Bachelor Thesis Astronomy
Impact of protoplanetary disks and their envelopes on X-ray observations of young
stars
Olivier van Bon
Supervisors:
Prof. Dr.
I.E.E. (Inga) Kamp
Dr.
C. (Christian) Rab
July 7, 2018
X-ray sources (on the order of 1028erg s−1 to 1031erg s−1). This has been observed to be the case for Class I, Class II (CTTS) and Class III (WTTS) stars. Class I stars are the youngest of the three and still have an envelope present. Because of their disk-like nature, protoplanetary disks are randomly inclined towards an observer.
Aims. The impact of X-ray radiation on inclined protoplanetary disks and their envelopes will be studied. Also ProDiMo’s capabilities to look at this impact are investigated.
Lastly, the methods used by X-ray projects on protoplanetary disks, e.g. XEST, will be put to the test with the new found results.
Methods. The two dimensional, radiation thermo-chemical disk code Protoplanetary Disk Model (ProDiMo) was used to model the disks and disks with envelope and produce SEDs under varying inclinations. The absorption decrements were correlated with the total hydrogen column densities. Results. With increasing inclination, absorp- tion/extinction increases (emission decreases). Soft stellar X-rays are largely absorbed in disks and even more so in disks with an envelope. Surprisingly, hard X-rays are actually absorbed less in disks with envelopes than in disks. The critical column densities (where the decrement in the SED at a certain energy steeply increases) line up well with the gas opacities for the corresponding energies.
Conclusions. The soft and hard X-rays behave very differently in the observed SEDs, but have a clear impact on them. Scattering does not seem to be negligible for the hard X-rays and may actually be able to increase emission by scattering photons that would otherwise be absorbed. ProDiMo’s grid may be too limited, because the absorption starts at almost the same angle as the grid. It is unknown if the extinction behaves properly below those inclinations. Further research may show if this is indeed the case.
The XEST project’s fits may be biased because the decrease in SED barely changes with column density before and after the gas becomes optically thick at the critical column densities. These results are the first step towards a better understanding of the impact that stellar X-ray radiation from inclined protoplanetary disks has on observed SEDs and opens doors to new research possibilities.
1.1 Protoplanetary disks . . . 1
1.1.1 Observations . . . 2
1.2 Structure of protoplanetary disks . . . 4
1.2.1 Disk structure . . . 6
1.2.2 Envelope structure . . . 7
1.3 X-ray radiation in disks and envelopes . . . 7
1.3.1 X-ray radiative transfer. . . 8
1.3.2 X-ray scattering and opacities . . . 9
1.4 Aim of this study . . . 10
2 Methods - Modelling embedded disks with ProDiMo 11 2.1 Model settings . . . 11
2.1.1 Reference model. . . 11
2.1.2 Gas disk structure . . . 11
2.1.3 Envelope structure . . . 13
2.2 Protoplanetary Disk Model. . . 15
2.2.1 Radiative transfer . . . 15
2.2.2 Spectral energy distributions . . . 15
2.2.3 Inclined models and ProDiMo’s grid structure . . . 16
3 Results 18 3.1 Spectral Energy Distributions . . . 18
3.1.1 Impact of inclination - disk structure . . . 18
3.1.2 Impact of inclination - disk and envelope structure . . . 20
3.1.3 Disk structure versus disk with envelope structure . . . 21
3.2 Correlating absorption with gas column densities . . . 23
4 Discussion 26 4.1 Impact of high inclinations on observed SEDs . . . 26
4.2 Impact of envelope structure on UV to mm SEDs . . . 27
4.3 Grid limitations at low inclinations . . . 27
4.4 Relating z/r to inclination . . . 27
4.5 Comparing results with XEST . . . 27
5 Conclusion and future outlook 28
References 30
1 Introduction - Protoplanetary Disks
It was in the year of his death in 1543 that Nicolaus Copernicus published his heliocen- tric theory in De revolutionibus orbium coelestium (On the Revolutions of the Heavenly Spheres), triggering the start of the Copernican Revolution. It is now generally accepted that the Earth and the other planets revolve in orbits with the Sun at their centre, sim- ilar to many other planetary systems just like it. What we are left with to do now, is to uncover the physics behind the evolution of these systems with the help of modern technology.
A method often used to investigate the history of the Solar System is to look at other such systems at varying stages of evolution. As their predecessors, protoplanetary disks pose an essential source of information on the formation of planetary systems. With a comprehensive theory on the physics, structure and evolution of protoplanetary disks, our knowledge of the history of the Solar System can be further refined. To verify these theories and compare them to observations, computer models are used to simulate the physical, thermal and chemical processes occurring in these disks. For protoplanetary disks there are several models available. In this thesis, the two dimensional, radiation thermo-chemical disk code Protoplanetary Disk Model (Woitke et al.,2009,2016) (from here on ProDiMo) was used to perform the modelling, which will be explained in more detail later.
The focus of this thesis is on the interaction of stellar X-rays with the protoplanetary disk and envelope structure and its impact on the observables. The rest of the introduction will be dedicated to elaborating on the theory of protoplanetary disks so the reader has a better understanding of all the processes at hand. At the end of the introduction, in Section 1.4, the aim of this study will be described in more detail.
After the introduction, Section2will explain the methods used to achieve these aims.
ProDiMo plays the leading role in this section, serving as the foundation of this thesis.
Relevant computational methods and the model settings will be described. The results will then be presented in Section 3 and discussed in Section 4 in the context of the aims of this study. Finally, the study will be concluded in Section 5.
1.1 Protoplanetary disks
This thesis deals with the modelling of protoplanetary disks on the basis of theory and observations. Before diving into the theory, the stage will be set with some general and observational context. What is the history behind protoplanetary disk research and what are the possible constraints that observations lay on the theory?
Protoplanetary disks (henceforth called PPDs - see Figure1) accompany young stellar objects (YSOs) after the collapse of a gas cloud. The generally accepted theory on the formation and evolution of the Solar System (and other planetary systems) is the nebular hypothesis, suggesting that they form from nebulous material. The first theories date back
Figure 1: An artist’s impression of a protoplanetary disk. Image credit: ESO/L. Cal¸cada.
to 1755 when Immanuel Kant published his Allgemeine Naturgeschichte und Theorie des Himmels (”Universal Natural History and Theory of the Heavens”). Kant argued that the stars and planets form from slowly rotating gaseous clouds, or nebulae, gradually collapsing due to gravity and flattening due to angular momentum. Many others expanded on Kant’s work, such as Pierre-Simon Laplace in his Exposition du systeme du monde, or had critique on these views, like James Clerk Maxwell, but the general idea persisted until it was eventually refined in the 20th century as the solar nebular disk model (SNDM) (see Woolfson,1993).
1.1.1 Observations
Eventually in 1983, with the launch of the IRAS satellite, it was found that YSOs fre- quently show more emission in the infra-red (IR) than one would expect from the photo- sphere of a pre-main-sequence star (Woitke, 2015). This so-called IR excess is attributed to the emission by dust in the circumstellar disk. Currently, one of the most frequently used techniques to detect and characterise the structure of PPDs is via the observation of the spectral energy distribution (SED) of stars and their disks. The SED shows the observed energy as a function of wavelength. Figure 2 shows the SED of Herbig Ae star HD 163296, with five different wavelength regions indicating the physical processes caus- ing the specific SED behaviour (Woitke,2015). See Section 1.2.1 about the disk contents for a description of these regions. A great part of the work in this thesis relies on SEDs and puts them into context with properties such as structure and inclination.
The central stars of PPDs are also luminous X-ray sources. A recent project dedicated to X-ray emission by young stellar objects is The XMM-Newton extended survey of the Taurus Molecular Cloud (XEST) (G¨udel et al., 2007). It observed the Taurus Molecular
Figure 2: Spectral energy distribution of the Herbig Ae star HD 163296. The black line is a disk model that fits the data points and error bars in blue, red, black and green. The green and orange lines show a de-reddened UV spectrum and an ISO/SWS spectrum, respectively. The red line is the stellar photospheric spectrum. Image credit: Woitke (2015)
Cloud (TMC), the nearest large star formation region, aiming to study (amongst others) the thermal structure, variability and long-term evolution of hot plasma. Especially their spectral analysis method is of interest here. To investigate X-ray properties from spectra, they performed a one- or two-component spectral fit. These components are defined by two plasmas with different temperatures and emission measures. One important parameter is the total hydrogen column density between the source and the observer. This parameter is usually a fitting parameter and can provide information on the gas surrounding young stars. However, in those models the complex structure of the circumstellar environment (i.e. disk and/or envelope) is usually not considered. Furthermore, this method does not discriminate between material bound to the star and other clouds that could be along the line of sight. Because the inclination of the disk and whether an envelope is present or not could have significant effects on the SED, this may bias their fits. This is one of the main drivers for this thesis, as this work makes a first attempt to study the impact of the disk’s inclination and presence of an envelope structure on the observed SED of the stellar X-ray emission (see Section 1.4).
Figure 3: Schematic view of the formation of the disk. The rotation of the sphere induces angular momentum in the gas. Depending on this angular momentum, the gas will either accrete onto the protostar or onto the disk that forms perpendicular to the rotation axis. Image credit: Dominik (2015)
1.2 Structure of protoplanetary disks
Protoplanetary disks are the natural by-product of newly formed stars from a rotating cloud of gas. Once a cloud of gas starts collapsing, gas from throughout the cloud will accrete either onto the protostellar core or onto the accretion disk. This depends on the specific angular momentum (the angular momentum per unit of mass) of such a gas parcel - see Figure 3. If it has a very low specific angular momentum it will accrete onto the forming star, otherwise it will eventually land on the accretion disk. Exactly where it reaches the accretion disk depends on its initial angular momentum as the collapse process starts. Assuming the cloud is in hydrostatic equilibrium before the collapse and rotating slowly as a solid body, the early stages of collapse are almost radial. This will result in two structures; an almost spherical outer envelope and an inner region that distorts due to the rotation, forming the accretion disk perpendicular to the rotating axis, supported by the gas pressure (Dominik, 2015).
After formation the disk will undergo certain evolutionary stages until ultimately forming a planetary system. The lifetime of PPDs is on the order of a few Myr (Williams and Cieza, 2011). During this time, the star will accrete mass from the disk and the disk itself will keep accreting mass from the envelope. After a while the envelope will have fully accreted onto the disk. As even more time passes, planetesimal formation kicks in and eventually what is left is a planetary system and little to no disk.
PPDs are mostly found around T Tauri stars or Herbig Ae/Be stars. T Tauri stars are a class of young (< 10M yrs) pre-main-sequence stars with a mass below 2 M . They are named after their prototype, T Tauri, found in the Taurus star-forming region. A distinction is made between Classical and Weak-line T Tauri stars (CTTS and WTTS,
Figure 4: Physical structure underlying the α classification of YSOs, evolutionary sequence and the fraction of sources falling into each class (Evans et al.,2009).
Image credit: Armitage(2015)
respectively). The main difference is that the former show strong Hα lines, signalling the presence of an accretion disk, whereas in the latter these lines are suppressed, indicating that accretion has ceased - CTTS evolve into WTTS as their disk disappears. For T Tauri stars, the central star of the PPD can be classified according to the slope (αIR) of its SED in the IR part, or the physical state of the PPD - see Figure 4. This slope is measured between different anchor points throughout literature, but these are typically around 2µm to 25µm. It is an indication of the amount of circumstellar matter (Armitage, 2015;
Telleschi et al.,2007) and given by
αIR = d log νFν
d log ν = d log λFλ
d log λ . (1)
The classification that follows from this was formalized by (Lada, 1987) as
• Class 0: a protostar that is still heavily enshrouded by the gas cloud from which it is forming, and therefore αIR can not be defined yet
• Class I: αIR > 0.3, a star with bipolar jets that still has an infalling envelope structure
• Flat spectrum sources: −0.3 < αIR < 0.3
• Class II: Classical T Tauri stars, −1.6 < αIR< −0.3
• Class III: Weak-line T Tauri stars, αIR < −1.6
Herbig Ae/Be stars are heavier analogues of T Tauri stars, with a mass of 2 < M < 8.
This thesis will, however, use a reference model based on T Tauri stars - see Section 2.1.
For more on the classification of Herbig Ae/Be disks as opposed to T Tauri disks, see e.g.
Agundez et al. (2018).
Figure 5: Schematic representation of the dust (left) and gas (right) contents of the disk. Image credit: Dullemond et al.(2007)
1.2.1 Disk structure
The physical disk structure of protoplanetary disks evolves over their lifetime of a few million years (Williams and Cieza,2011). Although they slowly yet continuously evolving, their global structure generally consists of an axisymmetric dust and gas distribution, with a smooth surface density (Armitage,2015). A mathematical description used by the model will be given in Section2.1.2. A lot of the physical processes, such as accretion rates, turbulence and in-depth disk structure (and evolution), can be described analytically, as outlined by e.g. Armitage(2015). Only the relevant processes and disk properties will be described briefly here.
The ratio of gas to dust in disks is greatly in favour of the gas. The amount of dust is only a small fraction of the total disk mass of PPDs, on the order of 1%. Though only scantly present, the dust plays a critical role in the disk structure because of its excellent absorption qualities of short-wavelength radiation (i.e. UV, optical). The dust particles re-emit this radiation in the longer wavelengths (IR to mm - see Figure 2). The dust structure schematically consists of a cold midplane and a hot surface layer, see Figure 5. The wavelength of the thermal radiation emitted by the dust globally increases with the radius of the disk. Dust in the inner sections is closer to the star and absorbs more radiation than the dust in the deep layers of the outer region, shielded by the warmer layer above it and the dust before it. The more radiation it absorbs, the warmer it is and the shorter the peak-wavelength of the black body emission of the dust will be. This effectively means that the near IR-excess mostly comes from the hot inner sections and the sub-mm to mm radiation comes from the colder outer regions. The most prominent feature in the SED is due to silicate emission. Most of the dust in disks is in the form of silicates. Stretching and bending modes of silicon-oxygen bonds lead to two prominent
peaks in the mid infra-red; at 10µm (stretching of Si-O) and 18µm (bending of O-Si-O).
The gas content follows a similar layered structure, see Figure5. Beyond the ice lines in the cold dust midplane, gas molecules freeze-out on dust particles. The gas close to the star is ionized thermally, whereas the atomic surface layers of the disk are ionized through non-thermal processes, such as stellar X-ray radiation (see Rab et al., 2017b). The gas opacities are very different from the dust opacities. The latter is a continuous function (dust generally absorbs at all wavelengths, though here we assume that it does not in the X-ray regime). The gas opacities come from the atomic and molecular content, which have sharp discontinuities due to e.g. electron transitions and ionization potentials.
1.2.2 Envelope structure
In the star-formation process, the initial molecular cloud is assumed to have some angular momentum in order to form the accretion disk. This angular momentum is also carried by the infalling envelope structure, causing it to flatten over time. The envelope model used here is the same as in Rab et al. (2017a) and assumed to be a flattened spherical cloud, again consisting of gas and dust. Gas densities in the envelope are much lower than in the disk, though, with a hydrogen number density on the order of n<H> ≈ 105− 107cm−3, compared to n<H> ≈ 1012 − 1014cm−3 in the disk. The maximum dust particle sizes are also assumed to be smaller compared to the disk, i.e. amax = 1µm in the envelope compared to amax = 1000µm in the disk (Rab et al., 2017a). However, the outer radius of envelopes can be a lot larger, with Rout up to 10.000 au (5000 au in this model). The central stars of PPDs have been found to emit strong jets of radiation from their rotation poles. These jets create hollow cavities in the envelope of varying structure. Though different structures have been used to model these cavities, the envelope structure here has a conical streamline cavity structure with an opening angle of 20o at 5000 au. Section 2.1.3 will further describe the specific envelope structure used in this thesis.
1.3 X-ray radiation in disks and envelopes
The YSOs in the centre of disks are luminous X-ray emitters (on the order of 1028erg s−1 to 1031erg s−1) (G¨udel et al., 2007; Rab et al., 2017b; Preibisch et al., 2005). This has been observed to be the case for Class I (Preibisch, 2004), Class II (CTTS) and Class III (WTTS) objects (Telleschi et al., 2007). This radiation is emitted by plasma in the neighbourhood of the star with temperatures over one million and up to 100 million degrees (G¨udel et al., 2007). It only directly reaches the surface layer of the disk, but it has far reaching implications for the thermal and chemical structure of PPDs.
The gas and dust in the disk both scatter, absorb and radiate energy at specific wavelengths. Because the dust does not radiate at these wavelengths, X-rays might pose a useful tool to deduce properties about the structure of observed PPDs. In this thesis the YSO in the centre of the disk will be treated as the single source of X-ray radiation.
Figure 6: Some of the physical processes occurring in the disk that determine the temperature and composition of the disk. Image credit: Armitage (2015)
1.3.1 X-ray radiative transfer
As a ray of radiation passes through matter, its intensity can increase or decrease due to the emissivity of or extinction by the matter, respectively. These are wavelength dependent processes and can be calculated through radiative transfer. Of course this does not only have effect on the radiation itself, but also on the matter it passes through.
Figure 6 shows some of the physical and radiative processes occurring in the disk that determine the temperature and composition of the disk. X-ray radiation is very important for the thermal and chemical structure of PPDs. A distinction is made between soft (0.2 keV < E < 1 − 3 keV) and hard X-rays (E > 1 − 3 keV). Throughout the literature different values are used for the transition between the two, as it is not an abrupt change.
They are usually identified based on their behaviour in a gas or dust environment. Soft X-rays are more easily absorbed, heating the surface disk layers to temperatures over 5000K and ionizing the atomic layer. Hard X-rays, on the other hand, are scattered more and can penetrate through the upper layers to ionize the molecular hydrogen below it, driving the molecular-ion chemistry (Rab et al., 2017b). The composition and thermal properties of the gas can be changed through e.g. photoionisation, photodissociation and the escape of line emission. The dust temperature is not affected by X-rays, because the dust does not absorb at these wavelengths. Section 2.2.1 will elaborate on how radiative transfer ProDiMo.
10
010
1energy[keV]
10
510
410
310
210
110
010
110
2/H [1 0
21cm
2H
1] C
N O Ne
Figure 7: X-ray photoelectric (absorption) gas cross-sections per hydrogen nucleus.
1.3.2 X-ray scattering and opacities
Evidently, not all X-ray radiation is absorbed in the disk; stellar light is also scattered, eventually leaving the medium. Both scattering and absorption are wavelength / energy dependent processes. The scattering is in the form of (inelastic) Compton scattering and (elastic) Rayleigh scattering. The former is dominant for X-ray radiation, reducing to Thomson scattering at low energies (Rab et al., 2017b). Differently to optical radiation, the main opacity source for X-rays is the gas. However, under certain circumstances the dust can also play a role (Rab et al., 2017b). In the models used here, the dust is not considered as X-ray opacity source.
The gas opacities used here are from the open source library xraylib1 - see Figure7.
The xraylib library is a compilation of different published works on the X-ray absorption and scattering cross-sections. In Figure 7, there are several absorption edges visible from different elements. These absorption edges are discontinuities in the absorption coefficients and arise from photon energies corresponding to the ionization potential (or other electron transitions) (Schoonjans et al., 2011). The values for the edges in xraylib are taken from Larkins (1977): 532.0 eV (oxygen), 870.1 eV (neon), 401.6 eV (nitrogen), 283.8 eV (carbon).
1xraylib website: https://github.com/tschoonj/xraylib
never been used to look at the impact of the disk and/or envelope on the observed SEDs of stellar X-rays.
In the long run, the goal is to relate this work to observations and methods used by e.g. XEST. The XEST project did not take the complex structure of the circumstellar material (i.e. disk and/or envelope) into consideration nor does it differentiate between this circumstellar material and possible other clouds along the line of sight. This thesis is a first step towards that goal.
2 Methods - Modelling embedded disks with ProDiMo
For this thesis, ProDiMo was used to model the physical, thermal and chemical structure of the protoplanetary disk. ProDiMo iteratively solves for the gas and dust temperatures and opacities and chemical abundances. In the following sections the model settings that were used are explained, as well as the disk and envelope geometries. In Section 2.2 relevant calculation mechanisms and outputs are explained that were used to find the results.
2.1 Model settings
The model used here employs several modules that were added to ProDiMo over the past, including an extension of the readily available radiative transfer module to cover the X-ray wavelength regime (Rab et al.,2017b) and a new module providing an envelope structure (Rab et al.,2017a).
2.1.1 Reference model
The model structure as described here is based on the reference model developed for the DIANA2 (DiscAnalysis) project. The parameters of this reference model are chosen to be (roughly) in accordance with real Class II T Tauri stars’ continuum and line fluxes. A list of the relevant disk parameters for the reference disk model are given in Table 1. The reference model is described in more detail by Woitke et al. (2016); Kamp et al. (2017).
In the next sections the disk structure geometry will be discussed in more detail.
2.1.2 Gas disk structure
The gas in the disk is described in a 2D axisymmetric form. The gas density structure in the disk can then be described as a function of the cylindrical coordinates r and z (the height of the disk), (e.g. Andrews et al. (2009);Woitke et al. (2016))
ρ(r, z) = Σ(r)
√2π · h(r) exp
− z2 2h(r)2
[g cm−3] (2)
where Σ(r) is the surface density in [g cm−2] and h(r) is the vertical disk scale height.
This scale height is given by a radial power-law
h(r) = H(100 au)
r
100 au
β
. (3)
2DIANA Project website: https://www.www.diana-project.com
envelope 0.6 au
tapering-off radius Rtap 100 au
column density power index 1.0
tapering parameter γ
disk 1.0
envelope -1.0
reference scale height H(100 au) 10 au
flaring power index β 1.15
centrifugal radius Rc 100 au
mass infall rate M˙if 5 × 10−6 M yr−1
outer radius Rout
envelope 5000 au
disk 600 au
cavity opening angle βcav 20o
Table 1: Main parameters for the reference disk model. A detailed description of the parameters and their meaning can be found in (Woitke et al.,2016;Rab et al.,2017a)
For the reference model the disk scale height at 100 AU is taken to be H(100 au) = 10 au. The flaring power index, β, is taken to be 1.15. The radial sur- face density in equation 2 is also taken to be a power-law, modified by an exponential tapering-off factor
Σ(r) = Σ0
r Rin
−
exp −
r Rtap
2−γ!
[g cm−2] (4)
where Rin = 0.07 au is the inner disk radius and Rtap = 100 au is the tapering-off radius.
The outer disk radius Rout Rtap is taken large enough so that the surface density there is small enough that it can be neglected, e.g. Σ(Rout) ≈ 1020 cm−2. In the case of the reference disk model, this is Rout = 620 au. The column density power index, , is taken to be 1.0. For the disk, the tapering parameter is the same as the column density power
Figure 8: Contour plot of the total hydrogen number density n<H> in the disk. The height of the disk z is scaled by the radius r. The white dashed contour lines correspond to the levels shown in the contourbar. Image credit: Rab et al. (2017b)
index , γ = 1.0.
Figure 8 shows the total hydrogen number density n<H> in the disk. The height of the disk z is scaled by the radius r. The white dashed contour lines correspond to the levels shown in the contourbar.
2.1.3 Envelope structure
The envelope structure byRab et al. (2017a) that is placed on top of the disk structure, is based on the model ofWhitney et al.(2003). Their model is built up of a disk model and a superimposed envelope model. Both structures are calculated separately and wherever the disk density is higher than the envelope density, the disk structure replaces the envelope structure.
The envelope gas density structure is given by ρ(r, µ) =
M˙if 4π
2GM∗r3−1/2 1 2+ µ
2µ0
−1/2
µ
µ0 + 2µ20Rc r
−1
[g cm−3] (5) M˙if is the mass infall rate of the envelope onto the disk, r is the radial distance to the central star, Rc is the centrifugal radius, µ = cos θ is the cosine of the polar angle of a streamline of infalling matter and µ0 is the value of µ at r → ∞. The streamline angles are calculated by
µ30+ µ0(r/Rc− 1) − µ(r/Rc) = 0 (6) The bipolar cavities of the envelope were accounted for with the same approach used by Whitney et al. (2003). An empty cavity with an opening angle βcav of 20o was used with
Figure 9: Contour plot of the total hydrogen number density n<H> in the disk. The height of the disk z is scaled by the radius r. The white dashed contour lines correspond to the levels shown in the contourbar. Image credit: Rab et al. (2017a)
a power-law to account for its curved shape, For the envelope, the tapering parameter γ in Equation 4 was taken to be γ = −1.0 to secure a smooth transition at the border of the disk. For further details on the envelope structure and its parameters, see Rab et al.
(2017a).
Figure 9shows the total hydrogen number density n<H> in the envelope model. The white dashed contour lines correspond to the levels shown in the contourbar. The second and third plot are zoomed-in versions of the plot directly above them (marked by the grey squares).
2.2 Protoplanetary Disk Model
2.2.1 Radiative transfer
ProDiMo carries out a 2D continuum radiative transfer, providing the local continuum radiation field Jν(r, z) and the local dust temperature Td(r, z) at every grid point in the disk. From each of these grid points a number of rays are traced back along the photon propagation direction. For every ray, the equation of radiative transfer is solved
dIν
dτν = Sν− Iν (7)
assuming LTE and assuming coherent isotropic scattering Sν = κabsν Bν(Td) + κscaν Jν
κextν . (8)
Iν is the spectral intensity, Jν = 4π1 R IνdΩ is the mean intensity, Sν is the source function, Bν is the Planck function and κabsν , κscaν and κextν = κabsν + κscaν are the dust absorption, scattering and extinction coefficients (Woitke et al., 2009). The dust grains in the disk are all assumed to have a unique temperature Td in radiative equilibrium
Γdust+ Z
κabsν Jνdν = Z
κabsν Bν(Td)dν (9)
where Γdust is the additional heating rate. Γdust accounts for non-radiative heating and cooling processes. In order to get converged results concerning Jν(r, z) and Td(r, z), an accelerated Λ scheme is used. The details of this can be found in (Woitke et al., 2009).
From the found dust temperatures and radiation field, the chemistry and cooling / heating balance can be determined, such as the species densities (assuming kinetic chemical equilibrium) and the kinetic gas temperatures Tg. The species abundances (of ions, atoms and molecules, including ices) can be integrated over to find the element abundances, which are kept constant throughout the disk. All element abundances (e.g.
C, O, N, . . . ) are therefore a constant fraction of H throughout the disk (and envelope).
2.2.2 Spectral energy distributions
Spectral energy distributions are plots of energy versus wavelength. In this work, the energy flux is shown in the form of νFν [erg/cm−2/s] and the x-axis shows the wavelength as well as the photon energy. It presents a clear view of how much energy is emitted across various wavelength regions. By knowing which contents (e.g. gas or dust) in each part of the disk (e.g. midplane, surface layer, inner disk radii, outer disk radii) emit, absorb or scatter radiation at certain wavelengths, the different characteristics of the SED can be explained. As explained in Section 1.1.1, SEDs are a powerful tool to
Figure 10: Sketch of an inclined protoplanetary disk with envelope structure. The disk is in black, envelope is in blue and lines of sight are in red under inclination i. The disk’s colour is more dense than the envelope’s colour to indicate the higher gas and dust densities. The image is not to scale. An inclination of 0o is regarded as face-on and an inclination of 90o as edge-on.
interpret observations. By fitting an SED to e.g. photometric or spectroscopic data, global properties of the disk can be inferred from the observations.
The SED output by ProDiMo contains the monochromatic flux Fν at each wave- length gridpoint, ranging from 6.1992 × 10−5 to 3.0 × 103µm.
2.2.3 Inclined models and ProDiMo’s grid structure
When calculating the SED for the base model, ProDiMo calculates the face-on SED.
The inclined SED is calculated by keeping the entire structure fixed, moving the observer or the target by an angle of inclination i and recalculating the flux along the new lines of sight - see Figure 10. An inclination of 0o is regarded as face-on and an inclination of 90o as edge-on.
Following from the theory of radiative transfer in Sections 1.3.1 and 2.2.1, increasing the inclination will result in a higher column density of the material the rays pass through, leading to more extinction. The angle of inclination i is related to the the zr value through
tan(i) = 1 z/r, z
r = 1
tan(i). (10)
This zr value is commonly used in protoplanetary disk content plots, because the disk is geometrically very thin (H r). By plotting zr versus r, instead of z versus r, the regions closer to the midplane are shown more prominently. Tracing a horizontal line
Figure 11: The ProDiMo grid with on the y-axis z (left) and z/r (right). Please note, the y- axis label of the z/r figure on the right falsely displays z instead of z/r. Image credit: ProDiMo (Woitke et al.,2009)
in a zr versus r plot will result in a radial line from the origin in a normal z versus r plot. This is not only useful to look at these radial lines and specific layers (e.g. surface or midplane), but also to inspect the contents at certain inclinations. According to the relations in Equation 10, a certain zr value will map to a corresponding inclination and vice versa. Although it is a simplistic view (the star is regarded as a point source and a straight ray is traced towards the star), it is still meaningful in this context. Depending on energy, X-rays are preferentially scattered forward. If there was significant scattering it would be a bad approximation, but as long as the emitting source is small and X-rays aren’t scattered efficiently it is fine.
The grid of ProDiMo lends itself perfectly for this purpose - see Figure 11. The grid points are spread logarithmically over r and about half of them are located in the inner 2.5 au (Woitke et al.,2009). The grid points are spread mostly linearly over zr, with a higher density for zr < 0.1 (the midplane). This way it is easy to use the zr strategy explained above to determine properties such as the total radial column density along lines of sight.
Section2.2.3 already explained the mechanism behind observing inclined disk structures.
Here, the resulting SEDs of models with only a disk structure will be presented and compared to that theory.
In Figure 12a the SED of the UV to mm regime is shown for a model with only a disk structure at varying inclinations. They are all based on the same disk model. The face-on SED and stellar input spectrum have been included for comparison. Figure 12b shows the SED for the same models and inclinations but in the X-ray regime. The colours in both plots coincide for easy comparison.
The profile of the face-on SED in the UV to mm regime is well in agreement with the one presented in Woitke (2015) (see Figure 2). As expected from Section 2.2.3, each successive increase in inclination decreases the SED. This occurs at all wavelengths, though not everywhere with the same factor. In the silicate emission region the decrease from face-on to edge-on covers 4 orders of magnitude, whereas in the sub-mm to mm region this is only 1 order of magnitude (see Figure 12a). In the optical part, where the only source of radiation is the star, the decrease only starts after i ≈ 73 deg. This indicates that the disk reaches a critical column density so that the column becomes slightly optically thick. The extinction starts to increase rapidly from here. In the FIR wavelength region and sub-mm to mm wavelength region, this decrease starts as soon as i ≈ 35 and 55o, respectively. This emission comes from cold dust in the middle to the outer edges of the disk midplane. Since the absorption opacities are wavelength dependent, these will always be lower in e.g. the mm regime than for shorter wavelengths. Also, when looking face-on at the disk, the midplane cannot be seen because it is optically thick. However, when looking edge-on the midplane can be seen better, because you are looking from the side.
In figure 12b, the SEDs of the same models are shown for the X-ray regime. Again, the stellar spectrum and face-on model are included. Unlike in the UV to mm regime, the face-on model and stellar spectrum completely overlap here because the star is the only source radiating in the X-ray regime. As opposed to the IR to mm regime, the dust does not emit X-ray radiation. The decrease in the SED only starts at i ≈ 65o for the softest X-rays and at i ≈ 75o for the higher X-rays. From there on the same principle holds again; every successive increase in inclination further decreases the SED. However, with
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Figure 12: Spectral energy distributions at varying inclinations (indicated in degrees in the legend) for a model with only a disk structure in (a) the UV to mm regime and (b) X-ray regime.
A face-on (dashed black) and stellar spectrum (bright red) are included for reference.
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Figure 13: Spectral energy distributions at varying inclinations (indicated in degrees in the legend) for a model with a disk and envelope structure in the X-ray regime. A face-on (dashed black) and stellar spectrum (bright red) are included for reference.
increasing energy the models start to converge towards the profile of the edge-on SED at higher inclinations. In other words, only at high inclinations will the disk absorb the hard X-rays (> a few keV), whereas the soft X-rays are already absorbed at lower inclinations.
Though even in edge-on disks, some of the hard X-rays will find their way through the whole disk.
Several absorption edges are clearly visible in the SEDs, the most prominent one being the oxygen absorption edge at 546 eV. The other prominent edge is that of neon, at 869 eV. Only upon close inspection can the small bumps of carbon and nitrogen be recognized, at 284 eV and 410 eV respectively. These coincide with the absorption edges found in the gas cross sections output (see Figure 7).
3.1.2 Impact of inclination - disk and envelope structure
In Figure 13 the X-ray SEDs are shown for a disk with envelope structure. As already described in Section 1.2.2, the envelope structure is associated with younger PPDs found around Class I T Tauri stars. The envelope structure is much larger than the disk structure (Rout up to 5000 au instead of 100 au).
The X-ray SEDs for the models with an envelope structure rapidly decrease after i ≈ 19o, which is expected because of the cavity opening angle of 20o. Even between 20 and 20.5 degrees, the build-up of absorption due to the envelope gas amounts to around 4 orders of magnitude for the soft X-rays. Just like for the disk-only structure, with
increasing energy the SEDs start to decrease at higher inclinations. The soft X-rays here actually experience more absorption for high inclinations than the disk-only models.
It makes sense that this happens only for the soft X-rays because they are more easily absorbed by the gas than the hard X-rays. Even though the gas densities in the envelope are orders of magnitude lower, its vast size makes up for this. The hard X-rays on the other hand, do not experience much absorption until inclinations of ≈ 75o, similar to the disk-only models. For an isolated envelope model without a disk, the scenario would be very different, as the hard X-rays would not be absorbed at all. This only happens in the disk, where the densities are much higher. This is interesting because it is not always obvious from observations if there is a disk in the embedded phases.
With the envelope structure included, the absorption edges are more well-defined than with only a disk. This is because the envelope models have a higher wavelength resolution (i.e. 1000 grid points (disk with envelope) over the whole wavelength range versus 500 grid points (disk only)) which is purely technical and has no further significant implications.
3.1.3 Disk structure versus disk with envelope structure
To further assess the different behaviour of the two structures, this section will present SEDs that include both model structures. Eventually these will be combined with the column densities to find a correlation between absorption and column density.
Figure14shows the SEDs from disk models and disk with envelope models together.
The envelope models are shown with a dashed blue line and the disk models are shown with a solid blue line. The stellar input spectrum is shown in solid black and the face-on model is shown with a dashed black line, for reference. The inclinations at which these are compared are selected based on the the envelope showing an effect on the SED in the X-ray (i ≈ 20 − 25o), the disk showing an effect of the SED in the X-ray regime (i ≈ 69o), an inclination to show the difference in extinction in the UV to mm regime (i ≈ 79o) and the fully edge-on models (i = 90o).
In the first row of Figure 14 (i = 25o) the disk with envelope model’s SED shows absorption in the soft X-ray part, but the disk model does not do so yet. This absorption is due to the gas, since the dust is not included as absorber of X-rays in these models. In the UV to mm, neither show signs of extinction yet.
What is noticeable is that the envelope models have a higher face-on SED in the IR to mm region than the disk models. The amount of extinction at higher inclination is similar for both structures, though the disk with envelope model has a higher energy than the disk only models in the entire right column. In the far-UV and optical part, the envelope SEDs could have a higher energy because scattering redirects photons that would otherwise be ’lost’ back into the line of sight. When considering a secluded disk model, once a photon leaves the disk in a direction that is not in the line of sight, it is
’lost’ and it does not end up in the observed flux. With the envelope included, this ’lost’
photon can be scattered back into the line of sight, because of the vast size of the envelope
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Figure 14: Spectral energy distributions for disk (solid blue) and disk + envelope (dashed blue) models at four different inclinations i. The stellar input spectrum is shown in bright red. The face-on models are included with black solid and dashed lines, for the disk and disk + envelope models respectively. The left column shows the X-ray regime, the right column shows the UV to mm regime. Note that the ranges on the y-axis differ for the left and right column.
structure. At ≈ 10 µm the envelope model converges to the disk model, after which it goes into the thermal emission region. A possible reason that the envelope SEDs generally have a higher energy than the disk SEDs at λ ≥ 0.8 µm (despite having more material that could scatter this radiation), is because the envelope also has a lot of dust that adds to the thermal emission and this radiation is not efficiently scattered. This could also indicate a problem with the ProDiMo code and maybe requires improvement (see the discussion in Section 4).
In the second row, at i = 69o, extinction starts to build up for both models in the UV to mm regime. In the soft X-ray regime the disk model starts experiencing noticeable absorption as well, indicating the start of the surface gas layer of the disk. In the X-ray column at i = 79o, the envelope SED shows a lower flux than the disk SED for the soft X- rays up to E ≤ 0.7 keV. At energies between 0.7 ≤ E ≤ 6 keV the envelope SED is above the disk SED (so less absorption). For the hard X-rays it converges to the disk model’s SED again, which indicates they are only absorbed in the disk. The intermediate effect (0.7 ≤ E ≤ 6 keV) can be seen even stronger in the edge-on models, where the envelope SED is completely above the disk SED for E ≥ 0.7 keV . This peculiar behaviour could not be explained at this point, but presents an interesting topic for further research (see Section 4).
3.2 Correlating absorption with gas column densities
In Figure 15 the total radial column densities are shown for hydrogen, carbon, nitrogen, oxygen and neon. They are the sum of all species containing those elements, as explained in Section 2.2.1. The envelope models are in dashed lines and the disk models are in solid lines for comparison. The disk model grid stops at z/r ≈ 0.5 and the envelope model grid stops at z/r ≈ 2.75. Until z/r ≈ 0.25 the envelope does not contribute significantly to the column densities. Over the range of z/r ≈ 0.25 to z/r ≈ 2.75, the column density of the envelope model slowly drops about two orders of magnitude in total.
In order to further assess the decreasing SED as a function of inclination and correlate that to the column densities of figure 15, the decrement between the face-on model and an inclined model at a certain energy (Edec) or wavelength (λdec) are calculated. This decrement at an energy Edec is calculated by
decrement(i, Edec) = log νFν,f ace−on(Edec) − log νFν,i(Edec)
log νFν,f ace−on(Edec) (11) where i is the inclination and Edec is the energy at which the decrements are calculated.
Figure 16 plots for each inclination the decrease/decrement against the total radial hydrogen column density that corresponds to that inclination via the z/r-inclination re- lation described in 2.2.3 (see Equation 10). Again, the disk is plotted in solid lines and the envelope is plotted in dashed lines, for comparison. The actual species on the x axis
0.0 0.5 1.0 1.5 2.0 2.5 z/r
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Figure 15: Total radial column densities for the total hydrogen, carbon, nitrogen, oxygen and neon populations plotted against z/r. The disk model (solid lines) grid only extends up to z/r ≈ 0.5, whereas the envelope (dashed lines) extends up to z/r ≈ 2.75.
is not relevant for the shape/profile of the curve. Any of the other elements (i.e. carbon, oxygen, nitrogen or neon) would have produced the same shape, because their abundances (fraction of hydrogen) are all a constant for every z/r value (see Figure 15). Hydrogen was used here because the gas and dust opacities and cross sections are also shown per hydrogen atom.
The envelope models all start on the x axis at the same column density because they are only calculated up to z/r ≈ 2.75, corresponding to an angle of i ≈ 20o. The inclinations up until that point will therefore lead to the same column density of N<H>,rad ≈ 1021 [cm−2] (it picks the closest z/r value that corresponds to that inclina- tion, which will just be the outermost value of zr ≈ 2.75). The same principle holds for the disk model up to i ≈ 63o, with a column density of N<H>,rad≈ 1019[cm−2]. This may be problematic, because the X-ray absorption in the disk with envelope models already starts at 19o and the soft X-ray absorption in the disk models starts at 65o. This means that absorption may not be observed in regions at lower inclinations because of the grid limitation. This will be discussed in the discussion in section 4.
The profiles all follow a similar structure. The decrement does not increase signifi- cantly until a certain critical column density is reached after which the gas quickly becomes optically thick. The decrement then steeply increases, after which it flattens again.
The critical column density for 2.0 and 6.0 keV can be correlated with the gas cross sections in Figure 7. From this Figure, a cross section of σ/H ≈ 2 × 10−23 cm2 H−1 and σ/H ≈ 6 × 10−25cm2 H−1 can be found for X-ray photons of 2.0 and 6.0 keV, respectively.
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Figure 16: SED inclination decrements relative to the face-on model in log units plotted against total radial hydrogen column density for four energies(Edec) / wavelengths (λdec). The disk models and envelope models are plotted with solid and dashed lines, respectively.
These absorption cross sections are roughly agreeing with the critical column density where the gas appears to become optically thick (τ = 1). This is at N<H>,rad ≈ 1023cm−2 and N<H>,rad≈ 1024cm−2, for the 2.0 keV and 6.0 keV energies respectively.
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Figure 17: Spectral energy distribution in the X-ray regime of a disk model with scattering, an envelope model with scattering and an envelope model without scattering. The stellar spectrum is included as well, with a dashed green line.
4 Discussion
4.1 Impact of high inclinations on observed SEDs
The results from Section3.1.2show that the hard X-rays are only absorbed by the disk and not by the envelope at 79o. This is an interesting result because from observations it is not always clear if a disk is present in the embedded phases. What this result indicates is that a developed disk structure at high inclinations (at least i > 79o) should be identifiable within an envelope through hard X-ray absorption. However, the next section showed that for an edge-on view the hard X-rays actually show more emission in the disk with envelope model than in the disk model itself. This is strange because the envelope itself should not cause more absorption for the hard X-rays at varying inclinations. It has a rather constant gas column density at inclinations above 35o (below z/r = 1.5, see Figure 15). Something that could be causing this is scattering. Figure 17 shows an SED in the X-ray regime with most importantly an envelope model that did not include X-ray scattering. What it shows is that scattering is indeed relevant for the hard X-rays and can produce more observable emission, which would otherwise be absorbed. At longer wavelengths, there are some differences between the envelope models with and without scattering. This might be because the cross-sections in those two cases are not the same, but this cannot be done differently in the current version of ProDiMo. This requires further improvement.
4.2 Impact of envelope structure on UV to mm SEDs
An surprising effect of the envelope structure on the UV to mm SEDs, is that they generally have more emission than the disk only model SEDs. This can be explained in the far-UV / optical regime (see Section 3.1.3, but is more difficult in the NIR to mm.
What is especially worrying there, is the big difference in the SEDs (about two orders of magnitude). This could not be explained yet as it might be a problem with ProDiMo as well and requires further research.
4.3 Grid limitations at low inclinations
Something possibly problematic that was found in Section 3.2, is that the decrements start at i ≈ 65o and i ≈ 19o while the grid starts at inclinations of i ≈ 63o and i ≈ 20o, respectively for the disk model and disk with envelop models. Because of this, it might be that no absorption can be seen in the lower inclination regions because of the grid limitations. To fix this, the grid should be expanded to make sure that the entire absorbing region falls into the grid.
4.4 Relating z/r to inclination
The issue denoted in Section 2.2.3 about the z/r-inclination relation is also worth dis- cussing. What was done in this thesis, is to strictly relate the z/r value to an inclination by regarding the star as a point source and one sight line to it. However, due to scattering we get extended emission, which might become significant in the envelope models. We made this assumption to quantify things and that is fine, but then again the models show that it might not be fine and need to be studied in more detail in the future.
4.5 Comparing results with XEST
From the results in Section 3.2, it can be seen that the decrements in the SED increase very steeply once a certain critical column density is reached. It shows the importance and difficulty of differentiating between structures. This is because the decrements are very low before and almost non-changing after becoming optically thick at the critical column density, which is the same for both models with and without an envelope. The only main difference is that the disks eventually reach higher decrements, showing that envelope models actually have more emission for a certain column density.
not considered to be an absorber nor an emitter of X-rays in this thesis. A final remark is on the methods by XEST. Since they did not take a possible envelope structure into consideration, their fits may be biased. The results that were found indicate that it is difficult to differentiate between models with and without an envelope based on column densities because the decrease in SED barely changes before and after the gas becomes optically thick at the critical column densities.
The results presented in this thesis are the first step towards a better understanding of the impact that stellar X-ray radiation from inclined protoplanetary disks has on observed SEDs and opens doors to new research possibilities.
Acknowledgements
As I’m nearing my deadline, I feel the need to write some words of thanks. I’ve taken my fair time with this bachelor’s degree but there are some people who have unmistakeably helped me along the way. First of all, I’d like to thank my thesis supervisor Inga Kamp for guiding me through this project. At times when I was really desperate and had no idea what I was doing, our weekly talks on Monday would always help me get back on track. Your everlasting optimism helped me stick through those times. I would also like to thank Christian Rab, my second supervisor, for his help with the project and writing tips. Lastly, I would like to thank my mother, who always told me to enjoy my time as a student but even more so taught me the value of education. Even though I may not always have liked your help or constant asking about my progress, it did help me pull through and get my degree in the end.
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