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Energy Scales of Physical Phenomena energyOrder scales: pressure of Magnitude Astrophysics Order of Magnitude Astrophysics Pressure exerted by the a system of particles as the rate of momentum tran Pressure exerted by the a system of particles as the rate of momentum transferred (per exerted by the aofsystem of !: particles as the rate of momentum transferred (pe area) from particles of energy : nitPressure area) unit from particles energy nit area) from particles of energy !: ! 2 2 ∞ < n! > if mc !! 1 3 ! 2 2=< n! > P 1= ∞ n(!)p(!)v(!)d! if mc !! 3 1 3 2 O P = n(!)p(!)v(!)d! = < n! > if mc "! 3 1 3 2 O 3 < n! > if mc " ! 2 Rest Mass Energy: We can associate the rest mass energy mc eachofparticle 2 est Mass Energy: We can associate the rest mass energy mc with eachwith particle mass m 2 m c2 ≈ 0.5MeV me c ≈ 0.5MeV e 2 2 mc = mc = m 2 pc ≈ m1GeV c2 ≈ 1GeV p the particles of the system have internal structure (molecular, atomic, nuclear, etc) then f the particles of the system have internal structure (molecular, atomic, nuclea articles of energy !: area) from particles of energy !: 1 = 3 ! ∞ O mc2 ! ! < n! > if scales: rest-mass energy ! ∞= 2 < n! > n(!)p(!)v(!)d! 1 1 < n! > 32 if mc "! P = n(!)p(!)v(!)d! = 3 3 O 2 3 1 3 < n! > if mc2 ! ! if mc2 " ! can associate the rest mass energy mc2 with particle of mass : We can We associate the rest mass energy witheach each particle of m. mass m. 2 Mass Energy: We can associate the rest mass energy mc with each particle me c2 ≈ 0.5MeV mc2 = m c2 ≈ 1GeV 2 p 2 mc = me c ≈ 0.5MeV m c2 ≈ 1GeV p the system have internal structure (molecular, atomic, nuclear, etc) then ergy scales that are characteristic of the interactions. The simplest is of the system internal structure (molecular, atomic, nuclear, etc) nuclear, particlesIf the of particles the system havehave internal structure (molecular, atomic, we get further energy scales that are characteristic of the interactions. The ng energy then of atoms and molecules, which arises from the electromagnetic simplest is the atomic binding energy of atoms and molecules, which arises from et furthertheenergy scalescoupling that are characteristic of the interactions. The sim electromagnetic between particles. particles. tomic binding energy of atoms and molecules, which arises from the electro Energy: Size and energy of the ground state of a hydrogen atom: ing between particles.2 ! −9 ≈ 5.2 × 10 cm 2 q and energy of the ground state of a hydrogen atom ic Binding Energy:mSize a0 = energy scales: rest-mass 2 we get further energy scales that of the The interactions. mc =are characteristic getwefurther energy scales that are characteristic of the of interactions. simplest get further energy scales that are characteristic the interactions. The sim 2 we get further energy scales that are characteristic of the interactions. atomic binding energy of atoms and molecules, arises from the electrom mp cwhich ≈ 1GeV atomic binding energy of atoms andofmolecules, arises from electromagne the atomic binding energy atoms andwhich molecules, whichthe arises from th the atomic binding energy of atoms and molecules, which arises from the electro energy scales: atomic the atomic binding energy of atoms and molecules, which arises from th pling between particles. pling between particles. coupling between particles. coupling between particles. between particles. Ifcoupling theEnergy: particles of the system have structure atomic, nuc mic Binding Size and energy of internal the ground state (molecular, of a hydrogen atom: mic Binding Energy: Size and energy theenergy ground ofstate a hydrogen atom: Atomic Energy: Sizeenergy and ofstate the ground state of a hydrog Size andBinding energy of the ground state ofof a hydrogen atom: Atomic Binding Energy: Size and of the ground of a hydrogen atom weAtomic get further energy scales that are characteristic of the interactions. The Binding Energy: Size2 and energy of the ground state of a hydrog −9 2! 2−9 a0 = ! 2 !≈ 10 cm 2 5.2 × ! 2 the atomic binding energy of atoms and molecules, a0 = a m= × 10 cm −9 which−9arises from the ele ! e q≈a5.2 = ≈ 5.2 × 10 2 ≈ 5.2 × 10 cm −9cm 0 2 0e q m 2 a = ≈ 5.2 × 10 cm 0 me q 2 4 m q e m q 1 e4 2 2 me q coupling between!particles. m q 1 4 4 e = = α m c ≈1 13.6 eV me qα22 m 1ecq2eq2≈4 13.6 mem 2= !a a= ! 2! 21 2eV 2 = 2αe m eV 213.6 == m 13.6 eV a 2= !a2!2== e c α≈ e c c2≈ 2! α m ≈ 13.6 eV Atomic Binding Energy: Size and energy of the ground state of a hydrogen a a e 2! 22!2 2 2 2! 2 e α = q 2 /(!c) is the fine structure constant. The wavelength corresponding to ! 2 wavelength corresponding to !a is eα = qα/(!c) fine structure constant. The 2 is the 2 is ! Here = q /(!c) the fine structure constant. The wavelength corresponding t is the fine structure constant. 2 −9wavelength Here α= is is thethefine The correspo Here α q= /(!c) q /(!c) finestructure structure constant. The wavelength corresp a0 = constant. ≈ 5.2 × 10 cm 2 m q e hc 2! hc = 2! 4 ≈ 3103 A λ = 2! ≈ 10 λ = ! = hc2α hc 2em m q 1 A hc 2! 33 c 2≈2! 23 A = 10 λ = a e ! α m c ≈ 10 AA = e =2= == α m c ≈ 13.6 The wavelength corresponding ato !a = is! λ λ e ≈ 10eV a 2!2α! m2 e c α2 m 2 a!a e ce c αm lies in in the theUV. UV.When Whenatoms atomsofofsize size aare closely packed: 0 are lies a closely packed: 0 2 UV. and lies inlies the When atoms ofstructure are closely packed: and lies inUV. When the fine atoms ofsize size a0 are closely packed: Here αlies = qin /(!c) is When the constant. The wavelength and inthe the UV. atoms ofofsize a0a0are closely packed: and the UV. When atoms size are closely packed:correspondin −3 −3 24 24 −3 −3 nsolid≈≈ (2a ) ≈ cm nsolid (2a ) ≈ 10 hc −310cm2! 24. . −3 3 nsolid 0≈0λ(2a ) ≈ 10 .24A −3 −3cm −3 −3 24 0 = ≈ 10 n= ≈ (2a ) ≈ 10 cm nsolid ≈ (2a ) ≈ 10 cm .. 2 solid 0 0 !a α me c and lies in the UV. When atoms of size a0 are closely packed: energy scales: atomic NGC 2237, the Rosette Nebula in Monoceros energy scales: atomic energy scales: molecular The simplest molecular structure consists of two atoms bound to each other in the form of a diatomic molecule: energy scales: molecular Molecular Binding Energy: In addition to the electronic internal binding energy of the Molecular Energy: In addition toto the binding energy of the InBinding addition to other the electronic internal binding energy ofinternal the atoms, there are twoVibrational other atoms, there are two contributions theelectronic energy a diatomic molecule. contributions to the energy of a diatomic molecule. Vibrational energy levels atoms, levels there are two other contributions to the energy of a diatomic molecule. Vibrational energy separated separated by by ! "1/2 energy levels separated by me Evib = !ωvib ≈! "1/2 "a ≈ 0.25 eV. mµe "a ≈ 0.25 eV. Evib = !ωvib ≈ µ The molecule can alsocan rotate. This will with The molecule also rotate. Thiscontribute will contribute withan anenergy energy of of approximately approximately ! " The molecule can also rotate. This Jwill 2 contribute with an energy of approximately me Evrot = 2 2 ≈! " "a ≈ 10−2 eV. Jµa mµe "a ≈ 10−2 eV. Evrot = 20 ≈ µa µ The wavelengths of radiation from 0vibrational transitions are ∼ 40 times larger than The wavelengths of radiation from vibrational transitions are ∼ 40 times larger than those of electronic transitions; similarly the rotational transitions lead to radiation with The wavelengths of radiation from vibrational transitions are about 40 times larger those of electronic transitions; similarly the rotational transitions lead to radiation with than those 3 of electronic transitions wavelengths ∼ 10 larger. wavelengths ∼ 103 larger. Nuclear Energy Scales: Atomic nuclei are bound by the strong interaction force that Nuclear Energy Scales: Atomic nuclei are bound by the strong interaction force that produces a binding energy per particle ∼ 8 MeV, which is the characteristic scale for nuclear produces a binding energy per particle ∼ 8 MeV, which is the characteristic scale for nuclear energy levels. In the astrophysical context, a more relevant energy scale is the one at which energy levels. In the astrophysical context, a more relevant energy scale is the one at which nuclear reactions can be triggered. For two protons to fuse together, while undergoing nuclear reactions can be triggered. For two protons to fuse together, while undergoing energy scales: molecular Energy levels of ammonia (NH3) in the lowest vibrational state (Wilson, T. L. et al. 1993, A&A, 276, L29). energy scales: molecular Multi-wavelength overview of the Orion nebula. energy scales: nuclear Atomic nuclei are bound by the strong interaction force that produces a binding energy per particle about 8 MeV, which is the characteristic scale for nuclear energy levels lear Energy Scales: Atomic nuclei are bound by the strong interaction force th duces a binding energy per particle ∼ 8 MeV, which is the characteristic scale for n energy scales: nuclear gy levels. In the astrophysical context, a more relevant energy scale is the one at ear reactions can be triggered. For two protons to fuse together, while undergo In the astrophysical context, a more relevant energy scale is the one at which nuclear reactions can be triggered. Forbrought two protons to fuse ear reaction, it is necessary that they are within the together, range ofwhile attractive undergoing nuclear reaction, it is necessary that they are brought within the range of attractive overcoming nuclear force. This overcoming the Coulomb repulsion: e. This requires therequires Coulomb repulsion: n q2 α "≈ = mp c2 ≈ 1 MeV. l 2π , however, possible for nuclear reactions to occur through quantum-mechanical tun n the de Broglie wavelength λdeB = h/(mp v) of the two protons overlap. This oc n the energy is approximately "nucl α2 ≈ 2 mp c2 ≈ 1 keV. 2π vitational Binding Energy: In the non-relativistic Newtonian theory of gravity, t × 10 W order Lof= 4magnitude 26 ! L! = 4 × 1026 W (12) (12) L! = 4 × 1026 W (12) This discussion assumes that the sunUis primary supported by pressure of an ideal (13) th ∼ kT gas of electrons and ions. Thus, in a star, the total (thermal) kinetic energy of Uth ∼ kT (13) particles is Uth ∼ kT k = 1.38 × 10−23 J/K An estimate of the typical temperature the Sun can be estimated by k = 1.38inside × 10−23 J/K combining the kinetic and gravitational potential energy per particle: −23 mpJ/K kL = 1.38 ×GM 10 !26 4 × 10 W ! kT= ∼ i R GM!!mp kTi ∼ R! GM! mp kT ∼ 7 i Ti ∼ 1.8∼×R 10 K Uth kT ! Ti ∼ 1.8 × 107 K where k is the Boltzman constant: √ 7 L! 8 10 K R! = √Ti√∼ 1.8 × = 7.1 −23 × 10 m 2 k =4πσ 1.38 J/K 8 L× T!10 !× R! = √ = 7.1 × 10 m 4πσ × T!2 √ L! 8 R! = √ = 7.1 × 10 m GM m 2 ! p 4πσ × T kTi ∼ ! R (13) (14) (14) (14) (15)(12) (15) (15) (16) (13) (16) (16) (17) (14) (17) (17) (15) sun’s structure nuclear reaction, it is necessary that they are brought within the range of attract q2 α 2 force. This "requires overcoming the Coulomb repulsion: ≈ = m ≈ 1 MeV. pc energy scales: l 2π nuclear 2 "≈ however, possible for nuclear reactions to q α = mp c2 ≈ 1 MeV. l 2πthrough quantum-mechanical occur It is, however, possible for nuclear reactions to occur through quantum-mechanica It is, however, possible for λ nuclear through quantum-mechanical the de Broglie wavelength h/(mptov)occur of the two protons overlap. Thi deB =reactions whenthe thededeBroglie Broglie wavelength λdeB = h/(mp v) of of thethetwo tunneling when wavelength twoprotons protonsoverlap. Th overlap. This occurs when the energy is approximately the energy is approximately when the energy is approximately "nucl α2 "nucl2 ≈ α2 mp c2 ≈ 1 keV. ≈ 2 mp c ≈2π 1 2keV. 2π Gravitational Binding Energy: In the non-relativistic Newtonian theory of grav tational Bindinggravitational Energy: binding In theenergy non-relativistic Newtonian theory of gravi tational binding energy Egrav Egrav GM 2 Gmp2 2 ≈ N . ≈ R R GM 2 Gmp2 2 ≈ N . ≈ R R "nucl α2 ≈ 2 mp c2 ≈ 1 keV. 2π energy scales: gravitational Gravitational Binding Energy: In the non-relativistic Newtonian theory of gravity, the In the non-relativistic Newtonian theory of gravity, the gravitational binding energy: gravitational binding energy Egrav Gm2p 2 GM 2 ≈ N . ≈ R R The potential energy per particle: The potential energy per particle varies as Gm2p Egrav 4π 1/3 !g = = N= Gm2p N 2/3 n1/3 . N R 3 otential energyGeneral per particle varies effects as relativistic become important when Rgm = Egrav /Emass ∼ 1. GeneralErelativistic important when 1/3 Gm2pbecome 4π grav Thermal and Degeneracy Energy: of the system depends on the origin of !g = = N= Gm2pThe N 2/3behavior n1/3 . N R 3 the momentum distribution of the particles. The familiar situation is the one in which al relativistic effects become important when Rgm = Egrav /Emass ∼ 1. short-range interactions between particles effectively exchange the energy so as to randomize mal and Degeneracy Energy: The behavior of the system depends on the origin of the momentum distribution. When such a system is in steady state, we can assume that the omentum distribution of the particles. The familiar situation is the one in which local thermodynamical equilibrium, characterized by a temperature T , exists in the system: range interactions between particles effectively exchange the energy so as to randomize omentum distribution. When such a system is in steady! state, ≈ kB Twe can assume that the Gmorigin Egrav 4π of 2 2/3 1/3 p the system depends !g = on the = N= Gmp N n . N R 3 Generalsituation relativistic effects become important when Rgm = Egrav /Emass ∼ 1. amiliar is the one in which energy scales: thermal Thermal and Degeneracy Energy: The behavior of the system depends on the origin of y exchange the energy so as to randomize the momentum distribution of the particles. The familiar situation is the one in which The behavior of the system depends on the origin of the momentum distribution of short-range interactions betweensituation particles the energyinteractions so as to randomize the particles. The familiar is effectively the one inexchange which short-range between particles effectively exchange the energy so as to randomize the momentum the momentum systemstate, is in we steady wethat canthe assume distribution.distribution. When such aWhen systemsuch is ina steady can state, assume local that the thermodynamical equilibrium, characterized by a temperature T, exists in the system: s in steady state, we can assume that the y a local temperature T , exists in the system: thermodynamical equilibrium, characterized by a temperature T , exists in the system: ! ≈ kB T ! " #2 $1/2 (2mkB T )1/2 ≈ k T /c if mc2 # kB T kB T mc2 2 1/2 if mc $ kB T B kB T ) if mc # kB T In this case, the momentum and the kinetic energy of the particles vanish when T → 0. The 2 /c mean energy mc $ofkelectrons In this if case, the momentum and thewill kinetic particle vanishes when BT of a system notenergy vanishofatthezero temperature because electrons 2kB T p ≈ mc + 2 mc 2 obey the Pauli exclusion principle, which requires that the number of electrons that can of the particles vanish when T → 0. The occupy any quantum state be two. Because the uncertainty principle requires ∆x∆px ≥ h, h atwezero temperature electrons can associate the number because of quantum states with a phase space volume: 2 1/3 energy scales: thermal #" !" $1/2 1/2 2 2 22 1/2 # (2mk T ) if mc # kBk2TB# 1/2 2 B 2k T k T (2mk T ) if mc # T kB T 1/2 2 2k kB B B T (2mk T ) if mc B B B T k T 2mkBppT≈ )≈ mc if mc # k T B B mc p ≈ mc + ≈≈ B+ + ≈ 2 2 2 2 mc mc 2 2 2 2 mc mcmc mc k T /c if mc $ kBk2TB$ B k T /c if mc T kB T kB T /c if $ mc B 2 energy scales: degeneracy kB T /c if mc $ kB T his case, case, thecase, momentum and kinetic vanish when Twh In this the momentum the energy kinetic energy of the particles vanish his the momentum and the theand kinetic energyof ofthe theparticles particles vanish when T→ rgy of the particles vanish when T → 0. The an energy ofenergy amean system will not vanish zero temperature because energy ofelectrons a system of electrons willnot not at vanish atat zero temperature meanThe of a of system of electrons will vanish zero temperature bee n energy of a system of electrons will not vanish at zero temperature becaus because electrons obey the Pauli exclusion principle, which requires that the number anish at zero temperature because electrons ofthe electrons thatexclusion can occupy principle, anywhich quantumrequires state be two. y the Pauli exclusion principle, that number ofofelectrons th obey Pauli which requires that the number of electr y the Pauli exclusion principle, which requires thatthe the number electrons res that the number of electrons that can occupy anythe quantum berequires two. Because the uncertainty principle require Because uncertainty principle upy any quantum state be state ∆x∆ py any quantum state be two. two. Because Becausethe theuncertainty uncertaintyprinciple principlerequires requires ∆ uncertainty principle requires ∆x∆p x ≥ h,states with a phase space volume: we can associate the number of quantum can associate the number of quantum states with a phase space volume: an associate the number of quantum states with a phase space volume: we can associate the number of quantum states with a phase space volume: with a phase space volume: 2 1/3 1/3 pF 2= !(3π pF = !(3π n) . n) . 2 1/3 pF = !(3π n) . n)1/3 . The The the quantum mechanical scale of the energy !F sets sets the quantum mechanical of the energy quantity quantity !F quantity sets the quantum mechanical scale ofscale the energy scale )of 2the * energy quantity !F sets the quantum mechanical 2 ) * p ! 2 2/3 2 ( F 2 2 p = (3π n) if mc # !F ! 2 2/3 2 ) * cale of the(energy F 2m 2m 2 = (3π n) if mc # ! 2 2 2 F 2 pF c2 4+ m2 c42 − mc p2m ≈ !2m 2 2/3 2 ( F 2 !F2 (3π n) 2 1/3 if mc # !2F !F = pF c + m c − mc ≈ 2m = * 2m 2 2 2 + m2 c4 − F c = (!c)(3π 2 1/3 n) 2if mc $ !F !2 2 p2/3 2 mc ≈ p c = p(!c)(3π ! c F F n) if mc $ !F (3π n) if mc # !F F m 2 1/3 2 = (!c)(3πifn) mc $ !F E F c dominant Quantum-mechanical effects willpbe !F # kB Tif(degenerate). antum-mechanical will!Fbe dominant if !F # kB T (degenerate). Electron !c)(3π 2 n)1/3 ifeffects mc2 $ ntum-mechanical !2F ∼ me c2 foreffects will be dominant if !F # kB T (degenerate). Electr ∼f m for !Fe c# kB T (degenerate). Electrons"have#−3 2 2 1/3 pF = !(3π 2 n)1/3 . p = !(3π n) . F 2 1/3 pF = !(3π n) . energy scales: degeneracy sets the quantum mechanical scale of!thesets energy The quantity the quantum mechanical scale of the ener F ) 2mechanical * quantity !F sets the quantum scale of the energy 2 2 ) 2* p 2 2/3 2 F = ! p (3π ( 2n) ) *if mc # !F F = ! (3π 2 n)2/3 2m 2m 2 2 2 4 2m pF2 2 c + m c(− mc ≈ 2 !2 2 4 2mc2/3 2 2m ! p c + m c − ≈ = (3π n) if mc # !F F Quantum-mechanical effects will be 22mFdominant 1/3 2m if 2 2 2 2 2 4 p c = (!c)(3π n) if mc $ ! 2 1/3 F mc ≈ F !F pF c + m c − p c = (!c)(3π n) F p c = (!c)(3π 2 n)1/3 if mc2 $ !F F nical effects will be dominant if !F # kB T (degenerate). Electrons have Quantum-mechanical effects will be dominant if !F # kB T (de ntum-mechanical effects will be dominant if !F # kB T (degenerate). Electrons hav 2 Electrons have !F ∼ me c for for " # −3 me c2 for " #−3 ! 31 −3 ! n= ≈ 10 cm# 31 −3 " −3 n = ≈ 10 cm me c ! me c 31 −3 n = ≈ 10 cm 7 −3 ρ ∼ nmp ≈ 10 gmcm . ec ρ ∼ nmp ≈ 107 g cm−3 . ρ ∼ nmp ≈ 107 g cm−3 . !F ∼ kB T occurs occursatat ∼ kB T occurs at nT −3/2 3/2 (mk ) B −3/2 16 2 3/2 nT = = 3.6 × 10 (cgs) for mc # !F . (mkB ) 3 16 2 = = 3.6 × !10 (cgs) for mc # !F . 3 ! The energy sclae of the indiviudal particles also characterizes the energy i e energy sclae of the indiviudal particles also characterizes the energy involved in the ! !3 particles also characterizes the energy involved nergy sclae of the indiviudal ons between particles. If thisscales: quantity isionization larger than the binding energy of th energy energy sclae of the indiviudal particles also he energy sclae of the indiviudalThe particles also characterizes the energy involved in cha the m, the atoms will be ionized and the electrons will be separated: particles. If this energy quantity is larger ollisions between particles. If thiscollisions quantity between is larger than the binding of the atom ystem, The energy scale of the individual particles also characterizes the energy involved in collisions particles. Ifthe quantity larger the binding energy system, the will be ionized and the ofelectrons kthis ≥ atoms !a .is will thethe atoms will between be ionized and electrons bethan separated: BT the atomic system, the atoms will be ionized and the electrons will be separated: w kB T ≥ !a . w kB T ≥ !half a. ransition temperature at which nearly of the atoms are ionized occurs 4 temperature atThe which nearly half ofof thethe atoms areatoms ionized occurs when ≈he!atransition /10 ∼The10transition K. The atoms can be stripped off the in another contex transition temperature at which nearly of temperature at which nearly half atoms are ionized occurshalf when ≈ !a /10 104 K. The off4are the atoms in together, another Thi s when the∼density is soatoms highkcan that atoms with elec T be ≈ the !stripped ∼ 10 K. so Theclose atoms can becontext. stripped o B a /10 BT ccurs whenThe the density is stripped so highoccurs the atoms so close with atoms can be offthat the atoms in another context. occurs when the when the are density isThis sotogether, high that theelectrons atoms a ng a common pool rming a density is so high that the atoms are so close together, with electrons forming a common pool common pool forming a common pool !F ≥ !a . !F ≥ !a . In this case, the electrons will be quantum mechanical. s case, the electrons will be quantum mechanical. !F ≥ !a . n this case, the electrons will be quantum mechanical. In this case, the electrons will be quantum mechanic Plasmas and Magnetic Fields: To treat a plasma as ideal, it is necessary tha deal Plasmas and Magnetic Fields: To Plasmas treat a plasma as ideal, Fields: it is necessary that the Ideal and Magnetic To treat a plas mb interaction energy of ions and electrons be negligible. The typical Coul !F ≥ !a . this case, the electrons will be quantum! mechanical. F ≥ !a . this case,and the Magnetic electrons will be quantum mechanical. energy scales: ideal plasma alInPlasmas Fields: To treat a plasma as ideal, it is necessary tha In this case, the electrons will be quantum mechanical. Ideal Plasmas and Magnetic Fields: To treat a plasma as ideal, it is necessary that th ulombIdeal interaction energy of ions and electrons be negligible. The typical Coulo Plasmas and Magnetic Fields: To treat a plasma as ideal, it is necessary that the CoulombTointeraction energy of ions and electrons be negligible. The typical Coulomb treat a plasma as ideal, it is necessary that the Coulomb interaction energy of ions tentialCoulomb energy interaction between ions and the electrons in the is given b interaction energy of ions and electrons be negligible. Theplasma typical and electrons be negligible. The typical Coulomb potential energy interaction betweenCoulomb potentialions energy between and and theinteraction electrons in the plasma ions is given by the electrons in the plasma is given by potential energy interaction between ions and the electrons in the plasma is given by !Coul ≈ Zq 22n1/3 1/3 . !Coul ≈ Zq 2n 1/3. !Coul ≈ Zq n . If the high temperature plasma is tobe betreated treated as as an ideal gas, this energy should be should heIfhigh temperature plasma is to an ideal gas, this energy the high temperature plasma is to be treated as an ideal gas, this energy should be If the high temperature plasma is to be treated as an ideal gas, this energy should be !Coul ≤ kkBT, !Coul T, ! ≤≤kB T, Coul B which requires requires which requires ichwhich requires !!!k k""3"3 3 BB 88 −3 −3−3 2.2 × 10 × 10 (cgs). nT 8Z −3(cgs). −3nT ≤ ≤kB 2 ≈≈2.2 2 ≈ 2.2 × 10 Z (cgs). nT ≤ Zq Zq 2 Zq energy scales: ideal plasma On the other to hand, to treat thehigh-density high density quantum gas as ideal n the other hand, treat the quantum gas as ideal the other hand, to treat the high-density quantum gas as ideal hich requires which requires ich requires !Coul ≤ !F = (!2 /2m)n2/3 , !Coul ≤ !F = (!2 /2m)n2/3 , 26 3 −3 n ≥ 8Z 3 a−3 ≈ 10 Z cm 0 3 −3 n ≥ 8Z a0 ≈ 1026 Z 3 cm−3 ρ ≈ nmp ≥ 102 g cm−3 . ρ ≈ nmp ≥ 102 g cm−3 . t us now go back to the tacit assumption we made in the above analysis, vi t us now go back to the tacit assumption we made in the above analysis, viz., th ysical interactions between the particles of ths system are capable of mantai ysical interactions between the particles of ths system are capable of mantaining ermal equilibrium. Determining the precise condition that will ensure this is n rmal equilibrium. Determining the precise condition that will ensure this is not a s sk; but naively we would require that (i) the mean free path for particles