The possible asymmetrical models for explaining the inner disk struc- ture of MWC480 

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Protoplanetary disk Models

The SED of a disk can be computed in first approximation assuming that each disk annulus emits as a black body at the effective temperature Td, so that the observed flux at any given wavelength λ is given by the relation (Malbet et al. 2005): cos i Rout Fλ = Bλ(Td)(1 − e−τλ )2π r dr (1.39). where Rin and Rout are the inner and outer disk radii, D is the distance from the observer, i is the disk inclination with respect to the line of sight (i=0 means face-on disks), Bλ is the black body emission. The optical depth τλ is, τλ = 1 κλ Σdust(r) (1.40).
Where κλ is the total (gas+dust) opacity at the wavelength λ and Σdust(r) is the surface density of the dust (see in details Sect. 1.6). Chiang & Goldreich (1997, CG97) computed a radiative transfer model based on a flared circumstellar disk, which explains the flaring geometry by means of vertical hydrostatic equilibrium and provided a temperature profile with index q = 0.5 and a SED at IR and longer wavelengths. CG97 have been quite successful in the interpretation of observed SEDs of disks around YSOs, and is still nowadays widely assumed as the starting base for studies of those objects.
Afterwards, many improvements have been suggested. For instance, Dullemond et al. (2001, henceforth DDN01) proposed a model based on CG97 with some further imple-mentations. The model DDN01 was basically based on CG97, plus the addition of an inner ’puffed-up’ rim. The idea of a puffed-up inner rim was proposed by Natta et al. (2001) and developed further by DDN01 for Herbig Ae stars, to account for the shape of the near-infrared excess of these stars (3-µm bump”). In the DDN01 model, the vertical structure of the circumstellar disk was set by a balance between disk gravity and the pressure gradient created from the heating of disk material by stellar photons (see. Fig. 1.9). One success of this model has been the correct prediction of the NIR size- luminosity diagram for YSOs (Monnier & Millan-Gabet 2002, Millan-Gabet et al. 2007).
According to Fig. 1.9, the inner rim in the DDN01 model casts a geometric shadow on the region behind it, preventing it from receiving direct star light. The disk in the shadow is heated by scattered photons from the rim edge and through radial heat diffusion. The size of the shadow can be several AU depending on the rim geometry, the mass of dust in the outer disk and dust grain properties in the outer disk. The shadow corresponds to the terrestrial planet forming region in circumstellar disks inside/close to the snow line.
When the outer disk is dominated by gas mass, the disk eventually emerges from the shadow and ’flares’. The flared disk can be a few hundred AU in size and emits radiation in the MIR and longer wavelengths (Fig. 1.9).
Although DDN01 succeed to explain a variety of observations, the structure of the inner rim was not explained. DDN01 assumed that the inner rim is ’vertical’, and that its photospheric height is controlled by radial heat diffusion behind the rim. This model is not consistent with the near-IR emission of Herbig stars where their disks for instance are face-on, for which the projection on the line of sight of the rim surface is null. (see. Fig. 1.10).

Transitional and Pre-transitional Disks

Several disks have been recently detected with a notable deficit of flux in the near- and mid- infrared (2 –20 µm) compared to disk of other Herbig stars, e.g. Bouwman et al. (2003). The lack of near- and mid-infrared flux in these disks could be indicative of the removal of the hot dust or warm dust close to the star which emits at near-IR and mid-IR wavelengths respectively, and the presence of inner disk holes or gaps.
A few years ago, a new class of disk was identified. These disks are so called ”pre-transitional disks” and they have gaps within their disks, which is different from the inner holes observed in transitional disks (Espaillat et al. 2007b). In these disks there is lack of mid-infrared flux (5 –20 µm) and significant emission beyond 20 µm (Fig. 1.14).
To create the inner disk holes and gaps in transitional disks and pre-transitional disks several mechanisms can be involved. These mechanisms could be due to planet forma-tion, stellar companions, grain growth, magnetorotational instability (MRI) and photo-evaporation. The MRI, grain growth and photoevaporation could not explain the gaps seen in the pre-transitional disks around for instance LKCa 15. Based on disk clearing theories, the planet formation is a more important factor to clear the inner disk of pre-transitional disks. HD 100546 is such a pre-transitional disk, which will be discussed in detail in Chapter 4.

Monte Carlo radiative transfer: MCFOST

MCFOST is a 3D continuum and line radiative transfer code based on the Monte Carlo method (Pinte et al. 2006). In a Monte Carlo method a probabilistic approach is taken to solve the radiative transfer equation. The photon packets from the central star are propagated through the disk to set its temperature (Fig. 1.15). Taking into account the scattering, absorption and emission of the photon packets in the disk, the SED and/or synthetic images are bulit.
MCFOST computes the temperature and the SEDs/images with a two-step processes:
-Step 1: determining the temperature distribution. When photon packets are produced at the stellar photosphere, they propagate through the disk until they exit the grid. By scattering, the orientation of the packet is changed but not its wavelength. By absorption, packets are re-emitted at once but at a different wavelength, calculated based on the temperature of the cell. For re-emission process, the temperature correction method proposed is used by Bjorkman & Wood (2001). In this step, all photon packets have a similar energy and are randomly scattered/absorbed within the disk. The concept of mean intensity proposed by Lucy (1999) is used to decrease noise in the temperature estimation for optically thin cases.
-Step 2: gives the SED and/or images from the temperature calculated in step 1. In this step, the number of photon packets is assumed to be constant at all wavelengths. This code uses a spherical or cylindrical grid, with an adaptive mesh filtering at the inner edge to properly sample the inner radius of the disk.
We used this code in order to reproduce our new PIONIER/VLTI data. This code is used in Chapter 4. For a comparison with other 2D radiative transfer codes, see Pinte et al. (2009).

The Electromagnetic field

When we look at the field of radiation, we may represent it as being composed of many trains of waves propagating with light velocity. For simplicity, we consider a monochromatic wave train, which is linearly polarized and propagating along the z direction. The electric field of such a radiation field can be represented by a cosine function as following: E(z, t) = a cos (2π(νt − z/λ)) (2.2).
where a is the real magnitude as is shown in Fig. 2.3, ν is the frequency of the waves and λ is the wavelength. If we consider a position of the observer along the z axis, so that assuming z=0, the radiation field only varies with time. In this case the distance between two successive maxima (see Fig. 2.3) is defined as a period and one period is equal to 1/ν. Now if we freeze the time (t=0), then the electric field varies as a function of distance. In this case, the distance between two successive maxima is one λ. The relation between λ and the period (T) is very simple. During the time of wave period, the light wavelength at the velocity of the light speed is: λ = cT (2.3). The electric function, Eq. 2.2, can be written in complex notation as: V (z, t) = Re(a exp [i2π(νt − z/λ)]) (2.4).

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Interferometric facilities around the world

The intent of this section is to give a short overview of the instruments available around the world for optical/infrared interferometric observations (Table 2.1). In the following, we focus on the two main interferometers currently offered to the international commu-nity: VLTI (Very Large Telescope Interferometer) and CHARA (Center for High Angu-lar Resolution). Since VEGA/CHARA (Visible spEctroGram polAmiter; Mourard et al. 2009), PIONIER (Precision Integrated Optics Near Infrared ExpeRiment; Le Bouquin et al. 2011) and MIDI (MID-infrared Interferometric instrument; Leinert et al. 2003) were used to retrieve the data for this thesis, they will be described in more detail in Sects. 2.2.1, 2.2.2, and 2.2.3, respectively.
VLTI is the ESO interferometer located on Cerro Paranal (Chile). It is equiped with 4 Unit Telescopes (UTs) with 8 m aperture and 4 Auxiliary Telescopes (ATs) with 1.8 m aperture. It allows the combination of 2 up to 4 apertures. The instruments offered to the community are AMBER and MIDI. AMBER (Petrov et al. 2007) is a 3-way beam Two second generation instruments are developed for VLTI: GRAVITY (Eisenhauer et al. 2008), and MATISSE (Multi-AperTure mid-Infrared SpectroScopic Experiment; Lopez et al. 2006). GRAVITY will combine 4 beams in the near-infrared and it will offer imaging and astrometric modes. The spectral resolution will be R = 22 ,500 ,4000. GRAVITY is expected to be started working for 2016. MATISSE will observe in the LMN-bands with 4 telescopes and at low- to mid-spectral resolutions (R = 30, 500, 1500–2000 and also 3000-5000 in L&M bands). MATISSE will be started working for 2017.
CHARA (Center for High Angular Resolution Astronomy) is an array of six 1-meter-telescopes for optical and infrared interferometry on Mount Wilson, California. It offers 15 different baselines with lengths ranging from 30 to 300 m. The instruments operating on CHARA are: CLASSIC, CLIMB, MIRC (Michigan Infrared Beam Combiner; Mon-nier 2006), FLOUR, PAVO and VEGA (Mourard et al. 2009, Visible spEctroGraph and polArimeter;). The first two instruments observe in H- or K-broad band and combine 2 and 3 apertures, respectively. I will explain in details the VEGA instrument that I used it in my thesis in Sect. 2.2.1.

The VEGA Instrument

VEGA observes in the optical (480-850 nm) with very high spectral resolutions R = 1700 to 30000. It combines 2 to 4 telescopes. The medium (6000) and high (30 000) spectral resolutions are well suited to perform kinematic analysis of the interferometric signal, providing resolution of 60 and 10 km s−1 respectively. These spectral resolutions are best dedicated to the extraction of differential spectral information. Radiative winds and fast rotating photospheres of hot stars can be probed efficiently with the medium spectral resolution. Another interesting possibility is the presence of a polarimeter that could be inserted into the beam. This gives new insight into many physical processes.
The low (1700) and medium resolutions are well suited for absolute visibility studies and are also well adapted for the study of binaries or multiple systems.

The PIONIER Instrument

PIONIER (Precision Integrated–Optics Near–infrared Imaging ExpeRiment, Le Bouquin et al. 2011) was commissioned at VLTI in Fall 2010. It is property of the Institute de Planetology et d’Astrophysique de Grenoble (IPAG) and partners. The instrument is optimised for imaging with its 4 way beam combiner working in the H band at low resolution. PIONIER combines either beams from the 8.2-m Unit Telescopes or from the 1.8-m Auxiliary Telescopes to provide visibilities of six different baselines, as well as four closure phase measurements, simultaneously. PIONIER features low resolution spectroscopic optics to measure at six different wavelengths within the H-band, increas-ing the (u,v) coverage, or can work in broad band light for sensitivity enhancement on faint targets.

The MIDI Instrument

MIDI was mounted on ESO/VLTI at Paranal (Chile). The conceptual design started in 1997. This instrument is the result of the efforts of several European institutions, and it detected the first fringes with the UTs on December 12, 2002. This instrument was decommissioned in the beginning of 2015, since the MATISSE instrument will be started working in 2017. MIDI combines the light coming from two apertures (UTs or ATs) using a two pupil- plane beam splitter. The wavelength range of observation is the N–band (8, 13 µm). Sixteen configurations are offered at the moment with a range in baselines between 11 and 130 m. The observations are dispersed in wavelength and two spectral resolutions are available: PRISM (R = 30), and GRISM (R = 230). The instrument is (as of Sept. 2011) offered in two modes: high-sensitivity (HIGH SENS), science and photometry (SCI PHOT). In HIGH SENS mode the photometry is recorded a few minutes after (or before) the fringes. This allows to observe fainter objects but is less accurate. In SCI PHOT mode the photometric channels are recorded simultaneously during the scanning of the fringes. The gain in accuracy in this mode is possible only for targets brighter than 200 Jy.

Table of contents :

1 Scientific overview 
1.1 Star formation
1.2 Protoplanetary disk phase
1.3 Planet formation
1.4 The radial structure of the disk
1.5 The vertical structure of the disk
1.6 Dust properties
1.6.1 Dust opacity
1.6.2 Dust scattering
1.6.3 Mie Theory
1.7 Protoplanetary disk Models
1.7.1 Transitional and Pre-transitional Disks
1.8 Radiative transfer
1.8.1 The radiative transfer problem
1.8.2 Monte Carlo radiative transfer: MCFOST
2 An introduction to optical/IR interferometry 
2.1 Why Spatial Interferometry?
2.1.1 The Electromagnetic field
2.1.2 The Young’s experiment
2.1.3 The Fizeau Interferometry
2.1.4 Light coherence
2.1.5 Complex degree of mutual coherence
2.1.6 Visibility
2.1.7 Phase
2.1.8 The differential phase
2.1.9 The closure phase
2.2 Interferometric facilities around the world
2.2.1 The VEGA Instrument
2.2.2 The PIONIER Instrument
2.2.3 The MIDI Instrument
3 The peculiar fast-rotating star 51 Oph probed by VEGA/CHARA 
3.1 Abstract
3.2 Introduction
3.3 Observations and data processing
3.4 Continuum emission: stellar photosphere
3.5 Hα emission line
3.5.1 Qualitative analysis
3.5.2 Kinematic model
3.6 Discussions
4 Probing the inner region of the pre-transitional disk of HD 100546 
4.1 Abstract
4.2 Introduction
4.3 Observations
4.4 Data description
4.4.1 Visibilities
4.4.2 Closure phase
4.4.3 The spectral energy distribution
4.5 Modeling
4.5.1 The radiative transfer code
4.5.2 Methodology
4.5.3 Evaluation of sensitivity to parameters
4.5.4 Results
4.6 Do we have evidence for a clumpy structure?
4.7 Conclusion
5 Study of the inner disk of the Herbig star MWC 480 
5.1 Abstract
5.2 Introduction
5.3 Observations
5.3.1 MIDI observations and data reduction
5.3.2 Spectroscopic observations
5.3.2.1 Variability
5.4 Modeling
5.4.1 Application to the one-component disk model
5.4.1.1 Results
5.4.2 Application to the two-component disk model
5.4.2.1 Results
5.5 Summary and perspectives
6 The possible asymmetrical models for explaining the inner disk struc- ture of MWC480 
6.1 Abstract
6.2 Introduction
6.3 Observations
6.3.1 MIDI observations
6.3.2 Spectroscopic observations with SpeX
6.4 Modeling
6.4.1 Azimuthally asymmetric models
6.4.1.1 Two-component disk model with wall
6.4.1.2 A two-component disk model with a bright feature
6.5 Discussion
6.6 Summary and perspectives
7 Conclusion and future work 
7.1 The visible point of view
7.2 The near- and mid-Infrared point of view
7.3 Conclusions
7.4 Future investigations
Bibliography 

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