Radiation Transfer in the Cavity and Shell of Planetary Nebulae

0.34 MB·English

by  M. D. Gray

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed20January2012 (MNLATEXstylefilev2.2)
Radiation Transfer in the Cavity and Shell of Planetary
Nebulae
M.D. Gray1, M. Matsuura2, A.A. Zijlstra1
2
1 1 Jodrell Bank Centre for Astrophysics, Alan Turing Building, Universityof Manchester, M13 9PL, UK
0 2 UCL-Instituteof Origins,Department of Physics and Astronomy, UCL, Gower Street,
2 London WC1E 6BT, UK

n
a
J Accepted ....Received...;inoriginalform...

8
1
ABSTRACT
] We develop an approximate analytical solution for the transfer of line-averagedradi-
A ation in the hydrogen recombination lines for the ionized cavity and molecular shell
G of a spherically symmetric planetary nebula. The scattering problem is treated as a
perturbation, using a mean intensity derived from a scattering-free solution. The an-
.
h alytical function was fitted to Hα and Hβ data from the planetary nebula NGC6537.
p The position of the maximum in the intensity profile produced consistent values for
- theradiusofthecavityasafractionoftheradiusofthedustynebula:0.21forHαand
o
r 0.20 for Hβ. Recovered optical depths were broadly consistent with observed optical
t extinction in the nebula, but the range of fit parameters in this case is evidence for a
s
a clumpy distribution of dust.
[
Key words: stars:AGBandpost-AGB–stars:evolution–Infrared:stars—(ISM:)

1 planetary nebulae: general – ISM: jets and outflows – ISM: molecules –
v
0
3
9
3 1 INTRODUCTION Aself-consistentradiativetransfercodeforthismixture
1. has been developed by Ercolano, Barlow & Storey (2005).
0 Shells with internal cavities are found in almost all PNe They have used a Monte Carlo method. Here, we use an
2 (Balick & Frank 2002). Such shells and cavities are related analytical solution, with some approximations, andconcen-
1 totheshaping ofPNebystellar winds. Lowand intermedi- trateontheconfigurationofcavitiesandshellsaroundthem.
: ate mass stars lose their mass at a very high rate, forming The aim of the paper is therefore to obtain detailed physi-
v
a circumstellar envelope. Fast stellar winds from the cen- calinsight throughasimplifiedanalyticalmodel,whichcan
i
X tral stars of PNe overtake the slower, denser, AGB ejecta, complement the more complex information from numerical
r resultingininteractionofthefastandslowstellarwindcom- solutions.
a ponents (Kwok, Purton & Fitzgerald 1978). The fast wind
sweeps the slow wind, and leaves a cavity inside. Cavities
are also found in supernova remnants, (Dwek et al. 1987;
Lagage et al. 1996) but these structures are more compli- 2 THE MODEL OF NGC6537
cated (Bilikova et al. 2007). Shell-cavitydimorphism is also
Inthissection,wedescribethephysicalandradiativetrans-
found in LBV stars and in AGN.
fer models that are used in our analysis of NGC 6573. Al-
In one of the well studied bipolar planetary nebulae, though certain parameters are definitely specificto this ob-
NGC6537,thecoreconsistsofbrightarcstracingashellsur- ject, taken from Matsuura et al. (2005), much of the model
rounding an elongated cavity (Matsuura et al. 2005). Arcs is generally applicable to any planetary nebula of similar
arebrightinHαandotherrecombination lines,andextinc- geometry, and in a similar state of evolution.
tion maps derived from Hα and Hβ suggests the presence
of dust grains in the arcs. Matsuura et al. (2005) suggested
that little dust exists in the cavity. Deriving accurate dust
2.1 Physical Model of the Nebula
densityfromHαandHβlinemapsisnotstraightforward,as
the light from the central star is scattered within the arcs. We assume that the nebula NGC6537 is accurately spheri-
In these arcs, both dust grains and gas are mixed together. callysymmetric.Thelayoutofthevariousradialzonesofthe
Therefore, wedevelopedaradiativetransfercodetoresolve object is summarised in Fig. 1. The central star is a white
cavity and shells for PNe, which includes scattered light in dwarfofnegligiblesolidangle,bothfromanobserver’spoint
a shell. of view, and from the modeller’s: no rays in the radiative
(cid:13)c 0000RAS
2 M.D. Gray, M. Matsuura et al.
2.2 Radiative Transfer Model
ThevolumeelementDinFig.1hasvolumeδV.Ifweletthis
C beunitvolume,thentheradiated powerperunitfrequency
and solid angle in theHα line is
hν
j (Hα)= 32A n (r)φ , (1)
ν 4π 32 3 ν,32
R
D whereν isthe‘laboratory’line-centrefrequencyoftheHα
32
line,andA istheEinsteincoefficientforspontaneousemis-
32
A sion. The subscripts refer to the principal quantum num-
r
bers of atomic hydrogen, so n (r) is the number density
3
of H-atoms in the upper state of the transition. With an
R
isothermal approximation, the lineshape function, φ is
2 B ν,32
not afunction of radius, butmaybea function of direction
if there are significant radial motions in the cavity or shell.
Giventhesedefinitions,asimilarexpressionmaybewritten
down for theemission coefficient of Hβ:
E j (Hβ)= hν42A n (r)φ . (2)
ν 4π 42 4 ν,42
Weassumecompletevelocityredistribution,sothatabsorp-
tionlineshapefunctionsareidenticaltothosespecifiedabove
4"
for emission.
The radiative transfer equation is
2M .2 1M .5 Extinction
nˆ ∇I = [κc(r)+σc(r)+κ (r)]I
ν ν ν
Figure1.Adiagramoftheplanetarynebulamodel.Thespher- · + −κc(r)B (T(r))+j (r)+σc(r)J (r), (3)
ically symmetric nebula is centered on the star, A, of negligible ν ν ν
solid angle. This is surrounded by the cavity, B, and molecular where nˆ is a unit vector along the ray of specific intensity
shell,C. An infinitessimal volume, D, of the cavity is marked at
I . Opacity is provided by radius-dependentabsorption co-
ν
radius, r. Optical radiation, E, escapes from the shell after suf- efficients, κ , κc, for the line and continuum, respectively,
feringsomedegreeofextinction.Theparticularvaluesshown(in ν
andthecontinuumscatteringcoefficient,σc.Kirchoff’sLaw
magnitudes)applytoNGC6537.
allows the continuum emission from dust to be written as
κc(r)B (T(r)), whilst the line emission coefficient is j (r),
ν ν
andwillbeoneofthespecificformsineq.(1)oreq.(2)forthe
appropriatetransition.Thefinaltermontheright-handside
ofeq.(3),equaltoσc(r)J ,isthescattering integralassum-
ν
ingisotropic,elasticscattering,andJ istheangle-averaged
ν
intensity.
transfer model are considered to start or end on the stellar
surface. Weexpandtheleft-handsideofeq.(3)insphericalpolar
coordinates (for example Peraiah (2002)). This introduces
The central star emits sufficient vacuum ultraviolet ra-
µ= cosθ where θ is the angle between the direction of ray
diation to ionize a surrounding cavity. Ions in this cav-
propagation andtheradialdirection.Ontheright,wecom-
ityundergofrequentrecombinationandphotoionizationcy-
binethecontinuumabsorptionandscatteringintoanextinc-
cles, and it is the main source of Hα and Hβ line radi-
tion coefficient, χc(r), and assume that the line absorption
ation. For NGC6537 specifically, the observed flux ratio
is negligible when compared to that in the continuum. The
F /F = 2.79, where these fluxes are averages over
Hβ Hα
transfer equation with thesemodifications is
the appropriate spectral-line bandwidth. From this ratio,
we adopt radially constant values of the electron temper- ∂I (1 µ2)∂I
natu=re,10T4ecm=−3.1.5 × 104K, and electron number density, µ ∂rν + −r ∂µν = −χc(r)Iν+κc(r)Bν(T(r))
e + σc(r)J +j (r) (4)
SurroundingtheHiicavityisaneutralshell,composed ν ν
mainlyofmolecularmaterial.Thisshellcontainsdust,which Wenowintegrateeq.(4)overthespectral-linebandwidthap-
weassume,forthecaseofNGC6537,tohaveamassfraction propriate to either the Hα or the Hβ line. We assume that
of0.01.Mostoftheshellmaterialis,ofcourse,intheformof this bandwidth is adequate to cover all the line radiation
molecular hydrogen. The shell produces approximately two in the two hydrogen lines studied, regardless of all Doppler
magnitudes of visual extinction in NGC6537, derived from shiftswithinthesource.Inthisconnexion,wenotethatthe
the measured excess, E(Hα Hβ). The shell is spatially velocity extentof thelineduetothebulkmotion of expan-
resolved, and the extinction −varies from 1.5magnitudes at sionisoforder20kms−1,whilstthethermalDopplerwidth
thecentre to 2.2magnitudes at 4′′ off centre. is21.4T1/2kms−1,withT equaltothekinetictemperature
4 4
At present, we leave the form of the density profile as in the ionized cavity in units of 104K. An unknown micro-
an unknown function in both the cavity and shell zones of turbulent width must be added in quadrature to the latter
thenebula. figure. It is therefore reasonable to suppose that even the
(cid:13)c 0000RAS,MNRAS000,000–000
Radiation Transfer in PNe 3
extremered-andblue-shiftedportionsofthelinearesignifi-
cantlyblendedbythethermalandmicroturbulentlineshape. r^
Inotherwords,thecombinationofalowterminalexpansion
velocity and alarge thermal plusmicroturbulent linewidth
Shell
allowsustoneglectvelocityfieldinducedDopplershiftsthat
would otherwise complicate the analysis considerably. Let ω
the spectral-line bandwidth be ∆ν, and the line-integrated
intensity is given by
R
∆ν/2
I = I dν (5)
ν s
Z
−∆ν/2
A similar equation to eq.(5) relates the angle-averaged in-
tensities J and Jν. If we assume also that thefunctions χc, r
κc, B (T) and σc vary only very slightly over ∆ν, these θ
ν
functions can be removed from thefilter integral when it is r^
applied to eq.(4). The result is
n^ θ
∂I (1 µ2)∂I
µ∂r + −r ∂µ = −χc(r)I+∆νκc(r)Bν(T(r))
+ σc(r)J+j(r), (6) Cavity
∆ν/2
where j(r) = j dν. The only frequency-dependent
−∆ν/2 ν
part of j is thRe appropriate lineshape function (see eq.(1)
ν
andeq.(2)).Weassumethatthefilterwidthissufficientfor
Figure 2.Arayentersthecavity,ofradiusR,atthetopofthe
thenormalisationconditionofthelineshapetohold,sothat
figurealongpathsindirectionnˆ.Therayisinitiallyatanangle
theintegral j(r) in eq.(6) is either,
ωtotheradialunitvector,rˆ.Thisrelationshipismodifiedalong
hν sasrˆchangesdirection:atanarbitrarypointatradiusrtheray
j(Hα)= 4π32A32n3(r) (7) unit vector nˆ and the radial unit vector cross at angle θ. Note
that whensissmallerthanthe distance tothemid-pointofthe
or theequivalent expression for Hβ.
chord,θ becomes anobtuseangle.
Finally,were-writeeq.(6)inseparateformsappropriate
tothecavity,andtotheshell,respectively.Inthecavity,we
maketheapproximationthatthereisnodust,andthatother
Thecavityandshellsolutions, forthespecificintensityofa
processes, such as free-free and bound-free emission, make
raythatmaypassthroughbothzones,arecombinedinSec-
a negligible contribution to the radiation flux within the
tion 6. Section 7 is devoted to developing a mean intensity
filter bandwidthsused, for example Matsuura et al. (2005).
in the shell, as a function of radius, by angle-averaging the
Thecavity therefore actseffectively as apuresourceof line
unperturbed solution from Section 6. This angle-averaged
radiation, and the radiation transfer equation in this zone
intensityisused todevelop explicit forms for thescattering
is,
perturbation in Section 8. The remaining sections are con-
∂I (1 µ2)∂I cernedwithobservationalpropertiesofthesolution,andfits
µ + − =j(r). (8)
∂r r ∂µ to HSTdata for NGC6537.
By contrast, the shell, which is rich in dust, and where al-
mostallthehydrogenismolecular,doesnotemitanyHαor
Hβlineradiation,isasiteofcontinuumabsorbtion,scatter-
3 THE CAVITY SOLUTION
ing and thermal emission by dust. The shell transfer equa-
tion is therefore,
Thelayout of a typical ray passing through thecavity zone
∂I (1 µ2)∂I ofthenebulaisdepictedinFig.2.Themodelcavityhasno
µ + − = χc(r)I innerradius.Allraysenterthecavityfrom thesurrounding
∂r r ∂µ −
+ ∆νκc(r)B (T(r))+σc(r)J (9) shellwithsomeboundaryvalueofthespecificintensity.This
ν value may differ from ray to ray, and, at present, we leave
Wenowproceedtosolveeq.(8)andeq.(9)inturn,coupling thisvalueas an unknown.The emission coefficient depends
the solutions via boundary conditions. The solution is pre- ontheradiusonlythroughthenumberdensityofHatomsin
sented in several stages, and we outline these briefly here. the upper level of the appropriate Balmer line from eq.(7).
InSection 3,wesolveeq.(8)forthespecificintensityofline Wemaketheapproximationthatthesenumberdensitiesare
radiationinthecavity,andthissolutionisangle-averagedin constants in the cavity, on the grounds that the cavity gas
Section4yieldingameanintensityinthecavityasafunction is in pressure equilibrium because of its high sound speed,
ofradius.Thisquantityisnotrequiredforlatercalculations, and that we have already assumed a constant temperature
soSection 4isonlyincludedfor completeness.InSection 5, in this region. Weignore the effect of overpressured regions
wesolveeq.(9)forradiativetransferintheshell:initiallythe near the edge of the cavity where the ionized medium is
specific intensity is obtained for dust extinction only, but a juxtaposed against the surrounding shell (Perinotto et al.
scattering source is then re-introduced as a perturbation. 2004). Therefore, we can re-cast eq.(8) as
(cid:13)c 0000RAS,MNRAS000,000–000
4 M.D. Gray, M. Matsuura et al.
∂I (1 µ2)∂I
µ∂r + −r ∂µ =j0, (10)
where j is the constant emission coefficient. The left-hand
0
side of eq.(10) is the spherical polar expansion of the dot
product,nˆ ∇I,butanalternativerepresentation isasthe
•
derivative,dI/ds,taken along theray characteristic. Anal- ω
ternativeway of writing eq.(10) is therefore, R s
dI
=j (11)
ds 0
which has thesimple linear solution, r^
r P
I(s)=I +j s, (12) θ
0 0
whereI istheunknownintensitywithwhichtherayenters n^
0
the cavity from the shell. The problem now consists only
of re-expressing the distance along the characteristic, s, in
terms of the independent variables r and µ=cosθ. To ob-
tain such a relation, we apply the sine rule to the triangle
in Fig. 2, which is boundedby r, s and R, yielding,
r R s
= = . (13) Figure 3.Theangleaveragedintensity,J,istobecomputedat
sinω sinθ sin(ω+θ) point P, at radius r from the centre. The average is over rays
NowbothRandω areconstants,soitfollowsfromthefirst (examples marked with arrows) which make angles, θ, with the
radiusvectoratP,whereθmaybeintherange0toπ.Apartic-
equality in eq.(13) that
ular ray is shown entering the cavity at angle ω, in direction nˆ,
rsinθ=Rsinω=K, (14) andpassingthroughadistancesinordertoreachP.
is a constant along the characteristic. Applying the cosine
rule to the same triangle yields a quadratic equation in s. Thesolidangleintegralforthemeanintensity,withthe
The geometry of Fig. 2 requires the positive root, so that integration over theazimuthal angle already carried out is
thedistance along thecharacteristic is,
1 1
s=rcosθ+(R2 r2sin2θ)1/2. (15) J(r)= 2Z I(r,µ)dµ. (18)
− −1
Usingtheresultineq.(14),andthedefinitionofµ,weobtain Aftersubstitutionofeq.(17)intoeq.(18),andapartialeval-
s=rµ+(R2 K2)1/2. The constant expression under the uation of theresulting integral, we find,
−
squarerootreducestoRcosω,sothecharacteristicdistance
iss=rµ+Rcosω.Theintensityalongthecharacteristic is J(r)=J + 1j r 1(α+µ2)1/2dµ, (19)
therefore, 0 2 0 Z
−1
I(r,µ)=I0+j0(rµ+Rcosω), (16) where α = [(R/r)2 1], and J0 is the angle average of I0.
−
The result is a standard integral in terms of the arcsinh
which is the required cavity solution for constant emission
function, and after converting this to logarithmic form, the
coefficient.Itistrivialtoprove,fromeq.(16)thatitsatisfies
angle-averaged intensity is
eq.(10).
j R R r2 r+R
J(r)=J + 0 1+ 1 ln . (20)
0 2 (cid:20) r (cid:18) − R2(cid:19) (cid:20)√R2 r2(cid:21)(cid:21)
4 MEAN INTENSITY IN THE CAVITY −
Wenotethattheapparentinfinityineq.(20)atr=0disap-
Toobtain theangle-averaged intensity,J(r),we needtoin- pearswhenthisequationisreplacedbyasuitableexpansion
tegrate over a number of different rays, which all meet at for thecase r R. The small radius form is,
≪
one point, where the radius, r, is a constant. At this point,
J(r) J +(1/2)j R(2 r2/R2). (21)
θ is the final angle along its path, and can be used in the → 0 0 −
required solid angle integral. However, now ω is a variable, Weplot,inFig.4,thefunctionQ=(J(r) J )/(j R)from
0 0
asweareintegratingovermanyrays,andω needstobeex- eq.(20), as a function of thedimensionless−radius x=r/R.
pressed in terms of θ, or µ. The situation is summarised in
Fig.3.Resultsfrom thecavitysolution,whichstillapplyto
Fig.3,arethatsinω=(r/R)sinθ ands=rcosθ+Rcosω.
5 THE SHELL SOLUTION
We use the first of these relations to eliminate ω from the
second. The expression for s can in turn be used to write Were-write the shell transfer equation as
thecavity solution, eq.(16) in theform,
∂I (1 µ2)∂I
I(r,µ)=I0(µ)+j0(rµ+[(R2 r2)+r2µ2]1/2, (17) µ∂r + −r ∂µ=−χc(r)I+f(r), (22)
−
noting that µ=cosθ is theonly variable on theright-hand where f(r) is an arbitrary function of the radius, which in-
side of eq.(17). corporates the continuum emission and line scattering. We
(cid:13)c 0000RAS,MNRAS000,000–000
Radiation Transfer in PNe 5
5.1 Shell Solution with Power-Law Density
To progress beyond eq.(24) we require some analytical ap-
proximationforthebehaviouroftheradius-dependentfunc-
tions.Westartwiththeextinction,χc.Weassumethatthis
depends on the radius only through the number density of
the shell dust, so that the functional form is the same for
both the absorption and scattering contributions. Further,
we assume a density dependence which follows the inverse
square behaviour of the singular isothermal sphere. There-
fore, we suppose that theextinction in the shell behavesas
χc(r)=χc(R)(r/R)−2, (27)
where χc(R) is the maximum value of theextinction coeffi-
cient,foundjustoutsidethecavityboundary.Withthehelp
of eq.(26), we can write the extinction as a function of s
ratherthanr. Integralsof thetypewhich appear in eq.(24)
Figure 4. A dimensionless form of the angle-averaged intensity
inthecavity,[J(r)−J0]/(j0R),seeeq.(20),plottedasafunction can nowbe written in theform,
ofdimensionlessradius,r/R.Equation21isusedforthepointat
ds
r/R=0. χc(r(s))ds=R2χc(R) . (28)
Z Z s2−2sR2cosω+R22
Introducing the new variables x = s R cosω and a =
2
−
R sinω, eq.(28) may be written as thestandard integral,
2
change the left-hand side of eq.(22), as in the cavity solu-
dx
tion,todescribeasolutionalongtheraycharacteristic.The χc(r(s))ds=R2χc(R) , (29)
equationcanthenbeputinstandardfirst-orderlinearform Z Z x2+a2
as
which has the solution (as an indefiniteintegral),
dI
+χc(r)I =f(r(s)). (23) R2χc(R) s R cosω
ds χc(r(s))ds= arctan − 2 . (30)
Z R2sinω h R2sinω i
Equation 23 may be integrated by the standard method of
Equation24containstwodefiniteformsofeq.(30).Thefirst
integratingfactors.Theboundaryconditionats=0isthat
of these has limits of 0 to s, and can be re-written, with
thespecific intensity enterstheshell from interstellar space
thehelpofadditionformulaeforthearctangentfromGrad-
withtheintensityI ,assumedequaltoatypicalvaluefor
BG
shteyn& Ryzhik(1965), as
Galactic starlight. The extinction coefficient at this same
Wpoistihtiotnhiissczoenrod,itrieognarimdlpesosseodf,itesq.v(a2r3i)athioans wthiethfionrmthaelsshoelull-. sχc(r(s′))ds′= β π +arctan ssinω ,(31)
tion, Z0 sinω h{ } (cid:16)R2−scosω(cid:17)i
where the π is to be included only for ray distances such
I(s) = IBGse−R0sχc(r(s′))ds′ s stehcaotnsd>defiRn2i/tecoisntωe,graanldfotrhme gfrroomupeβq.(=24)Rh2χasc(tRh)e/Rlim2.itTshse′
+ f(r(s′))exp χc(r(σ))dσ ds′. (24) to s, and it is convenient to write it without combining the
Z0 (cid:26)−Zs′ (cid:27) arctangents, since thepart with s can beremoved from the
′
integralovers.Overall,theshellsolution can nowbewrit-
The ray distance, s, is related to the radius r and direction
ten,
cosine µ by a modified form of the relation which appears
in eq.(16). If theshell has an outer radius R , then
2 β sinω
I(s)=I exp π +arctan
s=rµ+R2cosω, (25) BG (cid:26)−sinω (cid:20){ } (cid:18)(R2/s)−cosω(cid:19)(cid:21)(cid:27)
−βu(s) s ′ βu(s′) ′
where,forthemoment,weignorethepresenceofthecavity. +e sinω f(s)exp ds, (32)
Z (cid:26) sinω (cid:27)
Withthissamecaveat,rsinθ=R sinω isaconstantalong 0
2
a given ray if ω is the angle with which the ray enters the where
shellfrominterstellarspace.DefiningK =R sinω,wecan
2 2 s R cosω
expressµintermsoftheradiusasµ=(1 K2/r2)1/2,which u(s)=arctan − 2 (33)
− 2 h R2sinω i
may be substituted into eq.(25) to yield a relation between
randswithnoothervariablesinvolved.Thisrelation,with and thefunctional form of f remains to be determined.
theradius as the subject, is
r=(s2−2sR2cosω+R22)1/2. (26) 5.2 Scattering as a perturbation
The expression for the radius in eq.(26) can be used to ex- If we ignore continuum emission in the shell, the unknown
pandtheradiusintermsofsineq.(24),providedthatfunc- function,f ineq.(32)reducestolinescatteringalone,which
tional forms for χc(r) and f(r) are known. depends on the angle-averaged mean intensity, J(r). It is
(cid:13)c 0000RAS,MNRAS000,000–000
6 M.D. Gray, M. Matsuura et al.
then possible, in principle, to average eq.(32) over solid an-
gle,leadingtoanintegro-differentialequationforJ(r).How-
P
ever, owing to the difficulty in attempting to solve such an
equation analytically, we resort to thesimpler procedure of
ω
treating the scattering term as a perturbation. For this ap- s
p
proximation to be very good, the scattering contribution
to the extinction should be small, so that σc(r)/χc(r) is a
Q
small parameter for all radii. This is unlikely to be true for
realdust.Forexample,silicatedustmodelledbyOssenkopf, R2 ω’
Henning&Mathis(1992) hasanopticalefficiencyforscat- R
M
tering which is consistently 2-3 times that for absorption σ s’
∼
overopticalwavelengths.However,westilladoptthepertur-
bative approach as the only viable method of obtaining an Q’
approximate analytical solution. In this procedure, we first
r
solve the radiative transfer equation in the shell, asuming θ
that line extinction is the only contribution to the right- X
hand side of eq.(22). This means that we can set f = 0 in
s
eq.(32), and takeas the zero-order solution in theshell, shell
Z
βR
I(r,µ)=IBGexp(cid:26)r(1−µ2)21/2 [(π)
−
(1 µ2)1/2[rµ+(R2 r2(1 µ2))1/2]
+arctan − 2− − ,(34)
(cid:18) r(1 µ2) µ(R2 r2(1 µ2))1/2 (cid:19)(cid:21)(cid:27)
− − 2− −
which is just the first term of eq.(32), with θ and ω elimi- Figure 5. The complete ray path is shown divided into three
nated in favour of µ = cosθ, and with the help of the re- segments:between PandQadistancesp iscoveredintheshell,
lation sinω = (r/R )sinθ (see eq.(14), which still holds in havingentryangleω.Therayenters thecavityatQ,withangle
the shell). It is tedi2ous, but straightforward, to show that ω′, and covers the distance s′ between Q and Q’ before exiting
intotheshell.ThepointMisthemid-pointofs′,andtheclosest
eq.(34) is indeed a solution of eq.(22), for the case where
approach of the ray to the central star (at O). Once inthe shell
f(r) = 0. Equation 34 cannot be used alone, as along any
again, the ray reaches the general point X, at radius r, where
paththroughthenebula,araymayencounterasequenceof
its path makes an angle, θ, with the radial direction. The ray
shell,cavity,andagainshellconditions.Wethereforeneedto
directionvectorandradialdirectionvector(atvariouspositions)
determine some functional form for IBG for the case where aremarkedwithboldandlightarrowsrespectively.Inprogressing
a ray leaves the cavity and re-enters the shell, forming a fromQ’ to X, the ray covers the distance sshell. The ray finally
combined solution along a ray. leavesthenebulaatpointZ,wherer=R2.
As scattering is now to be treated as a perturbation,
it can be computed from a mean intensity which is derived
from thezeroth-ordercombinedsolution. Anequation fora therayentersthecavity.Theperturbationisthenconstant
perturbation in the intensity, δI, can be constructed from in the cavity, forming a finite value of I(s ) for the second
0
eq.(9) by expanding the specific intensity as I = I∗ +δI, shell segment, where s0 is now the path length where the
and then subtracting off the equation in the zeroth-order ray leaves the cavity.
estimate, I∗. The result is,
∂(δI) (1 µ2)∂(δI)
µ + − = χc(r)δI+σcJ(r), (35)
∂r r ∂µ − 6 THE COMBINED SOLUTION ALONG A
where we have assumed that continuum emission makes a RAY
negligible contribution, so that f(r) = σcJ(r). So, mathe-
Given the approximations made in the previous section, it
matically, the equation in the perturbation may be treated
isnowpossibletocombinethecavityandshellsolutionsfor
in the same way as the shell equation, eq.(22). The formal
a single ray. The geometry of the combined solution is set
solutionforδIthereforelookslikeeq.(24)withthescattering
out in Fig. 5. As we are ignoring, as this stage, rays which
term replacing the unknown function f. The perturbation
pass solely through the shell, each effective ray has three
solution is therefore given by
segments:thefirstisanabsorptivepassagethroughtheshell.
δI(s) = δI(s0)e−h(s) However, if the input background at r = R2 is very weak,
s we can ignore this segment, and treat the specific intensity
+ e−h(s) σc(r(s′))J(r(s′))eh(s′)ds′, (36) of theray at thecavity boundary,r=R as zero.
Z
s0 The second segment of the ray path, between points
where the function h is given by eq.(30). It is important to Q and Q’ in Fig. 5, passes through the cavity. Here the
notethat there are threepossible versions of eq.(36): A ray specificintensityisassumedtoincreasethroughspontaneous
which avoids the cavity has δI(s )=0 and a lower limit of emissionintheline,accordingtothecavitysolution,eq.(16).
0
s = 0 on the remaining integral. A ray which enters the Inparticular,wewantanexpressionforthespecificintensity
0
cavity has two shell segments. In the first, δI(s ) = 0 and atQ’,wherethecavitysolutionbecomestheinputvaluefor
0
s = 0, but the upper limit is the value of the path where the shell solution, which takes over as the third segment of
0
(cid:13)c 0000RAS,MNRAS000,000–000
Radiation Transfer in PNe 7
the ray path, which continues until point Z is reached, and When eq.(25) has been applied, and eq.(41) has been re-
theray passes into thevacuum. expressed entirely in terms of µ, theresult is,
At the point X in Fig. 5, which is as radius r, the ray s β µ
hastravelledadistances throughtheshell,anditmakes χc(x)dx = arctan
anangleθwiththeradiaslhvelelctor.Atthispoint,thesolution Zs′+sp (1−µ2)1/2 (cid:26) (cid:20)(1−µ2)1/2(cid:21)
will be the shell solution, with an input specific intensity (1 (r/R)2(1 µ2))1/2
arctan − − .(42)
from the cavity. − (cid:20) (r/R)(1 µ2) (cid:21)(cid:27)
Analysis of the triangle MOQ shows that the cavity −
′ ′ The first arctangent can beconverted to a simpler form, as
segmentoftheraypathhaslengths =2Rcosω .Thetotal
an arcsine, via therelation
distancefrompointPtopointMis,fromthetriangleMOP,
equaltosM =R2cosω.Therefore, the‘pre-cavity’distance x
′ arctan =arcsinx, (43)
(from P to Q) is sp = R2cosω Rcosω . The additional (cid:20)(1 x2)1/2(cid:21)
distance along the ray from M t−o X is sM′ = rcosθ = rµ. −
which is taken from Gradshteyn & Ryzhik (1965). The sec-
The total distance along the ray from P to X is therefore
ond arctangent in eq.(42) can be written as the reciprocal
s=R cosω+rµ,justasintheshellsolution,eq.(25).From
2
of the form which appears in eq.(43) by defining the new
theseresults,wefindthatthedistances inFig.5isgiven
shell variableq=r(1 µ2)/R.Anadditionalrelationlinkingarc-
by
−
tangents (Gradshteyn & Ryzhik 1965) for the case where
′
s =rµ Rcosω , (37) x>0, which is trueof q, is
shell
−
andbyapplyingthesineruletothetriangleOPQ,theangles arctan(1/x)=π/2 arctanx, (44)
−
′
ω and ω are related by andthisrelationallowsustouseeq.(43)directlyonthesec-
′ ondarctangentineq.(42).AthirdrelationfromGradshteyn
sinω =(R /R)sinω, (38)
2 & Ryzhik, arcsinx+arccosx = π/2 allows us to re-write
which can be used to eliminate ω′ from eq.(37), yielding eq.(42) as,
s β
s =rµ [R2 R2sin2ω]1/2. (39) χc(x)dx= [arcsinq θ]. (45)
shell − − 2 Zs′+sp (1−µ2)1/2 −
The limiting (maximum) value of ω, for a ray which just
Itisnowasimplemattertowriteoutthecombinedsolution
′
enters the cavity, occurs when ω = π/2, when sinω =
max foraraybyinsertingeq.(45)intotheshellsolution,eq.(24),
R/R .
2 with I given by eq.(40). With β and q fully expanded in
BG
terms of θ, theresult is
I(r,θ) = 2j R[1 (R /R)2sin2ω]1/2
0 2
−
6.1 Input Solution to the Shell Rχc(R) rsinθ
exp − arcsin θ ,(46)
As we are assuming that the specific intensity at point Q × (cid:26)(r/R)sinθ h (cid:16) R (cid:17)− i(cid:27)
in Fig. 5 is effectively zero, the input solution for the shell which is valid for R r R . For plotting, we re-write
2
is just the cavity solution evaluated at Q’. When this is eq.(46) in the form ≤ ≤
′
done, the cavity solution, eq.(16) applies, with ω replacing
oωb.tWaineitnagkeI(tRhi,sωs′o)lu=ti2ojnRwictohsIω0′.=W0,itrh=thRe haenldpµof=eqc.o(s3ω8)′,, I(x,a)=2j0R(1−a2)1/2exph−τa(arcsina−arcsinxa)i,(47)
0
this expression becomes an equation in two parameters, a = (R2/R)sinω, and the
opticaldepthparameter,τ =Rχc(R).Thelatterparameter
I(R,ω)=2j0[R2−R22sin2ω]1/2, (40) becomesequaltotheradialopticaldepthoftheshellinthe
limit where R R. The independent variable is x=r/R.
2
whichisthebackgroundforthesolutionintheshellbeyond ≫
Thislast definition implicitly introducesa thirdparameter,
point Q’.
theupperboundonx,equaltoR /R.Wenotethateq.(47)
2
reducesto a sensible limiting form, that is
I(x,a=0)=2j Rexp[ τ(1 R/r)], (48)
0
− −
6.2 Combined Solution in the Shell
for the case where a = 0. In Fig. 6 we show the function
As the limits of integration are modified for the combined U = I(x,a)/(2j0R) for five values of a between its mini-
solution, we generate the shell solution from the indefinite mum value of zero, and maximum of 1. We set the optical
integral, eq.(30). The upper limit is the distance along the depth parameter to be τ = 1.38 (see Section 10) to agree
raypath,s,butthelowerlimitisnow,fromFig.5,s′+s = approximately with the visual extinction of 1.5magnitudes
p
R cosω+[R2 R2sin2ω]1/2. The result for the extinction nearthecentreof NGC6537, and weplot thedimensionless
in2tegral is − 2 radius out to x=3.
s R2χc(R) s R cosω
χc(x)dx = arctan − 2
Zs′+sp R2sinω n h R2sinω i 7 MEAN INTENSITY IN THE SHELL
(R2 R2sin2ω)1/2 Theapproximationdiscussed inSection6.1-thatraysthat
arctan − 2 . (41)
− (cid:20) R2sinω (cid:21)(cid:27) do not cross the cavity contribute zero specific intensity -
(cid:13)c 0000RAS,MNRAS000,000–000
8 M.D. Gray, M. Matsuura et al.
suggests thesubstitution x=p2,yielding,
j Rρ2 1 1 x 1/2
J(ρ)= 0 −
2 Z (cid:18)1 ρ2x(cid:19)
0 −
1 1 3 1 1 3
exp γ F , ; ,x ρF , ; ,ρ2x dx(52)
× − 2 2 2 − 2 2 2
n h (cid:16) (cid:17) (cid:16) (cid:17)io
At this point, we expand the hypergeometric series which
appear in eq.(52). For the case here, where the first two
arguments are the same, the power series (Abramowitz &
Stegun 1965) is
a2z a2(a+1)2z2
F(a,a;g,z) = 1+ +
g1! g(g+1)2!
a2(a+1)2(a+2)2z3
+ +...., (53)
g(g+1)(g+2)3!
Figure 6.The function U =I(x,a)/(2j0R) (see eq.(47) plotted inthecaseofgenerala,g,andforthespecificcaseofa=1/2
asafunctionofdimensionlessradiusx=r/Rforvaluesofa=0.0 and g=3/2, we find,
(solidline),0.4(dottedline),0.6(dashedline),0.8(simplechain),
z 3z2 5z3
0.99(complexchain). F(1/2,1/2;3/2,z)=1+ + + +... (54)
6 40 112
Substitution of eq.(54) in eq.(52) leads to the expression
excludes all rays with negative values of cosθ. Rays with j Rρ2 1 1 x 1/2
non-zero specific intensity are limited to a subset of those J(ρ) = 02 Z (cid:18)1 −ρ2x(cid:19) exp{−γ[(1−ρ)
with positive values of cosθ, more precisely to those with 0 −
(1 ρ3)x 3(1 ρ5)x2 5(1 ρ7)x3
sinθ<R/r.Withtheadditionalrestriction thatanyradius + − + − + − +... ,(55)
where J(r) is computedhas R r R ,thesituation here 6 40 112 (cid:21)(cid:27)
2
≤ ≤
is similar to that shown in Fig. 3. The function to be aver-
which is as far as it is possible to proceed without making
agedisthecombinedsolutionspecificintensityinthezeroth-
furtherapproximations.
order (no scattering) approximation, given by eq.(46). The
above considerations allow us to write a formal integral for
themean intensity,recalling that rsinθ=R2sinω: 7.1 Approximate Forms
arcsin(R/r) Although the series in eq.(55) is convergent for all relevant
J(r) = j0RZ [1−(r/R)2sin2θ]1/2 valuesofx=[(r/R)sinθ]2,convergenceisratherslowwhen
0 x is close to 1, the condition for a ray in glancing contact
Rχc(R) rsinθ
exp arcsin θ sinθdθ(49) with the cavity.However,thecontribution of large x to the
× (cid:26)(r/R)sinθ h (cid:16) R (cid:17)− i(cid:27) integral is limited by the term in front of the exponential
which tends to zero as x 1 (for the particular case of
We simplify eq.(49) via a series of substitutions. The →
ρ = 1, see below). The physical reason for this is that rays
first of these is to let the impact parameter be p =
atlargeanglehaverelativelyshortpathsthroughthecavity,
(r/R)sinθ; we also define the new constant parameters,
andcorrespondinglysmallspecificintensitiesonentrytothe
γ = Rχc(R), and ρ = R/r. These definitions transform
shell.
eq.(49) to,
Althoughvaluesofρwhicharecloseto1(positionsclose
1 1 p2 1/2 to the shell/cavity boundary) can force the leading term in
J(r) = j0Rρ2Z p(cid:18)1 −ρ2p2(cid:19) eq.(55)to1,thisisnotaproblembecause,inthissituation,
0 − theargumentoftheexponentialapproacheszero,thewhole
γ exponential to 1, and integration of the leading term alone
exp − [arcsinp arcsin(ρp)] dp, (50)
× (cid:26) p − (cid:27) is sufficient for moderate accuracy. We evaluate eq.(55) to
twolevelsofaccuracyinx.Inthezero-orderapproximation,
notingthatpandρarealways 1.Giventhiscondition,we
≤ weabandonallbutthefirsttermintheseries,butthisdoes
canexpandtheinversesinesineq.(50)intermsofGausshy-
not depend on x, and can therefore be removed from the
pergeometricfunctions(forexample,Abramowitz&Stegun
integral, leaving,
(1965)). For complex argument z,
arcsinz=zF(1/2,1/2;3/2,z2), (51) J(ρ) j0Rρ2e−γ(1−ρ) 1 1−x 1/2dx. (56)
≃ 2 Z (cid:18)1 ρ2x(cid:19)
0 −
where the power series forming the Gauss hypergeometric
Onevaluationoftheintegralineq.(56),andrevertingtothe
function,F,isabsolutelyconvergentwithin,andon,theunit
original radius variable, r = R/ρ, we obtain the zero-order
circle for the arguments in eq.(51) (Gradshteyn & Ryzhik
approximation to themean intensity:
1965). The substitution of eq.(51) into eq.(50) has theben-
eficial consequence of cancelling p within the exponential, J(r)= j0Re−γ(1−R/r) 1 r2−R2 ln r+R . (57)
andleaving p2 everywhereexceptfor theproduct pdp.This 2 (cid:20) − 2rR (cid:16)r R(cid:17)(cid:21)
−
(cid:13)c 0000RAS,MNRAS000,000–000
Radiation Transfer in PNe 9
tousethemoreaccurateeq.(59).Therearevariousformsof
eq.(36)whichshouldbeusedfortheappropriatezoneofthe
source. For a ray which penetrates the cavity, we initially
consider the case where this ray has entered the shell, but
has not yet reached the cavity.Assuming a negligible input
intensityfromthevacuum,wehavetheboundarycondition
that s = 0 and δI(0) = 0, and therefore the problem for
0
thisray, and zone, reduces to solving theintegral,
s
δI(s)=e−h(s) σc(r(s′))J(r(s′))eh(s′)ds′. (60)
Z
0
′
The most useful form for the function h(s), which appears
in the integrating factor, for this zone is
′ R2χc(R)[π ω θ(s′)]
h(s)= − − , (61)
R sinω
2
Fofigduimreen7si.oTnlheessfurandctiuiosnxQ==1/Jρ(x=)/(rj/0RR)upsilnogttetdheazsearoftuhn-ocrtdioenr wanhiuchppiesrdliemriivtedoffsro′,mfolelqo.w(3ed1),bywittrhanasfloorwmeartiloimnsitsiomfi0larantdo
approximation,eq.(57),(solidline),thefirstorderapproximation, those used in working from eq.(43) to eq.(46). Note in par-
eq.(59),(dashedline),andexactintegralsfromeq.(50)(diamond ticular that for a ray in this case, ω is an acute angle, but
symbols). θ is obtuse, and the respective limits of these angles for a
radial ray are zero and π.
Wesubstituteforthescatteringcoefficient,σc(r(s′)),by
Wenotethatatthecavity/shellboundary,wherer=R,the
assuming that it has the same 1/r2 functional dependence
reductionofeq.(57)agreeswiththecavitysolution,eq.(20),
as the absorption (see Section 5.1). The mean intensity is
evaluated at the same radius. Both equations yield J(R)=
given by eq.(57). It is perfectly possible to express all these
j0R/2. functionsintermsofthedistance,s′,alongtheray,butthe
Itisalsopossibletoobtainananalyticintegralforafirst
integral in eq.(60) appears easier when working in terms of
order approximation, keeping the term in x in the series in
radius. Letting theintegral beΨ, we have
eq.(55). The integral in the first-order case is a standard
form in Gradshteyn & Ryzhik(1965). With theparameters Ψ= σc(R)R2j0exp[ǫ(π−ω−a)]
specific to thecurrent problem, the integral is 2
1 1 x 1/2 γ(1 ρ3)x 3 Pdρexp[−ǫ(arcsinaρ−aρ)] 1 (1−ρ2)ln1+ρ (,62)
− exp − − dx=B(1, ) ×Z (1 a2ρ2)1/2 (cid:20) − 2ρ (cid:18)1 ρ(cid:19)(cid:21)
Z (cid:18)1 ρ2x(cid:19) (cid:26) 6 (cid:27) 2 P0 − −
0 −
Φ (1,1,5;ρ2,z),(58) where ρ = R/r as before, a = (R2/R)sinω, ǫ = γ/a, and
× 1 2 2 thelimits are given by
wherez= γ(1 ρ3)/6.Thefirstfunction,B,ontheright- P0 =R/R2 (63)
− −
hand side of eq.(58) is an Euler beta-function, whilst the
and
secondisaconfluenthypergeometricseriesintwovariables,
ρ2andz.Furtherstandardformulaeallowthebeta-function P =R/[s2 2sRcosω+R2]1/2, (64)
− 2
B(1,3/2)tobeexpressedasaratiooffactorialsthatreduces
which reduces to 1 for a ray entering the shell from the
to 2/3, so that themean intensity to first order in x is
vacuum, and reaching the edge of the cavity, where r =R.
J(ρ)= j0R3ρ2e−γ(1−ρ)Φ1(cid:18)1,21,52;ρ2,−γ(16−ρ3)(cid:19) (59) Iρf,wweedciavnidcearerqy.(o6u2)tbayntinhteetgorpatliionne,bwyhpiachrtsis. Tinhdeepinetnedgernatteodf
part is
We plot, in Fig. 7, the mean intensity in the shell as
exp[ ǫarcsin(aρ)] exp[ ǫarcsin(aρ)]
a function of dimensionless radius, 1/ρ, computed from the V = − dρ= − , (65)
exact integral, eq.(50), and both approximations (eq.(57) Z (1 a2ρ2)1/2 − ǫa
−
and eq.(59)). For the value of the optical depth parameter whichremovestheproblematicsquare-rootterm.Theresult
used(γ=1.38,seeSection6.2),weseethateventhezeroth- of this integration is,
order formula is an excellent approximation. The graphs in
Fvaiglu.e7s sohforu/lRd b>e 1seeanndastoanQex<te0n.s5ioinnotfhethcaatseinwFhiegr.e4thtoe Ψj = −2(cid:20)e−ǫ(a−aρ+arcsin(aρ))(cid:18)1− (1−2ρρ2)ln11+ρρ(cid:19)(cid:21)P
− R/R2
input background is negligible.
P
+ g(ρ)exp ǫ(a aρ+arcsin(aρ)) dρ, (66)
Z {− − }
R/R2
8 PERTURBATION SOLUTION IN THE where the function g(ρ), now entirely composed of loga-
SHELL rithmsand rational functions of ρ, is defined by
Wenowproceed tosolveeq.(36) with themean intensityin 1 ǫa 1 1+ρ
g(ρ)=2ǫa 2 + 1 + +ǫaρ ln (67)
the scattering term given by eq.(57): it is too complicated − ρ (cid:20) − ρ ρ2 (cid:21) (cid:18)1 ρ(cid:19)
−
(cid:13)c 0000RAS,MNRAS000,000–000
10 M.D. Gray, M. Matsuura et al.
and Ψ is given by
j
4ǫaΨ
Ψ = . (68)
j σc(R)R2j eǫ(π−ω)
0
Notethat theargument of theexponentialin eq.(68) isdif-
ferentfromthatineq.(62)becausetheoriginalintegral,that
is the lower line of eq.(62), was multiplied by a factor of
2ǫae−ǫa in order to obtain Ψ in eq.(66).
j
Tocalculateanapproximationtotheintegralineq.(66),
we expand the argument of the exponential, noting that
aρ 1. The first term in the power-series expansion of the
arcs≤inecancelswithaρ,andthenexttermisin(aρ)3 which
we ignore, such that
e−ǫ(a+arcsin(aρ)−aρ) e−ǫa(1+(aρ)3/(6a)) e−ǫa, (69)
≃ ≃
whichisindependentofρ,andcanbemovedoutsidethein-
tegral,whichhasnowbeenreducedtotheintegrationofthe Figure 8. The function U = δI(s)/(2j0R) for complete path
function g(ρ) from eq.(67). The integration of g(ρ) breaks lengths s = s(R2) plotted as a function of entry angle into the
downintosixintegrals, fiveofwhich canbesolvedinterms nebula, ω. The optical depth parameter is χc(R)R = 1.38 as
of elementary functions and the sixth in terms of the Rie- before, and the ratio of the shell to cavity radii is R2/R = 3.0.
Theratioofthescatteringcoefficienttotheextinctioncoefficient
mann Φ(z,s,v) function (Gradshteyn & Ryzhik 1965). A
isσc(R)/χc(R)=0.25.
full form for the solution, Ψ, appears in AppendixA.
ζ =σc(R)R2j /(4ǫa), (72)
0
8.1 Rays which avoid the cavity
then we can re-write eq.(71) as
Araywhichdoesnotenterthecavityobeysthesameprop-
agationequation,eq.(60),buthasadifferentupperlimiton δI(s) = ζΨj eǫarcsin[(R2/r(s0))sinω]
the distance, s. Instead of integrating to the cavity bound- + eǫarc(cid:8)sin[(R2/r(s))sinω , (73)
ary, we integrate to the mid-point, where the radial vector
(cid:9)
in perpendicular to the direction of the ray. At this point, notingthatanyconstantsintheexponentialshavebeencan-
the value of s = R cosω, and the corresponding radius celledbetweenthee−htermoutsidetheintegrals,andtheeh
2
is R2sinω. We can therefore still use eq.(62), eq.(63) and inside. The equivalence of the integrals, Ψj, has been used,
eq.(64), but noting that the expression in eq.(64) now re- fromthesymmetryargumentabove.Wecannowwritedown
duces to P =R/(R sinω), rather than 1. The integrations twoversionsofeq.(73).Thefirstisforaraywhichavoidsthe
2
and approximations which follow still apply, providing we cavity, where r(s0) = R2sinω and r(s) = R2 on emerging
use thenew valueof P. from thecavity. In thiscase,
δI(s)=ζΨ eǫπ/2(1+e−ǫ(π/2−ω)). (74)
j
8.2 Outward bound rays Fortheothercase,wheretheraycrossesthecavity,thefirst
value changes to r(s ) = R; the second remains the same,
0
Provided that we are interested only in emergent rays, we
so we have,
can use the symmetry of the nebula to simplify matters
greatly here. The integral along the ray, Ψj, is indentical, δI(s)=ζΨj(eǫarcsina+eǫω), (75)
whetherwe are integrating inward to themidpoint (or cav-
where the integral Ψ is given in Appendix A (with appro-
j
ity),oroutwardagaintowardstheedgeofthenebula.These
priate limits).
integrals become formally the same because when we con-
In Fig. 8, we plot the emergent specific intensity for
vert from s′ to r as the integration variable, we must use
scattering, U = δI(s)/(2j R) over the full range of entry
0
different roots of the expression
anglestothenebula.Thevalueofsisthelargestpathlength
s′=R cosω (r2 R2sin2ω)1/2, (70) through the shell material for each angle. The same optical
2 ± − 2 depth parameter for extinction, 1.38, has been used as in
withthenegativerootrequiredfortheinwardsegment,and Fig.6andFig.7.Theratiooftheouter(shell)radiustothe
the positive root holding for the outward ray. The result inner(cavity)radiusis3.0:thishasnotbeenusedasaformal
is that the integrals over radius are identical. The form of parameter before, but the same value was also adopted for
eq.(60) which applies to outward rays is the above figures. The ratio of the scattering coefficient to
s theextinction coefficient has, rather arbitrarily, been set to
δI(s)=δI(s0)+e−h(s) σc(r(s′))J(r(s′))eh(s′)ds′, (71) 0.25.Fortheperturbativemethodtobeaccurate,thisvalue
Z
s0 should really be small, but is unlikely to be so for typical
where s is now eitherthemidpoint, for a ray which avoids dust models (see discussion in Section 5.2). This parameter
0
the cavity, or the outward cavity boundary. We assume in anywayactsasasimplescalingfactorwhichdoesnotchange
eq.(71) that there is negligible scattering within the cavity. the shape of the function in Fig. 8. The normalising factor
Theinput,δI(s ),isjusttheinboundsolution,evaluatedat of 2j R is the same as in the other specific intensity plot,
0 0
themidpoint or theinbound cavity boundary.If we define, Fig. 6.
(cid:13)c 0000RAS,MNRAS000,000–000