Full text - University of Amsterdam

University of Amsterdam
The effect of disk structure and dust
properties on scattered light brightness
profiles of protoplanetary disks
bachelor thesis by
Stijn van der Waal
#10247750
examined by
Prof. Dr. Carsten Dominik
Dr. Jean-Michel Désert
supervised by
Tomas Stolker
July 27, 2016
Populair wetenschappelijke samenvatting
Als ik google op ”meest gestelde vragen” is een van de eerste hits: ”Zijn wij alleen?”.
Met ”wij” niet doelend op jij en je maat wanneer jullie je afvragen of de kust veilig is
om ongegeneerd te gaan zitten ruften en boeren, maar op ”wij” als aardbewoners in ons
ogenschijnlijk leeg en gigantische universum. Deze vraag heeft de mens al sinds heugenis
bezig gehouden. Naast de onuitputtende nieuwsgierigheid die de mens van nature kado
heeft gekregen is het bij uitstek deze vraag geweest die ons heeft gemotiveerd astronomie
te gaan bedrijven.
Leven lijkt onlosmakelijk verbonden te zijn aan planeten. De studie naar buitenaards leven
is daarom op zijn beurt onlosmakelijk verbonden aan de studie naar extrasolaire planeten,
planeten anders dan die binnen ons zonnestelsel. Daarnaast vormen exoplaneten een schat
aan informatie en kunnen exoplaneten ons veel leren over ons eigen zonnestelsel.
Kortom, de studie naar exoplaneten is relevant en de moeite waard om aan bij te dragen
door middel van een bachelorproject.
Planeten ontstaan in zogenoemde protoplanetaire schijven. Deze schijf-vormige wolken
van stof en gas ontstaan tijdens de geboorte van een ster. Dankzij moderne technologien,
zoals de SPHERE-VLT (Spectro-Polarimetric High-contrast Exoplanet REsearch - Very
Large Telescope), is men in staat deze schijven heel nauwkeurig waar te nemen.
Gebruik makend van een computerprogramma hebben wij protoplanetaire schijven gesimuleerd. Het voordeel is nu: niet moeder natuur, maar wij hebben de touwtjes in handen.
Door het programma specifieke instructies te geven bepalen wij de eigenschappen van de
schijf.
Door heel nauwkeurig ieder foton te volgen dat vanuit de ster, door de schijf naar een
denkbeeldige observeerder reist, genereert het programma ook plaatjes identiek aan die
van de VLT-telescoop. De plaatjes kunnen hierdoor direct in verband worden gebracht
met de eigenschappen van de schijf.
Het computerprogramma luistert naar instructies in de vorm van input-parameters. De
waarden van deze parameters specificeren dus de eigenschappen van de schijf. In dit onderzoek hebben we het programma gedraaid voor duizenden unieke combinaties van waardes
voor de input-parameters. Op deze manier hebben we kunnen bepalen hoe de plaatjes
afhangen van de eigenschappen van de schijf.
We hopen dat we met dit onderzoek kunnen helpen bij de analyse van SPHERE waarnemingen en zo indirect te kunnen bij dragen aan de studie naar exoplaneten in het algemeen.
1
Abstract
CONTEXT SPHERE at the VLT provides us with high resolution spacially resolved disk
images of protoplanetary disks. Resolved disk images are a topic of recently increasing
interest.
AIMS We aim to define in a qualitative manner the relation between the parameters total
dust mass (Mdust ), maximum dust grain size (amax ), dimensionless turbulence parameter
(α), scale height flaring index (β) and the surface brightness profile of protoplanetary disks
and to inspire future quantitative in-depth research.
METHODS We use a 2-D radiative transfer code MCMax to generate disk models and
calculate corresponding spacial images. We run two grids of simulations. One grid assumes a self-consistent vertical structure and relies on the Monte Carlo Radiative Transfer
method and the notion of dust settling. The other grid assumes a fixed vertical structure
parameterized by β.
RESULTS We speculate on a direct relation between β and α. We suggest the existence of
a parameter acouple (α). We distinguish between two regimes. β < 1 and β > 1. In the former the notion of self-shadowing leads to steep slopes of the profiles. In the latter, a flaring
geometery leads to more gradual slopes of the profiles. Both processes can be amplified
by an increase in Mdust . Considering self-consistent disk models, we speculate a similar
process of amplification. Finally, we hypothesize the notion of turbulent saturization.
Contents
1 Introduction
1.1
Disk Physics
3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.1.1
From Cloud to Disk . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.1.2
Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2
Imaging of Scattered Light
. . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.3
Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.3.1
1.4
Monte Carlo Radiative Transfer . . . . . . . . . . . . . . . . . . . . . 10
Thesis Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Disk Models
12
3 Results
17
4 Discussion
20
1
Appendices
29
A Additional contour plots for values amax = 1 and amax = 100
30
B Imaging in R-band for none inclined disks (0.1 ◦ )
33
C Complete list input parameters MCMax self-consistent models
34
D Complete list input parameters MCMax fixed models
37
E Python script running grid MCMax simulations, generating brightness
profile, fitting brightness profile
40
2
CHAPTER
1
Introduction
Modern technology in the form of VLT/SPHERE allows us to observe protoplanetary
disks, natural byproducts of star formation, even as far as 145pc away from the earth in
very fine detail (Figure 1.1). This new form of data promotes new forms of research.
Planets form in protoplanetary disks. In this thesis we contribute to the study of planet
formation by aiming to link resolved scattered light images to the nature of protoplanetary
disks. We use the 2-D radiative transfer code MCMax developed by Michiel Min [8] to
model circumstellar dust disks. A disk model is generated for each specified set of physical
parameters. Also, MCMax provides us with a scattered light image for each disk model.
This allows us to link scattered light images to the physical nature of the disk.
The function of this chapter is to provide basic context. We note that the content lacks in
fine detail. Also, because scattered light images depend primarily on the vertical structure
of protoplanetary disks [10], the emphasis in this paper is on the vertical structure. For a
more thorough and wider context we refer to [2].
In section 1.1.1 we provide a simplified explanation on how protoplanetary disks form. In
section 1.1.2 we touch upon the physics that leads to the vertical structure of the gas and
we introduce the parameter β. In this section we also discuss interaction of dust and gas
within the disk which allows us to introduce the parameter α. Section 1.3 elaborates on
the numerical methods used in this paper and we conclude the chapter by formulating the
thesis objectives in section 1.4.
3
Figure 1.1: SPHERE observation of the protoplanetary disk around SAO
206462 [12]
1.1
1.1.1
Disk Physics
From Cloud to Disk
The space in between solar systems is not empty. Rather, it contains the interstellar
medium (ISM): matter in the forms of e.g. ionic, atomic and molecular gas, cosmic rays
and dust particles. The interstellar medium is turbulent and as a result its density, temperature and velocity fields are highly varied and cover a wide range of magnitudes. Stars,
objects that generate enough energy by nuclear fusion to balance the surface radiation
losses, originate from molecular clouds, regions within the ISM of densities high enough
to allow molecules to form. When the density of a cloud of molecular gas gets sufficiently
large for the inward orientated gravitational force to overpower the counteracting forces
resulting from gas pressure and magnetic fields, it starts to collapse. Although such a disturbance of a stable system can occur spontaneously, it can also be triggered by external
factors in the form of e.g. radiative pressure caused by nearby supernovae explosions or
collisions with other interstellar clouds [2].
Because of its turbulent nature the collapsing molecular cloud has nonzero angular momentum. As a result, the collapsing cloud will form into a star-disk system. A particle
situated on the rotational axis of the cloud wil not have any angular momentum. Gravity
will cause it to fall straight down onto the core. A particle in the collapsing cloud located
elsewhere however, will in addition to the gravitational pull, be affected by a centripetal
force (1.1):
GmM∗
mv 2
F⊥rot.axis = −
(1.1)
+
2
r
s
At some point during its infall, it will reach a point where the component of the gravitational pull in the direction perpendical to the rotational axis will be balanced by the
centripetal force. In the direction parallel to the rotational axis there is no force counteracting gravity. In other words, instead of accreting onto the core, it will accrete the
midplane and end up in a keplerian orbit around the star. Because the angular momentum
of the particle is conserved along its trajectory during its infall, the radius at which it will
accrete onto the disk depends on its initial location in the cloud. Particles initially located
on the midplane of the system will end up at the largest radius: the centrifugal radius.
4
Initially, the centrifugal radius is a measure for the size of the protoplanetary disk, but
due to redistribution of angular momentum through viscous spreading the disk will grow
to larger sizes [2].
1.1.2
Structure
Vertical structure gas
A protoplanetary disk consists of dust and gas. By assuming the gas is in hydrostatic
equilibrium, and Mdisk << M∗ so that the gravitational potential of the disk can be
neglected and only stellar gravity has to be considered, its vertical structure can be deduced
in the following way.
The vertical component of the gravitational acceleration is
gz = sin θ
GM∗
GM∗
=z 3 .
2
d
d
2
A thin
pdisk is assumed, so r >> z, and r ≈ d from which follows that gz ≈ zΩ , where
Ω = GM∗ /r3 is the Keplerian angular velocity. Balancing to the acceleration due to
the vertical pressure gradient of the gas, and writing the pressure as P = c2s ρ, with cs the
speed of sound gives:
dP
= −ρgz
dz
.
dρ
Ω2
= − 2 zdz
(1.2)
ρ
cs
The temperature of the gas is determined by the stellar radiation, so as a reasonable
approximation, the gas is assumed to be vertically isothermal, i.e. the speed of sound is
assumed to be independent of height. Integrating (1.2) then gives:
2
ρ(z, r) = ρ0 (r)e
z
− 2H(r)
2
,
(1.3)
where ρ0 (r) is the mid-plane density which can be written in terms of the surface density
(T )
Σ
ρ0 = √12π H
, and H(r) = csΩ
is the vertical disk scale height. The geometery of the disk
is determined by the radial dependence of the scale height, which can be parameterized
as H(r) ∝ rβ , where β is called the flaring index. From the ideal gas law the radial
dependence of the speed of sound is found to be cs ∝ T 1/2 . If we parameterize the radial
dependence of the temperature as T ∝ r−γ , the scale height varies as
H(r) ∝ r(3−γ)/2 .
(1.4)
For values of β > 1, or a temperature profile of T (r) ∝ r−1 or shallower, the disk will have
a concave, bowl-like shape (Figure 1.2a). All points on the surface of such a flaring disk
have a clear line of sight to the star. Also, as seen from the star, a flaring disk covers a
larger solid angle than shapes corresponding to lower values of β. A flaring disk in general
will therefore be exposed to more stellar radiation than flat disks (see Figure (1.2)).
5
(a) β > 1
(b) β = 1
(c) β < 1
Figure 1.2: A schematic representation of the radial dependence of the vertical scale height for different values of β.
Dust settling and fragmentation
Unlike a gas parcel, the vertical component of the gravitational force on a dust particle
is not balanced by an upward directed pressure gradient. If the only active forces are
gravity and the aerodynamic drag a dust particle experiences while traveling through a
gassy environment, the settling time, the time it would take a dust particle to sediment
out of the upper layers of the disk onto the mid-plane is tsettle = 1.5 × 105 yr [2]. Dust
particles can clot together after colliding. When this process of coagulation is also taken
into account, due to the fact that whereas the gravitational force grows with ∼ s3 , the
counter-acting drag force is proportional to the frontal area ∼ s2 , dust appears to settle
even faster (tsettle = 103 yr) [6].
However, there is another force in play. Turbulent mixing acts to oppose the vertical
settling of dust particles. In order to quantify turbulence, the timescales of the process
resulting in upward movement (the eddy turn-over time tedd ) and downward movement
(the friction time tf ric ) are compared to get an expression of the Stokes number (St).
mv
The friction time tf ric = |F
, where m is the grain mass and v is the relative velocity
D|
between the dust and gas, is the time scale in which a dust grain is being significantly
influenced by the gas. The drag force FD is dependent on the characteristics of the dust
grain. Calculating the drag force there are two different regimes to be considered, both
dependending on the relative grain size to consider. Grains of size s < λ, where s is the
grain radius and λ is the average distance travelled by a gas particle between succesive
colissions (mean free path), are in the Epstein regime. The Stokes regime considers grains
of size s > λ. As in this paper grains of sizes s < 1 cm are considered and λ is greater
than tens of centimters under most disk conditions [2], the Epstein regime is relevant. In
the Epstime regime the gas can be treated as a collisionless ensemble of molecules with a
Maxwellian velocity distribution. The drag force in the Epstein regime is:
FD = −
4π
ρgas s2 vcs ,
3
(1.5)
in which ρgas is the gas density. The friction time can then be written as:
tf ric =
3 m
,
4 σρgas cs
(1.6)
where σ = πs2 is the cross-sectional area of the dust grain. The eddy turn-over time
depends on the size of the largest eddies and their typical velocity and is taken to be
comparable to the orbital period tedd = Ω−1 . The Stokes number then becomes:
(1.7)
6
Figure 1.3: An artist’s impression of the effect of α on the dust structure.
Darker colours correspond to larger grain sizes. The value of α related to the
upper figure is higher than the on related to the lower figure. This illustrates
the phenomena that higher α leads to stronger vertical stirring.
St =
tf ric
3m Ω
=
.
teddy
4 σ ρgas cs
(1.8)
The Stokes number quantifies the coupling of the dust and gas. A Stokes number smaller
than one indicates strong coupling whereas Stokes numbers larger than one correspond
to decoupled dust particles. Note that the coupling strength depends on the dust grain
size. This leads to stratification (Figure 1.3). Assuming that the dust can be treated as a
’dust-fluid’ with density ρd the evolution of the dust is described by the advection-diffusion
equation:
∂
ρd
∂ρd
∂d
∂
=D
ρgas
(Ω2 tf ric ρd z),
(1.9)
+
∂t
∂z
∂z ρgas
∂z
where
cs H
,
(1.10)
1 + St2
the diffusion coefficient, with α the parametrization of the turbulent mixing strength.
D=α
Considering a solid spherical particle of radius s and constant density ρ. When traveling
through a gas disk, the force excerted by the gassy environment, decelerating the particle,
is proportial to the frontal area that the particle presents to the gas πs2 . The acceleration
caused by the gas drag is in turn equal to drag force devided by the mass of the particle
(ρ 43 πs3 ). The aerodynamc drag therefore decreases with the particle size, as s−1 , and
becomes negligable as the particle grows to planetesimal size, as s−1 . In short, the larger
dust grains grow, the less friction they experience. Turbulent mixing would not prevent the
dust from settling into a thin layer on the mid-plane if the particles would be able to grow
7
to limitless sizes. Due to the smaller friction force, larger particles however travel with a
higher velocity through the gas. Particle growth will be halted then when the velocity gets
high enough for a subsequent collision to be energetic enough to lead to fragmentation.
Because of limited particle growth and fragmentation the grain size distribution within
the disk contains both the smallest particles of the order of a micrometer, and the largest
particles reaching sizes of a few millimeter.
1.2
Imaging of Scattered Light
In this section we briefly describe some characteristics of light scattering by dust grains.
We then describe how polarization helps in the imaging of protoplanetary disks.
How light is scattered depends on the properties of the scattering grain. Large grains with
respect to the wavelength of the scattered light tend to forward-scatter. Relatively small
dust grains scatter more isotropic. [9]. Light scattered by large particles therefore tends
to dissipate into the disk. Light scattered by small particles however is more likely to be
scattered into the observer’s direction (Figure 1.4). A disk comprised of large dust grains
then is expected to appear fainter than a disk containing smaller particles. Lowering of
the wavelength too is expected to have an negative affect on the brightness of the disk.
In observing protoplanetary disks, the main obstacle is to detect the disk despite the
immense contrast created by the bright central star drowning the signal from the disk.
Fortunately, scattering leads to polarization. Light coming directly from the star is not
polarized. Making use of this distinction, the scattered light can be filtered from the total
signal. In this paper therefore, the disks models are analyzed using images of polarized
light. The technique of distracting the signal of scattered light from the total intensity is
described in depth in [7].
There are two sources of light that influence the appearance of a disk. One is stellar light
that is scattered. The other one is thermal emission by the disk itself. We finally note
that the disk’s thermal emission can also be scattered by the disk’s dust grains. An image
of a protoplanetary disk of polarized light therefore can also be affected by the thermal
emission of the disk [10].
8
Figure 1.4: An illustration of the notion of grain size dependent scattering.
Small grains scatter more light into the observer’s direction (vertically in this
figure) than larger, more foreward-scattering particles. This figure is taken
from [11].
1.3
Numerical methods
In this section a brief overview is given on the numerical techniques that are used in this
paper. As most of a protoplanetary disk’s features are influenced by stellar radiation,
in the study of protoplanetary disks it is relevant to understand radiative transfer: the
transfer of energy by photons. Radiative transfer problems concerning models of highly
symmetric objects can be solved analytically. For less spherical objects with more complex
geometeries like a protoplanetary disk however, we rely on numerical methods. For this
thesis we used the 2D-radiative transfer code MCMax. MCMax is able to resolve the
temperature structure and the vertical gas and dust structure of a circumstellar disk
for given parameters in an iterative manner. Effectively, through balancing the upward
turbulent mixing and downward gravitational settling as expained in section 1.1.2, MCMax
calculates the dust scale height using:
s
(1 + γ0 )−1/2 α
Hg (r)
(1.11)
Hd (r, a) =
τf (r, a)Ω(r)
τf (r, a) =
ρd a
,
ρmid (r)cs,mid (r)
(1.12)
where the parameter τ0 depends on the nature of the turbulence and in general takes
on a value 5/3 > γ0 < 3 [4]. For compressible turbulence, considered in this paper,
γ0 = 2. Alternatively, the vertical structure can be set manually by parameterizing the
9
radial dependence of the scale height through β as aforementioned (section 1.1.2). The
temperature structure throughout the disk is calculated using the probabilistic Monte
Carlo Radiative Transfer technique. MCMax uses the method of ray-tracing to provide
observable quantities such as spectral energy distributions and spatially resolved disk
images. In the following section we elaborate on Monte Carlo Radiative Transfer in more
detail.
1.3.1
Monte Carlo Radiative Transfer
The stellar radiation is split among Nγ photon packets emitted over a time interval δt of
equal and constant energy Eγ = Lδt
Nγ . Space is divided into grid cells of volume Vi constant
temperature Ti , where i is the cell index. Each of the photon packets is ’traced’ one after
the other. To begin with, a random number sets the direction in which a packet is emitted
by the star. It sets out to travel a distance L, again determined by a random number but
now sampled from the probability distribution
P (L) = e−nσL = e−τ
(1.13)
where n is the number density of scatterers and absorbers, or in this case dust particles,
RL
σ their cross section area, and τ = nσL, or in general τ = 0 nσds the optical depth.
Whether the photon will be scattered or absorbed depends on the albedo: the probability
a photon-matter event is a scattering event.
If the random number generation yields a scattering event, the direction of its path will
be altered. The change in direction is determined by a random number sampled from an
angular probability distribution, or phase function. The shape of the phase function is
dependant on the type of material that is scattering.
In case of absorption, the energy of the absorbed photon packet is deposited into the
grid cell. In order to conserve radiative equilibrium, the absorbed photon packet is to be
reemitted with a different frequency according to the temperature of the grid cell. By
equating the total absorbed energy by the grid cell with the energy it emits, Bjorkman &
Wood (2001) [3] provide the following relation for calculating the temperature of the grid
cell:
Ni L
,
(1.14)
κp σSB Ti4 =
4πmi N
with κp the Planck mean opacity, σSB Stefan-Boltzmann constant, Ni the total number
of absorbed packets by the grid cell with index i , mi its mass, L the luminosity of the star
and N the number of photon packets into which the stellar luminosity is devided. With
every step of absorbing and directly reemitting a photon packet, the temperature of the
grid cell increases.
A more in depth explanaition of the Monte Carlo radiative transfer method is provided
by Bjorkman and Wood (2001) in [3] and Whitney (2011) in [13].
10
1.4
Thesis Objectives
In this paper the slope of the surface brightness profile of a protoplanetary disk is studied
by using numerically generated disk models. Two different methods of modelling disks are
, differing in their calculation of the vertical structure.
The first method computes a self-consistent gas scale height. The method is reliant on
the Monte Carlo radiative transfer technique and grain size dependent process of dust
settling which is directly coupled to the turbulent parameter α. This method produces
disk models with a stratified dust component.
The second method takes a fixed scale height, parameterized by β. Dust is assumed to
be well coupled to the gas. The dust scale height Hd and the gas scale height Hg are
therefore taken to be equal. Moreover, the disk is assumed to be well-mixed implying no
stratification.
Both methods assume a Gaussian vertical distribution for both gas and dust:
z2
.
(1.15)
ρ(r, z) ∝ exp −
2H(r)2
Using these methods we aim to define in a qualitative manner the relation between the
parameters total dust mass (Mdust ), maximum dust grain size (amax ), dimensionless
turbulence parameter (α), scale height flaring index (β) and the surface brightness profile
of protoplanetary disks and to inspire future quantitative in-depth research.
11
CHAPTER
2
Disk Models
We distinguish between two different methods of generating disk models. The first method
calculates the a self-conssitent vertical structure in an iterative manner using the Monte
Carlo radiative transfer technique and the process of dust settling. The dust component
in the resulting disk models (from now on: self-consistent models) is stratified. The second method generates disk models with a fixed vertical structure. The radial scale height
dependence of both gas and dust is parameterized by a fixed β. The resulting disk models
(from now on: fixed models) are well-mixed meaning no stratification and a constant size
distribution throughout the disk.
The parameters are sampled from the sets: Mdust = {10−6.0 , 10−5.9 , ..., 10−3.0 }
M ; α = {10−4 , 10−3.9 , ..., 10−1 }; amax = {10−1 , 100 , ..., 103 }µm;
β = {0.9, 0.95, .., 1.3}. For each unique combination of the parameters Mdust , α and amax
a self-consistent model is generated. For each unique combination of the parameters Mdust ,
α and amax a fixed model is generated. The number of disk models is determined by the
product of the lengths of the relevant sets. Consequently, we analyzed images of 4805
self-consistent models and 1989 fixed models.
The values of some of the parameters used to calculate disk models of both free and fixed
parameters in this section are presented in table 2.1. For a full list of MCMax input parameters, see appendix C and D.
As a method of analyzing the images, we consider a surface brightness profile. The brightness profile is taken along the major axis of the disk. We fit each profile with a power-law.
The power-law indeces are plotted in a contour plot with fixed value of amax . For the
self-consistent models the axes of the contour plot are Mdust and α . For the fixed models
the axes are Mdust and β. λ Is the wavelength of the image produced and is thus not used
12
Parameter
Value
Self-consistent
Mstar
Tstar
Lstar
IncAngle
amin
amax
apow
Rin
Rout
denspow
sh
rsh
β
Mdust
α
λ
Unit
Fixed
1.6
1.6
6750
6750
7.8
7.8
30.0
30.0
0.1
0.1
−1
0
3
−1
0
{10 , 10 , ..., 10 }
{10 , 10 , ..., 103 }
−3.5
−3.5
20
20
200
200
−1
−1
10
10
100
100
1.25
{0.9, 0.95, ..., 1.3}
{10−6 , 10−5.9 , ...,10−3 } {10−6 , 10−5.9 , ..., 10−3 }
{10−4 , 10−3.9 , ...,10−1 }
10−1.25
1245
1245
M
K
L
◦
µm
µm
dimensionless
AU
AU
AU
AU
AU
AU
M
dimensionless
nm
Table 2.1: An overview of the parameters and corresponding values used by
MCMax to calculate disk models. Note in particular the difference between
the self-consistent and fixed models. β is kept fixed for the self-consistent
models whereas α is kept fixed for the fixed models. We elaborate on the
following parameters: IncAngle is the angle between the orbital plane of
the disk and the direction of observation; amin Is the minimum grain size
below which the grain abundance is set to zero; apow Defines the grain size
distrubution through the mrn-like distribution N (a) ∝ aapow ; Rin and Rout
specify the inner and outer radius of the disk respectively; denspow Is the
power-law index for the surface density Σ ∝ r−denspow ; sh And rsh are the
vertical and horizontal coordinates respectively that specify a scale height
reference point.
13
in the calculation of the disk. λ = 1245nm Corresponds to a SPHERE J-band filter [1].
See Figure 2.1.
14
Figure 2.1: A schematic representation of the method used to analyse the
large amount of disks. The figure corresponds to analyzing self-consistent
models. This choise is arbitrary as both fixed and self-consistent models are
analyzed in a similar fashion. The cube corresponds to a three-dimensional
parameter space with axes corresponding to Mdust , β and amax . Each grid
point represents a unique combination of the parameters within the sets as
specified in table 2.1. Each grid point can be linked to a disk image. The
white line in the disk image schematically represents the area of the image
considered in order to generate a brightness profile. The area is taken along
the disk’s major axis. A cut bound by the angles −2.5◦ and 2.5◦ is used. The
green line in the profile schematically represents a power-law fit. Effectively,
each unique combination of the three parameters can be linked to a powerlaw fit index. The power-law fit indices dependent on Mdust , α and amax
are presented in contour plots corresponding to constant values of amax as
schematically illustrated in Figure 2.2. Arbitrary, in this figure a power-law
fit index of −3.2 is chosen.
15
Figure 2.2: See caption Figure 2.1.
16
CHAPTER
3
Results
The dependence of the slope of the surface brightness profile of a disk model, represented
by the exponent of the power-law fit, on the parameters Mdust , β and amax is shown in
Figure (3.2) using three contour plots. Figure (3.1) shows the dependence of the exponent
on the parameters Mdust , α and amax . The data in Figure (3.2) are related to the disks
generated with a fixed vertical structure and the data in Figure (3.1) correspond to the
disks generated in an iterative manner as described in section 2. amax is held fixed in
both figures at the values 1 × 10−1 µm,1 × 101 µm and 1 × 103 µm for the upper, middle
and bottom plot respectively.
17
Figure 3.1: Contour plots of the power-law fit index (exponent) corresponding to the profile of scattered light intensity of a fixed disk model. The contour plots illustrate the dependence of the profile on the parameters Mdust ,
β and amax . The contour plots correspond to a value of amax = 0.1 µm,
10 µm, 1000 µm for the upper, middle and bottom contour plot respectively.
Interesting features are remarked and interpreted in section 3.
18
Figure 3.2: Contour plots of the power-law fit index (exponent) corresponding
to the profile of scattered light intensity of a self-consistent disk model. The
contour plots illustrate the dependence of the profile on the parameters Mdust ,
α and amax . The contour plots correspond to a value of amax = 0.1 µm,
10 µm, 1000 µm for the upper, middle and bottom contour plot respectively.
Interesting features are remarked and interpreted in section 3.
19
CHAPTER
4
Discussion
We have examined the dependence of the slope of the surface brightness profile of fixed and
self-consistent disk models by running one grid of simulations for both types in parameter
space Mdust , β and amax and Mdust , α and amax respectively. We fitted the profiles with a
power-law. The dependency of the slope, characterized by the power-law exponent, on the
parameters is presented in the form of contour plots in Figures 3.1 and 3.2, (from now on
fixed plots self-consistent plots respectively) . In this section we discuss the results given
in the preceding section. We identify and interpret interesting trends. We hypotesize on
their explanation.
Fixed models
In Figure 3.2 we identify the following features. For a value of β ∼ 1 the exponent
is independent of Mdust and amax and maintains a value of ∼ −4.2. The exponent is
inversely proportional to Mdust in the regime β < 1. The decrease of the exponent as a
result of an increase Mdust gets stronger for lower β.
In contrast, the exponent is proportional to Mdust in the regime β > 1. Here an increase
of the exponent as a result of an increase of Mdust gets stronger for higher β.
As an hypothesis, we attribute this feature to an amplification of the flaring geometery of
the dust component of the disk with increasing Mdust . On the other hand, an amplification
of self-shadowing (a phenomena elaborated on in [5]) as a result of increasing Mdust could
20
account for the decrease of the exponent with increasing Mdust in the non-flaring regime
β < 1.
Moreover, although not evident from Figure 3.2, we speculate an overall increase of the
exponent with increasing amax .
Self-consistent models
In Figure 3.1 we identify the following features. Like in Figure 3.2, Figure 3.1 shows a
straight isoline for which the exponent has a constant value of around ∼ −4.2. Arguably,
this value is constant with respect to a changing amax . The isoline is neither vertical nor
horizontal. From the straight isoline then, we deduce the following: for an exponent value
of ∼ −4.2, the change in α that is needed to compensate for the effect a change in Mdust
has on the exponent is directly proportional to the change of Mdust .
Hypothesizing on the explaination of this feature, we suggest that a decrease of scatterers
lead to a decrease of the exponent. Noting that lowering Mdust implies a shift along
the number density axis of the grain size distribution which in turn leads to less dust
grains, a decrease of Mdust leads to a decrease of the exponent. This decrease can be
compensated by increasing α, as an increase in vertical stirring leads to an overall higher
vertical structure. A thicker disk subsequently leads to scattering of a larger fraction of
the stellar radiation and we suggest a higher exponent.
In the regime above the straight isoline, the effect of increasing the value of alpha on the
exponent is not constant but dependent on the value of alpha. The effect on the exponent
by an increase of alpha is less significant for larger values of alpha. In the extreme case,
for high values of dust mass (Mdust > 10−5 ) and high values of alpha (α > 10−1.5 )
increasing alpha does not change the exponent. In other words: for high values of dust
mass, there seems to be a value of alpha for which the disk becomes turbulently saturized.
Increasing the value of alpha beyond this saturation value will not change the net effect
turbulent mixing has on the structure of the scatterers (dust particles).
Increasing Mdust in a fully mixed, or turbulently saturized disk results in a steady increase
in the exponent. This is in contrast with the dust mass dependence at lower values of α.
At lower values of α, corresponding to disks that are not fully mixed, the dust mass
dependence decreases with increasing dust mass. This feature results in a fanning-effect,
or diverging of the contour lines evident in particular at lower values of amax . For an
explanation we rely on further research.
We consider the regime below the isoline. In the contour plot corresponding to amax =
100 µm in particular (Figure A.1), for low values of α, there is a certain value of Mdust
depending on α below which a change in Mdust does not affect the exponent.
In general, the effect of α on the exponent appears to be decreasing with lower Mdust ,
opposite to the situation above the isoline.
To give a more quantitative explaination of some of the remarked features, a grain size
value dependent on α in a proportional manner, acouple (α) is hypothesized above which a
21
dust grain is not influenced by turbulent mixing. The notion of the existence of a quantitiy
acouple (α) is in line with :
1. An increase in Mdust for large α leads to a change in the exponent. Namely: for
large α, and thus large acouple (α) an increase of Mdust is distributed among particles
of which a large part is influenced by turbulent mixing. An increase in Mdust leads
to a change in the vertical distribution of dust. A change in the vertical distribution
leads to a change in the exponent.
2. An increase in Mdust leads to a change in the exponent throughout the contourplot for low values of amax only. A low value of α corresponds to a low value
of acouple (α). However, aslong as amax < acouple (α), all dust mass will be distributed among particles influenced by turbulence. This reasoning can be used
to explain the straight contour lines below the ∼ −4.2-isoline. We hypothesize
that amax = 0.1 µm < min(acouple (α)) for the range of α used in this paper
α ∈ [10−4 , 10−1 ]. See Figure 4.1.
3. At high values of amax , the effect of changing Mdust on the exponent becomes less
significant with lower α. Namely: for high amax , the total dust mass is distributed
among a wide range of grain sizes. acouple (α) Determines which fraction of this
size range is influenced by turbulence. The smaller α, the smaller acouple (α), the
smaller the fraction of dust particles influenced by turbulence for a given amax . For
high values of amax then, the lower α, the less significant is the effect on the vertical
dust structure and thus the exponent by an increase of Mdust .
4. At large values of α the contour lines appear to be horizontal. Horizontal contour
lines can still be explained by the notion of saturation of turbulent mixing.
Figure 4.1: Two mrn grain size distributions are presented in a logarithmic
plot. The upper, straight line corresponds to a certain value of Mdust . The
dashed line corresponds to a lower value of Mdust . The light grey shaded area
corresponds to the number of particles over which Mdust is distributed. The
figure illustrates that for values of amax < acouple all particles in the disk will
be influenced by turbulence.
22
Figure 4.2: Two mrn grain size distributions are presented in a logarithmic
plot. The upper, straight line corresponds to a certain value of Mdust . The
dashed line corresponds to a lower value of Mdust . The light grey shaded area
corresponds to the number of particles over which Mdust is distributed. The
darker shaded area corresponds to the fraction of particles that are influenced
by turbulence. Note that there are two phenomena leading to an increase in
the scattered light intensity with increasing α. First: an increase in α leads to
an increase of the fraction of particles that will be influenced by turbulence.
Second: an increase in turbulent mixing strength leads to greater flaring
because the dust grains reach greater heights because of a stronger upward
turbulent stirring force. For amax > acouple both phenomena are true. For
amax < acouple , the former is false.
However, an explanation in agreement with the existence of a quantity acouple (α) is not
obvious for the following traits:
1. The apparent overall decrease of the exponent in moving from amax = 0.1 µm to
am ax = 1 µm and the overall increase of the exponent with increasing amax for
values of amax > 1 µm.
2. The vertical contour lines in contour plots for high values of amax in the regime of
low α and low Mdust (evident in particular for amax = 100 µm). The relevant area
is highlighted (Figure C presented in the appendix))
We further note that for large values of α we see no strong amax dependence. We assume
therefore for α ∈ [10− 4, 10− 1] that max (acouple (α)) > 1000 µm, the largest considered
value of amax . In other words, for our disk models, the largest value of acouple (α) does not
exceed the largest value of amax . See Figure 4.1.
23
A Speculative Analogy
In this section we consider a probable similarity between the self-consistent plots and fixed
plots which could provide more insight in the nature of the examined parameters. Inspired
by the straight isoline corresponding to a exponent value of ∼ −4.2 present in both the
self-consistent and the fixed plots, we introduce a new set of orthogonal axes in the selfconsistent plots. A β-axis and y-axis (Figure 4.3). We hypothesize β to be the flaring
index of the dust scale height. Along the y-axis, β = 1.
An analogy as such would imply a direct relation between α and β. Moreover, moving
upward along the y-axis would lead to amplification of self-shadowing (for β < 1) and
flaring (for β > 1) analogous to increasing Mdust for fixed models.
Figure 4.3: One of the contour plots of Figure 3.1. As a possible analogy we
introduce the orthogonal axes β and y. The y axis is assumed to to be the line
β = 1. We hypothesize that β is the flaring index implying a direct relation
between α and β. As a second hypothesis, we state that moving upward in
the positive y-direction could lead to amplification of self-shadowing in the
regime β < 1 and flaring in the regime β > 1. amax Is chosen arbitrary as
this analogy is not dependent on amax .
Final Remarks
We list some final comments:
• We assumed the fitting algorithm used to be reliable. We have not studied the
fitting algorithm in detail. Bias introduced by unreliable fitting can not be ruled
out.
24
As a second remark on the fitting process we note that inconsistent fitting might
have introduced bias. We used a fixed range of radius over which a power-law fit
was calculated. However, we have observed that for larger Mdust the peak of the
surface brightness profile shifts to smaller radii. This shift could possibly lead to a
biased fit as the profile is not a perfect power-law.
• Although the inclination angle for both the self-consistent models and fixed models
is taken to be 30◦ . We assume that the brightness profile along the major axis
does not change with respect to the inclination angle. In the appendix we present
Figure B.1. The contour plots are produced with parameters as shown in table
2.1 excluding the values of λ and IncAngle. The former is set to λ = 658nm,
corresponding to a SPHERE IRDIS R-band filter [1]. The latter is set to 0.1◦ .
A profile is made for the azimuthally integrated intensity of the disk instead of
exclusively the intensity along the major axis. The contour plots supports our
assumption as there appears to be no significant discrepancy. Moreover, although
not evident, an overall increase in the exponent suggest brighter disks in R-band
which would conflict with the theory described in section 1.2.
• We propose the following idea for futher research. Scattered images of physical
protoplanetary disks can be analyzed in a manner similar to what is done in this
paper. If constraints on certain parameters are known, constraints can possibly be
introduced on one or more other parameters by comparing its surface brightness
profile to profiles of self-consistent disk models.
25
Acknowledgements
I dedicate this page to thank Carsten Dominik for his aid in form of - although sporadic inspiring discussions, and Tomas Stolker for never refraining from supporting and guiding
me (I count a total of 207 emails!).
Bibliography
[1] ESO - sphere filters.
https://www.eso.org/sci/facilities/paranal/
instruments/sphere/inst/filters.html. Accessed: 2016-07-10.
[2] Astrophysics of Planet Formation. Cambridge University Press, 2010.
[3] J. E. Bjorkman and Kenneth Wood. Radiative Equilibrium and Temperature Correction in Monte Carlo Radiation Transfer. The Astrophysical Journal, 554(1):615–623,
2001.
[4] B. Dubrulle, G. Morfill, and M. Sterzik. The dust subdisk in the protoplanetary
nebula, 1995.
[5] C P Dullemond and C Dominik. The effect of dust settling on the appearance of
protoplanetary disks. Astronomy and Astrophysics, 421:1075, 2004.
[6] C. P. Dullemond and C. Dominik. Dust coagulation in protoplanetary disks: a rapid
depletion of small grains. Astronomy and Astrophysics, 986:971–986, 2005.
[7] M Kim, D Keller, and C Bustamante. Differential polarization imaging. I. Theory.
Biophysical journal, 52(6):911–27, 1987.
[8] M. Min, C. P. Dullemond, C. Dominik, A. de Koter, and J. W. Hovenier. Radiative
transfer in very optically thick circumstellar disks. Astronomy and Astrophysics,
497(1):155–166, 2009.
[9] Michiel Min. Dust Opacities. EPJ Web of Conferences, 102:00005, 2015.
[10] Michiel Min. Modeling and interpretation of images. 6:1–12, 2015.
[11] Gijs Mulders. Radiative transfer models of protoplanetary disks: Theory vs. Observations. 2013.
27
[12] T. Stolker, C. Dominik, H. Avenhaus, M. Min, J. de Boer, C. Ginski, H. M.
Schmid, A. Juhasz, A. Bazzon, L. B. F. M. Waters, A. Garufi, J.-C. Augereau,
M. Benisty, A. Boccaletti, T. Henning, A.-L. Maire, F. Menard, M. R. Meyer, M. Langlois, C. Pinte, S. P. Quanz, C. Thalmann, J.-L. Beuzit, M. Carbillet, A. Costille,
K. Dohlen, M. Feldt, D. Gisler, D. Mouillet, A. Pavlov, D. Perret, C. Petit, J. Pragt,
S. Rochat, R. Roelfsema, B. Salasnich, C. Soenke, and F. Wildi. Shadows cast on
the transition disk of HD 135344B. ArXiv e-prints, March 2016.
[13] Barbara A. Whitney. Monte Carlo Radiative Transfer. Bulletin of the Astronomical
Society of India, pages 1–27, apr 2011.
28
Appendices
29
APPENDIX
A
Additional contour plots for values amax = 1 and
amax = 100
30
Figure A.1: Contour plots of the power-law fit index (exponent) corresponding to the profile of scattered light intensity of a self-consistent disk model.
The contour plots illustrate the dependence of the profile on the parameters
Mdust , α and amax . Highlighted in the amax = 100 contour plot is the regime
in which contour lines appear to be vertical for unknown reason. Excluded
from chapter 3 for technical editing reasons.
31
Figure A.2: Contour plots of the power-law fit index (exponent) corresponding to the profile of scattered light intensity of a fixed disk model. The contour plots illustrate the dependence of the profile on the parameters Mdust ,
α and amax . Excluded from chapter 3 for technical editing reasons.
32
APPENDIX
B
Imaging in R-band for none inclined disks (0.1 ◦)
Figure B.1: See chapter 3
33
APPENDIX
C
Complete list input parameters MCMax self-consistent
models
**** general setup ****
startype=’KURUCZ’
Tstar=6750d0
Mstar=1.6d0
Lstar=7.8d0
Distance=140d0
IncAngle=30d0
**** density setup ****
denstype=’ZONES’
Nrad=150
Ntheta=60
computepart01:standard=’DIANA’
computepart01:amin=0.01d0
34
computepart01:apow=3.5d0
computepart01:fmax=0.8d0
computepart01:ngrains=20
computepart01:blend=.true.
computepart01:fcarbon=0.3d0
computepart01:porosity=0.25d0
dirparticle=’particles’
zone1:rin=20d0
zone1:rout=200d0
zone1:rexp=100d0
zone1:denspow=1d0
zone1:amin=1d-2
zone1:apow=3.5d0
zone1:gamma_exp=1d0
zone1:fix=.false.
zone1:sizedis=.true.
zone1:sh=10d0
zone1:rsh=100d0
zone1:shpow=1.15d0
**** grid refinement ****
Nspan=5
Nlev=10
**** wavelength grid ****
lam1=0.1
lam2=3000
nlam=300
**** scattering ****
scattype=’FULL’
storescatt=.false.
nspike=2
**** disc structure ****
iter=.true.
maxiter=5
epsiter=3.0
fweight=1d0
fixmpset=.false.
mpset=.true.
35
**** output characteristics ****
outputfits=.true.
fastobs=.false.
**** diffusion and interaction limits ****
nphotdiffuse=30
randomwalk=.true.
factRW=4.0
multiwav=.true.
36
APPENDIX
D
Complete list input parameters MCMax fixed models
**** general setup ****
startype=’KURUCZ’
Tstar=6750d0
Mstar=1.6d0
Lstar=7.8d0
Distance=140d0
IncAngle=30d0
**** density setup ****
denstype=’ZONES’
Nrad=150
Ntheta=60
computepart01:standard=’DIANA’
computepart01:amin=0.01d0
computepart01:apow=3.5d0
computepart01:fmax=0.8d0
37
computepart01:ngrains=20
computepart01:blend=.true.
computepart01:fcarbon=0.3d0
computepart01:porosity=0.25d0
dirparticle=’particles’
zone1:rin=20d0
zone1:rout=200d0
zone1:rexp=100d0
zone1:denspow=1d0
zone1:amin=1d-2
zone1:apow=3.5d0
zone1:gamma_exp=1d0
zone1:fix=.true.
zone1:sizedis=.true.
zone1:sh=10d0
zone1:rsh=100d0
**** grid refinement ****
Nspan=5
Nlev=10
**** wavelength grid ****
lam1=0.1
lam2=3000
nlam=300
**** scattering ****
scattype=’FULL’
storescatt=.false.
nspike=2
**** disc structure ****
iter=.false.
maxiter=20
epsiter=3.0
fweight=1d0
fixmpset=.false.
alphaturb=5.62341d-2
mpset=.false.
**** output characteristics ****
38
outputfits=.true.
fastobs=.false.
**** diffusion and interaction limits ****
nphotdiffuse=30
randomwalk=.true.
factRW=4.0
multiwav=.true.
39
APPENDIX
E
Python script running grid MCMax simulations,
generating brightness profile, fitting brightness profile
# -*- coding: utf-8 -*#Gebruikt maar een output map om onder de opslag grens te blijven
import math, numpy, os, time, sys
import matplotlib.pyplot as plt
from astropy.io import fits
from scipy.optimize import curve_fit
#specify directories
Scripts = ’/home/stijn/Desktop/output/scripts’
MCMax_Out = ’/home/stijn/Desktop/output’
Data_Out = ’/home/stijn/Desktop/output’
Figures_Out = ’/home/stijn/Desktop/output/figures’
plt.ion()
i=0
os.chdir(Scripts)
file=open(’parameters_1jun_iter.txt’,’r’)
for line in file:
40
i+=1
line_element=line.split()
os.system("~/MCMax/MCMax ~/Desktop/output/scripts/input_it.dat 1e5 -o %s ~/Desktop/output/scrip
print line_element[0], line_element[1], line_element[2], line_element[3], line_element[4]
os.chdir(MCMax_Out)
while os.path.isfile("ImageQU_i%s_l00001.25_fov00490.0.fits" %line[9:13])==False:
pass
print "Waiting for ImageQU_i%s_l00001.25_fov00490.0.fits" %line[9:13]
time.sleep(10)
print "MCMax is klaar. ImageQU_i%s_l00001.25_fov00490.0.fits wordt geanalyseerd" %line[9:13]
hdulist = fits.open(’ImageQU_i%s_l00001.25_fov00490.0.fits’ %line[9:13])
data = []
data = hdulist[0].data[:,:]
hdulist.close()
def datadr(R,dR,theta0,theta1,data):
’R, dR, theta0, theta1’
schil = []
x0=len(data[0])/2
y0=x0
for y in range(y0-(R+int(dR)),y0+(R+int(dR))):
for x in range(x0-(R+int(dR)),x0+(R+int(dR))):
r = math.sqrt((x-x0)**2+(y-y0)**2) #afstand van midden (249,249) tot
theta = math.atan2(float(-(y-y0)),float(-(x-x0))) +math.pi
#geeft ho
if r**2 < ((R+dR)**2) and r**2 > (R**2) and theta >= theta0 and theta
schil.append(data[y,x])
if len(schil)==0:
meanschil = 0
else:
meanschil = numpy.mean(schil)
return meanschil
bindr = []
r = []
theta0 = 2.5/180.*math.pi
41
theta1 = -2.5/180.*math.pi
r0,r1 = 25,100
r_max = len(data[0])/2-2
for m in range(r_max):
getal = datadr(m,2,theta0,theta1,data)
bindr.append(getal)
r.append(m)
xdata=numpy.asarray(r[r0:r1])
ydata=numpy.asarray(bindr[r0:r1])
rlog=numpy.log10(xdata)
flog=numpy.log10(ydata)
def powerlaw(x,a,b):
return a*x**b
def linear(x,a,b):
return a+b*x
#guess = numpy.array([10000,-15,-2])
print xdata
print ydata
maxfev=800
while maxfev<50000:
try:
fit,cov = curve_fit(linear,rlog,flog,maxfev=maxfev)
break
except:
maxfev+=1000
continue
if maxfev >= 50000:
fit = [1,1]
print ’kan geen fit vinden voor parameters %s.’%line
xx=numpy.linspace(xdata.min(),xdata.max(),50)
yy=powerlaw(xx,10.**fit[0],fit[1])
##Zet fit coefficienten in bestandje.
fit_file = open("fit_1jun_iter.txt","a+")
fit_file.write(str(line))
for k in fit:
fit_file.write(str(k) + ’ ’)
fit_file.write(’\n’)
fit_file.close()
##Zet flux data in bestandje.
os.chdir(Data_Out)
42
z = numpy.array(zip(r,bindr))
f = open(’IA=%s_Md=%s_am=%s_sh=%s_at=%s.dat’ %(line_element[0].split(’=’,1)[-1],line_element[1]
numpy.savetxt(f,z)
f.close()
#
#
#
#
#
#
#
#
#
#
#
#
os.chdir(Figures_Out)
plt.figure(2)
plt.plot(r[25:100],bindr[25:100],’o’,markevery=5,label=’%s_%s_%s_%s_%s’ %(line_element[0].spl
plt.plot(xx,yy,label="fit: %.2f" %fit[1])
plt.ylabel("Surface Brightness "+r’($\frac{mJy}{arcsec^2}$)’)
plt.xlabel("Radius (AU)")
plt.legend(fontsize=’x-small’)
plt.pause(0.0001)
if i % 7 == 0:
plt.savefig(’%s.png’ %str(i))
plt.close(2)
# plt.close(1)
plt.figure(1)
plt.plot(r[25:100],bindr[25:100],’o’,markevery=5,label=’IA=%s_Md=%s_am=%s_sh=%s_at=%s’ %(line_e
plt.plot(xx,yy,label="fit: "+’$%.3fx+^%.3f$’ %(fit[0],fit[1]))
plt.ylabel("Surface Brightness "+r’($\frac{mJy}{arcsec^2}$)’)
plt.xlabel("Radius (AU)")
plt.legend()
plt.title(’Surface Brightness Profile (ImageQU)’)
# plt.pause(0.0001)
# plt.savefig(’IA=%s_Md=%s_am=%s_sh=%s_at=%s.png’ %(line_element[0].split(’=’,1)[-1],line_eleme
43