛQCDð∼ 300 MeVÞ is the QCD en- c ∼ GM2 but slightly unstable if general relativistic ergy scale (10), and 3 is its number of Pc R4 , where M is the total stellar mass and constituent quarks (11). The pion, how- R is the stellar radius, and we used the crude effects are included). Hence, the onset of ρ ∼ M relativistic electron motion throughout a ever, is a pseudo-Goldstone boson, and relation R3. the square of its mass is proportional to For a given star, its equation of state, large fraction of the star, the importance of Λ ð + Þ quantum mechanical degeneracy, and the QCD mu md , where mu and md are the connnecting pressure with temperature, up and down bare quark masses, re- density, and composition, is also known in- inexorable effects of gravity conspire to yield a maximum mass for a white dwarf (13, 14). spectively. mu + md is very approximately dependently. Setting the central pressure de- equal to 10 MeV (quite small). In princi- rived using hydrostatic equilibrium equal to This mass, at the confluence of relativity (c), ple, all the hadron masses and physical the central pressure from thermodynamics or quantum mechanics (Z), and gravitation dimensions can be determined from the statistical physics can yield a useful relation (G), is the Chandrasekhar mass, named bare quark masses and the QCD energy between M and R (for a given composition).
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