Absorption features caused by oscillations of electrons on the surface of a quark star

0.13 MB·English

by  R. X. Xu

Absorption features caused by oscillations of electrons on the surface of a quark star
R. X. Xu1, S. I. Bastrukov1,2, F. Weber3, J. W. Yu1 and I. V. Molodtsova2
1State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China
2Joint Institute for Nuclear Research, 141980 Dubna, Russia
3Department of Physics, San Diego State University, San Diego, California 92182, USA
(Dated: January 10, 2012)
Ifquarkstarsexist,theymaybeenvelopedinthinelectronlayers(electronseas),whichuniformly
surround theentire star. These layers will beaffected by themagnetic fieldsof quark stars in such
a way that the electron seas would transmit hydromagnetic cyclotron waves, as studied in this
paper. Particular attention is devoted to vortex hydrodynamical oscillations of the electron sea.
2 The frequency spectrum of these oscillations is derived in analytic form. If the thermal X-ray
1 spectra of quark stars are modulated by vortex hydrodynamical vibrations, the thermal spectra
0 of compact stars, foremost central compact objects (CCOs) and X-ray dim isolated neutron stars
2 (XDINSs), could be used to verify the existence of these vibrational modes observationally. The
n central compact object 1E 1207.4-5209 appears particularly interesting in this context, since its
a absorption features at 0.7 keV and 1.4 keV can be comfortably explained in the framework of the
J hydro-cyclotronoscillation model.

9
PACSnumbers: 26.60.-c,97.60.Jd,97.60.Gb

]
E
H I. INTRODUCTION magnetospheric activity. The best absorption features
(at ∼0.7 keV and ∼1.4 keV) were detected for the cen-
.
h tralcompact object (CCO) 1E 1207.4-5209in the center
ThespectralfeaturesofthermalX-rayemissionarees-
p
ofsupernovaremnantPKS1209-51/52(seeTableI).Ini-
sentialfor usto understandthe realnatureofpulsar-like
-
o compact stars. Calculations show that atomic spectral tially, these features were thought to be associated with
r
lines form in the atmospheres of neutron stars. From
t
s the detection and identification of atomic lines in ther-
a
[ malX-rayspectraonecaninferneutronstarmasses(M) TABLE I. Dead pulsars (CCOs and XDINSs) with observed
2 sapnedctrraadliilin(Res),dseipnecnedthoenrMed/sRhifatnadndMb/rRo2a,derensipngecotifvethlye. Bspetchteramlaagbnsoetripctifioenldlin(Bes10[1=3,B19/,102130],Gw)itdhePrivtehdebsypimnapgenreiotdo-,
v dipole braking,T theeffectivethermal temperaturedetected
Atomic features are expected to be detectable with the
6 spectrographs on board of Chandra and XMM-Newton. at infinity, and Ea the absorption energy. We do not list the
2 B-fields of XDINSs for which the propeller braking could be
2 No atomic features have yet been discovered with cer- significant because of theirlong periods.
1 tainty, however. This may have it origin in the very
. strongmagneticfieldscarriedbyneutronstars. Analter-
0 Source P/s B10 kT/keV Ea/keV
1 nativeexplanationcouldbethattheunderlyingcompact RXJ0822.0-4300 0.112 <98 0.4 −
1 staris notaneutronstarbutabarestrange(quarkmat- 1E 1207.4-5209 0.424 <33 0.22 0.7, 1.4
1 ter) star [1, 2]. The surface of such an object does not CXOUJ185238.6+004020 0.105 3.1 0.3 −
: consist of atomic nuclei/ions, as it is the case for a neu- RXJ0720.4-3125 8.39 0.085 0.27
v
i tron star, but of a sea of electrons which envelopes the RXJ0806.4-4123 11.37 0.096 0.46
X quark matter. RXJ0420.0-5022 3.45 0.045 0.33
r Strangestarsarequarkstarsmadeofabsolutelystable RXJ1308.6+2127 10.31 0.086 0.3
a RXJ1605.3+3249 − 0.096 0.45
strange quark matter [3–8]. They consist of essentially
RXJ2143.0+0654 9.43 0.104 0.7
equal numbers of up, down and strange quarks as well
as of electrons [9–11]. The latter are needed to neutral-
ize the electric charges of the quarks, rendering the in-
terior of strange stars electrically neutral. Quark matter the atomic transitions of ionized helium in a stellar at-
is bound by the strong interaction, while electrons are mosphere where a strong magnetic field is present [14].
bound to quark matter by the electromagnetic interac- Soon thereafter, however, it was noted that these lines
tion. Since the latteris long-range,someofthe electrons areofelectron-cyclotronorigin[15]. The spectrumof1E
in the surface region of a quark star reside outside of 1207.4-5209showstwomorefeaturesthatmaybecaused
the quark matter boundary, leading to a quark matter by resonant cyclotron absorption, one at ∼ 2.1 keV and
core which is surrounded by a fairly thin (thousands of another, but of lower significance, at ∼ 2.8 keV [16].
femtometer thick) sea of electrons [8, 9, 12]. Due to the These features vary in phase with the star’s rotation.
enormous advances in X-ray astronomy, more and more Although the detailed mechanism which causes the ab-
so-called dead pulsars are discovered, whose thermal ra- sorptionfeaturesisstillamatterofdebate,timingobser-
diation dominates over a very weak or negligibly small vationspredictaratherweakmagneticfieldforthisCCO,
2
in agreement to what is obtained under the assumption Thegoverningequation,Eq.(1),canberepresentedas
that the lowest-energy line at 0.7 keV is the electron-
∂δv
cyclotron fundamental, favoring the electron-cyclotron +ω [n ×δv]−ν∇2δv=0, (3)
c B
interpretation [17, 18]. Besides 1E 1207.4-5209, broad ∂t
absorption lines have also been discovered in other dead eB B η
pulsars (listed in Table I), especially in so-called X-ray ωc = , nB = , ν = . (4)
m c B ρ
dim isolatedneutronstars (XDINSs), between about0.3 e
and 0.7 keV [19]. where ω is the cyclotron frequency. In the Appendix,
c
In this paper, we re-investigate the physics of these we show that the electron sea can transmit macroscopic
absorption features. The key assumption that we make perturbations in the form of rotational hydro-cyclotron
hereisthatthesefeaturesoriginatefromtheelectronseas waveswhicharecharacterizedbythefollowingdispersion
on quarks stars rather than from neutron stars, whose relation,
surface properties are radically different from those of
strangestars[8,9,12]. Ofkeyimportanceisthemagnetic 1 (νk2/ω)
ω =±ω cosθ +i , (5)
field carried by a quark star, which critically affects the c (cid:20)1−(νk2/ω)2 1−(νk2/ω)2(cid:21)
global properties (hydrodynamic surface fluctuations) of
where ω and k denote the frequency and wave vector of
the electron sea at the surface of the star. We study
theperturbations,respectively. The Larmorradiusofan
this problemintheframeworkofclassicalelectrodynam-
ics in terms of cyclotron resonances of electrons in weak electron in a strong magnetic field, rL ≃ mec2/(eB) ∝
B−1, is very small for pulsar-like compact stars, and we
magnetic fields, since the magnetic fields of dead pulsars
are much lower than the critical field, B =4.414×1013 neglect the viscosity term in the following analysis of
q
the motion of collective electrons. In the collision-free
G, at which the quantization of the cyclotron orbits of
regime, ν = 0, the hydro-cyclotron electron wave is de-
electrons into Landau levels occurs.
scribed as a transverse, circularly polarized wave whose
dispersion relation and propagationspeed are given by
II. HYDRO-CYCLOTRON WAVES
ω =±ω cosθ and V =±(ω /k)cosθ, (6)
c c
A. Governing equations respectively. Here, θ is the angle between the mag-
netic field B and the wave vector k. If k k B one has
For what follows we restrict ourselves to a discussion ω = ±ω . In metals these kind of oscillations are ob-
c
ofthelarge-scaleoscillationsofanelectronseasubjected served as electron-cyclotron resonances. There are two
toastellarmagneticfield. We willbe applyingthe semi- possible resonance states, one for ω = ω and the other
c
classical approach of classical electron theory of metals for ω = −ω . These resonances correspond to the two
c
andmakinguseofstandardequationsoffluid-mechanics. opposite orientations of circularly polarized electron cy-
The electrons are viewed as a viscous fluid of uniform clotron waves.
density ρ = nm (where n is the electron number den-
e
sity) whose oscillations are given in terms of the mean
electron flow velocity δv. This implies that the fluctua- B. Hydro-cyclotron oscillations of electrons on
tion current-carrying flow is described by the density of bare strange quark stars
the convective current δj = ρ δv, where ρ = en is the
e e
electron charge density. The equations of motions of a We restrict our analysis to the collision-free regime of
viscous electron fluid are then given by [20] vortex hydro-cyclotron oscillations. Using spherical co-
∂δv 1 ordinates, equation (3) then takes the form
ρ = [δj×B]+η∇2δv, (1)
∂t c
∂δv
+ω [n ×δv]=0. (7)
j=ρeδv, ρ=men, ρe =en, (2) ∂t c B
where e and n are the change and number densities of Taking the curl of both sides of Eq. (7), we obtain
electrons, respectively. We emphasize that δj stands
for the convective current density and not for Amp´ere’s ∂δω =ω (n ·∇)δv, δω =∇×δv. (8)
j = (c/4π)∇ × δB, as it is the case for magneto- ∂t c B
hydrodynamics. This means that the hydrodynamic os-
Let the magnetic field B be directed along the z-axis,so
cillationsinquestionareofnon-Alfv´entype. InEq.(1),η
that in Cartesian coordinates n = (0,0,1). We then
denotestheeffectiveviscosityoftheelectronfluid,which B
have
originates from collisions of electrons with the magnetic
field lines at the stellar surface. It is worth noting that n =cosθ, n =−sinθ, n =0. (9)
r θ φ
the cyclotron waves can be regarded as an analogue of
the inertial waves in a rotating incompressible fluid, as From a mathematical point of view, the problem can be
presented in Eq. (III.56) of Chandrasekhar’s book [21]. considerably simplified if one expresses the velocity δv,
3
whichobeysthe condition∇·δv=0,interms ofStokes’
TABLE II. Comparison between single-particle (Landau
stream function, χ(θ,φ). This leads to
level) and hydro-wave results for the cyclotron frequencies
1 ∂χ(θ,φ) 1∂χ(θ,φ) (B12 = B/1012 G, ω1 = ω(ℓ = 1), ωc denotes the cyclotron
δvr =0, δvθ = , δvφ =− (.10) frequency).
rsinθ ∂φ r ∂θ
ω(ℓ=1) ω(ℓ=2) ω(ℓ=3) ~ω1/keV
The depth of the electron layer near the star is much
smaller than the stellar radius so that r ≈ R to a very Landau level ωc 2ωc 3ωc 11.6B12
good approximation. Equation (8) then simplifies to Hydro-wave ωc/2 ωc/6 ωc/12 5.8B12
∂δω n δv
r θ θ
=−ω , (11)
c
∂t R
From
with the radial component of the vortex given in terms
ω(ℓ) ℓ+2
of χ, = , ℓ≥1, (20)
ω(ℓ+1) ℓ
1 ∂δv 1 ∂δv 1
δωr = R(cid:20) ∂θφ − sinθ ∂φθ(cid:21)=−R2∇2⊥χ(θ,φ),(12) it follows that this ratio becomes a constant for ℓ ≫ 1.
Such a spectral feature is notably different from the one
1 ∂ ∂ 1 ∂2
∇2 = sinθ + . (13) of electron-cyclotron resonances of transitions between
⊥ sinθ∂θ (cid:18) ∂θ(cid:19) sin2θ∂φ2 different Landau levels.
From the energy eigenvalues, E , of an electron in a
Substituting Eq.(12) andEq.(13)into Eq.(11) leads to n
strong magnetic field, which are found by solving the
∂χ ∂χ Dirac equation (see Ref. [22]), one may approximate the
∇2 +ω =0. (14)
⊥(cid:18)∂t(cid:19) c∂φ valueofEn forarelativelyweakmagneticfield,B ≪Bq,
by
Thefactthatfreeelectronsundergocyclotronoscillations
intheplanesperpendiculartothemagneticfieldsuggests En =mc2+n~ωc, n≥0. (21)
thatthestreamfunctionχcanbewritteninthefollowing
Therefore, in the framework of a single-particle approx-
separable form,
imation, the emission/absorption frequencies, which are
χ(θ,φ)=ψ(θ)cos(φ±ωt). (15) given by ~ω(ℓ)=En+ℓ−En, should occur at
The “+” sign allows for cyclotron oscillations which ω(ℓ)=ℓω , ℓ≥1. (22)
c
are induced by the clockwise polarized wave, and the
“−” sign allows for cyclotron oscillations induced by the TableII comparestheresultsofEq.(22)withthe results
count-clockwise polarized wave. Substituting (15) into ofEq.(19)obtainedforthehydro-cyclotronwavemodel.
(14) leads to Mostnotably,itfollowsthatforthehydro-cyclotronwave
model one obtains ω(ℓ=2)/ω(ℓ=3)=2, in contrast to
ω
∇2 ψ(θ)± cψ(θ)=0. (16) the cyclotron resonance model for single electrons which
⊥
ω
predicts this ratio for ω(ℓ=2)/ω(ℓ=1)=2..
In the reference frame where the polar axis is fixed, Eq.
(16) is identical to the Legendre equationfor the surface
spherical function, III. 1E 1207.4-5209 AND OTHER COMPACT
OBJECTS
∇2 P (θ)+ℓ(ℓ+1)P (θ)=0, (17)
⊥ ℓ ℓ
where P (cosθ) denotes the Legendre polynomial of de- As already mentioned in the Introduction, 1E 1207.4-
ℓ
gree ℓ. Hence, setting ψ(θ)=P (θ) we obtain 5209(orJ1210-5226)inPKS1209-51/52isoneofthecen-
ℓ
tral compact objects in supernova remnants [23], where
ω eB
ω±(ℓ)=± c , ωc = , ℓ≥1. (18) broadabsorptionlines, near (0.7, 1.4)keV[14], and pos-
ℓ(ℓ+1) mec sibly near (2.1, 2.8) keV [16] were detected for the first
time. The interpretation of the absorption feature at
From this relation we can read off the frequency of a
∼2.8 keV is currently a matter of debate, in contrastto
surface hydro-cyclotronoscillation of a given order ℓ.
thefeatureat∼2.1keVwhichisessentiallyunexplained.
Intriguingly,anabsorptionfeaturewiththesameenergy,
C. Characteristic features of cyclotron frequencies 2.1 keV, has also been detected in the accretion-driven
X-ray pulsar 4U 1538-52 [24].
For what follows, we assume that 1E 1207.4-5209is a
Let us consider the spectrum of the positive branch
strange quark star and that (some of) these absorption
ω(ℓ)=ω+(ℓ) of Eq. (18),
featuresareproducedbythehydro-cyclotronoscillations
ω(ℓ) 1 of the electron sea at the surface of such an object. As-
= , ℓ≥1. (19)
ω ℓ(ℓ+1) suming a magnetic surface field of B ≃ 7×1011 G and
c
4
TABLE III. The frequencies, ω(ℓ), at which hydro-cyclotron TABLE IV. Comparison between the frequencies, ω(ℓ), and
oscillations occur for 1E 1207.4-5209 with effective tempera- the absorption frequencies detected, ωobs, for SGR 1806-20
ture T ≃0.2 keV,assuming a magnetic field of B ≃7×1011 (withanassumptionofB≃1.86×1013 G).Thedataofωobs
G. are from [29]. Both ω(ℓ) and ωobs are in keV.
ℓ 1 2 3 4 5 6 ℓ 1 2 3 4 5 6 7 8
ω(ℓ)/keV 4.2 1.4 0.7 0.4 0.3 0.2 ω(ℓ) 108 36 18 10.8 7.2 5.1 3.8 3.0
ωobs − − 17.5±.5 11.2±.4 7.5±.3 5.0±.2 − −
thus ω(ℓ = 3) = 0.7 keV, we obtain the oscillation fre-
quenciesshowninTableIII.Amagneticfieldof∼7×1011 of the hydro-cyclotron oscillation model. Assuming a
normal magnetic field of B ≃ 1.86 × 1013 G so that
G is compatible with the magnetic fields inferred for 1E
1207.4-5209 from timing solutions [18] (9.9×1010 G or ~ωc/12 = 18 keV, one sees that the oscillation model
2.4×1011 G), since 1E 1207.4-5209 shows no magneto- predicts hydro-cyclotronfrequencies whichcoincide with
spheric activity and the P˙-value would be overestimated the observed listed in Table IV.
ifoneappliesthespin-downpowerofmagnetic-dipolera-
diation [25, 26]. We note that the absorption feature at
IV. SUMMARY
ω(ℓ = 1) = 4.2 keV shown in Table III may not be de-
tectable since the stellar temperature is only ∼ 0.2 keV
In this paper, we study the global motion of the elec-
(see Table I), which will suppress any thermal feature in
tron seas on the surfaces of hypothetical strange quark
that energy range.
stars. It is found that such electron seas may undergo
Aside from 1E 1207.4-5209, one may ask what would
hydro-cyclotron oscillations whose frequencies are given
be the magnetic fields of other dead pulsars (e.g., radio-
by ω(ℓ) = ω /[ℓ(ℓ + 1)], where ℓ ≥ 1 and ω the cy-
quiet compact objects) if their spectral absorption fea- c c
clotron frequency. We propose that some of the absorp-
tures would also be of hydro-cyclotron origin? Intrigu-
tion features detected in the thermal X-ray spectra of
ingly, the hydro-cyclotronwavemodelpredicts magnetic
dead (e.g., radio silent) compact objects may have their
fields that are twice as large as those derived from the
originin excitations of these hydro-cyclotronoscillations
electron cyclotron model if the absorption feature is at
of the electron sea, provided these stellar objects are in-
ω(ℓ = 1) = ω /2; these fields could be ∼ 10 times
c
terpreted as strange quark stars. The central compact
greater (see Table II) if the absorption feature is at
object 1E 1207.4-5209 appears particularly interesting.
ω(ℓ = 2) = ω /6 or ω(ℓ = 3) = ω /12. The absorp-
c c
It shows an absorption feature at 0.7 keV which is not
tion lines at (0.3∼0.7) keV may indicate that the fields
of XDINSs are on the order of ∼1010 to 1011 G, if oscil- much stronger than the another absorption feature ob-
served at 1.4 keV. This can be readily explained in the
lation modes with ℓ≥4 are not significant.
frameworkofthehydro-cyclotronoscillationmodel,since
As noted in [15], unique absorption features on com-
pact stars are only detectable with Chandra and XMM- twolineswithℓandℓ+1couldessentiallyhavethesame
Newton if the stellar magnetic fields are relatively weak intensity. Thisisverydifferentforthe electron-cyclotron
(1010Gto1011G),sincethestellartemperaturesareonly model, for which the oscillator strength of the first har-
monic ismuchsmallerthanthe oscillatorstrengthofthe
a few 0.1 keV. The fields of many pulsar-like objects are
fundamental.
generally greater than this value, with the exception of
old millisecond pulsars whose fields are on the order of
108 G.Centralcompactobjects,ontheotherhand,seem
APPENDIX
to have sufficiently weak magnetic fields (see Table I) so
that absorption features originating from their surfaces
shouldbedetectablebyChandraandXMM-Newton. Ar- Here we derive the dispersion relation characterizing
the propagation of hydro-cyclotron electron wave in the
guments favoring the interpretation of compact central
slab-geometryapproximation. Thegoverningequationof
objects as strange quark stars have been put forward in
viscouselectronfluidunder the actionofLorentzforceis
[27],whereitwasshownthatthemagneticfieldobserved
given by
for some CCOs could be generated by small amounts of
differential rotation between the quark matter core and ∂δv ρ
ρ = e[δv×B]+η∇2δv,
the electron sea. ∂t c
Besidesdeadpulsars,anomalousX-raypulsars(AXPs)
which can be written as
and soft gamma-ray repeaters (SGRs) are enigmatic ob-
∂δv
jects which have become hot topics of modern astro- +ω [n ×δv]−ν∇2δv=0, (23)
c B
∂t
physics. Whether they are magnetars/quark stars is an
openquestion[28]. IncasethatAXPs/SGRSsshouldbe where
bare strange stars, the absorption lines detected from eB B η
ω = , n = , ν = , ρ=m n, ρ =en,
SGR 1806-20 could be understood in the framework c m c B B ρ e e
e
5
ω is the cyclotron frequency, and η stands for the effec- where (k·δv) = 0. It is convenient to rewrite the last
c
tive viscosity of an electron fluid originating from col- equation as
lisions between electrons. To make the problem ana-
ω 1+i(νk2/ω)
lytically tractable, we treat the electron sea as an in- [k×δv]= i c(n ·k)δv . (29)
compressible fluid and assume a uniform magnetic field. ω B (cid:20)1−(νk2/ω)2(cid:21)
Equation (23) can then be written as
Multiplication of both sides of Eq. (29) with k leads to
∂δv 1+i(νk2/ω)
=−ω [n ×δv]+ν∇2δv. (24) −ωδvk2 =iω (n ·k) [k×δv].(30)
∂t c B c B (cid:20)1−(νk2/ω)2(cid:21)
Inserting the left-hand-side of Eq. (29) into the right-
Upon applying to Eq. (24) the operator ∇×, we arrive
hand-side of Eq. (30) gives
at
(n ·k)2 1+i(νk2/ω) 2
∂ ω2 =ω2 B , (31)
[∇×δv]=ωc(nB ·∇)δv−ν∇×∇×∇×δv(,25) c k2 (cid:20)1−(νk2/ω)2(cid:21)
∂t
or
where ∇·δv=0, and
(n ·k) 1+i(νk2/ω)
B
ω =±ω ,
∂δω c k (cid:20)1−(νk2/ω)2(cid:21)
=ω (n ·∇)δv−ν∇×∇×δω, (26)
c B
∂t
which is Eq. (5).
where δω = ∇×δv. Considering a perturbation in the Acknowledgement. We would like to acknowl-
form of δv=v′exp[i(kr−ωt)], we have edge valuable discussions at the PKU pulsar group.
F.W. is supported by the National Science Foundation
(USA) under Grant PHY-0854699. This work is sup-
ω νk2
[k×δv]=i c(n ·k)δv+i [k×δv], (27) ported by the National Natural Science Foundation of
B
ω ω China(grants10935001and10973002),theNationalBa-
νk2 ω sic Research Program of China (grants 2009CB824800,
1−i [k×δv]= i c(n ·k)δv, (28) 2012CB821800),and the John Templeton Foundation.
B
(cid:18) ω (cid:19) ω
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