PERSPECTIVE PERSPECTIVE Astronomical reach of fundamental

Adam S. Burrows1 and Jeremiah P. Ostriker Department of Astrophysical Sciences, , 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 . 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 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- ples are 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). after one of its discoverers. Its value can scale (12), because the associated running For example, if the pressure is a power-law κ 3=2 be derived quite simply using G (Eq. coupling constant (which sets, for example, function of density alone (i.e., a polytrope), 3π2 Zc 4=3 γ 4)andκ = = Y , where we have set 2 = π = κρ ρ ∼ M 4 m4 3 e gπNN 4 ) perforce approaches unity (large such that P ,thenusing R3 and p κργ GM2 = = values) on the very spatial scales that set setting equal to R4 gives us NA 1 mp. The result is particle size. 1 = κ −γ 3 2 2 For our purposes, we assume there are two 2 ð4−3γÞ=ð2−γÞ Zc Y M ∝ R : [4] M ∼ e : [5] fundamental masses in the theory. These G Ch 2 G mp could be ΛQCD and mu + md, but they could also be mp and mπ. We opt for the latter. One notices immediately that M and R are Importantly, pion exchange mediates the decoupled for γ = 4=3. In fact, one can show Note that the electron mass does not occur ∼ strong force between nucleons, and the nu- from energy arguments that if γ is the adia- and that the proper prefactor is 3.1. For = : : 2 = cleon-nucleon interaction determines nuclear batic gamma and not merely of structural Ye 0 5, 3 1Ye is of order unity and MCh binding energies and nuclear energy yields. significance, then at γ = 4=3thestarisneu- 1:46 M⊙. We recovered our first significant Moreover, since the pion interaction is a de- trally stable—changing its radius at a given result. There is a maximum mass for a white rivative interaction (11), the corresponding mass requires no work or energy. In other dwarf, and its value depends only on fun- 2 π ∼ Z nuclear energy scale is not m ( 135 MeV/c ), words, the star is unstable to collapse for damental constants , c, G,andmp,mostof ð 2 = πÞ ðη =η Þ2 γ < = but is proportional to gπNN 4 mπ p π . 4 3 and stable to perturbation and pul- which are of the microscopic world. We can The upshot is that the binding energies of sation at γ > 4=3. For γ = 4=3, there is only 2 κ 3=2 rewritethisresultinnumerousways,some nuclei scale as mπðη =ηπÞ and the size of the one mass, and it is . p G of which are nucleon is ∼ Z=mπc, about one fermi for the Thesignificanceofthisisthatwhitedwarf ! measured value of mπ. Hence, the two fun- stars are supported by electron degeneracy 3=2 Zc mp damental masses, mπ and mp, determine the pressure, fermionic zero-point motion, which ∼ = MCh mp 2 3=2 density of the nucleus and nuclear binding is independent of temperature. Such de- Gmp α g [6] energies, both useful quantities. This fact is generate objects are created at the terminal ∼ η3 MCh mp p the reason we include mπ (or ηπ)inourlistof stages of the majority of stars and are what ’ 2 fundamental constants with which we de- remains after such a star sthermonuclearlife. MCh ∼ mplη : — p scribe the Cosmos reasons for including mp The pressure, which like all pressures re- η (or p) are more self-evident. An ultimate sembles an energy density, is approximately theory would provide the ratios of all the given by the average energy per electron Hence, the Chandrasekhar mass can be couplings (QCD, electro-weak, gravitational) ðhɛiÞ times the number density of electrons considered to be a function of the Planck mass and the ratio, η , or the proton mass and and mass ratios, as well as all other dimen- (ne). If the electrons are nonrelativistic, p hɛi ∼ 2 = the small dimensionless gravitational coupling sionless numbers of nature. Because we are PF 2me,wherePF is the fermi mo- 2 α = Gmp currently absent of such a theory, we proceed mentum, and simple quantum mechan- constant, g Zc . Such relationships are using the quantities described. ical phase-space arguments for fermions impressive in their compactness and in their

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implications for the power of science to explain ρ ∼ mp Of course, the pion mass, mπ,enters PERSPECTIVE nucleus nuc πλ3 = . Although nucleons nature using fundamental arguments. 4 π 3 through its role in the nuclear density, as arefermions,themaximummassofaneu- Z As an aside, we can use the arguments does and mp. c enters through GR and tron star is not determined by the special above to derive a relation for the character- through its role in setting the range of the relativistic physics that sets the Chan- istic radius of a white dwarf. To do this, we Yukawa potential. G enters due the perennial drasekhar mass (15). Rather, the maxi- return to Eq. 4,butinsertγ = 5=3, the non- combat of matter with gravity in stars. All but mummassofaneutronstarissetby mπ are actors in the Chandrasekhar saga, so relativistic value relevant for most of the a general relativistic (GR) instability. GR we should not be surprised to see MCh once family. The degeneracy pressure is then given gravity is stronger than Newtonian gravity, by P = κρ5=3,where again, albeit with a modifying factor. Note and at high enough pressures, pressure that, because neutron stars are formed when = 2 2 3 2 itself becomes a source, along with mass, ð3π Þ Z = a white dwarf achieves the Chandrasekhar κ = Y5 3; [7] to amplify gravity even more. The result is 5 5=3 e mass and critical neutron stars collapse to memp that increasing pressure eventually becomes black holes, the existence of stable neutron self-defeating when trying to support the stars requires that MNS > MCh,withmany, which is an expression easily shown to follow more massive neutron stars and the object many details. When those details are collapses. Whereas the collapse of a Chan- from the general fact that pressure is average accounted for we find that MNS (baryonic) is drasekhar mass white dwarf leads to particle energy times number density, but for about twice MCh, the ratio being comfortably a neutron star, the collapse of a critical neu- the nonrelativistic relation between momen- greater than, but at the same time un- tron star leads to a black hole. comfortably close to, unity. Suffice it to say, tum and energy. We will find this relation The critical mass of a neutron star is useful elsewhere, so we highlight it. Eq. 4 then stable neutron stars do exist in our universe. κ reached as its radius shrinks to near its ∼ 2GM Theradiusofaneutronstarissetbysome yields Rwd 1=3. We now use some sleight- Schwarzschild radius, , say to within a GM c2 multiple of the Schwarzschild radius. The of-hand. Noting that a white dwarf’s mass factor of two (but call this factor β ). GR n baryonic mass of a neutron star is set by its changes little as the electron’s fermi energy physics introduces no new fundamental con- formation and accretion history, but it is transitions from nonrelativistic to relativistic stants and involves only G and c. Therefore, 2GMβ bounded by the minimum possible Chan- and the mass transitions from ∼ 0:5 M⊙ to = n the condition for GR instability is R c2 . drasekhar mass (recall this is set by Y ). 7 e Mch,andusingEq. , we obtain We can derive the corresponding mass by Therefore, if we set a neutron star’smass ρ ∼ 3M Z using the relation 4πR3. The result is equal to its maximum, we will not be far off. R ∼ η ! = wd p 3=2 3 2 The upshot is the formula mec Zc 1 ηπ [8] M ∼ 3=2 ∼ η η NS 2 β η 2G Zc 1 Z Rpl e p G mp 2 n p ∼ ∼ η ∼ η2: Rns 2 2 2 p Rpl p 4 ! c G m mpc ∼ 10 km 3=2 p η ∼ π 8 MCh β η This equation is very much like Eq. for the ∼ 2 n p where we have set M MCh. This relation [9] radius of a white dwarf, with mp substituted ! = powerfully connects the radius of a white 3 2 for me. Therefore, the ratio of the radius of mp ηπ dwarf to the product of the electron Compton ∝ a white dwarf to the radius of a neutron star α3=2 2β η wavelength and the ratio of the Planck mass g n p is simply the ratio of the proton mass to the ! ðη =η Þ to the proton mass. It also provides the white 3=2 electron mass e p , within factors of order ηπ η =η ∼ ∝ η2 : unity. In our universe, e p is 1,836, but dwarf radius in terms of the Planck radius. mpl p β η 2 n p we can call this “three orders of magnitude.” Putting numbers in yields something of order Ten-kilometer neutron stars quite natu- a few thousand kilometers, the actual mea- β ð∼ ð ÞÞ 3 4 Dropping n O 1 , MNS is also even rally imply ∼ 5 × 10 to ∼ 10 -km white sured value. We will later compare this radius 3=2 ηπ dwarfs. By now, we should not be surprised more simply written as MCh η .The with that obtained for neutron stars and de- p that this ratio arises in elegant fashion from rive an important result. Planck mass and the Chandrasekhar mass fundamental quantities using basic physical appear again, but this time accompanied arguments. Maximum Mass of a Neutron Star by the proton-pion mass ratio and un- : 2 β ∼ Neutron stars are supported against gravity accompanied by 3 1Ye .For n 2, given the Minimum Mass of a Neutron Star by pressure due to the strong repulsive nu- values of mp and mπ with which we are fa- The minimum possible mass (Mns)forasta- clear force between nucleons. Atomic nuclei miliar, MNS is actually close to MCh.Detailsof ble neutron star may not be easily realized in are fragmented into their component sub- the still-unknown nuclear equation of state, nature. Current theory puts it at ∼ 0:1 M⊙ atomic particles and experience a phase GR, and the associated density profiles (16), and there is no realistic mechanism by ∼ : transition to nucleons near and above the yield values for MNS between 1 5and which a white dwarf progenitor near such density of the nucleus. As we have argued, ∼ 2:5 M⊙, where eventually we would need a mass, inside a star in its terminal stages or the interparticle spacing in the nucleus is to distinguish between the gravitational mass in a tight binary, can be induced to collapse set by the range of the strong force, which andthebaryonicmass.Wenoteherethatthe to neutron star densities. The Chandrasekhar is the reduced pion Compton wavelength fact that the two common types of degenerate instability is the natural agency, but MCh λπ = Z=mπc.Therefore,becausetherepulsive stars—end products of stellar evolution— is much larger than ∼ 0:1 M⊙. However, nuclear force is so strong, the average density havenearlythesamemassderivesinpart such an object might form via gravita- of a neutron star will be set (very ap- from the rough similarity of the pion and tional instability in the accretion disks of proximately) by the density of the proton masses. rapidly rotating neutron stars of canonical

Burrows and Ostriker PNAS | February 18, 2014 | vol. 111 | no. 7 | 2411 Downloaded by guest on October 1, 2021 η 3 3 mass. Mass loss after the collapse of a p mπ central pressure of a constant density object by the small factor η .Note Chandrasekhar core is possible by Roche- π mp in hydrostatic equilibrium. The result is : lobe overflow in the tight binary, but the = 4 5 = ∼ ð β Þ3 2 ∝ mπ that Mns MNS 2 nf m , which 1=3 associated tidal potential is quite different p 1 4π = = P = GM2 3ρ4 3 [11] from the spherical potential for which Mns fortunately is a number less than one. How- 2 3 0 is derived. ever, to derive the precise value for Mns re- Nevertheless, a first-principles estimate of quires retaining dropped factors and much The other method is to use the radius-mass Mns is of some interest. What is the physics more precision, but the basic dependence on that determines it? A white dwarf is sup- fundamental constants is clear and revealing. relations for both white dwarfs and constant- ported by the electron degeneracy pressure of density planets and to set the radius obtained Maximum Mass of a Rocky Planet free electrons, and its baryons are sequestered using one relation equal to that using the A rocky planet such as the Earth, Venus, in nuclei. A neutron star is supported by the other. Stated in equation form, this is Mars, or Mercury is comprised of silicates = repulsive strong force between degenerate ∼ κ ∼ ∼ 3M 1 3 and/or iron and is in hydrostatic equilibrium. Rwd 1=3 Rrock 4πρ . For both meth- free nucleons, and most of its nuclei are GM 0 If its composition was uniform, its density ρ dissociated. Both are gravitationally bound. ods, one needs the density, 0, and using ei- wouldbeconstantatthesolid’slaboratory One can ask the question: for a given mass, ther method one gets almost exactly the same value. This value is set by the Coulomb in- ρ = Z2 result. is set by the Bohr radius, aB 2, which of the two, neutron star or white 0 mee teraction and quantum mechanics (which and one then obtains dwarf, is the lower energy state? Note that establishes a characteristic radius, the Bohr to transition to a neutron star, the nuclei μ radius). As its mass increases, the interior ρ ∼ 3mp ; [12] of a white dwarf must be dissociated into ’ 0 π 3 pressures increase and the planet smatteris 4 aB nucleons, and the binding energy of the nu- compressed, at first barely and slowly (be- cleus must be paid. Note also that a neutron cause of the strength of materials), but later where μ is the mean molecular weight of the star, with its much smaller radius, is the more (at higher masses) substantially. Eventually, constituent silicate or iron (divided by m ). A gravitationally bound. Therefore, we see that for the highest masses, many of the electrons p bit of manipulation yields when the specific (per nucleon) gravitational in the constituent atoms would be released ! binding energy is equal to the specific nuclear into the conduction band, the material would 3=2 2 binding energy of the individual nuclei, we e 6 = = be metalized (17), and the supporting pres- M ∼ m Y5 2μ1 3 rock p 2 π e are at Mns. Above this mass, the conversion sure would be due to degenerate electrons. Gmp of an extended white dwarf into a compact Objects supported by degenerate electrons α 3=2 neutron star and the concomitant release of are white dwarfs. Therefore, at low masses ∼ [13] mp α gravitational binding energy can then pay the and pressures, the (constant) density of g necessary nuclear breakup penalty. However, = materials and the Coulomb interaction imply ∼ M α3 2 of course, a substantial potential barrier must R ∝ M1=3, whereas at high masses, the object Ch ∼ η2α3=2: be overcome. acts like a white dwarf supported by electron- mpl p In equation form, this can be stated as degeneracy pressure and R ∝ M−1=3.This 2 GM ∼ ΔMc2,whereΔMc2 is the binding behavior indicates that there is a mass at R Note that Mrock is actually independent of Z, energy of the nucleus. This equation can be which the radius is a maximum for a given c, and m . rewritten as GM ∼ ΔM = f ,wheref is the composition (18). It also implies that there e Rc2 M This equation expresses another extraor- binding energy per mass, divided by c2.From is a mass range over which the radius of the dinary result: M can be written to scale thefactthatthenucleonisboundinanu- object is independent of mass and dR ∼ 0. For rock dM with the Planck mass and η or the Chan- η 2 p 2 p hydrogen-dominated objects, this state, for cleus with an energy ∼ mπc η , we obtain drasekhar mass, but multiplied by a power of π which Coulomb and degeneracy effects bal- η 3 the fine-structure constant. The latter comes p ance, is where Jupiter and Saturn reside, but f ∼ η (an appropriate extra factor might π one can contemplate the same phenomenon from the role of the Coulomb interaction in ∼ ∼ – be 3). In our universe, f is 0.01 0.02. for rocky and iron planets. It is interesting to setting the size scale of atoms and also indi- Now, notice that we can rewrite this con- ask the following question: what is the max- cates that Mrock is much smaller than MCh,as 2GM ∼ 13 dition as Rc2 2f , which is the same as the imum mass ðMrockÞ of a rocky planet, above one might expect. As Eq. also shows, Mrock condition for the maximum neutron star which its radius decreases with increasing scales with mp, amplified by the ratio of the 1 β 9 mass, with 2f substituted for n in Eq. . mass in white-dwarf-like fashion, and below electromagnetic to the gravitational fine Using exactly the same manipulations, we which it behaves more like a member of a structure constants to the 3/2 power. Plug- then derive constant-density class of objects? This defi- ging numbers into either of these formulas, ∼ × 30 η 3=2 nition provides a reasonable upper bound to one finds that Mrock is 2 10 g ∼ 3=2 π ∼ ∼ Mns MChf η themassofarockyorsolidplanet.Withthe 300 MEarth 1 MJup. Detailed calcula- p anticipated discovery of many exoplanet tions only marginally improve on this es- η 3 = = p [10] “Super-Earths” in coming years, this issue 5 2μ1 3 ∼ M timate. Note that reinstating the Ye Ch η is of more than passing interest. π dependence yields roughly the same value η 3 There are two related methods to de- for both hydrogen and iron planets. ∼ η2 p : mpl p rive Mrock. The first is to set the expres- ηπ sion for nonrelativistic degeneracy pressure Maximum Mass of a Star = κρ5=3 κ 7 The Planck mass and MCh emerge again (P0 0 ,where is given by Eq. )equal The fractional contribution of radiation pres- π2k4 = ð = Þ 4 = B (quite naturally), but this time accompanied to the corresponding expression for the sure [Prad 1 3 arad T ,wherearad 15Z3c3

2412 | www.pnas.org/cgi/doi/10.1073/pnas.1318003111 Burrows and Ostriker Downloaded by guest on October 1, 2021 ’ ð Þ PERSPECTIVE and kB is Boltzmann s constant] to the total The ratio of MS to MCh Ye depends only energy from its surface occasions further μ β supporting pressure in stars increases with on , Ye, , and some dimensionless con- gradual quasi-hydrostatic contraction. In mass. At sufficiently large mass, the gas stants. Therefore, it is a universal number. parallel, the central temperature increases β = = = : μ ∼ “ becomes radiation dominated, and Prad For 1 2, Ye 0 5, and 1, we find via what is referred to as the negative spe- ρ kBT = ∼ β ” exceeds the ideal gas contribution P = . MS MCh 20, but for smaller s(agreater cific heat effect in stars. Energy loss leads to IG μmp The entropy per baryon of the star also rises degree of radiation domination), the ratio is temperature increase. From energy conser- 1 β ≡ = ∝ 2 β vation, the change in gravitational energy with mass. If we set P PIG,whereP larger β .Therelevant and actual ratio 2 + β (loosely −GM ) due to the shrinkage is equal PIG Prad, thereby defining ,wederive(19) depend on the details of formation and wind 2R to the sum of the photon losses and the mass loss (and, hence, heavy element abun- − β 1=3 1=3 increase in thermal energy, in rough equi- = 1 3kB kB ρ4=3: [14] dance), and are quite uncertain. Nevertheless, P 4 partition. Using either hydrostatic equilib- β μmparad μmp indications are that the latter ratio could range from ∼50 to ∼150 (21). rium or the Virial theorem and the ideal As an aside, we note that one can derive gas law connecting pressure with tempera- The upshot is that the effective polytropic ρ = kBT γ another (larger) maximum mass associated ture P μ , we obtain gamma (defined assuming P ∼ κρ )decreases mp with general relativity by using the fact that from ∼5/3 toward ∼4/3asthestellarmass μ the fundamental mode for spherical stellar ∼ GMmp ∼ μ 2=3ρ1=3; [16] increases. As we argued in the section on the kBTint G mpM pulsation occurs at a critical thermodynamic 3R Chandrasekhar mass, the onset of relativity γ † gamma ( 1). Including the destabilizing ef- makes a star susceptible to gravitational insta- γ fect of general relativity, this critical 1 equals where μ is the mean molecular weight (of = + GM ∼ bility. Photons are relativistic particles. A radia- 4 3 K 2 ,whereK 1. A mixture of ra- 1=3 Rc ∼ 3M tion pressure–dominated stellar envelope can diation with ideal gas for which radiation is order unity), and R 4πρ has been used. γ ∼ = + β= easily be ejected if perturbed and at the very dominant has a 1 of 4 3 6(6).One We note that Eq. 16 demonstrates that the least is prone to pulsation if coaxed. Such coax- could argue that, if the temperature is high temperature increases as the star shrinks. ingand/orperturbationcouldcomefromnu- enough to produce electron-positron pairs, However, the negative specific heat is an γ clear burning or radiation-driven winds. With then 1 would plummet and the star would indirect consequence of the ideal gas law. As radiation domination, and given the high opac- be unstable to collapse. Hydrostatic equilib- the protostar contracts and heats, its density rium suggests that the stellar temperature ity of envelopes sporting heavy elements with rises.Indoingso,thecoreentropydecreases times Boltzmann’s constant when pairs start high Z, radiation pressure–driven winds can (despite the temperature increase) and the to become important would then be some core becomes progressively more electron blow matter away and in this manner limit 2 GMmp fraction of ∼mec and would equal . the mass that can accumulate. This physics sets R degenerate. When it becomes degenerate, be- This correspondence gives us an expres- cause degeneracy pressure is asymptotically the maximum mass (MS) of a stable star on the GM ð ∼η =η Þ sion for Rc2 p e and, therefore, that independent of temperature, further energy hydrogen-burning main sequence. β= ∼ = = η =η 15 6 me mp p e.UsingEq. ,wefind loss does not lead to a temperature increase Using Eq. 4,withγ = 4=3, Eq. 14,andthe a maximum stellar mass due to general- but a decrease. Hence, there is a peak in the same arguments by which we derived the relativistic instability and pair production ∼ 6 core temperature that manifests itself at the Chandrasekhar mass, we find the only mass near 10 M⊙, close to the expected value onset of core degeneracy. The standard ar- κ (22). Because other instabilities intervene for a given gument states that if at this peak temperature before this mass is reached, this limit is not ð Þ κ 3=2 Tmax the integrated core thermonuclear ∼ rad relevant for the main sequence limit. MS power is smaller than the surface luminosity G However, the facts that (i) M is much S (also a power), then the star will not achieve 1=2 1=2 3=2 greater than M and (ii) their ratio is a large 1 − β 45 Zc 1 Ch the hydrogen main sequence. It will become ∼ constant are interesting. Therefore, we can β4 π2 G m2μ2 a brown dwarf, which will cool inexorably p take some comfort in noting that because M [15] S 1=2 1=2 into obscurity, but over Gigayear timescales. 1 − β 45 MCh is much larger than MCh,whitedwarfs, ∼ The mass at which Tmax is just sufficent for 4 neutron stars, and stellar-mass black holes, all β π2 μ2 core power to balance surface losses is the stellar “residues” that would be birthed in = = 1 2 1 2 minimum stellar mass (Ms). Below this mass 1 − β 45 2 stars are allowed to exist. ∼ m η =μ ; β4 π2 pl p istherealmofthebrowndwarf.Aboveit Minimum Mass of a Star reside canonical, stably burning stars (23). Stars are assembled from interstellar medium We can use Eq. 16 to derive the critical where MCh is our Ye-free Chandrasekhar gas by gravitational collapse. As they con- mass in terms of Tmax. A gas becomes de- mass. Because kB has no meaning indepen- tract, they radiate thermal energy and the generate when quantum statistics emerges to dent of temperature scale, it does not appear compact protostar that first emerges is in be important. For an electron, this is when 15 in Eq. . That MCh occurs in MS should not quasi-hydrostatic equilibrium. Before achiev- the deBroglie wavelength of the electron be surprising, because in both cases the onset ing the hydrogen-burning main sequence, the (λ = h ,wherev is the average particle speed) mev μ 1=3 of relativity in the hydrostatic context, for protostar becomes opaque and radiates from mp approaches the interparticle spacing ρ . electrons in one case and via photons in the a newly formed photosphere at a secularly evolving luminosity. The progressive loss of It is also when the simple expression for the other, is the salient aspect of the respective ρ ideal gas pressure = kBT equals the simple limits (20). In determining both MCh and μmp

Z † MS, relativity (c), quantum mechanics ( ), γ is the logarithmic derivative of the pressure with respect to the expression for the degeneracy pressure 1 = and gravity (G) play central roles. mass density at constant entropy. (P = κρ5 3,whereκ is given by Eq. 7). We use

Burrows and Ostriker PNAS | February 18, 2014 | vol. 111 | no. 7 | 2413 Downloaded by guest on October 1, 2021 the latter condition and derive an expression is determined by the method of steepest de- magnetism and gravity. The last expression 21 for Tmax scent, whereby the product of the Boltzmann for Ms in Eq. reveals something else. It is the same as the expression in Eq. 13 = and Gamow factors is approximated by the μ G2μm8 3m ∼ mp 2 4=3 ∼ p e 4=3: exponential of the extremal value of the argu- for the mass of the largest rocky planet kBTmax G M = M = κ Z2 5 3 ment: − E − pbffiffi. That Gamow peak energy α 3 2 Ye kBT m , but with an additional factor of E = p αg [17] ð Þ ð = Þ2 3 = Egam is bkBT 2 . The result is a term ðη =η Þ3 4 ∼ e p .Thisfactoris 300 and is, as one

− E −pbffiffi −3Egam would expect, much larger than one. As k T E → k T ; [19] ThetextbooksstatethatMs is then derived e B e e B a result, we find that, given our simplifying by setting Tmax to some ignition temperature μ 6 assumptions and specific values for Ye and , (frequently set to 10 K), and then solving for = ðη =η Þ3=4 in the thermonuclear rate. Although the surface Ms Mrock depends only on e p .For M in Eq. 17.Inthisway,usingmeasured = luminosity depends on metallicity and details of measured values of mp and me, Ms Mrock is constants and retaining a few dropped co- ∼ ∼ : the opacity, that dependence is not very strong then 100; the mass of the lowest mass star efficients, one derives Ms 0 1 M⊙ and this exceeds that of the most massive rocky number is rather accurate. (23). Furthermore, due to the Boltzmann and Gamow exponentials (Eq. 19), any characteristic planet. If we now scale Ms to the detailed However, this procedure leaves unex- theoretical value of M , we obtain a value ∼ 6 ‡ temperaturewemightderive(whichmightbe rock plained the origin of Tmax 10 K and here for M that is within a factor of 2–3ofthe set equal to T ) depends only logarithmically s we depart from the traditional explanation to max correct value. This quantitative correspon- on the problematic rate prefactor. This argu- introduceourown.Therearetwothingsto dence is a bit better than might have been ment allows us to focus on the argument of note. First, Ms is not determined solely by expected but is heartening and illuminating — 19 thermonuclear considerations photon opac- the exponential in Eq. . Therefore, we set nevertheless. ities, temperatures, densities, and elemental 3Egam in Eq. 19 equal to a dimensionless number kBT abundances (metallicity) at the stellar surface fg of order unity (but larger) and presume that Characteristic Mass of a Galaxy set the emergent luminosity that is to be it, whatever its value, is a universal dimension- For stars and planets, we were not concerned balanced by core thermonuclear power. Sec- less number, to within logarithmic terms. The with whether they could be formed in the ond, the ignition temperature is not some resulting equation for the critical T is universe (nature seems to have been quite fundamental quantity but is in part de- fecund, in any case), but with the masses (say, termined by the specific nuclear physics of = 27 α2 2 [20] maximum and minimum) that circumscribed kBTmax 3 mpc the relevant thermonuclear process, in this 16fg and constrained their existence. However, be- case the low-temperature exothermic proton- cause of the character of galaxies (as accu- 3 4 proton reactions to deuterium, He, and He. In Eq. 20, both α and m come from the 2e2 mulations of stars, , and gas), the Hence, we would need the details of the in- p Zv mass we derive for them here is that for the term associated with Coulomb repulsion in teraction physics of the proton-proton chain, typical galaxy in the context of galaxy for- Gamow tunneling physics. Setting Tmax in integrated over the Maxwell-Boltzmann dis- 20 17 mation, and we find this approach to be the tribution of the relative proton-proton ener- Eq. equal to Tmax in Eq. , we obtain most productive, informative, and useful. ! ! ! gies and over the temperature and density 2 3=4 3=4 First, we ask the following. Is it obvious on η η 27π2 Y5=4 profiles of the stellar core. However, a simpler M ∼ m p eα2 e simple physical grounds that a galaxy mass s pl 9=8 η μ3=4 approach is possible (20). We can do this fg p 8 will be much greater than a stellar mass, i.e. ! that a galaxy will contain many stars? The because the thermonuclear interaction rate is 3=4 η π2 3=4 5=4 greatest at large particle kinetic energies, MCh e 2 27 Ye distribution of the stellar masses of galaxies − = ∼ α ð∝ E kBT Þ 9=4 η μ3=4 ðM ; Þ is empirically known to satisfy the which are Boltzmann suppressed e , fg p 8 star gal and because Coulomb repulsion between ! ! ! so-called Schechter form with probabilities 3=2 3=4 3=4 the protons requires that they barrier pen- 2 η π2 5=4 distributed as ∼ e e 27 Ye mp = etrate (quantum tunnel) to within range of 2 η 3 μ3 4 −α p Gmp p 8fg p p − = dP ∝ M ; =M e Mstar;gal M the nuclear force. This fact introduces the ! ! star gal [22] 3=2 3=4 p Gamow factor (24) α η 3 d M ; =M ; ∝ e star gal mp α η −2e2 −2αc −pbffiffi g p Zv = v = E ; [18] e e e ! with α typically having the value of ∼1.2 and 3=4 p η p ∝ η2α3=2 e : M a constant with units of mass. Given this = = μ 2 μ mpl p η where E 1 2 pv , p here is the reduced p form, with a weak power law at low masses ð = Þ proton mass mp 2 , v is the relativeqffiffiffiffiffiffiffiffiffiffi speed, [21] and an exponential cutoff at high masses, α μ = and this equation defines b as c p 2. The there is a characteristic mass for galaxies, h i ∼ Mstar;gal , the value being somewhat greater reaction rate contains the product of the If we now set fg 5 and use reasonable values p μ than M , or roughly 1011 M⊙; galaxies much Boltzmann and Gamow exponentials and for Ye and , we finally obtain a value of ∼ : more massive than this are exponentially rare. the necessary integral over the thermal Max- 0 1 M⊙ for Ms. At the other extreme, there is very little mass well-Boltzmann distribution yields another Importantly, we tethered the minimum bound up in galaxies having masses much less exponential of a function of T. That function main sequence mass to the fundamental con-  7 ⊙ stants G, e, me,andmp,butinfactc and h than 10 M . The observed range of masses have cancelled! Ms depends only on mp and seems to be set by fundamental physics in ‡ 6 A better number for Tmax is ∼ 3:5 × 10 K. the ratio α=αg in a duel between electro- that it does not appear to be very dependent

2414 | www.pnas.org/cgi/doi/10.1073/pnas.1318003111 Burrows and Ostriker Downloaded by guest on October 1, 2021 on the epoch of galaxy formation or the en- galaxies near the observed lower bound is of Eq. 24 indicates that the cross-over tem- PERSPECTIVE vironment in which the galaxies are formed the order 30 km/s (26). perature, below which recombination cool- 2 (e.g., in clusters, groups, or the field). The physical argument for the upper ing predominates, is ∼ α2mec ,which,using kB The theory of galaxy formation is by now bound and the typical mass is somewhat measured numbers, is ∼ 3 × 105 K. The fairly well developed, with ab initio hydro- more complex. A collapsing proto-galactic temperatures of relevance during the incipient ρ dynamic computations based on the stan- clump has an evolving density ( )andtem- stages of galaxy formation are not much larger dard cosmological model providing reason- perature (T). Its collapse timescale (tf)isset than this, so we can neglect the Aff term in Eq. ably good fits to the formation epochs, by the free-fall timep dueffiffiffiffiffiffi to gravitation and is 23 Λ ∝ ρ2 = ρ and find that C T1=2. For an ideal gas, the masses, sizes and spatial distributions of gal- proportional to 1 G .Astheclumpcol- 3=2ρkBT internal energy density is μ .Therefore, lapses and the temperature and density rise, mp axies, although this theory still provides ρkBT 3=2 1=2 tc ∼ Λ ∝ T =ρ. Because tf ∝ 1=ρ , tc=tf rather poor representations of their detailed the gas radiates photons. Its associated char- mp C is proportional to T3=2=ρ1=2; this is propor- interior structures. For a recent review of acteristic cooling time (tc)istheratioofthe some of the outstanding problems, see internal energy density to the cooling rate. tional to MJeans, the Jeans mass! Therefore, = ∼ Ostriker and Naab (25). In the standard Because the cooling rate per gram (Eq. 23) we find that the tc tf 1 condition filters out a = ∼ ΛCDM cosmological model, the mean den- scales as density to a higher power than the specific mass. What is its value? From tc tf 1 ð= Þ sity of matter at high redshifts is slightly less free-fall rate, only the most overdense per- and the proper expression for MJeans Mgal , than the critical density for bound objects to turbations can cool on a time comparable to we derive ! form. However, there is a spectrum of per- the free-fall time. Those for which tc tf 1=2 σ will never form stars on either the free-fall α5 η turbations such that those that are several M ∼ m e gal pα2 η more dense than average are gravitationally time or the only somewhat (factor of 20) g p longer Hubble time. Moreover, because the bound and will collapse, with the dark matter ! = highest-density regions are exponentially η 1 2 and the baryons forming self-gravitating ∼ η α5 e [25] MCh p lumps of radius Rhalo, determined by the re- rare, perturbations having tc tf are un- η ∼ p quirement that the density of the self-gravi- common. Thus, the condition tf tc sets ! = tating system formed in the collapse is several a natural and preferred scale for galaxy for- η 1 2 ∼ η3α5 e : hundred times the mean density of the uni- mation. How does this scale compare with mpl p η ð Þ p verse at the time of the collapse (26). The the Jeans mass MJeans ,themassforwhich temperature of the gas (absent cooling) will gravitational and thermal energies are in 2 25 be the virial temperature (C ∼ GMtot =Rhalo, balance and which is proportional to Note that, with Eq. , we obtained Mgal not 3=2=ρ1=2 ∼ η η α where C is the speed of sound). If the gas can T ?Inprinciple,thetf tc condition only in terms of mpl, p, e, and , but should cool via radiative processes given its tem- will imply a relationship between T and ρ we wish to so express it, in terms of the familiar that might yield Mstar;gal sandMJeans that are  η η α perature and density, it will further collapse constants G,h,andc (as well as e, p,and ). functions of ρ or T and, hence, may not be η to the center of the dark matter halo within Note also that p is a very large number and which it had been embedded and will form universal. A wide range of values for Mgal more than compensates for the smallness of α5. would vitiate the concept of a preferred a galaxy, some fraction of the gaseous mass The appearance of m , c,andα is a natural mass scale. e being formed into stars and a comparable consequence of cooling’s dependence on elec- fraction ejected by feedback processes sub- However, here nature comes to the rescue. tromagnetic processes. As an indication of the sequent to star formation. The cooling rate (energy per volume per time) of an ideal gas of hydrogen can be ap- importance of quantum mechanics in determin- The lower bound for normal galaxy masses  is not well understood but is thought to be proximated(followingrefs.7,8,and28)by inggalaxycharacteristics,hdoes not cancel. regulated by mechanical energy input pro- the formula For measured values of the fundamental constants and retaining the prefactor drop- cesses such as stellar winds and supernovae, ρ2 Λ ∼ + ; [23] 25=2π7=2 all primarily driven by the most massive C Abf Aff T ped in Eq. 25 , but retained in Silk T1=2 27 ð≥20 M⊙Þ stars comprising approximately 11 (8), we find Mgal ∼ 10 M⊙, reassuringly one-sixth of the total stellar mass. One can close to the characteristic mass in stars ðMpÞ of easily show that roughly 1051 ergs in me- where Abf is the bound-free (recombination) p rate coefficient, and A is the corresponding the average L galaxy in our universe. More- chanical energy input per star is able to drive ff over, one can derive a Jeans length and, hence, free-free (bremsstrahlung) rate coefficient. For winds from star-forming galaxies with ve- a length scale for this average galaxy. It is locities of ∼300 km/s, and Steidel et al. (27) the exploratory purposes of this study, these ! ! 1=2 3=2 observed such winds to be common. Because coefficients are Z η η R ∼ α3η2 e ∼ R α3 e ; velocities of this magnitude are comparable = gal p η pl η 9=2π1=2 4α 1 2 mec p p ∼ 2 e kB to the gravitational escape velocities for sys- Aff = = 33 2 3 2 2 2 tems less massive than our Milky Way, gal- me mpc [24] an expression that scales with the Compton axy formation becomes increasingly inef- 2 2mec ficient for low-mass systems and essentially Abf ∼ α Aff : wavelength of the electron. Plugging in mea- kB ∼ ceases when the sound speed of gas photo- sured numbers gives Rgal 50 kiloparsecs, heated by the ultraviolet radiation from a number well within reason in our universe 23 24 α massive stars (10 → 20 km/s) approaches the In Eqs. and , me, e,and appear due to (and, in fact, for our own Milky Way). escape velocity of low-mass dark matter the importance of electromagnetic radiation Although it is reassuring that the critical halos.Theresultisalowerboundfornormal processes. As Eq. 23 suggests, free-free cooling mass that appears from our dimensional galaxies and, in fact, the escape velocity from exceeds bound-free cooling at high temperatures. analysis corresponds well to the upper mass

Burrows and Ostriker PNAS | February 18, 2014 | vol. 111 | no. 7 | 2415 Downloaded by guest on October 1, 2021 range of normal galaxies, we are left with two Eq. 26 reduces the maximum mass of a rocky 3=2 M=MCh < α ; questions. First, there exist galaxies that are planet ðM Þ, the minimum mass of a star ∼ p rock up to 10 times greater in mass than M ; (M ), the maximum mass of a star (M ), the how do these form? Second, what happens to s S which is comfortably smaller than the mini- maximum mass of a white dwarf (M ), the dark matter lumps that are much more Ch mum mass of stars. Finally, normal galaxies massive than this critical mass and are dense maximummassofaneutronstar(Mmax), have a characteristic mass enough to collapse? The answer to the first the minimum mass of a neutron star (Mns), = ∼ α5 = = 1=2; question is becoming observationally clear. and the characteristic mass of a galaxy (Mgal)to M MCh mpl mp mp me All of these supergiant galaxies are brightest only five simple quantities and makes clear cluster galaxies (BCGs). We now know that a natural mass hierarchy related only to mp, which is larger than the characteristic mass of they form at early times, reach a mass com- particle mass ratios, and the αs. We expressed both normal low mass stars and even the p parable to M , and cease star formation but the basics of important astronomical objects most massive stars by a very large factor. keep on growing in mass (by roughly a factor with only five constants, modulo some dimen- Everything astronomical is indeed con- – – of 2 3) and size (by roughly a factor of 4 8). sionless numbers of order unity. Eq. 26 summa- nected, and that the essence of an object can “ The process by which this happens is ga- rizes the interrelationships imposed by physics be reduced to a few central quantities is one lactic cannibalism” (29), by which gravita- between disparate realms of the cosmos. of the amazing consequences of the unifying tionally induced dynamical friction causes Alternatively, it is instructive to put these character of physical law. Indeed, in this ex- the inspiral of satellite galaxies to merge with same relations into a somewhat more familiar ercise, we focused on stars, planets, and gal- the central galaxy. Thus, the excessive mass form. We drop all dimensionless constants of axies, avoided complexity, eschewed any hint of BCGs is caused by a distinct process of order unity and summarize the relations de- that emergent phenomena might be of fun- mass growth. With regard to the second rived for the masses of astronomical bodies in damental import, and ignored topics such as question, the answer again lies in observa- unitsoftheChandrasekharmass(whichis life and the ubiquitous complexity that clut- tions of groups and clusters. If these giant, close to a solar mass), the three particle masses ters most experience. Rather, our goal here self-gravitating units (dark matter halos) have p (mp, mπ,andme), and the Planck mass, mpl, was to understand and articulate the simple total masses far above ðΩmatter =ΩbaryonÞ M , 3 with MCh ∼ mpðmpl =mpÞ ∼ 1M⊙. connections inherent in the universal opera- i.e. greater than 1012 M⊙, then they host not We found that neutron stars can exist tion of a small number of physical principles one giant mass galaxy but rather a distribu- within the range and fundamental constants and to identify tion of galaxy masses, the distribution of the ties between seemingly unrelated, but key, which is given by Eq. 22.§ Thus, we find it = 3 < = < = 3=2: astronomical entities. We hope we conveyed natural that most of the mass in the universe mπ mp M MCh mp mπ is in stellar systems containing roughly to the reader that not only are these con- 1011 stars, each with mass between the limits Normal stars can exist within the range nections knowable and quantifiable but that they are both simple and profound. Ms and MS, which both scale with the 2 3=4 Chandrasekhar mass. mp=me α < ðM=MChÞ < ∼ 50; ACKNOWLEDGMENTS. We thank Jeremy Goodman and Neta Bahcall for insightful comments on an earlier Conclusion draft of this manuscript and Paul Langacker for stimulat- We now recapitulate in succinct form most and rocky planets can exist with masses ing conversations. of the masses discussed in this paper. First, we express our results in the most funda- 1 Weisskopf VF (1975) Of atoms, mountains, and stars: A study in 16 Haensel P, Zdunik JL, Douchin F (2002) Equation of state of dense mental units – = qualitative physics. 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