Download Seismology of stellar envelopes: probing the outer layers of a star

Mon. Not. R. Astron. Soc. 322, 473±485 (2001)
Seismology of stellar envelopes: probing the outer layers of a star through
the scattering of acoustic waves
IlõÂdio P. Lopes1w and Douglas Gough1,2w
1
2
Institute of Astronomy, Madingley Road, Cambridge CB3 0HA
Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge CB3 9EW
Accepted 2000 August 2. Received 2000 August 2; in original form 1999 December 7
A B S T R AC T
The outer layers of Sun-like stars are regions of rapid spatial variation which modulate the
p-mode frequencies by partially reflecting the constituent acoustic waves. With the accuracy
that has been achieved by current solar observations, and that is expected from imminent
stellar observations, this modulation can be observed from the spectra of the low-degree
modes. We present a new and simple theoretical calculation to determine the leading terms
in an asymptotic expansion of the outer phase of these modes, which is determined by the
structure of the surface layers of the star. Our procedure is to compare the stellar envelope
with a plane-parallel polytropic envelope, which we regard as a smooth reference background state. Then we can isolate a seismic signature of the acoustic phase and relate it to
the stratification of the outer layers of the convection zone. One can thereby constrain
theories of convection that are used to construct the convection zones of the Sun and Sunlike stars. The accuracy of the diagnostic is tested in the solar case by comparing the
predicted outer phase with an exact numerical calculation.
Key words: Sun: interior ± Sun: oscillations ± stars: interiors ± stars: oscillations.
1
INTRODUCTION
As a result of the high accuracy of observations of solar acoustic
oscillations, helioseismology has become a powerful tool for
studying the Sun's internal structure and dynamics (for recent
reviews see Vorontsov & Zharkov 1989, Gough & Toomre 1991,
Libbrecht & Woodard 1991, Gough 1996, Christensen-Dalsgaard
et al. 1996 and Gough et al. 1996). The observations have yielded
oscillation frequencies with a precision as high as 1 part in 105.
Two basically different approaches are used to diagnose the
structure of the solar interior. The first is the forward approach, in
which observational frequencies are compared with corresponding
numerically computed eigenfrequencies of a set of theoretical
models (e.g. Christensen-Dalsgaard 1993). The second is inverse
analysis, which is aimed at seeking directly a representation of the
Sun that in some sense is favoured by the data. The inverse
procedures can be divided into two broad categories, one in which
one considers the (small) difference between the seismically
accessible component of the structure of the Sun and that of a
model, and the other in which one transforms the data directly into
a representation of that structure (e.g. Gough 1996). The latter is
non-linear, and has been tackled only by analytical asymptotic
methods. The former relies on a linearization, and is most
precisely tackled numerically, although asymptotic methods can
usefully be used in that case too.
w
E-mail: [email protected] (IPL); [email protected] (DG)
q 2001 RAS
Inverse analysis permits one to extract information from the
data pertaining to different, often local, regions of the star. It is
essential for answering unambiguously questions about the
physics of the stellar interior. In this paper we address an example
of such a separation of seismic information, based on an asymptotic method. Although the precision is not as high as that which
can be obtained numerically, we believe that the insight derived
from analytical procedures such as this amply justifies the effort.
The most detailed asymptotic analysis of stellar acoustic
oscillations has been applied to modes of low degree, partly
because it is perhaps for these modes that the analysis is the most
accurate, and partly because these modes penetrate the most
deeply and can be used to gauge physical mechanisms operating
in the nuclear-reacting region of the star. Several complementary
asymptotic expressions, based on the Liouville±Green expansion
of the adiabatic oscillation equations, have been presented (e.g.
Tassoul 1980, 1990; Gough 1984b, 1993; Vorontsov 1991). These
have motivated the construction of different seismic parameters to
diagnose certain global or local properties of the star. Data from
the ground-based helioseismological network GONG (Harvey
et al. 1996) and from the seismological instruments VIRGO
(Appourchaux et al. 1995; FroÈhlich et al. 1995), GOLF (Gabriel
et al. 1995) and SOI/MDI (Scherrer et al. 1995; Zayer et al. 1995),
together with anticipated asteroseismic data from space missions
such as COROT (Catala et al. 1995), MONS (Kjeldsen & Bedding
1998) and MOST (Matthews 1998) from which the frequencies of
low-degree acoustic modes will be obtained with great precision,
474
I. P. Lopes and D. Gough
have stimulated a more thorough development of the asymptotic
analysis.
Asymptotic representations of the modes of high order are valid
throughout most of both the outer convection zone and the
radiative interior of a star. However, they are liable to be invalid in
regions where seismically relevant aspects of the structure of the
background state vary on a scale comparable with the radial
wavelength of the mode. These regions are responsible for partial
wave reflexion, which is an essential ingredient for establishing a
resonant cavity. Reflexion occurs in regions of rapid variation of
the sound speed, such as that encountered in the transition
between the chromosphere and the corona, and, to a lesser extent,
in regions of ionization of abundant elements. It is also produced
by steep variation in the density, as occurs near the photosphere.
Near-discontinuities in the first or higher derivatives of density,
such as those occurring at the boundaries of convective envelopes
(Balmforth & Gough 1990) can also reflect the waves. In the case
of the Sun, such reflexions produce weak, but measurable, signatures. In somewhat more massive stars, however, the signatures
are stronger, particularly those associated with the region of
partial ionization of helium (Lopes et al. 1997). The reflexions are
not total, and energy propagates both forwards and backwards, in
some cases permitting leakage from the star and thereby contributing to the decay of the mode. Partial reflexion of acoustic
waves can occur also more deeply in the interior of a star, such as
at the edge of the convective core of a massive star (e.g. Audard,
Provost & Christensen-Dalsgaard 1995).
The effect of partial internal reflexion is to modify the
resonance condition for the constructive interference required to
form a standing wave. We represent it as a contribution to the
phase a which necessarily arises in any wave-like description of
resonance,
the condition for which we write here generically as
„
k dr ˆ …n ‡ a†p; where k is a vertical wavenumber, and r is a
radial coordinate (e.g. Gough 1993). This resonance condition is
valid only when k 21 is much less than any dynamically relevant
scaleheight deep in the stellar interior, which for acoustic waves is
the case for waves of high frequency v . For the purposes of this
discussion, we separate a into two components, an outer phase,
a o, which is determined by the outer layers of the convection
zone, and an inner phase, a i, which is determined by the structure
deeper down. In the case of the Sun, a i is small, because the
stratification of the deep interior is relatively smooth. However,
the outer phase exhibits a distinctive variation with frequency.
Because it is in the outer layers of the star that the usual
asymptotic approximations break down, several authors (e.g.
Christensen-Dalsgaard & Frandsen 1983; Brodsky & Vorontsov
1987; Christensen-Dalsgaard & PeÂrez HernaÂndez 1988, 1992;
Dziembowski, Pamiatnykh & Sienkiewicz 1991; Vorontsov 1991;
PeÂrez HernaÂndez & Christensen-Dalsgaard 1994) have adopted a
composite method for determining a o, matching a numerical
solution to the wave equation in the outer layers to an asymptotic
representation beneath. Our intention here is to progress with an
analytical expression for a o.
Our analysis enables us to identify different regions near the
stellar surface that contribute to this phase. The description is based
on a integral representation of the phase contribution due to the
reflexion (back-scattering) of acoustic waves in a residual potential.
We must point out that, in addition to the breakdown of
asymptotic analysis in the surface layers, there is also a breakdown of the simple adiabatic wave equation itself. In these layers,
non-adiabatic processes, resulting from radiative and, more
importantly, convective heat transfer, influence the oscillations
substantially (Balmforth & Gough 1990; Balmforth 1992a,b), as
does the Reynolds stress. The inhomogeneity associated with
convection and magnetic fields also plays a significant role. One of
our long-term tasks is to use the outer acoustic phase that can be
derived from the data to constrain physical processes such as these.
The next section contains a description of the acoustic
oscillations near the stellar surface, and the formulation of a
numerical procedure to calculate the phase a o. In Section 3 we
present a theoretical description of an expansion procedure, based
on an integral representation, and we verify the validity of the
description numerically. In Section 4 we compare the values of
this new expression both with a direct numerical determination of
the phase from a model solar envelope and with a calculation
of the phase from a table of eigenfrequencies. We conclude, in
Section 5, with a discussion of the method, and its pertinence to
asteroseismology.
2 AC O U S T I C O S C I L L AT I O N S N E A R T H E
S T E L L A R S U R FAC E
In determining the behaviour of non-radial adiabatic oscillations
in the surface layers, two approximations to simplify the analysis
can be made. The first is Cowling's approximation, which is to
neglect the Eulerian perturbation to the gravitational potential. It is
valid for high-order acoustic modes everywhere except near the
centre of the star. The second approximation is possible because in
the surface layers the sound speed is relatively low, and consequently, except very near the upper caustic surface, the vertical
component of the wavenumber of low-degree waves is much
greater than the horizontal component. Indeed, it was noted by
Gough (1986) that this is so over quite a wide range of values of
the degree. Consequently, the phase velocity is very nearly vertical
(as is also the group velocity), and the properties of the oscillations depend essentially on the frequency v alone. In particular,
this is so of the outer phase [cf. equation 17, and we shall assume
that ao ˆ ao …v†Š: Locally, the oscillations are barely distinguishable from radial modes. We note that this is no longer so when the
degree l is large; the deviation of the ray paths from the vertical
must have some influence on a o. However, for low and intermediate l, that influence is small. For example, if the lower turning
point is located at r ˆ 0:85 R( …l ˆ 75 for a 3-mHz oscillation),
then the ratio of the horizontal to the vertical component of the
wavenumber is about 1/4 in the middle of the He ii ionization
zone. However, the influence on a o evaluated in that ionization
zone is only about 0.13 per cent. Accordingly, we shall adopt the
radial-wave approximation.
Our analysis is carried out for stars whose background states are
spherically symmetrical; however, we note that, provided due care
is taken, the results can be applied to aspherical stars having a
relatively gentle horizontal variation in their surface layers.
2.1
Approximate wave equation for acoustic oscillations
The linearized adiabatic radial-wave equation for the vertical
displacement j can be obtained in the Cowling approximation by,
for example, setting l ˆ 0 in equations (15.5) and (15.6) of Unno
et al. (1989). One can then cast it into a standard form (in which
there is no first derivative), with acoustic depth t as independent
variable, by the following transformation:
tˆ
…R
r
dr
;
c
p
c ˆ r rcj;
…1†
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Seismology of stellar envelopes
where c is the adiabatic sound speed, r is the density, and R is a
fiducial radius of the star. The outcome is
d2 c
‡ …v2 2 U 2 †c ˆ 0;
dt2
in which the acoustic potential U(t ) is given by
2 2
2 g g d ln h
1 d
r h
1 d2
r h
2
ln
2
ln
‡
U ˆ
2
;
c c
dt
2 dt
c
2 dt2
c
and
21
h…r† ˆ r
…r
g
exp 22
dr ;
2
0c
…2†
…3†
…4†
where g is the local acceleration due to gravity. A similar
equation for radial oscillations has been obtained by ChristensenDalsgaard, Cooper & Gough (1983), without adopting the
Cowling approximation. One can reduce their equation to
equations (2)±(4) by (appropriately) ignoring v2J ; 4pGr compared with v 2 in their expression for the acoustic potential. Sekii
& Shibahashi (1989) used a similar type of representation to
improve the asymptotic inversion for sound speed originally
carried out by Christensen-Dalsgaard et al. (1985). It is possible
also to take account of the geometry of the ray paths by generalizing the acoustic depth to sound travel time along the ray, as
Brodsky & Vorontsov (1993) and Gough (1993) have done
(although in neither case was buoyancy and the acoustic cut-off
taken fully into account).
The outer phase is determined by matching the appropriate
solution of equation (2) to the asymptotic form of that solution
valid for large t :
… t
p
c , C0 …v2 2 U 2 †21=4 cos …v2 2 U 2 †1=2 dt 2 2 pa~ o ; …5†
4
0
where C0 and aÄ o are constants. We stress that deep in the
stellar envelope, v 2 is substantially greater than U 2 . Consequently, the argument of the cosine function in the expression
for
to vt 2 p=4 2 pao ; where ao ˆ a~ o ‡
„ 1 c reduces22almost
2 1/2
0 ‰1 2 …1 2 v U † Š dt: For that reason, in the next sections
we shall determine a o by casting the argument of the sinusoidal
component of the wave function into a similar form. The outer
boundary condition determining the solution is applied at the
surface r ˆ R …t ˆ 0† of the star.
There is some diversity amongst the boundary conditions that
have been adopted in the past, but provided the fiducial surface of
the star is taken to be high in the atmosphere the precise condition
is of little importance. In this study we take r ˆ R to be the
location of the temperature minimum …T ˆ T m †; and we adopt the
condition presented by Unno et al. (1989) that is obtained by
matching on to an evanescent oscillation in an isothermal
atmosphere of perfect gas at temperature Tm, namely
d ln c
v2 c 1 c
42gc
ˆ2
‡
2
; V…v†:
dt
g g 2H
g r
…6†
In this equation, g is the first adiabatic exponent …­ ln p=­ ln r†ad ;
and H is the scaleheight in the atmosphere.
2.2
Numerical determination of the acoustic outer phase
In regions of rapid variation of the structure of the star, the JWKB
approximation upon which the asymptotic wave function (5)
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475
depends is not valid. A representation of the form (5) could still be
adopted, however, if the amplitude C0 and outer phase a o were
considered to vary appropriately with t . Our aim is to find a
procedure to determine that variation. We do so using a method
introduced by PruÈfer (1926) to analyse the properties of the
solutions of a self-adjoint differential system, such as that which
arises in the Sturm±Liouville problem. The method involves
transforming the second-order linear differential equation into a
system of two non-linear first-order equations, one determining
the variation of the phase with acoustic depth, the other determining the variation of the amplitude in terms of that phase. The
advantage gained is that the first equation does not depend on the
second; consequently, the eigenfrequencies of the problem, which
depend only on phase and not on amplitude, are determined by an
equation of only first order. Moreover, the phase, which is
the object of our study, can be determined as a function of the
eigenfrequencies in a straightforward way.
The transformation can be accomplished by writing c in terms
of an amplitude A(s) and a phase u (s) thus:
c ˆ A cos u;
…7†
and relating the two by insisting that
dc
ˆ A sin u;
ds
…8†
in which we have introduced the dimensionless acoustic depth
s ˆ vt as a new independent variable. Substituting this representation into the wave equation (2) yields an equation for the
phase:
du
ˆ 21 ‡ v22 U 2 cos2 u;
ds
…9†
the criterion for equations (7) and (8) to be consistent yields the
equation for the amplitude in terms of the phase:
d ln A 1 22 2
ˆ v U sin 2u:
ds
2
…10†
Finally, for the eventual purpose of matching the wave function to
its asymptotic form (5), we write u in terms of a function a o(v ; s)
according to
uˆ
p
‡ pao 2 s:
4
…11†
Then
dao
U2
p
2
:
cos
s
2
a
ˆ
2
p
o
ds
pv2
4
…12†
We note that Brodsky & Vorontsov (1987, 1989) have obtained
this equation using a method of Barbikov (1976) developed for
quantum mechanical problems.
The amplitude function, normalized to unity at r ˆ R; is given
by
A ˆ exp…2I =2v2 †;
where
… vt
h
i
p
U 2 …s 0 =v† sin 2s 0 2 2 2pao …v; s 0 † ds 0 :
I ˆ
2
0
…13†
…14†
The condition on a o imposed by the boundary condition (6) is
1 1
V
:
…15†
ao …v; 0† ˆ 2 2 tan21 2
4 p
v
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I. P. Lopes and D. Gough
The exact eigenfunction
p
c ˆ A cos s 2 2 pao
4
…16†
is obtained by integrating the first-order equation (12) numerically
from s ˆ 0 to obtain a o, and then evaluating A using equations
(13) and (14). We discuss in Section 4 the result of doing so for a
model of the Sun.
2.3
Inversion for the total acoustic phase
The leading-order asymptotic eigenfrequency equation for modes
of low and intermediate degree can be written (e.g. Gough 1984b,
1986; Christensen-Dalsgaard et al. 1985; Brodsky & Vorontsov
1988; Vorontsov & Zharkov 1989; Lopes & Turck-ChieÁze 1994):
…R
…n ‡ a†p
; F…n; v; a† ˆ F…w; a† ; a21 …1 2 a2 =w2 †1=2 d ln r;
v
r1
…17†
where a…r† ˆ c=r; w ˆ v=L; L ˆ l ‡ 1=2; and the lower limit of
integration is the radius r1 at which a ˆ w: The different
functional dependences of F and F on the oscillation eigenfrequencies enables one to determine F and F from the frequency
data. The latter can be inverted to determine the sound speed c as a
function of r if the functions F and F are continued analytically
between the discrete values of n, w and v . The former defines
the phase a . That phase contains two principal components, one
arising from the surface layers, the other from regions of rapid
variation and, if the mode penetrates that far, from the deep
interior. In the case when l is sufficiently high for the modes to be
essentially confined to the convection zone, so that they experience no region beneath the outer layers in which the scale of
variation of the background state is small (such as the lower
boundary of the convection zone), the lower contribution to a is
very nearly zero, and therefore the variation of a with v is a
diagnostic of the surface layers alone.
Several techniques have been used to extract F and F from
frequency data, either directly using the leading-order equation
(17) (e.g. Gough 1986; Brodsky & Vorontsov 1993) or from more
accurate representations that take into account corrections to that
equation arising from the neglected buoyancy frequency and
gravitational potential perturbation (e.g. Gough 1984b; Vorontsov
& Zharkov 1989; Brodsky & Vorontsov 1993; Gough & Vorontsov
1995; Christensen-Dalsgaard & Thompson 1997). Here we follow
Brodsky & Vorontsov (1987), Lopes (1994) and Lopes & TurckChieÁze (1994), by differentiating equation (17) to yield
d a
­ ln v 21
­ ln v ­ ln v
ˆ
b…v; l† ; 2v2
12
2
;
dv v
­n
­ ln n
­ ln L
…18†
in which v is regarded as a differentiable function of the (now)
continuous variables n and L. (Later, we shall use equation 18
also to relate the outer phase signature b o to the outer phase a o.)
Thus it is b that is determined directly by the data. The phase a is
then determined from b by integration:
…
b
a…v† ˆ 2
dv ‡ C v;
…19†
v2
where C is a constant of integration. It is evident from equation
(17) that the value of C is related directly to the value of R that
is adopted for the fiducial radius (recall that a=w ! 1 where
j1 2 r=Rj ! 1†: In this context, the acoustic phase, a , absorbs the
deviation of the leading-order asymptotic eigenfrequency equation
(17) from the full eigenfrequency equation for linear non-radial
adiabatic oscillations. For practical reasons, a (or b ) is separated
into two components, as we have already pointed out, corresponding to the phase arising from the interior and that from the outer
layers. The second component, which has been named the outer
acoustic phase, a o, is the main subject of this paper. The phase
produced in the deep interior, the inner acoustic phase, a i, is
almost zero for all the acoustic modes except for those of low and
intermediate degree which penetrate sufficiently deeply into the
star to experience regions in which the density is high, and thus
where the Eulerian perturbation to the gravitational potential is no
longer negligible (cf. Fig. 6). An important contribution arises also
from the geometry of the spherical background state (Gough
1993). In contrast to the outer phase, a i is dependent on the degree
of the mode. Consequently, there is the potential of separating the
two components of a in an analysis of an acoustic spectrum, as
has already been discussed in the case of the Sun and Sun-like
stars (Lopes 1994; Lopes & Turck-ChieÁze 1994; Lopes et al.
1997).
3 D E T E R M I N AT I O N O F T H E O U T E R P H A S E :
I N T E G R A L R E P R E S E N TAT I O N
If the stratification of the surface layers of the star were
polytropic, a o would be a constant. Indeed, provided R is chosen
to be near to the point where the linear extrapolation of c2(r) from
the almost adiabatically stratified region of the convection zone
vanishes (cf. Fig. 1), the variation of a o as v varies is small
compared with the mean value of a o. This result motivates one to
express the structure of the outer layers of a star in terms of its
deviation from that of a representative polytrope (Gough 1986).
The deviation of the phase from that of the polytropic value can
then be calculated by perturbation theory. The usefulness of the
outcome is that it provides a representation in which one can
conveniently identify the contributions from the regions of
relatively rapid variation, such as the superadiabatic boundary
layer and the He ii ionization zone.
3.1
The background state
The solar convection zone extends from radius r . 0:713R to the
photosphere. Except in the upper superadiabatic boundary layer,
in which significant amounts of heat are transported by both
convection and radiation, the stratification is very close to being
adiabatic. This is particularly so beneath the ionization zones of
the abundant elements H and He …t * 1000 s†: As can be seen
from Fig. 1, however, the variation of c2 is quite close to the linear
polytropic behaviour even in the H ionization zone, because the
scaleheight of g exceeds the depth by a substantial margin. Thus
there are two prominent distinct regions evident in Fig. 1(a): the
almost polytropic adiabatically stratified interior …t * 100 s†; and
the nearly isothermal atmosphere above …t & 40 s†; in which c .
7 km s21 : The region between them is essentially the superadiabatic boundary layer and the photospheric transition into the
optically thin atmospheric layers. In this region supercritical
acoustic waves are partially reflected (Balmforth & Gough 1990).
Note that the `surface' of the reference polytrope (such as the
polytrope indicated by the dashed line in Fig. 1a) does not
necessarily coincide precisely with the fiducial surface, at r ˆ R;
q 2001 RAS, MNRAS 322, 473±485
Seismology of stellar envelopes
477
(a)
(a)
(b)
(b)
HeII
HeI
HI
Figure 1. (a) Variation of the sound speed with acoustic depth in a solar
model, from the temperature minimum t ˆ 0 …r ˆ 6:964 1010 cm† into
the adiabatically stratified body of the convection zone. The dashed linear
extrapolation vanishes at r ˆ 6:972 1010 cm; this corresponds to the
sound speed of a polytrope of index …m ‡ 1†=g0 ˆ 3:3 (after Balmforth &
Gough 1990). (b) Variation of the first adiabatic exponent g with acoustic
depth (continuous curve) and the corresponding local polytropic index
…g 2 1†21 (dotted curve). The horizontal bars indicate the locations of the
ionization zone of H and the first and second ionization zones of He,
extending from 10 per cent to 90 per cent ionization.
of the star. We shall find that the most convenient polytrope for
our purpose has a polytropic index m characteristic of the value of
…g 2 1†21 in the region of wave propagation immediately beneath
the upper turning point of the mode under consideration, where
U~ 2 is comparable with v 2.
In the upper layers of a star, where total or partial reflexion of
acoustic waves takes place, the rapid variation of g implies a rapid
variation of m , and the background state can be approximated well
by a single polytrope only in narrow regions. In the Sun, for
example, there is a sharp transition in m above the superadiabatic
layer …t & 80 s†; followed by a decrease from a value of about 5.0
immediately beneath the H ionization zone to 1.5 just above the
He ii ionization zone (cf. Fig. 1b). Consequently, one might expect
modes of different frequency, and consequently with different
upper turning points, to require different representative polytropes.
However, the differences are moderated by the fact that reflexion
of the waves occurs over a finite distance, comparable with the
inverse vertical wavenumber, and in the case of the Sun it is
possible to choose a single representative polytrope for the
reference state.
q 2001 RAS, MNRAS 322, 473±485
Figure 2. The continuous curve corresponds to the potential U 2 , obtained
from a solar envelope, the dashed vertical line defines the photosphere, and
the dot-dashed curve corresponds to U 20 ; the potential of the representative
polytrope. In this case, U 20 is given by …m2 2 1=4†=t~ 2 ; where m ˆ 3:01 and
t0 ˆ 111:0 s: (a) The dotted lines correspond to the frequencies of waves
with n ˆ 2:0 and 4.0 mHz; (b) the dotted lines correspond to frequencies
n ˆ 0:3; 0:5; 1:0; 2:0; 3:0; 4:0 and 5.0 mHz.
The outer turning points of acoustic waves with cyclic
frequencies between 2 and 4 mHz are located between 6:953 1010 cm …t ˆ 130 s† and 6:958 1010 cm …t ˆ 80 s† (Fig. 2). This
group of waves is highly sensitive to the stratification of the
convecting layers below 6:956 1010 cm …t ˆ 100 s† and above
6:910 1010 cm …t ˆ 400 s†; where the adiabatic stratification of
convection is determined by a polytrope whose index is about 3
(Figs 1b and 2). This is consistent with the results obtained from
the phase calibration procedure that we discuss later. In our
method, the index of the representative polytrope is estimated a
posteriori, by finding a value for which the difference between the
outer acoustic phase obtained from a theoretical model and that
obtained from a table of frequencies is small.
3.2 The acoustic wave function
We consider the acoustic potential to be composed of the potential
U 20 of a plane-parallel polytrope and a (dimensionless) perturbation U~ 2 …t; t0 ; v† ˆ v22 …U 2 2 U 20 †: The polytrope has index m
and its `surface' is at an acoustic height t 0 above the fiducial
surface of the star. Thus, density, pressure and sound speed are
2 2…m‡1†
given by r ˆ r0 t~ 2m ; p ˆ g21
and c ˆ c0 t~ ; respectively,
0 c0 t~
where t~ ˆ t ‡ t0 is the acoustic depth beneath the surface of the
478
I. P. Lopes and D. Gough
polytrope, r 0 and g 0 are constants, and c0 ˆ g0 g=2…m ‡ 1†: The
idea is to choose m such that U~ 2 is small, and t 0 such that the
representation of the eigenfunction is conveniently simple. Since
U 20 is inversely proportional to tÄ 2:
U 20 ˆ
m2 2 1=4
;
t~ 2
…20†
the wave equation (2) is of Riccati±Bessel type. We write it in the
form
d2 c
m2 2 1=4
Wc ; 2 ‡ 12
c ˆ U~ 2 c;
…21†
s~ 2
d~s
where s~ ˆ tv
~ ˆ s ‡ s0 : The right-hand side is responsible for the
deviation of c from the polytropic Bessel form. It can be regarded
as a source of forward and backward scattering. The value of t 0 is
determined by insisting that the solution c 0 of W c ˆ 0 is of the
simple form given by equation (22); it depends on the outer
boundary condition, and is approximately 111.0 s for our solar
model (Fig. 3).
In Fig. 2 we compare the acoustic potential U 2 in a solar model
envelope with the potential U 20 of a representative plane-parallel
(a)
polytropic envelope. The deviation of the solar envelope from a
polytrope as plotted in these figures is responsible for only part of
the seismic difference; there is also a geometrical contribution,
particularly at the greater depths, arising from the fact that the
envelope has spherical and not planar symmetry. Note, however,
that both U 2 and U 20 are small at greater depths, and therefore the
geometry does not contribute substantially to the surface phase
a o; this is certainly the case for the Riccati±Bessel potential U 20 ;
and is evidently also the case for the yet smaller potential U 2 : One
might have thought that it would have been better to have chosen a
smaller value of m to reduce that difference. However, it transpires
that to do so increases the difference in the upper reflecting layers,
and so leads to a deterioration in the accuracy of the predictions of
a o and b o (see Fig. 5).
It is evident from Fig. 2 that one can identify two sources of
reflexion. The more prominent is the rapid variation of U 2 with t
between t . 40 s and t . 90 s: It is associated with H ionization
and superadiabaticity at the top of the convection zone (and
produces an acoustic cut-off for 5-min waves at t . 75 s†: We are
thus led to surmise that a calibration of the structure of the star in
the vicinity of the photosphere might be accomplished using an
appropriate signature of the waves scattered from those layers.
The second source of reflexion is the ripple in the vicinity of
t . 600 s; produced by the second ionization of He. Both
ionization zones cause g to drop, as is evident in Fig. 1(b); the
first ionization of He is merged with the tail of the H ionization,
and is not easily distinguished from it by visual inspection.
Finally, we stress that the acoustic waves with cyclic frequency
greater than 3 mHz are weakly scattered by the interface between
the superadiabatic layer and the atmosphere.
The behaviour of acoustic waves propagating through the
polytropic background state is determined by the wave equation
(21) with the right-hand side set equal to zero. This differential
equation has a general solution of the form
s~1=2 ‰J m …~s† ‡ eY m …~s†Š;
(b)
where Jm and Ym are Bessel functions of the first and second kinds
respectively, and the constant e is determined from the outer
boundary condition. However, with the objective of obtaining a
simple asymptotic expression for the outer phase, we choose a
more convenient representation by casting the leading-order wave
function in the form
c . c0 ˆ s~1=2 J m …~s†;
Figure 3. Representation of the variation with the cyclic frequency of (a)
the dimensionless acoustic height of the top of the reference polytrope
above the fiducial surface of the star, s0, and (b) the actual acoustic height,
t 0. The continuous curves correspond to the exact solution of equation
(23), and the dot-dashed curves correspond to the approximate solution
given by equation (24).
…22†
our aim being to choose an appropriate value of t 0 to render this a
good approximation. We remark that had we chosen a background
state corresponding to the potential U0 with s0 ˆ 0; the wave
function c satisfying equation (21) (including the inhomogeneous
term) would not have taken adequately into account the scattering
in the upper layers. In fact, the singularity that occurs in U 20 at the
origin introduces a divergence of the potential U~ 2 at the surface of
the polytrope, where scattering of acoustic waves is high. This
means that the potential U~ 2 is not a small quantity in that region,
and a straightforward perturbation method cannot be used.
Nevertheless, we have succeeded in removing this deficiency by
introducing the non-zero value t 0, forcing the scattering potential
U~ 2 and the wave function not to be excessively susceptible to the
singularity. This improves significantly the convergence of the
asymptotic series, allowing us to obtain an approximate expression for the wave function c that takes into account almost all the
scattering produced. One way to choose the depth t 0 is to require
q 2001 RAS, MNRAS 322, 473±485
Seismology of stellar envelopes
that the outer boundary condition (6) be satisfied by the function
c 0 given by equation (22); it is then determined by the condition
J m‡1 …vt0 †
V
1
1
ˆ m‡
2
:
…23†
J m …vt0 †
v
2 vt0
We solve this equation for t 0 by Newton±Raphson iteration.
An explicit expression for s0 can easily be obtained when s0 is
small, for then the Bessel functions can be expanded near the
origin. In that case, s0 is given approximately by
0
s1
V
V2 2m ‡ 1 A
@
:
…24†
‡
s0 ; vt0 , …m ‡ 1† 2 ‡
v
v2
m‡1
In the case of the Sun, the dimensionless acoustic depth s0 of the
surface of the reference model exhibits a linear dependence on the
frequency, as expected from the behaviour of the right-hand side
of equation (24) for small values of v /V (Fig. 3a). We present also
the acoustic height t 0 of the surface reference polytrope (Fig. 3b),
which for certain purposes is more useful than s0. The analytical
expression (24) is a good approximation for cyclic frequencies
n ˆ v=2p smaller than about 2.5 mHz, and loses accuracy as n
increases. The variation of t 0 (Fig. 3b) is determined by the upper
layers in the vicinity of the fiducial surface of the star. However,
for certain purposes it is convenient to adopt a constant (characteristic) value of t 0, which in the solar case is about 111.0 s
(corresponding to r ˆ 6:964 1010 cm†: This value corresponds to
the acoustic height in the star at which the sound speed of the
reference polytrope vanishes (cf. Fig. 1).
Equation (21), together with the boundary condition (6),
describes the propagation of acoustic waves in the upper layers
of the star, as is determined by the original wave equation (2) and
boundary condition (6). A representative polytrope has been
chosen to approximate the background state. Taking advantage of
the fact that the potential difference U~ 2 between the star and the
polytrope is small in the envelope, the contribution of U~ 2 to the
wave propagation can be estimated using a perturbation method.
Accordingly, the wave function is written in the integral form:
c…v; s~† ˆ s~1=2 J m …~s†
… s~
p
‡
…~s 0 †1=2 J m …~s 0 † U~ 2 …~s 0 ; v†c…v; s~ 0 †d~s 0 s~1=2 Y m …~s†
2 s0
… s~
p
…~s 0 †1=2 Y m …~s 0 † U~ 2 …~s 0 ; v†c…v; s~ 0 †d~s 0 s~1=2 J m …~s†:
‡
2 s0
…25†
This wave function can be approximated asymptotically in the deep
interior of the stellar envelope where the scattering is negligible and
the values of the integrals in equation (25) approach constants. The
(renormalized) wave function is thus cast in the form
c ˆ s~1=2 ‰J m …~s† ‡ g*Y m …~s†Š;
…26†
where g * (which we trust will not be confused with the adiabatic
exponent introduced in Section 2) determines the contribution to c
from the scattered waves produced by the interaction of the Jm with
U~ 2 : We now consider g * to be an expansion parameter, and retain
only terms linear in g * in the expansion. Accordingly, we ignore the
effect of scattering on the wave function c where it appears in the
scattering integrals on the right-hand side of equation (25), which is
achieved by replacing c by the function c 0 given by equation (22).
In this linear approximation the second scattering term is neglected
q 2001 RAS, MNRAS 322, 473±485
479
because the correction to Jm , after normalizing the amplitude,
contributes a term that is quadratic in g *. We note that the
linearization is valid only for modes with intermediate values of n ;
as can be seen in Fig. 4a, jg*j & 0:1 when 1:2 mHz & n &
4:2 mHz:
It is possible to isolate two components of g * …ˆ g1 ‡ g2 †: The
first corresponds to the scattered acoustic waves generated
predominantly in the interior of the stellar envelope:
… tmax
p
g1 ˆ v2
~ 0 † dt~ 0 ;
…27†
U~ 2 …t 0 ; t0 ; v†t 0 J 2m …tv
2
t0
where t 0 ˆ t~ 0 2 t0 is the (dummy) acoustic depth beneath r ˆ R;
and the value of t max is large enough for the integral to be
insensitive to that value. Normally, we take t ˆ tmax to be just
above the base of the convective zone; in the case of the Sun,
tmax . 2000 s: The component
… tmax
p
U~ 2 …t 0 ; t0 ; v†J 2m …tv
g2 ˆ v2 t0
~ 0 † dt 0
…28†
2
t0
is sensitive predominantly to conditions in the outer layers of the
star.
In Fig. 4(a) is illustrated the amplitude g * of the scattered
component of the wave function c evaluated deep in the stellar
envelope. It is plotted against the cyclic frequency n . Acoustic
waves of lower frequency ± with n & 1:3 mHz ± are reflected in
the envelope of the star, and the tunnelling through the atmosphere
is very small; then g 1 dominates the main behaviour of g *. Modes
with the higher frequencies are more sensitive to the superadiabatic region, and to the atmosphere of the star; as g increases
above 3.5 mHz, g * becomes progressively more influenced by g 2.
There are two prominent features of g 2 worthy of mention. The
first is the dip centred at about 0.4 mHz; as will become evident,
this arises principally from the deviation of U 20 from U 2 beneath
the He ii ionization zone. The second is an oscillatory component
with a `period' T21 of about 0.7 mHz; it is caused by the modulation of U~ 2 by He ii ionization at an acoustical depth 12 T . 700 s:
Contributions to g * from contiguous intervals of t are depicted
in Fig. 4(b). From this figure it is evident that the most prominent
feature of g * in Fig. 4(a), namely the dip centred at about 0.4 mHz
coming from g 2, is produced by the deviation U~ 2 in the deep
adiabatically stratified layers beneath the He ii ionization zone,
where the stratification of the star is more nearly approximated by
a polytrope of index 1.5 than it is by the reference polytrope (see
Fig. 1b). It is also evident that high-frequency waves are particularly sensitive to the deviation of the convection zone from the
single reference polytrope just below the superadiabatic region,
100 & t & 500 s: The highest frequency waves are also sensitive
to the outer superadiabatic boundary layer of the convection zone
above t ˆ 100 s: Contributions from the intermediate intervals are
generally smaller, but are none the less not negligible. The oscillations result from the variation with frequency of the effective
phase of the eigenfunctions; they depend not only on U~ 2 ; but also
on the dissection of t .
3.3 The outer acoustic phase
To obtain an expression for the outer acoustic phase, we cast
equation (26) into the form (5):
h
i
p
c ˆ A …v; vt† cos vt 2 2 pao …v; vt† ;
…29†
4
480
I. P. Lopes and D. Gough
Figure 4. (a) Variation of g * (continuous curve) with cyclic frequency n ; the components g 1 (dashed curve) and g 2 (dot-dashed curve) are also depicted. (b)
Contributions to g * from intervals t ˆ 0±100 s (dotted), 100±500 s (short-dashed), 500±700 s (dot-dashed), 100±1000 s (3dot-dashed), 1000±1500 s (longdashed) and 1500±2050 s (continuous).
where A(v ; vt ) is the amplitude. We adopt the expressions
…p~s=2†1=2 J m …~s† ˆ P cos x 2 Q sin x;
…p~s=2†
1=2
Y m …~s† ˆ P cos x ‡ Q sin x;
which are valid for large x ˆ s~ 2 p=4 2 mp=2; in which
P…m; s~† ˆ
1
X
…m; 2k†
…21†k
;
…2~s†2k
kˆ0
1
X
…m; 2k ‡ 1†
…21†k
;
…2~s†2k‡1
kˆ0
ÿ
ÿ
where …m; k† ; G m ‡ k ‡ 12 =k!G m 2 k ‡ 12 is Hankel's symbol
Q…m; s~† ˆ
(e.g. Abramowitz & Stegun 1964). Then
m v
1
Q 2 g*P
:
ao …v; vt† ˆ 2 t0 2 arctan
2 p
p
P ‡ g*Q
…30†
The convergence of the series for P and Q is rapid, and for our
purposes it is adequate to retain only the first terms of each,
namely 1 for P and …4m2 2 1†=8tv
~ for Q. Therefore, in the solar
envelope near the base of convective zone, for example, where tÄ is
large, the outer phase can be approximated by
2
m vt0 1
4m 2 1
2 arctan
ao …v; vt† . 2
2 g* :
…31†
2
p
p
8tv
~
If we consider t~ ! ‡1; then the outer phase ao …v† ; ao …v; 1†
q 2001 RAS, MNRAS 322, 473±485
Seismology of stellar envelopes
can be written as
ao …v† .
m vt0 1
2
‡ arctan g*:
2
p
p
…32†
The seismic signature b o(v ; vt ) can be obtained by substituting
expression (31) for a o into the definition (18):
bo …v; vt† ˆ
m 1
v2 dt0 v
2
2 arctanQ ‡
2 p
p dv p
ÿ
21 4m2 2 1
1 dt0
dg*
1
‡
‡
;
1 ‡ Q2
t~ dv
8tv
~ 2
dv
where
Q.
…33†
4m2 2 1
2 g*:
8tv
~
In the next section we shall verify the accuracy of these approximations to the outer acoustic phase. The phase will allow us to
determine seismic signatures of the background state, especially in
the regions where the classical asymptotic expressions fail.
4 C O M P U TAT I O N O F T H E P H A S E I N T H E
SOLAR CASE
Our purpose in this section is to test the theoretical expressions
obtained for the outer acoustic phase and to discuss the limits of
their validity. In particular, we wish to ascertain whether the
approximate expression (30), based on the linearization (26) of the
solution to the exact equation (25), can provide a good
representation of the outer phase; we find that it can, provided
that m is chosen judiciously. We have chosen for our equilibrium
structure a solar model with standard microscopic physics,
corresponding to the solar reference model presented by TurckChieÁze & Lopes (1993).
481
The numerical tests of the expressions are performed in two
steps. First, we have determined numerically the outer phase
a o(v ) by integrating equation (12) subject to condition (15), and
from it computing the seismic parameter b o(v ) according to the
definition (18). The result is compared with the values obtained
from the analytical expression (30). In addition, we have computed b (n , l) from the theoretical frequency table, as described in
Section 2.3. The eigenfrequency table is determined by solving
numerically the fourth-order system of linear adiabatic non-radial
oscillations, with two regularity conditions at the centre and the
two boundary conditions at the surface determined by matching to
the upper layers of the solar model an isothermal atmosphere at
the temperature Tm (Unno et al. 1989). The mechanical condition
(6) applied at the outer boundary is the same as that adopted
earlier for determining the outer acoustic phase.
In Fig. 5 we present the seismic parameter b o(n ), computed by
the two different methods and plotted against cyclic frequency n .
(In this section we adopt the convention, not uncommon in the
astronomical literature, of using arguments as labels to denote the
variables on which we are regarding the quantities to depend,
rather than as being the values of the arguments of mathematical
functions; thus, for example, we take ao …n† ˆ ao …v† when
n ˆ v=2p, not when n ˆ v:† Also, we present b (n , l) obtained
from the theoretical eigenfrequencies for acoustic modes with
degrees between 0 and 20. The computations were performed over
a range of frequencies between 0.3 and 5.0 mHz. We point out that
when n is less than typically 1.5 mHz, the accuracy of the
asymptotic expressions can be poor, even in very simple models
such as polytropes.
We notice that the `observable' b (n , l) has two principal
components (Lopes 1994). The first is produced in the external
layers of the star, and has a characteristic bell shape, which can be
identified with b o (see Fig. 5). The second component, b i, is the
difference between b and b o, and is illustrated in Fig. 6. Its
dominant behaviour is a decline with frequency at fixed l
Figure 5. The acoustic signature b . The function b o(n ) was obtained by numerical differentiation of ao =v: The continuous curve was obtained from a o
obtained from equation (30), and the dashed curve from a o obtained by integrating equation (12) with m ˆ 3:15: The computation of b (n ) from the
theoretical frequency table is calculated for modes with degrees smaller than 20: l ˆ 0…‡†; l ˆ 1…†; l ˆ 2…e†; l ˆ 3…S† and up to 4 # l # 20 (´). The value
of b (n , l) obtained from the frequency table each comprises two terms: one related to the internal phase, b i(n ,l), to which only the most penetrating lowdegree modes are sensitive, and a second term related to the phase in the surface, b o(n , l), to which low- and intermediate-degree modes are sensitive (see
text).
q 2001 RAS, MNRAS 322, 473±485
482
I. P. Lopes and D. Gough
(a)
(a)
(b)
(b)
Figure 6. Variation of b i(v , l): (a) with the cyclic frequency n and (b) with
w(;v /L); b i(n ) is computed as the difference b…v; l† 2 bo …v†; where
b (v , l) is obtained from the theoretical frequency table and b o(v ) is
determined from a o by integrating equation (12). Values of b i(n ) have
been computed from modes with frequencies between 1.4 and 4.0 mHz,
and with degrees up to 20: l ˆ 0…‡†; l ˆ 1…†; l ˆ 2…e†; l ˆ 3…S†; 4 #
l # 20 (´).
proportional to n 21, and is due predominantly to the pulsational
perturbation to the gravitational potential and to buoyancy in the
radiative interior, both of whose leading effects are to add a term
of the form n 21G(w) to a and b . (In that case, equation 18 still
holds, provided the derivative of a /v with respect to v is
interpreted as a partial derivative at constant w.) The decline of its
magnitude with increasing l is a result of increasing far-field
cancellation.
Superposed on this behaviour is a decaying oscillatory feature,
which is characteristic of there being a localized region of rapid
variation of a dynamically significant property of the background
envelope which scatters the waves (Gough 1990). Its `period' T21
is about 0.70 mHz, implying that the scattering region is located at
an acoustical depth t . 12 T . 700 s; which is where helium
undergoes second ionization (Fig. 1). The decline with n of the
amplitude of this oscillatory feature arises from increasing
destructive interference between the scattered waves as the wavelength of the principal (incident) wave diminishes; it becomes
evident once the wavelength of the principal wave becomes
comparable with the thickness of the scattering region. At least
part of this feature should be regarded as being a component of the
outer phase; it arises in b 2 bo because equation (12) for a o is not
exact, partly because it disregards the horizontal variation of the
non-radial modes, and partly because it ignores the perturbation to
Figure 7. Variation with the cyclic frequency of (a) the outer phase a o(n )
and (b) the seismic parameter b o(n ). The dotted curves correspond to a o
computed from equation (30) with m ˆ 1:75; the dashed curves with m ˆ
3:01; and the dot-dashed curves with m ˆ 3:15: The continuous curves
correspond to the numerical determination of a o(v ), from equation (12).
In all cases, b o was obtained from a o by numerical differentiation
according to equation (18).
the gravitational potential. The contribution from the latter is
global, and cannot be assigned solely to either a i or a o.
We note that in the case of modes of low degree, the number of
eigenmodes used to determine b (n , l) from the frequency table is
relatively small; consequently, b (n , l) is not determined accurately.
The function b o(n ) determined from the stellar envelope is in
quite good agreement with the frequency dependence of b (n , l)
when l * 15; particularly when 1:3 & n & 4 mHz: The modes do
not penetrate sufficiently deeply into the star to be severely
influenced by the perturbation to the gravitational potential, nor
sphericity and buoyancy. However, when b o is determined from
equation (30) or (33), there can be disagreement at low frequencies. This is because the difference between U and U0 can be quite
large in the vicinity of the upper turning point of the mode (see
Fig. 2) where the outer phase is particularly susceptible to the
stratification, rendering linearization of c to determine g * inaccurate. Similarly, for acoustic waves with frequencies greater than
about 4 mHz, the discrepancy between b (n , l) and b o(n ) (determined from the solar envelope either analytically or numerically)
is again relatively large (see Fig. 5); in this case, acoustic waves
propagate almost up to the superadiabatic boundary layer of the
q 2001 RAS, MNRAS 322, 473±485
Seismology of stellar envelopes
(a)
(a)
(b)
(b)
Figure 8. The variation with cyclic frequency n of the acoustic outer phase
a o(n ) obtained from equation (30) and the seismic parameter b o(n ), for
the solar case. The index of the reference polytrope is m ˆ 3:01: The
dotted curves correspond to the same case, but ignoring the phase
produced in the solar envelope …g* ˆ 0†; in this case, only the contribution
from the outer boundary condition to the outer phase is included (see
Section 3.2).
convection zone, within which U 2 becomes negative, and
experience tunnelling and multiple reflexions. The discrepancy
increases as the frequency of the mode approaches the acoustic
cut-off frequency. To improve the determination of b o(n ) at high
frequency directly from the stellar envelope requires a more
realistic account of acoustic wave propagation in the subphotospheric layers to be taken.
The analytical approximation (33) to b o(n ) is obtained in terms
of a representative polytrope of index m . The value of m is
determined a posteriori by comparison of b o with b (n , l), the
latter having been obtained from a table of precise eigenfrequencies of the entire stellar model. It is influenced predominantly by
the stratification of the outer layers of the star, near the upper
turning points of the modes. In the case of the Sun, m takes a value
close to 3, which leads to a fair correspondence between b o(n )
and b (n , l) when l is not small. The sensitivity of b o to the index
of the representative polytrope is illustrated in Fig. 7, in which
both a o and b o determined by equation (30) are compared with
the values obtained by numerical integration of equation (12).
Since the parameter b o is a derivative of a o, it is much more
sensitive to the stratification, auguring the power of b to diagnose
the structure of the outer layers of a star.
Another important factor determining the outer acoustic phase
comes from the value of the acoustic height, t 0, of the top of the
q 2001 RAS, MNRAS 322, 473±485
483
Figure 9. Variation of the outer phase a o(n ) and of the seismic parameter
b o(n ) with the cyclic frequency, for the solar envelope. The continuous
curves were obtained from equation (30) with m ˆ 3:01: The dot-dashed
curves correspond to a first approximation given by equation (31), and the
dashed curves to the approximation (32).
reference polytrope, which is influenced at high frequencies by
the propagation properties of the atmosphere. The contribution to
the outer phase indices a o and b o is illustrated in Fig. 8 for the
solar case. The phase indices were computed from equation (30),
with g* ˆ 0 and with t 0 determined, as usual, by equation (23).
Thus they correspond to the modes of an incomplete polytrope of
index m supporting an isothermal atmosphere. In all cases the
outer phase index a o exhibits a single maximum, at a frequency
which increases as m increases.
For frequencies above about 1.5 mHz the simplifications
suggested in Section 3.3 capture much of the effect of scattering.
This is evident in Fig. 9, in which the values of expressions (31)
and (32) for a o, and the corresponding values of b o, are compared
with expression (30) from which they were derived.
5
SUMMARY AND CONCLUSIONS
It has been known since the first sound-speed inversions
(Christensen-Dalsgaard et al. 1985; Gough 1985) that the asymptotic description of wave propagation is invalid in the upper layers
of the stellar envelope, where the background state varies rapidly.
This is still a major difficulty for inferring the radial profile of
sound speed in those layers. Duvall (1982) had already demonstrated that the observed stationary-wave frequencies could be
represented by a single curve (given by equation 17), which relates
484
I. P. Lopes and D. Gough
the sound travel time along a ray to the sound speed at the lower
boundary of the acoustic cavity. This was accomplished by
assigning the value 1.5 to the phase a , which minimized the
scatter about a single curve. The relation was close to a power law
in w (there having been only high-degree modes in Duvall's
original analysis), implying that the sound speed c in at least the
outer layers of the Sun varies approximately as a power of depth z
(Gough 1984a). Indeed, c2 was found to be not far from a linear
function of z, which is a property of a plane-parallel polytrope, a
model which had been used earlier to perform a seismic
calibration of the depth of the convection zone (Gough 1977a)
and which was suggested subsequently as being a suitable
reference against which to compare a realistic model envelope
(Gough 1986), provided the index m is taken to be about 2a . 3:
Deviations from the polytropic structure cause deviations from the
Duvall law. This is what has led us to develop here a careful
analysis of the scattering processes occurring in the upper layers
of the star, for the purposes of inferring the structure of those
layers. The high accuracy of the observed acoustic frequencies of
the Sun, as well as the accuracy expected of imminent measurements of the frequencies of Sun-like stars from the space missions
COROT, MONS and MOST, is adequate to detect the scattering,
and so to provide the means for calibrating stellar models.
Our analysis has been carried out by the following two
procedures. First, we presented two methods to determine the
acoustic phase produced in the outer layers of the star, one by
integrating the wave equation numerically through those layers,
and the other as a perturbation about a reference polytropic state.
The methods complement one another, the first being the more
precise, and the second being physically more transparent. The
numerical method has been discussed by Lopes et al. (1997) for
obtaining the outer phase as a seismic signature of the outer layers
of the Sun and Sun-like stars. The second method uses a suitably
chosen reference polytropic envelope about which to perform a
perturbation expansion.
The total acoustic phase can also be inferred directly from the
eigenfrequencies of oscillation, and thus is essentially observable.
That phase can be split into an outer and an inner component,
which are particularly sensitive to the outer layers and the inner
layers, respectively, of the star. The inner component is influenced
by buoyancy, sphericity and the acoustic perturbation to the
gravitational potential, which are typically neglected in asymptotic analyses (Gough 1993; Lopes & Turck-ChieÁze 1994). In a first
approximation, the outer phase depends on frequency v alone,
whereas the inner phase depends also on l. Evidently, it is
necessary to have data from modes whose frequencies span
adequately wide ranges of both v and l for a separation of the two
components to be possible.
Our principal achievement is to have shown the manner by
which the outer acoustic phase is a signature of scattering in the
upper layers of the star, and to indicate how it might be extracted
from low-degree frequency data. That phase offers the means to
investigate properties of the outer envelope associated with rapid
spatial variation. In particular, it could be used to calibrate the
superadiabatic upper layers of the convection zone of a stellar
model, and thence the theory of convection that had been used to
construct that model. It can also be used to calibrate the helium
abundance, as have PereÂz HernaÂndez & Christensen-Dalsgaard
(1994) for the Sun, using the oscillatory component of the phase
parameter b o (see also Lopes et al. 1997).
We stress that the analysis has been carried out under the
adiabatic approximation, and that dynamical interactions with the
convection have been ignored. Gough (1984c) and Balmforth
(1992a,b), using a non-local formulation of mixing-length theory
(Gough 1977b) which takes account of the influence of the mean
turbulent pressure, estimate a decrease by up to 10 mHz in the
pulsation frequencies below 5 mHz. These effects are not insignificant, and should eventually be included in the analysis.
The application of this technique to the Sun and its extension to
other stars can improve our understanding of the physical
processes that occur in the upper layers of the envelopes, in
particular in the superadiabatic regions. In the case of the Sun, a
preliminary diagnostic of the solar envelope has been carried out
using data obtained with the MDI/SOI instrument (Lopes &
Gough 1998). Furthermore, the method has been used to calibrate
the outer envelopes of Sun-like stars of 1.35 M( (Lopes et al.
1997). In this case, the outer acoustic phase was successfully
determined from a table of artificial `observed' frequencies,
assuming that the observational noise is smaller then 0.5 mHz and
the identification of the modes has been carried out correctly.
AC K N O W L E D G M E N T S
We thank T. Sekii, S. Turck-ChieÁze and S. Vorontsov for stimulating discussions, and F. PeÂrez HernaÂndez for suggestions for
improving the presentation. IPL is grateful for support by a grant
from PPARC.
REFERENCES
Abramowitz M., Stegun I., 1964, Handbook of Mathematical Functions.
National Bureau of Standards, Washington DC
Appourchaux T. et al., 1995, in Ulrich K. R., Rhodes J. E. Jr, DaÈppen W.,
eds, GONG 1994: Helio- and Astero-seismology from the Earth and
Space. Astron. Soc. Pac., San Francisco, p. 408
Audard N., Provost J., Christensen-Dalsgaard J., 1995, A&A, 297, 427
Balmforth N. J., 1992a, MNRAS, 255, 603
Balmforth N. J., 1992b, MNRAS, 255, 632
Balmforth N. J., Gough D. O., 1990, ApJ, 362, 256
Barbikov V. V., 1976, Method of Phase Functions in Quantum Mechanics.
Nauka, Moscow
Brodsky M. A., Vorontsov S. V., 1987, SvA, 13, (3), 179
Brodsky M. A., Vorontsov S. V., 1988, in Christensen-Dalsgaard J., eds,
Proc. IAU Symp. 123, Advances in Helio- and Asteroseismology.
Reidel, Dordrecht, p. 137
Brodsky M. A., Vorontsov S. V., 1989, SvA, 15, (1), 27
Brodsky M. A., Vorontsov S. V., 1993, ApJ, 409, 455
Catala C. et al., 1995, Hoeksema J. T., Domingo V., Fleck B., Battrick B.,
ESA SP, Proc. 4th Soho Workshop, Helioseismology. European Space
Agency, Paris, p. 549
Christensen-Dalsgaard J., 1993, in Weiss W. W., Baglin A., eds, Proc. IAU
Colloq. 137, Inside the Stars
Christensen-Dalsgaard J., Frandsen S., 1983, Sol. Phys., 82, 165
Christensen-Dalsgaard J., PeÂrez HernaÂndez F., 1988, in Rolfe E., eds,
Seismology of the Sun and Sun-like Stars. ESA SP-286, Noordwijk,
p. 478
Christensen-Dalsgaard J., PeÂrez HernaÂndez F., 1992, MNRAS, 257, 62
Christensen-Dalsgaard J., Thompson M. J., 1997, MNRAS, 284, 527
Christensen-Dalsgaard J., Cooper A. J., Gough D. O., 1983, MNRAS, 203,
165
Christensen-Dalsgaard J., Duvall T. L.,Jr Gough D. O., Harvey J. W.,
Rhodes E. J. Jr, 1985, Nat, 315, 378
Christensen-Dalsgaard J. et al., 1996, Sci, 272, 1286
Duvall T. L. Jr, 1982, Nat, 300, 242
Dziembowski W. A., Pamyatnykh A. A., Sienkiewicz R., 1991, MNRAS,
249, 602
q 2001 RAS, MNRAS 322, 473±485
Seismology of stellar envelopes
Fridlund M. et al., 1995, in Ulrich K. R., Rhodes J. E. Jr, DaÈppen W., eds,
GONG 1994: Helio- and Astero-seismology from the Earth and Space.
Astron. Soc. Pac., San Francisco, p. 416
FroÈhlich C. et al., 1995, Solar Phys., 162, 101
Gabriel A. et al., 1995, Solar Phys., 162, 61
Gough D. O., Bonnet R. M., Delache Ph. G., 1977a, Proc. IAU Colloq. 36.
Clermont-Ferrand, p. 3,
Gough D. O., 1977b, ApJ, 214, 196
Gough D. O., 1984a, Mem. Soc. Astron. Ital., 55, 13
Gough D. O., 1984b, Phil. Trans. R. Soc. Lond., A313, 27
Gough D. O., 1984c, Adv. Space Res., 4, 85
Gough D. O., 1985, in Chen B., de Jager C., eds, Proc. Kunming Workshop
on Solar Physics and Interplanetary Travelling Phenomena, Vol. 1.
Science Press, Beijing, p. 137
Gough D. O., 1986, in Gough D. O., eds, Seismology of the Sun and the
Distant Stars. Reidel, Dordrecht, p. 125
Gough D. O., 1990, in Osaki Y., Shibahashi H., eds, Lecture Notes in
Physics Vol. 367, Progress of Seismology of the Sun and Stars.
Springer-Verlag, Berlin, p. 283
Gough D. O., 1993, in Zahn J. P., Zinn-Justin J., eds, Astrophysical Fluid
Dynamics. Elsevier, Amsterdam, p. 399
Gough D. O., 1996, in Roca CorteÂs T., eds, The Structure of the Sun.
Cambridge Univ. Press
Gough D. O., Toomre J., 1991, ARA&A, 29, 627
Gough D. O., Vorontsov S. V., 1995, MNRAS, 273, 573
Gough D. O. et al., 1996, Sci, 272, 1296
Harvey J. W. et al., 1996, Sci, 272, 1284
Kjeldsen H., Bedding T. R., 1998, in Kjeldsen H., Bedding T. R., eds, The
First MONS Workshop: Science with a Small Space Telescope. Aarhus
Universitet, Aarhus, Denmark, p. 1
q 2001 RAS, MNRAS 322, 473±485
485
Libbrecht K. G., Woodard M. F., 1991, Sci, 253, 152
Lopes I., 1994, PhD thesis, Universite de Paris VII
Lopes I., Gough D. O., 1998, ESA-SP, Structure and Dynamics of the
Interior of the Sun and Sun-like Stars, SOHO 6/GONG 98 Workshop,
Boston, Massachusetts, p. 485
Lopes I., Turck-ChieÁze S., 1994, A&A, 290, 845
Lopes I., Turck-ChieÁze S., Goupil M. J., Michel E., 1997, ApJ, 480,
794
Matthews J. M., 1998, ESA-SP, Structure and Dynamics of the Interior of
the Sun and Sun-like Stars, SOHO 6/GONG 98 Workshop, Boston,
Massachusetts, p. 395
PeÂrez HernaÂndez F., Christensen-Dalsgaard J., 1994, MNRAS, 267, 111
PruÈfer H., 1926, Mathematische Annalen 95
Roxburgh I. W., Vorontsov S. V., 1994, MNRAS, 267, 297
Scherrer P. H. et al., 1995, Solar Phys., 162, 129
Sekii T., Shibahashi H., 1989, PASJ, 41, 311
Tassoul M., 1980, ApJS, 43, 469
Tassoul M., 1990, ApJ, 358, 313
Turck-ChieÁze S., Lopes I., 1993, ApJ, 408, 347
Unno W., Osaki Y., Ando H., Saio H., Shibahashi H., 1989, Non-radial
Oscillations of Stars. Univ. Tokyo Press, Tokyo
Vorontsov S. V., 1991, SvA, 35, (4), 400
Vorontsov S. V., Zharkov V. N., 1989, Ap. Space Phys. Rev., 7, 1
Zayer I. et al., 1995, in Ulrich K. R., Rhodes J. E. Jr, DaÈppen W., eds,
GONG 1994: Helio- and Astero-seismology from the Earth and Space,
p. 456
This paper has been typeset from a TEX/LATEX file prepared by the author.