7 Model-fitting with a genetic algorithm

One of the major goals of our observations of GD 358 was to discover additional modes to help refine our seismological model fits. We were also interested in how much the globally optimal model parameters would change due to the slight shifts in the observed periods. With these goals in mind, we repeated the global model-fitting procedure of Metcalfe, Winget & Charbonneau (2001) on several subsets of the new observations.

Our model-fitting method uses the parallel genetic algorithm described by Metcalfe & Charbonneau (2003) to minimize the root-mean-square (rms) differences between the observed and calculated periods (P_k) and period spacings ( $\Delta P\equiv P_{k+1}-P_k$) for models with effective temperatures ( $T_{\rm eff}$) between 20 000 and 30 000 K, total stellar masses (M_*) between 0.45 and 0.95 $M_\odot$, helium layer masses with  $-\log(M_{\rm He}/M_*)$ between 2.0 and $\sim$7.0, and an internal C/O profile with a constant oxygen mass fraction ($X_{\rm O}$) out to some fractional mass (q) where it then decreases linearly in mass to zero oxygen at 0.95 m/M_*. This technique has been shown to find the globally optimal set of parameters consistently among the many possible combinations in the search space, but it requires between $\sim$200 and 4000 times fewer model evaluations than an exhaustive search of the parameter-space to accomplish this, and has a failure rate <10^-5.

We attempted to fit the 13 periods and period spacings defined by the m=0 components of the 14 modes identified as k=7 to k=20 in Table 9. Because of our uncertainty about the proper identification of k=18(see Sect. 4) we performed fits under two different assumptions: for Fit a we assumed that the frequency near 1233 $\mu$Hz was k=18 (similar to the frequency identified in 1990), and for Fit b we assumed that the larger amplitude frequency near 1255 $\mu$Hz was k=18. The results of these two fits led us to prefer the identification for k=18 in Fit a, and we included this in an additional fit using only the 11 modes from k=8 to k=18, which correspond to those identified in 1990 (Fit c). We performed an additional fit (Fit d) that included the same 13 periods used for Fit a, but ignored the period spacings. The optimal values for the five model parameters, and the root-mean-square residuals between the observed and computed periods ($\sigma_P$) and period spacings ( $\sigma_{ \Delta P}$) for the four fits are shown in Table 10.

 

 

Table 10: Optimal Fits to 2000 data.

Parameter	Fit a	Fit b	Fit c	Fit d
$T_{\rm eff} (K)$	24 300	23 500	24 500	22 700
$M_* (M_{\odot})$	0.61	0.60	0.625	0.630
$\log(M_{\rm He}/M_*)$	-2.79	-5.13	-2.58	-4.07
$X_{\rm O}$	0.81	0.99	0.39	0.37
q (m/M*)	0.47	0.47	0.83	0.42
$\sigma_P (s)$	2.60	3.65	2.12	1.72
$\sigma_{\Delta P} (s)$	4.07	4.92	2.21	$\cdots$

Our preferred solution from Table 10 is Fit a, because it includes our favored identification for the k=18 mode and the additional pulsation periods. The larger $\sigma_{ \Delta P}$ in Fit a compared to Fit c is dominated by the large period spacings between the k=19 and 20 modes (47.6 s) and the k=7 and 8 modes (49.4 s). Fit a has a mass and effective temperature that are essentially the same as the fit of Bradley & Winget (1994), and are consistent with the spectroscopic values derived by Beauchamp et al. (1999). The other structural parameters are otherwise similar to those found by Metcalfe et al. (2001) ( $T_{{\rm eff}}= 22~600$ K, $M_* = 0.650~M_{\odot}$, $\log(M_{\rm He}/M_*) = -2.74$, $X_{\rm O} = 0.84$, and q = 0.49). We caution, however, that the large values of $\sigma_P$ and $\sigma_{ \Delta P}$ for Fit a imply that our model may not be an adequate representation of the real white dwarf star. New and unmodeled physical circumstances may have arisen between 1994 and 2000 (e.g. whatever caused the forte in 1996), which may account for the diminished capacity of our simple model to match the observed periods.