# Presentation: 532338

## Population III Wolf-Rayet Stars in the CAK Regime

- Andrew Onifer

- First Stars III
- July 16-20, 2007

**Introduction**

- Low metallicity presents a challenge to the line driving process that is believed to be responsible for the large mass-loss rates seen in Wolf-Rayet (WR) winds. The small number of available lines, especially of Fe and other iron-group elements, lowers the overall line opacity and can open spectral gaps through which photons can leak.  At Z=0, the wind has only the burn products CNO, He, and in WC or WO-type stars, 22Ne.  In this poster, I will attempt to find parameter regimes that allow a CAK-type solution for WR mass loss rates at Z = 0.  I will focus on WC-type stars, where the large amount of line-rich carbon is likely to contribute most of the line opacity and dominate the line driving.  The following models are not WR models per se, rather than conditions under which a CAK-type driving can occur given the constraint of a “zero metallicity” WC-like abundance profile.

- CAK theory, first developed by Castor, Abbott, & Klein (1975), is a technique for analytically modeling radiative transfer in a line-driven wind with a large local velocity gradient such that the Sobolev approximation holds (*v**wind* >> *v**thermal*).  It reduces the complex, location-dependent opacity to an effectively static problem defined by an effective opacity, ***eff* which is a sum of 1-e-sob, where sob is the optical depth of a line over the Sobolev length, or the length over which a constantly doppler-shifting photon can interact with a given line.  A CAK model is characterized by the equation
- where *M*(*t*) is the force multiplier and is the ratio of line acceleration to electron acceleration, *t* is a parameter related to the electron optical depth, ** is effectively the of lines that are optically thick in sob (0 - all lines optically thin; 1 - all lines optically thick), and *k* is a parameter roughly related to the overall strength of the lines.

**CAK Theory**

- The mass-loss rate can be found by solving the momentum equation at the point of lowest force efficiency, called the critical point:

- where *g**grav* and *g**rad** *are the gravitation and radiative acceleration, respectively:

- where *M***** *is the stellar mass, ***e* is the electron cross-section per gram, *L**** is the stellar luminosity, *r* is the radial distance, *F**NID** *is a correction for the finite radius of the star, and E is a correction for photon redistribution via line branching.  Putting everything together with some algebra yields

- where *y* is the inertial term, scaled to the effective gravity and  is the Eddington parameter, the ratio of radiative acceleration on electrons to gravity:

- Finally, the mass loss rate itself is:

- where *v**th* is the thermal speed of electrons.  More details of the calculation can be found in Onifer & Gayley (2006).

- The bottom plots show the effect that zero metallicity has on the line opacity spectrum.  The left plot shows the Sobolev optical depth spectrum for a typical WC star at solar metallicity.  The right shows a similar model for Z = 0.  The Planck curve to the left shows that the most important lines are between log  of 15.6 and 16.1.

- The line opacities were calculated using data from the Opacity Project (Badnell & Seaton, 2003).  As CNO and Ne come from the He burning interior, their abundances should be similar to modern WR stars.  The abundances chosen are appropriate for a WC-type star and maximize the amount of C in the wind.

*M**** = 12 M
*v*** - *v**crit* = 1000 km s-1
*T**crit* = 1.3105 K
- electron = 2, 5, 10
-  = 0.3, 0.5, 0.7, 0.9
- The electron density *n**e* is set proportional to *t* such that log (*n**e* / *t*) = 15, 16, 17, 18.

| Ion | Mass Fraction |
| --- | --- |
| He | 0.43 |
| C | 0.43 |
| O | 0.12 |
| Ne | 0.02 |

- No solutions were found for  = 0.3 for any values of electron or *n**e*.
- The following values were found for the other  values

| electron |  | t | ne (cm-1) | M(t) |
| --- | --- | --- | --- | --- |
| 2 | 2.9x10-8 | 2.51x10-4 | 2.51x1014 | 2.4 |
| 10 | 1.4x10-7 | 1.00x10-3 | 1.00x1015 | 1.7 |

****** ****= 0.7**

| electron |  | t | ne (cm-1) | M(t) |
| --- | --- | --- | --- | --- |
| 5 | 1.5x10-8 | 7.84x10-5 | 7.84x1013 | 3.5 |

****** ****= 0.5**

| electron |  | t | ne (cm-1) | M(t) |
| --- | --- | --- | --- | --- |
| 2 | 8.1x10-8 | 4.25x10-3 | 4.25x1014 | 0.46 |
| 2 | 1.5x10-5 | 4.15x10-1 | 4.15x1017 | 0.55 |
| 5 | 1.2x10-8 | 7.30x10-4 | 7.30x10-12 | 0.26 |
| 5 | 2.9x10-7 | 1.00x10-2 | 1.00x1015 | 0.39 |
| 10 | 3.5x10-7 | 1.00x10-2 | 1.00x1015 | 0.39 |
| 10 | 2.3x10-5 | 4.32x10-1 | 4.32x1017 | 0.54 |

****** ****= 0.9**

- Plot of the effective opacity (in optical depth units) of a *Z* = 0 wind at  = 0.5 as a function of the frequency-like coordinate *f*, which is the frequency convolved with the Planck function at *T* = 1.3105 K.  The wind is optically thin in the lines over most of the spectral range.

**Conclusions**

- Due to the scarcity of lines in a Population III-type WR star, calculating a mass-loss rate via _x0005__x0005_CAK-type line driving is very difficult.
- Only models with   0.5 have thus far converged to a solution.  Solar-metallicity WR stars can have  ~ 0.5, and Population III WR stars can be more luminous, though  = 0.9 may be unrealistically large.
- Vink & De Koter (2005) found                                for *Z* = 10-5 *Z* and a similar WC-type abundance profile.  Thus it would be reasonable to assume similar or smaller mass-loss rates for *Z* = 0.
- Other effects can serve to enhance the line driving.  For example, large rotation rates (not included in this model) will effectively push the Eddington parameter closer to critical.

**References**

- Badnell, N. R., & Seaton, M. J., 2003, *Journal of Physics B*, 36, 4367.
- Castor J. I., Abbott, D. C., & Klein, R. I., 1975, *The Astrophysical Journal*, 195, 157.
- Onifer, A. J., and Gayley, K. G.,  2006, *The Astrophysical Journal*, 636, 1054.