figurec

A MM AND NEAR-IR STUDY OF YSOS:
FROM OUTBURSTING PROTOSTARS TO SATELLITES

A THESIS SUBMITTED BY

PEDRO HENRIQUE SOARES DA SILVA DE PINHO NOGUEIRA

TO THE

FACULTAD DE INGENIERÍA Y CIENCIAS
INSTITUTO DE ESTUDIOS ASTROFÍSICOS

In partial fulfillment of the requirements for the degree of Doctor of
Philosophy in Astrophysics.

Supervisor: Alice Zurlo
Co-Supervisor: Lucas Cieza

UNIVERSIDAD DIEGO PORTALES
Santiago, Chile
2024

###

©2024, Pedro Henrique Soares da Silva de Pinho Nogueira

All rights reserved

Figure 1: Poster secretly made by UDP friends to advertise others regarding my
defense.

##  Approval report

A copy of the approval report signed by the Thesis committee is displayed:

To and for clearer skies ahead

##  ACKNOWLEDGEMENTS

Acknowledging so many great people who have supported me on this journey is a
challenging task, but I will do my best.

First of all, I would like to express my gratitude to my scientific mentors
and advisors, Alice Zurlo and Lucas Cieza. Your guidance and support were
fundamental in making this achievement possible. I also extend my thanks to
the committee members for their help during the development of my thesis and
the defense itself. Special thanks go to Camilo, Cecilia, and Trisha. You
helped me tremendously behind the scenes, showing immense patience and
dedication while teaching and supporting me through the toughest moments.

Thanks especially to Trisha, my extra advisor and super duper friend. You
helped me a lot with everything—in work, and in life. Sometimes, just your
presence in the office was enough to give me strength and courage to advance
more, do more, and be more. You are engraved in my heart, Trishita!

Thanks to the UDP community. I was pleasantly surprised when I first arrived
at UDP. The Astronomy Institute is truly an oasis of gentle people. Everyone
there is so nice! I really wish all academic environments were like the one in
which I had the opportunity to see and live.

Eu queria especialmente agradecer, do fundo do meu coração, meus pais, Vitor e
Roseli, e minha irmã, Ana Clara, os quais sempre me apoiaram com esse sonho
doido, de entender e alcançar as estrelas, apesar de me levar ao ponto de
ficar mais velhinho e ainda viver de forma instável hahaha. É o preço de
sonhadores nesse mundo louco em que hoje vivemos, mas posso dizer, ainda hoje,
que não me arrependo da trilha que percorri. Agradeço muito todo o apoio que
tive para concluir esses suados 12 anos de estudo e dedicação. Para vocês, o
meu mais sincero OBRIGADO e AMO VOCÊS!

Agradeço aos amigos que fiz nessa minha longa jornada, alguns que ficaram
comigo até hoje e outros que deixaram suas marcas em uma página que já
folheei, porém foram fundamentais para mim. A mencionar alguns deles: grupinho
do rock de Amparo - a gente se fala e se encontra um dia a cada 1 ou 2 anos
mas sempre é muito prazerosa a companhia de vocês. É muito legal acompanhar a
nossa trajetória desde o período de crianças/adolescentes até jovens adultos.
Desejo o melhor para vocês! Também menciono meu grande amigo Victor.
Companheiro desde início/meio da graduação, foi e continua sendo uma ótima
pessoa, de grande coração e humor duvidoso (porém similar ao meu). Te agradeço
pela companhia e por tudo até hoje. Te desejo o melhor! Agradeço aos amigos da
graduação e do mestrado, alguns deles que por consequências da vida ME
SEGUIRAM até o Chile (Priscila, Camila, Douglas), e outros que ainda sigo os
passos por redes sociais e eventuais mensagens (Eduardo, Yanna, Stephane,
Aline, Natasha, Rodrigo, e muitos outros).

Agradeço ao BR chilean gang, grupo esse que reuniu brasileiros únicos porém
duvidosos (Priscilla, Camila, Douglas, Carol, Tuila, Danielle, Flor, Johnny,
Thallis e Larissa). Vocês fizeram com que minha estadia aqui não fosse tão
dolorosa como poderia ter sido. É normal que um estrangeiro sinta um grande
impacto ao estar longe de casa, cultura e família. Porém vocês reduziram
demais esse possível dano com sua maravilhosa companhia, risadas, diversões,
fofocas, memes e muito mais! Amo vocês, lindos!

Agradezco mucho las amistades que hice en Santiago, especialmente a los grupos
de tenis de mesa en los que participé. Recordé mi vieja pasión por el deporte
y realmente me dediqué mucho a mejorar. Esa experiencia fue increíble gracias
a la compañía de gente maravillosa y muy amable. Gracias a los grandes
maestros Juan Papic, Francisco Mendoza, Roberto, Luis Peña y Luis “Loco”
Fredes por ayudarme a mejorar y desarrollar esa pasión que tenía, y por la
paciencia que tuvieron con mi español limitado. Agradezco a mis amigos Milio,
Leo, Lucas y Gabriel. Agradezco muchísimo a Milio, que fue un gran amigo, me
invitó a su casa para jugar muchas veces, y es una persona increíble, de
corazón gigante. Agradezco de corazón a Gabriel, con quien además del tenis de
mesa, compartió sus logros, se interesó por los míos, me llevó a cafés y me
enseñó acerca de la cultura chilena y de cosas muy raras como películas
underground, weás filosóficas/políticas y la participación de Platón en todo
eso. Descubrí cuánto me gustaba la comunidad chilena gracias a él.
Lamentablemente, dejaré su compañía por ahora, pero dejo aquí constancia de
que fue el chileno que más me hizo apreciar su país y su cultura. Un día
volveré y te sacaré la chucha en un partido.

I profoundly thank the people from my office. From the veterans who have moved
on to other offices across lands and oceans (Aldo, Kacper, Rafi, Emily), to
the core members of the Why??? group (Keerthana, Pablo, Trisha, Tatevik,
Emilio, Florence, Kevin, and Simon). I don’t know if you guys realize how
important you were/are to me and how crucial you were for me to keep my sanity
during my Ph.D. The never-ending conversations, laughs, and gossips (while of
course keeping the working pace) made my stay and my working hours incredibly
enjoyable. Every day I was excited to go to the university because of your
presence. I love you guys! I will really miss you, and your close family
members who I also love (Ranajit and the beautiful Rohini, Karen and the long-
awaited Vaneh).

Agora agradeço à minha família mais próxima nesses anos: minha amada esposa,
Carol, e meu primogênito felino, Morpheu. Eu amo tanto vocês! São tão parte de
mim que não consigo imaginar uma vida sem suas presenças. Estiveram comigo nos
momentos difíceis, tristes, felizes, durante derrotas e conquistas! No filme
“Dança Comigo”, há um trecho que acho super pertinente a esse parágrafo. A
personagem de Susan Sarandon questiona, “Why do people get married?” e logo em
seguida diz: “There are billions of people on the planet. What does any one
life mean? In a marriage, you are promising to care about everything: the good
things, the bad things, the terrible things, the mundane things. All of it.
All the time, every day. You are saying, – Your life will not go unnoticed,
because I will notice it. Your life will not go unwitnessed, because I will be
your witness.” Portanto, deixo aqui palavras mais pertinente à Carol: Te amo,
meu amor! Obrigado por ser minha âncora quando precisei frear, meu motor
quando precisei acelerar, mas principalmente minha principal testemunha e
cúmplice! Agora, deixo palavras mais pertinentes ao Morpheuzinho: miau, miau,
miau, miau, miau, miau, miau, miau, miau. Você sempre foi e sempre será nosso
filhinho amado, nosso reizinho, o líder dos gatinhos.

Por fim, agradeço a mim mesmo. Deixo essa nota pessoal para relembrar-me que
no final das contas, o carro-chefe de mudanças e progresso é você mesmo. É
você quem colhe os frutos que planta, e você é quem deveria ser seu maior
motivador e quem mais acredita no seu potencial. Eu consegui! Sou doutor!

> “When you feel yourself hitting up against your limit, remember for what
> cause you clench your fists… remember why you started down this path, and
> let that memory carry you beyond your limit.”
>
> — All Might, in My Hero Academia – Kōhei Horikoshi

We are currently in a golden era of observing and characterizing Young Stellar
Objects (YSOs), protoplanetary disks, and substellar objects. This period
provides essential data that helps unveil the most complex aspects of the
formation and evolution of these celestial bodies. In this thesis, we explore
two binary systems at different stages of formation.

Firstly, we investigate an eruptive YSO system with ALMA band 6 (1.3 mm)
observations, revealing its binarity and dust/gas properties. HBC 494, a FUor
object in the Orion Molecular Cloud, displays disk asymmetry at
0.2${}^{\prime\prime}\thinspace$resolution, resolved into two components, HBC
494 N (primary) and HBC 494 S (secondary), at
$\sim$0.03${}^{\prime\prime}\thinspace$. The disks, quasi-aligned and with
similar inclinations give hints about its formation scenario. The system
exhibits a projected separation of $\sim$0.18${}^{\prime\prime}\thinspace$(75
au). HBC 494 N is $\sim$5 times brighter and $\sim$2 times larger than HBC 494
S. We notice that the northern component has a similar mass to the FUors,
while the southern has to EXors. The HBC 494 disks show individual sizes that
are smaller than single eruptive YSOs. The 12CO, 13CO, and C18O molecular line
observations reveal bipolar outflows and rotating, infalling envelopes. Cavity
features within the continuum disks’ area suggest continuum over-subtraction
or slow-moving jets and chemical destruction along the line-of-sight.

Secondly, we explored a young binary system with the advent of high-contrast
imaging observations performed with the instrument VLT/SPHERE in the H band.
We performed astrometric, photometric, and orbital analysis. $\eta$ Tel, a 18
Myr system, comprises a 2.09 M⊙ A-type star and a M7/M8 brown dwarf companion,
$\eta$ Tel B, separated by $\sim$4.2${}^{\prime\prime}\thinspace$(208 au). New
SPHERE/IRDIS coronagraphic observations with a contrast of 1.0$\times 10^{-5}$
at the location of the companion are presented in this work. Alongside with
previous literature astrometric measurements, a $\eta$ Tel B astrometric
follow-up of over a 19-year baseline is provided. The astrometric positions of
the companion are calculated with an uncertainty of 4 milliarcseconds (mas) in
separation and 0.2 degrees in position angle. The companion’s contrast is 6.8
magnitudes in the H band, with a separation of
4.218${}^{\prime\prime}\thinspace$and a position angle of 167.3 degrees.
Orbital analysis reveals a low eccentric orbit (e $\sim$ 0.34) with an
inclination of 81.9 degrees (nearly edge-on) and a semi-major axis of 218 au.
The mass of $\eta$ Tel B is determined to be 48 MJup, consistent with previous
literature. No significant residual indicative of a satellite or disk
surrounding the companion is detected, with limits ruling out massive objects
around $\eta$ Tel B at separations down to 33 au with masses as low as
1.6MJup.

The two studies presented in this thesis establish a couple of different
analyses and methodologies, ranging from sub-mm to near-IR wavelength
observations. I hope it can help elucidate how diverse, complex, and
intriguing the field of (sub)stellar formation/evolution can be.

###### Contents

  1.   2. Approval report
  3. 1 Introduction
    1. 1.1 Protoplanetary disks - theory and observations
      1. 1.1.1 YSOs classification and evolutionary stages
      2. 1.1.2 Eruptive stars - a “special” case
      3. 1.1.3 Protoplanetary disks - properties, structure, and principal physical mechanisms
    2. 1.2 Substellar objects: formation and detection via HCI
      1. 1.2.1 The main scenarios for substellar objects formation
      2. 1.2.2 (Sub)stellar multiplicity formation
      3. 1.2.3 High-Contrast Imaging: instrumentation, observing strategies and data-processing
      4. 1.2.4 High-Contrast Imaging: observing/characterizing substellar companions
    3. 1.3 Outline of this thesis
  4. 2 ALMA resolves the binary HBC 494
    1. 2.1 Introduction
    2. 2.2 Observations and data reduction
      1. 2.2.1 Continuum imaging
      2. 2.2.2 Line imaging
    3. 2.3 Results
      1. 2.3.1 Continuum analysis
      2. 2.3.2 Line analysis
    4. 2.4 Discussion
      1. 2.4.1 Multiplicity and triggering mechanisms
      2. 2.4.2 Disk sizes and masses
      3. 2.4.3 Binary formation and alignment
    5. 2.5 Conclusions
    6. 2.6 Appendix
      1. 2.6.1 Effects of continuum over-subtraction
      2. 2.6.2 12CO, 13CO and C18O channel maps with contours and the effect of cloud contamination
      3. 2.6.3 Extra channel maps
  5. 3 Characterization of the $\eta$ Tel system
    1. 3.1 Introduction
    2. 3.2 Observations and data reduction
      1. 3.2.1 Observations
      2. 3.2.2 Data reduction
    3. 3.3 Methodology and results
      1. 3.3.1 Astrometry and photometry
      2. 3.3.2 Orbital fitting analysis
    4. 3.4 Discussion
      1. 3.4.1 Orbital fitting constraints
      2. 3.4.2 Possible formation scenarios
      3. 3.4.3 The close surroundings of $\eta$ Tel B
    5. 3.5 Conclusions
    6. 3.6 Appendix
      1. 3.6.1 Cornerplot of the orbital fitting
  6. 4 Final remarks and conclusions
    1. 4.1 Specific questions/concepts addressed
      1. 4.1.1 What is the correlation between stellar multiplicity and accretion processes, and why are binary eruptive systems so infrequently observed?
      2. 4.1.2 How robust astrometric analysis of a system contribute to its characterization
      3. 4.1.3 How can one deduct the formation scenario for multiple systems?
      4. 4.1.4 Is it possible to search for satellites orbiting substellar companions via HCI?

###### List of Figures

  1. 1 Poster secretly made by UDP friends to advertise others regarding my defense.
  2. 1.1 Schematic representation of the stages of star/planetary formation
  3. 1.2 Four main mechanisms proposed to explain transition disks: grain growth, photoevaporation, clearing by a companion and dead zones
  4. 1.3 Classification of YSOs according to SED classification
  5. 1.4 Artistic representation of a YSO having an outbursting accretion and rise in luminosity
  6. 1.5 YSOs variabilities in magnitudes due to bursts and outbursts and their timescales
  7. 1.6 Schematic representation of regions of an accreting disk
  8. 1.7 Schematic representation of the structure of protoplanetary disks and how they can be observed depending on the wavelength
  9. 1.8 Gas surface density profiles for edge-on and large cavity transition disks, with higher CO-based gas-to-dust ratio
  10. 1.9 Substructures observed in PPDs
  11. 1.10 Proposed scenario of substructures evolution driven by giant planets formation
  12. 1.11 Representation of different geometries and structures observed in scattered light and sub-mm continuum
  13. 1.12 Illustration of the spectral index as a function of radius in a protoplanetary disk
  14. 1.13 Schematic representation of the outcomes of dust collisions in PPDs
  15. 1.14 Schematic representation of the grain growth towards planetary formation
  16. 1.15 Luminosity-time evolutionary tracks computed using the Marley et al., 2007b  methodology
  17. 1.16 Mechanisms for multiple (sub)stellar formation
  18. 1.17 Evolution of effective temperature for substellar and low mass stars
  19. 1.18 Exoplanets population and their respective discovery techniques
  20. 1.19 Schematic representation of how a contrast curve is interpreted
  21. 1.20 Diffraction-limited images of two point objects formed by a lens.
  22. 1.21 Rayleigh’s criterion for just-resolvable diffraction patterns.
  23. 1.22 Simplified AO configuration and its impact in observations
  24. 1.23 Schematic representation of how a classical Lyot coronagraphic works
  25. 1.24 Representative examples of possible coronagraphic designs with Lyot stops
  26. 1.25 Illustration outlining the key stages of ADI observing strategy
  27. 1.26 Images of the system around GJ 758
  28. 1.27 Schematic representation of SDI observing strategy coupled with PCA post-processing
  29. 1.28 Schematic representation of RDI sequence
  30. 1.29 Typical differential imaging pipeline for speckle/PSF subtraction observing strategies coupled with post-processing techniques
  31. 1.30 Diagram illustrating the ADI observing strategy with three distinct post-processing techniques for building the speckle model
  32. 1.31 Representation of the conventional establishment of the position angle (PA), along with the parameter separation ($\rho$) for companion astrometry
  33. 1.32 Representation of the six Keplerian orbital elements to uniquely describe the orbit of a companion
  34. 1.33 Demonstration of key properties inferred from direct imaging observations
  35. 1.34 Spectra of ultracool dwarfs from M9 to T7.5
  36. 1.35 Exoplanets discovered by HCI up to 25-02-2024
  37. 1.36 Mosaic of remarkable companions and protoplanet candidates detected via HCI
  38. 2.1 Small-scale moment maps for the analyzed CO isotopologues, 12CO (top), 13CO (middle) and C18O (bottom), without subtraction of the continuum
  39. 2.2 ALMA observations at 1.3 mm of the HBC 494 system
  40. 2.3 Radial brightness profile of HBC 494 N and HBC 494 S
  41. 2.4 12CO channel maps of the HBC 494 system
  42. 2.5 Large-scale 12CO moment maps split in each one of the bipolar outflow structures
  43. 2.6 Large-scale moment maps for the analyzed CO isotopologues and spectral fluxes
  44. 2.7 Small-scale moment maps for the analyzed CO isotopologues, after subtraction of the continuum, and their spectral profiles
  45. 2.8 Dust disk masses as a function of characteristic radii (FWHM/2)
  46. 2.9 12CO moment 0 maps before and after continuum removal, along with their spatial profiles
  47. 2.10 12CO channel maps of HBC 494 system with contours
  48. 2.11 13CO channel maps of HBC 494 system with contours
  49. 2.12 C18O channel maps of HBC 494 system with contours
  50. 2.13 13CO channel maps of HBC 494 system
  51. 2.14 C18O channel maps of HBC 494 system
  52. 2.15 12CO channel maps of HBC 494 system, small-scale, after removing the continuum contribution
  53. 2.16 13CO channel maps of HBC 494 system, small-scale, after removing the continuum contribution
  54. 2.17 C18O channel maps of HBC 494 system, small-scale, after removing the continuum contribution
  55. 3.1 ADI-processed from IRDIS coronagraphic sequence (epoch: 2015-05-05; filter: H2) for the $\eta$ Tel system. The red circle marks the position of $\eta$ Tel B. The star is masked under the coronagraph at the center of the image. North is up, East is left.
  56. 3.2 Contrast flux variations for the companion with respect to the star (filter: BB_H).
  57. 3.3 Contrast curve with respect to $\eta$ Tel A, using the datacube corresponding to the 2016-06-15 (BB_H) observation. The vertical blue line corresponds to the position of the companion.
  58. 3.4 Speckle noise estimation for $\eta$ Tel B in the data set of 2015-05-05. The histograms illustrate the offsets between the true position and flux of a fake companion and its position and flux obtained from the NEGFC technique. The dashed lines correspond to the 1D Gaussian fit from which we determined the speckle noise.
  59. 3.5 Relative astrometry for $\eta$ Tel B. The red star marks the position of $\eta$ Tel A. The blue-filled circles represent astrometric points extracted from the literature. The green-filled diamonds represent the astrometric points from this work. The astrometric point from epoch 2000.378 was not included due to its high uncertainties.
  60. 3.6 87 randomly selected possible orbits calculated by the Orvara fitting, after excluding orbits based on constraints described in Section 4.1. The black star indicates the position of $\eta$ Tel A. The best-fit orbit is shown in black.
  61. 3.7 Analysis of the close surroundings of $\eta$ Tel B to look for satellites and/or CPDs. In each row, the first panel shows the ADI image zoomed on $\eta$ Tel B, the second panel shows the model, and the last one the residuals from the subtraction of the two. The three IRDIS epochs are shown. The residuals are expressed in counts. All the panels were normalized to the peak of the central PSF.
  62. 3.8 Schematic representation of the placement of putative fake satellites around $\eta$ Tel B to calculate the contrast curves. In this illustration, just one annulus is represented.
  63. 3.9  Contrast curves with respect to the star $\eta$ Tel A, centered on the companion as depicted in Fig. 3.8. The three contrast curves, corresponding to different epochs, are showcased. The vertical dashed line indicates the Hill radius of $\eta$ Tel B under the assumption of an orbit with an eccentricity of $e$=0.34
  64. 3.10 Corner plot showing the results of the orbital fitting performed by Orvara, after excluding orbits based on constraints described in section 3.4.1. The best-fit values are also reported in Table 3.7.

###### List of Tables

  1. 2.1 List of the Cycle 4 ALMA observations of the target HBC 494, used in this work
  2. 2.2 ALMA imfit analysis results and physical parameters inferred from the HBC 494 continuum disks.
  3. 3.1 List of the SPHERE/IRDIS $\eta$ Telescopii observations used in this work.
  4. 3.2 IRDIS flux contrast in flux and magnitude with respect to the central star for $\eta$ Tel B.
  5. 3.3 List of astrometric positions of $\eta$ Tel B obtained from the IRDIS observations. The error budget is listed in Table 3.4.
  6. 3.4 Uncertainties used to calculate the error budget for the astrometry of the companion. The values correspond to errors associated with individual frames.
  7. 3.5 Astrometric positions of $\eta$ Tel B from the literature and our analysis included in the Orvara analysis.
  8. 3.6 Priors and initial and distribution of parameters for $\eta$ Tel system astrometrical fittinga.
  9. 3.7 Best orbital-fitting parameters for $\eta$ Tel B calculated from the Orvara orbital characterization.

##  CHAPTER 1 Introduction

Look up, and then look further. Astronomers are adventurers seeking to
discover and unveil the secrets that extend beyond our clouds. Embedded by
mighty minds and advanced tools, we are equipped with knowledge meant to be
shared.

In this thesis, I am pleased to elaborate on the work conducted during my
doctoral stage. My goal is to offer valuable perspectives on stellar and
planetary formation by studying a binary eruptive system. I also aim to
provide an exhaustive analysis of a young stellar system composed of a giant
star and a brown dwarf, on its first Myrs of life. In this chapter, I
introduce the scientific background required to provide a comprehensive basis
for the subsequent analysis of our research findings.

###  1.1 Protoplanetary disks - theory and observations

A protoplanetary disk is a rotating circumstellar disk composed of gas and
dust. This structure is a direct outcome of angular momentum conservation from
a rotating chunk of a molecular cloud, which undergoes flattening as part of
its evolutionary process. The protoplanetary disk is formed by the leftover
materials that were not accreted into the forming protostar at the center or
were evaporated/expelled through other physical mechanisms.

The investigation of protoplanetary disks plays a fundamental role in our
comprehension of planetary formation. In an era where observations offer
comprehensive insights into the structure and physical mechanisms of disks at
various developmental stages, the theoretical aspects of planetary formation
are continually refining, yielding physically more accurate models over time.
Nevertheless, many questions remain unanswered such as: How do physical and
chemical mechanisms vary across different environments and stellar masses? To
what extent is chemical complexity necessary for the formation of mature
planetary systems? At what frequency rate do protostars accrete and disks
expel gas and dust? How and when do planets form, and how do they interact
with disks? What processes contribute to the formation of a system like the
Solar System, and how common is such an occurrence? These and other questions
underscore the complexity of our understanding of this field.

####  1.1.1 YSOs classification and evolutionary stages

Young stellar objects (YSOs) can be denoted as objects composed of a protostar
and its surroundings, mainly the protoplanetary disk and its substructures. A
YSO delimits a phase between the formation of a rotating quasi-spherical
molecular core around a forming protostar and a main-sequence star and its
system.

The connections between stellar/planetary theory and observations allow us to
trace how a YSO evolves. According to the most recent compilation/description
of protoplanetary disk formation and evolution (Pineda et al.,, 2023), it can
be summarized as follows:

According to observations, bubbles of both neutral and ionized gas are
ubiquitously observed in the interstellar medium across the Galaxy. Also,
according to models (e.g., Padoan et al.,, 2017; Kim & Ostriker,, 2018;
Rathjen et al.,, 2021), bubbles can be formed by supernova explosions,
radiation, and winds from massive stars. Therefore, first, on a scale of 1-100
pc, neutral and ionized hydrogen gas in the shape of quasi-spherical bubbles
will expand. The shock between two or more bubbles will generate layers of
magnetized molecular gas ($\sim$100 cm-3), which can be described as filaments
(scales up to tenths of pc). The filaments will continuously accrete gas and
dust and compress it media transverse velocity gradients arising from large-
scale inflow, rotation, shearing motions, or a combination of these types of
motions. These filaments may further break down into filamentary
substructures, called fibers, which can also co-locate with dense cores. These
cores will collapse if their local mass per unit length ($M_{line}$) overcomes
the critical $M_{line/crit}$, equals to $2c^{2}_{s}/G$, where $c_{s}$ it is
the sound speed at the environment, and $G$ the gravitational constant. If the
core starts to collapse, it will trigger gas and dust infalling towards its
center. This process will form a hydrostatic core. Consequently, this core,
surrounded by a gas and dust thick quasi-spherical shell, will get more dense
and hot, thus forming a protostar. Then, due to the conservation of angular
momentum, the rotating shell will turn into a disk, becoming flattened and
flared. This stage is characterized by episodic accretion and magnetically
driven bipolar outflows that are launched from the inner disk. After, the
envelope will start to be cleared where the central star, now called pre-main
sequence star (PMS). Finally, the primordial disk (protoplanetary disk) will
dissipate due to accretion, winds, outbursts, and photoevaporation. However, a
debris disk, where second-generation dust will continuously be replenished by
the collision of planetesimals, will remain. The evolutionary path here
presented usually involves the formation of multiple planets and minor bodies
(see figure 1.1). But when and how these planets form still is not well
understood.

Figure 1.1: Schematic representation of the stages of star/planetary
formation, since the formation of filaments through the shock of expanding
bubbles mature star/planetary systems. Credits: Pineda et al., (2023).

The presence of circumstellar material was first inferred through infrared
excess observations due to the thermal emission of warm dust and through
Spectral Energy Distribution analysis (SEDs; Mendoza V.,, 1966; Cohen & Kuhi,,
1979; Kenyon & Hartmann,, 1987; Strom et al.,, 1989). This happened before
radio observations assumed a fundamental role. However, even in the current
landscape, where interferometry radio data is the primary means for observing
disks, allowing for the resolution of substructures, determination of
geometries, and identification of multi-components, as well as exploring the
involved physical and chemical mechanisms, there remains a common practice of
classifying evolutionary stages. This approach is reminiscent of the earlier
SED analysis era, measuring the slope between about 2 and 25 $\mathrm{\mu m}$
and first formalized by Lada, (1987) and Greene et al., (1994):

$\alpha_{\mathrm{IR}}=\frac{d\log vF_{v}}{d\log v}=\frac{d\log\lambda
F_{\lambda}}{d\log\lambda}$ (1.1)

These stages include111It is important to state that this classification can
lead to evolutionary misinterpretation depending on the disk’s
inclination/dust extinction. For example, a class II YSO at high inclination
has a similar SED as a class I YSO (Robitaille et al.,, 2006; Williams &
Cieza,, 2011):

  * •

Class 0: These sources are fully embedded by the envelope throughout this
short stage (t $<$ 104 yrs), and they show almost no emission in optical or
near-IR wavelengths. Their SEDs are similar to a cold black body and their
emission can be detected at longer wavelengths (far-IR to (sub)-mm).

  * •

Class I: In this stage, it starts the clearing of the envelope. The SEDs will
be constituted by the black body of the protostar, radiation scattered into
the line of sight from dust grains in the disk, emission from the collapsing
outer envelope, and thermal emission from a circumstellar disk. The infrared
excess slope is positive ($\alpha_{\mathrm{IR}}$ $>$ 0.3) in this stage and
the SED peaks in the mid to far infrared. The presence of the cold envelope
absorbing emission from the dust, gas, and protostar, will produce absorption
lines. In this stage, jets and bipolar outflows, playing a fundamental role in
clearing the remaining envelope, can be observed. The age of these objects is
between 104 and 105 years.

  * •

Flat-spectrum: This stage is an intermediate between Class I and II and has
slopes between -0.3 and 0.3. This stage shows less envelope obscuration and
has a greater contribution to the accretion into the SED.

  * •

Class II: In this phase, the disks exhibit a slope of -1.6 $<$
$\alpha_{\mathrm{IR}}$ $<$ 0.3. The dissipation of the envelope is a defining
characteristic, leading to an SED dominated by the stellar photosphere and the
circumstellar accretion disk. The initiation of clearing in the accreting
circumstellar disk surrounding the now-called pre-main sequence star (PMS)
shifts the SED towards the near-IR to the optical range. The SED, wider than a
blackbody, reflects the disk’s contribution. Concurrently, the YSO’s activity
begins to diminish, marked by fewer outbursts, jets, and accretion events.
These systems typically have an age range of approximately 106 to 107 years
and are commonly referred to as classical T Tauri stars (YSOs leading to a
lower mass stellar evolution, up to 2 M⊙), or Herbig Ae/Be stars (YSOS with
spectral type earlier than F0). Moreover, classical T Tauri stars are also
known to have strong H$\alpha$ and UV emission, while Herbig Ae/Be stars do
not show a so prominent UV excess.

  * •

Transition disks: The disks in this stage are known to have large inner gaps
due to pressure bumps and/or due to planet formation. They also represent the
last stages of inner disk clearing due to accretion onto the PMS. Transition
disks are known as having inner cavities between the PMS+inner disk and an
outer disk. The process of halting the dust and causing the cavities can be
many (see van der Marel, (2023) and Fig. 1.2 for further references).

  * •

Class III: In this phase, the disk exhibits near-complete passivity with
either no or very weak accretion, and the sources demonstrate negative slopes
($\alpha_{\mathrm{IR}}$ $<$ -1.6). The spectral energy distribution (SED)
closely resembles the blackbody spectrum of the pre-main sequence (PMS) or
main-sequence central star. During this stage, the gas content diminishes,
leading to the gradual disappearance of the circumstellar accretion disk.
These disks host newborn planets, comets, asteroids, and residual pebbles,
earning them the designation of debris disks. When the central star remains in
the pre-main sequence phase, it is termed a Weak-lined T Tauri, indicating a
minimal or absent accretion presence.

Figure 1.2: Four main mechanisms proposed to explain transition disks: grain
growth, photoevaporation, clearing by a companion and dead zones. Credits: van
der Marel, (2023)

A representation of Class 0, I, II, and III YSOs is presented in Fig. 1.3.

Figure 1.3: Classification of YSOs according to SED classification. The stages
represent the standard low-mass star formation. Credits: Adapted from Macías
Quevedo, (2016).

####  1.1.2 Eruptive stars - a “special” case

Young Stellar Objects (YSOs) have photometric and spectroscopic variability
across multiple evolutionary stages. Although understanding the precise
mechanisms driving these variations can be challenging in certain cases, it is
established that some fluctuations arise from changes in accretion rates.

When a protostar or pre-main sequence (PMS) star experiences a large enhanced
phase of accretion, that may last for months to decades, it can be called an
eruptive star. During this stage, the accretion disk will transit from a
magnetospheric accretion, typical from Class 0-2 YSOs, towards an outbursting
accretion. Roughly, the luminosity of the YSO can be expressed as:

$L_{\text{proto }}=L_{\text{photospheric }}+L_{\text{accretion
}}=L_{\text{phot }}+GM_{*}\dot{M}_{\mathrm{acc}}/R_{*},$ (1.2)

where $M_{*}$ and $R_{*}t$ are the mass and radius of the star, thus stating
that an enhancement of accretion will raise the total luminosity (see Fig.
1.4). This process will also trigger more violent outbursts and jets.

Figure 1.4: Artistic representation of a YSO having an outbursting accretion
and rise in luminosity, becoming an eruptive star. Credit: T. Pyle
(Caltech/IPAC).

Changes in the accretion rate can be measured through different physical
components and mechanisms of a YSO. If low-extinct and with a favorable
geometry, a direct diagnosis from the heated gas can be done through the UV
continuum and several emission lines in UV and optical wavelengths. In the
case of more embedded sources, the analysis also can be done up to near-IR. In
other cases, a reprocessed emission that occurs at infrared or millimeter
wavelengths needs to be used.

For classical T Tauri stars, the accreting gas is conducted by stellar
magnetic fields from the disk to the star. The accretion shock heats the
photosphere, mostly with X-rays. The heated photosphere and the accretion flow
will produce hydrogen recombination, $H^{-}$ continuum, and line emission,
processes observed in optical and UV wavelengths.

For passively heated dust disks (low-accretion, the heating of the dust is
essentially from the starlight), the innermost dust emits in the near and mid-
infrared. The outer disk, however, dominates long-wavelength emission.

For rapidly accreting disks, viscous processes heat efficiently the disk to
higher temperatures, which can surpass the stellar one. Therefore, the
luminosity of the YSO is dominated by the accretion, which can be measured at
optical and infrared wavelengths. If a typical low-state accretion disk
temporarily turns to be a high-state viscously heated disk, as expected for
eruptive disks, a large increase in brightness will be detectable at optical
and infrared wavelengths.

Historically, since the review from Audard et al., (2014), eruptive stars are
mainly classified into two groups, the EX-Lupi-like stars or EXors and the FU-
Ori-like stars or FUors. Nowadays, however, due to several eruptive stars
having different spectroscopic signatures, or having peak luminosities and
different light curves than the regular FUOrs and EXOrs, this classification
is being challenged (for an updated review, see Fischer et al., 2023). For the
sake of simplicity, I will proceed to explain these two categories.

EXors are YSOs that present 2.5-5 magnitude optical outbursts that last for
months to years and may recur after a few years. These objects were inspired
by Ex Lupi, a T-Tauri star in the constellation of Lupus. During the periods
of quiescence, Ex-Lupi, and other EXors can’t be distinguished from a regular
T-Tauri in optical and near-IR wavelengths, and spectroscopically in
absorption lines of Na, Ca, K, Fe, Ti, Si, and CO. However, in outbursts, most
absorption lines turn into emission and many emission lines become stronger.
Also, the emission line fluxes concerning accretion, as Paschen $\beta$ and
Brackett $\gamma$, show accretion rates on the order of 10-7 to 10-6 M⊙ yr -1.

FUors, from the precursor FU Orionis, are a type of Young Stellar Objects
(YSOs) distinguished by more intense and prolonged outbursts compared to EXor
variables. During an outburst, the bolometric luminosity of a FUor undergoes a
substantial increase, reaching tens to hundreds of solar luminosities. Their
accretion rates can vary from 10-6 to 10-4 M⊙ yr-1. Spectroscopically, FUors
in outburst exhibit features typical of a viscously heated accretion disk
overshadowing the central protostar across all wavelengths. However, their
behavior differs at various wavelengths—longer wavelengths reveal colder and
slower-rotating regions of the disk. In the optical, they display absorption
spectra resembling F or G supergiants, featuring broad lines indicating strong
winds, a P-Cygni profile, and notable Li absorption (Hartmann & Kenyon,,
1996). In the infrared, their characteristics are more similar to M or K
giants and supergiants, showing strong CO absorption, water, TiO, or VO
absorption bands, and prominent He I absorption (Connelley & Reipurth,, 2018).

In an early investigation of protostars within the Taurus-Auriga region,
Kenyon et al., (1990) introduced the ”luminosity problem.” This concept
pointed out a statistical anomaly, indicating that protostars displayed lower
luminosity than anticipated, considering a steady accretion scenario and the
expected duration for star formation. At the time, two explanations were
proposed: protostellar lifetimes last longer than $10^{5}$ yrs proposed by the
authors, or episodic bursts of accretion, akin to EXors and/or FUors,
accelerating the process. Later, it was inferred from mid-infrared surveys
that protostellar lifetimes are $\sim$5 times longer than expected (Evans et
al.,, 2009; Dunham et al., 2014a, ; Dunham et al.,, 2015), reducing the
luminosity necessary to be observed in YSOs in case of steady accretion.
Although some authors like Offner & McKee, (2011), consider the “problem”
solved, discrepancies in luminosity persist, ranging from 2 to 10 times lower
than expected from a steady accretion scenario. Following, Fischer et al.,
(2017) have shown that protostellar luminosities have different accretion
rates in different evolutionary stages, lowering the rate between Class 0 and
I YSOs. This reinforced that a variable scenario, not necessarily episodic or
with regular frequency in nature, can match observed protostellar
luminosities. Therefore, even though episodic accretion was detected in
several sources, it is hard to establish they occur in all YSOs and during all
of their evolution. A contemporary understanding of the “luminosity problem”
diverges from its initial meaning, now more fittingly used to emphasize the
wide range of protostellar luminosities observed in different sources,
environments, and stages of evolution. To better represent the variabilities
in magnitudes and timescales where a YSO can undergo, see Fig. 1.5.

Figure 1.5: YSOs variabilities in magnitudes due to bursts and outbursts and
their timescales. I call attention to the already mentioned EX Lup and FU Ori
outbursts, being V1647 Ori one intermediate case between both. Credits:
Fischer et al., (2023).

Up to now, a few tens of eruptive stars have been detected and
spectroscopically characterized. Considering that variability in accretion is
fundamental to connecting models to observations and inferring important
physical parameters such as lifetime and efficiency of mass bulking towards
the star, a lot of questions remain unanswered. One of the most crucial ones
refers to triggering mechanisms. What drives FUors/EXors accretion variability
in a YSO? And how do they stop? In literature, the triggering processes are
usually summarized in 3: 1) disk fragmentation (Vorobyov & Basu,, 2005, 2015;
Zhu et al.,, 2012); 2) magneto-rotational instability (MRI) (Armitage et al.,,
2001; Zhu et al.,, 2009); and 3) tidal interactions between a disk and a
companion (Bonnell & Bastien,, 1992; Lodato & Clarke,, 2004; Borchert et al.,
2022b, ).

The mechanism of disk fragmentation is based on gravitational instabilities
usually producing clumps, which will be rapidly inward driven towards the
inner disk, and possibly tidally disrupted replenishing material in the
viscously heated part. Disk fragmentation can also induce magneto-rotational
instabilities (MRI) in the inner disk, another important triggering process.

Magneto Rotational Instability (MRI) is an efficient triggering mechanism. As
YSOs undergo radial rotation at different angular velocities, the embedded
magnetic fields become twisted and distorted, thereby initiating the MRI. This
instability induces turbulence in the disk, enabling the efficient inward
transport of gas and dust. Consequently, the MRI redistributes angular
momentum, preventing its excessive accumulation in specific regions. However,
a certain level of ionization is necessary to couple the material to the
magnetic field, making it susceptible to MRI. The dynamically cold regions in
a disk, thus, stall the inward transport, accumulating self-gravitating
material in the outer disk (see Fig. 1.6). All of the potential solutions to
break this flow discontinuity point towards episodic mass accretion events,
generated by heat-inducing of the outer disk to the dynamically cold region,
re-engaging the transport of material to the inner disk.

Figure 1.6: Schematic representation of regions of an accreting disk, showing
the inner disk coupled to MRI; a quiescent layer, with a temperature lower
than the required to couple material to MRI, thus stalling inward transport;
and the outer disk which has its material transported by self-gravity.
Credits: Armitage et al., (2001).

The tidal disruption due to a companion is a mechanism, not as efficient as
MRI since depends on close encounters with external companions, which may be
or not periodic and may be less common to happen. The gravitational
disturbance, thus, can form spiral arms, density waves, or other features,
redistributing the material in the disk, or even causing disk fragmentation,
leading to another mechanism already mentioned and its consequent episodic
accretion.

All these mechanisms are not constant and, thus, could explain the return to
quiescence stages.

####  1.1.3 Protoplanetary disks - properties, structure, and principal
physical mechanisms

Here, I characterize the protoplanetary disk (PPD) as the circumstellar disk,
excluding consideration of the envelope. A PPD primarily consists of gas and
dust inherited from the natal envelope, with molecular hydrogen (H2) and
helium (He) dominating the gas composition, alongside (sub)micron-sized dust
particles. The gas constitutes approximately 99% of the PPD’s total mass
(Ollivier et al.,, 2009). Typically, the temperature distribution exhibits
higher temperatures in the innermost regions (closer to the star), gradually
decreasing radially towards the outermost regions. However, the disk’s
vertical structure also significantly influences temperature distribution and
the distribution of grain sizes. Additionally, different regions within the
disk exhibit variations in the movement velocity of both dust and gas, guided
by distinct physical mechanisms. Subsequently, I provide a more detailed
exploration of the principal parameters and physical mechanisms inherent in a
PPD. A schematic representation of the different disk components probed by
different wavelength ranges and instruments can be seen in Fig. 1.7, both
panels. Specifically, the temperature distribution is depicted at the top
panel, while the bottom panel shows different grain evolution processes.

Figure 1.7: Schematic representation of the structure of protoplanetary disks
and how they can be observed depending on the wavelength. Top panel: dust
temperature profile and gas density structure. On the left, the grain sizes
are represented according to the size of black-filled circles, when blue
contours representing icy water, and white contours CO-coat. On the right,
different gas emission regions of the main simple molecules are shown, and
dust thermal and scattered emissions are highlighted in purple and yellow,
respectively. Credits: Miotello et al., (2023). Bottom panel: dust evolution
main transport and collision mechanisms are represented on the left. On the
right side, areas of the disk that can be observed by different wavelengths
and current/future instruments to detect them are shown. Credits: Testi et
al., (2014).

##### Disk mass

Knowing the total content of dust in a PPD is crucial to determine accretion
rates, lifetime, and which substructures and objects can be formed within.
Additionally, the grain size distribution is important to determine which
grain size can be observed in different wavelengths and also helps to
disentangle problems in planetary formation models regarding where, when, and
how rocky planets/cores of giant planets are formed. Therefore, it is crucial
to determine masses and surface densities, which describe how the mass is
distributed per unit area on the observed surface (this distribution can vary
with depth depending on the wavelength used for observation).

Disk masses are typically derived from observations of thermally heated dust
grains. This involves measuring the dust mass and, based on an assumed dust-
to-mass ratio, inferring the total mass of the disk. A frequently adopted
assumption is a gas-to-dust ratio of 100, assuming a fixed Interstellar Medium
(ISM) (Andrews & Williams,, 2005, 2007). The continuum emission is optically
thin, except for the innermost regions of the disks (approximately less than
10 astronomical units, at 1 mm observations), where the column density becomes
too high. The determination of the disk mass from the optically thin
reprocessed emission involves employing the following equation:

$M(\text{gas}+\text{dust})=\frac{F_{\nu}d^{2}}{\kappa_{\nu}B_{\nu}(T)},$ (1.3)

where $F_{\nu}$ is the measured flux at frequency $\nu$, d is the distance to
the source, $\kappa_{\nu}$ is the dust opacity, and $B_{\nu}(T)$ is the Planck
function. At (sub)mm wavelengths, the Planck function approximates to the
Rayleigh-Jeans limit, and the following approximation can be done:

$B_{\nu}(T)=\frac{2h\nu^{3}}{c^{2}}\frac{1}{e^{\frac{h\nu}{k_{\mathrm{B}}T}}-1}\approx\frac{2h\nu^{3}}{c^{2}}\cdot\frac{k_{\mathrm{B}}T}{h\nu}=\frac{2\nu^{2}k_{\mathrm{B}}T}{c^{2}},$
(1.4)

where h in Planck’s constant, c is the speed of light and $k_{\mathrm{B}}$ is
the Boltzmann constant. Therefore, observing at these wavelengths provides a
dust temperature profile that linearly varies with radius, contrasting with an
exponential trend.

Although equation 1.3 is countless times used in literature, it is highly
dependent on the determination of average temperature and dust opacity. A good
and classical approximation for both is a temperature of 20 K, consistent with
IR-millimeter modeled SEDs, CO observations, and theoretical expectations
(Andrews & Williams,, 2005; Qi et al.,, 2004; Chiang & Goldreich,, 1997 and
references therein) and a dust opacity for mm wavelengths from Beckwith et
al., (1990), expressed as:

$\kappa_{\nu}=0.1\left(\frac{\nu}{10^{12}\mathrm{\leavevmode\nobreak\
Hz}}\right)^{\beta}\mathrm{cm}^{2}\mathrm{\leavevmode\nobreak\ g}^{-1},$ (1.5)

where the power-law index, $\beta$ is related to the size distribution and
composition of the dust grains (Ossenkopf & Henning,, 1994; Pollack et al.,,
1994). It is commonly assumed a $\beta$ is equal to 1 for 1.3 mm observations.
In addition, the assumptions do not provide good estimations for mass
accountability of larger grains, which remains a source of great uncertainty,
offering the idea that mass estimations through equation 1.3 may be
underestimated. A more complicated scenario is driven by the more recent
studies, stating that not only the spectral index $\beta$ can vary a lot
depending on the properties of the grains, such as how porous it is but also
estimates big mass content in disks arising from optical thick regions.
Additional analyses and methodologies have been explored and suggested for
estimating disk masses based on gas content, primarily utilizing CO
isotopologues and HD emission. This approach is taken because H2 lacks an
electric dipole moment as a symmetric molecule, resulting in rare transitions
capable of emitting radiation (for a review about mass measurements from gas,
see e.g. Bergin & Williams,, 2017; Miotello et al.,, 2023).

##### Disks structure and temperature distribution

As aforementioned, how disk material and temperature are distributed in a PPD,
is of major importance. During the lifetime of a disk, those distributions can
highly vary, as part of the disk will be accreted onto the protostar/PMS,
dissipated, and/or radially and vertically. In particular, which process
dominates will impact the surface density distribution, $\Sigma(R)$, and the
disk outer radius, $R_{out}$222There is no total consensus and a robust
physical definition for $R_{out}$, besides as being the radius that encloses a
chosen percentage of the emission. However many authors recently have been
using $R_{out}$ = $R_{68\%}$.. Therefore, these two parameters are priorities
whenever PPDs are been analyzed.

The surface density of a disk can be modelled and analyzed in total or split
into gas and dust counterparts, depending on the estimations you are
accounting for. A common prescription to describe the radial distribution of
the dust and the gas in a viscous disk, and which fits interferometric
observations is given by:

$\Sigma(R)=(2-\gamma)\frac{M_{\text{disk }}}{2\pi
R_{\mathrm{c}}^{2}}\left(\frac{R}{R_{\mathrm{c}}}\right)^{-\gamma}\exp\left[\left(-\frac{R}{R_{\mathrm{c}}}\right)^{2-\gamma}\right],$
(1.6)

where $M_{disk}$ is the disk mass, $R_{\mathrm{c}}$ is the characteristic
radius, and $\gamma$ is the power-law exponent that describes the radial
dependence of the disk viscosity (Lynden-Bell & Pringle,, 1974; Hartmann et
al.,, 1998). In this equation, the characteristic radius refers to the region
where the surface density starts to deeply steepen from the power law, where
it indicates at which part of the disk is viscously dominated. More recently,
mainly due to ALMA observations, a recurring observation of continuum
substructures, such as gaps and rings, has emerged in most disks observed at
sufficient spatial resolution. Additionally, faint optically thin molecular
lines could be detected in several cases, tracing disk masses and gas
distributions. These findings indicate considerable variability in surface
density profiles, deviating from the assumptions of a simple model (see an
example in Fig. 1.8). To improve surface density models, multi-wavelength
observations of resolved disks become crucial. Such observations may not only
disentangle the degeneracy among temperature, dust surface density, and dust
opacity but also provide valuable insights for accurately modeling radial
variations in dust particle sizes and scattering opacities (Miotello et al.,,
2023). As for gas direct detections to model the surface densities, it is
necessary to find optically thin emissions, which are usually faint and rarely
show emission from the outer disk in small disks ($R_{c}<$ 100 au; Williams &
McPartland,, 2016; Miotello et al.,, 2018). Detecting them implies
observations with high sensitivity and well-known ratios of the tracer to
other more abundant tracers. Usually CO gas mass is traced using 13CO or C18O,
and adopting a ratio CO-to-H2.

Figure 1.8: Gas surface density profiles for edge-on and large cavity
transition disks, with higher CO-based gas-to-dust ratio. The dashed black
lines provide the Minimum Mass Solar Nebula (extrapolated to 100 au) and a
viscous disk profile for comparison. Credits: Miotello et al., (2023).

Outer gas disk radii, as observed in many 12CO observations, are generally
larger than the radii of (sub)mm-sized dust particles. The reason behind this
difference is currently under discussion. It could be due to optical depth
effects, where CO remains optically thick and bright in the outer disk, while
the dust becomes optically thin, and thus faint and challenging to detect.
Another possibility is the efficient radial drifting of dust, leading to a
reduction in dust radii. The ongoing debate revolves around determining
whether optical depth effects, radial drifting, or a combination of both
factors contribute to this observed difference. Moreover, according to recent
models (e.g., Toci et al.,, 2021), if radial drifting is the main process, it
is extremely efficient, halting the disk size way more quickly than expected
from observations. Thus, it is needed in such cases to invoke substructures
that locally slow down or stop radial drifting, as gaps.

Analyzing a sample of 470 YSOs observed at NIR and/or mm wavelengths, Bae et
al., (2023) offered a comprehensive statistical and morphological overview of
substructures. From this examination and prior analyses, it is accurate to
assert that Class II-to-Class III disks can exhibit three primary classes of
substructures: rings and gaps, spirals, and crescents (see Fig. 1.9).

Figure 1.9: Example of substructures observed in PPDs. Black and white
contours show millimeter continuum observations, blue contours show millimeter
line observations and red contours show near-IR observations. Credits: Bae et
al., (2023).

Rings are generally defined as circular and azimuthally symmetric. They are
the most common type of substructure detected until the moment. Gaps are the
regions of reduced surface density between denser resolved rings. Rings and
gaps are most commonly observed in mm observations rather than near-IR. The
apparent discrepancy between both wavelengths can be explained by considering
the possibility that the observed rings correspond to pressure maxima within
the protoplanetary disk. In such regions, larger particles may be efficiently
trapped, leading to their enhanced detection in mm observations. Rings and
gaps play a crucial role in the study of planetary formation, as they can be
carved by the presence of giant planets forming and opening gaps in the
protoplanetary disk. In recent studies, there has been a focus on
understanding the evolution of these substructures over time. Some of the
proposed scenarios associate these substructures with the formation of giant
planets and their consequent migration (see e.g. Cieza et al.,, 2021 and Fig.
1.10).

Figure 1.10: Proposed scenario of substructures evolution driven by giant
planets formation using massive disks from ODISEA sample. Credits: Cieza et
al., (2021).

Spirals are more commonly detected at NIR than in mm observations, mainly
because they are linked to smaller grains that are coupled to the gas. While
several mechanisms have been suggested to explain the formation of these
substructures, the only one that has been conclusive confirmed involves tidal
interactions with companions, whether gravitationally bound or not (see
discussion and references in Bae et al.,, 2023). Detecting stronger spirals is
expected to be associated with higher companion masses, making it challenging
to observe these substructures in the presence of giant planets using current
observational capabilities. Therefore, spiral observations are more commonly
associated with brown dwarfs or larger objects. Another popular scenario that
arises to explain spirals is gravitational instability (GI), where pitch
angles are inversely proportional to disks’ mass. When the disk is
sufficiently massive ($M_{disk}/M_{*}\geq 0.25$, based on numerical
simulations to explain Elias 2-27 spirals morphology), such that the Toomre
parameter $Q=c_{s}\Omega/(\pi G\Sigma)$ is $\leq$ 1, perturbations at
intermediate scales are subject to GI (Toomre,, 1964; Kratter & Lodato,,
2016). This may lead to the formation of non-axisymmetric spirals which can
efficiently transport angular momentum through gravitational torques. Finally,
infalling streamers and sound wave propagation can also be invoked as possible
causes for spirals.

Crescents, the less commonly observed substructure, are rings that have an
azimuthal variation in intensity. Only two leading mechanisms can explain
crescents: vortices, created by hydrodynamic instabilities trapping particles
at the region, and lumps created by angular differential velocities in
eccentric disks or by trapping material in Lagrangian points. Different from
the other two substructures, there is no clear evidence that companions or
planet formation are connected to crescents.

Protoplanetary disks have finite thickness and are flared mainly due to
pressure support from the gas disk balanced by the gravitational force of the
star. In a disk, at a radius $r$ from the star, and at a height of $z$ above
the mid-plane, the pressure $p$ of the gas is given as:

$\frac{dp}{p}=\frac{\Omega_{\mathrm{K}}^{2}}{v_{\mathrm{s}}^{2}}zdz$ (1.7)

where $v_{s}$ is the speed of sound, and $\Omega_{K}$ is the keplerian angular
velocity defined by
$\Omega_{\mathrm{K}}=\left(\frac{GM_{\star}}{r^{3}}\right)^{1/2}$. From
equation 1.7, I derive the scale height $h_{\mathrm{g}}$ of the disk (Kenyon &
Hartmann,, 1987):

$\frac{h_{\mathrm{g}}}{r}=\left(\frac{v_{\mathrm{s}}^{2}r}{GM_{\star}}\right)^{1/2},$
(1.8)

or simply, $h_{\mathrm{g}}=v_{\mathrm{s}}/\Omega_{K}$. If the internal
temperature of the disk, which is proportional to $v_{\mathrm{s}}^{2}$
decreases more slowly than $r^{-1}$, the height scale will increase outwards.
And the warmer the disk, the higher will be its maximum height - it will be
thicker. How the dust vertical structure is deployed, however, depends on the
dust coupling to the gas, which also depends on the thermal speed of the gas,
grain sizes, and relative velocity between gas and dust and their interaction
timescales. The mechanism responsible for this process is aerodynamic drag,
which proves to be more effective with smaller dust grains, in less dense
regions. Consequently, to capture the emissions and flared surface
characteristics of a disk through dust observations, a more effective strategy
involves observing at shorter wavelengths, specifically in the infrared range.
This approach targets the scattering emissions of smaller grains. On the other
hand, sub-mm continuum emission access larger grains decoupled from the gas,
which are settled in the mid-plane. Thus, mm-sized grains settle to a thin
layer as smaller grains provide a greater vertical distribution (see Fig.
1.11).

Figure 1.11: Representation of different geometries and structures observed in
scattered light and sub-mm continuum. Top: Collage of observations of IM Lup,
where the white dashed lines represent the disk surface layer. Credits:
Avenhaus et al., (2018); Andrews et al., (2018). Bottom: Combined observations
of the edge-on disk Tau 042021. Credits: Villenave et al., (2020).

Understanding the distribution of temperature is crucial for modeling dust
dynamics, gas flow, and chemical mixing within protoplanetary disks. In disks
maintaining a steady state (radiative equilibrium), the balance between
cooling through radiation emission and heating through absorption of starlight
(in passive disks) or a combination of starlight absorption and accretional
heating due to viscous dissipation (in active disks) is essential. Theoretical
models suggest that 2D temperature profiles for both types of disks should
exhibit vertical increases from edges to mid-plane and radial decreases from
inner to outer regions (see Fig. 1.7). In the innermost part of the disk,
stellar radiation significantly affects the gas and dust at the surface.
Smaller grains at the surface scatter, absorb, and re-emit radiation to larger
grains in the mid-plane. For dust particles, thermal and scattered continuum
emission at a wavelength of $\lambda$ are dominated by grains with sizes
similar to $\lambda/$2$\pi$. Thus, at a certain distance from the star,
temperature profiles are dominated by reprocessed emission throughout the
vertical distribution. Radially, from the region where dust is not sublimated
towards the outer regions, the inner disk experiences a mixture of stellar
irradiation and viscous accretional heating until it encounters “dead zones”
where low ionization levels inhibit MRI, locally halting viscous transport and
heating toward the star. Further from the star, self-scattering and re-
emission dominate, implying that a higher scale height at a specific distance
results in a colder mid-plane. Vertically, uppermost disk layers with
sufficient stellar irradiation exhibit higher temperatures, while intermediate
layers, shielded by the upper layers, remain warm ($>$30-40 K). The heavily
shielded disk mid-plane is cold ($<$30 K) and harbors the highest gas
densities. Despite more complex models considering additional mechanisms and
deviating from standard vertically isothermal assumptions, these common
simplifications persist due to their practical simplicity.

An essential feature in protoplanetary disks (PPDs), driven by the radial
temperature distribution in the mid-plane, is the presence of ice lines – a
key element in the context of planetary formation. Icelines mark the radial
positions where volatiles undergo sublimation and deposition. These locations
are composition-dependent, with different volatile species exhibiting ice
lines at distinct radial distances due to their varying sublimation
temperatures. For example, N2, CO, CO2, and H2O have sublimation temperatures
of $\sim$10-20, 20-30, 60-70, and 130-160 K at typical disk densities,
respectively (Zhang et al.,, 2015). Ice formation within and beyond these
regions introduces a new dynamic to dust interactions and the distribution of
volatiles. The presence of icy caps can enhance the stickiness of grains,
promoting the coagulation and lumpiness essential for the formation of giant
gas planet cores in the outer regions of disks (see, e.g., Chambers,, 2014;
Banzatti et al.,, 2015). Furthermore, the inward transport of ice or icy caps,
upon sublimation, releases vapor that can be transported outward by
turbulence. Additionally, larger grains, upon losing their icy mantles, may
fragment, causing a sudden increase in optical depth in inner regions—a
scenario proposed as one hypothesis to explain gaps. As small fragments drift
more slowly than larger ones, they may accumulate near the ice lines (see,
e.g., Saito & Sirono,, 2011; Ida & Guillot,, 2016; Hyodo et al.,, 2019 and
Fig. 1.12). However, current observations do not show a clear correlation
between rings and the expected locations of ice lines, challenging the notion
of ice lines co-locating with these substructures (Bae et al.,, 2023).

Figure 1.12: Illustration of the spectral index as a function of radius in a
protoplanetary disk. Credits: Banzatti et al., (2015).

To determine disk temperature profiles from observations, optically thick
tracers as 12CO were used. More recently, with the help of high-resolution
observations, better results can be obtained. For example, to study the
molecular layer of the disk (the region between which molecules are shielded
from dissociating radiation but are warm enough to remain in the gas phase),
three main techniques can be used:

  * •

Chemistry: Through freeze-out regions for each molecule. For example,
observing sharp drops in a sub-mm emission may indicate a freeze-out of
specific molecules.

  * •

Line excitation: Through multiple transitions of optically thin lines, the
excitation temperatures can be inferred.

  * •

Line optical depth: If the spectral line emission is optically thick, the
brightness temperature at a certain region can be interpreted as gas
temperature.

For the mid-plane, the same techniques can be used. However, as this region
contains larger dust grains, additional techniques may come into play:

  * •

Dust optical depth: Millimeter dust grains may have optically thick emission
in the inner disk. In such a case, the brightness temperature reveals the dust
temperature, which is equal to the gas temperature where collisions
counterbalance these two components’ temperatures.

  * •

Polarization fractions: Temperature gradients can manifest in the observed
polarization fraction due to distinct polarization angles probing varying
line-of-sight optical depths. This methodology has been developed by Lin et
al., (2020), where conversions from polarized flux patterns to line-of-sight
temperature gradients.

  * •

Optically thick gas lines in highly inclined disks: This method utilizes
projection effects to derive temperature differentials between the near and
far sides of the disk (Dullemond et al.,, 2020). It is effective for warmer
regions of low-mass star disks or disks around more massive stars with
extended CO ice lines.

##### Grain growth

The grain growth process within a PPD is predominantly influenced by
collisions among smaller dust particles. Despite persistent efforts in
experiments and modeling, challenges arise in explaining the emergence of dust
particles exceeding (sub)$\mu$m sizes to mm/cm sizes or larger (see e.g. Tu et
al.,, 2022). The collisional processes leading to particles of this scale can
be categorized into four main mechanisms, dependent on factors such as the
relative size/velocities and composition of the grains involved (see, for
example, Testi et al.,, 2014, Fig. 1.7, bottom panel, and Fig 1.13):

  * •

Sticking: It is the dominant mechanism for smaller scales between grain sizes
and relative velocities, thus being the dominant outcome from collisions at
early stages. Assuming minimal influence from electrostatic charges or
magnetic materials during collisions between dust grains, their interaction
can lead to sticking through van der Waals forces (Heim et al.,, 1999;
Gundlach et al.,, 2011). At lower velocities, grains can grow up to mm sizes.

  * •

Bouncing: Even at the same relative velocities as the previous sticking
regime, if grains are of similar sizes and are too large (mm-cm sizes), their
collisional outcome will be bouncing instead of sticking. This stage/size
scale is also known as the ”bouncing barrier” (Zsom et al.,, 2010).

  * •

Fragmentation/erosion: If the grains surpass the bouncing barrier, getting
larger, or if the grains have higher relative velocities, they may undergo
fragmentation. This process is important to replenish the disk with smaller
grains and provide slower grain growth and slower inward migration.

  * •

Mass transfer: If grains with different sizes collide at a relative velocity
approaching the fragmentation velocity, the collision can lead to substantial
mass transfer from the smaller to the larger particle. This mechanism can
surpass the bouncing barrier, enabling grains to grow to the size of
planetesimals.

Therefore, small dust grains can efficiently grow up to mm sizes through
sticking. Subsequently, mass transfer becomes a dominant factor. Throughout
this process, fragmentation/erosion continually replenishes the population of
small grains in the disk (Birnstiel et al.,, 2009). However, this
straightforward model may exhibit discrepant size scales and relative
velocities depending on the composition, porosity, and the presence of ice or
icy shells.

Figure 1.13: Schematic representation of the outcomes of dust collisions in
PPDs. It displays collision velocities (denoted by contours, in units of cm/s)
between two non-fractal dust agglomerates in a minimum-mass solar nebula model
at 1 AU. The green, yellow, and red boxes represent the explored parameter
space and results of laboratory experiments, where green signifies sticking or
mass transfer, yellow bouncing, and red fragmentation or erosion. The arrow
labeled “Sticking” indicates the direct growth of mm-sized dust aggregates, as
observed in simulations by Zsom et al. (2010). Further growth is impeded by
bouncing. The arrow labeled “Growth by mass transfer” suggests a potential
direct path to the formation of planetesimals. Credits: Testi et al., (2014).

##### Coupling between gas and dust

Different processes are associated with dust transport and migration, mainly
driven by dust-to-gas interaction through aerodynamic drag forces and
turbulence (see Fig. 1.7, bottom panel).

Aerodynamic drag forces are caused by the relative motion between gas and dust
particles, which produces a friction force. When the grains radii $a$ are
smaller than the mean free path between gas particles, they will be subject to
Epstein drag (Epstein,, 1924). In the Epstein regime, the drag force $F_{D}$
is given by:

$F_{D}=-\frac{4}{3}\pi a^{2}\rho_{g}\Delta vv_{\mathrm{th}},$ (1.9)

where $\rho_{g}$ is the gas density, $\Delta v$ is the relative velocity
between the particle and the gas, $v_{\mathrm{th}}\equiv\sqrt{8/\pi}c_{s}$ is
the thermal speed of the gas molecules and $c_{s}$ is the sound speed.

The drag force acts opposite to $\Delta v$, so the solid particle loses
angular momentum. The timescale over which this happens is called stopping
time $t_{\text{stop }}\equiv m\Delta v/\left|F_{D}\right|$, where $m$ is the
mass of the solid particle of interest. The stopping time can be converted to
a dimensionless number, called Stokes number ($St$):

$\mathrm{St}=\Omega_{K}t_{\text{stop }},$ (1.10)

where $\Omega_{K}$ is the local Keplerian frequency. In the context of dust
dynamics in protoplanetary disks, the $St$ plays a crucial role in determining
the behavior of dust particles relative to the gas. Mathematically, when
$t_{\text{stop}}$ is less than the dynamical time $t_{\text{dyn}}$ or when
$St\ll 1$, the term $\Delta v$ vanishes, indicating that the dust particles
follow the gas motion (more aerodynamic drag force). When $St\gg 1$,
aerodynamic drag becomes negligible, and the particle decouples from the gas.
Physically, the Stokes number reflects how well particles are coupled to the
gas and how they respond to thermal pressure in less dense environments. In
the case of small particles (sub-micron sizes), they are significantly
influenced by gas thermal pressure and remain coupled to the gas, with little
acceleration produced by drag forces. On the other hand, large particles (mm -
cm sizes) experience higher drag force acceleration until they grow to
approximately km sizes, beyond which the drag force becomes negligible
(Weidenschilling,, 1977).

Aerodynamic drag forces are fundamental to explain two dynamical processes,
horizontal drift and vertical settling. Due to a balance between centrifugal
force, stellar gravity, and a radial gas pressure gradient that increases
towards inner regions, the gas in protoplanetary disks (PPDs) rotates at sub-
Keplerian velocities. Conversely, large enough particles, not entirely dragged
by the gas, would rotate at Keplerian velocities. Consequently, these
particles experience a headwind from the gas, leading to the removal of their
angular momentum and causing inward drift, commonly referred to as radial
drift. In terms of the vertical structure, grains located above the midplane
experience the vertical component of stellar gravity pulling them toward the
midplane. This gravitational force is counteracted by the pressure gradient
force pointing outward from the midplane. Unlike lighter and smaller particles
that are primarily influenced by gas pressure and are less affected by
gravity, larger particles undergo more efficient vertical settling due to the
gravitational forces acting upon them. Thus, there is a larger concentration
of larger dust particles in the mid-plane, while smaller particles are more
vertically scattered. Moreover, PPDs are marked by significant turbulence.
Turbulence is characterized by chaotic changes in pressure and flow velocity
of the gas, facilitating the mixing and diffusion of particles in all
directions. This turbulent setting proves especially efficient for grains with
stronger coupling, characterized by lower Stokes numbers.

The interplay between turbulent mixing, affecting principally smaller dust
particles, and drag forces, primarily influencing larger particles, governs
the overall size distribution of dust grains within the disk.

###  1.2 Substellar objects: formation and detection via High-Contrast Imaging

A star is defined as a self-luminous astronomical object, maintained by the
balance between gravity and gas pressure (known as hydrostatic equilibrium),
and possessing sufficient mass to initiate thermonuclear fusion of hydrogen
isotopes, particularly 1H.

In contrast, a substellar object is a celestial body with less mass,
insufficient for triggering 1H fusion. Its luminosity may vary, and this term
is commonly associated with brown dwarfs, possessing enough mass for deuterium
(2H) fusion, and giant planets. Importantly, giant planets do not exhibit
self-luminosity; they depend on reflected sunlight for visibility, or, when
young, on the formation heat.

In this Section, I intend to detail how substellar objects can be formed and
consequently detected and characterized by high-contrast imaging observations
and post-processing techniques.

####  1.2.1 The main scenarios for substellar objects formation

Among several possible scenarios for planetary formation, the two widely
discussed and accepted models are the gravitational instability (henceforth
GI) (Boss,, 1997 and references therein) and the core-accretion (henceforth
CA) (e.g., Pollack et al.,, 1996 and references therein). Both theories
require different initial conditions to explain the formation of planets and
minor bodies observed to date. Also, each one presents its pros and cons while
explaining observed systems; none can be discarded. Below, I highlight their
more important characteristics and their type of planetary outputs.

##### Core-accretion: the bottom-up process

The core accretion (CA) model serves as a paradigm for the formation of giant
gas planets, particularly for objects orbiting within the range of 10-50
astronomical units (au) (Perryman,, 2014). Moreover, it extends its
applicability to the formation processes of rocky planets and minor celestial
bodies. It works under the assumption that smaller grain interactions will
eventually form larger bodies and, possibly, planetary cores. The model steps
can be summarized as the following.

First, dust and ice particles (sub-micron to micron) will start to agglutinate
when they hit each other due to van der Wall forces. With these initial sizes,
the dust grains are still well coupled to the gas. Thus, as the solids grow,
they eventually will settle to the disk mid-plane (Kusaka et al.,, 1970).

The higher surface densities in mid-plane promote collisional growth, leading
to the formation of larger (m-km sized) rocky/metal structures. However, the
growth of meter-sized objects is yet to be fully explained, since they are
expected to quickly drift towards the star and also fragment when colliding at
high speed (meter-size barrier; Adachi et al.,, 1976; Weidenschilling,, 1977;
Blum & Wurm,, 2008 and references therein).

Next, km-sized objects, also known as planetesimals, will induce gravitational
interactions, quickly attracting more dust and other planetesimals. This
particular stage, heavily dependent on the masses of the objects, is called
runaway growth (Stewart & Wetherill,, 1988; Wetherill & Stewart,, 1989) and it
precedes the oligarchic growth, the second to last stage of planetary
formation in the CA model.

After reaching $\sim$100-1000 km, the bodies, now called planetary embryos,
will move on their planetary feeding zone, thus accreting matter while
perturbing their vicinities. They will achieve an isolation mass, thus,
causing a lower growth rate at the time.

The previous stages are thought to cease in less than 1-10 Myr. The last
stage, called post-oligarchic or chaotic growth, is a slow stage that lasts as
much as 100 Myr or more. This stage leads to the formation of planets larger
than Mars. Additionally, during this step, the remaining planetary embryos may
cross other orbits, leading to an increase in eccentricities, inclinations,
and numbers of catastrophic collisions.

From the continuous interactions and growth of some planetary embryos, a
critical core mass (Bodenheimer & Pollack,, 1986; Pollack et al.,, 1996),
thought to be between $\sim$3.5 Mearth and $\sim$8.5 Mearth (Piso & Youdin,,
2014), will trigger a gas accretion rate that exceeds the ones that planetary
embryos induce. Consequently, this process may lead to gas giant planets’
formation (for further details, see figure 1.14 and Perryman, 2014).

Finally, it is possible to cite some characteristics of CA: it can efficiently
produce metal-enriched objects, rocky planets, and minor bodies. However, the
classical CA process, fed principally by km-sized objects (planetesimals),
faces two major problems. Firstly, the model requires extremely high column
densities of planetesimals to construct the observed core masses of gas giants
within their respective feeding zones. This raises concerns about the fate of
the majority of planetesimals that are not utilized in core formation.
Additionally, the accretion rates of planetesimals decrease significantly with
distance from the star, since they migrate fast, making the formation of cold
giant cores and subsequent atmosphere formation particularly challenging
within the protoplanetary disk’s lifetime. Secondly, the assumption that cores
grow in isolation and can accrete planetesimals with negligible velocity
dispersion has been challenged by dynamic simulations. Consequently, these
challenges have been the inspiration to alternative perspectives.

To address the challenges encountered in the classical Core Accretion (CA)
through planetesimal accretion, a more efficient model has been developed –
pebble accretion coupled with streaming instabilities (see reviews by Chiang &
Youdin,, 2010; Johansen et al.,, 2014; Johansen & Lambrechts,, 2017). Pebbles,
typically ranging from millimeter to meter sizes, are scattered radially
throughout the entire disk, vertically confined on its majority in the mid-
plane, and exhibit partial decoupling from the gas (see subsection 1.1.3). The
aerodynamic drag can reduce the speed difference between pebbles and larger
bodies as they reach closer. This slowdown prevents some pebbles from escaping
the gravitational pull of the larger bodies or from having collisional speeds
leading to fragmentation (see Fig. 1.13). As a result, these pebbles spiral or
settle toward the surface of the larger body and become accreted. This entire
process enhances the effective cross-sectional area for the accumulation of
material by larger bodies, accelerating their growth. Conversely, streaming
instability is a process where pebbles within the disk undergo collective
gravitational collapse due to aerodynamic drag. This leads to the formation of
dense filaments, providing favorable sites for rapid particle growth and
subsequent planetesimal formation, up to objects of $\sim$ 100 km. According
to pebble accretion models, they thus become the dominant growth mechanism
over planetesimals accretion, at a starting point from the mass of Ceres/mass
of the Moon. Consequently, the process is halted at the pebble isolation
masses of approximately 10 M⊕, where contraction of the gas envelope to form
giant planets starts. Therefore, streaming instability, working in conjunction
with pebble accretion, contributes to the quicker and more efficient formation
of planetary building blocks, even at larger distances from the star.

Figure 1.14: Schematic representation of the grain growth towards planetary
formation, starting with sub-micron dust, and extending up to the terrestrial
planets in the inner disk, and the gas giants in the outer disk. Some
indicative time scales are given, although some intervals, especially around
the meter-size barrier, remain highly uncertain. Credits: Perryman, (2011)

##### Gravitational instability: the top-down process

The planetary formation by GI, derived from the Nebula Hypothesis (Kant,,
1755; Laplace,, 1796), works under the assumption of large clumps of gas and
dust collapsing and forming giant planets. This process, which starts
similarly to the formation of chunks by instabilities in molecular clouds, can
be summarized as follows.

During the protoplanetary disk evolution, instabilities can cause over-
densities and clumps. However, interactions with another over-density or tidal
shredding due to slow cooling/rapid migration can easily disrupt these clumps
and stop the process. Therefore, if a massive clump subject to self-gravity
survives to dispersal, a giant body (planet, brown dwarf, or low-mass star)
will begin to form.

In the context of planetary formation, GI modeling may be challenging to
explain observations. GI planets’ cores don’t need to be dense nor reach a
minimum value to explain the extended atmospheres that gas/ice giants show.
Nevertheless, disks and clumps susceptible to GI show that the initial disk
must be very massive ($M_{disk}/M_{star}\geq 0.1$; Kratter & Lodato,, 2016),
which is not so observed333For a list of references measuring the percentage
of observations that have disk masses prone to GI, check section 2.2.1,
Kratter & Lodato, (2016).. Also, it is known that the amount of gas on a clump
that lasts attached until the formation of a denser structure usually leads to
brown dwarfs or low-mass star objects (e.g., Kratter & Matzner,, 2006;
Stamatellos & Whitworth,, 2009; Kratter et al., 2010b, ; Zhu et al.,, 2012;
Forgan & Rice,, 2013).

Finally, when comparing GI and CA processes, it is possible to affirm that the
first is rather faster ($\sim$103-104 years) than the latter ($\sim$107-108
years), and young GI planets are hotter and irradiate more than recently
formed CA planets with similar conditions. It is common to say that GI planets
have a “hot start” and CA planets have a “cold start” (see figure 1.15).

Figure 1.15: Luminosity-time evolutionary tracks computed using the Marley et
al., 2007b  methodology. The hot start, credited to gravitational instability
planets (solid), and the cold start, representing the core-accretion planets
(dashed) are shown, according to the initial planetary masses (1-10
$M_{jup}$). Credits: Natasha Batalha;
https://natashabatalha.github.io/picaso/notebooks/workshops/SaganSchool2021/1_Spectroscopy.html

####  1.2.2 (Sub)stellar multiplicity formation

Multiple (sub)stellar systems can be defined as systems with two or more
gravitationally bound stars/brown dwarfs with separations $\lesssim$ 0.1 pc.
Multiple formation requires two or more events of gravitational collapse that
will thus form a bound system. The process by which self-gravitating objects
develop substructures that independently collapse is called fragmentation. If
formed at a certain distance, within time, they can gradually approach each
other and become gravitationally bound. Following, the main models for
multiple star formation are better detailed.

The multiple formation models can be categorized in three: 1) fragmentation of
a core or filament; 2) fragmentation of a massive accretion disk; or 3)
through dynamic interactions (see Fig. 1.16). Following, these processes are
better detailed:

Figure 1.16: Mechanisms for multiple (sub)stellar formation. The model (top
row) presents an approximate range of time and length scales for each process.
The middle row shows proposed observational examples for each case. The bottom
row shows examples of each formation model from numerical simulations.
Credits: Offner et al., (2022). The observations/simulations references are
listed in Figure 5 caption of that work.

  * •

Core and filament fragmentation: The processes of core and filament
fragmentation occur in dense embedded structures, with each object developing
and collapsing within the same parent structure. Filaments are the second
larger structures in the stellar formation sequence, formed by the encounter
of expanding shells (see subsection 1.1.1). However, hereafter I define cores
and filaments as morphological structures, rather than physical ones, which
explains the similar scales between the two modes of fragmentation in Fig.
1.16. Therefore, the two presented fragmentation modes are defined by an over-
density in a structure formed, before the formation of a disk. Two main
mechanisms trigger core/filamentary fragmentation, rotation, and turbulence:

    * –

Rotation-driven fragmentation:

When collapsing cores rotate rapidly, they become susceptible to fragmentation
if specific criteria, ensuring a balance among thermal, gravitational, and
rotational energies, are satisfied. Excluding fragmentation as the initial
trigger, the simplified expression by Inutsuka & Miyama, (1992) establishes
the relationship between the initial ratio of thermal to gravitational energy,
$\alpha_{\mathrm{vir}}$, and the ratio of rotational to gravitational energy,
$\beta_{\text{rot }}$:

$\alpha_{\text{vir
}}\beta_{\mathrm{rot}}=0.02\left(\frac{T}{10\mathrm{\leavevmode\nobreak\
K}}\right)\left(\frac{M}{1M_{\odot}}\right)^{-2}\left(\frac{R}{0.1\mathrm{pc}}\right)^{4}\left(\frac{\Omega}{10^{-14}\mathrm{\leavevmode\nobreak\
s}^{-1}}\right)^{2}.$ (1.11)

Here, $T$, $M$, $R$, and $\Omega$ represent the local temperature, mass,
radius, and Keplerian frequency of the collapsing structure, respectively. The
conditions for fragmentation in this expression are $\alpha_{\text{vir
}}\beta_{\text{rot}}>0.12$ and $\alpha_{\text{vir }}<0.5$. Rotational
fragmentation often results in the formation of multiple systems within the
same plane, tending towards spin alignment and coplanarity (Offner et al.,,
2016; Bate,, 2018). These characteristics serve as signatures of the formation
scenario, which can be used as an educated guess in the case of evolved
multiple systems (see chapters 2 and 3).

    * –

Turbulent fragmentation: Turbulence is an effective trigger for structure
formation, encompassing both stars and brown dwarfs within the same system.
Regions within a molecular cloud can fragment when the turbulence seeds
collapse and provide pressure support – when the local gravitational energy
exceeds the contribution of both turbulent and thermal energy. To initiate
fragmentation via turbulence, a critical density $\rho_{\text{crit}}$ must be
achieved or exceeded. For an isothermal cloud, a relatively straightforward
expression is provided by:

$\rho_{\text{crit
}}=\frac{\rho_{0}}{1+\mathcal{M}^{2}}\left(\frac{R}{R_{0}}\right)^{-2}\left[1+\mathcal{M}^{2}\left(\frac{R}{R_{0}}\right)^{p-1}\right],$
(1.12)

where $\rho_{0},R_{0}$ and $\mathcal{M}$ are the initial density, radius, and
Mach number of the parent cloud ( ratio of local flow velocity to the sound
speed of the medium), respectively (Guszejnov & Hopkins,, 2015; Guszejnov et
al.,, 2017). In this expression, p denotes a term equal to 2 for supersonic
turbulence and 5/3 for subsonic scales. Additional contributions can also
increase the complexity. For example, an isothermal filament collapses if its
mass per unit length, $M_{\mathrm{l}}$, exceeds a critical value. A filament
threaded laterally by a magnetic field, $B_{\|}$, will become unstable when
$M_{\mathrm{l}}$ exceeds (Inutsuka & Miyama,, 1997; Tomisaka,, 2014):

$\displaystyle M_{l,\mathrm{crit}}$
$\displaystyle\simeq\frac{2c_{s}^{2}}{G}\left(1+\beta_{\mathrm{mag}}^{-1}\right)$
(1.13) $\displaystyle\simeq 16.4\left(\frac{T}{10\mathrm{\leavevmode\nobreak\
K}}\right)+7.8\left(\frac{n_{\mathrm{H}}}{10^{4}\mathrm{\leavevmode\nobreak\
cm}^{-3}}\right)^{-1}\left(\frac{B}{10\mu\mathrm{G}}\right)^{2}M_{\odot}\mathrm{pc}^{-1},$

where $\beta_{\text{mag }}=8\pi c_{s}^{2}\rho/B_{\|}^{2}$ is the ratio of
thermal to magnetic pressure. Furthermore, within filaments, if turbulent
support after triggering the collapse is negligible, the critical mass can be
stated as:

$\displaystyle M_{crit}$ $\displaystyle\simeq
1.3\frac{c_{s}^{4}}{G^{2}\Sigma_{\mathrm{fil}}}$ (1.14) $\displaystyle\simeq
1.1\left(\frac{T}{10\mathrm{\leavevmode\nobreak\
K}}\right)^{2}\left(\frac{M_{1,\text{ crit
}}}{40M_{\odot}\mathrm{pc}}\right)^{-1}\left(\frac{W_{\mathrm{fwhm}}}{10^{4}\mathrm{au}}\right)M_{\odot},$

where the filament surface density, $\Sigma_{\mathrm{fil}}$, is the ratio of
the filament mass-per-length to the width of the filament
$\Sigma_{\mathrm{fil}}=M_{\text{line }}/W_{\text{fwhm }}$ (André et al.,,
2019). Multiple systems originating from the same parent structure through
turbulent fragmentation are expected to share similar ages and typically
exhibit a modest number of members, typically two to three within a core. They
are usually formed at wider separations (set to a scale of 102 au to 0.1 pc;
Offner et al.,, 2022) compared to objects formed through rotational-driven
fragmentation. As a result, they accrete gas at varying rates and have
distinct spin alignments and rotational speeds. Consequently, (sub)stars
resulting from turbulence-driven fragmentation tend to have misaligned stellar
spins, accretion disks, and protostellar outflows (Bate,, 2000; Offner et
al.,, 2016; Bate,, 2018; Lee et al.,, 2019).

  * •

Disk fragmentation: Circumstellar disks can undergo fragmentation at later
stages compared to the previously mentioned fragmentation modes. This
particular type of fragmentation is expected to occur in massive forming disks
through gravitational instability (GI), as discussed in subsection 1.2.1.
Referenced in 1.1.3 as a mechanism for triggering spirals in circumstellar
disks, GI can also cause instabilities that may lead to fragmentations when
the Toomre parameter $Q$ is $\leq$ 1 (Toomre,, 1964; Kratter & Lodato,, 2016)
in razor-thin disk. The Toomre parameter criteria can also be described in
terms of the mass of the protostar/PMS and the disk, aspect ratio and radial
distance:

$Q=\frac{c_{s}\Omega}{\pi G\Sigma}=f\frac{M_{*}}{M_{d}}\frac{H}{r}\leq 1$
(1.15)

where $c_{s}$ is the disk sound speed, $\Omega$ the disk orbital frequency,
$\Sigma$ the surface density, $M_{*}/M_{d}$ the star-disk mass ratio, $H$ the
disk scale height, $r$ the radial distance, and $f$ the order unity scale
factor which depends on the assumed surface density power law (Offner et al.,,
2022). However, gravitational instability (GI) does not inevitably lead to
fragmentations; it serves as the mechanism causing local instabilities. For
fragmentation to occur after GI is triggered, effective cooling must
counteract the inflation caused by higher temperatures. In the case of thick
disks, the critical $Q$ significantly drops below unity due to the extended
vertical distribution of matter (Lau & Bertin,, 1978). This implies that
gravitational instability (GI) in such disks can be triggered in outer
regions, specifically when the disk is not excessively flared and is located
at a distance with sufficient gas density – in these outer regions, lower
temperatures are associated with higher radial distances. However, the
probability of gravitational instability fragmentation increases during the
early stages of disk formation, particularly when the infall rate dominates
over viscous accretion (typically in the Class 0 phase) since during this
phase, an increase in surface density does not necessarily correspond to
higher temperatures that could inhibit gravitational instability (Kratter et
al., 2010a, ; Harsono et al.,, 2011; Zhu et al.,, 2012). This scenario is
likely linked to the rapid formation of binaries sharing approximately the
same mass (Bate & Bonnell,, 1997; Ochi et al.,, 2005; Young & Clarke,, 2015).
An alternative pathway to GI fragmentation arises when the disk temperature
decreases more rapidly than accretion can modify the surface density. A
plausible explanation for this scenario is variations in stellar luminosity
resulting from accretion changes (see section 1.1.2). Fragmentation may occur
suddenly if there is a significant decrease in central luminosity, provided
that the temperature drop does not coincide with an increase in the cooling
timescale (Dunham et al., 2014b, ; Kuffmeier et al.,, 2018). Regardless of the
process that triggers gravitational instability (GI) fragmentation, the binary
companion must overcome interactions with the disk and subsequent tidal forces
induced by inward migration.

  * •

Dynamic interactions: A more effective mechanism to elucidate the formation of
close binaries is through capture and subsequent orbital evolution.

The effectiveness of the capture mechanism depends on the environment, whether
it is gas-dense or has less gas. In gas-dense environments, objects within a
stellar cluster may transition between unbound and bound states due to energy
and angular momentum exchange with the surrounding gas cloud or individual
circumstellar disks. This process, known as gas-mediated capture, is observed
in various scales in numerical simulations and can lead to the formation of
bound systems. However, the net frequency of capture through this mechanism is
likely low, making these systems observationally indistinguishable from those
formed via fragmentation, except in cases where two disks seem to be on a
collision course. On the contrary, in regions of high stellar density without
much gas, interactions between truly unbound stars can lead to the formation
of new binaries and partner exchange. Recent work emphasizes the contributions
of higher order multiples to overall interaction rates in moderately dense
clusters, producing a more diverse set of outcomes, including the formation of
very compact binaries (Geller & Leigh,, 2015; Dorval et al.,, 2017; Hamers &
Samsing,, 2020). These processes are more relevant for dense open clusters and
globular clusters but are sub-dominant in gas-rich environments. In addition,
widely separated bound binaries can reduce their separations through the
dynamical friction force, caused by the torque provoked by accreting systems
trailing their material during close encounters (Bate & Bonnell,, 1997;
Stahler,, 2010; Lee & Stahler,, 2011), by numerical viscosity and mergers
(Vorobyov & Basu,, 2005; Zhu et al.,, 2012) or by Kozai-Lidov cycles coupled
with tidal friction (Kozai,, 1962; Lidov,, 1962; Fabrycky & Tremaine,, 2007).
The latter scenario occurs when binaries exchange angular momentum with an
inclined tertiary object, which thus leads to a periodic exchange between
eccentricity and inclination of the binary, therefore leading to
circularization and a hardening of their orbits.

####  1.2.3 High-Contrast Imaging: instrumentation, observing strategies and
data-processing

In the Galaxy, more than 5500 exoplanets have been identified to date444For
more information, refer to sources such as
https://exoplanetarchive.ipac.caltech.edu/ and
http://www.exoplanet.eu/catalog/, alongside a few thousand brown dwarfs555For
details, see resources like
http://www.johnstonsarchive.net/astro/browndwarflist.html and https://simple-
bd-archive.org (hereafter called BDs). While various techniques can unveil the
existence and characteristics of these objects by observing their interactions
with the brighter stars they orbit, high-contrast imaging (HCI), also known as
direct imaging (DI), stands out as the sole method capable of directly
revealing their presence. Thus, the HCI offers the advantages of extracting
the flux and astrometry of the detected source. With a few observations of the
latter, orbital characterization can also be performed. Lastly, HCI provides a
recovery of the photometric spectra, which can provide key parameters such as
effective temperature, surface gravity, mass of the object, and presence of
clouds and certain molecules.

Planck’s law defines that a black body, an object that is a perfectly
efficient energy absorber and emitter, will have its radiation characterized
by its temperature. Assuming that stars and substellar objects can emit
approximately black bodies, an observational instrument will be more
susceptible to detect them using broadband filters covering wavelengths close
to the peak emission. The detectability of such objects will also depend on
the instrument’s sensibility, resolution power, and possible interference from
background sources. Black-body objects with a radiation peak in the optical/IR
can emit in two different ways. They can produce their radiation through
nuclear reactions (as in the case of low-mass stars and deuterium-burning
objects) or they can emit thermal radiation due to atomic and molecular
agitation. The latter applies to newly formed planets and moons, which have
enough primordial heat derived from their origin. Brown dwarfs are a special
case, where their deuterium-burning phase does not last a long time and
therefore, similarly to giant planets, they will cool down gradually with time
(see Fig. 1.17).

Figure 1.17: Evolution of effective temperature for substellar and low mass
stars. The red lines trace stars (masses above the hydrogen-burning limit),
the orange track brown dwarfs, and the blue trace objects below the deuterium-
burning limit. Credits: Adaptation of Bailey, (2014).

The HCI technique relies on enhancing the contrast of substellar objects, by
suppressing the starlight from the host star. This technique focuses on
detecting objects that still retain heat from their formation, which are faint
in comparison with the host star. In summary, the technique is optimized for
detecting massive substellar objects on their first millions of years (Myrs)
and at wider separations from their host star using Optical/near-IR
wavelengths (see Fig. 1.18, where discovered by imaging planets (HCI/DI
planets) are compared to planets detected by other techniques).

Figure 1.18: The diagram depicts the population of exoplanets and the
corresponding discovery techniques. It illustrates masses in Jupiter masses
versus orbital periods in days. The high-contrast imaging/direct imaging
planets are specifically highlighted by blue-filled stars. Credits:
https://exoplanetarchive.ipac.caltech.edu/exoplanetplots/exo_massperiod.png.

Despite its potential, HCI effectivity has significant challenges as it
requires a certain level of contrast between star-companion, and also a
certain angular resolution between both. The contrast is quantified by the
flux ratio $F_{object}/F_{host\\_star}$, a value influenced by both the
angular separation of the sources and their intrinsic brightness corresponding
to the observed wavelength. In the most favorable cases, with the best
instruments and post-processing algorithms at the moment, contrasts can be
reached in the order of 10-6 to 10-5. Greater contrast values result in
brighter companions compared to the host star, hence aiming for lower values
(fainter objects – less massive or older companions) is the objective for the
new generation of telescopes and post-processing methodologies dedicated to
HCI (see Fig. 1.19). The angular resolution is determined by a combination of
the orbital separation between the objects, linked to the distance observer
system.

Figure 1.19: The chart illustrates how to interpret a contrast curve, where a
companion becomes detectable when positioned above the solid curve at a
particular contrast and separation. Notably, the achieved contrast strongly
depends on the separation from the central star, making only the brightest
planets detectable at close separations. The dashed line represents a deeper
contrast curve compared with the solid line. A deeper contrast curve improves
the effectiveness of high-contrast imaging, making it more probable to detect
faint objects at closer separations. Credits: Adapted from Follette, (2023).

##### Instrumentation

In order to perform HCI/DI, it is required a proper instrumentation,
characterized by adaptive optics (AO) in the case of terrestrial telescopes,
and coronagraphs for all cases.

The light coming from celestial objects, which travels approximately a quasi-
infinite distance, should be a perfect plane wave when entering the pupil of
the telescope. In such case, without interference from the atmosphere, these
spherical waves should ideally form an “Airy Disk” diffraction pattern at the
telescope focal plane (see Fig. 1.20). This diffraction pattern will thus form
the ideal Point Spread Function (PSF), with an angular resolution
$\theta_{diff}$ (angular radius of the central airy disk maximum) given by:

$\theta_{\text{diff }}=1.22\frac{\lambda}{D}$ (1.16)

where $D$ is the diameter of the telescope’s aperture and $\lambda$ is the
wavelength of the observation (see e.g., Pedrotti et al.,, 2017 and also Fig.
1.21). $\theta_{\text{diff }}$ is of the order of 30 mas for
$\lambda=1\mu\mathrm{m}$ and $D\sim 8\mathrm{\leavevmode\nobreak\ m}$.

Figure 1.20: Diffraction-limited images of two point objects produced by a
lens are illustrated. In the schematic depiction shown in Figure (a), two airy
disks are formed upon crossing a lens, originating from two distinct points
$S_{1}$ and $S_{2}$. Figures (b) and (c) respectively depict scenarios where
the sources are resolved and unresolved. Credits: Pedrotti et al., (2017).
Figure 1.21: Rayleigh’s criterion for just-resolvable diffraction patterns.
The dashed curve is the observed sum of independent diffraction peaks.
Credits: Pedrotti et al., (2017).

AO is the technology used in the instrument to compensate for the effects of
atmospheric turbulence. The wavefronts coming from the sources are susceptible
to unpredictable Earth’s atmospheric variations. The atmosphere is formed by
layers of turbulent “bubbles”. Before reaching the telescope’s pupil, the
incoming light is often divided into sub-apertures determined by the radius of
these bubbles. The interference of these sub-apertures creates fringes-
patterns of constructive and destructive interference. In regions where the
sub-apertures interfere constructively, bright regions (speckles) are formed.
On the contrary, in regions of destructive interference, dark regions appear.
The resulting pattern is the speckle pattern. The speckles exhibit dimensions
of approximately $\lambda/d$, determined by the separation $d$ between the two
sources of interference. When $d\sim D$, these speckles will share identical
dimensions with the PSF of the star, making it challenging to differentiate
them from an astrophysical source with equivalent brightness. To mitigate
these wavefront variations, a control system must continuously measure the
impact of the fluctuations at a high frequency during observations, and
subsequently provide corrections as/when needed. To simulate atmospheric
behavior, a reference point source, usually a star, is employed. This source
should be sufficiently bright to ensure a high signal-to-noise ratio (SNR) in
the wavefront sensor. Certain instruments offer the option of employing an
artificial reference, specifically the laser guide star (LGS) or natural point
sources when a bright star is unavailable, or even the host star. The
instrument responsible for quantifying these fluctuations is known as a
wavefront sensor, providing valuable information about phase variations. A
real-time calculator then analyzes this data and determines the precise
adjustments required to compensate for wavefront errors, transmitting
instructions to deformable mirrors in a closed-loop system (see Madec,, 2012
and Fig. 1.22). More recently, Extreme Adaptive Optics (ExAO or XAO) systems
were developed in high-performance high-contrast imagers as Spectro-
Polarimetric High-contrast Exoplanet Research (VLT/SPHERE, Beuzit et al.,,
2019) and Gemini Planet Imager (GPI, Macintosh et al.,, 2008). They are able
to provide Strehl ratios (ratio of maximum diffraction intensities of an
aberrated wavefront to a perfect wavefront) up to 95%.

Figure 1.22: Simplified AO configuration and its impact in observations. Upper
panel: a diagrammatic representation of an Adaptive Optics (AO) system
according to Rigaut, (2015). Real-time correction of the distorted wavefront
involves a wavefront sensor, a feedback loop, and a deformable mirror. Lower
panel: illustration of an image with a distorted wavefront before AO
correction (left) and the same image after AO correction (right), revealing
the Airy pattern (Sauvage et al.,, 2016). Credits: Bhowmik, (2020).

Coronagraphs are optical devices designed to suppress the brightness of stars,
thereby improving the contrast for observing faint substellar objects and
structures in the context of HCI. According to Guyon et al., (2006), the
classical coronagraphic designs can be classified into four, 1) the
interferometric coronagraphs, 2) the pupil apodized coronagraphs, 3) the Lyot
coronagraphs, and 4) hybrid designs combining elements from the aforementioned
categories. Modern coronagraphic designs often integrate elements from these
categories, with variations centered around the apodizer concept, mask
properties, and the complexity of the coupled Adaptive Optics (AO) system and
mirrors’ disposition.

The most simple description of such a system, following the incoming light
towards the detector focal plane unfolds as follows: to attenuate incoming
light, an apodizer may be introduced into the telescope pupil, modifying the
amplitude or intensity profile – typically gradually attenuating the intensity
towards the apodizer’s edges. Following the pupil, and potentially the
apodizer, the light advances through a lens, converging at the initial focal
plane. At this stage, a mask is implemented, serving either as a physical
occulting mask or a phase mask. The latter works through destructive
interference, thus also blocking the starlight as the former. Following
another passage through a lens and the alignment of the wavefront, a component
known as the Lyot stop is strategically positioned. This device partially
obstructs the light and serves to attenuate the diffraction caused by the
mask. The Lyot stop will thus permit the incoming of a predominant portion of
light from surrounding sources close to the limits of the mask, therefore
contributing to the final image as it converges on the detector plane. A
schematic representation of how a Lyot stop works is presented in Fig. 1.23.
Three representative examples of possible coronagraphic designs with Lyot
stops are presented in Fig. 1.24. The instrument used in the work presented in
chapter 3, VLT/SPHERE, provides a possibility of using three coronagraphic
designs, a classical Lyot coronagraph (CLC, without an apodizer), an apodized
pupil Lyot coronagraph (ALC or APLC), and a four-quadrant phase-mask (4QPM,
with a phase mask instead of an amplitude mask as the previous mentioned
configurations).

Figure 1.23: Schematic representation of how a classical Lyot coronagraphic
works. The figure also shows how a Lyot stop can enhance the signal from
regions closer to the borders of a physical mask. Credits: Ben R. Oppenheimer
– https://lyot.org/background/coronagraphy.html. Figure 1.24: Representative
examples of possible coronagraphic designs with Lyot stops. Top-left panel:
Schematic layout of an apodized pupil Lyot coronagraph (APLC). The incoming
light follows the sequence from A to D. The labels “-” and “+” represent
before and after the stage/instrument utilization, represented by the letters.
Credits: Por et al., (2020). Top-right panel: Schematic layout of a phase-
apodized pupil Lyot coronagraph (PAPLC). In that case, the apodizer has a
phase shift in different regions, and the mask only covers the brighter half-
side of incoming light. Credits: Por, (2020). Bottom panel: a schematic
representation of a four Quadrant Phase-Mask (4QPM). In this case, no apodizer
is presented, and a phase mask is utilized. Credits: Boccaletti et al.,
(2015).

##### Observing strategies

In addition to the speckle patterns produced by the atmospheric variations, a
second type of speckle pattern that changes relatively slowly over time and
arises from residual aberrations and imperfections in the optical system is
called quasi-static speckle. They are mostly permanent during an exposure of
tens of minutes (see e.g. Soummer et al.,, 2007). These speckles can be
partially suppressed using post-processing techniques after the observation
sequence is completed. To do so, an observing strategy, determining how data
is acquired at the instrument level, provides the raw data, which is
subsequently processed through a post-processing technique. The main observing
strategies to detect substellar companions can be categorized into three: 1)
angular differential imaging (ADI; Marois et al.,, 2006), 2) spectral
differential imaging (SDI; Racine et al.,, 1999; Sparks & Ford,, 2002), and 3)
reference differential imaging (RDI; e.g. Lafrenière et al.,, 2009; Soummer et
al.,, 2011; Ruane et al.,, 2019). They are briefly described as follows:

  * •

ADI: Angular differential imaging (ADI) is one of the most commonly used
observing strategies to mitigate quasi-static speckles in HCI. The ADI process
requires the science target to pass near the meridian passage. The procedure
follows as the pupil of the telescope is fixed in the host star, and its
derotator is switched off. Consequently, the coronagraphic frames will
maintain the center of the mask fixed with the host star position, while the
Field-of-View (FoV) will be noticed rotating. This will result in the
acquisition of frames with different parallactic angles throughout the
observation sequence. Moreover, the stabilization of the pupil ensures that
the aberrations it induces remain relatively constant across all parallactic
angles. This is crucial, as a significant portion of quasi-static speckles
originates from the pupil and its supporting structures (spiders). By having
the frame sequence, a model of the speckle halo will be created and then
subtracted from all frames, thus removing substantially the quasi-static
speckles. The frames will thus be derotate, aligning whichever possible
companion/structure at the same position, thus enhancing its signal (see a
general schematic representation of ADI sequence, independently of the post-
processing model used in, Fig. 1.25). The post-processing technique used will
define how the speckle field models will be created and how the signal of the
companion will be reconstructed (see subsection 1.2.3 and Fig. 1.30).

Figure 1.25: Illustration outlining the key stages of ADI observing strategy.
From left to right, the sequence includes the calibrated science images (A),
the frame-wise reference PSF (B), the residual images (C), the derotated
residual images, and the final residual map. Credits: Dahlqvist, (2022).

  * •

SDI: The spectral differential imaging (SDI) consists of decoupling the
companion/structure from the speckles using different wavelengths (rather than
azimuthal rotation as ADI). This observing strategy was first proposed by
Racine et al., (1999). The researchers relied on two images taken in adjacent
bands, aiming to exploit the presence/absence of an exoplanet’s signal in two
or more adjacent bands. Their idea was to carefully select two wavelengths to
ensure the exoplanet’s presence in one image while limiting its brightness in
the other, often utilizing the methane absorption line. This approach
effectively reduced speckle noise without causing self-cancellation of the
planetary signal during subtraction. One example, where the methane absorption
is more pronounced in one band than in the other is presented in Fig. 1.26.

Figure 1.26: Images of the GJ 758 system, highlighting the late T-dwarf GJ 758
B within the white circle. A background star is indicated in the red circle.
The methane absorption in the H3 filter is evident, showcasing lower flux in
the object. In the SDI combination of the two filters, the T-dwarf displays
two distinct peaks, while the neighboring star exhibits equal peaks. Credits:
Adapted from Vigan et al., (2016).

A more advanced SDI approach, also known as the spectral deconvolution
technique (SD), was introduced by Sparks & Ford, (2002). Their concept
exploits the fact that quasi-static speckles exhibit radial scaling
proportional to $\lambda$ (appearing to move radially from the image center),
while real objects remain stationary. This difference becomes noticeable when
frames observed with different bands are appropriately rescaled (see Fig.
1.27). To ensure a more robust analysis, it is essential to observe multiple
bands simultaneously, invoking the utilization of Integral Field Spectrograph
(IFS) datacubes, as suggested later by Thatte et al., (2007).

Figure 1.27: Schematic illustration of the Spectral Differential Imaging (SDI)
observing strategy in combination with Principal Component Analysis (PCA)
post-processing. The process involves scaling the images (A$\lambda$)
inversely proportional to their wavelength (B$\lambda$), followed by the
creation of a stellar PSF model (C) using PCA analysis. The stellar PSF model
(C) is then rescaled for each wavelength channel proportional to its
wavelength (D$\lambda$) to match the original images’ scaling (A$\lambda$).
This results in a reduced Field of View (FoV) marked by a red border, with the
outer region of small wavelength images not covered by the scaled stellar PSF
models (D$\lambda$). The companion is denoted by a red dot. Credits: Kiefer et
al., (2021).

  * •

RDI: Reference-star differential imaging (RDI) is an observing strategy that
consists of observing different stars to build a model of the speckle field.
This procedure uses the same setup as the one used for the target of interest.
After scaling the flux/brightness, the PSF model can be used to subtract the
quasi-static speckles of the science data frames, which are mainly created by
the instrumentation (see Fig. 1.28). RDI has the detectability advantage under
a small rotation of FoV in comparison to ADI or SDI. However, this technique
requires high stability of the instrument, short duty cycles, and relatively
similar magnitudes between the target and the reference star. Some
contemporary ground-based HCI systems have an observing mode called “star-
hopping” (see e.g. Wahhaj et al.,, 2021). This functionality enables the AO
loop to pause temporarily on a target, like the science target, and resume
when the telescope points to another nearby target, such as the PSF reference
star. This ensures a close match in their PSFs. Additionally, RDI can be
carried out to improve performance, especially using space telescopes like
Hubble space telescope (see e.g., Golimowski et al.,, 2006; Schneider et al.,,
2009).

Figure 1.28: The figure depicts the Reference Differential Imaging (RDI)
process using Gemini Planet Imager H-band images of HR 4796A. RDI involves a
library of images of stars without known disk or planet signals, serving as
references. These reference images can be straightforwardly combined or
employed to create a customized PSF for each target image in the sequence. In
cases where the instrument derotator is turned off, subtracted images are
rotated to a common on-sky orientation and combined, as in the ADI procedure.
Credits: Follette, (2023)

##### Post-processing techniques

Following the acquisition of data through an observing strategy executed in
the instrument/coronagraphic observation, a post-processing procedure designed
to enhance the contrast of faint companions/substructures will be employed.
Post-processing techniques can be categorized into three primary groups: (1)
speckle subtraction techniques, (2) maximum likelihood techniques, and (3)
supervised machine learning techniques (Dahlqvist,, 2022).

Speckle subtraction techniques, often referred to as PSF-subtraction
techniques, are widely utilized within the High-Contrast Imaging (HCI)
community. The process typically involves the following steps outlined in Fig.
1.29. Initially, the process involves estimating a model of the speckle field
(also referred to as the reference PSF), subtracting this model from each
science image, and performing de-rotation (ADI) and/or rescaling (SDI) of the
PSF-subtracted images. Subsequently, these processed images are combined to
produce a residual image.

The maximum likelihood post-processing techniques involve addressing the
potential distortion of the planetary signal due to the subtraction of the
speckle field model. Instead of subtracting the reference PSF or decomposing
it into multiple components, these techniques replace or complement reference
PSF subtraction with forward modeling of the planetary companion. This
approach includes a maximum likelihood-based estimation of the planetary flux.
By modeling and tracking the planetary signal based on knowledge of speckle
noise statistics, the expected planetary movement (depending on the observing
strategy), and the impact of speckle field modeling, these techniques employ a
maximum likelihood approach to provide estimated contrast values for each
position within the field of view. Additionally, they may provide a detection
map based on SNR.

The post-processing techniques employing supervised machine learning can be
categorized into two main methods: decision trees and deep neural networks.
These approaches, exemplified by the SODINN (Supervised exOplanet detection
via Direct Imaging with deep Neural Networks) algorithm proposed by Gomez
Gonzalez et al., (2018), redefine the task of exoplanet detection as a binary
classification problem (is it a astrophysical source or a speckle?). The
primary challenge lies in obtaining a substantial number of labeled samples
for effective training since few planets were detected with HCI. The negative
classes (speckle noise) can be generated by standard machine learning data-
augmentation techniques (Gomez Gonzalez et al.,, 2018) or through the use of
Generative Adversarial Networks (GANs; (Hou Yip et al.,, 2019)). On the
contrary, the positive classes can be generated by injecting synthetic
substellar companions at various SNR levels, using an off-axis PSF of the host
star as model.

Figure 1.29: Typical differential imaging pipeline for speckle/PSF subtraction
observing strategies coupled with post-processing techniques.Credits:
Dahlqvist, (2022).

Although an exhaustive list of post-processing techniques and their
variations, associated with various observing strategies could be provided and
described, I will narrow the focus to speckle subtraction techniques which can
be coupled with ADI. Specifically, I will briefly mention the median
subtraction, Principal Component Analysis (PCA), and Locally Optimized
Combination of Images (LOCI) techniques. These three were employed throughout
the course of the work outlined in Chapter 3:

  * •

Median subtraction: The most simple approach to create a speckle model is to
perform a median frame with the pre-processed frames from an observing
sequence. This technique, originally developed alongside ADI (Marois et al.,,
2006), relies on the fact that a moving planet is not visible in the median
image of the sequence, which can be used as an initial estimation of the
speckle model. Subsequently, after subtracting the median, the residual noise
averages incoherently when the images are aligned to a common north. This
initial approximation, however, is less effective in the immediate vicinity of
the host star. In this region, residual speckle noise is more concentrated and
intense, and the companion’s differential angular displacement is lower,
leading to more pronounced self-subtraction. Median subtraction, thus,
performs better for brighter and more angular separated sources.

  * •

Locally Optimised Combination of Images (LOCI): The LOCI technique, similar to
median subtraction, generates speckle model frames through a combination of
pre-processed frames. However, LOCI’s models are more robust as they are
constructed using a linear combination of selected images. In LOCI, reference
frames are segmented into subsections, both radially and azimuthally.
Coefficients, derived from a reference frame and through residual
minimization, are calculated to weight each section across all frames.
Consequently, the model frames utilize these weighted linear combinations of
images, resulting in an enhanced speckle model. Since its original development
by (Lafrenière et al.,, 2007), various adaptations of LOCI have been
introduced (see e.g., Pueyo et al., 2012a, ; Marois et al.,, 2014; Currie et
al.,, 2012).

  * •

Principal Component Analysis (PCA): PCA-based techniques (Soummer et al.,
2012a, ; Amara & Quanz, 2012a, ) are widely utilized as a statistical tool in
post-processing for HCI images. In HCI, PCA serves as the builder of speckle
models for each pre-processed frame. Mathematically, PCA techniques involve
the projection of images onto a lower-dimensional orthogonal basis. By
vectorizing pre-processed images along their uncorrelated dimensions, such as
position, rotation, wavelength, brightness, and other n-parameters, a
covariance matrix can be built. If there is more variation in specific aspects
of information across different images, more vectors for a complete
representation are needed. These vectors, within the orthogonal basis, are
termed principal components – the number of principal components needed to
express the required information is selected by the PCA user. The initial
components of a PCA encapsulate more information than higher-order components.
Therefore, the principal components can be defined as a truncation of the
orthogonal basis, where the higher the number of principal components, the
closer the information expressed to the original data. In the context of PCA
for HCI applications, the conceptual idea involves identifying patterns across
multiple pixels present in both the target image and a certain number of
reference images. The initial principal components typically capture large-
scale PSF structures, while the higher-order principal components resemble
various manifestations of the speckle pattern. For the aforementioned
observing strategies, it can be used as a tool to select just a part of the
more correlated information, taking advantage of the fact that the companion
rotates in the FoV but not the quasi-static speckles. The addition of
components to the model enhances its “aggressiveness”, increasing the
probability of a well-matched PSF model. However, this also raises the
likelihood of over-subtracting or self-subtracting the companion’s signal.

A diagram illustrating the ADI observing strategy and the functionality of the
three described post-processing techniques is presented in Fig. 1.30.

Figure 1.30: Diagram illustrating the ADI observing strategy with three
distinct post-processing techniques for building the speckle model: 1) median
subtraction (bottom-left), 2) LOCI (bottom-center), and 3) PCA (bottom-right).
The subsequent steps follow the procedure outlined in subsection 1.2.3 and
Fig. 1.25. Credits: Adapted from Gómez González, (2017).

####  1.2.4 High-Contrast Imaging: observing/characterizing substellar
companions

The HCI technique, as previously mentioned, is an adequate tool to detect and
study substellar companions. Usually, companions detected by HCI play a
fundamental role in connecting the formation of these objects in disks (see
section 1.1) and their later evolution into mature stellar/planetary systems.
This section provides a brief description of substellar companions
observations, encapsulating the most recent and remarking exoplanetary
observations, and highlighting how they were/can be characterized.

##### Taxonomy of substellar objects

Earlier in this chapter, substellar objects were classified as those lacking
sufficient mass to trigger 1H mass fusion. However, within this category,
further subdivisions can be made. Based on their nature and properties,
substellar objects can be categorized into planetary objects, and brown
dwarfs, which can exist either as free-floating objects or companions.

According to the “classical” definition, a BD can be defined as a deuterium-
burning object, whereas a threshold between planets and BDs should occur at
$\sim$ 13 $M_{Jup}$, and between a BD and star $\sim$ 80 $M_{Jup}$, with
metallicities involved on the uncertainties (see e.g., Chabrier & Baraffe,,
1997; Spiegel et al.,, 2011 and further references). When defining BDs and
planets, is still commonly used the definition proposed by the IAU Working
Group Definition (Boss et al.,, 2007), which states:

1) Objects with true masses below the limiting mass for thermonuclear fusion
of deuterium (currently calculated to be 13 Jupiter masses for objects of
solar metallicity) that orbit stars or stellar remnants are “planets” (no
matter how they formed). The minimum mass/size required for an extrasolar
object to be considered a planet should be the same as that used in our Solar
System.

2) Substellar objects with true masses above the limiting mass for
thermonuclear fusion of deuterium are “brown dwarfs”, no matter how they
formed nor where they are located.

3) Free-floating objects in young star clusters with masses below the limiting
mass for thermonuclear fusion of deuterium are not “planets”, but are “sub-
brown dwarfs” (or whatever name is most appropriate).

However, the classical mass classification has been challenged regarding the
deuterium burning criteria and formation scenario of these objects. For
example, some below the deuterium burning mass limit show indicative of cloud
fragmentation formation, including members of quadruple systems with inferred
masses as low as 5 Jupiter masses and free-floating objects with sub-
deuterium-burning masses (Todorov et al.,, 2010; Liu et al.,, 2013). Also,
some observations revealed systems with companions above the deuterium limit
but orbiting at low separations and exhibiting characteristics indicating
formation is a disk (see e.g. Quirrenbach et al.,, 2019). Therefore, more
recent criteria have been proposed.

It is possible to categorize substellar objects based on the presence and
distance from a host star, and the mass ratio $q=M_{*}/M_{sub\\_obj}$. By
definition, protoplanets and planetary objects form within a protoplanetary
disk. In contrast, a BD can either form within a circumstellar disk (without
being the object formed at the center) or exist as an isolated object, forming
separately from the clump that eventually becomes the closest star (refer to
subsections 1.2.1 and 1.2.2).

Given that the distributions of PPDs decline to low frequencies at
approximately 300 au (Andrews & Williams,, 2007), a tentative characterization
limit for the separation between substellar objects detected after the
dissipation of the disk can be established. Specifically, if an object
presents a semi-major axis $a_{sub\\_obj}>$ 300 au from a star, is probably a
brown dwarf, while lower separations can denote any of the mentioned objects,
including BDs.

Furthermore, a substellar object can be ejected from the formation system, or
in the case of BDs, they can form in isolation. In a detection, if no stellar
host is present or if the substellar object is not gravitationally bound to
any discernible star, it can be referred to as a free-floating or field
object. Conversely, if it is gravitationally bound to a star, it will be
classified as a companion. In the context of HCI, providing evidence of
companionship, especially for wide-separation objects, remains challenging as
it requires multi-year observations to disentangle it from a background
source.

Demographic analyses of substellar objects also offer criteria for
distinguishing planets from brown dwarfs. The substellar mass function, as
observed in both previous and recent radial velocity surveys (see e.g.