Table Of Content**FULL TITLE**
ASP Conference Series, Vol. **VOLUME**, **YEAR OF PUBLICATION**
**NAMES OF EDITORS**
Circumstellar rings, flat and flaring discs
M.L. Arias1, J. Zorec2, Y. Fr´emat3
1 Facultad de Ciencias Astrono´micas y Geof´ısicas, UNLP, Argentina
6 2Institut d’Astrophysique de Paris, UMR7095 CNRS, Univ. P&M Curie
0
3Royal Observatory of Belgium
0
2
n Abstract. Emission lines formed in the circumstellar envelopes of several
a type ofstarscanbe modeled using firstprinciples ofline formation. We present
J simplewaysofcalculatinglineemissionprofilesformedincircumstellarenvelopes
4 having different geometrical configurations. The fit of the observed line profiles
with the calculated ones may give first order estimates of the physical param-
1 eters characterizing the line formation regions: opacity, size, particle density
v distribution, velocity fields, excitation temperature.
9
6
0 1. Rings and flaring discs
1
0
In optically thick regions, more than 90% of the emitted energy in a given line
6
can be considered produced in a limited region of the circumstellar disc. This
0
/ is the case of Feii in Be stars (Arias et al. 2005), but also hydrogen Balmer,
h
Paschen series if their formation region has the aspect of an expanding ring. In
p
both circumstances, the emitted radiation can be estimated using an equivalent
-
o isothermal ring with uniform semi-height H, whose surface density equals the
tr radial column density. If the particle density has a distribution N(R)∼ R−β,
as the radius of the ring is then Rr/R∗=[(1−β)/(2 −β)][1−(Ri/Re)β−2]/[1−
: (R /R )β−1], where R are the internal and external radii of the line forming
v i e i,e
i region. The ring can be considered having expansion Vexp and rotation VΩ
X
velocities, both represent averages of the respective velocity fields (Arias 2004).
r Theemitted fluxis simplyF = I (x,y)dxdy whereS(i)is theaspect-angle
a λ RS(i) λ
effective emitting surface projected on the sky and I (x,y) is:
λ
Ia(x,y,v−vr) = I∗(x,y)e−τfµ((vx−,yv)r) +S[1−e−τfµ((vx−,yv)r)]
((sfrtoenlltars−iedmτet(ives−msvioris)nsiao−bnτs)fo(rv−bverd) by the fro−nτtfs(vid−ver))+ (1)
Ib(x,y,v−vr) = S[1−e µ(x,y) ]e µ(x,y) +S[1−e µ(x,y) ]
((rfreoanrtesmidisesieomnisasbiosonr)bed by the front side)+
withτv=τoΦ(v−vr)andvr(x,y)=±{Vexp(R)[1−(Rx)2]12±VΩ(R)Rx}sini,where
(f,r)standfor‘front’and‘rear’sidesofthering. Thesignsintheradialvelocity
v are chosen according to the quadrant; v is the Doppler displacement in the
r
observed emission line profile, µ(x,y)= cos(ring-normal;observer), i inclination
of the ring-star system, τ is the radial opacity of the ring in the line center and
o
1
2 Arias et al.
1.3 1.2
V =300km/s V =185km/s =-0.5
1.2 1.4 Vexp=150km/s 1.0 Vexp=0km/s = 0.1
=0.7 = 0.5
S/F=0.08 = 1.0
1.1 1.3 0.8 R=5.9
=15o
1.0 0.6
1.2
0.9 V =300km/s 0.4
Vexp=0km/s =0.25
0.8 R=3.5 =0.5 1.1 0.2
H=1.0 =1.0
S/F=0.17 =1.5
0.7 i==415.5o ==32..00 1.0 0.0 H ( Eri)1994
0.6 -0.2
-300 -200 -100 0 100 200 300-300 -200 -100 0 100 200 300 -400 -300 -200 -100 0 100 200 300 400
V(kms-1) V(kms-1) V(kms-1)
Figure 1. a)Line profilesfor a ring with V =0 andthe indicated param-
exp
eters. b) ’Steeple’ line profiles produced by rings with same parameters as
in a) but V 6=0. c) Line profiles due to flaring discs treated as rings, for
exp
several β and opening angle φ=15o. β= 0.1 closely fits Hα of α Eri in 1994
(a flat ring would require H=3.8; β=−0.1)
V =150km/s Vexp=150km/s V =Vexp
0.1 0.1 0.1
Vexp=0km/s
V =0km/s V=0km/s
60km/s 80km/s
80km/s
120km/s 140km/s
140km/s
240km/s 260km/s 260km/s
(So/F*)R2r=4.5
Ho==01..50R* 320km/s 340km/s 340km/s
-600 -400 -200 0 200 400 600-600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600
V(kms-1) V(kms-1) V(kms-1)
Figure 2. Three-peak line profiles produced by rotating+expanding rings
Φ(v) is the intrinsic absorption line profile. The line sourcefunction is: S (τ )=
λ o
S for τ ≤ 1; S τp for τ > 1 [p ≃ 1/2 for Gaussian Φ(v)] and B (T ) for
o o o o o λo e
τ ≥ (B /S )2 (Mihalas 1978, Chap.11). S depends on the nature of the line
o λo o o
transition: collision-, photoionization-, mixed-dominated. Using S as done in
o
Cidale & Ringuelet (1989) we can determine the excitation temperature (Arias
et al. 2005). Thus, the free parameters to fit the observed emission line profiles
are: So/F∗, τo, Rr, H, i, VΩ, Vexp and density distribution β (or α= 2−β). The
shape of the line profiles reduce strongly the space of free parameters, mainly
those of the velocity field and the ratio between τ and H. Line intensities
o
are sensitive to (So/F∗)Rr2 and Rr to the temperature, which in turn determines
So/F∗. Figure1ashowslineprofilesobtainedfortheindicatedparameters ofthe
ring with V =0, which are typical for ‘shell’ lines. The same ring parameters
exp
are used for Fig. 1b, but V = 150 km/s. The line profiles are of the ‘steeple’
exp
type seen frequently in Feii lines. Flaring discs can be treated in the same way
as cylindrical ones, except that the surface density, and hence τ depends on
o
the coordinate perpendicular to the equator (Vinicius et al. 2005). A fit of the
Hα line of α Eri in 1994 with a flaring disc of opening angle φ = 15o is shown
in Fig. 1c, where are also shown line profiles for different β-values. Several
three-peak emission line profile are shown in Fig. 2 produced in regions treated
Rings and discs 3
2.4 1.3
V =300km/s i=70o V =300km/s i=70o (c)
2.2 VRe=xp3=.5150km/s ii==3500oo 10 VRe=xp5=0km/s ii==3500oo 11..21
21..08 SHo/F===11..000.3 iii===51200ooo 8 So/=F=2=1.000.3 ii==1200oo 10..09
=1.5 6
1.6 0.8
V =300km/s i=85o
1.4 4 00..67 VRe=xp3=.50km/s ii==5700oo
1.2 H=1.0 i=30o
2 0.5 S/F=0.17 i=20o
1.0 (a) (b) 0.4 o==11.5.0 ii==51o0o
0.8 0 0.3
-300 -200 -100 0 100 200 300-300 -200 -100 0 100 200 300 -400 -300 -200 -100 0 100 200 300 400
V(km/s) V(km/s) V(km/s)
Figure 3. Emissionline profiles obtained with discs. a) P Cyg like profiles.
b) Bottle shaped. c) Broadening due to v at i→90o
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as equivalent expanding rings. Due to the Doppler shifts, the central emission
components are produced by the front and rear sides of ring sectors where the
radial velocity is v ≃ 0.
r
2. Discs
The main difference in the treatment of discs with respect to that for rings is in
thevelocitydependenceoftheintrinsicabsorptionlineprofile. Ithasbeenshown
by Horne & Marsh (1986) that for a Gaussian Φ(v) the Doppler width of the
profile is enlarged by a “turbulent” term due to the differential rotation in the
disctowards theobserver’s direction. Thewavelength dependentopacity is then
proportional to exp{−(1/2)[(λ−λ )/(∆ ×δ)]2}, where λ = λ (v /c) and:
D D D o o
v = [V (R)sinθ+V (R)cosθ]sini
o Ω exp
δ = [1+(λ v /c∆ )2]1/2 (2)
o 1 D
v = [1V (R)sinθ+V (R)(2−β)cosθ]cosθtanisini
1 2 Ω exp
whereθ is the azimuthal angle. For large values of H, v acts as anon-negligible
1
broadening agent of the effective Doppler line width. Different examples of
line profiles obtained with discs seen at several inclination angles are shown
in Fig 3. The parameters used in Fig 3a can suite for P Cyg type line pro-
files, while Fig 3b reproduce the bottle shaped emission line profiles seen fre-
quently in Be stars. The ‘bottle’ shaped profiles can be obtained also with rings.
They are produced by the τp opacity dependence of the source function due
o
to its non-local energy supply. The broadening of the effective Doppler line
by v is depicted in Fig 3c (i=85o). Other related subjects can be found in
1
http://www2.iap.fr/users/zorec/.
References
Arias, M.L. 2004,PhD Thesis, University of La Plata, Argentina
Arias, M.L., Zorec, J., Cidale, L., Ringuelet, A. 2005, A&A, submitted
Cidale, L.S, & Ringulete, A.E. 1989, PASP, 101, 417
Horne, K., Marsh, T.R. 1986, MNRAS, 218, 761
Mihalas, D. 1978, Stellar Atmospheres, Freeman
Vinicius, M.M.F., Zorec, J., Leister, N.V., Levenhagen, R. 2005,A&A, in press
