A&A 392, 353368 (2002)
DOI: 10.1051/00046361:20020912
Nonradial instabilities of isothermal Bondi accretion
with a
shock: Vorticalacoustic cycle vs. postshock acceleration
T. Foglizzo
Service d'Astrophysique, CEA/DSM/DAPNIA, CEASaclay, 91191
GifsurYvette, France
Received 17 May 2002 / Accepted 12 June 2002
Abstract
The linear stability of isothermal Bondi accretion with a shock is
studied analytically in the asymptotic limit of high incident Mach number
.
The flow is unstable with respect to radial perturbations as
expected by Nakayama (1993), due to postshock acceleration.
Its growthtime scales like the advection time from the shock
to the sonic point
.
The growth rate of nonradial perturbations
l=1 is higher by a factor
,
and is therefore intermediate
between the advection and acoustic frequencies.
Besides these instabilities based on postshock acceleration, our
study revealed another generic mechanism based on the cycle of acoustic and
vortical perturbations between the shock and the sonic radius,
independently of the sign of postshock acceleration.
The vorticalacoustic instability is fundamentally nonradial. It
is fed by the efficient excitation of vorticity waves by the isothermal
shock perturbed by acoustic waves. The growth rate exceeds the advection
frequency by a factor
.
Unstable modes cover a wide range of frequencies from the
fundamental acoustic frequency
up to a cutoff
associated with the sonic radius. The highest growth rate is reached
for l=1 modes near the cutoff. The additional cycle of acoustic waves
between the shock and the sonic radius is responsible for variations of the
growth rate by a factor up to 3 depending on its phase relative to the
vorticalacoustic cycle.
The instability also exists, with a similar growth rate, below the fundamental
acoustic frequency down to the advection frequency, as vorticity waves are
efficiently coupled to the region of pseudosound.
These results open new perspectives to address the stability of
shocked accretion flows.
Key words: accretion, accretion disks  hydrodynamics 
instabilities  shock waves  stars binaries: close  Xrays: stars
Hydrodynamic instabilities in accretion flows can help our understanding of
the variability observed in the luminosity of Xray binaries.
Numerical simulations have revealed the existence of such a
hydrodynamical instability in the accretion flow of a gas on a compact
accretor moving at supersonic velocity (BondiHoyleLyttleton
accretion). A first step towards understanding
the physical mechanism underlying this instability was made by Foglizzo
& Tagger (2000, hereafter FT00) who recognized the unstable cycle of
entropic and acoustic perturbations between the shock and the sonic surface.
This cycle is unstable if there is a large enough temperature difference
between the shock and the sonic surface. The academic case of shocked
Bondi accretion was studied by Foglizzo (2001, hereafter F01) who revealed the
importance of non radial perturbations, and vorticity in particular.
Both vorticity and entropy perturbations are advected from the shock
to the accretor, and both are coupled to the acoustic perturbations.
This coupling was formulated in a compact way by Howe (1975).
If the adiabatic index is in the range
,
entropy and
vorticity perturbations are intimately related in the shocked Bondi
flow (Foglizzo 2002, in preparation), so that it is difficult to identify their
respective roles in the instability mechanism. By contrast, their roles
are well separated in the isothermal limit (), where entropy
perturbations are absent from the problem. The present paper is
therefore dedicated to the study of the linear stability of shocked
accretion in the isothermal Bondi flow, where the incident Mach number is the
only parameter. The stability of shocked isothermal flows was studied
by Nakayama (1992, 1993) in the more general context of flows with
small angular momentum. The study of Nakayama was restricted to
axisymmetric perturbations, thus precluding any possible vortical
acoustic cycle. In this approximation, Nakayama analytically obtained
the result that the flow is unstable if the flow accelerates immediately
after the shock surface. In this respect the shocked Bondi flow should
be unstable. We are therefore interested here in an
extension of Nakayama's results to the case of non radial
perturbations, in order to include the effect of vorticity.
It should be noted that isothermal flows are very particular as far as
BondiHoyleLyttleton (hereafter BHL) accretion is concerned, with unsettled
issues concerning the influence of the numerical resolution.
The instability observed in numerical simulations seems very weak in 3D
according to Ruffert (1996), whereas it is violent in 2D (Ishii et al.1993;
Shima et al.1998). Pogorelov et al.(2000) discussed the possible responsability
of the numerical procedure in producing the instability. With a
different approach, the instability of the shocked Bondi flow described in
this paper could contribute to guide our physical understanding of more
complex flows involving isothermal shocks.
The paper is organized as follows. Perturbed equations are described in
Sect. 2 and eigenfrequencies are determined numerically in
Sect. 3.
Analytical methods are used to disentangle the effect of advection from the
effect of the boundary. In the spirit of F01, the
coupling between the vorticity and acoustic perturbations in the classical
isothermal Bondi flow, without a shock, is described in
Sect. 4. Boundary conditions are
taken into account to build a vortical acousticcycle in
Sects. 5 and 6. The instability due to postshock
acceleration is analysed in Sect. 7.
The relationship between the vorticalacoustic instability and existing
numerical simulations of BHL accretion and shocked discs is discussed in
Sect. 8.
2 Linearized equations of the shocked Bondi flow
2.1 Properties of the unperturbed shocked flow

Figure 1:
Radial profile of the Mach number in the unperturbed
shocked flow, for M_{1}=5 (full line). The subsonic cavity stands between
the sonic point and the shock radius. The dashed lines show the analytical
continuation of the solutions beyond the shock radius. 
Open with DEXTER 
The Mach number in the unperturbed transonic flow satisfies the following
equation deduced from the conservation of the mass flux and Bernoulli
constant B and the regularity at the sonic radius:

(1) 
The sonic radius is half of the Bondi radius GM/c^{2}. The
Bernoulli constant,

(2) 
is conserved along the flow lines. By contrast with the
adiabatic case ,
this quantity is not conserved through a shock.
The shock radius
corresponding to an incident Mach number
is determined from Eq. (1) together
with the Rankine Hugoniot jump condition
.
The Mach number profile is shown in Fig. 1. It should be
noted that the presence of a radial shock is not guaranteed a priori,
since the supersonic preshock flow (
)
could be continued without a
shock down to the accretor (dashed line in Fig. 1).
Several physical mechanisms could trigger the formation of a shock, such as the
heating of protons to temperatures at which the fluid becomes collisionless
(Mezaros & Ostriker 1983), the trapping of relativistic particules
(Protheroe & Kazanas 1983) or the dissipation of magnetic fields
(McCrea 1956; Scharlemann 1981).
In the context of BondiHoyleLyttleton accretion, the existence of a
stagnation point behind the accretor implies that a fraction of the supersonic
flow decelerates to subsonic velocities, and therefore naturally produces
a shock. The shock radius deduced from Eq. (1)
increases with
for spherical accretion:

(3) 
Compared to BHL accretion, the presence of an accretion shock at a distance
exceeding the accretion radius
is rather artificial and
due simply to the assumption of purely radial velocity. In the highly
supersonic limit
initially considered by Hoyle & Lyttleton (1939),
any trajectory with an angular momentum larger than 2
would
of course miss the accretor. Bearing this feature in mind, the case of strong
shocks studied in this paper is still very useful in order to understand the
mechanisms involved, because of the separation of timescales it enables.
2.2 Scaling of timescales in the shocked Bondi
flow
If the shock is strong, the timescale
associated with
advection is much longer than the sound crossing time
:

(4) 
The third fundamental timescale of the flow is related to the presence
of the sonic radius
.
Together with Eq. (3), the scaling of the three special
timescales of the problem is thus the following for strong shocks:

(5) 

Figure 2:
Schematic structure of the perturbed acoustic (full line) and
vorticity (dotted line) fields, depending on the frequency of the perturbation.
The curve beyond the shock radius shows the acousticperturbation in the Bondi
flow without a shock. The turning point
separates the
region of pseudosound from the region of propagation of acoustic
pertrbations. The vorticalacoustic coupling is most efficient in the
region between
and
. 
Open with DEXTER 
As in the case
studied by F01, the
vorticity equation can be directly integrated, so that the three quantities
(
)
are
conserved when advected.
The perturbations of the radial velocity
and density
in the isothermal Bondi flow are conveniently described by
the functions f, g defined by:
They satisfy the same differential equations as the functions f, g in
the Appendix B of F01, by suppressing the entropy terms:
where the function
is defined by:
The constant
is related to vorticity as follows

(12) 
By contrast with the case of adiabatic flows (F01),
is the unique
source term for the excitation of acoustic waves.
For radial perturbations (
), the equations
analysed by Nakayama (1992, 1993), with
,
are
recovered.
The position of the turning point
of nonradial acoustic
waves is defined by
in Eq. (10). This turning point
lies inside the subsonic cavity if
,
with

(13) 
Below
(or inside the turning point
if it
exists), the acoustic perturbation is not propagating. It was named
"pseudosound'' by Ffowcs Williams (1969).
The threshold between absorbed and trapped sound was introduced in F01
through the cutoff frequency
,
which is independent of the
shock strength:

(14) 
This cutoff frequency used to be
refered to as a "refraction'' cutoff in F01 for ,
due to the
nonhomogeneity of the sound speed which bends trajectories outwards.
By contrast, in the isothermal flow
considered here, acoustic trajectories are only bent inwards, due to the
sole effect of flow acceleration. The existence of a cutoff for
nonradial acoustic waves can be
understood by considering the frequency dependence of the direction
of propagation of the incoming acoustic flux relative to the radial
direction. This angle is approximated in Appendix E in the WKB approximation
(Eq. (E.11)). The higher the frequency, the smaller ,
so that
high frequency waves with a given order l are pointing towards the
accretor and are absorbed inside the sonic sphere.
The scaling of frequencies stressed in Eq. (5) for strong shocks
consequently separates three ranges of eigenmodes schematized
in Fig. 2:
(i) absorbed sound
,
(ii) trapped sound
,
(iii) pseudosound
.
The corresponding structure of the vorticity wave is also displayed in
Fig. 2, in each range of frequencies.
The boundary conditions at the shock associated with the variables
f,g can be computed with the same method as Nakayama (1992), extended to
the case of non radial perturbations.
Let us denote by
the radial displacement of the shock produced
by a sound wave propagating against the stream in the subsonic region, and
its velocity. The incident Mach number
in the frame of the shock is altered by the perturbations
of velocity and displacement as folllows:
where the parameter
measures the strength of the local
gradient of the Mach number immediately after the shock:

(17) 
The boundary conditions
,
written as functions of
,
are deduced from the conservation of the mass flux and impulsion
across the shock:
The non radial perturbation of the velocity, and the perturbed
vorticity computed in Appendix A enable us to relate
to :

(20) 
Equation (20) can be used together with
Eqs. (18)(19) in order to express
as functions of
for
nonradial perturbations.
In Appendix C the boundary value problem is reduced to a single
equation incorporating the boundary conditions both at the shock and
at the sonic point. The eigenfrequencies satisfy the following
equation, where f_{0} is the unique homogeneous solution which is
regular at the sonic point:



(21) 
Equation (22) is independent of the normalization of f_{0}.
It should be noted that the integral on the right hand side is
well defined despite the singularity of the phase near the sonic radius.
This equation describes the perturbation of the shock by the interplay of the
acoustic and vortical perturbations. The same calculation in any other
potential (e.g.the PaczynskiWiita potential) would lead to the same
system of Eqs. (21)(22), only the shape of
(and its derivative )
described by Eq. (1) would be affected.
The left hand side of Eq. (22) involves only the acoustic
perturbation f_{0}, and is independent of the vortical perturbation.
By contrast, the integral on the right hand side describes the acoustic feed back of the vortical
perturbation coupled to the acoustic field.
3 Spectrum of eigenfrequencies
The regularity of the solution at the sonic radius has already been discussed
in F01: the sonic point is a regular singularity if
.
For a given incident Mach number
,
numerical intergration
is performed from
the sonic point towards the shock in order to determine the eigenfrequencies
of perturbations with a latitudinal number l. For a
given value of ,
the unique solution which is regular at the
sonic point is expanded in a Frobenius series in order to start the
numerical integration away from the singularity. A RungeKutta
algorithm is then used to simultaneously integrate four functions,
namely
,
,
the integral on the right hand side of
Eq. (22) and the integral phase inside it.
is
varied and shooting is repeated until Eq. (22) is satisfied.

Figure 3:
Growth rate of the purely growing radial instability,
multiplied by the advection time (thick dashed line). The thin dashed lines
correspond to the lower and upper bounds determined by Nakayama (1993).
Anticipating on the calculations of the rest of the paper,
the four other curves are analytical estimates of the growth rate of
l=1 perturbations at high Mach number. The growth rate of the
entropicacoustic instability increases along the dotted line at low
frequency(Eq. (74)), and is bounded by the two thin lines
at high frequency (Eqs. (67)(68)). The thick full line describes the
most unstable branch (Eq. (85)). 
Open with DEXTER 
The radial instability found numerically in Fig. 3 is consistent
with the result of Nakayama (1992, 1993), who found that the local acceleration
of the flow immediately after the shock is a source of nonoscillating radial
instability.
The growth rate is shown as a function of the incident Mach number,
in units of the advection time
.
The lower and upper bounds found analytically by Nakayama (1993, Eq. (27))
are also displayed in Fig. 3:

(23) 
where the parameter
describes the acceleration of the flow and is
proportionnal to :
Note that Eq. (24) is corrected for a factor of 2 for spherical flows,
as noticed by Nakayama (1993). The lower and upper bounds of
Eq. (23) cover a wide range of frequencies from the advection
frequency
up to the acoustic frequency
.
In the present case of isothermal radial accretion, the growth rate computed
numerically converges precisely towards the value
.

Figure 4:
Real and imaginary parts of the eigenfrequencies of the modes
l=0,1,2,3 in the isothermal flow, for
(
). The real part is measured in units of
the cutoff frequency
(lower axis) and advection frequency
(upper axis). The imaginary part is normalized to the advection time.
Analytical estimates of the growth rates of the modes l=0,1,2 at
low freqency are indicated using big symbols (Eqs. (78), (83) and (85) in Sects. 6 and 7). 
Open with DEXTER 

Figure 5:
Real and imaginary parts of the eigenfrequencies of the
modes l=1 in the isothermal flow, for different Mach numbers.
Frequencies are multplied by the advection time. The strong l=1
instability due to postshock acceleration corresponds to the thick
line in both plots. In the upper plot, the growth
rate of the fastest of the mode l=0 is also displayed (thick dashed line).
The analytical approximation of the fastest growth rates
of the l=1instabilities due to the entropicacoustic cycle and to
postshock acceleration are shown as the thin dashed line (Eqs. (68))
and the thin dotted line Eq. (85).
In the bottom plot, the cutoff
frequency
,
the minimum acoustic frequency
,
and the frequency
of the most unstable mode
(Eq. (84)) are indicated as references. 
Open with DEXTER 
The effect of non radial perturbations ()
involves an integral on the right hand side of Eq. (22), which
reflects the cumulative excitation of acoustic waves by the vorticity
perturbations advected from the shock to the sonic radius.
Figure 4 shows the eigenspectrum of the isothermal Bondi flow for
M_{1}=100, l=0,1,2,3. This spectrum is characterized as follows:
(i) The most unstable mode is asymmetric (l=1)
and well separated from the other eigenmodes. The growth rate of this low
frequency mode is significantly larger than the advection time, nearly four times
faster than the unstable radial mode. The analytical estimates of the growth
rates at low frequency correspond to Eqs. (77)(78),
(81)(83) and (84)(85)
in Sects. 6 and 7.
(ii) Non radial unstable modes are very numerous and cover a wide range of
frequencies, from the advection frequency (
), up to the
cutoff frequency (
).
(iii) A striking feature of the eigenspectrum in Fig. 4 is
the apparent
oscillation of the imaginary part as a function of the real part of the frequency.
This behaviour is explained in Sect. 5 on the basis of the
vorticalacoustic cycle.
The variation of the growth rate with the incident
Mach number is shown in Fig. 5. The higher the Mach number,
the more unstable the non radial modes, while the radial instability
saturates. The growth rate of the most unstable l=1 mode increases
with the incident Mach number much faster than in the other modes.
The real part of its complex frequency is intermediate
between the advection and acoustic frequencies. It is
computed analytically in Sect. 7.
The rest of the paper is dedicated to understanding this
apparently complicated spectrum, and to questioning the role of postshock
acceleration.
4 The sound of vorticity

Figure 6:
Efficiency
of the excitation of
acoustic waves propagating against the stream by the advection of a vortex in an
isothermal flow, for l=1,2,3. The frequency is in units of the
cutoff frequency
.
The dotted lines correspond to the
asymptotic behaviour
at
low frequency determined analytically in Eq. (30). 
Open with DEXTER 
Let us assume that a vorticity wave is
advected from infinity towards the accretor: what is the sound
produced by the advection of this vorticity? This question is solved
in Appendix D by computing the coupling coefficient
measuring the outgoing acoustic flux F_{}
associated with a vorticity perturbation :

(26) 
The relationship between the acoustic flux
and the acoustic
perturbation
is deduced from the WKB approximation in Appendix B:

(27) 
Thus the complex efficiency
is defined at a
radius
,
in the region of propagation of acoustic waves,
as follows:
is approximately independent of R in the WKB region
of propagation of acoustic waves, i.e.away from their turning point.
This coefficient is approximated in Appendix D in the low frequency limit
,
i.e.below the cutoff frequency, using the
function
defined from an integral of the Spherical Bessel
function j_{l} (Eq. (D.7)):

(30) 
with
,
and
.
The direct integration of Eq. (29) is compared to the
approximation obtained in Eq. (30) for l=1,2,3 in
Fig. 6. The asymptotic behaviour
is correctly reproduced. The scaling factor of this power law
obtained analytically is too low by a factor 2 for l=1 perturbations,
which can be attributed to the roughness of the approximation.
This calculation clearly indicates that the vorticalacoustic coupling is
most efficient near the cutoff frequency. The same conclusion
was reached in F01 concerning the efficiency of the entropicacoustic
coupling.
4.2 Region of coupling
This approximation enables us to determine the region where the coupling
between the vorticity and the acoustic perturbations is most efficient.
It occurs mainly in the
subsonic region
where the wavelength of the vorticity
perturbation is largest (see Fig. 2):

(31) 
A similar result was obtained in F01 concerning entropy perturbations for
.
Due to the differences in velocity profiles in
Eq. (31),
for
at high frequency
(Eqs. (E.6), (E.11) and (E.12) of F01), whereas

(32) 
in the isothermal flow at low frequency. It is therefore important to note
that the most efficient coupling at low frequency lies in the region of
pseudosound well within the turning point (
)
of
acoustic waves:

(33) 
According to Eq. (31), the upper bound
of the
coupling region for perturbations near the advection frequency coincides with
the shock radius.
5 Vorticalacoustic instability at high frequency
The formalism developed in FT00 for the entropicacoustic cycle within the
WKB approximation can be transposed to the case of the
vorticalacoustic cycle. This formalism requires the frequency to be high
enough so that acoustic waves can be identified and their propagation time
measured. An initial perturbation of vorticity
at the
shock, advected towards the accretor, triggers the excitation of an
acoustic flux F_{} propagating outwards. As it reaches the shock
surface, this acoustic flux induces new vorticity perturbations
,
where
is the duration of
this cycle. The global efficiency
of the cycle is naturally

(34) 
and
depend a priori on the frequency
and spatial structure l of the perturbation considered.
The growth (or damping) rate of the cycle is identified with the imaginary
part of the eigenfrequency, given by:

(35) 
The timescale
is dominated by the
advection timescale
if the shock is strong
(
,
):

(36) 
The global efficiency
can be decomposed
in terms of the efficiencies of the coupling between
F_{} and
during advection (
)
and at the shock (
), defined immediately after
the shock according to Eq. (28) and:
 (37) 
 (38) 
The modulus of
is directly interpreted as a coupling efficiency
between F_{} and
:

(39) 
The efficiency
is computed in Appendix E,
following a method introduced by D'Iakov (1958) and Kontorovich (1958, 1959).
can be approximated
by a WKB analysis when the wavelength of the perturbation is small
compared to the lengthscale of the local gradients of the flow, by
keeping only the first order terms in
and
,
at high frequency and for strong shocks:

(40) 
This calculation proves that the flow reacceleration does not affect
the vorticalacoustic coupling at the shock for

(41) 
and that
is approximately independent of
frequency in the range
.
As a consequence of Eqs. (37) and (40), the
vorticity is produced at the shock with an efficiency proportional to
the shock strength:

(42) 
The global efficiency
should consequently peak near the
cutoff frequency, and drop abruptly above it. The frequency dependence
of
appears in Fig. 8 (top picture, full line),
as deduced from Fig. 6 and Eq. (40) in the WKB
approximation. Thus the instability should be strongest near the
cutoff frequency with the following scaling, obtained from
Eqs. (14), (30) and (40) for
:

(43) 
where the scaling factor
depends on l through both
and
:

(44) 
diverges for ,
thus favouring the instability
of low degree modes. According to Fig. 6, the
extrapolation of Eq. (43) to
gives a very
rough upper bound of
,
particularly
overestimlating it for high degree perturbations. Although
exceeds
and
,
the actual
efficiency
is maximal for the mode l=1, as checked in
Fig. 8 by multiplying the curve
(Fig. 6) by
(Eq. (40)).
Using Eqs. (35) and (36), the growth rate
at high frequency is deduced from Eq. (43):



(45) 
Equation (47) indicates that the range of unstable frequencies
below the cutoff frequency gets narrower for .
This is also confirmed by the eigenspectrum obtained numerically for
l=1,2,3 in Fig. 4. Note that the efficiency
was estimated using
which was computed for
perturbations with a purely real frequency. Thus the estimate of the
growth rate in Eq. (46) implicitly assumes that the imaginary
part is small compared to the real part. This is true since
.
This calculation proves that the vorticalacoustic cycle is unstable
for strong shocks, with a growth rate exceeding the growth rate of
the radial mode, and a mechanism which is independent of the
sign of the local flow acceleration immediately after the shock.
A more refined description of the entropicacoustic cycle is
developped in the next section
in order to understand the apparent oscillations in the
spectrum obtained numerically in Figs. 4 and 5.
5.2.1 Dispersion relation for the double cycle
The role of the purely acoustic cycle had been anticipated in FT00.
It is characterized by a time scale
and a global efficiency
.
Let us show how the simultaneous existence
of the two cycles can explain the apparent oscillation in the
eigenspectrum of Fig. 4. The following extension of the analysis
of FT00 is valid for both entropicacoustic and vorticalacoustic cycles,
by replacing "vortical'' by "entropic''.
A perturbation f is influenced by the two cycles as follows:

(48) 
A solution of the form
satisfies
Eq. (48) if the complex eigenfrequency (
)
is a solution of the following dispersion equation (Eq. (25) of FT00):

(49) 
This dispersion relation is recovered in Appendix F as a WKB
approximation of the exact dispersion relation (22).
The analysis of the dispersion relation (49) in Appendix G
enables us to extract physical information from the complicated
eigenspectrum in Fig. 4, such as the ratio of timescales
,
and the dimensionless efficiencies
and
.

Figure 7:
Number of eigenmodes in each oscillation of the
eigenspectrum in Fig. 4, identified as the ratio of the
timescales of the two cycles. For strong shocks,
. 
Open with DEXTER 
The timescale
of the acoustic cycle for strong shocks
can be written approximately:

(50) 
According to Eqs. (36) and (50), the ratio
for a strong shock is simply:

(51) 
As shown in Appendix G, this ratio is the average
number of eigenmodes in each oscillation of the eigenspectrum.
As an illustration, this number is reported in Fig. 7 for
the modes l=1,2,3 in the case
corresponding to Fig. 4,
in good agreement with Eq. (51).

Figure 8:
Global efficiencies
,
asociated to non radial
perturbations l=1,2,3 deduced from the eigenspectrum in Fig. 4
in the framework of the double cycle model (dotted line), compared
with the efficiencies deduced from the calculation of
and
. 
Open with DEXTER 
As noticed in FT00, the acoustic cycle can be neglected (
)
if the efficiencies and timescales
of the two cycles are such that
,
with

(52) 
If
is not negligible, the acoustic cycle can contribute to either
stabilize or destabilize the vorticalacoustic cycle. The most stabilizing effect of the acoustic cycle
is the effective reduction of
by a factor 2 (see Appendix G).
By contrast, its destabilizing contribution can be much larger
comparatively, if the acoustic time is short compared to the advection time.
This is the case for strong shocks. Depending on the
relative phases of the two cycles, the growth rate can then cover the following
range:

(53) 
Conversely, the values of the dimensionless physical parameters
can be extracted from the eigenspectrum,
as in Fig. 4, by measuring
(i) the extremal values
,
(ii) the period
of the oscillations
(iii) the average number n of modes per period.
As demonstrated in Appendix G, the timescale of
the acoustic cycle is simply
.
Thus the oscillations are well resolved in the eigenspectrum only if the
advection time is significantly longer than the acoustic time
(i.e.
).
The values of
are determined using the following
relations:
where
 (56) 
 (57) 
As an illustration, Fig. 8 is obtained
from the eigenspectrum of Fig. 4 using Eqs. (54)
and (55). The accuracy of the measurements of
and
is of course of the order 1/n.
As a check of consistency, the value of
obtained
from the product of
(Fig. 6) and
(Eq. (40)) is also displayed,
showing an excellent agreement except for the lowest frequency modes
where WKB approximation breaks down. It is remarkable that
seems
to be maximized at low frequency for l=1 whereas it is maximized near the
cutoff frequency
for .
The efficiency
of the acoustic cycle is bounded by
one and decreases to zero above the cutoff frequency, as could be expected
from FT00 and F01.
The global efficiency
can be decomposed
in terms of the efficiencies
and
of the coupling between the acoustic
fluxes ,
such that the
global efficiency
is

(58) 
The efficiency
measures the outgoing acoustic flux F_{} produced by the
deviation of an ingoing acoustic flux F_{+}:



(61) 
is simply the limit when
of the
efficiency
computed in F01: it is close to unity below the cutoff frequency
,
and decreases exponentially above this cutoff.
Conversely, an acoustic flux F_{} reaching a shock
produces a reflected ingoing acoustic flux F_{+} with the efficiency
:



(62) 



(63) 
Keeping only the first order terms in
and
,
the complex
efficiency is computed in Appendix E:

(64) 
Thus the global efficiency
should be close to unity below
the cutoff frequency. The product
is displayed in Fig. 8 (bottom picture), in good agreement
with the value of
deduced from Fig. 4 and Eq. (55).
The extremal values
of the growth rate of the
vorticalacoustic cycle are computed
in Appendix G (Eq. (G.7)) for strong shocks:

(65) 
The asymptotic scaling of
is deduced from
Eqs. (43), (51) and (64) :

(66) 
Together with Eq. (36), and despite the roughness of the
approximation in Eq. (30) near the cutoff frequency, the leading
order of the asymptotic values of
is
comparable to:
The values of
are thus responsible for a dispersion of the growth rate by a factor 3in Eqs. (67)(68) near the cutoff frequency, depending on
the relative phases of the two cycles. This factor 3 is consistent
with both Figs. 4 and 5.
6 Vorticalacoustic instability at low frequency
The region of efficient coupling between the vorticity and acoustic
perturbations lies in the region of pseudosound according to
Eq. (33). Thus it seems natural to expect an instability at
low frequency as long as the coupling region determined from
Eq. (32) lies inside the shock radius, i.e.down to the
advection frequency.
This case is similar to the hole tone instability (e.g.whistling kettle)
studied by Chanaud & Powell (1965), or in the oscillations of impinging
shear layers (see a review by Rockwell 1983).
In some of these cycles, the vorticity perturbations are coupled to the
acoustic field in the region of pseudosound. As stressed by Chanaud
& Powell (1965), this does not preclude the use of the term "acoustic feedback''
so that we may talk of a vorticalacoustic cycle even in this range of
frequencies.
Although the problem cannot be treated in the WKB
limit, the homogeneous solution can be approximated by a Spherical Bessel
function of the first kind in the region of pseudosound far from the sonic
point. The calculation in Appendix H shows that the case l=1 and
must be treated separately.
In the domain of pseudosound
,
Eq. (22) is reduced to:
The l=1 acoustic perturbation is approximated in Appendix H
as
,
and Eq. (69) is
transformed into

(70) 
A branch of solutions with
corresponds to a vorticalacoustic cycle between the shock and the sonic point:







(71) 







(72) 
The maximum growth rate of this branch of solution is reached for a
real frequency similar to the frequency of the local instability.



(75) 
This growth rate is compared to the results of numerical calculations
in Fig. 5. This growth rate exceeds the growth rate of
the vorticalacoustic instability at high frequency (Eq. (46)), but
is asymptotically smaller than the maximum growth rate
reached when the purely acoustic and vortical acoustic cycles are in phase
(Eq. (68)).
If ,
the acoustic feedback due to an isolated vortical perturbation near
the shock is negligible compared to the integral effect for strong shocks.
Eq. (69) is reduced to:



(76) 
The most unstable modes are therefore the modes l=2, with
These approximations are in excellent agreement with the numerical
result of Fig. 4.
By contrast with the case of high frequency perturbations,
it seems difficult to separate completely the effect of postshock
acceleration from the global vorticalacoustic cycle in this range of
frequencies.
7 Effects of postshock acceleration: A global instability
with a local criterion
The boundary condition (18) can be rewritten
for strong shocks as follows:

(79) 
This equation is similar to the argument of Nobuta & Hanawa (1994)
concerning the balance total pressures (thermal and dynamical)
on both sides of the shock. Rather than treating the shock surface
as a material surface pushed by the local pressures on both sides of it,
Eq. (79) is interpreted as follows: an excess of Bernoulli perturbation
f (which we may call "energy'') on the subsonic side of the shock is
associated with a displacement of the shock in the direction of increase of
the local Mach number. If
(Eq. (17)), an excess of
f produces an outward displacement of the shock. This statement alone is not
conclusive: instability occurs only if the flow is not able to evacuate this
excess of energy. Although the displacement of the shock indeed liberates
some potential energy locally, the instability depends
on the leakage of energy through the sonic radius (Nakayama 1993).
For radial perturbations (l=0), the only way of evacuating the
excess of energy is through acoustic perturbations.
Acoustic energy, however, is trapped in the subsonic region if the real
frequency of the mode is low enough. Thus a low frequency instability
is expected. Although the criterion of the instability is indeed local,
its growth rate depends on the spatial structure of
the acoustic perturbation from the sonic radius to the shock, as illustrated
by Eq. (22) for strong shocks:

(80) 
f_{0} is approximated as the acoustic perturbation of a uniform
medium in spherical coordinates (see Appendix B), using a Spherical Bessel
function
for
.
The solution of the dispersion relation (80)
at low frequency is a purely imaginary eigenfrequency:
The fact that the growth time scales like the advection time is not
obvious a priori.

Figure 9:
Interaction of the vortex with the shock. The contribution of a
vorticity perturbation
to the Bernoulli perturbation is
positive (
), and induces an increase of the local accretion
rate (g>0).Depending on the sign of the postshock acceleration, a
vorticity perturbation contributes to a local expansion ()
or
collapse ()
of the shock. 
Open with DEXTER 
Non radial perturbations generate vorticity, which contribute to the
energy balance in the subsonic part of the flow. Figure 9
illustrates the contribution of a vorticity perturbation to the
Bernoulli perturbation through the radial component of velocity
.
Thus the vorticity perturbation participates to
increase the local accretion rate (i.e.g>0) in the regions where the shock
moves inward (
). Comparing this statement with
deduced from Eq. (19), we
conclude that the vorticity contributes to the instability if .
A quantitative calculation of the growth rate requires taking into
account the acoustic perturbation in the region from the shock to the sonic
point. The fastest instability of the flow, well isolated from all the other
modes of instability, is obtained by neglecting the last term on the right
hand side of Eq. (70):
It is in excellent agreement with the numerical calculations in
Figs. 4 and 5.
This growth rate is intermediate between the advection and the
fundamental acoustic frequencies. The real and imaginary parts being
comparable, the growth rate is the time taken by the vorticity
perturbation to travel by one wavelength
.
8 Discussion
Most shock instabilities identified and studied in astrophysics are
related to the acceleration or deceleration of the shock itself. The
most famous is of course the RayleighTaylor instability of accelerated shocks,
analysed by Bernstein & Book (1978). Decelerated shocks are also
unstable to a rippling instability, studied by Vishniac (1983), Bertshinger
(1986) and Vishniac & Ryu (1989). By definition, stationary shocks are
stable with respect to these mechanisms, but can still be unstable.
Nakayama (1992, 1993) pointed out the radial instability of the shock if the
flow is immediately accelerated after the shock, in isothermal flows.
The validity of the postshock acceleration criterion for adiabatic flows
is still uncertain, even for radial perturbations (Nakayama 1994).
The vorticalacoustic cycle studied in the present paper resembles
by many aspects to the entropicacoustic cycle studied by FT00 and F01
in adiabatic flows. Two distinct mechanisms, however, are at work:
(i) the vorticalacoustic cycle is fed by the vorticity production
by a perturbed shock, which is highest for isothermal flows with a
strong shock,
(ii) the entropicacoustic cycle is fed by the temperature
increase from the shock to the sonic radius, which is highest if
(FT00, F01).
A calculation similar to Appendix E for adiabatic flows would show
that the vorticity production by a perturbed shock is large only in the
isothermal limit:

(86) 
Although shocked adiabatic flows with
are subject to both
entropicacoustic and vorticalacoustic cycles in principle, their stability
cannot be determined without a specific calculation.
The stability of BHL accretion can be addressed with new tools, using
the present results and the entropicacoustic cycle of FT00 and F01.
Although a detailed analysis of the existing numerical simulations of
BHL accretion is postponed to a forthcoming paper, let us outline the
possible consequences of the entropicacoustic and vorticalacoustic
mechanisms on this specific flow.
Numerical simulations of BHL accretion in 3D show a strong instability
for adiabatic flows with
and small accretors
(Ruffert & Arnett 1994; Ruffert 1994): this coincides nicely with the
properties of the entropicacoustic cycle,
which is most unstable if the temperature gradient in the subsonic part of
the flow is strongest (FT00, F01). By contrast, nearly isothermal flows are rather
stable in 3D numerical simulations (Ruffert 1996). This could seem puzzling
given the strong vorticalacoustic instability described in the present
paper: why would the entropicacoustic cycle be relevant for
the BHL instability with
,
and the
vorticalacoustic cycle be irrelevant to the BHL stability for ?
This apparent contradiction can be partly solved by remembering the
difference of topology between the spherical Bondi flow and the BHL
flow, following the remark of Sect. 2.1.
All the numerical simulations of isothermal BHL
accretion seem to agree with the fact that the shock is attached to the accretor,
whereas it is detached in high resolution simulations of adiabatic flows
with
.
This can be understood qualitatively in terms of
the weakness of pressure forces in isothermal flows compared to
adiabatic flows (
). More quantitatively,
Foglizzo & Ruffert (1997) proved that the shock in the isothermal BHL
flow cannot be detached unless the sonic surface extends up to distances
comparable to the Bondi radius GM/c^{2}, i.e.much larger than the
accretion radius. In view of this topological difference between
isothermal and adiabatic BHL flows, it seems easier to extend the
results obtained for shocked Bondi accretion to the subsonic region
ahead of the accretor, for detached shocks, than in the very
nonradial region of accretion behind the accretor for attached
shocks. Although true, however, this argument is not conclusive. Even in a non
radial isothermal flow, a small pressure perturbation of the shock is
able to generate vorticity perturbations very efficiently. The lack of
instability of isothermal BHL accretion in 3D must be sought for in the
lack of acoustic feedback from the advected vorticity perturbation.
Unfortunately we do not have quantitative arguments to explain why
the acoustic feedback is so weak, apart from noticing that geometric
compression of the vorticity perturbation from the shock to the sonic
radius might be insufficient in the BHL flow. Any coherent explanation
should of course also account for the strong instability observed in 2D
simulations of BHL accretion (Shima et al.1998).
Inviscid accretion flows with low angular momentum are much more
complex than the Bondi flow in the sense that the shock position is
not unique (Fukue 1987; Chakrabarti 1989).
The numerical simulations of Nobuta & Hanawa (1994) did not detect
the effects of vorticity perturbations since they were restricted
to axisymmetric motion. These simulations showed
the evolution of the radial instability due to postshock
acceleration: the unstable shock is either absorbed by the accretor
or inflates to reach the stable outer position.
Molteni et al. (1999) performed non axisymmetric numerical
simulations of shocked inviscid accretion flows with low angular momentum,
for
.
Unstable m=1 oscillations were observed, with no
physical explanation. The entropicacoustic and vorticalacoustic
cycles might provide a physical basis to understand this instability.
If the existence of the shock is postulated a priori, as in the
present study, the flow might simply converge towards another
solution, as illustrated by the dashed lines in Fig. 1. The shock
can either be absorbed by the accretor, leading to the fully supersonic
solution, or could expand towards infinity to establish the Bondi
solution if the outer boundary condition allows it.
In accretion flows stable with respect to the postshock acceleration
criterion, but unstable through the vorticalacoustic (or
entropicacoustic) cycle, the instability might be saturated by the effect of
the geometric dilution of the acoustic energy in the
subsonic cavity. Let us assume that advected vorticity perturbations
generate an acoustic flux propagating outward
(Eq. (26)), and that the acoustic perturbation produces in turn a
vorticity perturbation
(Eq. (42)). The amplitude of the pressure perturbation
depends on the volume in which the acoustic flux F_{}
is diluted through Eq. (27):
.
Thus the vorticalacoustic cycle is naturally
stabilized when the shock reaches a distance
defined by

(87) 
where
is the global efficiency of the vorticalacoustic
cycle in the linear regime. If the shock is not simply absorbed by
the accretor, the non linear evolution of the instability could
lead to quasi periodic oscillations of amplitude comparable to
.
However, given the number of unstable modes in a spectrum like
Fig. 4, the vorticalacoustic instability might as well
saturate into turbulence rather than be dominated by a single QPO.
This issue can only be solved with numerical simulations.
9 Conclusions
The linear stability of shocked isothermal Bondi accretion has been
studied by comparing the complex eigenfrequencies obtained through
a direct numerical integration (Sect. 3) to the analytical
results obtained for strong shocks by two different methods:
(i) an analytical estimate of the growth rate corresponding to
a cycle of perturbations with a purely real frequency, obtained
by separating the effects of advection from the boundary effects of
the shock. This WKB approximation, valid in the range
of acoustic waves (
), was used in
Sects. 4 and 5,
(ii) an analytical estimate of the complex eigenfrequencies in the range
of pseudosound (
)
using Spherical Bessel
functions (Sects. 6 and 7).
The results obtained by these methods are summarized as follows:
 As expected by the postshock acceleration of Nakayama (1993), the
isothermal Bondi accretion with a shock is unstable with respect to
radial perturbations. Its growth rate is comparable to the advection time
from the shock to the sonic point.
 The present analysis has revealed the existence of a new instability,
based on the cycle of vortical and acoustic perturbations in the subsonic
part of the flow.
The analytical study of this instability at high frequency
(
)
proves
that it is independent of the postshock acceleration criterion
established by Nakayama (1992, 1993). It is fed by the efficient
production of vorticity perturbations when the shock is perturbed non
radially (Eq. (42)), and by the
vorticalacoustic coupling in the region of the sonic radius which
enables the acoustic feedback. In this sense this non radial instability is
generic and can be expected in more complex situations such as
shocked flows with a weak angular momentum accreting into a black
hole, even if the flow is decelerated immediately after the
shock.
In the shocked Bondi flow the vorticalacoustic instability is faster than
the radial instability if the shock is strong, by a factor
.
The vorticalacoustic instability occurs for low degree perturbations
on a wide range of frequencies below the cutoff frequency
.
A branch of unstable l=1 eigenmodes corresponds to a
vorticalacoustic cycle in which the acoustic feedback is produced in the
pseudosound domain (
). The resulting growth
rate is comparable to the growth rate of the vorticalacoustic cycle at high
frequency.
The role of the purely acoustic cycle was pointed out at high
frequency in order to explain the large variations of the growth rate from
one mode to another. More generally, the formalism developped in Appendix G
concerning the simultaneous acoustic and vorticalacoustic cycles applies to
any context where the efficiencies
and timescales
can be defined.
 A strong l=1 oscillatory instability was found in the pseudosound domain,
at a frequency which is intermediate between the acoustic and advection
frequencies (
).
With comparable real and imaginary parts of the eigenfrequency, vorticity
perturbations are advected over a very short distance during one growth time.
On the basis of the contribution of a vortex to the Bernoulli perturbation
sketched in Fig. 9, this strong instability is a non radial
consequence of postshock acceleration.
 As outlined in Sect. 8, the vorticalacoustic
mechanism can be used as a tool in order to analyse the
instability observed in numerical simulations of more complicated accretion
flows. The specific application to BHL accretion or shocked flows with low
angular momentum will be developped in a forthcoming paper.
The non radial perturbation of velocity is deduced from the continuity
of the velocity parallel to the shock, as in Landau & Lifshitz (1987,
Chap. 90, p. 336):
The perturbed vorticity in the flow is deduced from the
Eqs. (A.1)(A.2) and the Euler equation:
w_{r}=0, 
(A.3) 
= 
(A.4) 
= 
(A.5) 
The conserved quantity
defined by Eq. (12)
is deduced from Eqs. (A.4)(A.5), resulting in Eq. (20).
If the wavelength of the perturbation is shorter than the lengthscale
of the flow inhomogeneity (
), a WKB approximation enables
us to describe the propagation of acoustic waves in the direction of the
flow (f_{0}^{+}) or against it (f_{0}^{+}):

(B.1) 
where we have chosen the normalization of
such that the
lower bound of the integral is the turning point .
The condition that
defines a minimum frequency
using Eq. (11) and
:

(B.2) 
Far from the accretor, the flow velocity decreases like
and the density of the gas is uniform. If the shock is strong
(
), the homogeneous equation associated with
Eq. (C.1) is approximated for
and
as follows:

(B.3) 
which is nothing more than the equation of acoustic waves in a uniform
steady medium, in spherical coordinates.
The solution f_{0} can therefore be approximated with a
spherical Bessel function of the first kind j_{l}, which is normalized
here as in Eq. (B.1):
The differential system (89) is written as a single
differential equation of second order:







(C.1) 
Following the method used in F01, the general solution of Eq. (C.1)
can be written using the solutions
f_{0},f_{1} of the homogeneous equation,
where f_{0} is the unique solution which is regular at the sonic radius.
Let us normalize f_{0} using the following definition of
the WKB solutions
(Eq. (B.1)).
The complex coefficient
is defined by:

(C.2) 
The wronskien of the couple of solutions
f_{0}^{+},f_{0}^{} is deduced
from Eqs. (B.1) and (C.1):

(C.3) 
Let us normalize f_{1} such that the wronskien of the couple of solutions
f_{0},f_{1} is the same as in Eq. (C.3). The general
solution is then:
f= 


(C.4) 
A Frobenius analysis of
f_{0},f_{1} near the sonic points leads to:
thus the integrals in Eq. (C.4) are converging when
if
.
The regularity at the sonic radius
therefore requires A_{1}=0.
A combination of Eq. (C.4) and its derivative at the shock
radius leads to eliminate A_{0} as follows:



(C.7) 
Equations (18)(20) provide the boundary
conditions at the shock in order to replace
,
and
in Eq. (C.7), resulting in Eq. (22).
The analytical expression for
can be determined
by writting the solution corresponding to zero acoustic flux F_{+}
at an outer boundary R, using the couple of homogeneous solutions (
f_{0}^{+},f_{0}^{}):
f= 

(D.1) 
The regularity at the sonic radius requires

(D.2) 
The condition of absence of an incoming acoustic flux at the outer
boundary is obtained by canceling the coefficient of f_{0}^{+} at
r=R, in the WKB limit of high frequency (
):

(D.3) 
Together with Eqs. (D.2) and (D.3), Eq. (D.1)
at the outer boundary becomes:
f(R)= 

(D.4) 
,
defined by Eq. (28), is deduced
from the asymptotic behaviour of f_{0}^{} in the WKB
approximation (Eq. (B.1)).
This calculation is formally similar to the calculation of
in F01,
corrected for a phase shift. The efficiency
involved
in the vorticalacoustic cycle is deduced from
Eq. (29), with
.
In the strong shock limit, below the cutoff frequency
(
), the acoustic efficiency
is
approximated using the Spherical Bessel function j_{l} (B.5):
For
,
a function
may be defined such that

(D.7) 
The main contribution to the integral in
Eq. (D.7) comes from the region
.
Equation (30) is obtained from Eqs. (D.5) and (D.7)
The perturbations f,g immediately after the shock, defined by
Eqs. (18) and (19) are decomposed as follows:

= 
f_{}+f_{+}+f_{K}, 
(E.1) 

= 
g_{}+g_{+}+g_{K}, 
(E.2) 
where
f_{K},g_{K} correspond to the vorticity wave
associated with the vorticity perturbation ,
and
correspond to the purely acoustic waves propagating
in the direction of the flow (index +) or against the flow
(index ).
An exact calculation can be made in the case of the reflexion
of an acoustic wave
with wavevector
(
)
on a plane shock in Cartesian
coordinates, in the absence of a gradient of .
The vorticity wave f_{K},g_{K} is advected at the velocity of the fluid:
Replacing these derivatives in Eqs. (8), (9), we obtain:
f_{K}= 
(E.5) 
g_{K}= 
(E.6) 

(E.7) 
Equations (E.5)(E.7) are used with
Eqs. (18), (19) in order to obtain Eq. (20) and
= 
(E.8) 
where
in Cartesian coordinates is also defined by
Eq. (10), but replacing
l(l+1)/r^{2} by the wavenumber
.
According to the definitions of
and
in Eqs. (37) and (62),
together with Eqs. (20) and (E.8):
These equations show the decrease of the vorticalacoustic
coupling for weak shocks (
), and the existence of maximal
efficiency at low frequency. Indeed, the maximum
is reached for a frequency such that
,
with
.
The angle
between
the direction of propagation of the wave and the vector orthogonal to the
shock surface is given by:

(E.11) 
As remarked by Kontorovich (1958), the reflected sound wave propagates away
from the shock with the same angle as the incident wave
(
), as in a classical reflexion.
The presence of a gradient of
in the Bondi flow precludes
the use of these formulae at low frequency. In the following
calculation we assume
and
,
and keep only
the first order terms in
and
.
Thus .
Neglecting the coupling between the vorticity and acoustic waves in
the vicinity of the shock, the vorticity wave f_{K},g_{K} is still advected
at the velocity of the fluid. Eqs. (E.5) and (E.6) are now
approximated at high frequency by
f_{K} 


(E.12) 
g_{K} 


(E.13) 
Acoustic waves are described by Eqs. (8), (9) in the
absence of vorticity perturbations, i.e.when
.
Using the WKB approximation of Eq. (B.1), the radial
derivative of
is approximated by:

(E.14) 
is deduced from Eqs. (8) and (E.14):

(E.15) 
Equations (E.12), (E.13) and (E.15) are used with
Eqs. (18), (19) in order to obtain Eq. (20) and
Combining Eq. (20) with Eq. (E.16), we obtain the
expressions for
in
Eqs. (40) and (64).
The pressure perturbations
associated with the acoustic waves
are deduced
from Eqs. (6)(7) and Eq. (E.15):
Using the integral expression of
(Eq. (29)),
Eq. (22) can be approximated as follows
for strong shocks (
):







(F.1) 
In the WKB limit
,
Eq. (F.1) can be simplified
into



(F.2) 
Using the expressions for
and
(Eqs. (40) and (64)), and the definition of
and
(Eqs. (38) and (58)), the global dispersion relation
(49) is recovered.
Let us introduce the new complex variable

(G.1) 
The resolution of the dispersion relation (49) is then equivalent to
finding the complex number z satisfying:

(G.2) 
where the dimensionless parameter
and
is defined by Eq. (52).
The minimum and maximum values
of z satisfy an equation
similar to Eq. (E3) of FT00:

(G.3) 
The minimum and maximum effect of the acoustic cycle on the growth
rate
are directly related to
through Eq. (G.1):

(G.4) 
From Eq. (G.3),
we deduce that the maximum stabilizing effect of the acoustic
cycle is to divide the efficiency
by a factor 2:

(G.6) 
If the acoustic time is much shorter than the advection time
(
), we deduce from Eq. (G.3) the values of :



(G.7) 
Thus the acoustic cycle may participate efficiently to the instability
if
and
.
In the more general case where
,
the following bounds on
x are obtained from Eq. (G.2):

(G.8) 
The resolution of Eq. (G.2) can be decomposed into a
phase condition applied to the points of a continuous curve ,
defined by:

(G.9) 
is a closed curve containing the point (1,0) in its
interior, and (0,0) in its exterior. It can be described in a univoque
way by the angle
.
Let us define the angle
.
The solutions of Eq. (G.2) are recovered by applying to the
solutions of Eq. (G.9) the following phase condition:

(G.10) 
where k,k' are two integers.
Since (0,0) is exterior to ,
the range of values covered
by
when
covers
is limited to
.
The discrete solutions of the phase Eq. (G.10) can be seen in a graphic way as the intersection of the
periodic curve
with the straight line
.
The number of solutions in each phase
is therefore equal to
on average.
Comparing Eq. (G.1), Eq. (G.9) with the definitions
of
,
one period of
corresponds to a
variation of
of
,
and one period of corresponds to
.
The values of
and
in Eqs. (54) and (55)
can be determined from the
measurement of
,
and the average number
of eigenmodes per period
,
by eliminating
from the set of Eqs. (G.3), (G.4), and (52).
The integral involved in Eq. (22) can be approximated in the low
frequency limit by introducing the complex variable z:

(H.1) 
The contour of integration is deformed in the complex plane
introducing a point on the real axis at ,
and performing two
integrations by parts:
The turning point in the Bondi flow would be at
.
For
,
the homogeneous solution f_{0} is
approximated by a Spherical Bessel function of the first kind
.
For
,
it is approximated as
follows:

(H.3) 
The first integral on the right hand side of Eq. (H.2)
is approximated using a Gamma function:
The second integral on the right hand side of Eq. (H.2) is
negligible if
.
In view of the particular case l=1, the differential equation
satisfied by f_{0} is used to sum up terms of same order:







(H.7) 
Using Eqs. (H.2), (H.6) and (H.7), the
dispersion relation (22) is approximated in the
pseudosound domain as follows:



(H.8) 
The left hand side of Eq. (H.8) is dominated
by the second derivative of f_{0} if :



(H.9) 
If l=1, the second derivative of f_{0} is approximated as follows:
 (H.10) 
 (H.11) 
Thus Eq. (H.8) becomes:



(H.12) 

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