Global Disk Oscillation Modes in Cataclysmic Variables and
Other Newtonian Accretors
Manuel Ortega-Rodŕıguez1
Escuela de F́ısica & Centro de Investigaciones Geof́ısicas,
Universidad de Costa Rica, San José, Costa Rica;
and Gravity Probe B, W. W. Hansen Experimental Physics Lab,
Stanford University, Stanford, CA 94305-4085
Robert V. Wagoner2
Dept. of Physics & Kavli Institute for Particle Astrophysics and Cosmology,
Stanford University, Stanford, CA 94305–4060
ABSTRACT
Diskoseismology, the theoretical study of small adiabatic hydrodynamical
global perturbations of geometrically thin, optically thick accretion disks around
black holes (and other compact objects), is a potentially powerful probe of the
gravitational field. For instance, the frequencies of the normal mode oscillations
can be used to determine the elusive angular momentum parameter of the black
hole. The general formalism developed by diskoseismologists for relativistic sys-
tems can be readily applied to the Newtonian case of cataclysmic variables (CVs).
Some of these systems (e.g., the dwarf nova SS Cygni) show rapid oscillations in
the UV with periods of tens of seconds and high coherence. In this paper, we
assess the possibility that these dwarf nova oscillations (DNOs) are diskoseismic
modes. Besides its importance in investigating the physical origin of DNOs, the
present work could help us to answer the following question. To what extent
are the similarities in the oscillation phenomenology of CVs and X-ray binaries
(XRBs) indicative of a common physical mechanism?
Subject headings: accretion, accretion disks — hydrodynamics — stars: indi-
vidual (SS Cygni) — stars: individual (VW Hyi) — stars: novae, cataclysmic
variables — white dwarfs
1mortega@cariari.ucr.ac.cr
2wagoner@stanford.edu
http://arxiv.org/abs/0704.1920v1
– 2 –
1. Introduction
During the outburst phase (timescale ∼ a few days) some non-magnetic (B ≪ 106 G)
CVs exhibit fluctuations with periods P & 10 s and Q ≡ 1/|dP/dt| ∼ 103 − 107. The
phenomenology of these DNOs is very rich (Warner 2004). Also observed are lpDNOs (long
period DNOs): P & 30 s and Q ∼ 103 − 107, as well as QPOs (quasi-periodic oscillations):
P & 100 s and Q ∼ 10. At times, all three features can be observed simultaneously.
The fact that P ∼ the Keplerian frequency at the radius of the white dwarf has moti-
vated several candidate theoretical explanations for DNOs, ranging from ‘hot blobs’ (Popham
1999) to star oscillation modes (Papaloizou & Pringle 1978) to explanations based on the dy-
namics of the boundary layer or some other type of transition region located where the disk
touches the white dwarf. Examples of the last type include magnetic accretion onto a slipping
belt at the equator (Warner & Woudt 2002) and ‘spreading layer’ models (Piro & Bildsten
2004). Hydrodynamic oscillations in and near the boundary layer have been investigated by
Carroll et al. (1985) and Collins, Helfer & Van Horn (2000). However, their analyses were
local (plane-wave WKB) in radius and height. Indeed, Carroll et al. (1985) mention the
need of a global analysis. None of these models is truly successful at the quantitative level,
although magnetic models appear to be favored.
Lately, other proposals based on the dynamics of the accretion disk have appeared.
For example, Kluźniak et al. (2005) propose a nonlinear hydrodynamical disk resonance
explanation for the oscillations. Part of the reason this approach is attractive is that it could
explain the similarity of the phenomenology in CVs and XRBs.
Some DNOs exhibit a 1:2 (e.g., the period halving in SS Cyg) and in one case (VW Hyi)
a 1:2:3 harmonic structure. In some sources, a 1:15 relation of the frequencies of a QPO
and DNO is observed (Mauche 2002; Warner & Woudt 2005), similar to that seen in XRBs.
Some high-frequency QPOs in black hole XRBs are in a 3:2 ratio (Abramowicz & Kluźniak
2001; McClintock & Remillard 2006). All the explanations listed above (except the hot blob
and resonance models) are not applicable to black holes, given the absence of a surface (even
though some could apply to neutron stars). Conversely, QPO explanations based on general
relativistic effects are not applicable to the CV case. The only common explanation is based
on hydrodynamical oscillations in the accretion disk (and/or corona).
With the intention of adding to this debate, this paper explores the possibility that the
DNOs in CVs are caused by global hydrodynamical oscillations (normal modes) in the inner
accretion disk. To this end we will use diskoseismology, a formalism previously developed
to study small adiabatic perturbations in relativistic accretion disks [see, e.g., Wagoner
(1999); Wagoner, Silbergleit, & Ortega-Rodŕıguez (2001)]. The DNOs would arise as the
– 3 –
fluctuations modulate the outgoing radiation. Although Yamasaki, Kato & Mineshige (1995)
mention global hydrodynamical p–modes (trapped acoustic waves) in accretion disks around
white dwarfs, these modes lie in a different region from the one studied in this paper; in
addition, the physical analysis of these authors is much simpler than ours, as they work with
vertically integrated variables and limit themselves to axisymmetric modes.
In section 2 we briefly discuss the unperturbed disk model and then outline diskoseis-
mology in the general case. The rest of the paper applies diskoseismology to cataclysmic
variables. An introduction (section 3) precedes the formal WKB solution in section 4. Fi-
nally, section 5 presents the results and discusses their implications.
2. Diskoseismology
We take c = 1, and express all distances in units of GM/c2 and all frequencies in units
of c3/GM (where M is the mass of the white dwarf) unless otherwise indicated. We work
in cylindrical coordinates r, ϕ, z.
The stationary (∂/∂t = 0), symmetric about the midplane z = 0, and axially symmetric
(∂/∂ϕ = 0) unperturbed disk is taken to be described by the standard thin disk models for
the Newtonian (Shakura & Sunyaev 1973) and the relativistic (Novikov & Thorne 1973)
cases. In particular, we note that the velocity components vr = vz = 0 (the disk fluid
moves in nearly circular orbits), the disk semi-thickness h(r) ∼ cs/Ω ≪ r (where cs and
Ω are the speed of sound and the Keplerian frequency, respectively), the disk is optically
thick and radiatively efficient, the viscous stress is given by αp (i.e., the pressure multiplied
by a constant parameter), and typical values for the disk density and central object mass
make the neglect of self-gravity, at both the unperturbed and the perturbed level, a good
approximation. Further properties of the unperturbed disk will be discussed in section 5.
Akin to helioseismology, diskoseismology is the theoretical study of the normal modes
of small adiabatic hydrodynamical global perturbations in geometrically thin, optically thick
accretion disks. In this section, we describe diskoseismology in general terms. In the following
sections, we will use Newtonian diskoseismology to study oscillations in white dwarf accretion
disks. Fortunately, the relativistic and nonrelativistic models are formally very similar.
Even though the unperturbed disk is not in equilibrium, this perturbative approach is
justified whenever the viscous time scales during which the disk evolves as a whole are much
longer than the modes’ periods, and whenever the turbulence has spatial scales which are
smaller than the modes’ spatial scales, and it is quasi-steady on the time scales of the modes.
(These conditions hold for the modes dealt with in this paper.)
– 4 –
We consider barotropic disks [p = p(ρ), giving vanishing buoyancy frequency] and as-
sume that the results thus obtained are generic, since typically the buoyancy frequency is
much less than the characteristic dynamical frequency Ω. In this case hydrostatic equilibrium
provides the vertical density and pressure profiles
ρ = ρ0(r)(1− y2)g , p = p0(r)(1− y2)g+1 , g ≡ 1/(Γ− 1) > 0 , (1)
where Γ > 1 is the adiabatic index. One has Γ = 4/3 within any radiation pressure dominated
region of the disk, and Γ = 5/3 within any gas pressure dominated region (such as the white
dwarf accretion disks). The disk surfaces are at y = ±1, with y related to the vertical
coordinate z by
To investigate the eigenmodes of the disk oscillations starting from these assumptions,
one applies the formalism that Ipser & Lindblom (1991, 1992) developed for perturbations
of purely rotating perfect fluids, starting from the equations of conservation of mass and
momentum. One can thus express the Eulerian perturbations of all physical quantities
through a single function δV ∝ δp/ρ (the Eulerian perturbation of the pressure divided by
the density) which satisfies a second-order partial differential equation. Due to the stationary
and axisymmetric background, the angular and time dependences are factored out as δV =
V (r, z) exp[i(mϕ+ σt)], where σ is the eigenfrequency, and the governing PDE becomes
ω2 − κ2
V = 0 , (2)
where β = dt/dτ , grr is the metric coefficient, cs stands for the speed of sound, the corotation
frequency
ω ≡ σ +mΩ , (3)
Ω(r) is the Keplerian frequency, and κ(r) is the radial epicyclic frequency (i.e., the frequency
of radial perturbations of free particle circular orbits).
An assumption of strong variation of modes in the radial direction (characteristic radial
wavelength λr ≪ r) ensures the approximate WKB separability of variables in this PDE,
V (r, z) = Vr(r)Vy(r, y). The ‘vertical’ part, Vy, of the functional amplitude V (r, z) varies
slowly with r. They obey (Perez et al. 1997, hereafter referred to as RD1)
(1− y2)
− 2gy
+ 2gω2
1− y2
Vy = 0 , (4)
(ω2 − κ2)
(ω2 − κ2)
+ β2grrc
2 − κ2)
Vr = 0 , (5)
– 5 –
where
ω∗ ≡ ω/Ω⊥ ; (6)
Ω⊥(r) refers to the vertical epicyclic frequency (i.e., the frequency of vertical perturbations
of free particle circular orbits), and the speed of sound cs is evaluated at midplane. In these
equations Ψ(r) is the slowly-varying separation function.
These equations have been used in the past to reveal the properties of different types of
modes in relativistic accretion disks. In this paper we solve the equations in the Newtonian
limit. We employ, as in the original papers, WKB methods in a straightforward manner.
3. Diskoseismic Modes for the White Dwarf Case: Physical Parameters and
Scope
We now apply diskoseismology to the accretion disks of white dwarfs. We are thus
working in the Newtonian limit (for a mass M = M⊙ white dwarf, its radius R ∼ 5 × 103,
corresponding to 7.4 × 103 km), which amounts to setting grr = β = 1, g = 3/2 in the
equations of the previous section. We note in particular that
Ω(r) = κ(r) = Ω⊥(r) = 1/r
3/2 , (7)
i.e., the rotational, radial epicyclic, and vertical epicyclic frequencies are all equal (up to
corrections of relative size ∼ 1/r or smaller). As we will see, this results in a qualitative
change in the nature of the solutions with respect to the relativistic case.
This is a straightforward problem with few free parameters. In addition to the white
dwarf mass M , there is ri [≥ R(M)], the inner radius of the accretion disk. Then, one
needs a functional form for the speed of sound. It is fortunate that the properties of the
unperturbed disk (viscosity and luminosity) enter into the formalism only through the speed
of sound, and that the dependence of the speed of sound on these properties is rather weak.
In this exploratory calculation, we ignore the complications related to the nature of the
transition from the disk to the star surface or magnetosphere and assume that DNO oscil-
lations are confined to the disk. We introduce a disk inner boundary condition parameter,
given our ignorance of the physical conditions in the (magnetic) boundary layer. As we will
see, the results are rather insensitive to this parameter. The approximate character of our
analysis should be stressed. A deeper study of boundary layer dynamics is needed in order
to understand better the physical conditions that would allow for the existence of the modes
described in this paper. For instance, one could ask how significant is the damping and
‘leakage’ of the modes into the magnetosphere, where the radial epicyclic frequency is given
– 6 –
by the more general expression κ2 = 2Ωr−1d(r2Ω)/dr, if the modes are still confined to a
small vertical extent h ≪ r.
As in the case of relativistic diskoseismology (with black holes or weakly magnetic
neutron stars), the formalism reveals the existence of different types of modes. In this first
diskoseismic investigation of cataclysmic variables, we choose to study the g–modes, since
they are trapped near the inner edge of the disk and are therefore the most luminous (and
robust). By definition, the g–modes have corotating eigenfrequencies which are smaller than
the gravitational radial oscillation (epicyclic) frequencies at the involved radii. In other
words, the (smaller) pressure restoring forces act against the gravitational restoring forces.
(The ‘g’ in the name comes from a mathematical similarity to the g–modes within slowly
rotating stars.) The acoustic (p–) modes extend to the outer region of the disk, thus involving
strong effects of viscosity and the uncertain boundary conditions at its outer radius.
4. Formal WKB Solution of the Eigenvalue Problem for G–modes
In this section we will proceed to solve formally the eigenvalue problem in order to
obtain equation (28), our ‘master’ equation, from which all further results in the paper are
derived.
Taking into account expressions (7), the separated equations (4) and (5) reduce in the
Newtonian limit to
(1− y2) d
− 3y dVy
+ 3ω2
1− y2
Vy = 0 , (8)
(ω2 − Ω2)
(ω2 − Ω2)
+ c−2s (ω
2 − Ω2)
Vr = 0 . (9)
(Recall from section 2 that g = 3/2 for white dwarf accretion disks.) Equations (8) and (9)
are two coupled eigenvalue equations, the eigenvalues being ω and Ψ (or equivalently, σ and
Ψ). We will solve these equations within a straightforward WKB formalism as described
by Ortega-Rodŕıguez, Silbergleit, & Wagoner (2002) (hereafter referred to as RD3). With
respect to boundary conditions, it turns out that it is sufficient to require that the vertical
eigenfunction and its derivative remain finite at the surface of the disk and to introduce an
inner boundary parameter for the radial equation, as mentioned.
By definition, g–modes exist wherever ω2 < κ2(= Ω2). This means that the mode exists
between an inner radius (which is ri, the inner radius of the accretion disk, for the cases
studied in this paper) and the radius r+, defined as the radius at which |ω| = Ω. The mode
decays exponentially for r > r+.
– 7 –
4.1. WKB Solution of the Radial Eigenvalue Problem
We will cast equation (9) in a form amenable to a WKB solution, using a procedure
which is similar to the one developed in RD3. We introduce the dependent variable
W ≡ 1
Ω2 − ω2
, (10)
and the independent variable
c−2s (r
Ψ(r′)
dr′ , (11)
so that equation (9) can be written in the simpler form
+ S(τ)W = 0 , S(τ) ≡
c2s(Ω
2 − ω2)
. (12)
The mode exists in the range ri < r < r+ or, equivalently, 0 < τ < τi ≡ τ(ri). The function
S(τ) changes sign at τ = 0 (where r = r+). The boundary conditions are that W decays
for τ < 0, while at the other extreme, at τ = τi, we parameterize our ignorance at the inner
boundary by means of the quantity θi:
(Ω2 − ω2) cos θiW − sin θi
= 0 , (13)
which is equivalent to
cos θi
− sin θiVr = 0 . (14)
A straightforward WKB solution can be obtained for equation (12) at all points except for
τ close to 0:
W ∝ S−1/4(τ) cos
S1/2(τ ′)dτ ′ + Φ
, (15)
where Φ is a constant to be determined. For τ near zero, the solution is given by the Airy
function of the first kind:
W ∝ Ai(−s1/3+ τ) , s+ ≡ dS(0)/dτ > 0 . (16)
The asymptotic matching of solutions (15) and (16) in the usual manner yields Φ = π/4+nπ,
where n is an integer, and
∫ r+(σ)
c−1s (r)
(Ω2 − ω2)(Ψ/ω2
− 1)dr = π(n± 1/4) (17)
for the cases θi = π/2 and 0, respectively. Thus n is the radial mode number constrained by
the fact that the integral in equation (17) is positive.
– 8 –
4.2. WKB Solution of the Vertical Eigenvalue Problem
Using the new independent variable
(1− y′2)−3/2dy′ , (18)
we can write equation (8) in the simpler form
+Q(ζ)Vy = 0 , Q(ζ) ≡ 3ω2∗
1− y2
1− y2
, (19)
which is amenable to a WKB solution.
We now discuss the issue of boundary conditions. In view of the symmetry of the
differential equation (8), one has
= 0 or Vy
= 0 (20)
for even and odd modes in the disk, respectively, and it is enough to consider the interval
0 < y < 1. Near the singular boundary point y = 1, equation (8) can be written
+ . . .
+ . . .
Vy = 0 , (21)
where the dots represent non-singular terms. According to the analytical theory of second
order ODEs [see, e.g., Olver (1982)], the general solution to equation (8) near y = 1 has the
Vy = C1v1(y) + C2(1− y)−1/2v2(y) , (22)
where C1 and C2 are arbitrary constants, and v1(y) and v2(y) are converging power series of
(1− y). Imposing the boundary condition that the eigenfunction and its derivative be finite
at y = 1 implies thus that C2 = 0.
As noted in RD1 and implied by equation (21) and the boundary condition, the vertical
eigenfunction near the boundary can be written in terms of the Bessel function of the first
kind as
Vy(y) ∝ (1− y)−1/4J1/2(2|ω∗|
(3/2)(1− y) ) , (23)
the asymptotic (large argument) expression of which is
Vy(y) ∝ (1− y)−1/2 cos
6(1− y)− (π/2)
. (24)
– 9 –
On the other hand, equation (19) can be solved with a WKB method for values of y not
close to 1:
Vy ∝ Q−1/4(τ) cos
Q1/2(ζ ′)dζ ′ − Iπ/2
, (25)
where I = 0, 1 for the even and odd eigenfunctions, respectively. Matching this with the
asymptotic expression (24) of the boundary solution (23), we arrive at
1 + (Ψ/ω2
− 1)(1− y2)
1− y2
3|ω∗|
. (26)
The parity (of the vertical eigenfunction) and the vertical mode number j are contained in
a single quantity J ≡ j + 3/4 ± 1/4 for the odd and even cases, respectively. Thus J takes
positive integer and half-integer values (see Table 1).
It should be stressed that, because of the method used to solve the equations, the above
results are accurate only when Ψ/ω2
− 1 ≫ 1 for all r in the range ri < r < r+.
4.3. Uncoupling
Equations (17) and (26) are two eigenvalue equations (the eigenvalues being Ψ and ω),
which need to be uncoupled. We start by inverting the integral I on the left hand side of
equation (26) to find Ψ/ω2
− 1 as a function of I:
− 1 = f(I) . (27)
Once this is done numerically, the function f(I) can be substituted in equation (17):
c−1s (r)
(Ω2 − ω2)f(πJ/
3|ω∗|)dr = π(n± 1/4) . (28)
The left–hand side of equation (28) now only depends on the eigenfrequency σ and not on
Ψ [recall definition (3)]. All further results in this paper are obtained from this equation.
5. Results and Discussion
In order to use equation (28) and look for solutions with integer values of n ≥ 0, we
need a value for ri and a functional form for the speed of sound cs. We will take ri = 5000,
corresponding to 7383(M/M⊙) km. The restriction ri ≥ R, the white dwarf radius, places a
– 10 –
limit on its mass M . Using a standard mass–radius relation (Nauenberg 1972), this value of
ri implies that M ≥ 0.76M⊙.
As for the speed of sound, there are different solutions given by Shakura & Sunyaev
(1973), depending on the region of the disk in which one is working. For the values of
interest in this paper, the relevant expression is the one corresponding to the outermost
solution, in which temperatures are low enough that pressure is dominated by gas (not
radiation) and opacity is dominated by photon absorption in the form of nonrelativistic
thermal bremsstrahlung (‘free-free’ transitions).
Their solution for the speed of sound is given by
(cs/c)
−1 = 300α1/10(M/M⊙)
1/4Ṁ
−3/20
3/8 , (29)
where c, α, M , and Ṁ16 stand, respectively, for the speed of light, the (dimensionless)
viscosity parameter, the mass of the white dwarf, and the mass accretion rate in units of
1016 g/s. The small powers of the three parameters render our results relatively insensitive
to them. We remind the reader that the interior structure of the unperturbed accretion
disk enters our formulation only through the speed of sound. As mentioned above, we
parameterize our ignorance about the radial inner boundary by means of the quantity θi
introduced in equation (14).
5.1. Axisymmetric Modes
The eigenmodes are described by the azimuthal, radial and vertical mode numbers m,
n, j, plus the value of the parity of the vertical eigenfunction. We start by studying the
m = 0 modes. For them, the condition Ω2 −ω2 > 0 becomes Ω2 − σ2 > 0, which means that
the modes are trapped between ri and r+ = |σ|−2/3.
Using the characteristic values ri = 5000, M = M⊙, α = 0.1 (Lasota 2001), and Ṁ16 = 1
(Frank, King & Raine 2002), we calculate σ for different values of n, j and parity for the
case θi = π/2. (Note that the values of j and parity are contained in J in the manner
described in section 4.2 and Table 1.) Table 2 shows the results. The dimensional frequency
f = 3.23 × 104(M⊙/M)|σ| Hz. The n = 0, J = 1 entry thus corresponds to a period of 12
seconds.
Table 3 shows ∆r/ri ≡ (r+ − ri)/ri for the same values of the parameters. As in the
case of axisymmetric g–modes in relativistic accretion disks (RD1), the value of cs/c is small
enough that: a) the small n eigenfrequencies are close to the maximum value of κ(r) [which
equals Ω(ri) = 2.83× 10−6 in the present case], approaching it as j increases; b) the modes’
– 11 –
radial width decreases as j increases and increases as n increases.
Self-consistency requires that the parameter Ψ/ω2
− 1 not be small, and the larger it
is, the better the WKB approximation. It turns out that Ψ/ω2
− 1 ∼ 1 when J = 1 and
− 1 ≫ 1 when J > 1, so the procedure passes this test. (Incidentally, it is interesting
that the minimum value of Ψ/ω2
− 1 over the allowed values of the radius does not depend
on n or m, only on J .)
There is no J = 1/2 entry in Table 2 because there is no WKB solution. A direct
numerical solution of equations (8) and (9) shows that the absence of a solution is not
an artifact of the WKB method, although it could result from the assumptions made in
obtaining these equations. The possibility remains open that there are modes not detected
by our methods.
Next we compute the behavior of the eigenfrequency σ as the various parameters change.
For the choice J = 1, n = 0, θi = π/2, we obtain
∂ log σ/∂ log ri = −1.5 , (30)
∂ log σ/∂ logα = 0.0041 , (31)
∂ log σ/∂ logM = 0.010 , (32)
∂ log σ/∂ log Ṁ = −0.0061 , (33)
which means that the dependence of σ on α, M , and Ṁ can be neglected since it is much
weaker than its (expected) dependence on ri. Moreover, a change of θi from π/2 to 0 implies
only a change of a few percent in σ. Note, however, that there can be no n = 0 solution
with θi = 0 because then the RHS of equation (17) would be negative.
Also note that equation (32) assumes a fixed value of ri. Care must be taken then
when comparing with observations, since usually the parameters are not independent. For
example, for a set of white dwarfs of fixed composition there is a relationship between their
radius (R) and mass. So since σ ∼= σ(ri) and ri = ri(M, Ṁ) (for fixed surface magnetic
field), one has thus, for fixed Ṁ ,
∂ log σ
∂ logM
∂ log ri
∂ logM
d log σ
d log ri
= 0.50 , (34)
for the case M ∝ r−3i (Nauenberg 1972) appropriate to low–mass white dwarfs with weak
magnetic fields (ri = R).
When computing the variation of σ with the luminosity L for any specific cataclysmic
variable, if one assumes that ri ∝ L−γ , where γ is a constant, one obtains
d logP
d logL
∂ log ri
∂ logL
d log σ
d log ri
= −1.5 γ . (35)
– 12 –
Such a dependence of ri on luminosity could be due to strong magnetic field effects (Shapiro & Teukolsky
1983). The idea is that inside the magnetosphere the accretion flow is controlled by magnetic
fields, so one takes ri to be the equatorial radius of the magnetosphere. Its value should be
only moderately greater than the white dwarf radius R for those magnetic field strengths
thought to be present in many of those white dwarf systems which exhibit DNOs. For exam-
ple, the value of d logP/d logL = −0.091 for the dwarf nova SS Cygni reported by Mauche
(1996) is reproduced by our formalism provided γ = 0.06.
In summary, the model may be consistent with observations. The radial width of the
modes may be large enough to make them visible, and the periods are consistent, although
this is hardly surprising as they are related to the Keplerian period at ri. This leads to
the limit P ≥ 10.94(M⊙/M)1/2[ri(km)/7383 km]3/2 seconds on the periods of axisymmetric
modes. However, the physics of the magnetosphere is at present too uncertain to provide a
means to test the model accurately. At this point it makes sense to look at the higher m
modes.
5.2. Non-Axisymmetric Modes
Here we discuss the m > 0 modes. Note that from definition (3) and the fact that only
|ω| enters in the equations, we can, without loss of generality, consider only the non-negative
m cases provided we allow for negative σ.
The condition Ω2 − ω2 > 0 implies that
−m− 1 < σ/Ω(r) < −m+ 1 . (36)
Therefore, g–modes are trapped between ri and r+ = (m+1)
2/3|σ|−2/3. The inner boundary
of the mode (r−) is ri, provided that |σ| ≥ (m− 1)Ω(r). The negative sign of σ reflects the
fact that the oscillations’ wavefronts travel in the same direction as the fluid.
Table 4 and Figure 1 show some properties of the n = 0, J = 1 modes for the m = 0, 1
and 2 cases. Interestingly, the results are consistent with the observation of a 1:2:3 harmonic
structure in the power spectra of the DNO VW Hyi, reported by Warner & Woudt (2005).
The observed peaks remain harmonic even during a threefold increase in the fundamental
frequency. As in other models, the increase in frequency is due to a decrease in ri induced
by an increase in Ṁ (and thus L).
If one expresses σ in terms of r+ and works to lowest order in ∆r/ri, from equation (28)
one obtains
∆r/ri ∼ (n± 1/4)2/3(m+ 1)−1/3ǫ2/3 . (37)
– 13 –
Since ǫ ≡ h(ri)/ri ∼ (cs/Ωr)ri, using equation (29) and our chosen set of parameters (which
only weakly affect the estimate) gives ǫ ≈ 4.2 × 10−3 r1/8i ≈ 1.2× 10−2. Then equation (37)
is seen to agree approximately with the values in Tables 3 and 4.
The m > 0 diskoseismic modes are in principle harder to observe than the axisymmetric
modes, since in the former the modulation of the outgoing radiation is not as efficient due to
cancelations over the entire disk (and their radial extent is somewhat smaller). However, the
fact that the white dwarf can eclipse parts of the inner disk renders these modes more visible.
We predict that the energy spectrum of the DNOs is characteristic of the very inner disk
(and not other regions such as the magnetosphere), and thus should be close to blackbody
at the highest disk temperature.
As in the case of g–modes in relativistic accretion disks, the eigenfrequency splittings
when m changes by unity are much larger than the n and j splittings. The g–modes that
are trapped by the general relativistic behavior of the radial epicyclic frequency κ have
m splittings that are not as harmonic as those found here (Table 4). In particular, their
eigenfrequencies are only proportional to m for m & 2 [an interesting difference between the
relativistic and the nonrelativistic eigenfrequencies is that, for m ≫ 1, |σ| → mΩ(ri) for the
former (see RD1) while |σ| → (m+ 1)Ω(ri) for the latter]. However, it must be noted that
it appears that g–modes with small mode numbers are viscously unstable, at least if the
viscosity acts in the usual hydrodynamic way. This result is proven for the relativistic case
in Ortega-Rodŕıguez & Wagoner (2000), where the exponential growth rate of the mode
is roughly αΩ. For the nonrelativistic g–modes studied in this paper, an application of
their method yields the same result. Furthermore, these low–lying modes do not contain a
corotation resonance [where ω(r) = 0].
6. Comments
Thus, if the ideas in this paper are correct and DNOs are linear diskoseismic modes
then the results could suggest a scenario in which white dwarf DNOs and black hole QPOs
are produced through somewhat different physical mechanisms. After all, as Kluźniak et al.
(2005) themselves point out, the phenomenology is similar but not identical. While in
white dwarf DNOs the simultaneous observation of two or even three of the harmonics is
well established, in black holes usually only one mode of oscillation is observed at a given
moment. Furthermore, white dwarf DNOs (and neutron star QPOs) have frequencies that
vary with time, while QPOs in black holes have fixed frequencies. In neutron star XRBs, the
(twin) high–frequency QPOs are analogous to the DNOs, but their frequency ratio is much
closer to unity (van der Klis 2006). It appears to be controlled (magnetically) by the neutron
– 14 –
star, since their frequency difference is close to the neutron star spin frequency or half of it.
We note that some DNOs are split at a QPO frequency. We also note that with increasing
radius of a neutron star magnetosphere, their g-modes should approach (for ri & 8) those of
this paper.
We hope to investigate the magnetic boundary layer in more detail in a future paper.
How much will the boundary layer contribute to the damping (or growth) of the g–modes?
Further work should try to explain why the value of Q is so high. Naively, one would expect
Q ∼ 1/α. (It is fortunate that most of our results are fairly insensitive to the value of α.)
The nonlinear growth and coupling of the modes must also be investigated.
This work was supported by grant 075-2002 (Incentivos) of Ministerio de Ciencia y
Tecnoloǵıa, Costa Rica, grant 829-A3-078 of Vicerrectoŕıa de Investigación, Universidad de
Costa Rica, and NASA grant NAS 8-39225 to Gravity Probe B. We are also grateful to the
Aspen Center for Physics for its support.
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Collins, T. J. B., Helfer, H. L. & Van Horn, H. M. 2000, ApJ, 534, 944
Frank, J., King, A., & Raine, D. 2002, Accretion Power in Astrophysics (Cambridge: Cam-
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Ipser, J. R. & Lindblom, L. 1991, ApJ, 379, 285
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G. Lewin & M. van der Klis (Cambridge: Cambridge Univ. Press), p. 157
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Nauenberg, M. 1972, ApJ, 175, 417
Novikov, I. D. & Thorne, K. S. 1973, in Black Holes, ed. C. DeWitt & B. S. DeWitt (New
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Ortega-Rodŕıguez, M., Silbergleit, A. S., & Wagoner, R. V. 2002, ApJ, 567, 1043 (RD3)
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Piro, A. L. & Bildsten, L. 2004, ApJ, 616, L155
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Shapiro, S. L. & Teukolsky, S. A. 1983, Black Holes, White Dwarfs, and Neutron Stars (New
York: Wiley)
van der Klis, M. 2006, in Compact Stellar X-Ray Sources, ed. W. H. G. Lewin & M. van der
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Wagoner, R. V. 1999, Phys. Rep., 311, 259
Wagoner, R. V., Silbergleit, A. S., & Ortega-Rodŕıguez, M. 2001, ApJ, 559, L25
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This preprint was prepared with the AAS LATEX macros v5.2.
– 16 –
Table 1. Value of J as a function of j and parity
j parity J
0 even 1/2
0 odd 1
1 even 3/2
1 odd 2
– 17 –
Table 2. Values of the eigenfrequency σ × 106 for m = 0 g–modes.
n 1 3/2 2
0 2.65 2.73 2.75
1 2.39 2.55 2.61
2 2.23 2.43 2.51
Table 3. Values of the modes’ fractional radial width ∆r/ri for m = 0 g–modes
n 1 3/2 2
0 0.044 0.024 0.019
1 0.12 0.072 0.055
2 0.17 0.11 0.083
Table 4. Harmonic structure of the J = 1, n = 0 g–modes
m −106 σm σm/σ0 ∆r/ri
0 2.65 1 0.044
1 5.39 2.03 0.033
2 8.14 3.07 0.028
– 18 –
1 1.05 1.1 1.15 1.2
m = 0 
m = 1
m = 2
Fig. 1.— The horizontal lines show the radial extent and the frequency of the three lowest
fundamental (n = 0, J = 1) axial g-modes.
	Introduction
	Diskoseismology
	Diskoseismic Modes for the White Dwarf Case: Physical Parameters and Scope
	Formal WKB Solution of the Eigenvalue Problem for G–modes
	WKB Solution of the Radial Eigenvalue Problem
	WKB Solution of the Vertical Eigenvalue Problem
	Uncoupling
	Results and Discussion
	Axisymmetric Modes
	Non-Axisymmetric Modes
	Comments

  Diskoseismology, the theoretical study of small adiabatic hydrodynamical
global perturbations of geometrically thin, optically thick accretion disks
around black holes (and other compact objects), is a potentially powerful probe
of the gravitational field. For instance, the frequencies of the normal mode
oscillations can be used to determine the elusive angular momentum parameter of
the black hole. The general formalism developed by diskoseismologists for
relativistic systems can be readily applied to the Newtonian case of
cataclysmic variables (CVs). Some of these systems (e.g., the dwarf nova SS
Cygni) show rapid oscillations in the UV with periods of tens of seconds and
high coherence. In this paper, we assess the possibility that these dwarf nova
oscillations (DNOs) are diskoseismic modes. Besides its importance in
investigating the physical origin of DNOs, the present work could help us to
answer the following question. To what extent are the similarities in the
oscillation phenomenology of CVs and X-ray binaries (XRBs) indicative of a
common physical mechanism?

Introduction
During the outburst phase (timescale ∼ a few days) some non-magnetic (B ≪ 106 G)
CVs exhibit fluctuations with periods P & 10 s and Q ≡ 1/|dP/dt| ∼ 103 − 107. The
phenomenology of these DNOs is very rich (Warner 2004). Also observed are lpDNOs (long
period DNOs): P & 30 s and Q ∼ 103 − 107, as well as QPOs (quasi-periodic oscillations):
P & 100 s and Q ∼ 10. At times, all three features can be observed simultaneously.
The fact that P ∼ the Keplerian frequency at the radius of the white dwarf has moti-
vated several candidate theoretical explanations for DNOs, ranging from ‘hot blobs’ (Popham
1999) to star oscillation modes (Papaloizou & Pringle 1978) to explanations based on the dy-
namics of the boundary layer or some other type of transition region located where the disk
touches the white dwarf. Examples of the last type include magnetic accretion onto a slipping
belt at the equator (Warner & Woudt 2002) and ‘spreading layer’ models (Piro & Bildsten
2004). Hydrodynamic oscillations in and near the boundary layer have been investigated by
Carroll et al. (1985) and Collins, Helfer & Van Horn (2000). However, their analyses were
local (plane-wave WKB) in radius and height. Indeed, Carroll et al. (1985) mention the
need of a global analysis. None of these models is truly successful at the quantitative level,
although magnetic models appear to be favored.
Lately, other proposals based on the dynamics of the accretion disk have appeared.
For example, Kluźniak et al. (2005) propose a nonlinear hydrodynamical disk resonance
explanation for the oscillations. Part of the reason this approach is attractive is that it could
explain the similarity of the phenomenology in CVs and XRBs.
Some DNOs exhibit a 1:2 (e.g., the period halving in SS Cyg) and in one case (VW Hyi)
a 1:2:3 harmonic structure. In some sources, a 1:15 relation of the frequencies of a QPO
and DNO is observed (Mauche 2002; Warner & Woudt 2005), similar to that seen in XRBs.
Some high-frequency QPOs in black hole XRBs are in a 3:2 ratio (Abramowicz & Kluźniak
2001; McClintock & Remillard 2006). All the explanations listed above (except the hot blob
and resonance models) are not applicable to black holes, given the absence of a surface (even
though some could apply to neutron stars). Conversely, QPO explanations based on general
relativistic effects are not applicable to the CV case. The only common explanation is based
on hydrodynamical oscillations in the accretion disk (and/or corona).
With the intention of adding to this debate, this paper explores the possibility that the
DNOs in CVs are caused by global hydrodynamical oscillations (normal modes) in the inner
accretion disk. To this end we will use diskoseismology, a formalism previously developed
to study small adiabatic perturbations in relativistic accretion disks [see, e.g., Wagoner
(1999); Wagoner, Silbergleit, & Ortega-Rodŕıguez (2001)]. The DNOs would arise as the
– 3 –
fluctuations modulate the outgoing radiation. Although Yamasaki, Kato & Mineshige (1995)
mention global hydrodynamical p–modes (trapped acoustic waves) in accretion disks around
white dwarfs, these modes lie in a different region from the one studied in this paper; in
addition, the physical analysis of these authors is much simpler than ours, as they work with
vertically integrated variables and limit themselves to axisymmetric modes.
In section 2 we briefly discuss the unperturbed disk model and then outline diskoseis-
mology in the general case. The rest of the paper applies diskoseismology to cataclysmic
variables. An introduction (section 3) precedes the formal WKB solution in section 4. Fi-
nally, section 5 presents the results and discusses their implications.
2. Diskoseismology
We take c = 1, and express all distances in units of GM/c2 and all frequencies in units
of c3/GM (where M is the mass of the white dwarf) unless otherwise indicated. We work
in cylindrical coordinates r, ϕ, z.
The stationary (∂/∂t = 0), symmetric about the midplane z = 0, and axially symmetric
(∂/∂ϕ = 0) unperturbed disk is taken to be described by the standard thin disk models for
the Newtonian (Shakura & Sunyaev 1973) and the relativistic (Novikov & Thorne 1973)
cases. In particular, we note that the velocity components vr = vz = 0 (the disk fluid
moves in nearly circular orbits), the disk semi-thickness h(r) ∼ cs/Ω ≪ r (where cs and
Ω are the speed of sound and the Keplerian frequency, respectively), the disk is optically
thick and radiatively efficient, the viscous stress is given by αp (i.e., the pressure multiplied
by a constant parameter), and typical values for the disk density and central object mass
make the neglect of self-gravity, at both the unperturbed and the perturbed level, a good
approximation. Further properties of the unperturbed disk will be discussed in section 5.
Akin to helioseismology, diskoseismology is the theoretical study of the normal modes
of small adiabatic hydrodynamical global perturbations in geometrically thin, optically thick
accretion disks. In this section, we describe diskoseismology in general terms. In the following
sections, we will use Newtonian diskoseismology to study oscillations in white dwarf accretion
disks. Fortunately, the relativistic and nonrelativistic models are formally very similar.
Even though the unperturbed disk is not in equilibrium, this perturbative approach is
justified whenever the viscous time scales during which the disk evolves as a whole are much
longer than the modes’ periods, and whenever the turbulence has spatial scales which are
smaller than the modes’ spatial scales, and it is quasi-steady on the time scales of the modes.
(These conditions hold for the modes dealt with in this paper.)
– 4 –
We consider barotropic disks [p = p(ρ), giving vanishing buoyancy frequency] and as-
sume that the results thus obtained are generic, since typically the buoyancy frequency is
much less than the characteristic dynamical frequency Ω. In this case hydrostatic equilibrium
provides the vertical density and pressure profiles
ρ = ρ0(r)(1− y2)g , p = p0(r)(1− y2)g+1 , g ≡ 1/(Γ− 1) > 0 , (1)
where Γ > 1 is the adiabatic index. One has Γ = 4/3 within any radiation pressure dominated
region of the disk, and Γ = 5/3 within any gas pressure dominated region (such as the white
dwarf accretion disks). The disk surfaces are at y = ±1, with y related to the vertical
coordinate z by
To investigate the eigenmodes of the disk oscillations starting from these assumptions,
one applies the formalism that Ipser & Lindblom (1991, 1992) developed for perturbations
of purely rotating perfect fluids, starting from the equations of conservation of mass and
momentum. One can thus express the Eulerian perturbations of all physical quantities
through a single function δV ∝ δp/ρ (the Eulerian perturbation of the pressure divided by
the density) which satisfies a second-order partial differential equation. Due to the stationary
and axisymmetric background, the angular and time dependences are factored out as δV =
V (r, z) exp[i(mϕ+ σt)], where σ is the eigenfrequency, and the governing PDE becomes
ω2 − κ2
V = 0 , (2)
where β = dt/dτ , grr is the metric coefficient, cs stands for the speed of sound, the corotation
frequency
ω ≡ σ +mΩ , (3)
Ω(r) is the Keplerian frequency, and κ(r) is the radial epicyclic frequency (i.e., the frequency
of radial perturbations of free particle circular orbits).
An assumption of strong variation of modes in the radial direction (characteristic radial
wavelength λr ≪ r) ensures the approximate WKB separability of variables in this PDE,
V (r, z) = Vr(r)Vy(r, y). The ‘vertical’ part, Vy, of the functional amplitude V (r, z) varies
slowly with r. They obey (Perez et al. 1997, hereafter referred to as RD1)
(1− y2)
− 2gy
+ 2gω2
1− y2
Vy = 0 , (4)
(ω2 − κ2)
(ω2 − κ2)
+ β2grrc
2 − κ2)
Vr = 0 , (5)
– 5 –
where
ω∗ ≡ ω/Ω⊥ ; (6)
Ω⊥(r) refers to the vertical epicyclic frequency (i.e., the frequency of vertical perturbations
of free particle circular orbits), and the speed of sound cs is evaluated at midplane. In these
equations Ψ(r) is the slowly-varying separation function.
These equations have been used in the past to reveal the properties of different types of
modes in relativistic accretion disks. In this paper we solve the equations in the Newtonian
limit. We employ, as in the original papers, WKB methods in a straightforward manner.
3. Diskoseismic Modes for the White Dwarf Case: Physical Parameters and
Scope
We now apply diskoseismology to the accretion disks of white dwarfs. We are thus
working in the Newtonian limit (for a mass M = M⊙ white dwarf, its radius R ∼ 5 × 103,
corresponding to 7.4 × 103 km), which amounts to setting grr = β = 1, g = 3/2 in the
equations of the previous section. We note in particular that
Ω(r) = κ(r) = Ω⊥(r) = 1/r
3/2 , (7)
i.e., the rotational, radial epicyclic, and vertical epicyclic frequencies are all equal (up to
corrections of relative size ∼ 1/r or smaller). As we will see, this results in a qualitative
change in the nature of the solutions with respect to the relativistic case.
This is a straightforward problem with few free parameters. In addition to the white
dwarf mass M , there is ri [≥ R(M)], the inner radius of the accretion disk. Then, one
needs a functional form for the speed of sound. It is fortunate that the properties of the
unperturbed disk (viscosity and luminosity) enter into the formalism only through the speed
of sound, and that the dependence of the speed of sound on these properties is rather weak.
In this exploratory calculation, we ignore the complications related to the nature of the
transition from the disk to the star surface or magnetosphere and assume that DNO oscil-
lations are confined to the disk. We introduce a disk inner boundary condition parameter,
given our ignorance of the physical conditions in the (magnetic) boundary layer. As we will
see, the results are rather insensitive to this parameter. The approximate character of our
analysis should be stressed. A deeper study of boundary layer dynamics is needed in order
to understand better the physical conditions that would allow for the existence of the modes
described in this paper. For instance, one could ask how significant is the damping and
‘leakage’ of the modes into the magnetosphere, where the radial epicyclic frequency is given
– 6 –
by the more general expression κ2 = 2Ωr−1d(r2Ω)/dr, if the modes are still confined to a
small vertical extent h ≪ r.
As in the case of relativistic diskoseismology (with black holes or weakly magnetic
neutron stars), the formalism reveals the existence of different types of modes. In this first
diskoseismic investigation of cataclysmic variables, we choose to study the g–modes, since
they are trapped near the inner edge of the disk and are therefore the most luminous (and
robust). By definition, the g–modes have corotating eigenfrequencies which are smaller than
the gravitational radial oscillation (epicyclic) frequencies at the involved radii. In other
words, the (smaller) pressure restoring forces act against the gravitational restoring forces.
(The ‘g’ in the name comes from a mathematical similarity to the g–modes within slowly
rotating stars.) The acoustic (p–) modes extend to the outer region of the disk, thus involving
strong effects of viscosity and the uncertain boundary conditions at its outer radius.
4. Formal WKB Solution of the Eigenvalue Problem for G–modes
In this section we will proceed to solve formally the eigenvalue problem in order to
obtain equation (28), our ‘master’ equation, from which all further results in the paper are
derived.
Taking into account expressions (7), the separated equations (4) and (5) reduce in the
Newtonian limit to
(1− y2) d
− 3y dVy
+ 3ω2
1− y2
Vy = 0 , (8)
(ω2 − Ω2)
(ω2 − Ω2)
+ c−2s (ω
2 − Ω2)
Vr = 0 . (9)
(Recall from section 2 that g = 3/2 for white dwarf accretion disks.) Equations (8) and (9)
are two coupled eigenvalue equations, the eigenvalues being ω and Ψ (or equivalently, σ and
Ψ). We will solve these equations within a straightforward WKB formalism as described
by Ortega-Rodŕıguez, Silbergleit, & Wagoner (2002) (hereafter referred to as RD3). With
respect to boundary conditions, it turns out that it is sufficient to require that the vertical
eigenfunction and its derivative remain finite at the surface of the disk and to introduce an
inner boundary parameter for the radial equation, as mentioned.
By definition, g–modes exist wherever ω2 < κ2(= Ω2). This means that the mode exists
between an inner radius (which is ri, the inner radius of the accretion disk, for the cases
studied in this paper) and the radius r+, defined as the radius at which |ω| = Ω. The mode
decays exponentially for r > r+.
– 7 –
4.1. WKB Solution of the Radial Eigenvalue Problem
We will cast equation (9) in a form amenable to a WKB solution, using a procedure
which is similar to the one developed in RD3. We introduce the dependent variable
W ≡ 1
Ω2 − ω2
, (10)
and the independent variable
c−2s (r
Ψ(r′)
dr′ , (11)
so that equation (9) can be written in the simpler form
+ S(τ)W = 0 , S(τ) ≡
c2s(Ω
2 − ω2)
. (12)
The mode exists in the range ri < r < r+ or, equivalently, 0 < τ < τi ≡ τ(ri). The function
S(τ) changes sign at τ = 0 (where r = r+). The boundary conditions are that W decays
for τ < 0, while at the other extreme, at τ = τi, we parameterize our ignorance at the inner
boundary by means of the quantity θi:
(Ω2 − ω2) cos θiW − sin θi
= 0 , (13)
which is equivalent to
cos θi
− sin θiVr = 0 . (14)
A straightforward WKB solution can be obtained for equation (12) at all points except for
τ close to 0:
W ∝ S−1/4(τ) cos
S1/2(τ ′)dτ ′ + Φ
, (15)
where Φ is a constant to be determined. For τ near zero, the solution is given by the Airy
function of the first kind:
W ∝ Ai(−s1/3+ τ) , s+ ≡ dS(0)/dτ > 0 . (16)
The asymptotic matching of solutions (15) and (16) in the usual manner yields Φ = π/4+nπ,
where n is an integer, and
∫ r+(σ)
c−1s (r)
(Ω2 − ω2)(Ψ/ω2
− 1)dr = π(n± 1/4) (17)
for the cases θi = π/2 and 0, respectively. Thus n is the radial mode number constrained by
the fact that the integral in equation (17) is positive.
– 8 –
4.2. WKB Solution of the Vertical Eigenvalue Problem
Using the new independent variable
(1− y′2)−3/2dy′ , (18)
we can write equation (8) in the simpler form
+Q(ζ)Vy = 0 , Q(ζ) ≡ 3ω2∗
1− y2
1− y2
, (19)
which is amenable to a WKB solution.
We now discuss the issue of boundary conditions. In view of the symmetry of the
differential equation (8), one has
= 0 or Vy
= 0 (20)
for even and odd modes in the disk, respectively, and it is enough to consider the interval
0 < y < 1. Near the singular boundary point y = 1, equation (8) can be written
+ . . .
+ . . .
Vy = 0 , (21)
where the dots represent non-singular terms. According to the analytical theory of second
order ODEs [see, e.g., Olver (1982)], the general solution to equation (8) near y = 1 has the
Vy = C1v1(y) + C2(1− y)−1/2v2(y) , (22)
where C1 and C2 are arbitrary constants, and v1(y) and v2(y) are converging power series of
(1− y). Imposing the boundary condition that the eigenfunction and its derivative be finite
at y = 1 implies thus that C2 = 0.
As noted in RD1 and implied by equation (21) and the boundary condition, the vertical
eigenfunction near the boundary can be written in terms of the Bessel function of the first
kind as
Vy(y) ∝ (1− y)−1/4J1/2(2|ω∗|
(3/2)(1− y) ) , (23)
the asymptotic (large argument) expression of which is
Vy(y) ∝ (1− y)−1/2 cos
6(1− y)− (π/2)
. (24)
– 9 –
On the other hand, equation (19) can be solved with a WKB method for values of y not
close to 1:
Vy ∝ Q−1/4(τ) cos
Q1/2(ζ ′)dζ ′ − Iπ/2
, (25)
where I = 0, 1 for the even and odd eigenfunctions, respectively. Matching this with the
asymptotic expression (24) of the boundary solution (23), we arrive at
1 + (Ψ/ω2
− 1)(1− y2)
1− y2
3|ω∗|
. (26)
The parity (of the vertical eigenfunction) and the vertical mode number j are contained in
a single quantity J ≡ j + 3/4 ± 1/4 for the odd and even cases, respectively. Thus J takes
positive integer and half-integer values (see Table 1).
It should be stressed that, because of the method used to solve the equations, the above
results are accurate only when Ψ/ω2
− 1 ≫ 1 for all r in the range ri < r < r+.
4.3. Uncoupling
Equations (17) and (26) are two eigenvalue equations (the eigenvalues being Ψ and ω),
which need to be uncoupled. We start by inverting the integral I on the left hand side of
equation (26) to find Ψ/ω2
− 1 as a function of I:
− 1 = f(I) . (27)
Once this is done numerically, the function f(I) can be substituted in equation (17):
c−1s (r)
(Ω2 − ω2)f(πJ/
3|ω∗|)dr = π(n± 1/4) . (28)
The left–hand side of equation (28) now only depends on the eigenfrequency σ and not on
Ψ [recall definition (3)]. All further results in this paper are obtained from this equation.
5. Results and Discussion
In order to use equation (28) and look for solutions with integer values of n ≥ 0, we
need a value for ri and a functional form for the speed of sound cs. We will take ri = 5000,
corresponding to 7383(M/M⊙) km. The restriction ri ≥ R, the white dwarf radius, places a
– 10 –
limit on its mass M . Using a standard mass–radius relation (Nauenberg 1972), this value of
ri implies that M ≥ 0.76M⊙.
As for the speed of sound, there are different solutions given by Shakura & Sunyaev
(1973), depending on the region of the disk in which one is working. For the values of
interest in this paper, the relevant expression is the one corresponding to the outermost
solution, in which temperatures are low enough that pressure is dominated by gas (not
radiation) and opacity is dominated by photon absorption in the form of nonrelativistic
thermal bremsstrahlung (‘free-free’ transitions).
Their solution for the speed of sound is given by
(cs/c)
−1 = 300α1/10(M/M⊙)
1/4Ṁ
−3/20
3/8 , (29)
where c, α, M , and Ṁ16 stand, respectively, for the speed of light, the (dimensionless)
viscosity parameter, the mass of the white dwarf, and the mass accretion rate in units of
1016 g/s. The small powers of the three parameters render our results relatively insensitive
to them. We remind the reader that the interior structure of the unperturbed accretion
disk enters our formulation only through the speed of sound. As mentioned above, we
parameterize our ignorance about the radial inner boundary by means of the quantity θi
introduced in equation (14).
5.1. Axisymmetric Modes
The eigenmodes are described by the azimuthal, radial and vertical mode numbers m,
n, j, plus the value of the parity of the vertical eigenfunction. We start by studying the
m = 0 modes. For them, the condition Ω2 −ω2 > 0 becomes Ω2 − σ2 > 0, which means that
the modes are trapped between ri and r+ = |σ|−2/3.
Using the characteristic values ri = 5000, M = M⊙, α = 0.1 (Lasota 2001), and Ṁ16 = 1
(Frank, King & Raine 2002), we calculate σ for different values of n, j and parity for the
case θi = π/2. (Note that the values of j and parity are contained in J in the manner
described in section 4.2 and Table 1.) Table 2 shows the results. The dimensional frequency
f = 3.23 × 104(M⊙/M)|σ| Hz. The n = 0, J = 1 entry thus corresponds to a period of 12
seconds.
Table 3 shows ∆r/ri ≡ (r+ − ri)/ri for the same values of the parameters. As in the
case of axisymmetric g–modes in relativistic accretion disks (RD1), the value of cs/c is small
enough that: a) the small n eigenfrequencies are close to the maximum value of κ(r) [which
equals Ω(ri) = 2.83× 10−6 in the present case], approaching it as j increases; b) the modes’
– 11 –
radial width decreases as j increases and increases as n increases.
Self-consistency requires that the parameter Ψ/ω2
− 1 not be small, and the larger it
is, the better the WKB approximation. It turns out that Ψ/ω2
− 1 ∼ 1 when J = 1 and
− 1 ≫ 1 when J > 1, so the procedure passes this test. (Incidentally, it is interesting
that the minimum value of Ψ/ω2
− 1 over the allowed values of the radius does not depend
on n or m, only on J .)
There is no J = 1/2 entry in Table 2 because there is no WKB solution. A direct
numerical solution of equations (8) and (9) shows that the absence of a solution is not
an artifact of the WKB method, although it could result from the assumptions made in
obtaining these equations. The possibility remains open that there are modes not detected
by our methods.
Next we compute the behavior of the eigenfrequency σ as the various parameters change.
For the choice J = 1, n = 0, θi = π/2, we obtain
∂ log σ/∂ log ri = −1.5 , (30)
∂ log σ/∂ logα = 0.0041 , (31)
∂ log σ/∂ logM = 0.010 , (32)
∂ log σ/∂ log Ṁ = −0.0061 , (33)
which means that the dependence of σ on α, M , and Ṁ can be neglected since it is much
weaker than its (expected) dependence on ri. Moreover, a change of θi from π/2 to 0 implies
only a change of a few percent in σ. Note, however, that there can be no n = 0 solution
with θi = 0 because then the RHS of equation (17) would be negative.
Also note that equation (32) assumes a fixed value of ri. Care must be taken then
when comparing with observations, since usually the parameters are not independent. For
example, for a set of white dwarfs of fixed composition there is a relationship between their
radius (R) and mass. So since σ ∼= σ(ri) and ri = ri(M, Ṁ) (for fixed surface magnetic
field), one has thus, for fixed Ṁ ,
∂ log σ
∂ logM
∂ log ri
∂ logM
d log σ
d log ri
= 0.50 , (34)
for the case M ∝ r−3i (Nauenberg 1972) appropriate to low–mass white dwarfs with weak
magnetic fields (ri = R).
When computing the variation of σ with the luminosity L for any specific cataclysmic
variable, if one assumes that ri ∝ L−γ , where γ is a constant, one obtains
d logP
d logL
∂ log ri
∂ logL
d log σ
d log ri
= −1.5 γ . (35)
– 12 –
Such a dependence of ri on luminosity could be due to strong magnetic field effects (Shapiro & Teukolsky
1983). The idea is that inside the magnetosphere the accretion flow is controlled by magnetic
fields, so one takes ri to be the equatorial radius of the magnetosphere. Its value should be
only moderately greater than the white dwarf radius R for those magnetic field strengths
thought to be present in many of those white dwarf systems which exhibit DNOs. For exam-
ple, the value of d logP/d logL = −0.091 for the dwarf nova SS Cygni reported by Mauche
(1996) is reproduced by our formalism provided γ = 0.06.
In summary, the model may be consistent with observations. The radial width of the
modes may be large enough to make them visible, and the periods are consistent, although
this is hardly surprising as they are related to the Keplerian period at ri. This leads to
the limit P ≥ 10.94(M⊙/M)1/2[ri(km)/7383 km]3/2 seconds on the periods of axisymmetric
modes. However, the physics of the magnetosphere is at present too uncertain to provide a
means to test the model accurately. At this point it makes sense to look at the higher m
modes.
5.2. Non-Axisymmetric Modes
Here we discuss the m > 0 modes. Note that from definition (3) and the fact that only
|ω| enters in the equations, we can, without loss of generality, consider only the non-negative
m cases provided we allow for negative σ.
The condition Ω2 − ω2 > 0 implies that
−m− 1 < σ/Ω(r) < −m+ 1 . (36)
Therefore, g–modes are trapped between ri and r+ = (m+1)
2/3|σ|−2/3. The inner boundary
of the mode (r−) is ri, provided that |σ| ≥ (m− 1)Ω(r). The negative sign of σ reflects the
fact that the oscillations’ wavefronts travel in the same direction as the fluid.
Table 4 and Figure 1 show some properties of the n = 0, J = 1 modes for the m = 0, 1
and 2 cases. Interestingly, the results are consistent with the observation of a 1:2:3 harmonic
structure in the power spectra of the DNO VW Hyi, reported by Warner & Woudt (2005).
The observed peaks remain harmonic even during a threefold increase in the fundamental
frequency. As in other models, the increase in frequency is due to a decrease in ri induced
by an increase in Ṁ (and thus L).
If one expresses σ in terms of r+ and works to lowest order in ∆r/ri, from equation (28)
one obtains
∆r/ri ∼ (n± 1/4)2/3(m+ 1)−1/3ǫ2/3 . (37)
– 13 –
Since ǫ ≡ h(ri)/ri ∼ (cs/Ωr)ri, using equation (29) and our chosen set of parameters (which
only weakly affect the estimate) gives ǫ ≈ 4.2 × 10−3 r1/8i ≈ 1.2× 10−2. Then equation (37)
is seen to agree approximately with the values in Tables 3 and 4.
The m > 0 diskoseismic modes are in principle harder to observe than the axisymmetric
modes, since in the former the modulation of the outgoing radiation is not as efficient due to
cancelations over the entire disk (and their radial extent is somewhat smaller). However, the
fact that the white dwarf can eclipse parts of the inner disk renders these modes more visible.
We predict that the energy spectrum of the DNOs is characteristic of the very inner disk
(and not other regions such as the magnetosphere), and thus should be close to blackbody
at the highest disk temperature.
As in the case of g–modes in relativistic accretion disks, the eigenfrequency splittings
when m changes by unity are much larger than the n and j splittings. The g–modes that
are trapped by the general relativistic behavior of the radial epicyclic frequency κ have
m splittings that are not as harmonic as those found here (Table 4). In particular, their
eigenfrequencies are only proportional to m for m & 2 [an interesting difference between the
relativistic and the nonrelativistic eigenfrequencies is that, for m ≫ 1, |σ| → mΩ(ri) for the
former (see RD1) while |σ| → (m+ 1)Ω(ri) for the latter]. However, it must be noted that
it appears that g–modes with small mode numbers are viscously unstable, at least if the
viscosity acts in the usual hydrodynamic way. This result is proven for the relativistic case
in Ortega-Rodŕıguez & Wagoner (2000), where the exponential growth rate of the mode
is roughly αΩ. For the nonrelativistic g–modes studied in this paper, an application of
their method yields the same result. Furthermore, these low–lying modes do not contain a
corotation resonance [where ω(r) = 0].
6. Comments
Thus, if the ideas in this paper are correct and DNOs are linear diskoseismic modes
then the results could suggest a scenario in which white dwarf DNOs and black hole QPOs
are produced through somewhat different physical mechanisms. After all, as Kluźniak et al.
(2005) themselves point out, the phenomenology is similar but not identical. While in
white dwarf DNOs the simultaneous observation of two or even three of the harmonics is
well established, in black holes usually only one mode of oscillation is observed at a given
moment. Furthermore, white dwarf DNOs (and neutron star QPOs) have frequencies that
vary with time, while QPOs in black holes have fixed frequencies. In neutron star XRBs, the
(twin) high–frequency QPOs are analogous to the DNOs, but their frequency ratio is much
closer to unity (van der Klis 2006). It appears to be controlled (magnetically) by the neutron
– 14 –
star, since their frequency difference is close to the neutron star spin frequency or half of it.
We note that some DNOs are split at a QPO frequency. We also note that with increasing
radius of a neutron star magnetosphere, their g-modes should approach (for ri & 8) those of
this paper.
We hope to investigate the magnetic boundary layer in more detail in a future paper.
How much will the boundary layer contribute to the damping (or growth) of the g–modes?
Further work should try to explain why the value of Q is so high. Naively, one would expect
Q ∼ 1/α. (It is fortunate that most of our results are fairly insensitive to the value of α.)
The nonlinear growth and coupling of the modes must also be investigated.
This work was supported by grant 075-2002 (Incentivos) of Ministerio de Ciencia y
Tecnoloǵıa, Costa Rica, grant 829-A3-078 of Vicerrectoŕıa de Investigación, Universidad de
Costa Rica, and NASA grant NAS 8-39225 to Gravity Probe B. We are also grateful to the
Aspen Center for Physics for its support.
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This preprint was prepared with the AAS LATEX macros v5.2.
– 16 –
Table 1. Value of J as a function of j and parity
j parity J
0 even 1/2
0 odd 1
1 even 3/2
1 odd 2
– 17 –
Table 2. Values of the eigenfrequency σ × 106 for m = 0 g–modes.
n 1 3/2 2
0 2.65 2.73 2.75
1 2.39 2.55 2.61
2 2.23 2.43 2.51
Table 3. Values of the modes’ fractional radial width ∆r/ri for m = 0 g–modes
n 1 3/2 2
0 0.044 0.024 0.019
1 0.12 0.072 0.055
2 0.17 0.11 0.083
Table 4. Harmonic structure of the J = 1, n = 0 g–modes
m −106 σm σm/σ0 ∆r/ri
0 2.65 1 0.044
1 5.39 2.03 0.033
2 8.14 3.07 0.028
– 18 –
1 1.05 1.1 1.15 1.2
m = 0 
m = 1
m = 2
Fig. 1.— The horizontal lines show the radial extent and the frequency of the three lowest
fundamental (n = 0, J = 1) axial g-modes.
	Introduction
	Diskoseismology
	Diskoseismic Modes for the White Dwarf Case: Physical Parameters and Scope
	Formal WKB Solution of the Eigenvalue Problem for G–modes
	WKB Solution of the Radial Eigenvalue Problem
	WKB Solution of the Vertical Eigenvalue Problem
	Uncoupling
	Results and Discussion
	Axisymmetric Modes
	Non-Axisymmetric Modes
	Comments
