A&A 368, 311324 (2001)
DOI: 10.1051/00046361:20000506
Entropicacoustic instability of shocked Bondi accretion
I.
What does perturbed Bondi accretion sound like?
T. Foglizzo
Service d'Astrophysique, CEA/DSM/DAPNIA, CESaclay, 91191
GifsurYvette, France
Received 17 October 2000 / Accepted 21 December 2000
Abstract
In the radial flow of gas into a black hole (i.e.Bondi accretion), the
infall of any entropy or vorticity perturbation produces acoustic waves
propagating outward. The dependence of this acoustic flux on the shape of
the perturbation is investigated in detail. This is the key process in the
mechanism of the entropicacoustic instability proposed by Foglizzo & Tagger
(2000) to explain the instability of BondiHoyleLyttleton accretion. These
acoustic waves create new entropy and vorticity perturbations when they
reach the shock, thus closing the entropicacoustic cycle. With an adiabatic
index
,
the linearized equations describing the perturbations
of the Bondi flow are studied analytically and solved numerically. The
fundamental frequency of this problem is the cutoff frequency of acoustic
refraction, below which ingoing acoustic waves are refracted out. This
cutoff is significantly smaller than the Keplerian
frequency at the sonic radius and depends on the latitudinal number l of
the perturbations.When advected adiabatically inward, entropy and
vorticity perturbations trigger acoustic waves propagating outward,
with an efficiency which is highest for non radial perturbations l=1.
The outgoing acoustic flux produced by
the advection of vorticity perturbations is always moderate and peaks at
rather low frequency. By contrast, the acoustic flux
produced by an entropy wave is highest close to the refraction cutoff.
It can be very large if
is close to 5/3.
These results suggest that the shocked Bondi flow with
is
strongly unstable with respect to the entropicacoustic mechanism.
Key words: accretion, accretion disks  hydrodynamics 
instabilities  shock waves  binaries: close 
Xrays: stars
The time variability of the emission from Xray binaries may be
related to hydrodynamic instabilities in the process of accretion onto
a compact star. The simplest accretion flow is the
stationary, spherically symmetric accretion of an adiabatic gas onto a
compact object, first studied by Bondi (1952).
The stability of Bondi accretion was established by Garlick (1979)
and Petterson et al.(1980) in the Newtonian limit, and by Moncrief (1980)
in the framework of general relativity. Kovalenko & Eremin (1998)
suggested that non radial perturbations might reach non linear amplitudes in
the supersonic part of the flow if the size of the accretor is small enough.
This effect was invisible in the 3D numerical simulations of the Bondi flow
by Ruffert (1994a), which confirmed its stability. By contrast, a strong
instability appeared in the numerical simulations of BondiHoyleLyttleton
accretion when a bow shock is present (e.g.Ruffert 1994b for
). The BondiHoyleLyttleton accretion is named after the
pioneering works of Hoyle & Lyttleton (1939) and Bondi & Hoyle (1944)
concerning an accretor moving
at supersonic speed with respect to the gas. Foglizzo & Tagger (2000)
(hereafter FT2000) suggested that this instability might be explained
by what they called the entropicacoustic instability. This generic
mechanism of instability in shocked converging flows is based on the cycle
of entropic and acoustic waves in the subsonic region of
the flow between the shock and the sonic surface surrounding the accretor.
The advection of entropy perturbations towards the accretor produces
outgoing acoustic waves which propagate towards the shock. Perturbed
by these acoustic waves, the shock produces new entropy perturbations, thus
closing the entropicacoustic cycle. According to FT2000, this cycle is
unstable (i.e.the amplitude of the new entropy perturbation exceeds
the initial one) if the sound speed at the sonic point is much larger than
the sound speed at the shock. This mechanism might play a destabilizing
role in various
astrophysical environments such as wind accretion or disc accretion.
In this series of two papers, we describe the entropicacoustic instability
in the Bondi flow, including a spherical shock as an outer boundary (as in
Foglizzo & Ruffert 1997). The study of this rather
academic configuration
serves two purposes:
 (i)
 it is the first detailed description of the entropicacoustic
instability in a specific flow, beyond the general concepts and
approximations of FT2000.
The Bondi flow is simple enough to allow for an analytical treatment of 3D
perturbations, including non radial perturbations of entropy and
perturbations of vorticity;
 (ii)
 understanding the instability of the Bondi flow with a spherical
shock might be the first step towards a comprehension of the instability of
non radial shocked accretion flows such as supersonic BondiHoyleLyttleton
accretion.
In this first paper, we investigate the key part of the entropicacoustic
instability: the excitation of outgoing acoustic waves by the advection of
entropy and vorticity perturbations. As explained in FT2000, the efficiency
of this process is directly linked to the efficiency of the refraction of
acoustic waves, which is also studied. The efficiencies of these processes
depend a priori on the frequency
of the perturbation, its
wavenumbers in spherical coordinates (l,m), and the adiabatic index
of the gas. The accretor is a compact object such as a
black hole, absorbing all the matter falling inside the sonic radius. The
shock, which plays no role in this first paper, is introduced in the second
paper of this series (Foglizzo 2001) in order to close the entropicacoustic
cycle and enable a description of the eigenmodes of the global
instability.
The present paper is organized as follows. Linearized equations are
established in Sect. 2. The refraction of acoustic waves is studied in
Sect. 3. The excitation of acoustic waves by the advection of entropy and
vorticity perturbations is examined in Sect. 4.
We consider the radial accretion flow of a gas with constant sound
velocity
and density
at
infinity, accelerated towards a pointlike accretor of mass M,
which is totally absorbing. The
equations describing the unperturbed Bondi flow are recalled in
Appendix A. In what follows, densities and velocities are normalized by
and
,
and distances are normalized to the Bondi
radius
.
After this normalization, the unperturbed flow
depends on a single parameter ,
the adiabatic index of the gas.
The sound velocity
at the sonic radius
in the
Bondi flow diverges for
:
The "natural" sonic frequency
is directly related to the
Keplerian frequency
at the sonic radius:

(3) 
The Euler equation, the equations of conservation of entropy and mass
in the Bondi flow are linearized for small perturbations in
Appendix B. After a Fourier transform in time, the perturbations are
projected onto a basis of spherical harmonics Y_{l}^{m}, where l,m are
the numbers associated to the latitudinal ()
and longitudinal
()
angles. Kovalenko & Eremin (1998) studied the same
set of equations in order to analyze the asymptotic behaviour of their
solutions near the origin r=0. By contrast, we are interested in the
effect of linear perturbations on the subsonic part of the flow (
).
A single differential equation of second
order is obtained in Appendix B.2, with the following structure:

(4) 
The coefficients
a_{0},a_{1},b_{0},b_{1} are functions of r which depend
on the flow velocity and sound speed radial profiles in the
unperturbed Bondi flow.
Their rather lengthy expressions are given in Appendix B,
Eqs. (B.21) to (B.24).
The homogeneous equation corresponding to the left hand side of
Eq. (4) describes the propagation of acoustic waves in the
Bondi flow and their refraction by the sound speed gradients. The
important feature of Eq. (4) is the presence of source terms
on the right hand side:
(i)
is the perturbation of entropy. For the sake of
simplicity, the ratio of the molecular weight
to the gas constant
is set to
throughout this paper, with no loss
of generality.
(ii)
involves the curl of the vorticity vector
projected
along the flow velocity:

(5) 
The radial dependence of the three components of the vorticity vector
is explicitely integrated in Appendix B.1
(Eqs. (B.5) to (B.7)). Nevertheless, the vorticity
contributes to the excitation of acoustic waves only through the
radial part of its curl, as it appears in Eq. (5).
In an adiabatic flow, both
and
are conserved when
advected (see Appendix B.1). A more compact mathematical
formulation is obtained in Appendix B.2.2 by writing the
differential equation satisfied by the perturbation f of the Bernoulli
constant.
where the frequency
is defined as:

(7) 
The structure of this differential equation is the same as
Eq. (4), with much simpler coefficients. The right hand side of
Eq. (6) contains two source terms proportional to
and
.
The pressure perturbation is related
to f by the following equation derived in Appendix B.1:

(8) 

Figure 1:
Refraction coefficient
for
,
l=0,1,2,3. The dotted line corresponds to the leading behaviour of
for
(Eq. (15)) 
Open with DEXTER 
The homogeneous equation associated with Eq. (6),
describing the propagation of acoustic waves is:







(9) 
This equation is of second order, which reflects the fact that pressure
perturbations can be decomposed into ingoing and outgoing acoustic waves.
Such an identification is possible only in the region of the flow where the
lengthscale of the flow gradients is longer than the wavelength of the
perturbation,
where a WKB treatment of the wave propagation can be used (see Appendix C).
This is always the case far from the accretor in the Bondi flow,
since the density and sound speed are uniform at infinity. We may
define
and
as the solutions of the
homogeneous
equation propagating respectively inward and outward at infinity.
Following FT2000, the index + (resp. ) refers to
waves propagating in the same (resp. opposite) direction as the flow.
The acoustic flux carried by these acoustic waves is defined by
Eq. (6) of FT2000:

(10) 
Let us normalize
so that they carry the same acoustic
flux at infinity. Any linear combination
of the two independent
solutions
,
with complex
coefficients
,
is also a solution of Eq. (9):

(11) 
There is, however, only one ratio
such that
is regular at the sonic radius. Indeed, a local analysis of the solutions of
Eq. (6) for
,
at the sonic point
using
Frobenius series, reveals that there is only one solution crossing the
sonic point regularly, the other exhibiting a logarithmic singularity
(Bender & Orszag 1978, Chap. 3.3, p. 68). This regularity condition
reflects the fact that an incoming acoustic wave
is only partially transmitted to the accretor: an outgoing acoustic wave of
amplitude
is refracted out.
Using the same notations as FT2000, the fraction
of refracted acoustic flux is deduced from
Eqs. (10) and (11):
The refraction coefficient is therefore directly related to the regularity
of the solution at the sonic point.
The differential Eq. (9) is integrated using a
RungeKutta implicit method, from the sonic point to the WKB region.
The refraction coefficient is then computed by identifying the
ingoing and outgoing waves in the WKB region.
If
,
a Frobenius expansion is used in the vicinity of
to
start the integration away from the singularity.
The calculation for
and l=0 is similar to the case
,
except that the sonic point is at r=0.
If
and ,
the homogeneous solution presents an essential
singularity at the sonic point, where one branch converges to zero,
and the other diverges. Nevertheless, an asymptotic expansion of the
converging solution (see Eqs. (C.32) and (C.33))
allows a numerical integration from the vicinity of the origin
towards infinity. As expected by Kovalenko & Eremin (1998), no
overreflection occurs (
). The typical frequency dependence
of
is shown in Fig. 1 for
.
If
,
the accretor is surrounded by a supersonic region
from which acoustic waves cannot escape. Only those waves with a high
enough frequency may penetrate this region, because lower frequency
waves are refracted out before the sonic radius. This leads us to expect
the refraction
to decrease to zero at high frequency, and be
close to unity at low frequency (see Fig. 1). The refraction
cutoff
of a wave of latitudinal order l is defined as the
intermediate frequency corresponding to a refraction of half of the incoming
acoustic flux:

(14) 
Bessel functions are used in Appendix D to approximate the homogeneous
solution at high frequency, in order to obtain the leading order of
the asymptotic behaviour of
.
This latter decreases exponentially
above
(see Fig. 1):

Figure 2:
Typical radial dependence of
(here
),
showing the turning point
of a non radial acoustic wave l=1
below the cutoff frequency

Open with DEXTER 
The turning point of non radial perturbations appears
explicitly on Eq. (9). It corresponds to the radius
solution of
,
where
is
defined by Eq. (7) (see Fig. 2). For
,
this turning point is double when the frequency equals the
maximum of the function
.
The refraction
coefficient is then equal to 1/2, according to Bender &
Orszag (1978, Chap. 10.6, p. 524):
This result was checked by measuring numerically the refraction of
acoustic waves. Figure 3 shows the dependence of
on
the adiabatic index .
should scale as
according to the asymptotic
behaviour of
when
,
deduced from Eqs. (A.12 to A.14). A refined comparison with
the results of numerical calculations of
leads to

(19) 
over the range
considered numerically.
The radius r_{1} such that
is the minimum
value of the turning radius r_{t} (see Fig. 2).
is independent of l,m and lies just outside the sonic
surface, and is well fitted by
for
close to 5/3.

Figure 3:
The cutoff frequencies
and
correspond to
for l=0 and l=1 respectively. They
cannot be distinguished from their fittings (Eqs. (19) and
(21)) on this plot. These frequencies are compared to the sonic
frequency
,
the Keplerian frequency at the sonic radius
,
and
.
Frequencies are expressed in
units of
,
as functions of
(or ,
or )

Open with DEXTER 
is computed numerically in Fig. 3, and corresponds
to a lower frequency than
.
The value of
can be estimated analytically for
close to 5/3 from the criterion of validity of the WKB
approximation of acoustic waves, derived in Appendix C:

(20) 
Since the acoustic flux is conserved in the region
of validity of the WKB approximation, any refraction of acoustic waves
must occur outside this region. We deduce that
scales as the maximum of the function on the right hand
side of Eq. (20). When
is close to 5/3, this
maximum scales as
.
This is confirmed by numerical calculations of
which indicate that
in the range of sound velocities
studied
numerically, the ratio
is constant within the
numerical errors:

(21) 
The scaling of
should be compared to the two other
particular frequencies of the flow near the sonic point,
in
Eq. (19) and the local Keplerian frequency
:

(22) 
and thus
for
close to 5/3. The
values of
computed numerically are
compared to
and to the Keplerian frequency in Fig. 3.
In a Bondi flow with
,
acoustic waves are perfectly
refracted at any frequency:

(23) 
This can be deduced
from the limit of flows with
close to 5/3, because the
cutoff frequency
tends to infinity when
.
This is not surprising, since an ingoing acoustic wave of any frequency
always ultimately meets "a wall", i.e.a subsonic region
where the scale of the flow gradients are much shorter than the
wavelength.
The WKB analysis at high frequency (Appendix C) indicates that ingoing and
outgoing acoustic waves are coupled together at
for
l=0,
and at their turning point
if .
The continuity between the flow
and flows with
close
to 5/3 is checked by remarking that
for
and
for
.
Comparing this
to
in Eqs. (21) and (19),
the cutoff frequency is simply interpreted as the
maximum frequency such that the coupling of acoustic perturbations
occurs in the subsonic flow.
According to Eq. (6), it is natural to define
two dimensionless complex acoustic efficiencies
,
associated respectively to the two source terms
and
.
These definitions are made so that the acoustic flux F^{}of an outgoing sound wave triggered by the perturbations of
entropy and vorticity, measured far from the accretor is:

(24) 
The integral expressions for
and
are derived in
Appendix D:
It is remarkable that both acoustic efficiencies are independent
of the longitudinal number m.
and
depend only on
and l.

Figure 4:
Acoustic efficiencies
and
depending
on the frequency for
,
l=0,1,2,3 
Open with DEXTER 

Figure 5:
The frequencies which maximize
are
displayed in units of the refraction cutoff
,
for l=0,1,2,3as functions of

Open with DEXTER 
The step of the numerical integration must be able to resolve both
acoustic and entropy waves. Far from the accretor, the wavelength of
the acoustic perturbation is asymptotically constant (
), whereas the wavelength of the entropy perturbation
decreases to zero as
.
It is therefore
useful to transform the integrals in Eqs. (25) and
(26) to accelerate their convergence. This is done
in Appendix D by successive integrations by parts, thus obtaining
Eqs. (D.6) and (D.19).
If
,
the integrals are transformed near the origin in
order to correctly treat the singularity.
4.3 Asymptotic expansion of
and
at high
frequency for
=

Figure 6:
Acoustic efficiency
of entropy perturbations for
,
l=0,1,2,3. The full line corresponds to the numerical
calculation, the dotted line corresponds to the asymptotic estimate
at high frequency (Eqs. (28) and (29)) 
Open with DEXTER 

Figure 7:
Acoustic efficiency
of vorticity perturbations for
,
l=1,2,3. The full line corresponds to the numerical
calculation, the dotted line corresponds to the asymptotic estimate
at high frequency (Eq. (30)) 
Open with DEXTER 
Special properties of the Bondi flow
are related to
the position of the sonic radius, at the central singularity.
In particular, both cutoff frequencies
and
are infinite, which means that at any
frequency, incoming sound waves are totally refracted outward:

(27) 
is estimated in Appendix E.1 at high frequency for
radial perturbations:

(28) 
where the Gamma function satisfies
.
The case of non radial perturbations is derived in
Appendix E.2, where we define
:
The asymptotic expansions in Eqs. (28), (29)
and (30), which involve no free scaling parameter, are
successfully compared to numerical calculations of
and
in
Figs. 6 and 7.
The radial case of Eq. (28) is recovered by taking the
continuous limit
in Eq. (29).
Note that the asymptotic behaviour of the Bessel function
for large arguments (),

(31) 
makes the acoustic efficiencies decrease rapidly for high order l.
Remembering that the frequency of the entropy perturbation is directly
related to its radial wavelength (
),
our calculation of
indicates that the
sound produced by the advection of an object of given mass in the
Bondi flow depends a lot on its shape. Virtually no sound comes out
if l is large and
is small (e.g.a stretched spaghetti).
By contrast, the shape producing the highest acoustic flux corresponds to
a small l and a large
(e.g.a wide thin saucer).
At high frequency, the coupling of entropy
and vorticity perturbations to acoustic waves comes essentially from the
region
(i.e.the main contribution to the
integrals in Eqs. (25) and (26), computed
in Appendix E). The radial scaling at
high frequency is therefore
.

Figure 8:
Maximum acoustic efficiency
as a
function of
(i.e.of
)
for l=0,1,2,3. The
dotted lines are deduced from the curves
,
(Fig. 6), where
is replaced by
.
Curves corresponding to l=0
are marked with circles 
Open with DEXTER 

Figure 9:
Acoustic efficiencies
and
depending on the
frequency for
(i.e.
), l=0,1,2.
Frequencies are expressed in units of
.
These efficiencies are very close to those obtained for
(dotted lines), truncated after the cutoff frequency

Open with DEXTER 
For
close to 5/3 and at frequencies lower than the cut off
frequency, we can use the approximation of
obtained for
.
According to Fig. 5, the maximum of
is reached
at the frequency
,
with
.
The asymptotic behaviour of
the maximum value of
is therefore estimated by extrapolating Eqs. (28) and
(29) to the cutoff frequency:
It is interesting to note in Fig. 8 that the most efficient
entropy perturbations at a given frequency when
are non radial
(l=1), whereas radial ones are more efficient for
(Fig. 6). This is directly related to
the fact that the critical refraction frequency is much higher for l=1
than for l=0 (Eqs. (19) and (21)). This is illustrated in
Fig. 9, where the acoustic efficiencies computed for
(
)
are very close to
their values for
,
up to the refraction cutoff. In this
figure, the efficiency
is higher for l=0 than for l=1 in the
limited range of frequencies
,
but the overall maximum is
reached near
for l=1.
5 Conclusions
The acoustic response of the Bondi flow to pressure, entropy and
vorticity perturbations has been studied thoroughly.
Let us summarize the main results:
(i) the critical refraction frequency for acoustic waves
is quite different from the "natural" frequency
one might think of at first glance. Moreover
is also very
different for radial and non radial perturbations.
The two fundamental frequencies
of this flow near the
sonic point were accurately determined (Eqs. (19) and (21)), and
are ordered as follows:
,
with
(ii) two quantities conserved in the linear approximation
of the perturbed Bondi flow are source terms for acoustic waves: the
entropy perturbation ,
and
a quantity
related to both vorticity and entropy
perturbations defined by Eq. (5).
(iii) the acoustic efficiencies
and
of
entropy and vorticity perturbations are independent of
the longitudinal number m.
(iv)
and
are highest for l=1 perturbations
if
.
(v)
is highest at relatively low frequency, and is
always rather moderate (
).
(vi)
is highest at high frequency near the
refraction cutoff, and can be much larger than one if
is
close to 5/3, i.e.if
.
This confirms the physical
argument of FT2000 which stressed the importance of the enthalpy increase
between the outer part of the flow and the sonic point.
(vii) asymptotic scalings of
were obtained at
high frequency for
(Eqs. (28) to (30)),
in excellent agreement with numerical calculations.
The analytical expressions obtained for
,
and for
and
at high frequency for
,
are accurate enough (see Figs. 3, 6
and 7) to be used as tests of the accuracy of 3D hydrodynamical
codes used for Bondi accretion, and particularly their limitations at
high frequency.
These results suggest that if a shock were present as outer
boundary, the entropicacoustic cycle would be unstable with respect
to high frequency waves if the sound speed at the sonic point is high
enough, i.e.if
is close to 5/3. Indeed, an entropy
perturbation
advected towards the accretor triggers an
outgoing acoustic flux
.
These acoustic waves propagate against the flow until they reach the shock,
where they produce new entropy perturbations
(see FT2000), with
since
.
The eigenmodes corresponding to this unstable cycle will be investigated
in the second paper of this series (Foglizzo 2001). The fact that the
most unstable simulations of BondiHoyleLyttleton accretion
also correspond to
(Ruffert & Arnett 1994; Ruffert 1994b)
already encourages us to look for an extrapolation of these results to the
case of nonradial shocked accretion flows.
In the flow
,
the fact that the outgoing acoustic flux
produced by entropy perturbations diverges at high frequency
(Eqs. (28) and (29)) is rather
surprising, and directly related to the hypothesis of a pointlike
Newtonian accretor (
).
In this section we show that
is naturally bounded
if the finite size of the accretor is taken into account.
Simple arguments are used in order to roughly estimate this upper bound in
the case of an accreting black hole, without computing the exact relativistic
corrections near its horizon.
Let us denote by
the Schwarzschild radius
of the black hole, where
is the speed of light.
With a sound speed at infinity
,
Eq. (1)
guarantees that relativistic effects are negligible except for
close to 5/3. Indeed Petrich et al.(1989) checked numerically
that relativistic corrections are small for
and
.
If
,
the sonic point is bound to lie
at a few Schwarzschild radii:

(38) 
Our description of non radial acoustic waves in the flow
is not
affected by relativistic corrections as long as their turning point
is far enough from the horizon. Since
at high
frequency (Eq. (C.34)), a natural relativistic cutoff is
introduced for
:

(39) 
The efficiency
of the generation of non radial acoustic
waves close to this cutoff frequency is deduced from
Eq. (29):

(40) 
The numerical values of the cutoff frequency and maximum acoustic
efficiency of the entropy perturbations l=1 are thus typically:
Therefore relativistic effects set an upper bound on the apparent
divergence of
at high frequency for
.
Equation (42) indicates that entropy perturbations may excite
low degree acoustic waves very efficiently, up to non linear amplitudes
if the component l=1 of the entropy perturbations exceeds a few percent.
This efficiency is much smaller for inhomogeneities with a small
angular scale, since
decreases exponentially when l increases (Eq. 40).
Non adiabatic processes are discussed by Chang & Ostriker (1985) in the
context of Bondi accretion. The efficiency
would be strongly
affected by a strong thermal conduction which smoothes out
entropy perturbations. This effect would be more pronounced for short
wavelength perturbations, which are also the most efficient for
close to 5/3. Without
going into the details of such processes, our understanding of the
acoustic efficiency allows the following remark: if the inner
region of the flow were dominated by non adiabatic processes, the optimal
frequency for
would be at most reduced to the maximum frequency of
acoustic waves refracted inside the adiabatic region of the flow.
Even when there is no shock, the mechanism of excitation of acoustic waves
from the advection of entropy perturbations opens interesting perspectives
to the problem of time delay in Cygnus X1, approached numerically by
Manmoto et al.(1996). Interpreting their results in terms of acoustic
efficiency of entropy perturbations suggest that this physical process
survives the inclusion of rotation, radiative cooling and viscous
heating. The study of effects deserve a careful analysis which is beyond
the scope of the present paper.
Acknowledgements
The author acknowledges stimulating discussions with Michel Tagger.
The RungeKutta algorithm used in this work for the numerical calculations
was kindly provided by Roland Lehoucq.
Appendix A: Description of the unperturbed Bondi flow
The Bernoulli equation and the conservation of mass are written after
normalizing velocities to the sound speed
at infinity, and
distances to the Bondi radius
By taking the radial derivative of these equation, we obtain:
At the sonic point,
are defined by Eqs. (1) and (2).
A Taylor expansion of Eqs. (A.3) and
(A.4) leads to:
A.2
In accretion flows where
,
Eqs. (A.1) and
(A.2) become:

= 

(A.8) 
r^{2}vc^{3} 
= 

(A.9) 
r,v,c can be expressed as explicit functions of the variable x defined by:
x 


(A.10) 

= 

(A.11) 
r 
= 

(A.12) 
v^{2} 
= 

(A.13) 
c^{2} 
= 

(A.14) 
B.1 Second order differential system
The Euler equation is written as follows:

(B.1) 
Projecting the Euler equation onto the flow velocity, we obtain the
equation of evolution of the Bernoulli constant:

(B.2) 
By combining the curl of Euler equation and the mass conservation, we
obtain the equation for the evolution of the vorticity w:

(B.3) 
We make a Fourier transform in time of the linearized equations
describing the radial flow perturbed in spherical coordinated
.
The linearized equation of entropy conservation can be directly
integrated:

(B.4) 
The vorticity Eq. (B.3) can also be integrated when
linearized:
w_{r} 
= 

(B.5) 

= 

(B.6) 

= 

(B.7) 
Thus the product
r^{2}w_{r} is conserved during advection, as
remarked by Kovalenko & Eremin (1998).
In order to write the linearized Euler equation in the simplest form,
let us define the two functions f,g as follows:
f is the perturbation of the Bernoulli constant, which is directly
related to the pressure variations according to Eq. (B.2). g
is related to the perturbation of the mass accretion rate and entropy
through:

(B.10) 
By linearization of Eq. (B.1), we express the velocity
components
as follows:
The radial part of the Euler equation, together with the continuity
equation, lead to the following differential system:
where
is the non radial part of the
Laplacian in spherical coordinates. The constant
is
defined as follows:

(B.15) 
We deduce from Eqs. ((B.6)(B.7)) that
is conserved
when advected:

(B.16) 
can be written independently of the system of coordinates,
as the first order perturbation of the quantity K defined by:

(B.17) 
but the conservation of K is established only in the linear approximation.
By projecting
and the radial component of the curl
of the vorticity
onto the spherical
harmonics
,
which are the eigenvectors of the
Laplacian, we obtain:
where we have defined
for the sake of the simplicity
of the equations.
B.2 Second order differential equation
B.2.1 Pressure perturbation
The most natural function to describe the behaviour of acoustic waves
is the pressure perturbation
,
which is related to the sound
speed perturbation through:

(B.20) 
Using this equation in Eqs. ((B.8)(B.9)) enables us to
write (f,g) as functions of
and
in
Eqs. ((B.18)(B.19)), and thus obtain Eq. (4) with:
a_{1} 


(B.21) 
a_{0} 










(B.22) 
b_{0} 






(B.23) 
b_{1} 


(B.24) 



(B.25) 
The function
is introduced by our choice of writing the
differential equation satisfied by the pressure perturbation, which
is justified from the physical point of view. Note that
never vanishes as long as the real part of the frequency is different
from zero.
B.2.2 A more compact mathematical formulation
It is convenient to define (
)
as:
The differential system Eqs. ((B.18)(B.19)) is then simpler:
The homogeneous differential Eq. ( 9) can
be transformed into a more compact form using the new variable X:
where
is defined by Eq. (7).
With this new variable, the sonic point
corresponds to
,
while the spatial infinity corresponds to X=0.
C.1 WKB approximation far from the accretor
The general solution f of the homogeneous equation
associated to Eq. (B.32) is a linear
combination of ingoing (f_{+}) and outgoing (f_{}) waves, which can
be approximated by the WKB method:

(C.1) 
This approximation is asymptotically valid for

(C.2) 
which is satisfied at high frequency or far from the accretor.
The Wronskien
associated to the couple
(
)
of solutions is:
The pressure perturbation
associated to a perturbation f
is deduced from Eq. (B.2):

(C.5) 
The normalization factor
in
Eq. (C.1) is such that the pressure perturbations
associated to these solutions by Eq. (B.2)
have the following asymptotic behaviour:

(C.6) 
Both carry the same acoustic flux
according
to Eq. (10).
C.2 Definition of the refraction coefficient
and estimate at high frequency
A Frobenius expansion in the vicinity of the sonic point (
)
shows that
any solution of the homogeneous equation can be projected onto the
couple f_{1},f_{2} of solutions, where f_{1}(r) is regular and f_{2}(r)
is singular:
f_{1}(r) 


(C.7) 
f_{2}(r) 


(C.8) 
This defines a unique coefficient
such that the solution

(C.9) 
is regular at the sonic point. The coefficient
is the
refraction coefficient of sound waves. The linearity of the equations
guarantees of course that the solution
is also regular.
An expansion of W in the vicinity of the sonic point leads to:
W 
= 

(C.10) 
q_{0} 


(C.11) 
q_{1} 


(C.12) 



(C.13) 
The WKB approximation of
is valid at high frequency
,
for
:

(C.14) 
The asymptotic solution to the differential Eq. (9)
can also be approximated by a Bessel function in the vicinity of the sonic
point, for
.
The regularity of f_{0} at the sonic radius
implies:
The Bessel and WKB approximations are both valid in the region where
.
In this region, the matching of the two
approximations gives the normalization constant
and the
refraction coefficient
.
The asymptotic behavior of the Bessel
function, when both the imaginary order
and the argument
tend to infinity, is obtained from Watson (1952, Chap. 8.6, p. 262).
The Bessel function
is written in terms of the Hankel
functions
and
.
Let x,z be real numbers, with ,
such that the order
is purely imaginary
:
Applying these asymptotic expansions to Eq. (C.15), with
we obtain
and
to first order at high frequency:
The exponential decrease of
at high frequency is deduced from
Eqs. (C.18) and (C.19), by identifying the Hankel functions
with the ingoing and outgoing acoustic waves:

(C.24) 
The WKB approximation is valid in the range
.

(C.25) 
where the normalization is chosen in agreement with
Eq. (C.1).
In the region
where the WKB approximation is not
valid, we approximate the solution by the Bessel function J_{0}, which
is regular at the origin:




(C.26) 



(C.27) 
The other Bessel solution, Y_{0}, is singular with a logarithmic
divergence, as in the case of
.
The Bessel approximation for
matches the WKB approximation:
From this we deduce that:
The case of non radial perturbations needs to be treated separately,
because the behaviour of the solution is quite different near the
sonic point. If ,
there is always a turning point beyond which
the solution is evanescent, such that it decays exponentially on the
sonic point. The leading behaviour of
f_{0},g_{0} on this essential
singularity is the following, within a multiplicative constant:
f_{0} 


(C.32) 
g_{0} 


(C.33) 
The solutions of the other branch diverge exponentially on the sonic
point: as in the case of a regular sonic point, there is a unique
combination of ingoing and outgoing waves such that the solution does
not diverge at the sonic point.
The turning point
corresponds to the radius where W=0
in Eq. (B.32):

(C.34) 
The WKB approximation is valid for
:
The outgoing pressure perturbation produced by the advection of a
perturbation
is computed by solving
Eq. (B.32), imposing the regularity at the sonic point and the
absence of an incoming sound wave from infinity. The general solution of
Eq. (B.32) with
can be written as:
According to Eqs. (C.6) and ((C.7)(C.8)), the
integrals involved in Eq. (D.2) are well defined both at
infinity and at the sonic radius. The condition that no sound wave
comes from infinity leads to:

(D.3) 
The regularity at the sonic radius can be written:

(D.4) 
Thus the efficiency of sound wave emission by the advection of
vorticity perturbations is:

(D.5) 
This integral converges as 1/r^{3} at infinity. It can be transformed
using integrations by parts together with the homogeneous differential
system satisfied by
f_{0},g_{0}, so that it converges rapidly at infinity,
thus allowing an easier numerical computation. The following expression
converges as 1/r^{5}:
The outgoing pressure perturbation produced by an incoming entropy
perturbation
is also computed by solving
Eq. (B.32), imposing the regularity at the sonic point and
the absence of an incoming sound wave from infinity. The general
solution of Eq. (B.32) with
can be written as:
Since the integrals of Eq. (D.8) converge at the sonic
point, the regularity of the solution at the sonic point can be
written:

(D.9) 
In Eq. (D.8) the integrals diverge at infinity. Two
integrations by parts leads to define the two functions
as follows:
The pressure perturbation associated to the entropy perturbation
is:
Far from the accretor, the pressure perturbation can be projected
onto the ingoing and outgoing sound waves.
is chosen such
that there is no sound wave coming from infinity:

(D.13) 
Using Eqs. (D.9) and (D.13), and the fact that
,
we take the limit
,
in
Eq. (D.12) and obtain:




(D.14) 
This integral can be written with the following format, using the
homogeneous differential system (Eqs. (B.28), (B.29))
satisfied by
f_{0},g_{0}:

(D.15) 
where the functions
A_{k}(r),B_{k}(r) depend only on the unperturbed flow,
and the integral converges as 1/r^{k} far from the accretor:
A_{1} 


(D.16) 
B_{1} 


(D.17) 
Two successive integrations by parts make the convergence of the
integral much faster, thus allowing an easier numerical calculation:
A_{5} 




+ 


(D.18) 
B_{5} 



(D.19) 
Appendix E: Asymptotic estimate of
at high frequency
for
= 5/3
Equation (D.14) can be estimated in the limit
,
when
.
We treat the case of
radial (l=0) and non radial ()
perturbations separately.
E.1 Radial perturbations
We use a local expansion of
where :

(E.1) 
Writing
,
we obtain
for
and :
The range of validity
implies that
and
.
The contribution of the region
is negligible.

(E.3) 
The contribution of the region
is estimated using the
asymptotic behaviour of the Bessel function J_{1}:



(E.4) 



(E.5) 
Thus we deduce the acoustic efficiency at high frequency:

(E.6) 
The main contribution to the integral comes from the region where
,
i.e.
.
Equation (E.6) is rewritten in Eq. (28) using the Gamma
function.
E.2 Non radial perturbations
As in the case of radial perturbations, the main contribution to the
integral comes from the region
accessible to the WKB approximation.
The outgoing wave is the main contributor to the integral
(D.14). Using the asymptotic development of v,c for
in Appendix A.2,







(E.7) 
With
,
the acoustic
efficiencies at high frequency are deduced from Eqs. (C.35),
(D.5), (D.14) and (E.7):
Using the new variable

(E.10) 
the integrals in Eqs. (E.8)(E.9) can be written using
,
the modified Bessel function of order 2/3
(Gradshteyn & Ryzhik 1980, p. 430):
Thus we obtain:
Here again, the main contribution to the integrals comes from the
region
.
By contrast with
,
decreases at high frequency.

Bender, C. M., & Orszag, S. A. 1978, Advanced Mathematical Methods for
Scientists and Engineers (McGrawHill International Editions)
In the text

Bondi, H. 1952, MNRAS, 112, 195
In the text
NASA ADS

Bondi, H., & Hoyle, F. 1944, MNRAS, 104, 273
In the text
NASA ADS

Chang, K. M., & Ostriker, J. P. 1985, ApJ, 288, 428
In the text
NASA ADS

Foglizzo, T. 2001, A&A, in preparation, Paper II
In the text

Foglizzo, T., & Ruffert, M. 1997, A&A, 320, 342
In the text
NASA ADS

Foglizzo, T., & Tagger, M. 2000, A&A, 363, 174 (FT2000)
In the text
NASA ADS

Garlick, A. R. 1979, A&A, 73, 171
In the text
NASA ADS

Gradshteyn, I. S., & Ryzhik, I. M. 1980 (Academic Press, inc.)
In the text

Hoyle, F., & Lyttleton, R. A. 1939, Proc. Cam. Phil. Soc., 35, 405
In the text

Kovalenko, I. G., & Eremin, M. A. 1998, MNRAS, 298, 861
In the text
NASA ADS

Manmoto, T., Takeuchi, M., Mineshige, S., Matsumoto, R., & Negoro, H. 1996,
ApJ, 464, L135
In the text
NASA ADS

Moncrief, V. 1980, ApJ, 235, 1038
In the text
NASA ADS

Petrich, L. I., Shapiro, S. L., Stark, R. F., & Teukolsky, S. A. 1989,
ApJ, 336, 313
In the text
NASA ADS

Petterson, J. A., Silk, J., & Ostriker, J. P. 1980, MNRAS, 191, 571
In the text
NASA ADS

Ruffert, M. 1994a, ApJ, 427, 342
In the text
NASA ADS

Ruffert, M. 1994b, A&AS, 106, 505
In the text
NASA ADS

Ruffert, M., & Arnett, D. 1994, ApJ, 427, 351
In the text
NASA ADS

Watson, G. N. 1952, A Treatise on the Theory of Bessel Functions
(Cambridge University Press)
In the text
Copyright ESO 2001