5 Results

In total, the values $R(\vec\Theta,\mathcal T,\mathcal N,B)$ and $S(\vec\Theta,\mathcal T,\mathcal F,B)$ were calculated as a function of projected baseline length for various sets of target and nuisance parameters at more than 10⁵ points of the parametric space. Table 1 presents the principal characteristics of the model grids. The ranges of variations of model parameters on the grids were chosen so as to include the values of parameters from earlier works on P Cyg cited in Sect. 4.1.

 

 

Table 1: Grids of models: the number of points in the parametric space $N_{\rm m}$; dimension, that is the number of varied parameters, of the grid D; the values of parameters that were fixed on the grid; and the ranges of variation for variable parameters

Grid	$N_{\rm m}$	D	$\alpha$	$\dot M$	$v_{\infty}$	$v_{\rm c}$	T1	R2	$\Delta _2T$	R3	$\Delta _3T$	R4	$\Delta _4T$	R5	$\Delta_5T$
no.				$M_\odot~\mbox{yr}^{-1}$	km s-1	km s-1	K		K		K		K		K
1	10000	3	3.5	1.010-5	200	15.0	11000
			4.5	2.010-5		55.0	15000
2	10000	3	3.5	1.010-5	200	15.0	11000	2.0	1300	3.4	2100	5.0	200	10.0	3000
			4.5	2.010-5		55.0	15000
3	32805	4	3.5	1.010-5	150	15.0	11000
			4.5	2.010-5	250	55.0	15000
4	15625	6	3.5	1.010-5	200	15.0	11000	2.0	0	5.0	0
			4.5	2.010-5		55.0	15000		1300		2300
5	46656	6	3.25	1.010-5	200	15.0	11000	2.0	0	3.4	0
			4.25	2.010-5		55.0	15000		1300		2100
6	6561	8	3.5	1.010-5	200	15.0	11000	2.0	0	3.4	0	5.0	0	10.0	0
			4.5	2.010-5		55.0	15000		1300		2100		200		3000
7	65536	8	3.5	1.010-5	200	15.0	11000	2.0	0	3.0	0	4.0	0	5.0	0
			4.5	2.010-5		55.0	15000		1300		2100		200		3000
8	6561	8	4.0	1.510-5	180	20.0	14000	2.0	0	3.0	0	4.0	0	5.0	0
			4.5	2.310-5	220		16000		1000		1000		1000		1000
9	19683	9	3.5	1.510-5	220	15.0	16000	2.0	0	3.0	0	4.0	0	5.0	0
			4.5	2.310-5	300	25.0	18000		1300		2100		200		3000

The main result of the present study is that the increase in accuracy provided by OLBI strongly depends on the number of free parameters in the model: it varies from almost negligible for models of low dimension to very significant for models of high dimension. Since in our case the boundary between "low'' and "high'' is nearly coincident with division into models with fixed and adjustable thermal structure, we discuss the results pertaining to those two classes of models separately.

5.1 Models with fixed thermal structure

Among our grids, the grid 1 most closely corresponds to the models often used in interpreting H$\alpha$ observations of P Cyg stars, where $v_{\infty}$ is excluded from the set of adjustable parameters of the model (see e.g. Scuderi et al. [1994]) and its value is taken from the analysis of absorption lines of metals in the ultraviolet (Casatella et al. [1979]; Lamers et al. [1985]).

The typical results for an individual point in the parametric space are illustrated by Fig. 1, where the random error ratio $R^{{\rm IS}}= C^{\rm I}(\vec\Theta,\mathcal T,\mathcal N,B) /C^{\rm S}(\vec\Theta,\mathcal T,\mathcal N,B)$is plotted as a function of the normalized baseline length B/B₀ for various $\mathcal T$ and $\mathcal N$. Here B is the projected baseline length, $B_0=\lambda/2\pi\delta$, and $\delta$ is the angular radius of the central star. For the adopted values of $R_*=76\,R_\odot$, $d=1800\,{\rm pc}$, and $\lambda=6265$Å one gets $B_0=110\,{\rm m}$.

Figure 1: Error ratio for isothermal model with $\alpha =3.94$, $\dot M=1.44\,10^{-5}\,M_\odot~\mbox{yr}^{-1}$, $v_{\rm c}=32.8{\mbox{ km s$^{-1}$ }}$, $T_1=12778\,{\rm K}$, $v_\infty=200{\mbox{ km s$^{-1}$ }}$, and the set of adjustable parameters $\mathcal T\cup\mathcal N=\{\alpha$, $\dot M$, $v_{\rm c}$, $T_1\}$. Target set $\mathcal T=\{\alpha\}$ (solid line), $\{\dot M\}$ (dashed line), $\{T_1\}$ (short-dashed line), and $\{v_{\rm c}\}$ (dotted line)

The principal features of dependence of $R^{\rm IS}$ on B are independent of details of the physical model used and can be easily explained qualitatively. As shown in Sect. 3.3, inequality $R^{\rm IS}<1$ holds for all $0<B<\infty$. When the projected baseline length is very small, the interferometry does not provide any additional information as compared with the spectroscopy, for the object gets unresolved. Consequently, $R^{\rm IS} \to 1$ for $B \to 0$. In the opposite case of large B the fringe contrast is close to 0 for all physically realistic intensity distributions and its dependence on model parameters is impossible to measure because of observational errors. That is, in that case the interferometry again provides no additional information and $R^{\rm IS} \to 1$ for $B \to \infty$.

It follows from the above stated general properties of $R^{\rm IS}$ that for any $\mathcal T$ and $\mathcal N$ it has a (possibly non-unique) minimum at a certain finite values of B. The location of the minimum indicates the projected baseline length at which interferometric observations are most informative, and the corresponding value of $R^{\rm IS}$ characterize to what degree the interferometry can reduce the errors in model parameters determination as compared to spectroscopy.

Let us note that it may not be correct to compare the merits of the methods using values of $R^{\rm IS}$ obtained for an isolated point or even for a few arbitrarily chosen points of parametric space. Since the model parameters are not known a priori, it is necessary to use integral characteristics describing the behaviour of $R^{\rm IS}$ on the whole grid. We will use the following three values calculated as a function of B_j/B₀ for various $\mathcal T$ and $\mathcal N$:

  

$$R_{\min}(\mathcal T,\mathcal N,B_j) \min_{1\le m\le N_{\mathrm m}} R^{\mathrm{IS}}(\vec\Theta_m,\mathcal T,\mathcal N,B_j)$$ = (24)

$$R_{\max}(\mathcal T,\mathcal N,B_j) \max_{1\le m\le N_{\mathrm m}} R^{\mathrm{IS}}(\vec\Theta_m,\mathcal T,\mathcal N,B_j)$$ = (25)

$$P_\mathrm{opt}(\mathcal T,\mathcal N,B_j)=N_\mathrm{opt}/N_{\mathrm m},$$ (26)

where $N_{\rm opt}$ is the number of points on the grid for which function $R^{{\rm IS}}(\Theta,\mathcal T,\mathcal N,B)$ reaches its minimum at B=B_j.

The function defined in Eq. (26) is immediately related to the problem of optimal choice of the baseline length: for given $\mathcal T$ and $\mathcal N$, the higher the value $P_{\rm opt}(\mathcal T,\mathcal N,B)$, the higher the probability that the interferometric observations at the baseline length B would yield the most accurate model parameters. The dependence of $R_{\min}$ and $P_{\rm opt}$ on the normalized baseline length and set of target parameters for grid 1 is shown in Fig. 2. For this grid, the value of $R_{\max}$ is close to unity for all $\mathcal T$ and $\mathcal N$ and is not shown in the plot.

Figure 2: The lower limit of random error ratio $R_{\min}$ (solid line) and distribution of optimal baseline lengths $P_{\rm opt}$ (dashed lines, plotted in an arbitrary scale) for grid 1. The set of adjustable parameters $\mathcal T\cup\mathcal N=\{\alpha,~\dot M$,  $v_{\rm c},~T_1\}$, target parameters are indicated on the plots

It can be easily seen that for this grid of models OLBI provides only a relatively small reduction of the random error in parameter determination as compared with spectroscopy. This conclusion remains also valid if we add the terminal velocity $v_{\infty}$ to the set of adjustable parameters (grid 3).

As $R^{{\rm IS}}(\vec\Theta,\mathcal T,\mathcal N,B)\approx1$ for all $\mathcal T$ and $\mathcal N$, the same approximate equality is evidently valid for the robustness ratio $S(\vec\Theta,\mathcal T,\mathcal F,B) =R(\vec\Theta,\mathcal T,\mathcal F,B)/R(\vec\Theta,\mathcal T,\emptyset,B)$, which indicates (see Sect. 2.3) that for models with a priori fixed thermal structure, interferometric data will not reduce appreciably the level of systematic errors.

5.2 Models with adjustable thermal structure

The number of possible target sets is an exponentially increasing function of the dimension D of the model parameter space, so that when the values $\Delta_iT$ are added to the set of unknowns, it becomes impossible to make an exhaustive presentation of even the integral results, i.e. the functions $R_{\min}(\mathcal T,\mathcal N,B)$, $R_{\max}(\mathcal T,\mathcal N,B)$, and $P_{\rm opt}(\mathcal T,\mathcal N,B)$.

In Fig. 3, we show the dependence of $R_{\min}$, $R_{\max}$, and $P_{\rm opt}$ on the number of nuisance parameters for one of our simplest grids of models with adjustable thermal structure, the target set consisting of one parameter, the mass loss rate $\dot M$. From many points of view, it is the most important parameter of the stellar wind, characterizing both its effect on the evolution of the star and the influence of the outflowing matter on the surrounding interstellar medium.

Figure 3: The lower and upper limits of random error ratio (solid lines, $R_{\max}$ differs noticeably from unity only in the bottom plot) and distribution of optimal baseline lengths $P_{\rm opt}$ (dashed lines, plotted in an arbitrary scale) for grid 5. Target set $\mathcal T=\{\dot M\}$, nuisance parameters are indicated on the plots

As it can be seen, the lower limit of the random error ratio $R_{\min}$ rapidly decreases as the number of adjustable temperature parameters increases, so that when the thermal structure of the envelope becomes a nuisance parameter, interferometry can reduce the random error in $\dot M$ by as much as an order of magnitude.

Note, that the decrease in $R_{\min}$ is caused by the fact that parameters $\Delta_iT$ are adjustable, rather then by deviation from isothermicity. This can be seen from the upper two plots in Fig. 3, where $R_{\min}>0.8$ for a grid containing a high proportion of models with a noticeable temperature gradient, and is also supported by the results obtained for grid 2, which differs from grid 1 only in that its thermal structure approximates that of Drew's model B, which corresponds to $\alpha=4$, $v_{\rm c}=15$ km s^-1, $v_\infty=300$ km s^-1, and $\dot M=1.5\,10^5\,\,M_\odot~\mbox{yr}^{-1}$ in our notations, instead of being isothermal.

For other single-parameter target sets the dependencies of $R_{\min}$ and $R_{\max}$ on $\mathcal N$ are qualitatively the same. Figure 4 displays the results for the most important case when all the parameters of the model except for $v_{\infty}$ are adjustable. As one can see, again, in favorable circumstances interferometry can increase accuracy by an order of magnitude.

Figure 4: The lower and upper limits of random error ratio (solid lines) and distribution of optimal baseline lengths $P_{\rm opt}$ (dashed lines) for grid 5, $\mathcal T\cup\mathcal N$=$\{\alpha$, $\dot M$, $v_{\rm c}$, T1, $\Delta _2T$, $\Delta _3T\}$, and target sets indicated on the plots

In contrast to $R_{\min}$, the value $R_{\max}$ is close to unity for all projected baseline lengths and combinations of sets $\mathcal T$ and $\mathcal N$ that we have studied. This signifies that for any projected baselength there exists a combination of model parameters for which interferometry practically do not provide reduction in error.

It is interesting to notice that for target sets containing several parameters (see Fig. 5), the influence of individual parameters on $R_{\min}$ and $R_{\max}$ to some extent averages out: $R_{\min}$ is systematically higher, and $R_{\max}$ is systematically lower than the corresponding values for individual parameters. For this reason, when more than one model parameter is to be determined, interferometry at a single projected baselength is unlikely to yield overall gain in accuracy in excess of a factor of two even at optimal baselines.

Figure 5: The lower and upper limits of random error ratio (solid lines) and distribution of optimal baseline lengths $P_{\rm opt}$ (dashed lines) for grid 5, $\mathcal T\cup\mathcal N$=$\{\alpha$, $\dot M$, $v_{\rm c}$, T1, $\Delta _2T$, $\Delta _3T\}$, and target sets comprising several parameters (indicated on the plots)

We studied further the effect of increasing the number of sublayers M_R, which permits us to investigate the influence of fine details of the thermal structure of the envelope. This appears to be justified because Drew ([1985]) showed that in certain cases this structure is quite complex, and the gradient of $\log T(r)$ varies strongly with radial distance. Consequently, a model aiming to closely approximate the realistic envelope should be parameterized using a large M_R (see Sect. 4.1) and hence requires computations on grids of high dimension D.

Since the number of mesh points of the grid depends on D exponentially, for computations based on such complex models (grids 5-9 from Table 1) the limited computer resources dictate sparser mesh points along each axis of the parameter space and a tuning of parameters controlling the computational process in such a way as to increase the speed of computation at the expense of precision.

Although numerically less precise than computations for simpler models, our results for more realistic grids indicate that the conclusions derived for low M_R can be safely extrapolated for higher M_R. This point is illustrated in Fig. 6 showing the results for single-parameter target sets for grid 7.

Figure 6: The lower and upper limits of random error ratio (solid lines) and distribution of optimal baseline lengths $P_{\rm opt}$ (dashed lines) for grid 7, $\mathcal T\cup\mathcal N$=$\{\alpha$, $\dot M$, $v_{\rm c}$, T1, $\Delta _2T$, $\Delta _3T$, $\Delta _4T$, $\Delta _5T\}$, and target sets consisting of a single parameter (indicated on the plots)

As can be seen in Figs. 3-6 and is confirmed by other results not presented here, the distribution of optimal projected baseline lengths in the majority of cases has a maximum in the range 0.4-0.8 in dimensionless units. In some cases this maximum shifts to higher baseline lengths, but this occurs only for a very particular choice of $\mathcal T$ that correspond to measurements of wind temperature gradient at distances of several stellar radii from the star with all other model parameters considered nuisance (see e.g. bottom plots in Figs. 4 and 6). Thus, the aforementioned range, corresponding to 45-90 m in linear scale, can be recommended as the optimal choice for interferometric observations aimed at determination of global characteristics of the P Cyg wind.

Figure 7: Robustness ratio for the model $\alpha =4.00$, $\dot M=1.5\,10^{-5}\,M_\odot/\rm{yr}$, $v_{\rm c}=35.0{\mbox{ km s$^{-1}$ }}$, $T_1=13000\,{\rm K}$, $\Delta _2T=650\,{\rm K}$, $\Delta _3T=1050\,{\rm K}$, $\Delta _4T=100\,{\rm K}$, $\Delta _5T=1500\,{\rm K}$ from grid 6 and $\mathcal T$={$\dot M$}. Set of fixed parameters $\mathcal F=\{\alpha$, $v_{\rm c}$, T1, $\Delta _2T\}$ (solid line), $\{\alpha$, $v_{\rm c}$, T1, $\Delta _2T$, $\Delta _3T\}$ (dashed line), $\{\alpha$, $v_{\rm c}$, T1, $\Delta _2T$, $\Delta _3T$, $\Delta _4T\}$ (dash-dotted line), and $\{\alpha$, $v_{\rm c}$, T1, $\Delta _2T$, $\Delta _3T$, $\Delta _4T$, $\Delta _5T\}$ (dotted line)

The typical dependence of the robustness ratio $S^{{\rm IS}}(\vec\Theta,\mathcal T,\mathcal N,B) = U^{\rm I}/U^{\rm S}$on the baseline length and the set of fixed parameters for a model with adjustable thermal structure is shown in Fig. 7. At nearly all points of parametric space $S^{{\rm IS}}<1$, and $S^{\rm {IS}}$ is generally lower when the set of fixed parameters $\mathcal F$ includes the values $\Delta_iT$. For a small fraction of models and for short projected baselines where interferometry can not significantly reduce random error, the value $S^{{\rm IS}}>1$ by a negligible amount.

Thus, our results indicate that if the baseline length and the model used for interpretation are chosen in such a way as to reduce random errors, OLBI also appears able to reduce the systematic errors.