THE LINEWIDTH-SIZE RELATIONSHIP IN THE SIMULATIONS

One important application of PCA to the study of interstellar structure has been given by HB99, who compute a velocity increment-size relation from the PCA results for a number of ``pseudo-simulations'', in which random velocity fields with prescribed energy spectra of index $-\beta$ are constructed. In this section we derive an analogous correlation for the simulation data. The velocity increment-size relation can be extracted from the PCA results by computing the autocorrelation function of each eigenvector, and for its corresponding eigenimage. For each of these, a characteristic (velocity or spatial) scale can be defined as the lag at which the correlation has decayed to a prescribed value. A velocity increment-size relation can then be obtained by plotting the pairs of characteristic velocity and spatial scales for all eigenvectors. In practice, only the data for the most statistically significant principal components are retained. The spatial autocorrelation is defined by

$$\omega_l (\tau)={\Biggl\langle \Bigl(\xi_l(r)-\langle \xi_l\r... ...igr) \Bigl(\xi_l(r)-\langle \xi_l\rangle\Bigr)\Biggr\rangle},$$ (1)

where $\xi_l(r)$ is the intensity of the l-th eigenimage at spatial position r, and $\tau$ is the spatial lag (or ``scale'') at which the correlation is evaluated. The brackets denote averages over all positions. Analogously, the autocorrelation function for the l-th eigenvector is defined by

$$\psi_l (\tau_v)={\Biggl\langle \Bigl(u_l(v)-\langle u_l\rangl... ...gle\Bigr)\Bigl(u_l(v)-\langle u_l\rangle\Bigr)\Biggr\rangle},$$ (2)

where u_l(v) is the value of l-th eigenvector at velocity channel v, and $\tau_v$ is the velocity difference at which the correlation is evaluated. We respectively define the characteristic size Rand velocity difference $\Delta v$ as the lags $\tau$ and $\tau_v$ for which the corresponding correlation has decreased to a certain value. In figs. 4a and 4b we respectively show the autocorrelations for the eigenvectors and eigenimages for the six most statistically significant principal components. For reference, in fig. 5 we reproduce the $\Delta v$-Rrelation obtained by HB99 for their pseudo-simulations. HB99 note that the slope $\alpha$ of the curve appears to depend on $\beta$, the spectral index, as $\alpha = \beta/3$, for $\beta \le 3$.

Figures 6a and 6b show the $\Delta v$-R relation obtained for the simulation data, for two different values of the fraction $\epsilon$. HS97 and HB99 use $\epsilon=0.125$ (fig. 6a), but since our correlations exhibit stronger fluctuations than theirs, we also use $\epsilon=0.6$(fig. 6b). We note that the slope $\alpha$ of the velocity dispersion-size curve for the simulations depends on $\epsilon$. At $\epsilon=0.125$ (fig. 6a), a least-squares fit gives $\alpha \sim 0.5$, while at $\epsilon=0.6$ (fig. 6b) we find $\alpha~\sim 1.1$. Interestingly, these values seem to follow the relation $\alpha = \beta/3$, since the energy spectrum of the simulation, shown in fig. 7, is not a perfect power law (because of the low resolution). Instead, the slope is shallower ($\sim -1.5$) at small wave numbers (i.e., larger scales) and steeper ($\sim -3$) at larger wavenumbers (smaller scales). On the other hand, usage of a lower threshold $\epsilon$ on the correlation function implies including the contribution from larger scales, so the appropriate spectral slope should be that for the large scales. Indeed, in this case, $\alpha = 1.1$ and $\beta \sim 3$. Conversely, using the larger threshold $\epsilon=0.6$ takes into account the contribution of the small scales only, and in this case $\alpha \sim 0.5$ and $\beta \sim 1.8$. In both cases, $\alpha \sim \beta/3$. Higher spatial and velocity resolutions are necessary to confirm this result.