{'fraction_non_alphanumeric': 0.027269986570559377, 'fraction_numerical': 0.015813851508756544, 'mean_word_length': 4.7770740975300825, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Statistical analysis of molecular line emission from T Tauri disk models

In this work we model the expected emission from the molecular line C17O(J=3-2) in protoplanetary disks, modifying different physical parameters to obtain distinctive observational signatures. Our aim is to determine the kind of observations that will allow us to extract information about the physical parameters of disks. With this purpose we perform a statistical analysis of principal components and a multiple linear correlation on our set of results from the models. We also present prospects for future molecular line observations of protoplanetary disks using SMA and ALMA.


INTRODUCTION
The mass of disks around young stars is ∼ 99% gas and only 1% dust. However, since the dust opacity is large compared to the gas opacity in a wider range of wavelengths, the dust component dominates the absorption and reprocessing of stellar radiation and the emergent spectral energy distributions (SEDs) of disks around Classical T Tauri (CTTS) and Herbig Ae stars (HAe). Most of the models constructed to explain observed SEDs have taken into account only the dust component in the calculation of the disk temperature (Chiang & Goldreich 1997;D'Alessio et al. 1998D'Alessio et al. , 1999D'Alessio et al. , 2001. These models have been successful to explain the observed SEDs. However, SEDs are not sensitive to kinematical information, details of the radial and vertical temperature distribution, the chemistry of the gaseous component (Aikawa et al. 1996(Aikawa et al. , 1997Aikawa & Herbst 1999;van Zadelhoff et al. 2001;Aikawa & Nomura 2006), the possibility that the gas in the upper layers is hotter than the dust in lower layers by absorption of UV and X-rays (i.e., Glassgold et al. 2004;Jonkheid et al. 2004;Kamp & Dullemond 2004) and/or photoelectrical effect (i.e., Nomura & Millar 2005), etc.
The study of molecular line emission from disks around young stars is an important tool to infer physical characteristics of disks (e.g., Dartois et al. 2003;Carr et al. 2004;Piétu et al. 2005;Qi et al. 2006;Raman et al. 2006;Dutrey et al. 2007). An advantage of a spectral line is that emission at different frequencies/velocities might be probing different disk regions, making lines an important test for disk models. However, this is also a disadvantage in some sense, since one observes intensities convolved with the beam of the telescope and with a finite spectral resolution, and therefore the information of the different regions is mixed up in a complex way. Thus, the analysis and relationship between observations and model properties might become very complicated and difficult to disentangle. There are some previous works that compare molecular line emission from protoplanetary or circumbinary disks with specific models of such an emission (e.g., Koerner et al. 1993;Guilloteau & Dutrey 1998;Qi et al. 2003Qi et al. , 2004 and, in general, agreement between model and observations is fairly good, at least in the general appearance of the maps. However, given the great deal of physical parameters involved in the resulting molecular line emission, it is not straightforward to determine those parameters from a particular observation just by fitting an emission model. Our main aim in this paper is to identify a set of observational characteristics that give most information on the physical properties of the disk. Such observational characteristics should then be given the heaviest weight in a fit between observations and model aiming to determine physical parameters in a disk. This paper is structured as follows: in §2 we show the outline of this work. In §3 we describe the assumptions to calculate the disk structure models and the initial input parameters. In §4 we explain the radiative transfer calculation and we discuss the selection of the C 17 O (J=3 → 2) as the molecular line to make our study. In §5 we outline the network of models and the general trends of the line emission maps. In §6 we describe the statistical study that identify the set of observational characteristics that give most information on the physical properties of the disk, and we comment the results derived in §7. In §8 we show the comments and prospects for these studies. Finally, in §9 we perform a study of the detectability of our modeled disks with SMA and ALMA and we summarize the conclusions in §10.

GENERAL CONSIDERATIONS
In order to achieve the goal described in the introduction, we have developed a set of molecular line emission models calculated for various mass accretion rates, radius, viscosities and maximum dust grain radius distributions. The small size scale of a protoplanetary disk (∼100 AU) and their low temperatures (∼ 100 K; Beckwith et al. 1990;Miyake & Nakagawa 1995;Beckwith et al. 2000) require observations with high sensitivity and subarsecond highangular resolution, since 100 AU subtends 0. ′′ 7 at 140 pc (the distance to the Taurus cloud). This makes interferometric observations necessary to carry out this kind of studies. With the intention of reproducing a real interferometric observation of a protoplanetary disk with different physical parameters, we integrated the radiative transfer equation and convolved each model with a beam of 0. ′′ 4, as a compromise between resolution and sensitivity. From each resultant map, we have measured different observational signatures, as if they were data from a real interferometric observation. Finally, in order to obtain the best combinations of such observational parameters that yield more information about the physical characteristics of disks, we have undertook a novel statistical approach to link observational properties of the expected molecular line emission with the underlying physical properties of the disk, by means of a principal component and multiple linear correlation analysis. We show that this is a promising type of analysis to prepare the observations with the new generation of millimeter and submillimeter interferometers.
In this study it is important to choose an appropriated molecular transition sensitive to the physical parameters in which we may be interested. To carry out our study we have selected the C 17 O(J=3 → 2) transition at 337 GHz. This line is a high excitation transition of a CO isotope with very low abundance, which makes it less susceptible to be affected by absorption and/or the emission from the surrounding cloud material. This transition is also a suitable candidate to be observed in protoplanetary disks using the Submillimeter Array (SMA) and the Atacama Large Millimeter Array (ALMA), as shown by Gómez & D'Alessio (2000).

Assumptions
We base our calculations of molecular line emission on structure models of accretion disks irradiated by the central star, which have been previously used to explain different observations of classical T Tauri Stars. The assumptions and calculation method of such models are described in D' Alessio et al. (1998Alessio et al. ( , 1999Alessio et al. ( , 2001. In summary, the disk is assumed to be in steady state, with a constant mass accretion ratė M and an α−viscosity (Shakura & Sunyaev 1973), with a constant value of the viscosity parameter α. The disk is in vertical hydrostatic equilibrium in the gravitational potential well of the star, and we neglect the disk self-gravity. We assume that gas and dust are thermally coupled, having the same temperature everywhere. This dust/gas temperature enters in the calculation of the disk volumetric density distribution through the integration of the hydrostatic equilibrium equation. The main heating mechanisms considered are viscous dissipation and stellar irradiation. The viscous dissipation is important in heating the inner regions (close to the star and close to the midplane), the direct stellar irradiation heats the disk atmosphere, and the stellar radiation scattered and reprocessed by the disk upper layers heats the whole vertical structure. The transfer of radiation through the disk is calculated taking into account that the dust scatters and absorbs stellar and disk radiation, implying that the temperature structure depends on the dust properties. The viscous irradiated disk models used in the present study show the temperature inversion previously found by Calvet et al. (1991Calvet et al. ( , 1992, i.e., at the outer disk, R ∼ > 10 AU, the upper layers are hotter than the disk midplane, because they are heated by direct stellar irradiation (see also Chiang &Goldreich 1997 andD'Alessio et al. 1998).
The dust opacity is calculated using the Mie theory for compact spherical grains. We consider a distribution of sizes given by n(a) = n 0 a −p , where a is the radius of the grains, n 0 is a normalization constant, and p is a free parameter. In this work, we have adopted p = 3.5 from Draine & Lee (1984) and the model of dust composition proposed by Pollack et al. (1994) with the variations introduced by D' Alessio et al. (2001). To account for the possibility of dust growth, we adopt different values of maximum grain sizes. Dust grains of different sizes have different continuum opacity at mm wavelengths, affecting the molecular line emission in different ways. The existence of bigger grains in disks than in the interstellar medium was proposed to explain the observed slope of the continuum SED at millimeter wavelengths Miyake & Nakagawa 1995). Large grains could be depleted from higher layers of the disk, but could be well mixed with gas below a few gas scale heights. For simplicity, the disk models adopted here (from D' Alessio et al. 2001), assume that gas and dust are well mixed. This seems a reasonable assumption if the emission from the molecular line arises mainly from areas closer to the midplane than to the upper layers. This point will be discussed later (subsection 4.1).

Input parameters
For the present study we have considered the following input parameters: maximum disk radius (R d ), maximum radius of dust grains (a max ), disk mass accretion rate (Ṁ ), and viscosity parameter (α) (see values in Table 1). We have adopted typical parameters of a T Tauri star from Gullbring et al. (1998), i.e., M * = 0.5 M ⊙ , R * = 2 R ⊙ , and T * = 4 000 K for all the models. The disks are assumed to be at 140 pc, the distance of the Taurus molecular cloud (Kenyon et al. 1994), with a typical inclination angle i = 60 o . It is important to mention that each disk structure is self-consistently calculated given these input parameters. This means that the whole disk structure is affected by all the parameters, consequently affecting the line properties. This might complicate the analysis of the resulting line properties, but we think this gives a more realistic description of the interplay between the different variables.

RADIATIVE TRANSFER AND MOLECULAR LINE EMISSION
The model of the disk structure provides a detailed density and temperature distribution through the disk as a function of the height and the distance to the disk center. To derive the line intensity for a given molecular transition, we must resolve the transfer equation. We have used the same assumptions and formalism that Gómez & D'Alessio (2000), and we summarize them briefly here. We assume local thermal equilibrium for the population of the molecular energy levels and we consider thermal line profiles. We divide the disk in a grid of cells considering isovelocity lines and their perpendicular lines (see Appendix in Gómez & D'Alessio 2000). We integrate the transfer equation dI ν /ds = κ ν ρ(S ν − I ν ) through the line of sight at the center of each cell, where I ν is the intensity, s is the length along the line of sight, κ ν is the absorption coefficient that consider the contributions from the line and continuum, i.e., κ ν = κ l + κ c , ρ is the mass density of the gas, and S ν the source function. The coefficient κ c is dominated by dust and we consider pure absorption opacity of each kind of dust grains size.
In order to derive the flux density, we convolved all our models with a beam of 0. ′′ 4 of HPBW, as a compromise between resolution and sensitivity. Finally we subtract the continuum emission to isolate the flux density of the molecular line, obtaining a set of model results that reproduce a real observation of a protoplanetary disk with different physical parameters.

Selection of the emission line transition
It is very common that young stellar objects like T Tauri stars are still embedded in the material of the parental cloud. The envelope that surrounds the disk-star system is composed by cold gas and dust that could hide the disk emission.
On one hand, the surrounding material could absorb the emission from the hotter, innermost part of the disk structure. This problem can be solved by selecting a molecular transition whose energy is high enough to trace the hotter gas from the disk, while few molecules in the colder envelope are in the states involved in this transition, thus avoiding line absorption at the envelope. However, the frequency of the transition must not be too high, because otherwise the dust in the envelope would become optically thick. On the other hand, the molecular line emission from the whole cloud could hide the deeper emission from the disk, as it is usually the case for the more abundant CO isotopes. To make sure that the observed emission comes only from the disk structure, we must select a molecular species of low abundance for which the envelope is optically thin and therefore its emission would be negligible.
In order to fulfill these conditions, we have chosen the (J=3 → 2) transition of the C 17 O molecule at 337 GHz. Its molecular abundance relative to H 2 in molecular cores is low, 5.0×10 −8 (White & Sandell 1995) and the frequency transition at 337 GHz is still low enough to avoid being much absorbed by the dusty envelope.
In a recent work, Dartois et al. (2003) have studied the disk vertical temperature structure using different isotopes of CO. Being characterized by different opacities, different lines trace the gas at distinct depths. Dartois et al. (2003) find a good agreement between the inferred temperature for each transition/isotope and the temperature where τ ∼ 1 in irradiated (dusty) disk models. Following Dartois et al. (2003), we have studied the formation regions of molecular lines from different isotopes of CO (CO, 13 CO, C 18 O, and C 17 O) in our set of disk models, resulting that the C 17 O is formed closer to the midplane than the rest of the isotopes. This fact makes the LTE assumption acceptable, since the region that mainly contribute to the line emission shows higher densities than the critical density of the selected transition (≃ 5 × 10 4 cm −3 ).
In addition, given the high gas densities in the disk, we expect that the abundance of this isotope, would be less affected by photodisociation produced by the incident radiation, since it would be shielded against it, specially in the deepest layers. Moreover, theoretical models of the evolution of molecular abundances in protoplanetary disks, predict depletion of CO from the gas phase for temperatures below 20 K (Aikawa et al. 1996) and therefore, depletion is probably not significant for the chosen molecule and for the disk radii we are considering here (< 150 AU). Thus, for simplicity, we adopt a constant abundance for C 17 O relative to H 2 , given the molecular core value (White & Sandell 1995).

RESULTS
We have calculated the expected emission in the C 17 O(J=3 → 2) transition for a typical T Tauri star with a disk inclination angle of 60 o , and for all possible combinations of the physical parameters shown on Table 1. We have 241 different models for which we have integrated the radiative transfer equation at 12 different velocities, from -2.5 to 2.5 km s −1 at steps of 0.5 km s −1 . The results are maps like those shown in Fig 1, in which we only show the positive velocities, since the maps are similar and almost symmetrical in negative velocities to the ones shown here (there are slight differences though, due to the hyperfine structure of C 17 O transition). From each map we have measured the following observational signatures as if they were data from a real interferometric observation: intensity of the principal (the more distant from the observer, to the north in our maps) and secondary (closer to the observer, to the south in our maps) peak at each velocity, distance from the disk center to principal peaks, half power size of the emission distribution, and velocity at which the maximum intensity is present. These represent a total of 43 different observational parameters for each input model.

Line emission maps
In all maps obtained from our set of models we observe the same tendencies as Gómez & D'Alessio (2000). Summarizing, we observe an asymmetry on both sides of the major axis, as expected from optically thick emission. The areas further away from the observer (positive declination in Fig. 1) show line emission of higher intensity because the line of sight intercepts areas of the disk closer to the central star, where the gas is warmer. This result is confirmed in DM Tau spectral line observations by Dartois et al. (2003), where an inclination angle i ≃ −37 o was assumed. We notice another asymmetry on both sides of the minor axis, that it is more pronounced when approaching to the velocity of the cloud (v = 0 km s −1 ). It is caused by the asymmetry of the hyperfine structure in the C 17 O molecular transitions.
We find that the maximum intensity of the line at the systemic velocity (v = 0 km s −1 ) traces the outer edge of the disk. As previously discussed by  and Gómez & D'Alessio (2000), this is a consequence of the fact that the effective area emitting at a given velocity within the beam increases with the distance to the central star more steeply than the decrease of brightness temperature with distance.
Moreover, at the center of the disk the emission intensity diminishes, and we can even see absorption lines in some of our maps. The high dust continuum opacities typical in the central parts of the disk, reduces the contrast between the emission lines and the continuum, and, if it is high enough, lines could show up in absorption.

STATISTICAL TOOLS
As we mentioned in section 5, we chose a set of 43 different observational parameters to characterize the maps resulting from our model calculations. These parameters are, in principle, somewhat arbitrary. If we want to simulate a real observation, in which we would like to extract information about the underlying physical characteristics of disks, it is obvious that some of these 43 parameters will have more informative power, while some may turn out to be irrelevant. Moreover, it is likely that not all the chosen observational parameters will be independent.
Here, we have undertook a statistical analysis to try to identify a set or a combination of observational parameters that could render more information about the disks properties. First, we used a principal component analysis to reduce the number of observational parameters to a small, informative set. Later, we investigate whether we can obtain quantitative values of physical magnitudes from observational parameters, by means of multiple linear regression.

Principal Components
The principal component analysis is a statistical technique that provides a dimensional reduction of a set of variables. In our case, each of the initial 43 observational parameters would be an axis in a system of coordinates in a multidimensional space. The method consists of finding a set of orthogonal axes in which the variance (heterogeneity) of our data is maximum (see Thurstone 1947;Kaiser 1958). This is solved through a linear and orthogonal transformation that corresponds to a rigid rotation of the original data into a new set of coordinates. The eigenvalues obtained provide information about the variances of the data in the new space, and the eigenvectors represent the direction of the axes in the new space of representation of our data. In our study, this analysis reduces the number of observational signatures necessary to derive information about the physical parameters, to a smaller set of linearly independent parameters: the principal components.

Multiple linear correlation
In order to quantitatively estimate each physical parameter from the set of observational variables, we have also carried out a multiple regression analysis (Pearson 1908). In our case, the obtained principal components will be considered the independent variables, and the physical properties of the disk will be the dependent ones. The multiple linear correlation analysis will then try to make the best fit, to derive a linear function of the form: where y i is the dependent variable, and x j are the independent ones. In an ideal case (correlation coefficient ≃ 1), we would obtain a function with a good predictive power for the dependent variable. For our particular problem, we would like to obtain a formula with which, from a set of observational parameters, we could calculate the physical characteristics of the disk.
Moreover, to test the signification of the multiple regression we applied the F-Snedecor test (Snedecor 1934) as hypothesis testing. The ratio of two chisquares divided by their respective degrees of freedom follows an F distribution. The test consist of comparing the relation between the variance of the predicted values for the dependent valuable and the error variance with the value of the F distribution.

Eigenvalues and Eigenvectors of Principal Components
To reduce the number of relevant components in the principal component analysis, we adopted the Kaiser criterion (Kaiser 1960), which only retains factors with eigenvalues greater than 1. In our case, the criterion selects four factors, which account for 91% of the total variance of the system. In fact, the first and the second factors alone represent 82% of the total variance. Therefore, most of the following analysis is based on these two first principal components.
We have represented the eigenvectors in Table 2, for each principal component. The numerical entries in this table show the linear combination coefficients for each observational parameter, used to build the corresponding principal component. Therefore, they indicate the relative weight of each observational parameter on the components.
For the first principal component (PC1) the observational parameters with a larger weight in its definition are (in order of decreasing relative weights) the velocity of the peak emission, the half power sizes for principal peaks at intermediates velocities, and the distance from principal peaks to center. The parameters that define the second principal component (PC2) are the half power sizes of principal peaks at 1.5 km s −1 , the velocity of the peak emission, and the half power sizes of secondary peaks at 0.0 and 1.5 km s −1 . The third principal component (PC3) is defined by the velocity of the peak emission and the half power sizes for principal peaks at 0.5 and 2.0 km s −1 , and for secondary peaks at 0.0 and 1.5 km s −1 . Finally, the physical parameters that define the forth principal component (PC4) are the half power sizes for principal peaks at 1.5 km s −1 and for secondary peaks at 0.0 and 1.5 km s −1 . We have marked in boldface these most representative observational parameters in Table 2.

PC1(kinematical component)-PC2(spatial component) diagrams
Due to the fact that the first and the second principal component represent the 82% of the total variance of the system, we have represented all our models in a PC1-PC2 diagram, to check whether such diagrams can be used to discriminate disks with particular physical characteristics, if we see some clustering related to the physical properties. Considering the parameters with the larger weights in each of these principal components, we have named them as "kinematical component" and "spatial component" in the case of PC1 and PC2, respectively.
The most evident result is shown when we represent disks with different radii in the PC1(kinematical)-PC2(spatial) diagram (see Fig 2). The disks with radius 50 AU are distributed on the right part of the diagram, the ones with radius 100 AU are located at the central part and the disks with radius 150 AU are located on the left of the plot. This means that the kinematic principal component (x axis) is qualitatively good to discriminate among disks with different radii. This result indicates that we can get information about the radius of a protoplanetary disk (physical parameter) from the velocity of the peak emission, the half power sizes for principal peaks at intermediates velocities and from the distance from principal peaks to the center (observational signatures). That the disk radii can be discriminated relatively well by maps of line emission may seem a relatively obvious result, but it illustrates the power of this kind of statistical analysis.
Another interesting trend is seen from the representation in the diagram of disks with different mass accretion rate (see Fig 3). Disks with higher mass accretion rates tend to populate the upper parts of the PC1(kinematical)-PC2(spatial) diagram. Therefore, in this case it is the second principal component (y axis) the one that better discriminates among disks with different mass accretion rates. Considering the parameters that give rise to this component, we can say that the half power sizes of principal peaks at 1.5 km s −1 , the velocity of the peak emission and the half power sizes of secondary peaks at 0.0 and 1.5 km s −1 velocities provide information about the mass accretion rate of the disks.
Other trends relating principal components and the rest of the physical parameters are also present, but qualitatively they are not as clear as the two we have mentioned. In the case of other physical parameters, it is the second principal component the one that provides more information about the α parameter and the maximum radius of dust grains. Further analysis including models of more molecular transitions will certainly be useful to obtain refined principal components that can better discriminate these physical parameters.

Multiple correlation
We have seen in the previous section that diagrams of principal components can be useful to discriminate among disks with different physical characteristics. It would be interesting, however, to have a mathematical tool to easily obtain numerical values for these physical characteristics based on the observed maps. As a first approach, we have here performed a multiple linear correlation with the observational parameters as independent variables, and the physical characteristics as dependent ones. The resulting coefficients for such a fit are shown in Table 3.
As it was deduced in the previous subsection, the linear correlation coefficients of each principal component (see r(PC i ) values in Table 3) show that PC1(kinematical component) is the best one to derive information about the disk radii and PC2(spatial component) provide the most information in the determination of the rest of the physical parameters,Ṁ , α, and maximum radius of dust grains.
The most important piece of information that can be retrieved with this kind of study is the deduction of the linear combination of observational signatures that provide quantitative information about physical characteristics. This linear combination can be derived from λ i coefficients in Table 3 as follows: where P is the physical parameter. As an example, in the case of the radius of the protoplanetary disk, this combination would be the following:  With these sets of linear combinations, we could estimate physical parameters from observations, provided that the fit is good enough.
The strongest correlation (see R values in Table 3) is obtained for radius (correlation coefficient R = 0.97), followed by mass accretion rate (R = 0.57), viscosity parameter (R = 0.31) and maximum radius of dust grains (R = 0.19). The multiple correlation coefficients alone are not good statistical indicators of the goodness of linear fits. To assess the validity of our fitted functions we carried out an F test to derive the signification of the linear regression. We assume a confident limit of 95%, which means that we could admit as good fits with values F < 1.44. However in all our results the values exceeded this critical value (see Table 3). This result suggests that our variables are far from the linear regime, which is certainly reasonable.

Effects of measurements errors
In our method, the derived value of a physical parameter depends linearly on the principal components, which are linear combinations of the observational data. Therefore we must consider on one hand the error in the calculation of the principal components, and on the other hand the error derived from the calculation of the multiple correlation.
In a real observation, the error of the intensity will depend on the rms noise reached during the observations, and, in particular, the achieved signalto-noise ratio will determine the positional accuracies (1σ error in position ≃ HP BW/(2 SNR)), assuming a compact emitting source). In our method, for the determination of the most significant principal components, the most important sources of error are related to the observational parameters that contribute with a larger weight, i.e., the half power sizes of emission for the principal and secondary peaks of intensity (whose error can be roughly estimated as ≃ HP BW/SNR), and the velocity at which the maximum intensity is present (with 1σ error ∼ ∆v/(2 SNR), where ∆v is the linewidth). Since principal components are linear combinations of observational parameters, the weight of the errors associated with each measured observational parameters will depend on theirs particular weight in the definition of each principal component (see Table 2). Nevertheless, we must point out that, in the case of calculations of the physical parameters of disks, the error is dominated by the uncertainties in the fits of the multiple regression. Since our derived variables are far from linear regime (see F test above),measurements errors are not significant in comparison.

COMMENTS AND PROSPECTS FOR THESE STATISTICAL STUDIES
The results derived from the statistical study presented in this paper show that this method could be a powerful tool to obtain information related to the physical characteristics in protoplanetary disks from observational parameters.
The most important information of our study is related to the radius of the disks. This preliminary study results in a way to obtain a first approximation of the disks radii that depends only on the obser- vations, avoiding the application of χ 2 fitting techniques.
Since this is a promising method of study, and considering the future set of observations in protoplanetary disks that will be carried out with the development of ALMA, we plan to complete and improve our models and analysis.
Nowadays it is not possible to test observationally our method because there is no observation of protoplanetary disks in our selected molecular transition, C 17 O(J=3→2). But, considering the excellent characteristics of this transition/isotope to observe protoplanetary disks (see subsection 4.1), one should be able to test our study in the near future with ALMA observations. Meanwhile this moment comes, we will improve or study.
First, to better determine the tendencies in the principal component diagram, we plan to increase the sample, building a more extended set of models with a wider variety of initial physical and observational parameters. Calculations in other molecular transitions or isotopes would further constrain the information on temperature and density. An update in the calculation of the disk structure models would be done, considering in detail the photodisociation or depletion effects in the line of study, along with dust grain growth and settling. We also plan to study the application of our method towards nonaxial-symmetric disks, for what the development of a new disk structure that takes into account the lack of symmetry is needed. Finally, the determination of a more complex method to derive the physical parameters of disks from the principal components is desirable, since the application of multiple correlation provides variables far form the linear regime and, introducing the greater source of error in our study.

DETECTABILITY OF MOLECULAR LINES WITH NEW INTERFEROMETERS
It would obviously be important to be able to test our results observationally in the future. With this mind, we have also studied the detectability of the calculated models, when observed with interferometers that will be able to reach subarcsecond resolution at the frequency of the C 17 O(J=3 → 2) transition, i.e., SMA and ALMA.
With the SMA, a sensitivity of ∼85 mJy beam −1 is expected 4 , considering the 8 antennas of the array working at 337 GHz, with 1 km s −1 velocity resolution and 10 hours integration time, under standard values of precipitable water vapor (∼2.0 mm for 300−355 GHz).
On the other hand, in the case of ALMA, considering 64 antennas, 1 km s −1 velocity resolution, only 1 hour integration time and 1.5 mm of precipitable water vapor (median at the site over all hours and seasons), the sensitivity expected is 1.2 mJy beam −1 (Butler & Wooten 1999).
Considering 5σ emission as detections, we conclude that all our disks modeled, even the faintest, will be detected with one hour integration time with ALMA. Nevertheless, it will be extremely difficult of observe any of our modeled disks with the SMA, at least the ones with the physical characteristics showed in this work (see Table 1), since the SNR obtained for the modeled disks with the highest intensities (∼180 mJy beam −1 ) in 10 hours of integration time, would only be 2σ.
Therefore, ALMA will be a crucial instrument to observationally test the predictions and assumptions of our models.

CONCLUSIONS
In this paper we have presented a statistical method to derive physical parameters from observational characteristics in a protoplanetary disk. To make this study we modeled the expected emission of the C 17 O(J=3 → 2) transition from protoplanetary disks with different physical properties. We then applied a principal component and a multiple correlation analysis, to obtain a set of linear combinations of observational parameters that may provide relevant information of the physics of the disks. The main conclusions are the following: • The most significant results of our analysis are related to disk sizes. We can discriminate among disks with different radii using a principal component composed mainly of the velocity of the peak emission, the half power sizes for principal peaks at intermediates velocities and the distance from principal peaks to center.
• Moreover, some information about the mass accretion rate could be obtained from a principal component made of the half power sizes of principal peaks at 1.5 km s −1 , the velocity of the peak emission and the half power sizes of secondary peaks at 0.0 and 1.5 km s −1 velocities, although the results are much less significant than in the case of radii.
• Although with preliminary results, the statistical method presented here seems to be promising and useful, and will be improved and completed in a future with studies in other transitions.
• We have performed a study of detectability with ALMA and SMA. All our modeled disks could be detected with ALMA, using one hour integration time, nevertheless the sensitivity reached by the SMA is not enough to detect our disks with reasonable integration times. We conclude that ALMA will play an important role to test observationally our models and our statistical results.
We wish to thank the LAEFF staff for allowing the use of their personal computers to carry out the calculations of the molecular line emission models. PD acknowledges grants from CONA-CyT and PAPIIT, México. The model calculations were performed in the cluster at CRyA-UNAM acquired with CONACYT grants 36571-E, 47366-F, and UNAM-PAPIIT grant 110606. FG and IdG are supported by Junta de Andalucía grant FQM-1747. JFG also acknowledges partial support from grants AYA2005-08523-C03-03 of the Spanish MEC (including FEDER funds) and TIC-126 of Junta de Andalucía.