Dynamics of extended AGB star envelopes
***Volume Title**ASP Conference Series, Vol. **Volume Number****Author** c (cid:13) **Copyright Year** Astronomical Society of the Pacific Dynamics of extended AGB star envelopes
Claudia Dreyer, Michael Hegmann, and Erwin Sedlmayr
Zentrum f¨ur Astronomie und Astrophysik, Technische Universit¨at Berlin,Berlin, Germany
Abstract.
The dust formed in extended circumstellar envelopes of long-period vari-ables and Miras has a strong influence on the envelope dynamics. A radiatively driveninstability caused by the formation of dust leads to the development of an autonomousdynamics characterised by a set of distinct frequencies. We study the interplay betweenthe envelope’s internal dynamics and an external excitation by a pulsating star.
1. Introduction
Long-period variables (LPVs) and Miras are radially pulsating, highly evolved starson the asymptotic giant branch (AGB). Their cool, extended atmospheres are excellentsites for the formation of complex molecules and dust particles. The interplay betweendust formation and stellar radiation results in circumstellar envelopes (CSEs) generat-ing slow mass loss, which finally enriches the interstellar medium with processed mate-rial. The observed light curves exhibit irregularities over intervals of several pulsationcycles, which are at least partially caused by the phenomenon of dust formation.To investigate the complex dynamical behaviour of carbon-rich envelopes in detail,CSEs can be considered as nonlinear multi-oscillatory systems, whose eigenfrequenciesand normal modes are controlled by the intrinsic timescales of various coupled physicaland chemical processes. This is done by applying established methods of non-lineardynamics such as Fourier Analysis and the study of stroboscopic and Poincar´e mapsupon the results of self-consistent model calculations (Fleischer et al. 1992).
2. Eigendynamics
The envelopes around highly luminous AGB stars, e.g. L ∗ = × L (cid:12) , M ∗ = (cid:12) , T ∗ = C / O = .
25 (henceforth model A) develop a self-maintaining os-cillation caused by dust formation (exterior κ -mechanism) even without the additionalinput of mechanical momentum by an underlying stellar pulsation (cf. Fleischer et al.1995; H¨ofner et al. 1995). In their power spectrum (Fig. 1, left) one can clearly see theeigenfrequency f κ ≈ − and its overtones (Dreyer et al. 2009)).In the case of lower stellar luminosities, e.g. L ∗ = × L (cid:12) , M ∗ = (cid:12) , T ∗ = C / O = .
75 (henceforth model B), the system generates no self-induced oscillation. An additional input of energy and momentum is needed to deter-mine the CSE eigendynamics. We excite such an envelope by white noise. Therebywe adopt a normal noise distribution with zero mean and intensity σ (see Dreyer et al.1 a r X i v : . [ a s t r o - ph . S R ] N ov Dreyer, Hegmann, and Sedlmayr ff erent eigenmodes. Theposition of the minima (marked in grey) coincides with the dust nucleation zone locatedat r ≈ .
25 R ∗ . This confirms that the dust is responsible for creating the characteristicfrequencies of the envelope. Figure 1. Power spectra of the radial outflow velocity u for a high luminous LPVpurely caused by the exterior κ -mechanism (model A, left) and caused by a stochasticexcitation for a standard luminous LPV (model B, right). The position of the dustformation zone in the envelope is highlighted.
3. Interaction with external excitation
Once the eigenfrequencies have been found, the question arises as to how these modesinteract with an external excitation caused by the stellar pulsation. For this purpose,we studied the response of our reference models for a series of harmonic excitationswith di ff erent periods P and strength ∆ u . Figure 2 shows the most dominant envelopefrequencies over the excitation period at constant strength. For model A (left) we canidentify three di ff erent dynamic domains, namely eigenmode -dominated, irregular andpulsation-dominated domain. The same study for model B (right) reveals a similarbehaviour. In contrast to model A we have not found an eigenmode dominated domain.Figure 3 shows the maps of our reference models for a typical oscillation period ofLPVs. In the maps it can be seen that for a constant phase angle (black) the system tendsto stay inside a finite number of stripes. Similar to the orbital resonance in celestialmechanics, we can find integers i and j which fulfil the resonance condition iP = jP CSE .This suggest that the systems returns to the same physical state after i excitation periods P or j resonance periods P CSE . For model A, we find that i =
27 and j =
4. For modelB the CSE response is double periodic, which means that the envelope period equalstwo times the excitation period (Dreyer et al. 2009, 2010).
4. Synthetic light curves
Figure 4 presents synthetic light curves for model A obtained for various excitations atseveral wavelengths. The light curves in the first and the last panels show monoperi-odic behaviour but with di ff ering periods. For the unexcited CDS (left) the dominanttimescale is the dust formation scale P κ = P = ynamics of extended AGB star envelopes Figure 2. Most dominant frequencies of a CSE f CSE normalised to its eigenfre-quency f κ (triangles) excited with various pulsation periods P near the dust nucle-ation zone for model A (left) and model B (right). The set of CSE-eigenmodes isdepicted by circles, the eigenperiod is labelled with P κ . The excitation frequencies(solid line) and harmonics (dash-dotted lines) are also shown.Figure 3. Maps of the ( u , ρ , T ) phase space in the dust nucleation zone formodel A (left) and model B (right). The stroboscopic maps were obtainedby sampling [ u ( n . P ) , ρ ( n . P ) , T ( n . P )] (grey) and the Poincar´e maps by[ u ( nP ) , ρ ( nP ) , T ( nP )] (black) for 1 ≤ n ≤ (cid:98) t max / P (cid:99) , n ∈ N . In the curves in the second panels the eigenperiod of the CSE P κ = P =
388 d interfere with each other. Nevertheless, the dominant timescaleis the slightly shifted dust formation timescale P CSE ≈ P CDS = P =
5. Special application: IRC + Winters et al. (1996) have found that the fundamental parameters: L ∗ = × L (cid:12) , M ∗ = (cid:12) , T ∗ = C / O = . P =
650 d, and ∆ u = − wellreproduce observations of IRC + P =
650 d (e.g. Menten et al. 2006), we excite the model exactly with this
Dreyer, Hegmann, and Sedlmayr
Figure 4. Synthetic light curves of model A for various excitations at di ff erentwavelength. The magnitudes are related to the mean magnitude. frequency. As can be seen in the power spectrum (Fig. 5, left) the CSE is domi-nated by the pulsation frequency and its first harmonic close to the star. With onsetof dust formation at r ≈ . ∗ these frequencies disappear and the envelope frequency f CDS = . × − d − ≈ (3900 d) − controlled by the timescale of the dust forma-tion process gains strength. The corresponding Poincar´e map (Fig. 5, right) shows fourclearly distinct stripes (black). A resonance such as in the case of the previously studiedmodels A and B is not found. Figure 5. Power spectrum of velocity u (left) and maps of the ( u , ρ, T )-phase space(right) of the periodically excited IRC + Figure 6 compares specific observations and synthetic light curves in di ff erent NIRbands. The stellar pulsation period P =
650 d can be seen in all bands. An additionalperiod of six times the stellar pulsation period is also visible (dashed-dotted line). Thisis the dust-determined eigenperiod of the CSE P CSE ≈ ynamics of extended AGB star envelopes Figure 6. Comparison between calculated (line) & observed (points) light curvesof IRC + ff erent NIR-wavelengths. The magnitudes are related to themean magnitude. The CSE-eigenperiod P CSE is marked with a dashed-dotted line.
6. Summary