3 The relative figure of merit of OLBI vs. spectroscopy

In this section we obtain the expressions for the elements of the covariance matrix ${\tens D}\vec{\hat \Theta}$ in the form that can be directly used to compare the OLBI and spectroscopy.

First, the measurement errors have to be specified. For this first trial application, the only considered source of random errors is the photon noise. Other sources of error, in particular those in OLBI arising from atmospheric seeing and calibrations of the modulation transfer function (see Roddier & Léna [1984]; Mourard et al. [1994]) are important and have to be incorporated in future work.

Also, the spectral coverage and the spectral resolution have to be specified. We assume that they are the same for both kinds of observations; the covered spectral region is $[\lambda_1,\lambda_2]$, the spectral resolution is sufficiently high, i.e. the monochromatic intensity received from each point of the observed object remains nearly constant across each spectral channel. In other words, we assume that $\Delta\lambda$, the width of the spectral channels, in terms of velocity is lesser than the thermal velocity of the emitting atoms. For the P Cyg type envelopes this corresponds to Å, which is easily satisfied with modern spectrographs. Then one can formally consider the convenient limit case $\Delta\lambda\to 0$. In this case, the resulting covariance matrices tend to a limit which is independent of the spectral resolution (cf. Eqs. (11) and (18)).

   
3.1 Spectroscopy

The physical quantity provided by spectroscopy is the flux density as a function of wavelength. In terms of observables, it is given by the vector ${\vec{\hat Y}}\!{}^{\rm S}$ with the components

$$\hat Y^{\mathrm S}_i=\hat N_i,\ i=1,2,\ldots,L\,,$$ (6)

where $\hat N_i$ is the number of photons recorded in the i-th spectral channel during the exposure, and L is the number of channels. The random values $\hat N_i$ are related to the physical parameters of the object by

$$\hat N_i=N_i+n_i\,,$$ (7)

where

$$N_i=F(\vec\Theta,\lambda_i) E \Delta_i \lambda,$$ (8)

$F(\Theta,\lambda)$ is the monochromatic flux from the object at wavelength $\lambda$, $\Delta_i \lambda$ is the width of i-th spectral channel, and n_i is the measurement error due to the photon noise, with the variance given by

$${\tens D}{n_i}=N_i\,.$$ (9)

Finally the factor E is given by

$$E=A_\mathrm{eff}t\,,$$

where $A_{\rm eff}$ is the instrument effective aperture and t is the exposure time. Hereafter we suppose that $E^{\rm S} = E^{\rm I}$, where $E^{\rm S}$ and $E^{\rm I}$pertain to the spectrometer and the interferometer respectively. Since the values $E^{\rm S}$ and $E^{\rm I}$ enter the final results only through the ratio $E^{\rm S}/E^{\rm I}$, we can for the sake of simplicity set $E^{\rm S} = E^{\rm I} = 1$.

Thus, $\varepsilon^{\rm S}_i=n_i$and the elements of the covariance matrix of measurement errors are given by

$${\tens M}{(\varepsilon^{\mathrm S}_i \varepsilon^{\mathrm S}_... ...q j $ } \\ N_i & \mbox{if $ i=j $ } \end{array} \right.\,.$$ (10)

Substituting Eqs. (6)-(10) into Eq. (1) and proceeding to the limit $\Delta_i \lambda\to 0$ we obtain:

$$\left( \tens C^{\mathrm S}\right) ^{-1}_{\rm pq} = \int_{\... ...\lambda)}} {\partial \Theta_{\rm q}} \,{\mathrm d}\lambda\,.$$ (11)

For spherically symmetric objects considered in the present paper,

$$F(\Theta,\lambda) =2\pi\int_0^\infty I(\Theta,\lambda,p)\,{\mathrm d}p\,,$$ (12)

where $I(\vec\Theta,\lambda,p)$ is the monochromatic intensity at the angular distance p from the center of the object.

   
3.2 OLBI

We consider the case of a two-aperture OLBI with an adjustable baseline. The physical quantity eventually provided by OLBI measurements is the fringe visibility V as a function of the baseline vector. The visibility V is equal to the real part of the complex degree of coherence, and therefore is the Fourier transform of the brightness distribution in the focal plane by virtue of the Van Cittert - Zernicke theorem (e.g. Mariotti [1998], textbooks of Perina [1972], and Goodman [1985]). However, the value estimated in practice is V². We refer the reader for details to the thorough discussion by Mourard et al. ([1994]). In the present analysis we consider the estimates $\hat W_i$ of the quantity

$$W_i=W(\vec\Theta,\lambda_i,B)=V^2(\vec\Theta,\lambda_i,B),$$ (13)

where i refers (as previously) to the i-th spectral channel, and B is the projected baseline length of the interferometer. In the case of spherically symmetric objects the intensity distribution is circularly symmetric and V is given by the normalized Hankel transform as follows (e.g. Bracewell [1978]):

$$V(\vec\Theta,\lambda,B) =\frac {2\pi\int_0^{\infty} I(\vec\Theta,\lambda,p)J_0(kp)\,{\mathrm d}p} {F(\Theta,\lambda)}\,,$$ (14)

where J₀ is the Bessel function, and $k=2 \pi B/\lambda$.

Statistical errors $w_i=\hat W_i - W_i$ obey the equations

$${\tens M}{(w_i w_j)}= \left\{ \begin{array}{ll} 0 & \mbox{... ...W_i}{N_i}(2-W_i) & \mbox{if $ i=j $ } \end{array} \right.,$$ (15)

and

$${\tens M}{(w_i n_j)}=0\,.$$ (16)

The validity of Eqs. (15) and (16) for $i \neq j$ is evident. The case i=j is treated in Appendix A.

As a rule, the spectrum of the object, i.e. the values $\hat N_i$, is also recorded. Consequently, the vector of observables for the OLBI and the elements of its covariance matrix of errors are given by

$$\vec{\hat Y}^{\mathrm I} =(\hat{N_1},\hat{W_1},\hat{N_2},\hat{W_2},\ldots,\hat{N_L},\hat{W_L})$$ (17)

and

$${\tens M}(\varepsilon^{\mathrm I}_i\varepsilon^{\mathrm I}_j)... ...{N_k}(2-W_k) & \mbox{ if $ i=j=2k $ } \end{array} \right.\,.$$

Substituting these expressions into Eq. (1), we obtain in the limit $\Delta_i \lambda\to 0$ that

$$\left(\tens C^{\mathrm I}\right) ^{-1}_{{\rm pq}} = \left(... ...tial {W}} {\partial \Theta_{{\rm q}}} \,{\mathrm d}\lambda\,,$$ (18)

where $\left(\tens C^{\rm S}\right) ^{-1}_{{\rm pq}}$ is defined in Eq. (11).

   
3.3 The relative figure of merit

As discussed in Sect. 2.2, the relative figure of merit of OLBI as compared to spectroscopy is the random error ratio:

$$R^\mathrm{IS}(\vec\Theta,\mathcal T,\mathcal N,B) =C^{\mathr... ...athcal N,B)/C^{\mathrm S}(\vec\Theta,\mathcal T,\mathcal N)\,.$$ (19)

From Eqs. (18) and (19), it follows that $R^{\rm IS}<1$, which merely reflects the fact that the observables of spectroscopy constitute a subset of the OLBI observables. Consequently, in the adopted comparison, the statistical errors in parameter determination by OLBI are always less than those obtained when only spectroscopic data are used, if however the spectroscopic data are of equal precision.

As explained in Sect. 3.1, the Eq. (19) is obtained in the assumption that $E^{\rm S} = E^{\rm I}$. To compare a pair of instruments for which that equality does not hold, the RHS of Eq. (19) should be multiplied by $(E^{\rm S}/E^{\rm I})^{1/2}$.