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The relativistic Doppler broadening of the line absorption profile
I Quant Spectrosc Radmt Transfer Vol 27, No 6, pp 653-655, 1982
0022..407318210t~0653.-035030010
Printed m Great Britain
© 1982PergamonPressLtd
THE RELATIVISTIC DOPPLER BROADENING OF THE LINE ABSORPTION PROFILE S KICHENASSAMY Laboratoire de Physique Th~orique, E R A No 533 assoclt[eau C N R S, Institut Henri Poincare, I l, rue P & M Curie, 75231 Paris Cedex 05, France R KRIKORIAN Laboratolre d'AstrophyslqueTh~orique du Coll~ge de France, Instltut d'Astrophysique,98 his Bd Arago, 75014 Paris, France and A NIKOGHOSSIAN Byurakan AstrophysicalObservatory,Arm~nie,U S S R
(Recewed 30 March 1981) Abstract--The classzcal results of Doppler broadening of the hne absorption profile are generalized to a relativistic gas in thermal eqmhbnum by taking into account the relativistic variance of the volume absorption coefficientsof the gas, as derived by L H Thomas This variance produces a small correction, even in the non-relativisticapproximation 1 INTRODUCTION We conszder the absorption of radmtzon by a relativistic gas m thermal equilibrium Let S be the rest-frame of the gas and I, the intensity of a v-radiation beam wah respect to S An atom of the gas moving with the velocity u = #c, in a direction making an angle O with the beam, receives tMs radiation with the frequency
v' = 3"v(1 - [3 cos 0), y = (1 - 32) -./2
(1)
The volume absorption coefficient X~ of the gas is obtained by summing the absorption profiles of the atoms over all velocities v and directions 0 At the non-relatzvlStlC level, the determination of X~ is well known (see Refs 1-3) On the other hand, as far back as 1930, Thomas 4 estabhshed by a direct study of the radiation transfer equations that X,v
= constant
(2)
under a Lorentz transformation. This result was later confirmed by Synge 5 and extended to general relativity by Llndquist 6 However, although Eq (2) may be used to define the variance of the atomic absorption coefficient a~, the determination of X~, when it is due to an assembly of relatlVlStm particles, is unavadable We present this derivation in Section 2 and show m Section 3 the llmzts of the non-relativistic expressions, as well as corrections induced by Eq (2) 2 DETERMINATION OF X~ The relativistic gas of atoms of proper mass m and momentum p~ at temperature T is described by the lnvarlant distribution function 5
p2 no m2K2(m~) ~: e -'~er, N ( p ) = -~s
3' = 4 ( 1 + ( m ) ),
(3)
where ~: = c2/kT, c = hght velocity, k = 1 380 x 10-i6 erg-deg -t, no = density of the gas in its rest-frame, the Bessel funcnon K2(m~) is defined by
K2(m~) = f : e -m~cht ch2tdt 653
q K1CHEN~SS~M~et al
654
The volume absorptmn coefficient X. of the gas is (5)
X~ = ;o ~ N(p)a.dfl,
where a. is the atomic absorption coefficient relative to the frame considered and dIl = dp'Adp2Adp 3= Ip°ldw, mdw is the 3-volume of a 3-ceU on the pseudo-sphere m = const (cf Synge 5) Let a'~ be the atomic absorption coetfioent relative to the rest-frame of an atom. t e , I" 1 a', = fro/77"(v' - vo)2+ F2,
zre" 0% = me c'
(6)
where v0 is the mvarlant proper frequency of the atom, v' the frequency of the incident as received by the atom, F the natural hne-wldth, e and me the charge and the mass of an electron, / the oscdlator strength of the atom A s t h e L o r e n t z m v a n a n c e o f x . v l m p l i e s t h a t o f N ( p ) a ' . , v ' d ~ o , we obtain from the preceding definmons no m , F ff fo ~ fo2'~ e""c(l+U2)uZ(1-flcosO)smO X. = 4,r K2(m~) °r°f-~ du dO d~p [yv(1 - / 3 cos 0 ) - Vo]2 + F 2 '
(7)
where u = Iplrnl = y~ Integrating over ~o and 0, we get
X,
=
no ~ro.f F f ~ 2 K2(mO ~ J _ e_,.a:.+:)/2.~ (x - xdx vo)2 + F 2'
(8)
where x = yv(l + [3) For vanishing F, Eq (8) gives I'tO -m~:(v2+Vo2/2VVol Xo. = noo'of 2K2(~¢)v2 e
(9)
We observe then that the mean atomic absorption coefficient So. = xodno satisfies the nortmlizatlon condition
fo
~S°~ dv = ~ [. /2 V0
(lO)
as it should, ff we assume the radmtion density to vary slowly with frequency 3 NON-RELATIVISTIC APPROXIMATION We assume the temperature T of the gas to be such that
me-"
(
(,,
)
while the velocmes of the atoms are small, i e . y= 1
(12)
Equatmn (8) may be written
x.
-
n o a ( v o ) o- a fj _ _~
e-
,~.
1 - f l o y: - dy, v---y +a 2 vo
(13)
The relalmStlC Doppler broadenmgof the hne absorption profile
655
where
[2kT~ u~ ~o = ~,m-~c ]
F
v - ~o
' b = ~o~o, a = - b , v =
b
' y = ~o ' a ( . o ) v
=
~°I b~(Tr)
(14)
Equation (9) yields for the atomm mean absorption coefficient its classical value So, -- a(v0)o e
-v2
, when v -~ vo
(15)
In the central part of the line where v is small, the greatest contribution to the integral in Eq (13) comes, when a is small, from values of y near vovlv, setting y = Vo(V+ z)/v, we get
X,,(centre)~-- noa( VO)D(1-- v --vvO) ~ e -('~'~2,
06)
which gives, when v = v0, Xv(centre) ~
noOt(Vo)De-~: = noso~.
(17)
In the wings of the line where v >> 1, the mare contribution to the same integral comes from terms ( y d v 0 ) ~ v; neglecting a 2 and expanding ( 1 - (v/vo)(y[v))2, we obtain
X'~w'"~') = n°aO'°)D~
[l- ~°(Iv ~°)]'
(18)
which reduces to its classical value, when the second term IS negligible Equations (16) and (18) show that the Lorentz (and therefore Gahlean) mvanance of XY and not of X~ produces at the non-relativistic level, small corrections, the second term In Eq (16) introduces a slight asymmetry in the central part of the line, whereas the second term in Eq (18) is a temperature-dependent small correction In the wings Because light propagation xs not Gahlean lnvanant, this result could not be obtained in prerelatlvlStlC Physics 4 CONCLUDING REMARKS This determination of the Doppler broadening of the hne absorption profile of a relativistic gas described by the Juttner-Synge distribution function provides a satisfactory generalizll~iJ P of classical results; however, small correcUons are present even in the n o n - r e l a f i v l approxlmahon; they are essentially due to the non-Lorentz mvarlance (and consequently tl~ non-Gahlean mvarlance) of X~A similar treatment can be extended to the case of absorption due to pure scattering, taking into account our ¢arher results on the relatlvlsttc variance of the scattering cross section (Klchenassamy and Krlkorlan 7) REFERENCES D W N Stlbbs and R v d R Wolley, The OuterLayers of a Star, Clarendon Press, Oxford (1953) V Ambartsumlan,Ed, TheoreticalAstrophysws,Pergamon Press, Oxford (1958) D Mihalas, StellarAtmospheres Freeman, San Francisco (1978) L H Thomas, Quart J Math, 1,239 (1930) J L Synge, The RelatwlsttcGas North Holland, Amsterdam(1957), also F Juttner, Ann d Phys 35, 145 (1911)and 3 Hazlehurst and W L W Sargent, Ap J 130, 276 (1959) 6 R W Lmdqmst,Ann Phys 37, 341 (1966), also J L Andersonand E Spiegel,Ap J 171 127 (1972) 7 S Kichenassamyand R Knkonan, C R Acad ScL Pans, 2~IB, 127 (1978)
1 2 3 4 5
0022..407318210t~0653.-035030010
Printed m Great Britain
© 1982PergamonPressLtd
THE RELATIVISTIC DOPPLER BROADENING OF THE LINE ABSORPTION PROFILE S KICHENASSAMY Laboratoire de Physique Th~orique, E R A No 533 assoclt[eau C N R S, Institut Henri Poincare, I l, rue P & M Curie, 75231 Paris Cedex 05, France R KRIKORIAN Laboratolre d'AstrophyslqueTh~orique du Coll~ge de France, Instltut d'Astrophysique,98 his Bd Arago, 75014 Paris, France and A NIKOGHOSSIAN Byurakan AstrophysicalObservatory,Arm~nie,U S S R
(Recewed 30 March 1981) Abstract--The classzcal results of Doppler broadening of the hne absorption profile are generalized to a relativistic gas in thermal eqmhbnum by taking into account the relativistic variance of the volume absorption coefficientsof the gas, as derived by L H Thomas This variance produces a small correction, even in the non-relativisticapproximation 1 INTRODUCTION We conszder the absorption of radmtzon by a relativistic gas m thermal equilibrium Let S be the rest-frame of the gas and I, the intensity of a v-radiation beam wah respect to S An atom of the gas moving with the velocity u = #c, in a direction making an angle O with the beam, receives tMs radiation with the frequency
v' = 3"v(1 - [3 cos 0), y = (1 - 32) -./2
(1)
The volume absorption coefficient X~ of the gas is obtained by summing the absorption profiles of the atoms over all velocities v and directions 0 At the non-relatzvlStlC level, the determination of X~ is well known (see Refs 1-3) On the other hand, as far back as 1930, Thomas 4 estabhshed by a direct study of the radiation transfer equations that X,v
= constant
(2)
under a Lorentz transformation. This result was later confirmed by Synge 5 and extended to general relativity by Llndquist 6 However, although Eq (2) may be used to define the variance of the atomic absorption coefficient a~, the determination of X~, when it is due to an assembly of relatlVlStm particles, is unavadable We present this derivation in Section 2 and show m Section 3 the llmzts of the non-relativistic expressions, as well as corrections induced by Eq (2) 2 DETERMINATION OF X~ The relativistic gas of atoms of proper mass m and momentum p~ at temperature T is described by the lnvarlant distribution function 5
p2 no m2K2(m~) ~: e -'~er, N ( p ) = -~s
3' = 4 ( 1 + ( m ) ),
(3)
where ~: = c2/kT, c = hght velocity, k = 1 380 x 10-i6 erg-deg -t, no = density of the gas in its rest-frame, the Bessel funcnon K2(m~) is defined by
K2(m~) = f : e -m~cht ch2tdt 653
q K1CHEN~SS~M~et al
654
The volume absorptmn coefficient X. of the gas is (5)
X~ = ;o ~ N(p)a.dfl,
where a. is the atomic absorption coefficient relative to the frame considered and dIl = dp'Adp2Adp 3= Ip°ldw, mdw is the 3-volume of a 3-ceU on the pseudo-sphere m = const (cf Synge 5) Let a'~ be the atomic absorption coetfioent relative to the rest-frame of an atom. t e , I" 1 a', = fro/77"(v' - vo)2+ F2,
zre" 0% = me c'
(6)
where v0 is the mvarlant proper frequency of the atom, v' the frequency of the incident as received by the atom, F the natural hne-wldth, e and me the charge and the mass of an electron, / the oscdlator strength of the atom A s t h e L o r e n t z m v a n a n c e o f x . v l m p l i e s t h a t o f N ( p ) a ' . , v ' d ~ o , we obtain from the preceding definmons no m , F ff fo ~ fo2'~ e""c(l+U2)uZ(1-flcosO)smO X. = 4,r K2(m~) °r°f-~ du dO d~p [yv(1 - / 3 cos 0 ) - Vo]2 + F 2 '
(7)
where u = Iplrnl = y~ Integrating over ~o and 0, we get
X,
=
no ~ro.f F f ~ 2 K2(mO ~ J _ e_,.a:.+:)/2.~ (x - xdx vo)2 + F 2'
(8)
where x = yv(l + [3) For vanishing F, Eq (8) gives I'tO -m~:(v2+Vo2/2VVol Xo. = noo'of 2K2(~¢)v2 e
(9)
We observe then that the mean atomic absorption coefficient So. = xodno satisfies the nortmlizatlon condition
fo
~S°~ dv = ~ [. /2 V0
(lO)
as it should, ff we assume the radmtion density to vary slowly with frequency 3 NON-RELATIVISTIC APPROXIMATION We assume the temperature T of the gas to be such that
me-"
(
(,,
)
while the velocmes of the atoms are small, i e . y= 1
(12)
Equatmn (8) may be written
x.
-
n o a ( v o ) o- a fj _ _~
e-
,~.
1 - f l o y: - dy, v---y +a 2 vo
(13)
The relalmStlC Doppler broadenmgof the hne absorption profile
655
where
[2kT~ u~ ~o = ~,m-~c ]
F
v - ~o
' b = ~o~o, a = - b , v =
b
' y = ~o ' a ( . o ) v
=
~°I b~(Tr)
(14)
Equation (9) yields for the atomm mean absorption coefficient its classical value So, -- a(v0)o e
-v2
, when v -~ vo
(15)
In the central part of the line where v is small, the greatest contribution to the integral in Eq (13) comes, when a is small, from values of y near vovlv, setting y = Vo(V+ z)/v, we get
X,,(centre)~-- noa( VO)D(1-- v --vvO) ~ e -('~'~2,
06)
which gives, when v = v0, Xv(centre) ~
noOt(Vo)De-~: = noso~.
(17)
In the wings of the line where v >> 1, the mare contribution to the same integral comes from terms ( y d v 0 ) ~ v; neglecting a 2 and expanding ( 1 - (v/vo)(y[v))2, we obtain
X'~w'"~') = n°aO'°)D~
[l- ~°(Iv ~°)]'
(18)
which reduces to its classical value, when the second term IS negligible Equations (16) and (18) show that the Lorentz (and therefore Gahlean) mvanance of XY and not of X~ produces at the non-relativistic level, small corrections, the second term In Eq (16) introduces a slight asymmetry in the central part of the line, whereas the second term in Eq (18) is a temperature-dependent small correction In the wings Because light propagation xs not Gahlean lnvanant, this result could not be obtained in prerelatlvlStlC Physics 4 CONCLUDING REMARKS This determination of the Doppler broadening of the hne absorption profile of a relativistic gas described by the Juttner-Synge distribution function provides a satisfactory generalizll~iJ P of classical results; however, small correcUons are present even in the n o n - r e l a f i v l approxlmahon; they are essentially due to the non-Lorentz mvarlance (and consequently tl~ non-Gahlean mvarlance) of X~A similar treatment can be extended to the case of absorption due to pure scattering, taking into account our ¢arher results on the relatlvlsttc variance of the scattering cross section (Klchenassamy and Krlkorlan 7) REFERENCES D W N Stlbbs and R v d R Wolley, The OuterLayers of a Star, Clarendon Press, Oxford (1953) V Ambartsumlan,Ed, TheoreticalAstrophysws,Pergamon Press, Oxford (1958) D Mihalas, StellarAtmospheres Freeman, San Francisco (1978) L H Thomas, Quart J Math, 1,239 (1930) J L Synge, The RelatwlsttcGas North Holland, Amsterdam(1957), also F Juttner, Ann d Phys 35, 145 (1911)and 3 Hazlehurst and W L W Sargent, Ap J 130, 276 (1959) 6 R W Lmdqmst,Ann Phys 37, 341 (1966), also J L Andersonand E Spiegel,Ap J 171 127 (1972) 7 S Kichenassamyand R Knkonan, C R Acad ScL Pans, 2~IB, 127 (1978)
1 2 3 4 5