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Home , Chandrasekhar limit, Human scale

The Planck and the Chandrasekhar limit ͒ David Garfinklea Department of Physics, Oakland University, Rochester, Michigan 48309 ͑Received 6 November 2008; accepted 10 March 2009͒ The Planck mass is often assumed to play a role only at the extremely high scales where quantum gravity becomes important. However, this mass plays a role in any physical system that involves gravity, , and relativity. We examine the role of the Planck mass in determining the maximum mass of . © 2009 American Association of Physics Teachers. ͓DOI: 10.1119/1.3110884͔

I. INTRODUCTION ery part of the gas is in equilibrium and thus must have zero net on it. In particular, consider a small disk of the gas The Planck mass, length, and time can be constructed with base area A and thickness dr at a distance r from the from the G, the Planck con- center of the ball. Denote the mass of the disk by m , and the ប 1 d stant , and the c. In particular, the Planck mass of all the gas at radii smaller than r by M͑r͒. Newton mass, MP is given by showed that the disk would be gravitationally attracted by បc the gas at radii smaller than r, but unaffected by the gas at M = ͱ . ͑1͒ radii larger than r, and the force on the disk would be the P G same as if all the mass at radii smaller than r were concen- The Planck mass is very small on the human scale because trated at the center. In other words, the inward force due to Ϸ −8 MP 10 kg. It is quite large on the scale of elementary gravity on the disk has a magnitude / Ϸ 19 particles because MP mH 10 , where mH is the mass of a GM͑r͒m d ͑ ͒ . This large ratio is often referred to as the Fin = 2 . 2 “hierarchy problem;” the “problem” is that such a large ratio r seems unnatural and thus requires an explanation. Because ␳ The volume of the disk is Adr, and so we have md = Adr, present day particle accelerators probe correspond- where ␳ is the of the gas at radius r. Thus using Eq. 2 ing to a few hundred mHc , present day accelerators are very ͑2͒ we find 2 far from the much higher scale of MPc , the energy scale at GM͑r͒␳Adr which quantum gravity becomes important. F = . ͑3͒ One might expect that the Planck mass occurs only in very in r2 exotic, highly speculative, and completely theoretical areas of physics. However, because the Planck mass is constructed If gravity is pulling the disk inward, then why does it not fall G ប c and the whole gas ball collapse? Because differ- from only , , and , it occurs in cases where gravity, ͑ ͒ quantum mechanics, and relativity all play an essential role. ences push the disk outward. Let P r be the pressure of the One such problem is the Chandrasekhar limit, the maximum gas as a function of radius. Then the outward force exerted mass of a white dwarf .2 All stars are held in equilibrium on the inner surface of the disk is P͑r͒A. This force is almost by the opposing of self-gravity and pressure. Thus but not exactly balanced by the inward force of P͑r+dr͒A. gravity plays a role in the properties of stars. In white dwarf The net outward force on the disk due to pressure is stars the pressure is supplied by the motion of due dP ͑ ͒ ͑ ͒ ͑ ͒ to the uncertainty principle and the Pauli exclusion principle. Fout = P r A − P r + dr A =−A dr. 4 Thus quantum mechanics plays a role. In the most extreme dr white dwarf stars, those nearing the maximum possible mass, Because the inward force and outward force must cancel and the electrons are moving so fast that their speeds are compa- thus must have equal magnitudes, it follows that Fin=Fout, rable to the speed of light, and thus relativity plays an essen- and therefore tial role. In this paper we will give a simple derivation of the Chan- dP GM͑r͒␳ =− . ͑5͒ drasekhar limit and show its connection with the Planck dr r2 mass. We define N ϵM /m and call this large dimension- h P H ͑ ͒ less number the hierarchy number. also presents us As we increase r to r+dr, M r increases because the mass with another large dimensionless number: the maximum contained within that radius is augmented by a thin shell of number of nucleons ͑protons and neutrons͒ in a white dwarf radius r and thickness dr. Such a shell has volume 4␲r2dr, star, which we will call the Chandrasekhar number and de- and because the density at that radius is ␳, it follows that Ϸ 3 M͑r+dr͒−M͑r͒=4␲␳r2dr,or note as NC. We can show that NC 0.774Nh. We will show how this result comes about and specifically the roles of G, dM͑r͒ ប, and c in the calculation. =4␲r2␳. ͑6͒ dr

II. GRAVITATION Suppose that the matter that makes up the star is so simple that the pressure is a function only of the density ͑and not the We start with some 19th century physics—the equilibrium temperature͒. If we know the P͑␳͒,wecan configurations for spherical balls of self-gravitating gas.3 Ev- rewrite Eq. ͑5͒ as

683 Am. J. Phys. 77 ͑8͒, August 2009 http://aapt.org/ajp © 2009 American Association of Physics Teachers 683 dP d␳ GM͑r͒␳ N ␳ ͑ ͒ ͑ ͒ =− 2 . 7 = . 12 d␳ dr r V 2mH We then multiply Eq. ͑7͒ by r2 /͑G␳͒, differentiate with re- Hence, we can write Eq. ͑11͒ as ͑ ͒ 2Uץ spect to r, and use Eq. 6 to obtain dP V = . ͑13͒ V2ץ d r2 dP d␳ d␳ ␳ 1 ͩ ͪ + ␳ =0. ͑8͒ 4␲Gr2 dr ␳ d␳ dr ͑ ͒ ϰ −1/3 From Eq. 9 it follows that pmax V and therefore that ץ Equation ͑8͒ allows us to make a mathematical model of the pmax 1 pmax ␳ =− . ͑14͒ V 3 Vץ star. Choose a value of the central density c and integrate Eq. ͑8͒ outward until the pressure vanishes at the surface of the star. Complete the model by assuming vacuum at all radii We apply Eq. ͑13͒ to Eq. ͑10͒ and use Eq. ͑14͒ and straight- ␳ larger than this stellar surface. Each value of c yields one forward algebra and obtain stellar model. dP 1 d⑀ = p4ͯ ͯ . ͑15͒ d␳ 9␲2ប3␳ dp p=pmax III. QUANTUM MECHANICS IV. Matter in a white dwarf consists of electrons and nuclei, ͑ ͒ ͑ ͒ / ␳ mostly of and . Because the matter is electri- Equation 15 with pmax given by Eq. 9 gives dP d ,if cally neutral, there is one for every . Also we have an expression for ⑀͑p͒. In nonrelativistic mechanics ⑀ 2 /͑ ͒ because the number of equals the number of pro- =p 2me , where me denotes the mass of the electron. In tons for both carbon and oxygen nuclei, there is one relativistic mechanics for each electron. Usually the pressure of a gas depends on ⑀ ͱ 2 2 2 4 ͑ ͒ its . However, the pressure at low is = p c + mec , 16 a purely quantum mechanical effect that is independent of and thus temperature. Recall that the uncertainty principle implies that 2 particles confined to a box must have a momentum. Because d⑀ c p = . ͑17͒ faster moving particles have larger energy, it takes work to dp ͱp2c2 + m2c4 make the box smaller. This work is due to the pressure. e Consider N electrons along with N and N neutrons We can gain insight by expressing Eq. ͑15͒ in terms of combined in carbon and oxygen nuclei confined to a cubical dimensionless quantities that have factored out the relevant box of side L. The electrons are free particles except for the physical scales. We define the dimensionless quantity X as confining walls of the box. Because of the Pauli exclusion p principle, each state can have no more than two electrons. X = max . ͑18͒ Thus for N electrons the ground state energy U is obtained mec / by putting the N electrons in the N 2 states of lowest energy. It follows from Eq. ͑9͒ that As shown in standard textbooks on statistical mechanics, the ␳ maximum momentum of an electron in the ground state pmax X3 = , ͑19͒ ͑which is the Fermi momentum͒ is given by ˜␳ ប ␳ ͑ ␲2 ͒1/3 ͑ ͒ where the constant ˜ is given by pmax = 3 N , 9 L 2 m ˜␳ = H , ͑20͒ ␲2 ␭3 and the ground state energy is given by 3 e 3 p ␭ 1 L max and e is the Compton wavelength of the electron given by U = ͩ ͪ ͵ ⑀͑p͒p2dp, ͑10͒ ␲2 ប ប 0 ␭ = . ͑21͒ e m c where ⑀͑p͒ is the energy of a single particle. e What we want to calculate is not U itself, but dP/d␳. It is straightforward to see the physical meaning of the scale -V, it follows that ˜␳: By the uncertainty principle, we expect electrons to acץ/Uץ−=Because P quire relativistic speeds at number on the order of 2Uץ dP =− . ͑11͒ one electron per cubic Compton wavelength. Because there -V are two nucleons per electron, this number density of elecץ␳ץ d␳ trons occurs at a mass density ˜␳ that is of the order of m per ͑ 3 ͒ H Here V=L is the volume of the box. Recall that in addition cubic Compton wavelength. In other words, ˜␳ is the mass to the N electrons the box contains N protons and N neu- density of white dwarf matter at which the electrons attain trons. Both the proton and the neutron have a mass approxi- relativistic speeds. ͑ ͒ mately equal to mH the mass of a hydrogen atom while the We can use Eq. ͑17͒ in Eq. ͑15͒ to find mass of the electron is much less. Hence the total mass of N dP 1 m X2 protons, N neutrons, and N electrons is approximately 2NmH. = e c2 . ͑22͒ ␳ / ␳ ͱ 2 Thus we find that =2NmH V or d 6 mH 1+X

684 Am. J. Phys., Vol. 77, No. 8, August 2009 David Garfinkle 684 V. PUTTING IT ALL TOGETHER 2.5

Given the expression for dP/d␳ we can construct stellar models. We use Eq. ͑22͒ in Eq. ͑8͒, and obtain 2 ␲ 3 2 2 1 d / dX ␭ N ͩr2͑1+X−2͒−1 2 ͪ + X3 =0. ͑23͒ 1.5 16 e h r2 dr dr R

Making a stellar model involves choosing a value Xc for X at 1 the center of the star and then integrating Eq. ͑23͒ outward until the value of r at which X vanishes, which is the surface of the star. To streamline this integration we define Y 0.5 / =X Xc so that Y =1 at the center of the star. We also intro- duce a dimensionless spatial coordinate in place of r as fol- 0 lows. Define the length a by 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 M ͱ3␲ a = N ␭ , ͑24͒ Fig. 1. White dwarf radius ͑in units of Earth radii͒ versus white dwarf mass h e ͑ ͒ 4Xc in solar . and the dimensionless coordinate ␨ by

r 3 ͑ ͒ ␨ = . ͑25͒ NhmH = 1.849M᭪. 33 a That is, the mass scale is of the same order as the mass of the Then Eq. ͑23͒ becomes , and the length scale is of the same order as the radius of the Earth. 1 d dY ͩ␨2͑1+X−2Y−2͒−1/2 ͪ + Y3 =0. ͑26͒ We can make the notion of length and mass scales more ␨2 d␨ c d␨ precise by asking how given a solution of Eq. ͑26͒ we would use it to calculate the radius and mass of the star. The solu- Though not quite as intuitive as the density ˜␳, the physical ͑␨͒ ␨ tion Y vanishes at some 0, which allows us to immedi- meaning of the length scale a is clear. If we have white dwarf ately calculate the radius R of the star as matter at mass density m /␭3, then because gravity is so H e ␨ ͑ ͒ weak, it must take a large amount of such matter for the R = a 0. 34 gravity to be able to balance the enormous pressure. Thus The mass of the star is given by integrating Eq. ͑6͒: there must be some large length scale such that the cube of that length scale is the volume needed to contain that large R ␲␳ 2 ͑ ͒ amount of matter at the given density. The expression for a M = ͵ 4 r dr, 35 ␭ 0 tells us that the relevant length scale is of the order of Nh e. Given a length scale and a density scale, we can combine or them to make a mass scale. This mass scale must be of the ␨ 0 ͑ /␭3͒ 3␭3 3 3 order of mH N =N mH. In other words, because the ͵ ␲␳ 3 3␨2 ␨ ͑ ͒ e h e h M = 4 ˜XcY a d , 36 main contribution to the mass of the star is nucleons of mass 0 mH, it follows that the relevant scale for the number of nucle- 3 which using Eqs. ͑20͒ and ͑24͒ becomes ons is Nh. ␨ What are these length and mass scales in more ordinary ͱ3␲ 0 ប M N3m ͵ Y3␨2d␨ ͑ ͒ units? From the values of G, , c, me, and mH we find that = h H . 37 8 0 N = 1.300 ϫ 1019, ͑27͒ h This discussion of length and mass scales does not actu- ally give us the sizes and masses of any white . To ␭ ϫ −13 ͑ ͒ e = 3.863 10 m, 28 do that we must solve Eq. ͑26͒ and use the results in Eqs. ͑34͒ and ͑37͒. Equation ͑26͒ cannot be solved in closed form, and therefore but it is easy to treat numerically using, for example, the 4 ␭ ϫ 6 ͑ ͒ fourth-order Runge–Kutta method. For each value of Xc we Nh e = 5.022 10 m, 29 ͑ ͒ ␨ integrate Eq. 26 outward to find the value 0 at which Y vanishes and then use Eqs. ͑34͒ and ͑37͒ to find the mass and 3 ϫ 30 ͑ ͒ NhmH = 3.678 10 kg. 30 radius of the star. Thus each Xc yields a point on a plot of R versus M for white dwarf stars. The result of such a calcula- ϫ 6 The radius R of Earth is R =6.378 10 m, and the mass tion is shown in Fig. 1 ͑see also Ref. 5͒. Here R is measured M᭪ of the Sun is in units of R, and M is measured in units of M᭪. It is clear 30 from Fig. 1 that there is a limiting mass of about 1.43M᭪. M᭪ = 1.989 ϫ 10 kg. ͑31͒ What is not clear from the figure is why such a limiting mass Hence, the mass and length scales of a white dwarf are exists. An answer can be found by an examination of Eq. ͑26͒. As the central density is increased without bound, Eq. ␭ ͑ ͒ ͑ ͒ Nh e = 0.7874R , 32 26 has a limit given by

685 Am. J. Phys., Vol. 77, No. 8, August 2009 David Garfinkle 685 1 d dY istic quantum mechanics to describe them, yet the derivation ͩ␨2 ͪ + Y3 =0. ͑38͒ ͑ ͒ ␨2 d␨ d␨ of Eq. 15 seems to use only nonrelativistic quantum me- chanics. ͑Eddington did raise an objection of this sort to 7͒ To find the limiting mass, the Chandrasekhar mass MC, the Chandrasekhar’s result. The only part of quantum mechan- mass in the limit of large central density, we need to numeri- ics ͑other than the Pauli exclusion principle͒ used in deriving ͑ ͒ ͑␨͒ ␨ ͑ ͒ cally solve Eq. 38 and use the resulting Y and 0 in Eq. Eq. 15 is that free particles with momentum pជ have the ͑37͒. The result is that wave function exp͓ipជ ·xជ /ប͔. This result holds in both nonrel- 3 ͑ ͒ ativistic and relativistic quantum mechanics. One might also MC = 0.774NhmH, 39 worry that the electrons are highly relativistic, but standard or the limiting number of nucleons is N =0.774N3. We use nonrelativistic fluid mechanics is used in the derivation of C h ͑ ͒ the conversion factor of Eq. ͑31͒ to find Eq. 8 . What makes this treatment permissible is that the nucleons are much more massive than electrons, so that even ͑ ͒ MC = 1.43M᭪. 40 when the electrons are moving relativistically, the protons To understand why this limiting mass exists we return to and neutrons, which make up most of the mass of the fluid, the central fact of stellar structure: stars are held in equilib- are not. Thus the Chandrasekhar limit is one of the simplest rium by the opposing forces of gravity and pressure. Because uses of the Planck mass in that it involves all three funda- compressing a star raises its pressure, a star will be com- mental constants, and yet can be treated in a way that in- pressed until its pressure is large enough to balance its self- volves only Newtonian gravity, quantum mechanics, and gravity. Because gravity is a 1/r2 force, compressing a star special relativity. also increases the force of its self-gravity. Thus it is not ob- Finally, we consider whether the Chandrasekhar limit can vious that an equilibrium will exist. In particular, once the be attained. Recall that the only physics used in the deriva- ͑ ͒ electrons in a white dwarf star become highly relativistic, it tion of the Chandrasekhar limit other than gravity is the turns out that as the star is compressed, the forces of self- quantum mechanical properties of the electrons. Because the gravity and pressure increase at the same rate. Thus the lim- approach to the Chandrasekhar limit involves arbitrarily high iting mass can be compressed to arbitrarily high densities density, we might expect that other physical effects will be- without ever reaching equilibrium. come important as this limit is approached. And that turns This argument can be turned into an estimate of the limit- out to be the case. The two important effects are nuclear ing mass, as was first done by Landau.6 We can somewhat fusion and . White dwarf stars form because crudely approximate the white dwarf star as a uniform den- low mass stars become hot and dense enough to fuse hydro- sity sphere of radius R containing N electrons and 2N nucle- gen into helium and then helium into carbon and oxygen. But ons. The total energy of the star consists of a kinetic part due they do not become hot and dense enough to fuse carbon or to the quantum mechanical motion of the electrons and a oxygen. If a white dwarf accretes material from another star potential part due to the mutual gravitational attraction of the and if that extra material brings the white dwarf close to the nucleons. ͑The kinetic part is what we referred to earlier as Chandrasekhar limit, then the white dwarf becomes dense the ground state energy of the electrons.͒ The equilibrium enough for fusion of carbon and oxygen to take place. What configuration can be thought of as the star adjusting its ra- results is not a white dwarf at the Chandrasekhar limit, but a dius to attain the lowest total energy. Because the mass is thermonuclear explosion that destroys the entire star. Such an 2NmH, a standard result of Newtonian gravity yields a po- explosion is called a type Ia . Electron capture be- 2 2 / tential energy of approximately −GN mH R. The kinetic en- comes important in another type of explosion: a type II su- ergy is given by Eq. ͑10͒. In the nonrelativistic limit; ⑀ pernova, which is brought about by the collapse of the 2 /͑ ͒ ϳប2 5/3 /͑ 2͒ =p 2me , we obtain U N meR . Thus, as long as core of a high mass star. In high mass stars in the electrons remain nonrelativistic, there is always an equi- the core proceeds all the way to iron, which is the most librium because as the star shrinks the kinetic energy grows tightly bound nucleus. The iron core is supported by degen- faster ͑as R−2͒ than the potential energy, which grows as R−1. eracy pressure in much the same way as is a white dwarf star. However, in the ultrarelativistic limit, ⑀=pc, Eq. ͑10͒ yields As the core approaches the Chandrasekhar limit and its elec- UϳបcN4/3 /R. Thus the kinetic energy and potential energy trons become relativistic, conditions become favorable for grow at the same rate as the star shrinks. The energy can be electron capture, in which an electron and proton combine to made arbitrarily large and negative when the coefficient of form a neutron and a . Because a neutron has more 1/R in the potential term is larger than the coefficient of 1/R mass than a proton and electron combined, electron capture in the kinetic term, so the limiting mass is estimated to occur does not take place unless something supplies extra energy. when these two coefficients are equal, that is, when As the electrons in the stellar core become relativistic, their 2 2 ប 4/3 ϳ 3 ϳ 3 GN mH = cN , which yields N Nh and thus MC NhmH. kinetic energy becomes sufficiently large for electron capture We now consider the nature of the calculation that has led to proceed. Because electron capture removes electrons that to this limiting mass, and in particular the roles of G, ប, and were the source of the pressure holding up the core, electron c. Although the end result depends on all three fundamental capture leads to a catastrophic collapse of the core of the star. constants, it does so in a somewhat compartmentalized way. Note that the products of electron capture are neutrons and The derivation of Eq. ͑8͒ uses only Newtonian gravity and . The core of neutrons collapses, but neutrons, like depends only on G. Equation ͑15͒ is derived using only electrons obey the Pauli exclusion principle, and thus have a quantum mechanics and depends only on ប. Finally Eq. ͑17͒ degeneracy pressure of their own. When the core becomes is found using only special relativity and depends only on c. sufficiently dense, it is prevented from further collapse by At first sight, the compartmentalized nature of the calculation neutron degeneracy pressure and becomes a . The seems strange: if the electrons are moving at speeds compa- neutrinos stream outward and cause the ejection of all the rable to the speed of light, we would expect to need relativ- rest of the star in a huge explosion: a type II supernova. Note

686 Am. J. Phys., Vol. 77, No. 8, August 2009 David Garfinkle 686 ͒ that just as there is a maximum mass, the Chandrasekhar a Electronic mail: garfi[email protected] 1 limit, that can be supported by electron degeneracy pressure, M. Planck, “Uber irreversible Strahlungsvorgange,” Sitzungsber. Deut. Akad. Wiss. Berlin, Kl. Math.-Phys. Tech. 5, 440–480 ͑1899͒. so there is a maximum mass for a neutron star. If the neutron 2 S. Chandrasekhar, “The maximum mass of ideal white dwarf stars,” As- star exceeds this mass, nothing will prevent its complete trophys. J. 74, 81–82 ͑1931͒. , and it becomes a . Thus we 3 J. H. Lane, “On the theoretical temperature of the Sun under the hypoth- see that the Chandrasekhar limit cannot be attained, but esis of a gaseous mass maintaining its volume by its internal heat and rather its near conditions become sufficiently extreme to pro- depending on the laws of gases known to terrestrial experiment,” Am. J. duce a catastrophic explosion. Sci. Arts, 2nd Series 50, 57–74 ͑1870͒. 4 W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Reci- pes in Fortran, 2nd ed. ͑Cambridge U. P., Cambridge, UK, 1992͒. 5 S. Chandrasekhar, “The highly collapsed configurations of a stellar ACKNOWLEDGMENTS mass,” Mon. Not. R. Astron. Soc. 95, 207–225 ͑1935͒. 6 L. Landau, “On the theory of stars,” Phys. Z. Sowjetunion 1, 285–287 The author would like to thank Alberto Rojo for helpful ͑1932͒. discussions. This work was partially supported by NSF Grant 7 A. S. Eddington, “Relativistic degeneracy,” Observatory 58, 37–39 No. PHY-0456655 to Oakland University. ͑1935͒.

Cardboard Steam Locomotive Model. The Millington/Barnard Collection in the University of Mississippi Museum collection includes a unique set of three cardboard steam engine models by Salleron of Paris. This locomotive model still works properly, with the piston going back and forth, the driving wheel rotating and the slide valve mechanism operating. ͑Photograph and Notes by Thomas B. Greenslade, Jr., Kenyon College͒

687 Am. J. Phys., Vol. 77, No. 8, August 2009 David Garfinkle 687