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The scattering of line radiation—II. Velocity-noncorrelated scattering
J. Quonr.
Specmsc.
Radial.
Transfer. Vol.
I I. pp.
1633-1646.
Pergamon
Press
1971. Printed
in Great
Britain
THE SCATTERING OF LINE RADIATION-II. VELOCITY-NONCORRELATED SCATTERING J. D. ARGYROS*and D. MUGGLESTONE Department of Physics, University of Queensland, St. Lucia, Queensland, Australia (Received 16 February 1971)
Abstract-An investigation is carried out into the effects of using different atomic models of the radiation scattering event with velocity-noncorrelated scattering (a physical process introduced in a previous paper). Special emphasis is placed on those models of the scattering event which lead to complete frequency redistribution in the radiation viewed by an observer at a large distance from the atmosphere being investigated. Of the three different models of the atomic radiation scattering event used, two were found to produce complete frequency redistribution. It is pointed out that the significance of the derivation of the mathematical forms describing complete frequency redistribution does not lie in the functional forms themselves, which have been known and used with great effect for many years, but rather in the physical interpretation of the radiation scattering event which must be used to obtain the particular mathematical expressions.
1. INTRODUCTION
which will be presented in this paper continues the research reported in a previous paper (I) hereafter referred to as Paper I. In this earlier paper, it was shown that the theory available for describing the frequency redistribution of radiation scattered by an atmosphere is incomplete because it does not allow scattered radiation to undergo complete redistribution; this earlier investigation was an attempt to generalize the theory of redistribution in such a way that this weakness in the existing theory could be overcome. In order to achieve the required generalization a condition usually applied, namely that the velocity of the scattering atom remains unchanged during the scattering event, was relaxed ; this allowed a general expression for the redistribution function to be deduced (equation (3.5) in Paper I). From this general result two limiting cases were obtained. In one limit, the velocity of the atom remained unchanged during the scattering event (velocity-correlated Q) in the other limit, the velocity scattering) which has been discussed at length by HUMMER, of the atom at the moment of absorption was taken to be completely uncorrelated with the velocity of the atom at the instant of emission (velocity-noncorrelated scattering) which will be the subject of further investigation in this paper. The centre of mass of the atmosphere in which the radiation is scattered is taken to be stationary relative to a distant observer ; in this atmosphere the individual atoms undergo continuous random thermal motions. It was suggested in Paper I that radiation which THE WORK
* Present address : University of London Observatory, Mill Hill Park, London N. W.7. 1633
J. D. ARGYROSand D. MUGGLE~TONE
1634
undergoes velocity-noncorrelated scattering in such an atmosphere could be completely redistributed for certain specific forms of the atomic scattering event. The various forms of radiation scattering that a single atom is capable of performing are described by different distribution laws which give the probability of occurrence of a particular scattering event. Of the many distribution laws which can be chosen to describe the radiation scattering event in the rest frame of the atom, the three cases described in Section 2 will be used. It will be seen that two of the three models of the radiation scattering event chosen for discussion lead to complete redistribution. Consider the scattering of line radiation by an elemental atmosphere, as observed by a distant observer, to consist of the absorption of a photon with frequency lying in the frequency range between x and x + dx followed by the emission in the same spectral line of another photon with frequency in the range x’ to x’ + dx’. The frequency variables x and x’ represent arbitrary measures of the frequencies of the radiation involved in the scattering event. In Paper I the redistribution function R(x, x’) was introduced as the probability function associated with this scattering event. It is possible to define two additional probability functions $(x) and 1+9(x’)which are related to the redistribution function in the following way :
$6-4 =
5R(x, x’) dx’
(1.1)
$(x’) =
1R(x, x’) dx.
(1.2)
and
Consequently the quantity 4(x) dx represents the probability of absorption by the atmosphere of a photon with frequency in the range x to x + dx while the quantity $(x’) dx’ represents the probability that the atmosphere will emit a photon with frequency between x’ and x’ + dx’. For complete redistribution to occur the observed frequencies x and x’ must be completely uncorrelated. This means that the probability of occurrence of the joint process ‘absorption at x followed by emission at x” must equal the product of the probabilities for the occurrence of the separate processes ‘absorption at x’ and ‘emission at x”, i.e. for complete redistribution we must have the condition R(x, i’) Therefore scattering
EE
&x)$(x’).
(1.3)
if the functions R, 4 and I/I satisfy this identity (1.3), it will be understood process represented by these functions leads to complete redistribution. 2.
MODELS
OF
THE
ATOMIC
The redistribution function used to observer is controlled by the functional the corresponding single atomic radiation atomic distribution law will be discussed
RADIATION
SCATTERING
that the
EVENT
describe the scattering event seen form of the distribution law which scattering event. Several alternative in this section. Using the notation
by a distant characterizes forms for this introduced in
Scattering of line radiation-II.
Velocity-noncorrelated
scattering
1635
Paper I where (l, A) is the label given, in the rest frame of the atom, to a photon with frequency between r and 5 + dr travelling within a solid angle dR in a direction A, the probability that such a photon (<, A) will be absorbed by an atom is denoted by f(<) dl dW4n,
(2.1)
where f(r) is assumed to be independent of the direction of the photon and is normalized according to the equation a, s
f(5) dtl = 1.
(2.2)
Let ~(5, <‘) be defined as an emission probability function with normalization m p&5’) d5’ = 1, I -Co
(2.3)
where P(<, &(A, A’)d5’ da
(2.4)
represents the probability that if a photon (5, A) has been absorbed then a photon (<‘,A’)is emitted ; g(A,A’)expresses the relationship between the directions A and A’for the absorbed and emitted photons respectively and is normalized to unity 4n
g(A,A’)dR’ = 1.
i.e.
(2.5)
s 0
If the scattering is assumed to be isotropic, then g(A,A’) = (47r- i.
(2.6)
Having defined the functions f(l), ~(5, 5’) and g(A,A), it is now possible to express the probability that a photon (5, A)is absorbed and that a photon (<‘, A) is subsequently emitted as M(<, A; t’, A) dr d<’ dR dR’ = f(&(& = f(&G
[‘)g(A, A’)(47c)-1 d< d<’ dR dQ 5’)(4~)-’ d5 dtl’ da dR’,
(2.7)
where it has been assumed that the scattering will be isotropic for all cases considered in this paper. Particular models used to describe the radiation scattering event, in the rest frame of the atom, are defined in terms of the functions f(t) and p(& r’). The following descriptions of the atomic radiation scattering event, in the rest frame of the atom, will be used : I Coherent scattering with zero line width If the spectral line has zero width, then it must be assumed that the atom performing the scattering has infinitely sharp energy levels. Therefore, the absorption profile must be given by f(5) = a(5 -vex
(2.8a)
J. D. ARGYROSand D. MUGGLESTONE
1636
where hv, is the energy of the radiation associated with transitions between the two sharp levels; the function 6 represents the Dirac delta function. Since there is only one energy for which transitions between these atomic energy levels can occur, the radiation emitted will have the same frequency as the radiation absorbed (coherent scattering). This requirement means that P(L
5’) = a(<’- 0
(2.8b)
II Coherent scattering with natural line broadening When the spectral the atomic absorption
line is broadened naturally (often referred to as radiation profile is given by the Lorentz distribution. Therefore,
damping),
where ‘a’ is referred to as the ‘line broadening parameter’ and is a measure of the width of the spectral line in the reference frame of the stationary atom. The frequency vO represents the central frequency of the spectral line. As in the previous case, the condition that the scattering be coherent requires that P(425’) = III
S(5’- 0.
(2.9b)
Complete redistribution in the atom’s frame of reference
If an identity similar to (1.3) is applied to the line scattering event in the reference of a stationary atom, then the atomic redistribution function is given by 1 l 7~’ (~-vo)2+2’(~‘-vO)2+a2’
f‘(5)P(4,5') = ft ____
frame
(2.10)
where it has been assumed that the individual atomic events of photon absorption and photon emission both follow a Lorentz distribution. Physically it is required in this case, that the atom be disturbed by some unspecified mechanism while it is in its excited state in such a manner that the emission process is completely independent of the absorption process. Future reference to any of the events described above will be made through the Roman numeral (I, II or III) appropriate to the particular scattering event as designated above. A pictorial representation of the three atomic scattering events just discussed appears in Table 1. 3. THE INTEGRATED REDISTRIBUTION FOR VELOCITY-NONCORRELATED
Since only the integrated The procedure It has been to be measured
FUNCTION SCATTERING
the case of velocity-noncorrelated scattering will be considered in this paper, redistribution function appropriate to this form of scattering is required. used to obtain this function has been outlined in Paper I, Section 4. found to be convenient in this work to choose the frequency variables x and x’ as the frequency departure from the central frequency vO of the spectral line
Scattering of line radiation-II.
Velocity-noncorrelated
scattering
1,
x
P
1637
1638
J. D. ARGYROSand D. MUGGLFSTONE
and in units of the Doppler half-width A where A = GI(V&);
(3.1)
the quantity c1represents the most probable value of the speed of a group of particles possessing a Maxwellian distribution of velocities and is given by ci = (2kT/m)“’
(3.2)
with T the temperature of the groups of particles and m the mass of an individual particle. Therefore, if v and v’ are the frequencies of the photons involved in the scattering process, as measured by a distant observer, then x = (v-vJA
and
x’ = (v’-vO)/A.
(3.3)
It is possible to simplify the resultant expression for the integrated redistribution function if a reduced velocity u is introduced. The reduced velocity is defined by u = v/CL
(3.4)
where v is the velocity of a particle relative to the distant observer. By substituting for the function M from the expression (2.7) from this paper into equation (4.12) of Paper I for the velocity-noncorrelated redistribution function, we obtain m
+1
0
-1
co
R&,x’)=$n/duSdu.
s
0
+1
dp 1
d$u2uf2exp(-u2-uf2)
-1
. f(Ax + v. - Apu)p(Ax + v. - Apu, Ax’ + v,, - A$#‘)
(3.5)
in terms of the frequency variables x and x’. Here the subscript N refers generally to the velocity-noncorrelated redistribution functions considered in the present paper. 4. PURE
DOPPLER
BROADENING
(CASE
1)
The description of the atomic radiation scattering event for the case of pure Doppler broadening is given by equations (2.8), i.e. we are discussing the case of coherent scattering in the atom’s frame of reference of photons of a spectral line of zero line width (case I above). Making the substitutions from equations (2.8) for f(Ax + v. - A,uu)and p(Ax + v. -Apu, Ax’ + v. - Ap’u’)in equation (3.5) for the redistribution function calculated by a distant observer, we obtain in this case Ri_,&,x’)
= ildnj,,’
1 0
0
dy 7
-u
dy'uu'exp(-u2--tJ2)
-II'
(4.1)
.6(x-y)h(x'-x-y'+y).
On integrating equation (4.1) with respect to the variables y and y’, we obtain the expression m
cc
R, -AI(x,x') = z
u
s
X
exp(-u2)
du.
s
u’ exp( - u’~)du’,
X’
Scattering of line radiation-II.
Velocity-noncorrelated
scattering
1639
which may be further reduced to RI _N(x, x’) = 6
II2 exp( - x2) . n- 1/Zexp( - x’2).
(4.2)
It follows directly from equation (4.2) using the expressions (1.1) and (1.2), that 4(z) = i&z) = n- 1/2exp( - z’).
(4.3)
Therefore, according to the identity (1.3) which is satisfied by R, -N(~, x’), C/J(X)and It/(x’),the redistribution function R, -N(~, x’) given by equation (4.2) describes a form of scattering which produces complete redistribution in the radiation field seen by a distant observer. For an observer viewing a single atom during a single radiation scattering event, the frequency of absorption x and the frequency of emission x’, as measured by the observer, are given by x = y+ucosB
for absorption
(4.4)
x’ = y’ + U’cos 0’
for emission,
(4.5)
and
where 8 represents the angle between the direction of the incident photon and the direction of the reduced velocity u at the moment of absorption (0’ being the corresponding angle between the direction of the scattered photon and the direction of the reduced velocity u’). Here y and y’ are the frequency differences measured from line centre in the atom’s frame of reference, in units of the Doppler half-width A, for the absorbed and emitted photons respectively. In the case presently being considered the spectral line in the atom’s frame of reference has zero width i.e.
y=y'=O
(4.6)
and equations (4.4) and (4.9, in this case, reduce to x =
ucoso
for absorption
(4.7)
and x’ = a’ cos 8’ for emission.
(4.8)
Since the radiation scattering has been assumed to be isotropic.(note equation (2.6)) 0 and 8’ are uncorrelated. Further, the definition of velocity-noncorrelated scattering requires that the velocities u and u’ are also uncorrelated so that consequently the frequencies x and x’ are uncorrelated in the present case. This last statement is expressed mathematically by the identity (1.3) so that it is not surprising to find that this particular example ofscattering gives complete redistribution in the radiation viewed by a distant observer. 5. COMBINED
NATURAL
AND
DOPPLER
BROADENING
(CASE
II)
The atomic model used in the present case to represent the radiation scattering event is the one described earlier (case II) as coherent scattering with natural line broadening where the absorption of the radiation, in the atom’s frame of reference, occurs according to the
woyuty
uoynq
-!Jls!pal paylap dlsno!AaJdaylJo dut?ql!~ @no!Aqo pay!ssep aq ~ouue3(g'S)uo~KXIn~uo!$ -nqpls!palay3 1t3qj savwpuy ~y~~@a~!p palaprsuo3 sywep ;e3!Jawnu aql uayM sno!Aqo alotu STq~!q~ &adord w(c,,~a~oao~ puv cc,3~~~~ pu13ss31~1~3fAq8u!law3s pavyalJo3 -dj!30laAJoasw aq3 uy IuaAa %u!Jawzs uopsypv~ cy.110~~ atut aq3 JOJ pauyqo IlnsaJayl) uo!ldlosqeJo d~uanbaqay~~e~ou(uoy~nq~~~s!pa~a~a~duIo~Ou!]uasa~da~ssuop~un~aylql!M N-*1v uoy3utq aqi laqi saiw!pur 1 %d~o uoy ~SIX~UO~W)U@!JO ayl le rayllausqt?ad(,x‘x) -~8psaAu!asol3'1OO.o‘T.o= 1,laiau1~led%u!uap~olqaql~osanIsn~o~put?x~suanbaquo!~ -dJosqt?aqiJosanlehsno!.n?h .10~,xd~1anba.1~uo~ss~u1aw.1~~8e pauoIds! uoyunjaqi alaqM (qI)pue(e1)3%d
u! uMoqss~(,x‘~)~-~~~uo~~~un~uo~inq~~~s~pa~aq~6quay~~uI~o~aq~
(L’S)
Lq ua@
s!‘uoy3un~ $!?!o~aql‘(a%)H alaqm
(9'S)
0
“-
(S’S)
p+,v-xx)
.npkp (,(iC+x_,x)_)dxa
n ($-)dxanwG s s
= (lX‘X)N-"~
01 sa~npa~(@'s)uo!ssa.Idxa aql ‘,nalqe!lenaqllaAo sl.wdAq uaq~‘,ka~qc!.w~aql.taAoiC~l"!l!u!suo~lt?l%a~u!aql%uru~~o~~ad ,I,+#-x) '(A+,,i-X-,X)&
(P'S) ,n-
(z,n-,n -)dxa
n-
,nn,iCpJ icp j,~pjinpj:~=(,x~x)N-~I~ ,n
uognqgs!pal %uy-tlgsqns (f’s)
"
CD
co
am luasaldaql LI!uyqo aM ‘uo!lwnJ paielallo3uou-/Clr3olaA aq] .IOJ (s’f) uogenba olur (f’s) pm (1’s) suogenba ‘pasqgn uaaq scq n/(x& = (xn)~ uogcmy ellap Del!a aq]Jo Allado.Id aql alaqM ‘(wf + ,n,rl- x - ,x)qr _ v = (,n,rlv - OA+ ,xv ‘mfv - OA+ xv)d sp]a!/( (96.2) uogenba
@~~~LLI~s~paXIpo.I~u~ uaaq wq
v/n = D
(Z’S)
Jalaumed
%!uapeo.Iq
aql alaqM
p + ,(nrl- x) vu _______~~~- = (nrlv-%t+xv)j
(I’S)
I
D
leqluMoqs aq ~23 1!‘(efj’~)uo!lenba
u101d 'uoyIqys!p zlLIa.Io~ OP91
Scattering of line radiation-II.
Velocity-noncorrelated
scattering
1641
0.3 -
‘x”
0.2-
X-
-7 & 0.1 -
0.0
-3.0
-2.0
-1.0
0.0
X'
1.0
2.0
3.0
FIG. la. The redistribution function R,,_,(x, x’) as a function of x’ for various values of x with the parameter 0 = 0.1.
0.3 -
.X x‘ -2
0.2-
I
lx0.1 -
0.0 -3.0
-1.0
-2.0
0.0
X'
1.0
2.0
3.0
FIG. lb. The redistribution function R,,_,(x, x’) as a function of x’ for various values of x with the parameter u = 0401.
A much more interesting set of curves is shown in Fig. 2 where, for x = 2, the ratio R,I_N(~, x’)/[c#~(x)lc/(x’)]is plotted as a function of x’. The profiles 4(x) and t&x’) can be
calculated from equation (5.6) by using the definitions (1.1) and (1.2); they are both found to be Voigt profiles. From the identity (1.3) we know that complete redistribution occurs whenever this ratio becomes unity. It is apparent from Fig. 2 that as r~decreases, a larger region about the core (i.e. x’ N O-0)is found for which this ratio approximates unity. This is not altogether surprising since it is readily shown that lim R,,-&, S-r0
x’) = R,_Jx,
x’)
(5.8)
and the function RI_N(~, x’) has already been shown in Section 4 to represent complete redistribution.
1642
J. D. ARGYROSInd D. MUGGLESTONE
FIG. 2. The change in R,,_,&, x’) towards complete redistribution with decreasing o for x = 2.
It is also of interest to note how the ratio R ,,-N(~, x’)/[$(x)$(x’)] plotted as a function of x’ varies for different values of x but for a given constant value of the parameter rr. Such a variation is shown in Fig. 3 for r~ = O@Ol. It should be noticed that for x I 2, there again exists a region of x’ (-2 I x’ I 2) where the ratio is unity. However, for x > 2, over that same region of x’ lying in the range - 2 I x’ I 2 the ratio assumes values very different from unity; indeed once this deviation from unity takes place, the ratio can be very sensitive to changes in frequency. This result indicates that for absorption in the wings of the spectral lines (x > 2 in the present example), there no longer exists a region of the redistribution function which represents complete redistribution. It is instructive to suggest physical interpretations of the effects observed in Figs. 2 and 3 in terms of the probability of occurrence of individual scattering events as seen by a distant observer. Since the spectral line in the atom’s frame of reference has a finite width, there exists a finite probability that the frequencies of absorption and emission in the atom’s frame of reference (y and y’ respectively in equations (4.4) and (4.5)) will occur at other than the central line frequency. Therefore, as stated earlier, the absorption and emission profiles measured by a distant observer are represented by the Voigt function. Consequently the larger the value of the broadening parameter G, the larger is the probability that such a distant observer will detect an absorption or an emission in the wings of the spectral line (see for example FINN and MUGGLESTONE’~)); in other words, as c decreases both absorptions and emissions will occur preferentially towards line centre. Thus to a very large extent the behaviour of the physical system for small 0 in this present case is very similar to the behaviour of the physical system discussed previously in Section 4. The consequence of this behaviour is seen in Fig. 2 where the region of x’ over which complete redistribution occurs increases as 0 decreases for a given x (typically 1x1 < 2 in this example). The interpretation of the results shown in Fig. 3 is made most readily with the aid of equations (4.4) and (4.5) which relate the frequencies in the atom’s frame of reference to the
Scattering of line radiation--II.
Velocity-noncorrelated
scattering
1643
FIG. 3. The change in J&-,(x,x’) towards complete redistribution with decreasing x for (T= 0401.
frequencies seen by a distant observer for a single scattering event. In Fig. 3, the region of complete redistribution starts to disappear only when the absorption starts to occur in the wings (say (xl > 2). From equation (4.4) we see that this condition corresponds to y being comparable with the velocity term u cos 8 so that there does not exist a region of complete redistribution (we may compare this line of reasoning with that used in case I where y = y’ = 0). The same argument, using equation (4.5), when applied to the wings of the existing curves explains the deviations from unity of the ratio R II-N(x, x’)/[c#~(x)l(/(x’)] for large x’. However, it must also be remembered that the probability of absorption in the wings of a spectral line is much lower than the probability of absorption in the core (for example, in the above case where cr = OXIOl,absorption at the core is of the order of 5000 times more probable than at x = 3).w Consequently, it would be reasonable to conclude that for the majority of scattering events which are redistributed according to the function (5.6), there exists a region about the origin (in the example considered lx’1 I 2) for which conditions appropriate to complete redistribution do exist. To conclude this section on the redistribution function RII-N, given by equation (5.6), we consider its asymptotic limit. Using the series expansion for the Voigt function,@’ H(a, u) = g
(
l+
3v2-a2 2(a2_v2)2+
.a.
(5.9)
7 I
to the first order in (a2 + 02)- ’ (since u = (x + xl)/,/2 is large in the asymptotic limit), we see that 2 r(lx -
x'l),
Ix-x'1
>>
1
(5.10)
where r(/3) = (2~)~ iI2 exp( - /12/2).
(5.11)
For certain order of magnitude calculations, HUMMER (‘) has shown that a straight line approximation to r(9) provides sufficient accuracy. Therefore for completeness, we provide
1644
the following
J. D. ARGYROSand D. MUGGLESTONE
least-square
fit to equation
(5.1 l), viz.
-0.1568fl+O.4016,0 r(B) =
< fl < 2.561
(5.12)
0, otherwise.
6. COMPLETE
(CASES
REDISTRIBUTION
III
AND
I)
Before discussing complete redistribution generally, the redistribution function for velocity-noncorrelated scattering is derived in the case when there is complete redistribution in the atom’s rest frame (case III). If the appropriate form of f(Ax + v0 - Apu)p(Ax + v0 - Ap.4 ; Ax’ + v0 - Ap’u’) (i.e. equation (2.10)) is substituted into equation (3.5) for the redistribution that we may write the redistribution function in this case as 5
RIIl_.(x,x')
=
%
function,
we find
”
dy du s ((x -y)2 + G”} P” U’ o(, dy’ u’ exp( - u’~) _u, d”’ s ((x -y’)2+02f uexp(-U2)
-
,
(6.1)
which can readily be shown to reduce to the form R,,,_.(x,
x’) = [H(o, x)/TT”~] . [H(a, x’)/Tc”~],
(6.2)
where H(a, v), the Voigt function, has been defined previously in equation (5.7). It is obvious that this redistribution function is of the form required by the identity (1.3) (the condition which must be satisfied in order that complete redistribution occurs); consequently complete redistribution in the atom’s rest frame also produces complete redistribution in the observer’s frame of reference in this case. Although this result appears trivial, it has been shown by FINN ~3) that complete redistribution in the atom’s frame of reference does not necessarily lead to complete redistribution in the observer’s frame of reference. It should be noted once again that lim R,I,_N(x, O-r0
x’) = R,_.(x,
x’).
(6.3)
The limit (6.3) would seem to imply that case I is a special case of case III as well as being a special case of case II. However, strictly speaking this is not so. Case I is the limit of case II for a vanishingly small value of the broadening parameter u (or by equation (5.2) the broadening parameter a). The fact that R, _ ,,,(x, x’) is the limit of both R,, _ N(~, x’) and R,,, _ Jx, x’) as CJ+ 0 is due to the fact that equation (4.2) can be deduced for two different functional forms of ~(5, 5’). By using equation (2.8b) i.e.
P(L5’) = S(5 - 4’)>
as we have done in Section 4, coherent the alternative functional form
scattering
has been assumed.
P(L5’) = 6(vo - 5’)
(6.4) On the other hand, if
(6.5)
Scattering of line radiation--II. Velocity-noncorrelated scattering
is used, the same result (4.2) can be deduced by factorizing the redistribution
1645
function
R,_N(~, x’) in a manner similar to that used to evaluate R,,,_N(~, x’) in equation (6.1). The
distinction here is that in the form given by equation (6.4) the emission frequency 5’ must be the same as the absorption frequency 5, while in the form given by equation (6.5) the emission frequency 5’ must be the same as the central frequency vO.However, equation (6.5) is the form of p(<, 5’) which results from an assumption of complete redistribution in the atom’s frame of reference with zero line width. Although the zero width of the spectral line in the atom’s rest frame renders the distinction between coherent scattering and complete redistribution, both being considered in the rest frame of the atom, rather academic (since there is only one frequency into which the radiation can be ‘redistributed’ completely, viz. the central frequency v,), the behaviour of the redistribution functions R , , _ N(~, x’) and RI II _ N(~, x’) is so different in the limit of small, but non-zero, values of g that it proves convenient to retain the distinction between coherent scattering and complete redistribution even in the limit of fJ -+ 0. Before leaving this section on complete redistribution, it should be pointed out that the deduction of the redistribution functions describing complete redistribution is not important in itself (the results of complete redistribution have been used for a very long time with great effect). Rather it is the physical interpretation of the radiation scattering events, which must be used in the derivation of the redistribution functions, which is important. By using the conditions for velocity-noncorrelated scattering established in Paper I, there now exists a method of determining, a priori, whether or not an assumption of complete redistribution is a reasonable one in any physical situation. If the conditions in an atmosphere are such that velocity-noncorrelated scattering can be expected to occur in conjunction with complete redistribution in the atom’s frame of reference, then a distant observer viewing this atmosphere will observe radiation which has undergone complete redistribution in that atmosphere by means of the scattering process. Finally it should be mentioned that because the model used for velocity-noncorrelated scattering results in complete redistribution for certain types of atomic scattering, this does not preclude the possibility that some other mechanism may also produce complete redistribution. What we have shown is that, given the conditions for velocity-noncorrelated scattering, complete redistribution can be expected to occur for certain types of atomic radiation scattering events. 7. CONCLUDING
REMARKS
The scattering processes discussed in the present paper are of particular importance in the case of the scattering of photons travelling through a stellar atmosphere. A detailed discussion of the properties and behaviour of the radiation field in uniform model atmospheres which results when complete redistribution is assumed has been presented by AVRETT and HUMMER.(~) Recently HUMMER(‘) has shown the effects, on the radiation field, of allowing a more complex form of redistribution to occur (viz. redistribution arising during velocity-correlated scattering). The redistribution functions which have proved to be most interesting in this latter paper are those functions belonging to the class which can be deduced by assuming that the atomic radiation scattering event occurs according to the equations (2.9). The functional forms of the redistribution functions obtained by HUMMER(~) for velocity-correlated scattering (in the various classes discussed in the present paper) are compared in Table I with the corresponding velocity-noncorrelated redistribution functions
1646
J. D. ARGYROSand D. MUGGLESTONE
derived in the present investigation ; for clarity a pictorial model of each of the atomic radiation scattering events, giving rise to both the velocity-correlated and velocity-noncorrelated redistribution functions, is also shown. The radiation emanating from a uniform semi-infinite model atmosphere in which the scattering of the radiation is described by the velocity-noncorrelated redistribution function of equation (5.6), has been investigated by ARGYROS.(lo) It was found there that the resulting radiation field exhibited certain characteristic properties also noted by HUMMER(‘) for radiation from the same model atmosphere and for the same type of atomic radiation scattering event but for velocity-correlated scattering. Further investigations into the effects of redistribution on the radiation emerging from various types of atmosphere is continuing and the results obtained will be published in a future paper of this series. Acknowledgements-One of the authors (JDA) was, during the period of the present research, a recipient of a Commonwealth of Australia Postgraduate Award for which support he wishes to express his appreciation. This research was supported financially by the Australian Research Grants Committee.
REFERENCES 1. J. D. ARGYROSand D. MUGGLESTONE, JQSRT 11, 1621 (1971). D. G. HUMMER,Mon. Not. R. Astr. Sot. 125,21 (1962). J. T. JEFFERIES and 0. R. WHITE,Astrophys. J. 132,767 (1960). V. V. SOeOLEV,Vestnik LmingraLkogo Gosudarstuennogo Universteta, No. 5, p. 85 (1955) G. D. FINN and D. MUGGLESTONE, Mon. Not. R. Astr. Sot. 129,221 (1965). B. J. O’MARA, Ph.D. Thesis, University of California, Los Angeles (1967). 7. D. G. HUMMER,Mon. Not. R. Astr. Sot. 145,95 (1969). 8. G. D. FINN,Astrophys. J. 147, 1085 (1967). 9. E. H. AVRETTand D. G. HUMMER,Mon. Not. R. Astr. Sot. 130,295 (1965). IO. J. D. ARGYROS,Proc. Astron. Sot. Amt. 8,387 (1970). 2. 3. 4. 5. 6.
Specmsc.
Radial.
Transfer. Vol.
I I. pp.
1633-1646.
Pergamon
Press
1971. Printed
in Great
Britain
THE SCATTERING OF LINE RADIATION-II. VELOCITY-NONCORRELATED SCATTERING J. D. ARGYROS*and D. MUGGLESTONE Department of Physics, University of Queensland, St. Lucia, Queensland, Australia (Received 16 February 1971)
Abstract-An investigation is carried out into the effects of using different atomic models of the radiation scattering event with velocity-noncorrelated scattering (a physical process introduced in a previous paper). Special emphasis is placed on those models of the scattering event which lead to complete frequency redistribution in the radiation viewed by an observer at a large distance from the atmosphere being investigated. Of the three different models of the atomic radiation scattering event used, two were found to produce complete frequency redistribution. It is pointed out that the significance of the derivation of the mathematical forms describing complete frequency redistribution does not lie in the functional forms themselves, which have been known and used with great effect for many years, but rather in the physical interpretation of the radiation scattering event which must be used to obtain the particular mathematical expressions.
1. INTRODUCTION
which will be presented in this paper continues the research reported in a previous paper (I) hereafter referred to as Paper I. In this earlier paper, it was shown that the theory available for describing the frequency redistribution of radiation scattered by an atmosphere is incomplete because it does not allow scattered radiation to undergo complete redistribution; this earlier investigation was an attempt to generalize the theory of redistribution in such a way that this weakness in the existing theory could be overcome. In order to achieve the required generalization a condition usually applied, namely that the velocity of the scattering atom remains unchanged during the scattering event, was relaxed ; this allowed a general expression for the redistribution function to be deduced (equation (3.5) in Paper I). From this general result two limiting cases were obtained. In one limit, the velocity of the atom remained unchanged during the scattering event (velocity-correlated Q) in the other limit, the velocity scattering) which has been discussed at length by HUMMER, of the atom at the moment of absorption was taken to be completely uncorrelated with the velocity of the atom at the instant of emission (velocity-noncorrelated scattering) which will be the subject of further investigation in this paper. The centre of mass of the atmosphere in which the radiation is scattered is taken to be stationary relative to a distant observer ; in this atmosphere the individual atoms undergo continuous random thermal motions. It was suggested in Paper I that radiation which THE WORK
* Present address : University of London Observatory, Mill Hill Park, London N. W.7. 1633
J. D. ARGYROSand D. MUGGLE~TONE
1634
undergoes velocity-noncorrelated scattering in such an atmosphere could be completely redistributed for certain specific forms of the atomic scattering event. The various forms of radiation scattering that a single atom is capable of performing are described by different distribution laws which give the probability of occurrence of a particular scattering event. Of the many distribution laws which can be chosen to describe the radiation scattering event in the rest frame of the atom, the three cases described in Section 2 will be used. It will be seen that two of the three models of the radiation scattering event chosen for discussion lead to complete redistribution. Consider the scattering of line radiation by an elemental atmosphere, as observed by a distant observer, to consist of the absorption of a photon with frequency lying in the frequency range between x and x + dx followed by the emission in the same spectral line of another photon with frequency in the range x’ to x’ + dx’. The frequency variables x and x’ represent arbitrary measures of the frequencies of the radiation involved in the scattering event. In Paper I the redistribution function R(x, x’) was introduced as the probability function associated with this scattering event. It is possible to define two additional probability functions $(x) and 1+9(x’)which are related to the redistribution function in the following way :
$6-4 =
5R(x, x’) dx’
(1.1)
$(x’) =
1R(x, x’) dx.
(1.2)
and
Consequently the quantity 4(x) dx represents the probability of absorption by the atmosphere of a photon with frequency in the range x to x + dx while the quantity $(x’) dx’ represents the probability that the atmosphere will emit a photon with frequency between x’ and x’ + dx’. For complete redistribution to occur the observed frequencies x and x’ must be completely uncorrelated. This means that the probability of occurrence of the joint process ‘absorption at x followed by emission at x” must equal the product of the probabilities for the occurrence of the separate processes ‘absorption at x’ and ‘emission at x”, i.e. for complete redistribution we must have the condition R(x, i’) Therefore scattering
EE
&x)$(x’).
(1.3)
if the functions R, 4 and I/I satisfy this identity (1.3), it will be understood process represented by these functions leads to complete redistribution. 2.
MODELS
OF
THE
ATOMIC
The redistribution function used to observer is controlled by the functional the corresponding single atomic radiation atomic distribution law will be discussed
RADIATION
SCATTERING
that the
EVENT
describe the scattering event seen form of the distribution law which scattering event. Several alternative in this section. Using the notation
by a distant characterizes forms for this introduced in
Scattering of line radiation-II.
Velocity-noncorrelated
scattering
1635
Paper I where (l, A) is the label given, in the rest frame of the atom, to a photon with frequency between r and 5 + dr travelling within a solid angle dR in a direction A, the probability that such a photon (<, A) will be absorbed by an atom is denoted by f(<) dl dW4n,
(2.1)
where f(r) is assumed to be independent of the direction of the photon and is normalized according to the equation a, s
f(5) dtl = 1.
(2.2)
Let ~(5, <‘) be defined as an emission probability function with normalization m p&5’) d5’ = 1, I -Co
(2.3)
where P(<, &(A, A’)d5’ da
(2.4)
represents the probability that if a photon (5, A) has been absorbed then a photon (<‘,A’)is emitted ; g(A,A’)expresses the relationship between the directions A and A’for the absorbed and emitted photons respectively and is normalized to unity 4n
g(A,A’)dR’ = 1.
i.e.
(2.5)
s 0
If the scattering is assumed to be isotropic, then g(A,A’) = (47r- i.
(2.6)
Having defined the functions f(l), ~(5, 5’) and g(A,A), it is now possible to express the probability that a photon (5, A)is absorbed and that a photon (<‘, A) is subsequently emitted as M(<, A; t’, A) dr d<’ dR dR’ = f(&(& = f(&G
[‘)g(A, A’)(47c)-1 d< d<’ dR dQ 5’)(4~)-’ d5 dtl’ da dR’,
(2.7)
where it has been assumed that the scattering will be isotropic for all cases considered in this paper. Particular models used to describe the radiation scattering event, in the rest frame of the atom, are defined in terms of the functions f(t) and p(& r’). The following descriptions of the atomic radiation scattering event, in the rest frame of the atom, will be used : I Coherent scattering with zero line width If the spectral line has zero width, then it must be assumed that the atom performing the scattering has infinitely sharp energy levels. Therefore, the absorption profile must be given by f(5) = a(5 -vex
(2.8a)
J. D. ARGYROSand D. MUGGLESTONE
1636
where hv, is the energy of the radiation associated with transitions between the two sharp levels; the function 6 represents the Dirac delta function. Since there is only one energy for which transitions between these atomic energy levels can occur, the radiation emitted will have the same frequency as the radiation absorbed (coherent scattering). This requirement means that P(L
5’) = a(<’- 0
(2.8b)
II Coherent scattering with natural line broadening When the spectral the atomic absorption
line is broadened naturally (often referred to as radiation profile is given by the Lorentz distribution. Therefore,
damping),
where ‘a’ is referred to as the ‘line broadening parameter’ and is a measure of the width of the spectral line in the reference frame of the stationary atom. The frequency vO represents the central frequency of the spectral line. As in the previous case, the condition that the scattering be coherent requires that P(425’) = III
S(5’- 0.
(2.9b)
Complete redistribution in the atom’s frame of reference
If an identity similar to (1.3) is applied to the line scattering event in the reference of a stationary atom, then the atomic redistribution function is given by 1 l 7~’ (~-vo)2+2’(~‘-vO)2+a2’
f‘(5)P(4,5') = ft ____
frame
(2.10)
where it has been assumed that the individual atomic events of photon absorption and photon emission both follow a Lorentz distribution. Physically it is required in this case, that the atom be disturbed by some unspecified mechanism while it is in its excited state in such a manner that the emission process is completely independent of the absorption process. Future reference to any of the events described above will be made through the Roman numeral (I, II or III) appropriate to the particular scattering event as designated above. A pictorial representation of the three atomic scattering events just discussed appears in Table 1. 3. THE INTEGRATED REDISTRIBUTION FOR VELOCITY-NONCORRELATED
Since only the integrated The procedure It has been to be measured
FUNCTION SCATTERING
the case of velocity-noncorrelated scattering will be considered in this paper, redistribution function appropriate to this form of scattering is required. used to obtain this function has been outlined in Paper I, Section 4. found to be convenient in this work to choose the frequency variables x and x’ as the frequency departure from the central frequency vO of the spectral line
Scattering of line radiation-II.
Velocity-noncorrelated
scattering
1,
x
P
1637
1638
J. D. ARGYROSand D. MUGGLFSTONE
and in units of the Doppler half-width A where A = GI(V&);
(3.1)
the quantity c1represents the most probable value of the speed of a group of particles possessing a Maxwellian distribution of velocities and is given by ci = (2kT/m)“’
(3.2)
with T the temperature of the groups of particles and m the mass of an individual particle. Therefore, if v and v’ are the frequencies of the photons involved in the scattering process, as measured by a distant observer, then x = (v-vJA
and
x’ = (v’-vO)/A.
(3.3)
It is possible to simplify the resultant expression for the integrated redistribution function if a reduced velocity u is introduced. The reduced velocity is defined by u = v/CL
(3.4)
where v is the velocity of a particle relative to the distant observer. By substituting for the function M from the expression (2.7) from this paper into equation (4.12) of Paper I for the velocity-noncorrelated redistribution function, we obtain m
+1
0
-1
co
R&,x’)=$n/duSdu.
s
0
+1
dp 1
d$u2uf2exp(-u2-uf2)
-1
. f(Ax + v. - Apu)p(Ax + v. - Apu, Ax’ + v,, - A$#‘)
(3.5)
in terms of the frequency variables x and x’. Here the subscript N refers generally to the velocity-noncorrelated redistribution functions considered in the present paper. 4. PURE
DOPPLER
BROADENING
(CASE
1)
The description of the atomic radiation scattering event for the case of pure Doppler broadening is given by equations (2.8), i.e. we are discussing the case of coherent scattering in the atom’s frame of reference of photons of a spectral line of zero line width (case I above). Making the substitutions from equations (2.8) for f(Ax + v. - A,uu)and p(Ax + v. -Apu, Ax’ + v. - Ap’u’)in equation (3.5) for the redistribution function calculated by a distant observer, we obtain in this case Ri_,&,x’)
= ildnj,,’
1 0
0
dy 7
-u
dy'uu'exp(-u2--tJ2)
-II'
(4.1)
.6(x-y)h(x'-x-y'+y).
On integrating equation (4.1) with respect to the variables y and y’, we obtain the expression m
cc
R, -AI(x,x') = z
u
s
X
exp(-u2)
du.
s
u’ exp( - u’~)du’,
X’
Scattering of line radiation-II.
Velocity-noncorrelated
scattering
1639
which may be further reduced to RI _N(x, x’) = 6
II2 exp( - x2) . n- 1/Zexp( - x’2).
(4.2)
It follows directly from equation (4.2) using the expressions (1.1) and (1.2), that 4(z) = i&z) = n- 1/2exp( - z’).
(4.3)
Therefore, according to the identity (1.3) which is satisfied by R, -N(~, x’), C/J(X)and It/(x’),the redistribution function R, -N(~, x’) given by equation (4.2) describes a form of scattering which produces complete redistribution in the radiation field seen by a distant observer. For an observer viewing a single atom during a single radiation scattering event, the frequency of absorption x and the frequency of emission x’, as measured by the observer, are given by x = y+ucosB
for absorption
(4.4)
x’ = y’ + U’cos 0’
for emission,
(4.5)
and
where 8 represents the angle between the direction of the incident photon and the direction of the reduced velocity u at the moment of absorption (0’ being the corresponding angle between the direction of the scattered photon and the direction of the reduced velocity u’). Here y and y’ are the frequency differences measured from line centre in the atom’s frame of reference, in units of the Doppler half-width A, for the absorbed and emitted photons respectively. In the case presently being considered the spectral line in the atom’s frame of reference has zero width i.e.
y=y'=O
(4.6)
and equations (4.4) and (4.9, in this case, reduce to x =
ucoso
for absorption
(4.7)
and x’ = a’ cos 8’ for emission.
(4.8)
Since the radiation scattering has been assumed to be isotropic.(note equation (2.6)) 0 and 8’ are uncorrelated. Further, the definition of velocity-noncorrelated scattering requires that the velocities u and u’ are also uncorrelated so that consequently the frequencies x and x’ are uncorrelated in the present case. This last statement is expressed mathematically by the identity (1.3) so that it is not surprising to find that this particular example ofscattering gives complete redistribution in the radiation viewed by a distant observer. 5. COMBINED
NATURAL
AND
DOPPLER
BROADENING
(CASE
II)
The atomic model used in the present case to represent the radiation scattering event is the one described earlier (case II) as coherent scattering with natural line broadening where the absorption of the radiation, in the atom’s frame of reference, occurs according to the
woyuty
uoynq
-!Jls!pal paylap dlsno!AaJdaylJo dut?ql!~ @no!Aqo pay!ssep aq ~ouue3(g'S)uo~KXIn~uo!$ -nqpls!palay3 1t3qj savwpuy ~y~~@a~!p palaprsuo3 sywep ;e3!Jawnu aql uayM sno!Aqo alotu STq~!q~ &adord w(c,,~a~oao~ puv cc,3~~~~ pu13ss31~1~3fAq8u!law3s pavyalJo3 -dj!30laAJoasw aq3 uy IuaAa %u!Jawzs uopsypv~ cy.110~~ atut aq3 JOJ pauyqo IlnsaJayl) uo!ldlosqeJo d~uanbaqay~~e~ou(uoy~nq~~~s!pa~a~a~duIo~Ou!]uasa~da~ssuop~un~aylql!M N-*1v uoy3utq aqi laqi saiw!pur 1 %d~o uoy ~SIX~UO~W)U@!JO ayl le rayllausqt?ad(,x‘x) -~8psaAu!asol3'1OO.o‘T.o= 1,laiau1~led%u!uap~olqaql~osanIsn~o~put?x~suanbaquo!~ -dJosqt?aqiJosanlehsno!.n?h .10~,xd~1anba.1~uo~ss~u1aw.1~~8e pauoIds! uoyunjaqi alaqM (qI)pue(e1)3%d
u! uMoqss~(,x‘~)~-~~~uo~~~un~uo~inq~~~s~pa~aq~6quay~~uI~o~aq~
(L’S)
Lq ua@
s!‘uoy3un~ $!?!o~aql‘(a%)H alaqm
(9'S)
0
“-
(S’S)
p+,v-xx)
.npkp (,(iC+x_,x)_)dxa
n ($-)dxanwG s s
= (lX‘X)N-"~
01 sa~npa~(@'s)uo!ssa.Idxa aql ‘,nalqe!lenaqllaAo sl.wdAq uaq~‘,ka~qc!.w~aql.taAoiC~l"!l!u!suo~lt?l%a~u!aql%uru~~o~~ad ,I,+#-x) '(A+,,i-X-,X)&
(P'S) ,n-
(z,n-,n -)dxa
n-
,nn,iCpJ icp j,~pjinpj:~=(,x~x)N-~I~ ,n
uognqgs!pal %uy-tlgsqns (f’s)
"
CD
co
am luasaldaql LI!uyqo aM ‘uo!lwnJ paielallo3uou-/Clr3olaA aq] .IOJ (s’f) uogenba olur (f’s) pm (1’s) suogenba ‘pasqgn uaaq scq n/(x& = (xn)~ uogcmy ellap Del!a aq]Jo Allado.Id aql alaqM ‘(wf + ,n,rl- x - ,x)qr _ v = (,n,rlv - OA+ ,xv ‘mfv - OA+ xv)d sp]a!/( (96.2) uogenba
@~~~LLI~s~paXIpo.I~u~ uaaq wq
v/n = D
(Z’S)
Jalaumed
%!uapeo.Iq
aql alaqM
p + ,(nrl- x) vu _______~~~- = (nrlv-%t+xv)j
(I’S)
I
D
leqluMoqs aq ~23 1!‘(efj’~)uo!lenba
u101d 'uoyIqys!p zlLIa.Io~ OP91
Scattering of line radiation-II.
Velocity-noncorrelated
scattering
1641
0.3 -
‘x”
0.2-
X-
-7 & 0.1 -
0.0
-3.0
-2.0
-1.0
0.0
X'
1.0
2.0
3.0
FIG. la. The redistribution function R,,_,(x, x’) as a function of x’ for various values of x with the parameter 0 = 0.1.
0.3 -
.X x‘ -2
0.2-
I
lx0.1 -
0.0 -3.0
-1.0
-2.0
0.0
X'
1.0
2.0
3.0
FIG. lb. The redistribution function R,,_,(x, x’) as a function of x’ for various values of x with the parameter u = 0401.
A much more interesting set of curves is shown in Fig. 2 where, for x = 2, the ratio R,I_N(~, x’)/[c#~(x)lc/(x’)]is plotted as a function of x’. The profiles 4(x) and t&x’) can be
calculated from equation (5.6) by using the definitions (1.1) and (1.2); they are both found to be Voigt profiles. From the identity (1.3) we know that complete redistribution occurs whenever this ratio becomes unity. It is apparent from Fig. 2 that as r~decreases, a larger region about the core (i.e. x’ N O-0)is found for which this ratio approximates unity. This is not altogether surprising since it is readily shown that lim R,,-&, S-r0
x’) = R,_Jx,
x’)
(5.8)
and the function RI_N(~, x’) has already been shown in Section 4 to represent complete redistribution.
1642
J. D. ARGYROSInd D. MUGGLESTONE
FIG. 2. The change in R,,_,&, x’) towards complete redistribution with decreasing o for x = 2.
It is also of interest to note how the ratio R ,,-N(~, x’)/[$(x)$(x’)] plotted as a function of x’ varies for different values of x but for a given constant value of the parameter rr. Such a variation is shown in Fig. 3 for r~ = O@Ol. It should be noticed that for x I 2, there again exists a region of x’ (-2 I x’ I 2) where the ratio is unity. However, for x > 2, over that same region of x’ lying in the range - 2 I x’ I 2 the ratio assumes values very different from unity; indeed once this deviation from unity takes place, the ratio can be very sensitive to changes in frequency. This result indicates that for absorption in the wings of the spectral lines (x > 2 in the present example), there no longer exists a region of the redistribution function which represents complete redistribution. It is instructive to suggest physical interpretations of the effects observed in Figs. 2 and 3 in terms of the probability of occurrence of individual scattering events as seen by a distant observer. Since the spectral line in the atom’s frame of reference has a finite width, there exists a finite probability that the frequencies of absorption and emission in the atom’s frame of reference (y and y’ respectively in equations (4.4) and (4.5)) will occur at other than the central line frequency. Therefore, as stated earlier, the absorption and emission profiles measured by a distant observer are represented by the Voigt function. Consequently the larger the value of the broadening parameter G, the larger is the probability that such a distant observer will detect an absorption or an emission in the wings of the spectral line (see for example FINN and MUGGLESTONE’~)); in other words, as c decreases both absorptions and emissions will occur preferentially towards line centre. Thus to a very large extent the behaviour of the physical system for small 0 in this present case is very similar to the behaviour of the physical system discussed previously in Section 4. The consequence of this behaviour is seen in Fig. 2 where the region of x’ over which complete redistribution occurs increases as 0 decreases for a given x (typically 1x1 < 2 in this example). The interpretation of the results shown in Fig. 3 is made most readily with the aid of equations (4.4) and (4.5) which relate the frequencies in the atom’s frame of reference to the
Scattering of line radiation--II.
Velocity-noncorrelated
scattering
1643
FIG. 3. The change in J&-,(x,x’) towards complete redistribution with decreasing x for (T= 0401.
frequencies seen by a distant observer for a single scattering event. In Fig. 3, the region of complete redistribution starts to disappear only when the absorption starts to occur in the wings (say (xl > 2). From equation (4.4) we see that this condition corresponds to y being comparable with the velocity term u cos 8 so that there does not exist a region of complete redistribution (we may compare this line of reasoning with that used in case I where y = y’ = 0). The same argument, using equation (4.5), when applied to the wings of the existing curves explains the deviations from unity of the ratio R II-N(x, x’)/[c#~(x)l(/(x’)] for large x’. However, it must also be remembered that the probability of absorption in the wings of a spectral line is much lower than the probability of absorption in the core (for example, in the above case where cr = OXIOl,absorption at the core is of the order of 5000 times more probable than at x = 3).w Consequently, it would be reasonable to conclude that for the majority of scattering events which are redistributed according to the function (5.6), there exists a region about the origin (in the example considered lx’1 I 2) for which conditions appropriate to complete redistribution do exist. To conclude this section on the redistribution function RII-N, given by equation (5.6), we consider its asymptotic limit. Using the series expansion for the Voigt function,@’ H(a, u) = g
(
l+
3v2-a2 2(a2_v2)2+
.a.
(5.9)
7 I
to the first order in (a2 + 02)- ’ (since u = (x + xl)/,/2 is large in the asymptotic limit), we see that 2 r(lx -
x'l),
Ix-x'1
>>
1
(5.10)
where r(/3) = (2~)~ iI2 exp( - /12/2).
(5.11)
For certain order of magnitude calculations, HUMMER (‘) has shown that a straight line approximation to r(9) provides sufficient accuracy. Therefore for completeness, we provide
1644
the following
J. D. ARGYROSand D. MUGGLESTONE
least-square
fit to equation
(5.1 l), viz.
-0.1568fl+O.4016,0 r(B) =
< fl < 2.561
(5.12)
0, otherwise.
6. COMPLETE
(CASES
REDISTRIBUTION
III
AND
I)
Before discussing complete redistribution generally, the redistribution function for velocity-noncorrelated scattering is derived in the case when there is complete redistribution in the atom’s rest frame (case III). If the appropriate form of f(Ax + v0 - Apu)p(Ax + v0 - Ap.4 ; Ax’ + v0 - Ap’u’) (i.e. equation (2.10)) is substituted into equation (3.5) for the redistribution that we may write the redistribution function in this case as 5
RIIl_.(x,x')
=
%
function,
we find
”
dy du s ((x -y)2 + G”} P” U’ o(, dy’ u’ exp( - u’~) _u, d”’ s ((x -y’)2+02f uexp(-U2)
-
,
(6.1)
which can readily be shown to reduce to the form R,,,_.(x,
x’) = [H(o, x)/TT”~] . [H(a, x’)/Tc”~],
(6.2)
where H(a, v), the Voigt function, has been defined previously in equation (5.7). It is obvious that this redistribution function is of the form required by the identity (1.3) (the condition which must be satisfied in order that complete redistribution occurs); consequently complete redistribution in the atom’s rest frame also produces complete redistribution in the observer’s frame of reference in this case. Although this result appears trivial, it has been shown by FINN ~3) that complete redistribution in the atom’s frame of reference does not necessarily lead to complete redistribution in the observer’s frame of reference. It should be noted once again that lim R,I,_N(x, O-r0
x’) = R,_.(x,
x’).
(6.3)
The limit (6.3) would seem to imply that case I is a special case of case III as well as being a special case of case II. However, strictly speaking this is not so. Case I is the limit of case II for a vanishingly small value of the broadening parameter u (or by equation (5.2) the broadening parameter a). The fact that R, _ ,,,(x, x’) is the limit of both R,, _ N(~, x’) and R,,, _ Jx, x’) as CJ+ 0 is due to the fact that equation (4.2) can be deduced for two different functional forms of ~(5, 5’). By using equation (2.8b) i.e.
P(L5’) = S(5 - 4’)>
as we have done in Section 4, coherent the alternative functional form
scattering
has been assumed.
P(L5’) = 6(vo - 5’)
(6.4) On the other hand, if
(6.5)
Scattering of line radiation--II. Velocity-noncorrelated scattering
is used, the same result (4.2) can be deduced by factorizing the redistribution
1645
function
R,_N(~, x’) in a manner similar to that used to evaluate R,,,_N(~, x’) in equation (6.1). The
distinction here is that in the form given by equation (6.4) the emission frequency 5’ must be the same as the absorption frequency 5, while in the form given by equation (6.5) the emission frequency 5’ must be the same as the central frequency vO.However, equation (6.5) is the form of p(<, 5’) which results from an assumption of complete redistribution in the atom’s frame of reference with zero line width. Although the zero width of the spectral line in the atom’s rest frame renders the distinction between coherent scattering and complete redistribution, both being considered in the rest frame of the atom, rather academic (since there is only one frequency into which the radiation can be ‘redistributed’ completely, viz. the central frequency v,), the behaviour of the redistribution functions R , , _ N(~, x’) and RI II _ N(~, x’) is so different in the limit of small, but non-zero, values of g that it proves convenient to retain the distinction between coherent scattering and complete redistribution even in the limit of fJ -+ 0. Before leaving this section on complete redistribution, it should be pointed out that the deduction of the redistribution functions describing complete redistribution is not important in itself (the results of complete redistribution have been used for a very long time with great effect). Rather it is the physical interpretation of the radiation scattering events, which must be used in the derivation of the redistribution functions, which is important. By using the conditions for velocity-noncorrelated scattering established in Paper I, there now exists a method of determining, a priori, whether or not an assumption of complete redistribution is a reasonable one in any physical situation. If the conditions in an atmosphere are such that velocity-noncorrelated scattering can be expected to occur in conjunction with complete redistribution in the atom’s frame of reference, then a distant observer viewing this atmosphere will observe radiation which has undergone complete redistribution in that atmosphere by means of the scattering process. Finally it should be mentioned that because the model used for velocity-noncorrelated scattering results in complete redistribution for certain types of atomic scattering, this does not preclude the possibility that some other mechanism may also produce complete redistribution. What we have shown is that, given the conditions for velocity-noncorrelated scattering, complete redistribution can be expected to occur for certain types of atomic radiation scattering events. 7. CONCLUDING
REMARKS
The scattering processes discussed in the present paper are of particular importance in the case of the scattering of photons travelling through a stellar atmosphere. A detailed discussion of the properties and behaviour of the radiation field in uniform model atmospheres which results when complete redistribution is assumed has been presented by AVRETT and HUMMER.(~) Recently HUMMER(‘) has shown the effects, on the radiation field, of allowing a more complex form of redistribution to occur (viz. redistribution arising during velocity-correlated scattering). The redistribution functions which have proved to be most interesting in this latter paper are those functions belonging to the class which can be deduced by assuming that the atomic radiation scattering event occurs according to the equations (2.9). The functional forms of the redistribution functions obtained by HUMMER(~) for velocity-correlated scattering (in the various classes discussed in the present paper) are compared in Table I with the corresponding velocity-noncorrelated redistribution functions
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J. D. ARGYROSand D. MUGGLESTONE
derived in the present investigation ; for clarity a pictorial model of each of the atomic radiation scattering events, giving rise to both the velocity-correlated and velocity-noncorrelated redistribution functions, is also shown. The radiation emanating from a uniform semi-infinite model atmosphere in which the scattering of the radiation is described by the velocity-noncorrelated redistribution function of equation (5.6), has been investigated by ARGYROS.(lo) It was found there that the resulting radiation field exhibited certain characteristic properties also noted by HUMMER(‘) for radiation from the same model atmosphere and for the same type of atomic radiation scattering event but for velocity-correlated scattering. Further investigations into the effects of redistribution on the radiation emerging from various types of atmosphere is continuing and the results obtained will be published in a future paper of this series. Acknowledgements-One of the authors (JDA) was, during the period of the present research, a recipient of a Commonwealth of Australia Postgraduate Award for which support he wishes to express his appreciation. This research was supported financially by the Australian Research Grants Committee.
REFERENCES 1. J. D. ARGYROSand D. MUGGLESTONE, JQSRT 11, 1621 (1971). D. G. HUMMER,Mon. Not. R. Astr. Sot. 125,21 (1962). J. T. JEFFERIES and 0. R. WHITE,Astrophys. J. 132,767 (1960). V. V. SOeOLEV,Vestnik LmingraLkogo Gosudarstuennogo Universteta, No. 5, p. 85 (1955) G. D. FINN and D. MUGGLESTONE, Mon. Not. R. Astr. Sot. 129,221 (1965). B. J. O’MARA, Ph.D. Thesis, University of California, Los Angeles (1967). 7. D. G. HUMMER,Mon. Not. R. Astr. Sot. 145,95 (1969). 8. G. D. FINN,Astrophys. J. 147, 1085 (1967). 9. E. H. AVRETTand D. G. HUMMER,Mon. Not. R. Astr. Sot. 130,295 (1965). IO. J. D. ARGYROS,Proc. Astron. Sot. Amt. 8,387 (1970). 2. 3. 4. 5. 6.