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Astron. Astrophys. 334, 935-942 (1998)

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Appendix A: spectral line emission from a circumstellar disk

We consider a simple model of an inclined, slightly flared disk in Keplerian rotation. Using Cartesian coordinates, and with the y -axis in the plane of the sky, the disk midplane (xy -plane) is inclined with respect to the line of sight by an angle i. The disk is rotating about the z -axis, along the azimuthal angle [FORMULA]. The vertical flaring of the disk, [FORMULA], is considered a constant. The physical variables are assumed not to change significantly in the z -direction, but to have only radial dependencies, where the radial distance from the star is [FORMULA] and [FORMULA], the stellar radius. The component of the velocity along the observer's line of sight (the `radial' velocity field) is then given by

[EQUATION]

where G is the gravitational constant and [FORMULA] is the mass of the central star. Eq. (A1) describes a set of iso-velocity surfaces (e.g. Gómez & D'Alessio 1995) and, by virtue of the Doppler effect, determines the frequency scale of the spectral line emission, the emergent intensity of which is found from

[EQUATION]

where [FORMULA] is the Planck function. Background radiation fields are specified through [FORMULA], whereas the intrinsic disk field is determined by the excitation temperature, [FORMULA]. Thus, [FORMULA] is the line source function, [FORMULA]. Hence, with obvious notations

[EQUATION]

[FORMULA] is obtained from the solution of the rate equations for statistical equilibrium of radiative and collisional excitations, i.e. from the (fractional) energy level populations, [FORMULA], where [FORMULA] corresponds to the number of energy levels considered. The level populations are found by inverting the matrix equation for the transition probabilities, [FORMULA] (e.g., Spitzer 1978). Hence, in the steady state,

[EQUATION]

where [FORMULA] is a matrix of dimension [FORMULA] and [FORMULA] is a vector containing [FORMULA] elements. In our models, [FORMULA] = 50 for CO with Einstein A -values from Chackerian & Tipping (1983) and collision rate constants from Green & Thaddeus (1976) and Green & Chapman (1978). For SiO, [FORMULA] = 21 with molecular data from Tipping & Chackerian (1981) and Turner et al. (1992).

To obtain the line intensity to be received by the observer, the source brightness distribution is convolved, at each frequency, with the diffraction pattern of the telescope (the `beam'). For simplicitity, the beam is here represented by a circular Gaussian with the full width at half power b. Hence, at any relative position [FORMULA] in the sky

[EQUATION]

[EQUATION]

[EQUATION]

The optical depth at frequency [FORMULA] is obtained as the sum of (molecular) line and (dust) continuum optical depth, i.e. [FORMULA]. The line optical depth is given by

[EQUATION]

where the column density of hydrogen

[EQUATION]

[FORMULA] is the Einstein transition probability and [FORMULA] is the molecular abundance with respect to hydrogen, H2. [FORMULA] is the line center velocity (systemic radial velocity) and the broadening is assumed purely thermal at the local (dust) temperature, i.e. [FORMULA], where [FORMULA] is the molecular mass number.

The continuum optical depth is assumed constant over the line and computed as

[EQUATION]

where µ is the mean molecular weight of the gas, assumed molecular (µ = 2.4). For the [FORMULA] disk, we adopt the density and temperature distributions of the dust of Chini et al. 1991, which are expressed as radial power laws. To obtain the distribution of the molecular hydrogen gas, n (H2), the dust density law has been scaled by the (constant) gas-to-dust mass ratio. For computational convenience, the frequency dependence of the dust opacity is expressed in power law form (Hildebrand 1983), with the parameters [FORMULA] and [FORMULA] adjusted such that the disk model of the dust emission fits the mm/submm observations by Chini et al. (1991) and by Zuckerman & Becklin (1993), i.e. [FORMULA] = 4 cm2 g-1 at [FORMULA] = 250 µm with [FORMULA] = 1.5.

We have computed an analogue of and compared with the model results for HL Tau obtained by Gómez & D'Alessio (1995) and our line to continuum ratios are in very good agreement with what can be estimated from the profiles in their Fig. 3. We also found very good agreement with Yamashita et al. (1993) regarding the profile shape of the CO (1-0) line for [FORMULA] Eri [however, our result for the integrated line intensity (in the [FORMULA] scale) for their model is a factor of about 4 below that given in their Table 2].

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© European Southern Observatory (ESO) 1998

Online publication: June 2, 1998

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