``The only real voyage of discovery consists not in seeking new landscapes but in having new eyes.''

--Marcel Proust

The results of forward modeling with one adjustable point in the chemical profile make it clear that information about the internal structure is contained in the data, and we just need to learn how to extract it. If we want to test more complicated adjustable profiles, forward modeling quickly becomes too computationally expensive as the dimensionality of the search space increases. We need to devise a more efficient approach to explore the myriad possibilities.

The natural frequency that dominates the determination of pulsation periods in white dwarf models is the Brunt-Väisälä (BV) frequency, which we calculate using the Modified Ledoux treatment described in Tassoul, Fontaine & Winget (1990). To get a sense of how the pulsation periods depend on the BV frequency in various regions of the model interior, we added a smooth perturbation to the best-fit model of GD 358 from Table 4.2, moving it one shell at a time from the center to the surface. The perturbation artificially decreased the BV frequency across seven shells, with a maximum amplitude of 10 percent. We monitored the effect on each of the pulsation periods as the perturbation moved outward through the interior of the model. The results of this experiment for the pulsation periods corresponding to those observed in GD 358 are shown in Figure 5.1. Essentially, this experiment demonstrates that the pulsation periods are sensitive to the conditions all the way down to the inner few percent of the model. Since the observational uncertainties on each period are typically only a few hundredths of a second, even small changes to the BV frequency in the model interior are significant.

5.2 Model Perturbations

The root-mean-square (r.m.s.) residuals between the observed pulsation
periods in GD 358 and those calculated for the best-fit from forward
modeling are still much larger than the observational noise. This suggests
that either we have left something important out of our model, or we have
neglected to optimize one or more of the parameters that could in
principle yield a closer match to the observations. To investigate the
latter possibility, we introduced * ad hoc* perturbations to the BV
frequency of the best-fit model to see if the match could be improved.
Initially, we concentrated on the region of the Brunt-Väisälä curve
that corresponds to the internal chemical profile.

If we look at the BV frequency for models with the same mass, temperature, helium layer mass, and central oxygen mass fraction but different internal chemical profiles (see Figure 5.2) it becomes clear that the differences are localized. In general, we find that changes in the composition gradient cause shifts in the BV frequency. Moving across an interface where the gradient becomes steeper, the BV frequency shifts higher; at an interface where the gradient becomes more shallow, the BV frequency shifts lower. The greater the change in the gradient, the larger the shift in the BV frequency.

We began by generating a model with the same mass, temperature, helium layer mass, and internal composition as the best-fit from Table 4.2, but using a uniform internal chemical profile with constant 20:80 C/O out to the 95 percent mass point. We identified a sequence of 100 shells in this model spanning a range of fractional mass from 0.20 to 0.97 and perturbed the BV frequency to try to produce a better match to the observations. We parameterized the perturbation as a linearly varying multiplicative factor applied to the BV frequency over a range of shells, described by four parameters: (1) the innermost shell to perturb, (2) the magnitude of the perturbation at the innermost shell, (3) the number of shells in the perturbation range, and (4) the magnitude of the perturbation at the outermost shell. These 4 parameters are sufficient to describe a profile with two abrupt changes in the composition gradient.

The innermost shell was allowed to be any of the 100 shells, and the number of shells in the perturbation range was allowed to be between 0 and 100. If the chosen range would introduce perturbations outside of the sequence of 100 shells, the outermost of these shells was the last to be perturbed. The magnitude of the perturbation was allowed to be a multiplicative factor between 1.0 and 3.0 at both the innermost and outermost perturbed shells, and was interpolated linearly in the shells between them.

By using a GA to optimize the perturbation, we can try many random possibilities and eventually find the global best-fit with far fewer model evaluations than a full grid search of the parameter-space. We demonstrated this by introducing a particular perturbation to the model and determining the pulsation periods. Using the same unperturbed model, we then attempted to find the parameters of the perturbation by matching the periods using the GA. In 9 out of 10 runs (500 generations of 64 trials), the GA found a first-order solution within two grid points of the input perturbation. Thus, the GA combined with a small (625 point) grid search yields a 90 percent probability of success for an individual run. By repeating the process several times, the probability of finding the correct answer quickly exceeds 99.9 percent, even while the number of model evaluations required remains hundreds of times lower than a full grid of the parameter-space.

Having demonstrated that the method works on calculated model periods, we
applied it to the observed pulsation periods of GD 358. We began with a
model similar to the 5-parameter best-fit determined from forward
modeling, but again using a uniform internal chemical profile (constant
16:84 C/O out to 0.95 *m*/*M*_{*}). After the GA had found the best-fit
perturbation for GD 358, we reverse-engineered the corresponding chemical
profile.

To accomplish this, we first looked in the unperturbed model for the fractional mass corresponding to the innermost and outermost shells in the perturbation range. We fixed the oxygen abundance to that of the unperturbed model from the center out to the fractional mass of the innermost perturbed shell. The size of the shift in the BV frequency is determined by how much the composition gradient changes at this point, so we adjusted the oxygen abundance at the fractional mass of the outermost perturbed shell until the change in the gradient produced the required shift. Finally, we fixed the oxygen abundance to that value from the outermost perturbed shell out to a fractional mass of 0.95, where it abruptly goes to zero.

After we found the C/O profile of the best-fit perturbation in this way, we fixed this reverse-engineered profile in the models and performed a new fit from forward modeling with the GA to re-optimize the mass, temperature, helium layer mass, and central oxygen mass fraction. The BV curve of the final model differs slightly, of course, from that of the original uniform composition model with the perturbation added. But the approximate internal structure is preserved, and leads to a better match to the observed pulsation periods than we could have otherwise found.

The calculated periods and period spacings
(*P* = *P*_{k + 1} - *P*_{k})
for the best-fit models from the 5-parameter forward modeling and from the
reverse approach are shown in the bottom two panels of Figure
5.3
along with the data for GD 358. The best-fit models of Bradley & Winget (1994b) and
Metcalfe, Nather & Winget (2000) are shown in the top two panels for comparison. The data in
Figure 5.3
for the observations and our best-fit models are given
in Table 5.1. Some of the improvement evident in the panels of Figure 5.3
is certainly due to the fact that we have increased
the number of free parameters. To evaluate whether or not the new models
represent a * significant* improvement to the fit, we use the Bayes
Information Criterion (BIC), following Montgomery, Metcalfe & Winget (2001).

Observed | 3-par fit | 5-par fit | 7-par fit | ||||||||

k |
P |
P |
P |
P |
P |
P |
P |
P |
|||

08... | 423.27 | 40.96 | 422.31 | 42.26 | 422.36 | 40.92 | 422.75 | 39.69 | |||

09... | 464.23 | 37.36 | 464.57 | 36.77 | 463.28 | 38.01 | 462.43 | 37.46 | |||

10... | 501.59 | 40.16 | 501.35 | 35.88 | 501.29 | 37.50 | 499.90 | 39.70 | |||

11... | 541.75 | 35.01 | 537.23 | 39.04 | 538.79 | 36.77 | 539.60 | 36.65 | |||

12... | 576.76 | 41.52 | 576.27 | 42.79 | 575.56 | 43.33 | 576.25 | 42.10 | |||

13... | 618.28 | 40.07 | 619.06 | 39.79 | 618.89 | 39.85 | 618.36 | 40.25 | |||

14... | 658.35 | 42.29 | 658.85 | 42.97 | 658.74 | 42.36 | 658.61 | 42.90 | |||

15... | 700.64 | 33.66 | 701.82 | 32.76 | 701.10 | 34.33 | 701.51 | 33.44 | |||

16... | 734.30 | 36.37 | 734.58 | 36.92 | 735.42 | 36.99 | 734.95 | 36.59 | |||

17... | 770.67 | 40.03 | 771.50 | 39.30 | 772.41 | 37.34 | 771.54 | 39.60 | |||

18... | 810.7 | 44.1 | 810.80 | 44.34 | 809.75 | 44.63 | 811.14 | 44.15 |

The fits listed in Table 4.2
used *n*_{p} = 3 completely free
parameters, sampling several combinations of two additional parameters.
This amounts to a partial optimization in 5 dimensions. To make a fair
comparison we use the best carbon-core model from Table
4.2,
which represents the best truly 3-parameter fit. This model had r.m.s.
residuals of
(*P*) = 2.30 seconds for the periods and
(*P*) = 2.65 seconds for the period spacings. For *N* = 11 data points, the BIC
leads us to expect the residuals of a *n*_{p} = 5 fit to decrease to
(*P*) = 1.84 and
(*P*) = 2.13 just from the addition of the
extra parameters. In fact, the fit from the forward modeling presented in
§4.6 has
(*P*) = 1.28 and
(*P*) = 1.42, so we
conclude that the improvement is statistically significant.

The results of the reverse approach presented in §5.2 are
harder to evaluate because we are perturbing the BV frequency directly,
rather than through a specific parameter. We consider each additional
point in the internal chemical profile where the composition gradient
changes to be a free parameter. Under this definition, the perturbed
models are equivalent to a 7-parameter fit since there are three such
points in the profiles, compared to only one for the 5-parameter case. If
we again use the BIC, we expect the residuals to decrease from their
*n*_{p} = 5 values to
(*P*) = 1.03 and
(*P*) = 1.14 seconds.
After re-optimizing the other four parameters using the profile inferred
from the reverse approach, the residuals actually decreased to
(*P*) = 1.11 and
(*P*) = 0.71 seconds. The decrease in the
period residuals is not significant, but the period spacings are improved
considerably. This is evident in the bottom panel of Figure
5.3.

The internal chemical profiles corresponding to the best-fit models from
the 5-parameter forward modeling and from the reverse approach are shown
in Figure 5.4
with the theoretical profile for a 0.61 *M*_{}
model from Salaris et al. (1997), scaled to a central oxygen mass fraction of 0.80.
The profile from the best-fit forward modeling matches the location and
slope of the initial shallow decrease in the theoretical profile. The
reverse approach also finds significant structure in this region of the
model, and is qualitatively similar to the Salaris et al. (1997) profile to the
extent that our parameterization allows. It is encouraging that both
approaches agree with each other and bear some resemblance to the models.