Self-Trapping of G-Mode Oscillations in Relativistic Thin Disks, Revisited

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by  Shoji Kato

MNRAS000,1–??(2016) Preprint9January2017 CompiledusingMNRASLATEXstylefilev3.0
Self-Trapping of G-Mode Oscillations in Relativistic Thin
Disks, Revisited
Shoji kato 1⋆
12-2-2 Shikanodai Nishi, Ikomashi, Nara, Japan, 630-0114
7
AcceptedXXX.ReceivedYYY;inoriginalformZZZ
1
0
2
ABSTRACT

n We examine by a perturbation method how the self-trapping of g-mode oscillations
a in geometrically thin relativistic disks is affected by uniform vertical magnetic fields.
J
Disks which we consider are isothermal in the vertical direction, but are truncated

6 at a certain height by presence of hot coronae. We find that the characteristics of

self-trapping of axisymmetric g-mode oscillations in non-magnetizeddisks is kept un-
]
E changedinmagnetizeddisksatleasttillastrengthofthe fields,dependingonvertical
thicknessofdisks.Thesemagneticfieldsbecomestrongerasthediskbecomesthinner.
H
This result suggests that trapped g-mode oscillations still remain as one of possible
.
h candidatesofquasi-periodicoscillationsobservedinblack-holeandneutron-starX-ray
p binaries in the cases where vertical magnetic fields in disks are weak.
-
o Key words: accretion, accretion discs – finite disk-thickness – general relativity –
r g-mode oscillations – self-trapping – waves
t
s
a
[

1
1 INTRODUCTION propagationregionoftheoscillationsisboundedbythetwo
v
radii where ω2 = κ2 is realized. At these two radii the
6 Inblack-holeandneutron-starX-raybinariesquasi-periodic max
5 oscillations are reflected. This is the self-trapping of the g-
oscillations (QPOs)areoccasionally observedwithfrequen-
5 mode oscillations in relativistic disks. Eigen-frequencies of
cies close to the relativistic Keplerian frequencies of the in-
1 thesetrappedg-modeoscillationsingeometricallythindisks
nermostregionofdiskssurroundingthecentralsources(e.g.,
0 arecalculatedindetailbyPerezetal.(1997),usingtheKerr
van der Klis 2000; Remillard & McClintock 2006). One of
. geometry.Nowaketal.(1997) suggestedthatthesetrapped
1 possible origins of these quasi-periodic oscillations is disk
oscillations might betheorigin ofthe67Hzoscillations ob-
0 oscillations in the innermost region of the disks (discoseis-
7 served in theblack-hole source GRS1915+105.
mology).
1 In the above studies, however, effects of magnetic
The innermost regions of relativistic disks are subject
: fields on oscillations were outside considerations. Fu and
v tostrong gravitational field of centralsources, and thusbe-
i haviors of disk oscillations in these relativistic regions are Lai (2009) examined these effects and found important re-
X sults. That is, they showed that the g-mode oscillations
differentfrom thosein Newtoniandisks(see,e.g.,Katoand
r are strongly affected if poloidal magnetic fields are present
Fukue 1980). One of the differences is related to the dif-
a
in disks and the self-trapping of g-mode oscillations is de-
ference of radial distribution of (radial) epicyclic frequency
stroyed even when the fields are weak. For example, they
κ(r). In relativistic disks the epicyclic frequency does not
showed that c /c ≥0.03 (i.e., c2/c2 ≥ 10−3) is enough to
monotonically increase inwards, buthas a maximum,κ , A s A s
max
destroytheself-trappinginthecaseofaxisymmetricg-mode
at a certain radius, say r , and inside the radius it de-
max
oscillations, wherec andc areAlfv´enandacousticspeeds
creases inward to vanish at the radius of ISCO (innermost A s
in disks, respectively.
stable circular orbit). This radius is roughly the inner edge
of thedisks. Their results are very interesting in considering what
The g-mode oscillations1 are oscillation modes whose kinds of disk oscillations are possible candidates of quasi-
frequencies, say ω, are lower than κ , i.e., ω2 < κ2 . periodic oscillations, but theiranalyses are rough and more
max max
Okazaki et al. (1987) showed that in relativistic disks the carefulexaminationsarenecessarytoevaluateaquantitative
conditionofdestructionoftheself-trappingofg-modeoscil-
lations. For example, they do not take into account quanti-
⋆ E-mail:[email protected] tatively theeffects of vertical structureof disks. In geomet-
1 See,forexample,Kato(2016)forclassificationofdiskoscilla- rically thin disks, thedisk density decreases apart from the
tionsingeometricallythindisks.Terminologyinclassificationby equatorialplane,whiletheAlfv´enspeedincreasesgreatlyin
Kato(2016) isslightlydifferentfromthatbyKatoetal(2008). the vertical direction, if the strength of poloidal magnetic
(cid:13)c 2016TheAuthors
2 Shoji Kato
fields is nearly constant in disks. These sharp decrease of et al. 2008)
density and sharp increase of Alfv´en speed in the vertical
z2
directionshouldbecarefullytakenintoaccountinexamina- ρ (r,z)=ρ(r) exp − , (2)
0 00 (cid:18) 2H2(cid:19)
tion of wave motions.
To avoid mathematical complication and difficulty re- where the scale height, H(r), is related to the isothermal
lated to the above situations, it is worthwhile to consider acousticspeed,cs(r),andverticalepicyclicfrequency,Ω⊥(r),
diskswithfiniteverticalthickness.Thisis,however,notonly by
for avoidance of mathematical complication. Consideration
ofsuchdisksisimportantfromtheobservationalandphysi- H2(r)= c2s . (3)
cal viewpoints. Many black-holeand neutron-starX-raybi- Ω2⊥
naries show both soft and hard components of spectra, and
In relativistic disks under the Schwarzschild metric, Ω⊥ is
recordasoft-hardtransition.Usuallythreestatesareknown:
equalto therelativistic Keplerfrequency.
high/soft state, very high/intermediate state (steep power-
law state), and the low/hard state. Remillard (2005) shows
that detection of high frequency QPOs in black-hole X-ray 2.1 Termination of disk thickness by hot corona
binaries is correlated with power-law luminosity, and their
Theabove-mentionedverticallyisothermaldiskscanextend
frequencies are related to soft components. This suggests
that at the state where high frequency QPOs are observed, infinitelyintheverticaldirection,althoughtheyarederived
undertheassumption thatthedisksare thinin thevertical
thesourceshavebothgeometrically thindisksandcoronae.
direction.Here,weassume thatthedisksare terminatedat
In this context, by using the above-mentioned disk ge-
afiniteheight,sayz ,bypresenceofhotcorona.Theheight
ometry (geometrically thin cool disk and a hot corona sur- s
z istakentobeaparameter.Asadimensionlessparameter
rounding the disk), we examine in this paper the effects of s
we adopt η ≡z /H hereafter.
verticalmagneticfieldsong-modeoscillationsbyaperturba- s s
tionmethod.Theresultsshowthattheperturbationmethod
is applicable till a critical strength of magnetic fields which
2.2 Equations describing perturbations
dependson disk thickness. In thecase where disk thickness
is three times the vertical half-thickness, H, of disks, for Weconsidersmall-amplitudeadiabaticperturbationsinthe
example, the critical strength of magnetic fields where the above-mentioned vertically isothermal disks. The perturba-
perturbationmethodisapplicableisgivenbyc2A0/c2s ∼0.14, tionsareassumedtobeisothermalintheverticaldirection.
where cA0 and cs are Alfv´en speed on the disk equator and Since the magnetic fields, B0 are perturbed to B =
acousticspeed,respectively.Ourresultsshowthattillthese (b ,b ,B +b ), we have
r ϕ 0 z
critical magnetic fields where the perturbation method is
∂b ∂b ∂b ∂b
applicable, axisymmetric g-mode oscillations are still self- curlb×B = B r − z ,−B z − ϕ ,0 . (4)
trapped. This result is different from that obtained by use 0 (cid:20) 0(cid:18)∂z ∂r (cid:19) 0(cid:18)r∂ϕ ∂z (cid:19) (cid:21)
ofaroughlocal approximationforverticallyextendeddisks Hence,thetimevariationsofEulerianvelocityperturbations
byFu and Lai (2009). over the rotation, (u , u , u ), are expressed by use of the
r ϕ z
r-,ϕ-andz-componentsofequationofmotion,respectively,
as
∂ ∂ ∂h c2 ∂b ∂b
+Ω u −2Ωu =− 1 + A r − z , (5)
2 UNPERTURBED DISK MODEL AND BASIC (cid:18)∂t ∂ϕ(cid:19) r ϕ ∂r B0(cid:18)∂z ∂r (cid:19)
EQUATIONS DESCRIBING
PERTURBATIONS ∂ ∂ κ2 ∂h c2 ∂b ∂b
+Ω u + u =− 1 − A z − ϕ , (6)
Weareinteresteding-modeoscillationsinrelativisticdisks. (cid:18)∂t ∂ϕ(cid:19) ϕ 2Ω r r∂ϕ B0(cid:18)r∂ϕ ∂z (cid:19)
The general relativistic treatments of dynamical equations
are, however, very complicated. Hence, following a conven- ∂ ∂ ∂h
+Ω u =− 1, (7)
tional way, we treat Newtonian equations, but adopt the (cid:18)∂t ∂ϕ(cid:19) z ∂z
generalrelativisticexpressionfor(radial)epicyclicfrequency
κ(r). where κ(r) is the epicyclic frequency, Ω(r) is the angular
velocity of disk rotation, h is
We consider vertically isothermal, geometrically thin 1
disks. Thedisks are assumed to besubject tovertical mag- h = p1 =c2ρ1, (8)
netic fields. We adopt the cylindrical coordinates (r, ϕ, z) 1 ρ sρ
0 0
whose origin is at the center of a central object and the z-
and c is the Alfv´en speed definedby
A
axis is perpendicular to the disk plane. Then, the magnetic
fieldsin theunperturbedstate, B ,are B2 z2
0 c2(r,z)= 0 =c2 (r)exp , (9)
A 4πρ A0 (cid:18)2H2(cid:19)
0
B (r)=[0,0,B (r)]. (1)
0 0
c beingtheAlfv´en speed on theequator.
A0
Since the magnetic fields are purely vertical, they have no Thetimevariationofmagneticfieldsisgovernedbythe
effectsontheverticalstructureofdisks,andthehydrostatic inductionequation, which is
balancein thevertical direction in disksgives thedisk den-
∂b ∂
sity,ρ0,stratified in thevertical direction as (e.g., seeKato ∂tr = ∂z(urB0), (10)
MNRAS000,1–??(2016)
Self-trapping of g-mode oscillations 3
This is the well-known wave equation in disks in a simpli-
∂b ∂ fiedsituation(e.g.,Kato2001, Kato2016).Weexaminethe
∂tϕ = ∂z(uϕB0), (11) effects of magnetic fields on the wave motions described by
equation(18)byaperturbationmethodinthenextsection.
∂b ∂ ∂
z =− (ru B )− (u B ). (12)
∂t r∂r r 0 r∂ϕ ϕ 0 2.3 Boundary condition
Finally,thetimevariationofdensityisgovernedbytheequa- As mentioned before we are interested in disks which are
tion of continuity,which is terminatedatcertainheightbypresenceofhotcorona.The
disk thickness is assumed to be terminated at z = z (i.e.,
∂ ∂ ∂ ∂ ∂ s
+Ω ρ + (rρ u )+ (ρ u )+ (ρ u )=0. at η = η ≡ z /H). The disk thickness, η , is a parameter,
(cid:18)∂t ∂ϕ(cid:19) 1 r∂r 0 r r∂ϕ 0 ϕ ∂z 0 z s s s
and we impose a boundarycondition at η .
s
(13) It will be relevant to assume that at the deformed
boundarysurfacebetweendiskandcoronaρu2+p+B2/8π
Hereafter,weconcentrateourattentiononlyonaxisym- n t
metricoscillations,i.e.,∂/∂ϕ=0.Thisisbecausewearein- is continuous2, where un and Bt are the normal compo-
nentoffluidvelocityandthetangentialcomponentofmag-
terested in g-mode oscillations in this paper,and asymmet-
ric g-mode oscillations are damped bycorotation resonance netic fields. By the definition of the boundary surface we
(Kato2003,Lietal.2003).Inaddition,theradialvariations have un = 0 at the surface. Furthermore, since the unper-
of unperturbedquantities,such asΩ(r), ρ (r),H(r),c2(r) turbed magnetic fields have no horizontal components, the
and c2(r), are neglected, assuming that0t0he radial wasve- Lagrangian variation of Bt2 vanishes there, i.e., δBt2 = 0,
A
lengths of perturbations are shorter than the characteristic whereδ representsLagrangianvariation.Hence,thebound-
ary condition we should adopt is continuation of δp at the
radial lengths of unperturbed quantities (local approxima-
tions in the radial direction). The radial variation of κ(r) surface. Furthermore, we assume that in the corona, pres-
is, however, carefully taken into account, because its radial sureperturbationsresultingfrom wavemotionsindisksare
quickly smoothed out by high temperature (large acoustic
variation is sharp near to the disk edge. Furthermore, ∂/∂t
is written as iω, since we are interested in normal mode speed), i.e., we assume δp=0 in the corona. The timescale
oscillations whose frequency is ω. of the smoothing in corona is shorter than the timescale
of oscillations, because the corona temperature will be on
Then, combiningequations (5) and (6) we have
theorderofvirialtemperature.Consequently,theboundary
∂h c2 ∂b ∂b ∂b condition we adopt is
−(ω2−κ2)u +iω 1 = A iω r − z +2Ω ϕ .
r ∂r B (cid:20) (cid:18)∂z ∂r (cid:19) ∂z (cid:21)
0 δp=p +ξ·∇p =0 at η=η , (19)
1 0 s
(14)
where ξ is displacement vector associated with pertur-
The righthand side of this equation shows how the relation bations. In the case of axially symmetric perturbations
between h1 and ur is affected by magnetic fields. There is (∂/∂ϕ=0), equation (19) is written as
another relation between u and h . This is obtained from
r 1
1 ∂p
equations(7) and (13) byeliminating uz, which is h1+ξrρ ∂r0 −Ω2⊥zξz =0. (20)
0
∂2 z ∂ ω2 ∂u
− + h −iω r =0. (15) Combination of equations (20) and (13) leads the
(cid:18)∂z2 H2∂z c2(cid:19) 1 ∂r
s boundarycondition to
If u is eliminated from the lefthand side of equation
r ∂u ∂u
(15) byusing equation (14),we have ∂rr + ∂zz =0 at η=ηs. (21)
∂2 −η ∂ + ω2 h +H2 ∂ ω2 ∂h1 This boundary condition is further reduced by using equa-
(cid:18)∂η2 ∂η Ω2⊥(cid:19) 1 ∂r(cid:18)ω2−κ2 ∂r (cid:19) tions (13) and (7) in a form expressed in termsof h1 alone,
∂ c2/B ∂b ∂b ∂b
=−iωH2 A 0 iω r − ϕ +2Ω ϕ , (16)
∂r(cid:20)ω2−κ2(cid:26) (cid:18)H∂η ∂r (cid:19) H∂η(cid:27)(cid:21) 2 Theequationofmotioncanbewritteninsuchaconservative
wheretheverticalcoordinatez hasbeenchangedtodimen- formas
dsiuocntlieosns ceoqouradtiinonatsewηedceafinnwedritbeytηhe=rizg/hHth.aBndy usisdinegoftheequina-- ∂∂ρuti + ∂∂rj(cid:20)ρuiuj+δij+ B8π2δij− 41πBiBj(cid:21)=0.
tion (16) in terms of u. Then, equation (16) is expressed Thenormalcomponentofthisequationshowsthat
as 1
ρu2 +p+ B2
∂2 ∂ ω2 ∂ ω2 ∂h n 8π t
−η + h +H2 1
(cid:18)∂η2 ∂η Ω2 (cid:19) 1 ∂r(cid:18)ω2−κ2 ∂r (cid:19) must be continuous at the boundary, where the subscript n and
⊥
t denote, respectively, the normal and tangential components at
=−iω ∂ c2A H2∂2ur + ∂2ur + 2Ω∂2uϕ . (17) the boundary, as is known in the field of hydromagnetic shocks.
∂r(cid:20)ω2−κ2(cid:18) ∂r2 ∂η2 iω ∂η2 (cid:19)(cid:21) In deriving this equation continuity of Bt has been used. The
tangentialcomponentoftheaboveequationofmotiongivesthat
In the case of no magnetic fields, the righthand side of
1
equation (17) vanishes,and we have ρunut− BnBt
4π
∂2 −η ∂ + ω2 h +H2 ∂ ω2 ∂h1 =0. (18) isalsocontinuousattheboundary.Thislatterconditionis,how-
(cid:18)∂η2 ∂η Ω2 (cid:19) 1 ∂r(cid:18)ω2−κ2 ∂r (cid:19) ever,notusedhere.
⊥
MNRAS000,1–??(2016)
4 Shoji Kato
which is
∂ ω2
−η + h =0 at η=η . (22)
(cid:18) ∂η Ω2 (cid:19) 1 s
⊥
Our subject is thus to solve equation (17) with boundary
condition (22).
3 PERTURBATION METHOD
Here,wesolveequation(17)byaperturbationmethod.That
is,westartfromthelimitofnomagneticfields,andexamine
theeffects of magnetic fields bya perturbation method.
3.1 Zeroth-order solution
Inthelimitofnomagneticfields,equation(17)isreducedto Figure 1.Eigenvalues ofthefundamental(n=1)g-modeoscil-
equation(18)andthislatterequationcanbesolvedeasilyby lations as functions of ηs. Eigenvalues depend on frequencies of
avariableseparationmethod.Thatis,byseparatingh1(r,η) oscillations, ω2, normalized by Ω2⊥ as shown by boundary con-
as h1(r,η)=g(η)f(r)and dividingequation (18) by h1, we dition (22). In the case of ηs = ∞, eigenvalue tends to unity,
can separate equation (18) into r- and η- dependent parts. correspondingtothecaseofinfinitelyextendedisothermaldisks.
Then,byintroducingaseparationconstant,K,wehavetwo
equations:
nolongerzeronorpositiveintegers, andequations(25) and
d2 d (26)areinfiniteseries(notterminated).Asboundarycondi-
−η +K g(η)=0, (23)
(cid:18)dη2 dη (cid:19) tion(22)shows,Kn andgn(η)dependonηs andω2/Ω2⊥.By
usingexpressionsforg givenbyequations(25)and(26),we
and
have numerically calculated eigenvalue K and eigenfunc-
n
d ω2 d ω2−KΩ2⊥ tiong (η).Theresultsinthecaseofn=1areshowninfig-
f(r)+ f(r)=0. (24) n
dr(cid:18)ω2−κ2dr(cid:19) c2s ures1and2.Figure1isforK,andfigure2isforfunctional
Theseparationconstant,K,isdeterminedbytheboundary formsofg(η).Figure1showsthatinthelimitofηs =∞,K
condition (22). tendstoK =1 asexpected.Asthisfigureshows, deviation
Equation(23)hastwoindependentsolutions,whichare from K =1 is large when the disk is thin (ηs is small) and
plane-symmetric(evenfunctionofη)andplane-asymmetric ω2/Ω2⊥ is small. The value of ω2/Ω2⊥ is a parameter at the
(odd function of η).Their formal solutions are present stage. It is determined after wave equation (24) in
the radial direction is solved with the boundary condition.
g(η)=1− 1Kη2−1K(2−K)η4−1K(2−K)(4−K)η6−... The eigenfunction, g(η), in figure 2 shows that eigenfunc-
2! 4! 6! tionsareclosetog(η)=η,andthedeviationfrom g(η)=η
(25) occurs only near totheboundary at η=η .
s
and
1 1
g(η)=η+ (1−K)η3+ (1−K)(3−K)η5 3.2 Quasi-orthogonality of zeroth-order
3! 5! eigenfunctions
1
+ (1−K)(3−K)(5−K)η7+... (26)
7! Inthecaseof ηs =∞,theseriesof theeigenvalues,Kn,are
K = 0, 1, 2,..., and the corresponding eigenfunctions, say
In the case where the disk extends infinitely in the vertical n
g , are orthogonal in thesense that
direction,i.e.,η =∞,equations(25)and(26)showthatthe n
s
seriesintheseequationsmustbeterminatedatfiniteterms. ∞ η2
exp − g g dη=n!(2π)1/2δ , (27)
Otherwise, the boundary condition that energy density of Z−∞ (cid:18) 2 (cid:19) n m nm
perturbationsdo not diverge at infinitycannot be satisfied.
This requires that the separation constant K [which is also becausegn is theHermite polynomial of order n.
eigenvalue of equation (23)] needs to bezero or positive in- The presence of orthogonality among the zeroth order
tegers, i.e., the set of K’s are K , where n= 0, 1, 2, .. and solutionsishelpfulinapplyingperturbationmethods.Inthe
n
Kn = 0, 1, 2,... The eigen-function corresponding to Kn, present case of ηs 6= ∞, however, there is no such orthogo-
say g (η), is the Hermite polynomial, i.e., g (η) = H (η) nality.Inspiteofthis,wehavequasi-orthogonalityasshown
n n n
(Okazakiet al. 1987). below, unless ηs is too small.
We are interested in this paper g-mode oscillations If the eigenfunction corresponding to Kn is written as
whicharefundamentalintheirbehaviorintheverticaldirec- gn, thezeroth-orderwave equation is expressed in theform
tion,i.e.,n=1.Hence,K andgwhichwillappearhereafter d η2 dg η2
without subscript are K1 and g1, respectively. dη(cid:20)exp(cid:18)− 2 (cid:19) dηn(cid:21)+exp(cid:18)− 2 (cid:19)Kngn =0. (28)
Distinct from the case of disks which extend infinitely
in the vertical direction, we impose in this paper a bound- Multiplying both sides of equation (28) by g (η) (m 6= n)
m
ary condition (22) at a finite height η . In this case K is and integrating the resulting equation over −η to η , we
s n s s
MNRAS000,1–??(2016)
Self-trapping of g-mode oscillations 5
4 WAVE EQUATION WHEN C2/C2 IS TAKEN
A S
INTO ACCOUNT
In the case of c2/c2 6=0, thesolution of equation (17) does
A s
nothavesuchaseparableformash (r,η)=g(η)f(r).Ifthe
1
effects of c2/c2 6= 0 on oscillations are weak, however, the
A s
righthand side of equation (17) can be treated as a small
perturbation in solving equation (17). That is, h (r,η) can
1
be approximately separated as h (r,η) = g(η,r)f(r) with
1
weak r-dependence of g. This weak r-dependence of g can
beexaminedbyaperturbationmethod.Itisnotedherethat
in what cases the effects of c2/c2 can be treated as small
A s
perturbationscan be found in thefinal results.
To proceed tothis direction, we write
h (r,η)=f(r)g(η,r),
1
u (r,η)=f (r)g (η,r),
r r r
Figure 2.Functional formsofeigenfunction,g(η), ofthefunda- u (r,η)=f (r)g (η,r), (32)
ϕ ϕ ϕ
mental(n=1)g-modeoscillations.Theirdependencesonηs and
ω2/Ω2⊥ are shown. Two cases where disk boundary ηs is at 3.0 anddivideequation(17)byfg.Then,theresultingequation
and5.0areshownforsomevaluesofω2/Ω2⊥.Itisnotedthatthe can be approximately separated into two equations with a
eigenfunctionsareterminatedatη=ηs,buttheirdifferencefrom weakly r-dependentseparation constant K(r) as
the eigenfunction for ηs = ∞ (i.e., g(η) = H1(η) = η), is quite 1 ∂2 ∂
small. −η g
g(cid:18)∂η2 ∂η(cid:19)
1 ω2 df ∂g ∂ ω2 ∂g
+ H2 + f
have fg (cid:20)ω2−κ2dr∂r ∂r(cid:18)ω2−κ2 ∂r(cid:19)(cid:21)
iω ∂ c2 ∂2 ∂2g 2Ω ∂2g
−Z−ηηssexp(cid:18)− η22(cid:19)(cid:18)ddgηnddgηm −Kngngm(cid:19)dη =+f−gK∂(rr(cid:20))ω2−Aκ2(cid:18)H2∂r2(frgr)+fr ∂η2r + iωfϕ ∂η2ϕ(3(cid:19)3(cid:21))
2ω2 η2 1
+Ω2nexp(cid:18)− 2s(cid:19)η gn(ηs)gm(ηs)=0, (29) and
⊥ s ω2 1 d ω2 df
+ H2 =K(r). (34)
where integration by part has been applied by using the Ω2 f dr(cid:18)ω2−κ2dr(cid:19)
⊥
boundarycondition (22).
Wenow expand g(η,r) as
We have an equation similar to equation (29) for g .
m
Thatis, startingfrom equation(28)for g andmultiplying g(r,η)=g(0)(η)+g(1)(η,r)+.... (35)
m
gn to both sides of the equation, we have a similar equa- Here,g(0) isthezeroth-ordersolutionobtainedintheprevi-
tion as equation (29) after performing integration by part. ous section with c2/c2 = 0, and g(1) is the perturbed part
A s
Then, taking the difference between the resulting equation of g(0) due to c2/c2 6= 0. It is noted that g(η) and K in
A s
and equation (29),we have the previous section are hereafter denoted by attaching su-
perscript(0)asg(0)(η)andK(0) inordertoemphasizethat
ηs η2
(K −K ) exp − g (η)g (η)dη theyare thezeroth-order quantities.
n m Z−ηs (cid:18) 2 (cid:19) n m In the lowest order approximation where the terms of
+2(ωn2 −ωm2)exp − ηs2 1g (η )g (η )=0. (30) c2A/c2s are neglected, g(0) is independent of r and equations
Ω2 (cid:18) 2 (cid:19)η n s m s (33) and (34) are reduced to equations (23) and (24). Let
⊥ s
usnowproceedtothenextorderapproximationswherethe
The term resulting from surface integral in equation (30) terms of c2/c2 are considered. Then, g(1) depends weakly
A s
(i.e., the second term) does not vanish in general, since
on r, and also the separation constant also dependsweakly
ωn2 6= ωm2 . Hence, there is no orthogonal relation such as on r as K(r) = K(0)+K(1)(r). Then, from equations (33)
equation (27) in the present case of η 6= ∞. However, in
s and (34), we have,respectively,
the case of g-mode oscillations, the difference between ω2
and ωm2 is much smaller than Ω2⊥ in the case of ηs 6= ∞n ∂2 −η ∂ +K(0) g(1)
(i.e., Okazaki et al. 1987). Differences between K and K (cid:18)∂η2 ∂η (cid:19)
n m
are on the order of unity. Furthermore, the surface value, ω2 1 df ∂g(1) ∂ ω2 ∂g(1)
exp(−ηs2/2)(1/ηs)gn(ηs)gm(ηs), is smaller than the value of =−H2(cid:20)ω2−κ2f dr ∂r + ∂r(cid:18)ω2−κ2f ∂r (cid:19)(cid:21)
integrationofthefirsttermofequation(30),whenη isnot
s iω ∂ c2 d2f d2g(0) 2Ω d2g(0)
too small. Hence,we haveapproximately − A H2 rg(0)+f r + f ϕ
f ∂r(cid:20)ω2−κ2(cid:18) dr2 r r dη2 iω ϕ dη2 (cid:19)(cid:21)
ηs exp − η2 g (η)g(η) dη∼0, (31) −K(1)(r)g(0). (36)
Z−ηs (cid:18) 2 (cid:19) n m and
when n6=m. We shall use this quasi-orthogonal relation in ∂ ω2 df ω2
H2 + f =(K(0)+K(1)f. (37)
thefollowing sections. ∂r(cid:18)ω2−κ2dr(cid:19) Ω2
⊥
MNRAS000,1–??(2016)
6 Shoji Kato
Equation(36)isaninhomogeneousdifferentialequation
Table1.ValuesofA,B,andtheconditionsofAc2 /c2<1and
withrespecttog(1),andthelefthandsideofequation(36)is ofself-trappingforthreecasesofηs withsomevaluAe0sosfω2/Ω2⊥.
written in thesame form as thezeroth-orderequation (23).
In the followings we solve equation (36) by a standard per-
ηs ω2/Ω2⊥ A,B conditionof
turbation method. The standard perturbation method re- Ac2 /c2<1
A0 s
quires that the zeroth order eigenfunctions are orthogonal.
In the present problem we do not have exact orthogonal- 2.5 0.4 4.517,-0.506 c2A0/c2s <0.22
ity, but orthogonal relations are roughly realized as shown 3.0 0.4 7.258,-0.549 c2 /c2<0.14
A0 s
in equation(31). We shall be satisfied by using this quasi- 0.6 7.305,-0.376 c2 /c2<0.14
A0 s
orthogonality, since theresulting errors seem to besmall. 0.8 7.356,-0.198 c2 /c2<0.14
A0 s
As in the standard perturbation method, g(1) is now
5.0 0.4 32.97,-1.111 c2 /c2<0.030
expandedintermsofthesetofeigenfunctionsin thezeroth A0 s
order equation (23), say g(0), where m=0, 1, 2,... as 0.6 33.09,-0.743 c2A0/c2s <0.030
m 0.8 33.20,-0.372 c2 /c2<0.030
A0 s
g(1)(η,r)= am(r)gm(0)(η) (38) ∞ ∞,0.0 c2A0/c2s =0.0
mX=0
andthecoefficientsa (r)’s(m6=1)aredeterminedbyuseof
m
quasi-orthogonalityofeigenfunctionsg(0).Theperturbation
m This shows that g(0) can be taken to be equal to g(0) and
methodremainsthecoefficienta (r)(m=1)undetermined, r
1 leads to
butrequiresthattherighthandsideof(36)isorthogonalto
g(0). This is the solvability condition of equation (36). It is df
(ω2−κ2)f =iω (44)
notedthatthecoefficienta1(r)canbetakentobezero,since r dr
theterma (r)g(0)inexpansionofg(1)canbeincludedinthe
1 1 inthelimitofc2 =0.Furthermore,inthelimitofc2 =0the
zerothordersolution ofh =f(r)g(0) (i.e.,normalization of A A
1
ϕ-componentofequationofmotiongivesarelationbetween
f)3.
f and f , which is
Usingthequasi-orthogonality(31),weseethatthesolv- r ϕ
ability condition that therighthand side of equation (36) is i κ2 κ2 1 df
orthogonal to g(0) is approximately written as fϕ =−ω2Ωfr = 2Ωω2−κ2dr. (45)
K(1)f =−iω d c2A0 AH2d2fr +B f + 2Ωf , Substitution of equations (44) and (45) into equation
dr(cid:20)ω2−κ2(cid:26) dr2 (cid:18) r iω ϕ(cid:19)(cid:27)(cid:21) (39) givesanexpression forK(1)f intermsoff alone. Sub-
(39) stitutionofthisexpressionforK(1)f intoequation(37)leads
finally to
where
A=Z−ηηssg(0)g(0)dη(cid:30)Z−ηηssexp(cid:18)− η22(cid:19)g(0)g(0)dη, (40) H2ddr(cid:20)ω2ω−2κ2(cid:18)1−Bcc2A2s0Ωω2⊥2 (cid:19)ddfr(cid:21)+ ω2−ΩK2⊥(0)Ω2⊥f
=H4 d Ac2A0 Ω2⊥ d2 ω2 df . (46)
B= ηs g(0)d2g(0)dη ηs exp − η2 g(0)g(0)dη. (41) dr(cid:20) c2s ω2−κ2dr2(cid:18)ω2−κ2dr(cid:19)(cid:21)
Z−ηs dη2 (cid:30)Z−ηs (cid:18) 2 (cid:19) This is a wave equation expressed in terms of f alone. The
In derivingsolvability condition (39) we haveused termswithAandB representtheeffectsofmagneticfields,
which are taken into account as perturbations. In treating
g(0)(η)=g(0)(η)=g(0)(η), (42) thisequation,however,weneedcarefulconsiderations,since
r ϕ
theorder of derivativewith respect to r hasbeen increased
whichwillbefoundlater,andc istheAlfv´enspeedonthe
A0 by the term with A from that of the unperturbedone. The
equator[seeequation(9)].ThevaluesofAandB areshown
unperturbedequation with nomagnetic fieldsis
in table 1 for three cases of η = 2.5, 3.0 and 5.0. Values
s
oωf2/AΩ2⊥an=d B0.4a,ls0o.6d,eapnednd0.8weaarkelyshoonwnω2f/oΩr2⊥ηs. =Th3r.e0eacnadses5.o0f. H2ddr(cid:20)ω2ω−2κ2ddfr(cid:21)+ ω2−ΩK2(0)Ω2⊥f =0. (47)
⊥
For comparison, values of A and B in the case of η = ∞
s
are also shown. Theincreaseoftheorderofdifferentialequationmeansthat
Tosimplifyequation(39)further,letusexpressfu and unless the terms resulting from the effects of c2A0/c2s 6= 0
fϕ by f. In thelimit of cA0=0, equation (14) gives are fully taken into account in equation (46), there is the
possibility that solutions of (46) do not tend to those of
∂h
(ω2−κ2)ur =iω ∂r1. (43) equation (47) in the limit of c2A0/c2s = 0. This is due to
thefact thatthecharacteristics of differentialequationsare
changed bythechange of order of equations.
To avoid this difficulties, we should remember that the
3 Itiseasilyshownthatifa1(r)istaken tobeanon-zeroarbi-
fourth order term in equation (46) is a term resulting from
traryfunctionofr,theresultsbecomethesameasthecasewhere
h1 is written formallyas h1 =f˜(r)g with f˜=f(r)+a1(r) and perturbations.Hence,bysubstitutingthezerothordersolu-
a1(r) in the expansion of g(1)(η,r) is taken to be zero. In this tion(47) intotherighthandsideof equation(46)wereduce
casef˜isfoundtofollowthesameequationasequation(37). the fourth order term to a second order term. Then, we re-
MNRAS000,1–??(2016)
Self-trapping of g-mode oscillations 7
duceequation (46) to tworadiiwhereω2=κ2 isrealized,whereκ isthemaxi-
max
mumvalueof epicyclicfrequency.Furthermore,at theradii
H2 d ω2 1−Bc2A0Ω2⊥ df + ω2−K(0)Ω2⊥f where ω2 = κ2 is realized the wavenumber of oscillations
dr(cid:20)ω2−κ2(cid:18) c2s ω2 (cid:19)dr(cid:21) Ω2⊥ vanishes: Waves approached there are reflected back. That
=−H2 d Ac2A0 Ω2⊥ d ω2−K(0)Ω2⊥f . (48) is, g-mode oscillations are trapped by cavity due to radial
dr(cid:20) c2 ω2−κ2dr(cid:18) Ω2 (cid:19)(cid:21) distribution of epicyclic frequency. The above results show
s ⊥
thatthecharacteristicsconcerningthewavetrappingregion
This equation is the final wave equation describing the ax-
of g-mode oscillations in the case of no magnetic fields re-
isymmetric g-modeoscillations.
mains unchanged even in magnetized disks, as long as con-
Therighthandsideofequation(48)comesfromtheper-
dition (49) holds.
turbation. Thus, comparison of the righthand side of this
It will be instructive to show that the trapping is also
equation with the first term on the lefthand side roughly
derivedfrom equation(46)bytakingd/dr=−ik .That is,
r
shows that equation (48) is valid only when
by taking d/dr = −ik under the local approximation and
r
Ac2A0|ω2−K(0)Ω2⊥| <1. (49) oeqthueartiaopnp(r4o6x)im: ationsadoptedabove,wehaveformallyfrom
c2 ω2
s
That is, we can studytrappingof axisymmetric g-modeos- ω2 (kH)2−ω2−K(0)Ω2⊥+Ac2A0 ω2Ω2⊥ (kH)4=0.
cillations byuseof equation(48),as longas c2A0/c2s issmall ω2−κ2 Ω2⊥ c2s (ω2−κ2)2
in thesense that inequality (49) is satisfied. Unless we con-
(53)
sider very low-frequency oscillations, theabove condition is
roughly Ac2 /c2 < 1. The values of c2 /c2 required for This equation needs to be solved by regarding the last
A0 s A0 s
Ac2 /c2 < 1 are shown in table 1 for disks with some fi- term as a perturbation term. That is, (kH)2 is written as
A0 s
nitedisk thickness(i.e., ηs 6=∞). (kH)2=(kH)20+(kH)21+.... Then,we have
Itisnotedthatthepresentperturbationmethodcannot
be applied when ηs = ∞ (see table 1), since in this case, (kH)20= (ω2−κ2)ω(ω2Ω2−2 K(0)Ω2⊥) (54)
A=∞ and condition (49) requiresc2 /c2 =0. ⊥
A0 s
and
(kH)2=−Ac2A0 Ω2⊥ (kH)4. (55)
5 PROPAGATION REGION OF G-MODE 1 c2 ω2−κ2 0
s
OSCILLATIONS AND SELF-TRAPPING
From theabove two equations,we have
Starting from equation (48), we examine the propagation
region of g-mode oscillations. In equation (48) there is an (kH)2= (ω2−κ2)(ω2−K(0)Ω2⊥) 1−Ac2A0ω2−K(0)Ω2⊥ .
apparent singularity at the radius of ω2 = κ2. To avoid ω2Ω2⊥ (cid:20) c2s ω2 (cid:21)
this inconvenience we introduce a new dependent function (56)
f˜definedby Thisexpressionfor(kH)2 isthesameasthatobtainedfrom
ω2 df equation (52) by taking d2f˜/dr2 = −k2f˜, when condition
f˜= . (50)
ω2−κ2dr (49) holdes.
This function f˜ is related to u [see equation (44)]. Fur-
r
thermore, for simplicity, the radial variation of (ω2 −
6 SUMMARY AND DISCUSSION
K(0)Ω2⊥)/Ω2⊥ in the wave propagation region is neglected.
ThetermwithB inequation(48)isalsoneglected,because Innon-magnetizedrelativisticdisks,g-modeoscillations are
it is a small term. Then, from equation (48) we have self-trappedintheinnermostregionofthedisks(Okazakiet
al.1987). Thisself-trappingof oscillations isof importance,
df˜+ω2−K(0)Ω2⊥f+Ac2A0 d ω2−K(0)Ω2⊥f˜ =0. (51) becausesuchdiscreteoscillationsininfinitelyextendeddisks
dr c2s c2s dr(cid:18) ω2 (cid:19) might be one of possible causes of high-frequency quasi-
Taking the radial derivative of this equation, we have ap- periodicoscillationsobservedinblack-holeandneutron-star
proximately X-ray binaries. Among g-mode oscillations, however, non-
axisymmetriconesareoutsideofourinterest,becausethese
1+Ac2A0ω2−K(0)Ω2⊥ d2f˜+(ω2−κ2)(ω2−K(0)Ω2⊥)f˜=0. oscillations havea corotation point in their propagation re-
(cid:18) c2 ω2 (cid:19)dr2 c2ω2 gionexceptforspecialcasesandarestronglydampedbythe
s s
corotation resonance (Kato 2003, Li et al. 2003).
(52)
Fu and Lai (2009), however, suggested that all g-mode
This equation tends to the wave equation describing oscillations arestrongly affected bythepresenceofpoloidal
oscillations in disks with no magnetic fields in the limit of magnetic fields and their self-trapping is easily destroyed
c2 /c2 =0.Thisequationclearlyshowsthatthewaveprop- evenif thefieldsare weak. Their results are of interest,but
A0 s
agation region is specified by (ω2−κ2)(ω2−K(0)Ω2⊥)>0, are based on a rough examination of wave motions in disks
whencondition(49)issatisfied.Inthecaseofg-modeoscil- which extend infinitely (i.e., η = ∞) in the vertical di-
s
lationswehaveω2−K(0)Ω2⊥ <0,andthustheirpropagation rection. Wave motions in vertically thick disks with verti-
region is ω2−κ2 <0. In relativistic disks, the propagation calmagneticfieldshavecomplicatedverticalbehaviors.The
region of g-mode oscillations with ω < κ is bounded by complications come from the fact that the kinetic energy
max
MNRAS000,1–??(2016)
8 Shoji Kato
density of waves, for example ρ u2, and that of magnetic tions, which gave the author an opportunity to much im-
0 r
energy density, for example b2/4π, have quite different z- provetheoriginal manuscript.
ϕ
dependences. This complication is related to the fact that
theacousticspeed,c ,isassumed tohavenoz-dependence,
s
while the Alfv´en speed, cA, increases with z and becomes REFERENCES
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such disks is, however, not temporizing. This is important
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cated at a certain height and sandwiched by hot coronae Kato,S. 2013, PASJ,65,75
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of epicyclic frequency. The magnetic fields required for
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A0 s
zero. That is, in vertically extended disks characteristics of
trappingof g-mode oscillations are strongly affected even if
the magnetic fields are weak. Our analyses cannot say how
the trapping is affected in such disks with strong vertical
magnetic fields.
Quasi-periodic oscillations are not always observed in
every black-hole and neutron-star sources, and also their
frequencies are not always robust except for those in some
black-hole sources. High frequency QPOs may not be a ho-
mogeneous class. Hence, trapped g-mode oscillations will
still remain as one of possible candidates of QPOs at least
in sources with weak vertical magnetic fields.
An issue to be mentioned here is whether axisymmet-
ric g-mode oscillations can be really excited in disks, since
theywillbedampedbyviscousprocessesunlikethep-mode
oscillations (see, e.g., Kato 1978, 2016). We think that the
most promising excitation process of axisymmetric g-mode
oscillations is stochastic one by turbulence, although any
numerical simulations do not suggest the presence of this
possibility in disks yet. Non-radial oscillations in the Sun
and stars are believed now to be excited by stochastic pro-
cesses by turbulence (Goldreich and Keeley 1977, see also
Kato1966).Similarprocessesarealsonaturallyexpectedin
disks,sincetheturbulenceindisksaremuchstrongerthanin
the Sunand stars (see Nowak and Wagoner 1993 and Kato
2016). Another possible excitation process of axisymmetric
g-mode oscillations is a wave-wave resonant excitation pro-
cess in deformed disks (Kato 2013). In this case another
oscillation which becomes a set of the g-mode oscillation
is excited. A question concerning this wave-wave resonant
excitation is whether a disk deformation required for the
resonance process is really expectedin disks.
The author thanks the referee for invaluable sugges-
MNRAS000,1–??(2016)