Vortex buoyancy in superfluid and superconducting neutron stars

0.21 MB·

by  Vasiliy A. Dommes

MNRAS000,000–000 (0000) Preprint25January2017 CompiledusingMNRASLATEXstylefilev3.0
Vortex buoyancy in superfluid and superconducting
neutron stars
V. A. Dommes, M. E. Gusakov
Ioffe Institute, Polytekhnicheskaya 26, 194021 St. Petersburg, Russia
7
1
0
2
ABSTRACT

n Buoyancyofprotonvorticesisconsideredasoneoftheimportantmechanismsofmag-
a neticfieldexpulsionfromthesuperconductinginteriorsofneutronstars.Hereweshow
J thatthegenerallyacceptedexpressionforthebuoyancyforceisnotcorrectandshould

4 be modified. The correct expression is derived for both neutron and proton vortices.
2 It is argued that this force is already contained in the coarse-grained hydrodynamics
of Bekarevich & Khalatnikov and its various multifluid extensions, but is absent in
]
the hydrodynamics of Hall. Some potentially interesting buoyancy-related effects are
E
briefly discussed.
H
. Key words: stars:neutron–stars:magneticfields–stars:interiors–hydrodynamics
h
p
-
o
r
t 1 INTRODUCTION drodynamics of Bekarevich & Khalatnikov (1961) (and its
s
multifluid extensions). But is it true?
a
According to theoretical calculations (e.g.,
[ We address these issues in the present note. The work
Lombardo & Schulze 2001; Page et al. 2013), protons
is organized as follows. Section 2 contains a brief discus-
1 in the neutron star cores condense into a superconducting
sion of the vortex structure, together with the Muslimov
v stateattemperaturesbelowT ∼108−1010 K.Iftheyform
& Tsygan derivation of thebuoyancyforce, and alternative
0
a type-II superconductor, the magnetic flux penetrates
derivation leading to a different result. The new expression
7
the core in the form of quantized fluxtubes (also called
8 forthebuoyancyforceisthencomparedtotheresultsofmi-
Abrikosov vortices or, simply, proton vortices), each carry-
6 croscopiccalculationsbasedonthesolutiontotheconserva-
ing a single flux quantum φ ≡ π~c/e , where ~, c, and e
0 0 p p tiveGinzburg–Landau (Gross–Pitaevsky) equation.Section
are the Planck constant, speed of light, and proton electric
. 3 clarifies whether the buoyancy effects are already present
1 charge, respectively.
in the hydrodynamics of Bekarevich & Khalatnikov (1961).
0
Since the magnetic field in the neutron star core is Section 4 considers a few applications of our results. Fi-
7
1 locked to proton vortices, the problem of its evolution re- nally,wesum upin section 5.Forsimplicity,weneglect the
: duces to that of vortex motion. This motion is, in turn, relativistic and entrainment effects in this note. They are
v
determined bythebalance of forces acting on vortices. One thoroughly discussed in Gusakov & Dommes (2016).
i
X of these forces is the so called buoyancy force, introduced
into the neutron-star literature by Muslimov & Tsygan
r
a (1985) and used subsequently in many works studying the
magnetic field expulsion from the superconducting interi-
2 BUOYANCY FORCE ON A VORTEX
ors of neutron stars (e.g. Harveyet al. 1986; Jones 1987;
Srinivasan et al. 1990; Chau et al. 1992; Ding et al. 1993; 2.1 Vortex structure
Chau 1997; Jahan-Miri 2000, 2002; Konenkov& Geppert
Consider first a proton vortex. It consists of a non-
2000,2001; Elfritz et al. 2016). Notethat thisforce should,
superconducting core with the radius of the order of the
in principle, act on both proton and neutron (Feynman– proton coherence length ξ ≡~p /(πm∗∆ ),
Onsager) vortices. p Fp p p
(1985T)hefrbomuoytahnecypuforreclyewhaysddroedriyvneadminicMcuosnlsimidoevra&tioTn.syOganne ξp ≈1.7×10−12cm 0.1n8pn 1/3 mmp∗ 0.76∆MeV , (1)
may ask whetherit is legitimate toapply suchan approach (cid:18) 0(cid:19) (cid:18) p(cid:19)(cid:18) p (cid:19)
onmicroscopiclength-scalescomparabletotheradiusofthe surrounded by superconducting currents with the velocity
vortex core. Another issue is related to the fact that the field (e.g., De Gennes1999; Landau & Lifshitz 1980)
buoyancy force is always added in the equation of vortex
e φ r
motion as some ‘external’ force. This implies that it should v(r)= p 0 K e . (2)
not be contained, for example, in the smooth-averaged hy- 2πmpcδp 1(cid:18)δp(cid:19) ϕ
(cid:13)c 0000TheAuthors
2 V. A. Dommes, M. E. Gusakov
Herep ,n ,m ,m∗,∆ aretheprotonFermimomentum, density) decreases because of local magnetic field and su-
Fp p p p p
number density, mass, effective mass, and energy gap, re- percurrents associated with the vortex. The pressure drop
spectively; n =0.16 fm−3 is the nuclear matter density; r ∆P(r) at a distance r from the vortex line is given by
0
isthedistancetothevortexcore;e istheunitvectorinthe
ϕ v2(r) B2(r)
Fazinimalulyth,δald≡ir[emct2iocn2/1(;4aπned2ρKi()x]1)/2isisthteheMLaocnDdoonnapldenfuetnrcattiioonn. ∆P(r)=ρsp 2 + 8π . (7)
p p p sp
depth for protons, Therefore, the buoyancy force per unit length fB, arising
dueto thedensity drop ∆ρ(r),equals
δ ≈4.24×10−12cm 0.18n0 1/2, (3) ∞ g
p n f =−g ∆ρ(r)rdrdϕ=−Eˆ , (8)
(cid:18) sp (cid:19) B Vpc2
Zξp s
where ρ and n = ρ /m are, respectively, the ‘super-
sp sp sp p whereg isthelocalgravitationalacceleration.Hereweused
conducting’ proton mass density and number density. At
r ≪ δ the velocity v(r) ≈ κ /(2πr) [κ ≡ π~/m is the therelation∆ρ=c−s2∆P,wherec2s ≡dP/dρisthesquared
p p p p
speedofsound.Usingthehydrostaticequilibriumcondition
circulation quantum], while at r ≫ δ it decays exponen-
p ρg=∇P, onecan rewrite thebuoyancyforce as
tially,v(r)∝exp(−r/δ ).Theprotonsupercurrentproduces
p
a magnetic field, g ∇ρ
f =−Eˆ =−Eˆ . (9)
B Vpc2 Vp ρ
φ r s
B(r)= 0 K ννν, (4)
2πδ2 0 δ Theproposedsimplederivationpredictsthatthereisanon-
p (cid:18) p(cid:19)
zero force acting on a vortex in an external gravitational
whereννν ≡ezistheunitvectorinthevortexdirection(along field.Onemayseethatthisforceisderivedfrom thepurely
z). The vortex energy per unit length, EˆVp, is given by the ‘hydrodynamic’ arguments. However, the validity of such
formula (Landau & Lifshitz 1980) a hydrodynamic approach is questionable at microscopic
v2(r) B2(r) κ2 δ length-scales (e.g., use of the notion of pressure in the very
EˆVp= ρsp 2 + 8π rdrdϕ≈ρsp4πp lnξp. (5) vicinityofthevortexcorelooksdoubtful).Belowwedemon-
Z (cid:20) (cid:21) p strate that equation (9) is correct only in one special case
To allow for the entrainment effect, ρ in equation (5) and,generally, it should be replaced by a different formula.
sp
shouldbereplaced with theelementρ oftheentrainment
pp
matrix ρ (Mendell 1991).
ik 2.3 Alternative derivation and different result
Similar formulas can also be written for vortices in an
unchargedsuperfluids,e.g.,forneutronvortex2.Itsvelocity The force acting on a vortex in inhomogeneous medium
field is v(r)=κn/(2πr)eϕ, and energy per unit length can also be derived from simple energetic arguments. As-
sumefirstthatthevortexisdirectedalongtheaxisz,while
κ2 b
Eˆ ≈ρ n ln n, (6) the background superfluid density ρ depends on x only
Vn sn4π ξ sp
n [here and below we use the standard Cartesian coordinates
whereκ ≡π~/m ;ξ istheneutroncoherencelength,given (x,y,z)].Now,ifweshiftthevortexfromxtox′ =x+δx,
n n n
by the formula similar to equation (1); and b is some ‘ex- its energy (5) will change according to
n
ternal’ radius of the order of the typical length-scale of the
problem (e.g., intervortex spacing or radius of a vessel). To Eˆ (x′)−Eˆ (x)≈ ρ (x′)−ρ (x) κ2p lnδp, (10)
Vp Vp sp sp 4π ξ
account for the entrainment, ρ in equation (6) should be p
sn
replaced withthecombination (ρnnρpp−ρ2np)/ρpp (Mendell sothatthebuoyancy(cid:2)force, given byt(cid:3)he(minus)vortexen-
1991;Glampedakis et al. 2011). ergygradient,willbef =−∇Eˆ (r).Inthemoregeneral
B Vp
Note that in inhomogeneous medium (when ρsn slowly case,when∇ρsp isnotnecessarilydirectedalongx,onehas
varies in space) the neutron vortex velocity differs from
Svh(ree)h=y&κnR/a(2dπzirh)oevϕsk(ysee2,00e4.g).,,bRuutbtinhsetevinor&texPiesnmeergny1E9ˆ94; fB =−∇⊥EˆVp(r)=−EˆVp∇ρ⊥sρpsp, (11)
Vn
still has the same form (6) up to the terms quadratic in where∇⊥ ≡∇−ν(ν∇) isthecomponent ofgradient per-
|∇ρsn|. The same, of course, applies to proton vortices. pendiculartothevortexlinedirectionν.Theappearanceof
∇⊥ instead of ∇ in this formula reflects the fact that the
translation of the vortex along the axis z does not change
2.2 Buoyancy force: standard derivation itsenergy.Ofcourse,similarformula(withindexpreplaced
by n) also applies to neutron (Feynman–Onsager) vortices
The concept of the buoyancy force on a vortex was intro-
(Sheehy& Radzihovsky 2004; see also section 2.4).
ducedbyMuslimov & Tsygan(1985)(seealsoHarvey et al.
Generally, the formula (11) differs from the ‘classical’
1986) who considered proton fluxtubes in superconducting
result (9). They are equivalent only for a one-component
interiors of neutron stars. These authors noted that in the
liquid in the hydrostatic equilibrium and at zero tempera-
vicinityofthevortexcorethefluidpressure(and,therefore,
ture(T =0), when ρ =ρ.
sp
1 We use the cylindrical coordinate system (r, ϕ, z) with the 2.4 Comparison with the microscopic theory
axisz directedalongthevortexline.
2 In this paper we assume, for simplicity, that neutrons pair in Itisinterestingtoconfrontourformula(11)withtheresults
thespin-singlet(1S0)state. of microscopic theory. For example, Rubinstein & Pismen
MNRAS000,000–000(0000)
Vortex buoyancy in superfluid and superconducting neutron stars 3
(1994) analysed, within the conservative Ginzburg–Landau Consideraneutralsuperfluid(e.g.,neutronliquid)con-
(Gross–Pitaevsky) equation, the motion of a neutral (e.g., taining vortices moving with some local velocity V . For
Ln
neutron) vortex in the system at T = 0 with a slightly in- simplicity, we shall be interested in theso called weak-drag
homogeneousbackgroundsuperfluiddensityρ (x,y).They limit,whenthemutualfrictionforcescanbeneglected(note
sn
considered a situation in which the vortex is straight and that, effectively, Rubinstein& Pismen 1994 also worked in
directed along the axis z. In such formulation the prob- thatlimit,seesection2.4).Inthisapproximationthehydro-
lem is two-dimensional – the vortex moves in the xy- dynamicsofBekarevich & Khalatnikov(1961)givesthefol-
plane.Usingthemethod ofmatched asymptoticexpansion, lowing expression for V (see equation16.45 in themono-
Ln
Rubinstein & Pismen (1994) found the following analytical graph by Khalatnikov 2000 with β =β′ =0 4),
expression for thevortex velocity V :
Ln 1 1 Eˆ
VLn=Vsn− 4κπn ν× ∇ρρsn lnRξn. (12) V(LBnK) =Vsn+ ρsncurlλnν =Vsn+ ρsncurl κVnnν!
sn n
Eˆ ∇ρ Eˆ
Here Vsn is the background superfluid velocity and Rn is a =Vsn− κ Vρn ν× ρ sn + κ Vρn curlν, (15)
typical length-scale of superfluid density variation, n sn sn n sn
where λ ≡ Eˆ /κ and in the second line we make use of
Rn =2a1e1/2−C(cid:30)s(cid:18)∇2ρρssnn(cid:19)2− 21∇2lnρsn (13) ethqeuaftaicotnnth(6a)t].EˆVIVnnntinsuranp,ptrhoeximhyadterolydypnroapmoirctsioonfalHtaollρ(sn19[6se0e)
predictsthat in the weak-drag limit V equals
where C ≈0.577 is the Euler constant and lna ≈0.405. Ln
1
One can try to derive an equation similar to (12) from Eˆ
V(H) =V + Vn curlν. (16)
theformula(11)(seealsoSheehy& Radzihovsky2004).The Ln sn κ ρ
n sn
onlyforcesactingonavortexintheconservative2Dproblem
Letuscompareequations(15),(16),andequation(14)from
considered by Rubinstein & Pismen (1994) are the Magnus
theprevioussection. First of all, thelast terms in ther.h.s.
force,f =ρ κ ν×(V −V ),andthebuoyancyforce,
M sn n Ln sn of equations (15) and (16) coincide and appear due to the
f .Neglectingvortexinertia,oneshouldhavef +f =0,
B B M localvortexcurvature(andrelatedtension),whichgenerates
which gives 3,
small contribution to V , making it a bit different from
Ln
V =V − EˆVn ν× ∇ρsn =V − κn ν× ∇ρsn lnbn. the average ‘transport’ velocity Vsn (see Donnelly 2005 for
Ln sn ρ κ ρ sn 4π ρ ξ details; note, however, that the corresponding discussion in
sn n sn sn n
(14) thatreferencecontainsanumberofmisprints).Becausethis
term vanishes for straight vortices, it does not appear in
This equation has exactly the same structure as (12) pro-
equation (14).
vided that we identify b =R , which is natural, since this
n n Next, thesecond term in the r.h.s. of equation (15) in-
is theonly ‘external’ length-scale in the problem. It follows
dicates that thedifferencebetween thevortex velocity V
Ln
from (14) that in the absence of background superfluid ve-
andthetransport velocity V isdrivennot only byvortex
locity(V =0)vortexmovesalongthesurfaceρ =const, sn
sn sn curvature, but also by inhomogeneity of the background,
i.e. its energy is conserved. ∇ρ . Itexactly coincides with thesimilar ‘buoyancy’term
sn
in equation (14),but is absent in theHall’s hydrodynamics
[see equation (16)].
3 IS THERE A BUOYANCY FORCE IN
Wecometoconclusionthatthesuperfluidhydrodynam-
SUPERFLUID HYDRODYNAMICS?
ics of Bekarevich & Khalatnikov (1961) implicitly contains
all buoyancy effects, whereas one has to include the buoy-
In thecontext of neutron stars thevortex buoyancy(in the
ancy force ‘by hands’ if one prefers to use the superfluid
form8)wasroutinelyintroducedassomeexternalforce(but
hydrodynamicsof Hall (1960).
see Jones 2006) in order to explain expulsion of the mag-
It remains to note that the buoyancy effects are
netic field (confined to Abrikosov vortices) from the super-
automatically included in both non-relativistic and rel-
conductinginteriors of neutronstars. Herewe shall demon-
ativistic extensions of Bekarevich and Khalatnikov hy-
stratethatthisforceisabsentintheoftenusedversion(Hall
drodynamics, describing superfluid/superconducting mix-
1960) of superfluid hydrodynamics of Hall & Vinen (1956)
tures (see, e.g., Mendell & Lindblom 1991; Mendell 1991;
(see also a monograph by Donnelly 2005) but is contained
Sedrakian & Sedrakian 1995; Glampedakis et al. 2011;
in the hydrodynamics of Bekarevich & Khalatnikov (1961)
Gusakov 2016; Gusakov & Dommes 2016).
(anditsvariousextensions).Thus,introductionofthisforce
‘byhands’mayleadtodouble-counting.Notethat,thefact
that the buoyancy is already incorporated in the hydrody-
namic equations of Bekarevich & Khalatnikov (1961), has 4 SOME APPLICATIONS
not beenwell recognized intheliterature (see, e.g., p.94 of
4.1 Vortices in a rotating neutral superfluid
the authoritative review by Sonin 1987, where this force is
described as ‘external’). Let us discuss, within the hydrodynamics of
Bekarevich & Khalatnikov (1961), the equilibrium con-
figuration of a rotating neutral superfluid with non-zero
3 NotethatVLnisdefineduptoanarbitrarytermparalleltothe
vortex line. Such term does not affect the vortex dynamics (see,
e.g., Khalatnikov 2000) and is chosen here such that VLnν = 4 Note a misprint in this equation: the second term in its r.h.s.
Vsnν. shouldhave‘+’signinfrontof1/ρscurlλν.
MNRAS000,000–000 (0000)
4 V. A. Dommes, M. E. Gusakov
∇ρ .Forsimplicity,werestrict ourselvestothecylindrical where W can be presented as a sum of two terms related,
sn p
geometry (e.g., cylindrical vessel rotating at a frequency respectively,to buoyancy and vortex tension,
Ω around its symmetry axis), and consider the case of a
Eˆ ∇ρ Eˆ
one-component liquid at T = 0 (hence ρsn equals the total WWWp =curl(λpν)=− κVp ν× ρ sp + κVp curlν. (22)
mass density ρ ). Generalization of these results to finite p sp p
n
temperatures is straightforward and does not change our Inequations(21)and(22)λ =Eˆ /κ ;ν =B/B;α and
p Vp p p
main conclusions.
β arethe(poorlyknown)mutualfrictioncoefficients.These
p
The superfluid is governed by the ‘superfluid’ and
equationshavebeenrecentlyderivedinGusakov & Dommes
continuity equations (see equations 16.24 and 16.40 in
(2016)underassumptionthatallthenormal(non-superfluid
Bekarevich & Khalatnikov1961),whichcanberewrittenin
andnon-superconducting)liquidcomponentsmovewithone
theframe, rotating with thevessel, as and the same ‘normal’ velocity V 6. This assumption
norm
∂V˜ canberelaxed(Gusakov&Dommes,inpreparation),which
∂tsn +2[Ω×V˜sn]+(V˜sn∇)V˜sn+∇(µ˘+φ) modifies VLp byadding a diffusion-induced term to Wp.
It is instructive to estimate typical time-scales appear-
1 1
=−ρ [curlV˜sn×curl(λnν)]− ρ [2Ω×curl(λnν)] (17) ing in the evolution equation (20). Following Graber et al.
sn sn (2015), one gets
∂ρ
∂tsn +div(ρsnV˜sn)=0, (18) κ L2 4πL2 δ −1
τ = p = ln p , (23)
whereV˜sn =Vsn−[Ω×r]ismeasuredintherotatingframe; cons |αp|EˆVp |αp|ρspκp (cid:20) (cid:18)ξp(cid:19)(cid:21)
µ˘ is thechemical potential per unit mass; and φ is the sum κ L2 4πL2 δ −1
τ = p = ln p . (24)
ofcentrifugalpotentialpluspotentialofsomeexternalforce diss βpEˆVp βpρspκp (cid:20) (cid:18)ξp(cid:19)(cid:21)
producing the density gradient (in neutron stars it is the
HereListhecharacteristiclength-scaleoftheproblem,char-
gravitationalpotential).Intheabsenceofdensitygradients,
whentheright-handsideofequation(17)vanishes5,onehas acterizing variation of the magnetic induction B, or super-
V˜ =0inequilibrium,thatissuperfluidmimicssolid-body fluid density ρsp, or vortex direction ν. Note that, near the
sn normal-superconducting boundary, the ratio ∇ρ /ρ [see
rotation. However, when the density gradients are allowed sp sp
the‘buoyancy’terminequation(22)]canbeverylarge,since
for, thesolution to equations(17) and (18) describes differ-
thereρ →0, while∇ρ isfinite.Also, ∇ρ canbelarge
ential rotation, sp sp sp
(i.e.,Lissmall)nearthecrust-coreinterfaceand,generally,
V˜ = λn ν× ∇ρsn. (19) in thevicinity of any place in thecore where thesuperfluid
sn ρsn ρsn proton density varies sharply. 7
The time-scales (23) and (24) are called, respectively,
Atthesametime,using(15)oneobtainsthatvorticesmove
‘conservative’and‘dissipative’becauseitcanbeshown(e.g.,
with the vessel (and with the normal component if T 6=0):
Mendell & Lindblom 1991; Gusakov & Dommes 2016) that
V = [Ω × r]. For neutron stars the typical precession
Ln the terms ∝ β create entropy in the system, while those
frequency of vortices with respect to the superfluid com- p
∝ α do not. The mutual friction coefficients α and β
ponent, ω ≡ |V˜ |/r = κ /(4πr) ln(b /ξ )|∇(lnρ )|, is p p p
extremelyPsmall,ωsn∼10−15ns−1,butitinsmnuchlargesrn(and are rather uncertain, especially at temperatures compara-
P ble to the nucleon critical temperatures, and there are no
is observed) in Bose–Einstein condensates (Anderson et al.
agreement in the literature on their actual values (e.g.,
2000). It is interesting that a similar formula for ω has
P Alparet al. 1984; Jones 2006; Andersson et al. 2006; Jones
beenobtainedinSheehy & Radzihovsky(2004)bydifferent
2009).AtT =0thesecoefficients canbeexpressed through
means. thedrag coefficient R ∼2×10−4 (Graber et al. 2015) as
p
1 1 1 R
α =− , β = p . (25)
p ρ 1+R2 p ρ 1+R2
4.2 Magnetic field evolution in superconducting sp p sp p
neutron-star cores Adoptingthese values, oneobtains, to leading order in R ,
p
Ifprotonsinaneutronstarcoreformatype-IIsuperconduc- R −1
τ ≈2×108L2 yr, τ ≈2×1012L2 p yr,
tor,thenalmostallmagneticfieldisconfinedtoprotonvor- cons 6 diss 6 10−4
(cid:18) (cid:19)
tices(e.g.,Glampedakis et al.2011).Therefore,evolutionof (26)
the magnetic induction B in the superconducting region is
governedbytheequation(e.g.,Konenkov & Geppert2001), where L6 =L/(106 cm). Clearly, these estimates should be
taken with caution. Still, they indicate that the time-scale
∂B
∂t =curl[VLp×B], (20) fiτceolndsecvaonlubteiocnominpaprualbsalerst.othetypicaltime-scaleofmagnetic
which describes transport of B with the velocity VLp of Note that τcons is smaller than the corresponding esti-
proton vortices. The latter is given by theexpression mateinGraber et al.(2015)by7ordersofmagnitude,which
is a consequence of a bit different evolution equation used
VVV =VVV −α WWW −β ννν×WWW , (21)
Lp norm p p p p
6 Graberetal.(2015)proposedamagneticfieldevolutionequa-
tionwhichisdifferentfrom(21);wediscussitbelow.
5 Notethatcurlν=0becauseν iscollinearwithΩ. 7 WethankA.I.Chugunovforpointingouttousthispossibility.
MNRAS000,000–000(0000)
Vortex buoyancy in superfluid and superconducting neutron stars 5
by these authors. The evolution equation of Graber et al. reviewbySonin1987,wherethebuoyancyforceisexplicitly
(2015) (theirequation 67) reads 8 considered as ‘external’).
(iv) However,thisforcedoesnotappearinsuperfluidhy-
∂B =curl V − EˆVp curlν ×B , (27) drodynamicsof Hall(1960) (seealso Donnelly2005),which
∂t " Lp ρspκp ! # is equivalent to that of Bekarevich & Khalatnikov in (al-
most) all other aspects.
where theexpression for thevortex velocity,
(v) Webrieflydiscussedanumberofapplicationsinwhich
Eˆ thebuoyancyforcemaylead tonoticeableeffects.Thesein-
V =V + Vp (curlν−R ν×curlν),
Lp norm ρ κ (1+R2) p cludehydrostaticequilibriumofneutralrotatingsuperfluids,
sp p p
as well as the magnetic field evolution in superconducting
(28)
neutronstars.Ourestimatesindicatethatthemagneticfield
coincideswithequation(21)ifαp andβp aregivenbyequa- canevolveonamuchshortertimescalethanitwasassumed
tion(25)and,additionally,∇ρsp =0(whichmeansthatthe by Graber et al. (2015) (and,subsequently,by Elfritz et al.
authors ignore thebuoyancyforce). Equation (27) is rather 2016), who used a different (and incorrect, as we argue in
puzzling:itstatesthatthemagneticfieldistransportedwith Section 4) magnetic field evolution equation.
the velocity that differs from the vortex velocity V , in
Lp
contradiction to the (explicitly made) assumption that all
themagneticfieldisstoredinvortices.Notethatthisequa-
tion has been used subsequently by Elfritz et al. (2016) for 6 ACKNOWLEDGEMENTS
detailed modelling of the magnetic field evolution in neu-
We are very grateful to Elena Kantor, Andrey Chugunov,
tronstars(seeequation16inthatreference).Theseauthors
Kostas Glampedakis, and D.A. Shalybkov for valuable dis-
also accounted for the buoyancy force, but in the incorrect
cussions and comments. This study was supported by the
form (9).
Russian Science Foundation (grant number14-12-00316).
Now the reason for the seven-order difference in the
estimates for τ can beeasily pinpointed.To thisaim we
cons
expandthesecondterminequation(28)intheTaylorseries
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