Limb darkening, rotation, and variability D. John Hillier University of Pittsburgh Financial support: STScI/NASA
Collaborators Jean-­‐Claude Bouret (Marseille) Thierry Lanz (Nice) Joseph Busche (Wheeling) Eps Ori Martins, M
arcolino, Hillier, of
et O9-B0.5
al. 2014, Martins
et al.: Variability
stars A&A in press. (a) Hα
(b) Hβ
(c) Hγ
(d) He I 4026
(e) He I 4471
(f) He I 4712
Eps Ori (d) He I 4026
Martins, Marcolino, Hillier, et al. 2014, A&A in press. (e) He I 4471
(f) He I 4712
(g) He I 5876
(h) He II 4542
(i) He II 4686
(j) He II 5412
(k) O III 5592
(l) C IV 5802
Fig. 1. Variability of $ Ori on a daily timescale (from 16th to 25th October 2007). Upper panels: individual spectra in black (each spectrum
corresponds to a night average) and global average spectrum in red. Lower panels: Temporal Variance Spectrum, together with the one sigma
Rotation
(1)  Broadens photospheric lines
(a) Generally done by convolution (e.g., Gray 1992).
(b) Formal integral across disk is better,
(2)  ‘’Weakly’’ broadens wind lines
(3)  Introduces multiple resonance zones into the wind.
For simplicity assume:
(a) No change in density structure
(b) V(ϕ) scales as 1/r above sonic point.
Studies:
Bjorkman and Cassinelli (1993, ApJ, 409, 429)
Busche & Hillier (2005, AJ, 129,454)
Perentz & Puls (1996, A&A, 312, 195)
Perentz & Puls (2000, A&A, 358,956)
Convolution
1. 
Fast
2. 
Designed for photopheric lines
3. 
Does not work for wind lines, since rotation rate decreases with height.
4.  Only works for photospheric lines if the line profile does not vary across
the disk (i.e., line and continuum have the same limb darkening law).
Works very well for the majority of photospheric lines, but IMPORTANT
exceptions.
Rotation Broad proUile allows surface to be resolved (modulo intrinsic proUile and macroturbulence). Doppler Imaging: Ap stars N IV 4058 Zeta Puppis Model (convolution) Model (formal int.) Hillier et al., 2012, 426,1043 Si IV λλ4089, 4116 Zeta Puppis Model (convolution) Model (formal int.) He II 4686 Zeta Puppis Model (convolution) Model (formal int.) Hα Wind p
z p = 0 => looking at disk center p ~ R★ => looking near limb N IV 4058
Continuum
NIV 4058 – does not follow the same limb darkening law as the adjacent continuum Center-­‐to-­‐limb variation N IV 4058 p/R(τ=2/3) = 0.0, 0.039, 0.733, 0.965, 0.983, 0.997, 1.007, 1.0127, 1.062 Cumulative spectrum (p=0 to pi) pi/R(τ=2/3) = 0.039, 0.733, 0.965, 0.983, 0.997, 1.007, 1.0127, 1.062, 100.0 Vs τ=2/3 N IV 4058 Origin of continuum Origin of N IV 4058 (line center) Center-to-limb variation
N III λ4634.
p/R(τ=2/3) = 0.0, 0.039, 0.733, 0.965, 0.983, 0.997, 1.007, 1.0127, 1.062
N III λ4634 (N III λλ4641, 4642 at 420 km/s)
C IV λ5801
p/R(τ=2/3) = 0.0, 0.039, 0.733, 0.965, 0.983, 0.997, 1.007, 1.0127, 1.062
pi/R(τ=2/3) = 0.039, 0.733, 0.965, 0.983, 0.997, 1.007, 1.0127, 1.062, 100.0
He II 4686 p/R(τ=2/3) = 0.0, 0.039, 0.733, 0.965, 0.983, 0.997, 1.007, 1.0127, 1.062 He II 4686 pi/R(τ=2/3) = 0.039, 0.733, 0.965, 0.983, 0.997, 1.007, 1.0127, 1.062, 100.0 Conclusions
A.  Rotation crucial for Ha formation: f(vsini, β, mass loss rate)
A. 
B. 
Hα variability problem in many stars (factor of 2 in mass loss, β=?)
Are winds intrinsically unstable?
B.  Be carful using convolution for spectral analyses
C.  Non-LTE effects can strongly modify “limb-darkening” laws, even for
pure absorption lines.
D.  Some weak features may be the result of cancellations
E.  I(p) & line formation zones useful for understanding line strengths, line
formation, and line variability etc.
F. 
Line formation mechanism will influence variability:
A. 
B. 
C. 
D. 
LTE
continuum pumping
dielectronic recombination/ recombination
collisional excitation
Najarro et al. (2006, A&A, 456, 1159) Fe IV lines affect transfer in He I resonance lines: * Alters 2p 1Po populations. * Affects strength of He I singlet transitions. As the Fe atomic data is uncertain, need to use He I triplets for analyses. f, N(Fe)
f/2, N(Fe)
f, N(Fe)/2
f, f/2
f/5, f/10
VD=13.5 km/s
VD=10.0 km/s
THE
END