Table Of ContentMon.Not.R.Astron.Soc.000,1–25(2007) Printed1February2008 (MNLATEXstylefilev2.2)
Condensed surfaces of magnetic neutron stars, thermal
surface emission, and particle acceleration above pulsar
polar caps
8
0 Zach Medin⋆ and Dong Lai†
0 Department of Astronomy, Centerfor Radiophysics and Space Research, Cornell University,Ithaca, NY 14853, USA
2
n
a Accepted2007September 18.Received2007September 17;inoriginalform2007August28
J
8
1
ABSTRACT
Recent calculations indicate that the cohesive energy of condensed matter increases
]
h withmagneticfieldstrengthandbecomesverysignificantatmagnetar-likefields(e.g.,
p 10 keV at 3×1014 G for zero-pressure condensed iron). This implies that for suffi-
-
ciently strong magnetic fields and/or low temperatures, the neutron star surface may
o
r be in a condensed state with little gas or plasma above it. Such surface condensation
t
s cansignificantlyaffectthethermalemissionfromisolatedneutronstars,andmaylead
a totheformationofacharge-depletedaccelerationzone(“vacuumgap”)inthemagne-
[
tosphere abovethe stellar polar cap. Using the latest results onthe cohesiveproperty
2 of magnetic condensed matter, we quantitatively determine the conditions for surface
v
condensationandvacuumgapformationinmagneticneutronstars.We findthatcon-
3
densation can occur if the thermal energy kT of the neutron star surface is less than
6
8 about8%ofitscohesiveenergyQs,andthatavacuumgapcanformifΩ·Bp <0(i.e.,
3 the neutron star’s rotation axis and magnetic moment point in opposite directions)
8. and kT is less than about 4% of Qs. For example, at B = 3×1014 G, a condensed
0 Fe surface forms when T . 107 K and a vacuum gap forms when T . 5×106 K.
7 Thus, vacuum gap accelerators may exist for some neutron stars. Motivated by this
0 result, we also study the physics of pair cascades in the (Ruderman-Sutherland type)
:
v vacuumgapmodelforphotonemissionby acceleratingelectronsandpositronsdue to
i both curvature radiation and resonant/nonresonantinverse Compton scattering. Our
X
calculations of the condition of cascade-induced vacuum breakdown and the related
r
a pulsardeathline/boundarygeneralizepreviousworksto the superstrongfield regime.
We find that inverse Compton scatterings do not produce a sufficient number of high
energy photons in the gap (despite the fact that resonantly upscattered photons can
immediatelyproducepairsforB &1.6×1014G)andthusdonotleadtopaircascades
for most neutron star parameters (spin and magnetic field). We discuss the implica-
tions of our results for the recent observations of neutron star thermal radiation as
well as for the detection/non-detection of radio emission from high-B pulsars and
magnetars.
Key words: radiation mechanisms: non-thermal – radiationmechanisms: thermal –
stars: magnetic fields – stars: neutron – pulsars: general.
1 INTRODUCTION
Recentobservations of neutronstars haveprovidedawealth of information on theseobjects, but theyhavealso raised many
new questions. For example, with the advent of X-ray telescopes such as Chandra and XMM-Newton, detailed observations
of the thermal radiation from the neutron star surface have become possible. These observations show that some nearby
⋆ Email:[email protected]
† Email:[email protected]
2 Zach Medin and Dong Lai
isolated neutron stars (e.g., RX J1856.5-3754) appear to have featureless, nearly blackbody spectra (Burwitz et al. 2003;
van Kerkwijk & Kaplan 2007). Radiation from a bare condensed surface (where the overlying atmosphere has negligible
opticaldepth)hasbeeninvokedtoexplainthisnearlyperfectblackbodyemission(e.g.,Burwitz et al.2003;Mori & Ruderman
2003; Turolla et al. 2004; van Adelsberg et al. 2005; Perez-Azorin et al. 2006; Hoet al. 2007; but see Ruderman 2003 for an
alternative view). However, whether surface condensation actually occurs depends on the cohesive properties of the surface
matter (e.g., Lai 2001).
Equally puzzling are the observations of anomalous X-ray pulsars (AXPs) and soft gamma-ray repeaters (SGRs) (see
Woods & Thompson2005forareview).Thoughthesestarsarebelievedtobemagnetars,neutronstarswithextremelystrong
magnetic fields (B & 1014 G), they mostly show no pulsed radio emission (but see Camilo et al. 2006, 2007; Kramer et al.
2007) and their X-ray radiation is too strong to be powered by rotational energy loss. By contrast, several high-B radio pul-
sarswith inferred surfacefield strengthssimilar tothoseofmagnetars havebeendiscovered (e.g.,Kaspi & McLaughlin 2005;
Vranevsevic,Manchester, & Melrose2007).Adeeperunderstandingofthedistinctionbetweenpulsarsandmagnetarsrequires
furtherinvestigationofthemechanismsbywhichpulsarsandmagnetarsradiateandoftheirmagnetosphereswherethisemis-
sion originates. Theoretical models of pulsar and magnetar magnetospheres depend on thecohesive properties of the surface
matterin strong magnetic fields(e.g., Ruderman& Sutherland1975; Arons& Scharlemann 1979; Cheng & Ruderman1980;
Usov & Melrose1996;Harding & Muslimov1998;Gil, Melikidze, & Geppert2003;Muslimov & Harding2003;Beloborodov & Thompson
2007). For example, depending on how strongly bound the surface matter is, a charge-depleted acceleration zone (“vacuum
gap”) above the polar cap of a pulsar may or may not form, and this will affect pulsar radio emission and other high-energy
emission processes.
The cohesive property of the neutron star surface matter plays a key role in these and other neutron star processes and
observed phenomena. The cohesive energy refers to the energy required to pull an atom out of the bulk condensed matter
at zero pressure. A related (but distinct) quantity is the electron work function, the energy required to pull out an electron.
Formagnetized neutronstarsurfaces thecohesiveenergy andwork functioncan bemanytimesthecorresponding terrestrial
values, dueto thestrong magnetic fields threadingthe matter (e.g., Ruderman1974; Lai 2001).
In two recent papers (Medin & Lai 2006a,b, hereafter ML06a,b), we carried out detailed, first-principle calculations of
thecohesivepropertiesofH,He,C,andFesurfacesatfieldstrengthsbetweenB=1012 Gto2 1015 G.Themainpurposeof
×
thispaperistoinvestigateseveralimportantastrophysicalimplicationsoftheseresults(somepreliminaryinvestigationswere
reportedinMedin & Lai2007).Thispaperisorganizedasfollows.InSection2webrieflysummarizethekeyresults(cohesive
energy and work function values) of ML06a,b used in this paper. In Section 3 we examine the possible formation of a bare
neutron star surface, which directly affects the surface thermal emission. We find that the critical temperature below which
a phase transition to the condensed state occurs is approximately given by kT 0.08Q , where Q is the cohesive energy
crit s s
∼
ofthesurface. InSection4weconsidertheconditionsfortheformation ofapolarvacuumgap inpulsarsandmagnetars.We
findthat neutron stars with rotation axis and magnetic moment given byΩ Bp >0are unableto form vacuum gaps (since
·
the electrons which are required to fill the gaps can be easily supplied by the surface), but neutron stars with Ω Bp < 0
·
can form vacuum gaps provided that the surface temperature is less than kT 0.04Q (and that particle bombardment
crit s
∼
does not completely destroy the gap; see Section 6). In Section 5 we discuss polar gap radiation mechanisms and the pulsar
death line/boundary in thevacuum gap model. Wefindthat when curvatureradiation is thedominantradiation mechanism
in the gap, a pair cascade is possible for a large range of parameter space (in the P–P˙ diagram), but when inverse Compton
scattering (either resonant or nonresonant) is the dominant radiation mechansim, vacuum breakdown is possible for only a
very small range of parameter values. Implications of our results for recent observations are discussed in Section 6. Some
technical details (on our treatment of inverse Compton scattering and vacuum gap electrodynamics of oblique rotators) are
given in two appendices.
2 COHESIVE PROPERTIES OF CONDENSED MATTER IN STRONG MAGNETIC FIELDS
Itis well-known that theproperties of mattercan bedrastically modified by strongmagnetic fields. Thenaturalatomic unit
for the magnetic field strength, B , is set by equating the electron cyclotron energy ¯hω = ¯h(eB/m c) = 11.577B keV,
0 ce e 12
where B =B/(1012 G), to thecharacteristic atomic energy e2/a =2 13.6 eV (where a is theBohr radius):
12 0 0
×
m2e3c
B = e =2.3505 109G. (1)
0 ¯h3 ×
For b = B/B & 1, the usual perturbative treatment of the magnetic effects on matter (e.g., Zeeman splitting of atomic
0
energy levels) does not apply. Instead, theCoulomb forces act as a perturbation to the magnetic forces, and theelectrons in
an atom settle intotheground Landau level. Because of theextremeconfinement of the electrons in thetransversedirection
(perpendicular to the field), the Coulomb force becomes much more effective in binding the electrons along the magnetic
field direction. Theatom attainsa cylindrical structure.Moreover, it is possible for theseelongated atoms to form molecular
Magnetic neutron star surfaces and polar caps 3
chainsbycovalent bondingalong thefield direction. Interactions between thelinear chainscan then lead to theformation of
three-dimensional condensed matter (Ruderman 1974;Ruder et al. 1994; Lai 2001).
The basic properties of magnetized condensed matter can be estimated using the uniform electron gas model (e.g.,
Kadomtsev 1970). The energy per cell of a zero-pressure condensed matter is given by
120Z9/5B2/5 eV, (2)
Es∼− 12
and thecorresponding condensation density is
ρ 560AZ−3/5B6/5g cm−3, (3)
s∼ 12
whereZ, Aarethechargenumberandmassnumberoftheion(seeLai2001andreferencesthereinforfurtherrefinementsto
theuniformgasmodel).Althoughthissimplemodelgivesareasonableestimateofthebindingenergyforthecondensedstate,
itisnotadequatefordeterminingthecohesivepropertyofthecondensedmatter.Thecohesiveenergyisthe(relativelysmall)
difference between the atomic ground-state energy and the zero-pressure condensed matter energy , both increasing
a s
E E
rapidlywith B.Moreover, theelectron Fermienergy (includingbothkineticenergy andCoulomb energy)in theuniform gas
model,
ε =(3/5Z) 73Z4/5B2/5 eV, (4)
F Es ∼− 12
may not give a good scaling relation for the electron work function when detailed electron energy levels (bands) in the
condensed matter are taken into account.
There have been few quantitative studies of infinite chains and zero-pressure condensed matter in strong magnetic
fields. Earlier variational calculations (e.g., Flowers et al. 1977; Mu¨ller 1984) as well as calculations based on Thomas-Fermi
type statistical models (e.g., Abrahams& Shapiro 1991; Fushikiet al. 1992), while useful in establishing scaling relations
and providing approximate energies of the atoms and the condensed matter, are not adequate for obtaining reliable energy
differences (cohesive energies). Quantitative results for the energies of infinite chains of hydrogen molecules H over a wide
∞
range of field strengths (B B ) were presented in Lai et al. (1992) (using the Hartree-Fock method with the plane-wave
0
≫
approximation;seealsoLai2001forsomeresultsforHe )andinRelovsky & Ruder(1996)(usingdensityfunctionaltheory).
∞
For heavier elements such as C and Fe, the cohesive energies of one dimensional (1D) chains have only been calculated at
a few magnetic field strengths in the range of B = 1012–1013 G, using Hartree-Fock models (Neuhauseret al. 1987) and
densityfunctional theory(Jones 1985).Therewere somediscrepancies between theresultsof theseworks, and someadopted
acrudetreatmentforthebandstructure(Neuhauseret al.1987).Anapproximatecalculation of3Dcondensedmatterbased
on density functional theory was presented in Jones (1986).
Ourcalculations of atoms andsmall molecules (ML06a) andof infinitechains andcondensed matter (ML06b) are based
on a newly developed density functional theory code. Although the Hartree-Fock method is expected to be highly accurate
in the strong field regime, it becomes increasingly impractical for many-electron systems as the magnetic field increases,
since more and more Landau orbitals are occupied (even though electrons remain in the ground Landau level) and keeping
track of the direct and exchange interactions between electrons in various orbitals becomes computational rather tedious.
Compared to previous density-functional theory calculations, we used an improved exchange-correlation function for highly
magnetized electron gases, and we calibrated our density-functional code with previous results (when available) based on
other methods. Most importantly, in our calculations of 1D condensed matter, we treated the band structure of electrons in
different Landau orbitals self-consistently without adopting ad-hoc simplifications. This is important for obtaining reliable
results for the condensed matter. Since each Landau orbital has its own energy band, the number of bands that need to be
calculated increases with Z and B, making the computation increasingly complex for superstrong magnetic field strengths
(e.g.,thenumberofoccupiedbandsforFechainsatB=2 1015 Greaches155;seeFig.16ofML06b).Ourdensity-functional
×
calculations allow ustoobtaintheenergiesofatomsandsmallmolecules andtheenergyofcondensedmatterusingthesame
method, thusprovidingreliable cohesive energy and work function valuesfor condensed surfaces of magnetic neutron stars.
In ML06a, we described our calculations for various atoms and molecules in magnetic fields ranging from 1012 G to
2 1015 GforH,He,C,andFe,representativeofthemost likelyneutronstarsurface compositions. Numericalresultsofthe
×
ground-state energies are given for H (up to N = 10), He (up to N = 8), C (up to N = 5), and Fe (up to N = 3),
N N N N
as well as for various ionized atoms. In ML06b, we described our calculations for infinite chains for H, He, C, and Fein that
same magnetic field range. For relatively low field strengths, chain-chain interactions play an important role in the cohesion
ofthree-dimensional(3D)condensedmatter.Anapproximatecalculationof3DcondensedmatterisalsopresentedinML06b.
Numerical results of the ground-state and cohesive energies, as well as the electron work function and the zero-pressure
condensed matter density,are given in ML06b for H and H(3D),He and He(3D),C and C(3D), and Fe and Fe(3D).
∞ ∞ ∞ ∞
SomenumericalresultsfromML06a,bareprovidedingraphicalforminFigs.1,2,3,and4(seeML06a,bforapproximate
scaling relations for differentfield ranges based on numerical fits).Figure 1shows thecohesiveenergies of condensedmatter,
Q = , andthemolecular energy differences,∆ = /N , for He,Fig. 2for C, and Fig. 3for Fe; here isthe
s 1 s N N 1 1
E −E E E −E E
atomic ground-state energy, is the ground-state energy of the He , C , or Fe molecule, and is the energy per cell
N N N N s
E E
4 Zach Medin and Dong Lai
10
) 1
V
e
k
(
y
g
r
e
n 0.1
E
Q
εs
∆
ε3
∆
2
0.01
1 10 100 1000
12
B (10 G)
Figure 1.Cohesive energyQs=E1−Es andmolecular energydifference ∆EN =EN/N−E1 forheliumas afunctionofthe magnetic
fieldstrength.
of the zero-pressure 3D condensed matter. Some relevant ionization energies for the atoms are also shown. Figure 4 shows
theelectron work functions φ for condensed He, C, and Feas a function of thefield strength. Wesee that the work function
increases muchmoreslowly withB compared tothesimplefreeelectron gas model[seeEq.(4)],and thedependenceon Z is
also weak. The results summarized herewill be used in Section 3 and Section 4 below.
3 CONDENSATION OF NEUTRON STAR SURFACES IN STRONG MAGNETIC FIELDS
AsseenfromFigs.1,2,and3,thecohesiveenergiesofcondensedmatterincreasewithmagneticfield.Wethereforeexpectthat
forsufficientlystrongmagneticfields,thereexistsacriticaltemperatureT belowwhichafirst-orderphasetransitionoccurs
crit
betweenthecondensateandthegaseousvapor.Thishasbeeninvestigatedindetailforhydrogensurfaces(seeLai & Salpeter
1997; Lai 2001), but not for other surface compositions. Here we consider the possibilies of such phase transitions of He, C,
and Fe surfaces.
AprecisecalculationofthecriticaltemperatureT isdifficult.WecandetermineT approximatelybyconsideringthe
crit crit
equilibriumbetweenthecondensedphase(labeled“s”)andthegaseousphase(labeled“g”)intheultrahighfieldregime(where
phase separation exists). The gaseous phase consists of a mixture of free electrons and bound ions, atoms, and molecules.
Phaseequilibriumrequiresthetemperature,pressureandthechemicalpotentialsofdifferentspeciestosatisfy theconditions
(herewe consider Feas an example; He and C are similar)
P =P =[2n(Fe+)+3n(Fe2+)+ +n(Fe)+n(Fe )+n(Fe )+ ]kT, (5)
s g 2 3
··· ···
1 1
µ =µ +µ(Fe+)=2µ +µ(Fe2+)= =µ(Fe)= µ(Fe )= µ(Fe )= , (6)
s e e ··· 2 2 3 3 ···
where we treat thegaseous phase as an ideal gas. Thechemical potential of thecondensed phase is given by
µ = +P V , (7)
s s s s s,0
E ≃E
where is the energy per cell of thecondensate and is theenergy per cell at zero-pressure (we will label this simply as
s s,0
E E
Magnetic neutron star surfaces and polar caps 5
10
)
V
e 1
k
(
Q
y
s
g
er Qε∞
∆
n
E 0.1 ε3
∆
2
I
1
I
2
0.01
1 10 100 1000
12
B (10 G)
Figure2.CohesiveenergyQs=E1−Esandmolecularenergydifference∆EN =EN/N−E1forcarbonasafunctionofthemagneticfield
strength. The symbol Q∞ represents the cohesive energy of a one-dimensional chain, and I1 and I2 arethe firstand second ionization
energiesoftheCatom.
). We have assumed that the vapor pressure is sufficiently small so that the deviation from the zero-pressure state of the
s
E
condensate is small; this is justified when the saturation vapor pressure P is much less than the critical pressure P for
sat crit
phase separation, or when the temperatureis less than the critical temperature bya factor of a few.
For nondegenerate electrons in a strong magnetic field the numberdensity is related to µ by
e
1 ∞ n ¯hω ∞ dp p2
ne ≃ 2πρ20eµe/kTnXL=0gnLexp(cid:16)− kLT ce(cid:17)Z−∞ hz exp(cid:18)2m−ekzT(cid:19) (8)
1 ¯hω
eµe/kTtanh−1 ce (9)
≃ 2πρ20λTe 2kT
1 (cid:16) (cid:17)
eµe/kT, (10)
≃ 2πρ20λTe
where g = 1 for n = 0 and g = 2 for n > 0 are the Landau degeneracies, λ = (2π¯h2/m kT)1/2 is the electron
nL L nL L Te e
thermal wavelength, and the last equality applies for kT ¯hω . The magnetic field length is ρ =(¯hc/eB)1/2. For atomic,
ce 0
≪
ionic, or molecular Fethenumberdensity is given by
1 K2
n(FeA) ≃ h3eµA/kT exp −EkAT,i d3K exp 2M−AkT (11)
Xi (cid:16) (cid:17)Z (cid:18) (cid:19)
1 µ
≃ λ3 exp −EAk−T A Zint(FeA), (12)
TA
(cid:16) (cid:17)
with theinternal partition function
∆
Zint(FeA)= exp − kETA,i . (13)
Xi (cid:16) (cid:17)
and∆ = .Here,thesubscriptArepresentstheatomic, ionic,ormolecularspecieswhosenumberdensityweare
A,i A,i A
E E −E
6 Zach Medin and Dong Lai
100
10
)
V
e
k
(
Q
y
g 1 s
er Qε∞
∆
n
E ε3
∆
2
0.1 I
1
I
2
10 100 1000
12
B (10 G)
Figure3.CohesiveenergyQs=E1−Es andmolecularenergydifference∆EN =EN/N−E1 forironasafunctionofthemagneticfield
strength. The symbol Q∞ represents the cohesive energy of a one-dimensional chain, and I1 and I2 arethe firstand second ionization
energiesoftheFeatom.Below5×1012 G,ourresultsforQ∞andQsbecomeunreliableasQ∞andQsbecomeverysmallandapproach
numericalerrorsforEN andEs.
calculating (e.g., Fe or Fe+) and the sum is over all excited states of that species. Also, λ =(2π¯h2/M kT)1/2 is the
2 i Te A
Fe particle’s thermal wavelength, where M = NAM is the total mass of the particle (N is the number of “atoms” in the
A
P
molecule, A is the atomic mass number, and M =m +m ). The vector K represents the center-of-mass momentum of the
p e
particle.NotethatwehaveassumedherethattheFe particlemovesacross thefieldfreely;thisisagood approximation for
A
large M . Theinternalpartition function Z representstheeffect of all excited states of thespecies on thetotal density;in
A int
this work we will use the approximation that this factor is the same for all species, and we will estimate the magnitude of
thisfactor later in this section.
The equilibrium condition µ = µ(Fe) for the process Fe +Fe = Fe yields the atomic density in the saturated
s s,∞ s,∞+1
vapor:
AMkT 3/2 Q
n(Fe) exp s Z , (14)
≃ 2π¯h2 −kT int
(cid:16) (cid:17) (cid:16) (cid:17)
where Q = is the cohesive energy of thecondensed Fe. The condition Nµ =µ(Fe ) for theprocess Fe +Fe =
s 1 s s N s,∞ N
E −E
Fe yields the molecular density in the vapor:
s,∞+N
NAMkT 3/2 S
n(Fe ) exp N Z , (15)
N ≃ 2π¯h2 −kT int
(cid:16) (cid:17) (cid:16) (cid:17)
where
S = N =N[Q ( /N)] (16)
N N s s 1 N
E − E − E −E
isthe“surfaceenergy”and /N istheenergyperioninthemolecule.Theequilibriumconditionµ(Fen+)=µ +µ(Fe(n+1)+)
N e
E
for the process e+Fen+ =Fe(n+1)+, where Fen+ is thenth ionized state of Fe,yields the vapordensities for theions:
b m kT I
n(Fe+)n e exp 1 n(Fe), (17)
e ≃ 2πa2 2π¯h2 −kT
0r
(cid:16) (cid:17)
Magnetic neutron star surfaces and polar caps 7
700
) He
V
e C
(
φ 500 Fe
n
o
i
t
c
n
u
F 300
k
r
o
W
100
1 10 100 1000
12
B (10 G)
Figure4.Numericalresultfortheelectronworkfunctionasafunctionofthemagneticfieldstrength,forHe,C,andFeinfinitechains.
b m kT I
n(Fe2+)n e exp 2 n(Fe+), (18)
e ≃ 2πa2 2π¯h2 −kT
0r
(cid:16) (cid:17)
and so on. Here, b = B/B and a is the Bohr radius, and I = represents the ionization energy of the nth
0 0 n E(n−1)+−En+
ionizedstateofFe(i.e.,theamountofenergyrequiredtoremovethenthelectronfromtheatomwhenthefirstn 1electrons
−
havealready been removed). The total electron density in thesaturated vapor is
n =n(Fe+)+2n(Fe2+)+ . (19)
e
···
The number densities of electrons [Eq. (19)] and ions [e.g., Eqs. (17) and (18)] must be found self-consistently, for all ion
species that contribute significantly to the total vapor density. The total mass density in the vapor is calculated from the
numberdensities of all of the species discussed above, using the formula
ρ =AM n(Fe)+2n(Fe )+ +n(Fe+)+n(Fe2+)+ . (20)
g 2
··· ···
Figure(cid:2) 5 (for Fe) and Fig. 6 (for C) show the the den(cid:3)sities of different atomic/molecular species in the saturated vapor
in phase equilibirum with the condensed matter for different temperatures and field strengths. These are computed using
the values of /N, , and presented in ML06a,b and depicted in Figs. 2 and 3. As expected, for sufficiently low
N s n+
E E E
temperatures, thetotal gas densityin thevaporis muchsmaller than thecondensation density,and thusphaseseparation is
achieved.ThecriticaltemperatureT ,belowwhichphaseseparationbetweenthecondensateandthegaseousvaporoccurs,
crit
is determined by thecondition ρ =ρ . Wefind that for Fe:
s g
T 6 105, 7 105, 3 106, 107, 2 107 K for B =5, 10, 100, 500, 1000, (21)
crit 12
≃ × × × ×
for C:
T 9 104, 3 105, 3 106, 2 107 K for B =1, 10, 100, 1000. (22)
crit 12
≃ × × × ×
and for He:
T 8 104, 3 105, 2 106, 9 106 K for B =1, 10, 100, 1000. (23)
crit 12
≃ × × × ×
8 Zach Medin and Dong Lai
In terms of thecohesive energy,these results can beapproximated by
kT 0.08Q . (24)
crit s
∼
Note that in our calculations for theiron vapordensity at B =5-500 we haveestimated themagnitudeof theinternal
12
partition function factor Z ; the modified total density curves are marked on these figures as “ρ Z ”. To estimate Z
int g int int
×
we useEq. (13) with a cutoff tothe summation abovesome energy.For B =5,10,100, and 500 we calculate or interpolate
12
the energies for all excited states of atomic Fe with energy below this cutoff, in order to find Z . The energy cutoff is
int
necessary because the highly excited states become unbound (ionized) due to finite pressure and should not be included in
Z (otherwise Z would diverge). In principle, the cutoff is determined by requiring the effective size of the excited state
int int
to be smaller than the inter-particle space in the gas, which in turn depends on density. In practice, we choose the cutoff
suchthatthehighestexcitedstatehasabindingenergy significantlysmaller thantheground-statebindingenergy
A,i A
|E | |E |
(typically 30% of it). As an approximation, we also assume that the internal partitions for Fe molecules and ions have the
N
same Z as the Fe atom. Despite the crudeness of our calculation of Z , we see from Fig. 5 that the resulting T is only
int int crit
reduced bya few tens of a percent from theT valueassuming Z =1.
crit int
Wenotethat ourcalculation ofthesaturated vapordensityisveryuncertain aroundT T ,sinceEqs.(14)–(18)are
crit
∼
derivedfor ρ ρ while thecritical temperatureof thesaturated vapordensity is found bysetting ρ =ρ . However,since
g s s g
≪
the vapor density decreases rapidly as T decreases, when the temperature is below T /2 (for example), the vapor density
crit
becomes much less than thecondensation density and phase transition is unavoidable. When thetemperature dropsbelow a
fraction of T , the vapor density becomes so low that the optical depth of the vapor is negligible and the outermost layer
crit
of the neutron star then consists of condensed matter. The radiative properties of such condensed phase surfaces have been
studied using a simplified treatment of thecondensed matter (see vanAdelsberg et al. 2005 and references therein).
4 POLAR VACUUM GAP ACCLERATORS IN PULSARS AND MAGNETARS
A rotating, magnetized neutron star is surrounded by a magnetosphere filled with plasma. The plasma is assumed to be an
excellent conductor, such that the charged particles move to screen out any electric field parallel to the local magnetic field.
The corresponding charge density is given by (Goldreich & Julian 1969)
Ω B
ρGJ ≃− 2π·c (25)
where Ω is therotation rate of theneutron star.
The Goldreich-Julian density assumes that charged particles are always available. This may not be satisfied everywhere
in themagnetosphere. Inparticular, charged particles traveling outward along theopen field lines originating from thepolar
cap region of theneutron star will escape beyondthe light cylinder. To maintain therequired magnetosphere charge density
theseparticleshavetobereplenishedbythestellarsurface.Ifthesurfacetemperatureandcohesivestrengtharesuchthatthe
required particles are tightly boundto thestellar surface, thoseregions of thepolar cap through which thecharged particles
areescapingwillnotbereplenished.Avacuumgap willthendevelopjustabovethepolarcap(e.g.,Ruderman & Sutherland
1975; Cheng & Ruderman 1980; Usov & Melrose 1996; Zhang, Harding, & Muslimov 2000; Gil, Melikidze, & Geppert 2003).
Inthisvacuumgapzonetheparallelelectricfieldisnolongerscreenedandparticlesareacceleratedacrossthegapuntilvacuum
breakdown (via pair cascade) shorts out the gap. Such an acceleration region can have an important effect on neutron star
emissionprocesses.Wenotethatintheabsenceofavacuumgap,apolargapacceleration zonebasedonspace-charge-limited
flow may still develop (e.g., Arons & Scharlemann 1979; Harding & Muslimov 1998; Muslimov & Harding 2003).
InthissectionwedeterminetheconditionsrequiredforthevacuumgaptoexistusingourresultssummarizedinSection2.
Thecohesiveenergyandelectronworkfunctionofthecondensedneutronstarsurfaceareobviouslythekeyfactors.Weexamine
thephysicsof particle emission from condensed surface in more detail than considered previously.
4.1 Particle Emission From Condensed Neutron Star Surfaces
WeassumethattheNSsurfaceisin thecondensedstate,i.e., thesurfacetemperatureT islessthanthecritical temperature
T forphaseseparation (seeSection 3).(IfT >T ,thesurfacewill beingaseous phaseandavacuumgap will notform.)
crit crit
Weshall see thatin orderfor thesurface not toemit too large afluxof chargesto themagnetosphere (anecessary condition
for the vacuumgap to exist), an even lower surface temperature will be required.
4.1.1 Electron Emission
ForneutronstarswithΩ B >0,whereB isthemagneticfieldatthepolarcap,theGoldreich-Julianchargedensityisnega-
p p
·
tiveatthepolarcap,thussurfaceelectronemission(oftencalledthermionicemissioninsolidstatephysics;Ashcroft & Mermin
Magnetic neutron star surfaces and polar caps 9
6
4
B =5
) 12
3
m
g/c 2 x Zint
ρ ( ρ g
0 0
1
g ρ
o g
l Fe
-2
+
Fe
2+
Fe
-4
5.4 5.5 5.6 5.7 5.8 5.9 6
6
6
4 B =10 B =100
12 12
3) 4
m
ρ (g/c 2 ρ gx Zint 2 ρ gx Zint ρ
og 10 0 ρFge 0 FFgee2
l -2 Fe2 Fe+
Fe+ Fe2+
Fe2+ -2 Fe3+
-4
5.5 5.6 5.7 5.8 5.9 6 6.2 6.4 6.6 6.8
8 8
3m) 6 B12=500 6 B12=1000
c
ρlog (g/10 24 ρ gx Zint ρFFFFgeeee23+ 24 FFFFρgeeee23+
0 FFee2≥3++ 0 FFee2≥3++
6.6 6.8 7 7.2 7.4 6.8 7 7.2 7.4 7.6
log T (K) log T (K)
10 10
Figure 5. The mass densities of various atomic/ionic/molecular species and the total density (ρg) of the vapor in phase equilibrium
with the condensed iron surface. The five panels are for different field strengths, B12 = 5,10,100,500,1000. The horizontal lines give
the densities of the condensed phase, ρs. All the vapor density curves are calculated assuming Zint = 1, except for the curve marked
by“ρg×Zint”,forwhichthe total vapor density iscalculated taking intoaccount thenontrivial internal partitionfunctions ofvarious
species.Thecriticaltemperature Tcrit forphaseseparationissetbytheconditionρg =ρs.
1976) is relevant. Let be the number flux of electrons emitted from the neutron star surface. The emitted electrons are
e
F
accelerated to relativistic speed quickly, and thus the steady-state charge density is ρ = e /c. For the vacuum gap to
e e
− F
exist, we require ρ < ρ . (If e /c > ρ , thecharges will berearranged so that the charge density equals ρ .)
e GJ e GJ GJ
| | | | | F | | |
To calculate the electron emission flux from the condensed surface, we assume that these electrons behave like a free
electron gasin ametal, wheretheenergy barriertheymustovercomeisthework functionofthemetal. Inastrongmagnetic
field,the electron flux is given by
∞ p 1 dp
= f(ǫ) z z , (26)
Fe Zpmin me2πρ20 h
wherep = 2m U ,U isthepotentialenergyoftheelectronsinthemetal,ǫ=p2/(2m )istheelectron kineticenergy,
min e| 0| 0 z e
and
p
10 Zach Medin and Dong Lai
6
4
4
3m) 2 B12=1 B12=10
c 2
g/
( 0
ρ ρ
10 ρ 0 Cg
og -2 Cg C2
l -4 CCC23+ -2 CCC34+
-4
4.5 4.6 4.7 4.8 4.9 5 5.1 5 5.2 5.4 5.6
8
6
6
3m) 4 B12=100 B12=1000
c ρ 4 ρ
g/ g g
( 2 C C
ρ C C
g 10 C23 2 C23
lo 0 CC4 CC4
5 0 5
+ +
C C
-2 2+ ≥2+
C C
-2
6 6.2 6.4 6.6 6.8 7 7.2 7.4
log T (K) log T (K)
10 10
Figure 6. The mass densities of various atomic/ionic/molecular species and the total density (ρg) of the vapor in phase equilibrium
withthecondensed carbonsurface.Thefourpanelsarefordifferentfieldstrengths,B12=1,10,100,1000. Thehorizontallinesgivethe
densities of the condensed phase, ρs. All the vapor density curves are calculated assuming Zint =1. The critical temperature Tcrit for
phaseseparationissetbythecondition ρg =ρs.
1
f(ǫ)= (27)
e(ǫ−µ′e)/kT +1
istheFermi-Diracdistributionfunctionwithµ′ theelectron chemicalpotential(excludingpotentialenergy).Integratingthis
e
expression gives
kT kT
= ln 1+e−φ/kT e−φ/kT, (28)
Fe 2πhρ2 ≃ 2πhρ2
0 0
(cid:2) (cid:3)
whereφ U µ′ istheworkfunctionofthecondensedmatterandthesecondequalityassumesφ kT.Thesteady-state
≡| 0|− e ≫
charge density supplied by thesurface is then
e
ρ = =ρ exp(C φ/kT), (29)
e −cFe GJ e−
with
e kT
C =ln 31+ln(P T ) 30, (30)
e (cid:18)c2πhρ20|ρGJ|(cid:19)≃ 0 6 ∼
where T = T/(106 K) and P is the spin period in units of 1 s. For a typical set of pulsar parameters (e.g., P = 1 and
6 0 0
T = 0.5) C 30, but C can range from 23 for millisecond pulsars to 35 for some magnetars. Note that the requirement
6 e e
∼
φ kT is automatically satified here when ρ is less than ρ . The electron work function was calculated in ML06b and
e GJ
≫ | | | |
is depicted in Fig. 4.
4.1.2 Ion Emission
For neutron stars with Ω B < 0, the Goldreich-Juliam charge above the polar cap is positive, so we are interested in ion
p
·
emissionfromthesurface.Unliketheelectrons,whichformarelativelyfree-movinggaswithinthecondensedmatter,theions
