University of Groningen
The future of protoplanetary disk models
Greenwood, Aaron James
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Publication date: 2018
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Greenwood, A. J. (2018). The future of protoplanetary disk models: Brown dwarfs, mid-infrared molecular spectra, and dust evolution. Rijksuniversiteit Groningen.
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1
I N T R O D U C T I O N
The simplest ideas about how our solar system formed date back to Kant in 1755 and Laplace in 1796, with their independent but similar “nebular theories”. Their ideas of an interstellar cloud of gas and dust that collapsed and flattened into a disk were a good explanation for the fact that the planets all orbit the Sun in the same direction along similarly-inclined planes. These ideas were remarkably prescient. We know that our Sun formed during the collapse of a molecular cloud, and that the Earth formed in a protoplanetary disk around the Sun. Our interest in understanding these disks is motivated by wanting to understand how, where, and when planets form, and how the mechanism(s) of formation can influence their size and composition.
To create a planetary system like our own, capable of supporting and evolving life as we know it, requires a combination of circumstances. Primarily, the disk must be capable of forming planets of the right size and with a suitable composition, in a stable orbit around a long-lived star at just the right radius. Where in the disk a planet forms is largely determined by the hydrodynamics of the disk; however, the composition of that planet is influenced by the chemical composition of the disk. While there may be many planets in our galaxy that are
capableof supporting life, we do not know how common it is for life to form.
Protoplanetary disks are an important stepping stone between the collapse of a cloud of interstellar gas and dust, and the formation of a planetary system. Such disks begin from a collapsing molecular cloud (or cloud fragment) that has a non-negligible amount of net rotation. The collapsing cloud forms a flattened disk of circumstellar material as a simple consequence of the conservation of momentum. Collisions within the cloud act to confine the distribution of matter to a disk of thickness determined by the vertical component of the star’s gravity, that rotates along the plane of the cloud’s total angular momentum. While most of the circumstellar matter is accreted by the central protostar, a small fraction (∼10%) of matter persists in the disk – in many or most cases for long enough (a few Myr) to form a planetary system.
2 introduction
The primary goal of protoplanetary disk research is to understand the physics and chemistry of these disks, and how the properties that we observe and model can influence their evolution and the creation of different types of planets. 1.1 goals of the thesis
Young stellar objects (YSOs) of T Tauri-type and lower masses(.2 M ),
includ-ing brown dwarfs, are of significant interest both theoretically and observationally. The crucial difference between a T Tauri star and a brown dwarf is that the
lat-ter’s mass is below the hydrogen-burning mass limit (about 0.08 M ). Thus,
the definition of a brown dwarf is a star that cannot fuse hydrogen into helium (Oppenheimer et al. 2000). The lower mass limit for brown dwarfs, as adopted by the IAU, is defined as the threshold of deuterium fusion at 0.013 M 1.
Solar- to lower-mass stars make up a significant majority of stars in our galaxy, yet there are many unsolved problems in describing the formation and evolution of these stars and their planetary systems. For example, although disks around brown dwarfs are just as common as those around T Tauri stars, there are no known exoplanets around a non-binary brown dwarf. We need further to understand the disks around brown dwarfs, how they may differ from T Tauri disks, and whether the lack of observed planets around brown dwarfs is an observational bias or if there is some process that impedes their formation.
The first step towards understanding this is to understand how brown dwarf disks differ in their physical and chemical structure. What do their surface density profiles and scale heights look like? What is the gas-to-dust ratio, and how does the dust evolve? What do observations of simple molecules tell us about their distribution in the disk, and where might the water or CO ice lines be located? More sub-mm ALMA observations of brown dwarf disks are currently in progress (e.g. the Cycle 5 proposal by K. Öberg), which may spatially resolve these small objects and allow us to measure these gas and dust properties in the
outer disk (outwards from∼10 AU). Unfortunately, observations with a good
spatial resolution and of molecular species other than CO do not yet exist. Until then, we seek to address these questions from a more theoretical perspective, and to see what differences there may be between prototypical brown dwarf and T Tauri disks.
The upcoming JWST and E-ELT observatories, as well as the proposed SPICA mission, will be able to observe in the infrared at much higher sensitivities and resolutions than were previously possible. A high-quality infrared spectrum will allow us to observe the line shapes and fluxes of several molecular species,
namely CO2, C2H2, H2O, NH3, OH, and HCN. Each of these species has a
1 An object less massive than 0.013 M is a sub-brown dwarf, unless it satisfies the definition of a planet. In that case, it is a planet. An object between the lower and upper(0.08 M )mass limits is always a brown dwarf, even if it orbits a main-sequence star and otherwise satisfies the definition of a planet.
line-emitting region that traces different radial and vertical regions in the disk, owing to their different formation mechanisms and the different gas temperatures and densities at which they prefer to emit. A higher spectral resolution helps better to distinguish these species my minimizing the effects of line blending. Together with these increased observational capabilities, we must also improve our modelling. Being able to accurately model the infrared spectra of disks is key to understanding the inner few AU of disks, where terrestrial planets are thought to form. In order to capture the complex radial and vertical structure of each species, we need a similarly complex 2D model.
The work in this thesis makes further progress towards understanding the protoplanetary disks around brown dwarf and T Tauri stars through thermo-chemical modelling. Our models encompass both the distribution of dust in the disk, the gas and dust temperatures, the distributions and level populations of key molecules, and even their infrared spectra. By computing models of these disks and changing parameters such as the amount of flaring in the disk or the UV flux, we can build our understanding of the effects of each parameter and of how to interpret observations.
1.2 disk formation
A circumstellar disk has its beginnings in a collapsing molecular cloud of in-terstellar medium (ISM) material. These cold molecular clouds have typical temperatures of 10 K to 20 K (Ferrière 2001), with the canonical approximation that the gas-to-dust mass ratio is 100 (Bohlin et al. 1978). Disk models are sensitive to both this ratio and the total disk mass. Mitchell et al. (2001) show from a survey of the Orion B molecular cloud that the gas-to-dust ratio can have significant local variations, but in cases where the gas has frozen onto dust grains these
measurements can be deceptive.2
By around 1972 it was largely believed that the massive HAeBe stars, along with their lower-mass T Tauri counterparts, were very young objects still in their protostellar stages of evolution and likely accompanied by accretion disks (Strom et al. 1972). The canonical infrared (IR) surveys of T Tauri stars by Bertout et al. (1988) and of HAeBe stars by Hillenbrand et al. (1992) cemented this hypothesis, by presenting strong statistical arguments that the observed infrared excesses are well-explained by the presence of circumstellar disks.
A simple and popular model for the collapse of a molecular cloud is the so-called “inside out” model by Shu (1977). The radius of the eventual protoplanetary
2 For example, Pani´c et al. (2008) find for the Herbig Ae/Be (HAeBe) disk HD 169142 that although the effects of freeze-out are likely insignificant for this disk, a gas-to-dust ratio of 100 is nevertheless only possible for high dust opacities and low CO abundances. It is probable that the gas-to-dust ratio in this disk is lower than 100. Later surveys, for example by Ansdell et al. (2016), find significant variations in the gas-to-dust ratio of Lupus disks, ranging from 103to 1:1. It seems likely that the
4 introduction
disk is dependent on the kinematics of its formation. A simplified scenario is that the initial properties of the molecular cloud influence the mechanism of collapse, giving the core rotation rateΩc and infall time t, which then largely determine
the eventual size and mass of the protoplanetary disk, as its centrifugal radius grows as R(t) ∝ Ω2ct3 (Terebey et al. 1984). Although such simple models do have their uses, the consensus is that there exists a polychotomy of formation mechanisms. All brown dwarfs are thought to form as part of a larger multiple system, but cannot accrete enough matter to cross the sub-stellar threshold. Thus a simple and singular model of collapse is not sufficient for understanding the origins of brown dwarfs.
Luhman (2012) has written a very lucid review of these formation mechanisms. They are, briefly: gravitational compression and fragmentation of a collapsing molecular cloud (Bonnell et al. 2008), turbulent compression and fragmentation (Padoan & Nordlund 2002), premature ejection due to dynamical interactions (Boss 2001), photoevaporation from a nearby massive (OB) star (Hester et al. 1996), and gravitational fragmentation (and sometimes ejection) of a much more massive circumstellar disk (Bate et al. 2002, 2003).
Although when studying the chemistry and structure of an individual disk we may not yet be greatly concerned with the method of cloud collapse, these initial formation conditions can clearly influence the disk’s later structure. The variety of multiple systems seen is evidence that the initial collapse is a very important stage; but the meaning of this simplification is that our static disk models are ignorant of these dynamics. Some observational evidence of the power of surveys is where Testi et al. (2016) find that brown dwarf disks in Ophiuchus appear to have been dynamically truncated during their formation. Although it is still difficult to determine the exact formation mechanism of these disks, the fact that there may be a large population of disks which formed in the same manner could tell us a lot about the history of the Ophiuchus star-forming region.
1.3 disk classification
Circumstellar disks are broadly categorized into several classes based on their evolutionary stage. Williams & Cieza (2011) summarize neatly the classification scheme defined by Lada & Wilking (1984) and later refined by Greene et al. (1994) with the introduction of the flat spectrum class as an intermediary between Class I and Class II. We classify YSOs into five broad groups corresponding to their evolutionary status. They trace the transition of the YSO’s outer material, from a protostellar envelope, to a protostellar disk, to a protoplanetary disk. The classifications are based on the slope of the spectral index, αIR, between about
2 µm and 25 µm.
Class 0 objects are deeply-embedded and extremely young YSOs that have no emission in the optical or near-infrared (NIR).
Class I objects have a shallow, positive infrared spectral energy distribution (SED) slope of αIR>0.3 and are generally invisible at optical wavelengths. Class
FS, or “flat spectrum” YSOs after Greene et al. (1994), have a relatively flat slope of 0.3>αIR>−0.3. Class II YSOs are accreting material from the circumstellar
disk, and have SED slopes−0.3>αIR >−1.6. The YSOs in these classes usually show strong and easily-detectable infrared excesses.
In the transition between classes II and III, there exists a class called transitional disks. Past literature has not agreed on an exact definition of a transitional disk, but researchers observe them to have large gaps or clearings in the dust structure that may be caused by planets (Kim et al. 2013). The final class, Class III, has all but ceased accretion and has a steeply declining SED slope αIR<−1.6.
One important aspect is the timescales of the processes that lead to the for-mation of an extrasolar planetary system. How long does each evolutionary stage last? We can infer these figures from population studies. In comparison to the main-sequence lifetime of stars.2 M , the lifetime of a protoplanetary
disk is exceedingly short. A cluster survey by Haisch Jr et al. (2001) shows that
85%±8% of YSOs in the 0.3 Myr NGC 2024 cluster show an L-band infrared
excess indicative of a circumstellar disk. That figure drops to 52%±10% in the
3.2 Myr old NGC 2264 cluster, and down to 12%±4% in the 5 Myr old NGC
2362 cluster. This research clearly suggests that the lifetime of YSOs in the Class II phase is on the order of a few or several million years.
The disk surrounding a Class II YSO is what we call a protoplanetary disk. Most of the circumstellar material is now in a disk around the protostar, allowing us to observe the infrared emission emanating from the disk that is prominent in its SED. Nevertheless, there may still be some accretion onto the protostar – for example, Herczeg & Hillenbrand (2008) use measured ultraviolet (UV) excesses to infer the accretion rates for a survey of low-mass and brown dwarf stars. These rates can still be quite considerable – as high as∼10−8M
yr−1, as measured
for the∼0.15 M dwarf CIDA 1 by measuring the UV and optical excesses in
Balmer and Paschen continua emission. However, this is an atypical result. The reported rates for the other surveyed low-mass objects are one to four orders of magnitude lower.
This classification scheme has only been strictly defined for T Tauri disks, and it remains to be seen if changes to the scheme are necessary in order to accom-modate brown dwarf disks. However, it does seem likely that this classification scheme will translate well to brown dwarf disks: for example, van der Plas et al. (2016) find that the established relationship between disk mass and stellar mass
6 introduction
Where do lines originate?
100 AU rcond ices [OI] 63 µm CO low J sub-mm CO high J HD 56, 112 µm H2O ro-vib CO ro-vib 2-5 µm ices [CII] 157 µm H2O low Tex 10 AU H2O high Tex C2H2, HCN e.g. [NeII] 12.8 µm H2 17.0 µm 1 AU every wavelength range probes a different part of the disk planetary systems1. The power of line observations
Inga Kamp, DIANA Summer School on Protoplanetary Disks, 15-20.6.2014, Ameland, The Netherlands
Figure 1.1: The structure of a typical T Tauri protoplanetary disk, showing the typical line-emitting regions of some important species. Image credit: Inga Kamp.
1.4 disk observations
Different regions of a protoplanetary disk are best probed at different wavelengths by different observatories. Figure 1.1 shows the typical layout of a T Tauri-type disk model. The disk begins at the dust condensation radius (but there may be gas inside this radius that accretes onto the star), and inside about 1 AU, the emission is dominated by near- and mid-infrared rotational and ro-vibrational
lines from species such as C2H2, HCN, CO, CO2, and H2O. Further out, the
disk becomes colder and less optically thick at mid-infrared wavelengths (e.g. 20 µm). The surface layers of the disk are warm enough to emit in the mid- and far-infrared, e.g. the[O i] 63 µm line, the brightest cooling line in disks, has been targeted in the large Herschel Open Time key programme GASPS (Gas Evolution in Protoplanetary Disks, Pinte et al. 2010; Woitke et al. 2010b; Kamp et al. 2011; Dent et al. 2013). Closer towards the midplane, we can observe the rotational lines from species such as CO and HCN. At sub-mm wavelengths, the rotational lines of these species have been studied extensively (Thi et al. 2001; Öberg et al. 2010a, 2011b; Guilloteau et al. 2013) for disks around T Tauri- and Herbig-type stars.
Within the next decade, we expect to see significant progress in the observation of protoplanetary disks. For example, the ALMA array of sub- or mm-wavelength interferometers is almost fully operational, and recent ALMA science verification observations of HL Tauri – at a baseline up to 15 km – have revealed remarkable dust structure at 35 mas resolution (ESO 2014). The observations show clearly resolved concentric gaps that may have been cleared by large planetary or protoplanetary bodies. Given that HL Tauri is estimated to be no older than
1 Myr, the apparent rate of planet formation in this system is surprisingly rapid. It has recently been suggested that the gas and dust structures may be decoupled in a disk that has been partially cleared by planets, because gas may be cleared from the outer regions more quickly than dust (Meru et al. 2014). HL Tauri will be an interesting system to see if this holds true.
Beyond focussing on individual disks, ALMA has matured enough that it is increasingly being used for surveys of disks in nearby molecular clouds. For example, Ansdell et al. (2017) surveyed disks in σ Orionis with M∗&0.1 M .
Although limited to 1.3 mm continuum and CO J=2−1 data, they measure dust
masses for 37 detected sources and detect CO in only 6 sources. The relatively low CO fluxes detected suggest that the disks have low gas masses; suggesting that either giant planet formation is uncommon (due to insufficient amounts of gas) or usually happens faster than the few Myr age of the cloud. Other surveys have a more focussed goal, aiming to observe fewer disks with much greater sensitivites: for example, Huang et al. (2017) measure deuterated species in six disks (two T Tauri disks, two T Tauri transition disks, and two Herbig disks). Their observations show diverse structures in the deuterated species, suggesting that they follow complex formation pathways that may in the future be able to reveal a lot about disk chemistry.
With large ALMA surveys concentrating mostly on dust continuum and CO lines, perhaps the most important global disk property measurements that can be obtained are the disk radii, dust masses, and gas masses. Although there remain considerable uncertainties in measuring the mass or column density of an optically-thick line or continuum, observing a large number of disks enables some valuable statistics to be done. For example, Ansdell et al. (2016, 2018) measure the gas and dust masses and radii for T Tauri disks in Lupus. They find that the gas disks are significantly larger than the dust disks, and also that most disks have gas-to-dust ratios less than 100. Woitke et al. (2016) show that the dust properties have a significant effect on the mm and cm dust continuum slopes, and on the CO lines. Thus, if the dust properties such as the grain size distribution can vary between disks, accurate determination of the gas-to-dust ratio becomes difficult. As well as difficulties in determining the dust mass, gas mass determinations can also be very uncertain. For example, using HD as a tracer can find gas masses that are orders of magnitude greater than CO, a difference that we cannot yet reconcile (McClure et al. 2016; Bergin & Williams 2018). The most significant survey of brown dwarf disks is by Testi et al. (2016), who measure Ophiuchus dwarfs in the dust continuum, determine dust masses, and find that the small disks are likely to have been dynamically truncated.3
ALMA is not the only tool for observing gaps in disks: SPHERE on the VLT is a spectro-polarimetric instrument with a coronagraph that is capable of resolving the dust emission in disks at AU-scale spatial resolution. Although we do not
3 No survey of gas in brown dwarf disks has yet been published, however observations by various researchers are underway.
8 introduction
know whether the gaps are formed by planets or by some other hydrodynamical process, directly imaging a planet within the gap of a protoplanetary disk would prove that at least some gaps can be carved by planets. For example, the disk TW Hya has been observed to have three gaps (Nomura et al. 2016; Boekel et al. 2017). Boekel et al. (2017) calculate the depths of the gaps from SPHERE data, and find that the planets could be no more than a few tens of Earth masses. The gaps are wider and shallower than might be expected for planet-disk interactions with relatively low-mass planets, and if such planets do exist then they should be observable with the E-ELT. One of the most exciting possibilities for the formation of planets in disk gaps is the case of HD100546. Quanz et al. (2015) find a possible exoplanet in the disk at a radius of 47 AU. Unfortunately, the follow-up observations by SPHERE failed to detect a gap associated with the candidate companion (Garufi et al. 2016). Although an extremely powerful technique, the scattered-light observations done by SPHERE suffer from sensitivity limits which restrict observations to Herbig and brighter T Tauri disks.
Moving into the infrared, the launch of JWST will allow us to study the inner few AU of disks in unprecedented detail. All of its instruments will be useful in achieving one of the main science goals of the mission: to image the disks around YSOs in the infrared and study organic molecules in the disks. We expect that imaging protoplanetary disk with MIRI will be technically possible, with the main constraint being high demand for telescope time. Additionally, MIRI’s spectral resolution of R∼2 800 provides a significant boost over Spitzer’s R∼600, while also greatly increasing sensitivity. This boost in spectral resolution and sensitivity is important because with Spitzer, lines are significantly blended and easily lost in the continuum because they are unresolved. For example, JWST may be able to detect water lines in Herbig disks that could not be detected with Spitzer(Antonellini et al. 2016).
Other current and future instruments and observatories of interest include
CRIRES+ on the VLT, METIS on the E-ELT, and the proposed SPICA space
mission. The upgraded CRIRES+ instrument on the VLT is significant, because
CRIRES will become a cross-dispersed echelle spectrograph with a significantly enhanced wavelength coverage and a new near-infrared detector array. It is expected to achieve a radial velocity precision of 2 to 3 m s−1. This very high
spectral resolution is expected to allow radial velocity detections of super-Earth
planets around nearby M dwarf stars. CRIRES+ will also be used for exoplanet
atmosphere characterization using transit spectroscopy, and its ability to take circularly polarized spectra can be used to analyze magnetic fields in M dwarf and brown dwarf disks. All of these possibilities, and technical details about the instrument, are summarized in Dorn et al. (2014).
METIS is a mid-infrared imager and spectrograph that can operate at a spectral
resolution of R ∼ 100 000 at wavelengths shorter than 5 µm, with reduced
resolutions at longer wavelengths. Although the Earth’s atmosphere renders it somewhat less sensitive than JWST, the E-ELT’s enormous 39.3 m aperture
means that METIS is expected to resolve the inner few AU of protoplanetary disks (Brandl et al. 2014). METIS is one of the three first light instruments on the E-ELT, expected to begin observations in 2024.
Finally, SPICA is a proposed infrared space telescope for the ESA’s M5 round of missions (Roelfsema et al. 2018). It is currently on the shortlist with only two other missions, so there is a substantial chance that the SPICA mission concept will be funded. It is a mid- and far-infrared space telescope with a modest 2.5 m mirror, as something of a successor to the late Herschel space telescope. Perhaps the most interesting instruments on SPICA are the SMI-HRS module, a spectrograph with a resolution of 28 000 between 12 µm and 18 µm, and SAFARI, a far-infrared imaging spectrometer (Sibthorpe et al. 2015; Roelfsema et al. 2018). SMI-HRS has a spectral resolution 10 times higher than JWST in the mid-infrared regions
that contain substantial emission from simple molecules such as CO2, C2H2,
H2O, and HCN. Figure 1.2 shows the differences between spectra convolved to
different resolutions, up to R=28 000. At this very high resolution, the effects of line blending on the spectra will be very significantly reduced. Although denser spectral regions will still be affected, we should be able to isolate lines such as the
H2O line at 17.754 µm, which may allow us to locate the H2O snow line (Notsu
et al. 2017).
1.5 brown dwarfs and their disks
Because some of this thesis discusses disks around brown dwarf disks, and they are not as well-studied as their higher-mass counterparts, it is worthwhile to briefly discuss the history of brown dwarf research.
The “lithium test” is a useful method to determine if a star is sub-stellar: a main-sequence star will burn its lithium within 100 Myr, while a brown dwarf may never reach the required temperature, leading to lithium enhancements in brown dwarfs (Basri 1998). Unfortunately, this discriminant is unreliable for objects that are much younger than the∼100 Myr lifetime of lithium in a stellar photosphere. Distinguishing and classifying brown dwarfs in young star-forming regions remains difficult: due to visual extinction and reddening, spectroscopy is generally required for an accurate determination.
The study of protoplanetary disks has been rapidly ramping up over the past few decades, giving a somewhat compressed timeline of major discoveries. The cool, embedded, and often highly-extincted nature of these objects means we must use infrared and sub-mm wavelength observations to study them in detail. Modern infrared observatories – ISO, Spitzer, Herschel, WISE, and large ground-based observatories such as SMA, CARMA, and ALMA have been instrumental in rapidly improving our base of observations.
Leading up to the discovery of the first brown dwarf, there were many tentative detections. For example, Rieke & Rieke (1990) found three possibly sub-stellar
10 introduction 13 13.2 13.4 13.6 13.8 14 14.2 14.4 14.6 14.8 15 wavelength (micron) 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 Flux (Jy)
blend of 2 C2H2 9 CO2 6 NH3 1 o-H2O 1 p-H2O
---blend of 2 C2H2 1 HCN 11 NH3 2 o-H2O 1 OH 1 p-H2 1 p-H2O
---blend of 7 C2H2 1 HCN 5 NH3 1 o-H2O 1 OH ---blend of 7 C2H2 10 NH3 1 o-H2O 2 OH ---blend of 8 C2H2 8 NH3 3 p-H2O
---blend of 8 C2H2 1 HCN 2 NH3 2 o-H2O 1 p-H2O
---blend of 72 C2H2 1 CO2 3 NH3 1 o-H2O 2 OH 2 p-H2O
---blend of 14 C2H2 1 CO2 6 HCN 1 NH3 1 o-H2O 3 OH
---blend of 5 C2H2 1 CO2 5 NH3 2 o-H2O 3 OH 2 p-H2O
---blend of 5 C2H2 1 CO2 1 HCN 9 NH3
---blend of 5 C2H2 1 CO2 5 NH3 2 o-H2O 6 OH 2 p-H2O
---blend of 5 C2H2 1 CO2 5 NH3 1 o-H2O 2 p-H2O
---blend of 4 C2H2 1 CO2 1 HCN 3 NH3 1 o-H2 1 o-H2O 3 p-H2O
---blend of 3 C2H2 1 CO2 1 HCN 5 NH3 2 o-H2O 2 OH 1 p-H2
---blend of 7 C2H2 1 CO2 1 HCN 6 NH3 1 o-H2O
---blend of 6 C2H2 1 CO2 1 HCN 6 NH3 2 o-H2O
---blend of 7 C2H2 5 CO2 7 NH3 4 o-H2O 2 OH 1 p-H2O
---blend of 6 C2H2 7 CO2 1 HCN 7 NH3 2 o-H2O 1 p-H2O
---R600 R2800 R28000
Figure 1.2: FLiTs spectra of a T Tauri disk model, convolved to spectral resolutions of 600, 2 800, and 28 000. A small number of line blends are annotated such that “blend of 5 CO2 8 NH3” means that at the internal resolution of FLiTs, five CO2and eight NH3lines are blended together. The full list of line blends is exhaustive.
objects in the ρ Ophiuchi cloud. However, the discovery lacked spectroscopic confirmation, concluding only that the objects fell below the theoretical luminosity
for a 0.08 M star. Later, Greene & Young (1992) surveyed ρ Ophiuchi and
estimated that 48% of sources in the cloud have near-infrared excesses indicative of circumstellar disks.
The first unambiguous brown dwarf is GI229 B, confirmed in the release of two papers in quick succession (Nakajima et al. 1995; Oppenheimer et al. 1995). The first tell-tale signs of circumstellar disks around brown dwarfs were detected only a few years later, in the IC 348 cluster by Luhman (1999): using an array
of four CCD detectors4on the 1.2 m telescope at the Fred Lawrence Whipple
observatory, they found an excess in the H and K photometric bands in some sources, which could possibly arise from a protoplanetary disk.
Moving from ground-based to space-based observatories, Bontemps et al. (2001) used ISOCAM on the ISO satellite to perform a survey of YSOs in ρ Ophiuchi, including low-mass stars and brown dwarfs at 6.7 µm and 14.3 µm. From this survey, 65 % of the Class II objects were found to have mid-infrared excesses indicating they had a protoplanetary disk, including the brown dwarf ρ Oph 102 discussed later in Chapter 2.
Even more recent are the first spatially resolved images of dust in a brown dwarf disk, of the object 2MASS J044427+2512, by Ricci et al. (2013). For further reading, a comprehensive review of early brown dwarf observations was written by Basri (2000). Luhman (2012) provide a more recent perspective that focusses on their formation and evolution.
Later, Herschel surveyed brown dwarf disks at wavelengths up to 160 µm. Harvey et al. (2012b) publish the first results of their survey: significant amounts of cold dust in the outer disks of three brown dwarfs. The full survey measures
disk masses of 47 M-type objects, including one disk as heavy as∼3×10−3M
around an M7.5 dwarf (Harvey et al. 2012a).
Around a similar time period, brown dwarf disks were also being observed by Spitzer: Pascucci et al. (2013) published infrared spectra of eight very low-mass and brown dwarf disks, with detections of C2H2, HCN, CO2, and H2O.
Further expanding the wavelength regime into the sub-mm, Ricci et al. (2013) used the CARMA interferometer to (tentatively) spatially resolve the dust emis-sion from a brown dwarf disk. Around the same time, ALMA began operations, which especially in its fully-functional state can operate at very high sensitivities and spatial resolutions. Ricci et al. (2012, 2014) presented the first brown dwarf disk observations using ALMA, reporting in 2014 a trio of spatially-resolved brown dwarf disks with outer radii much larger than are thought to be normal. These larger disks may be interesting to study because they have very low masses for their size (on the order of 1 MJup). Figure 1.3 shows the differences in CO 4 HgCdTe-based infrared detectors had only recently been developed into 2D arrays. Most near-infrared
12 introduction 0.1 1.0 10.0 r [AU] 0.0 0.1 0.2 0.3 0.4 z / r logχ/n=-3.5 Tdust=20K -12 -10 -8 -6 -4 log ε (CO) 0.1 1.0 10.0 100.0 r [AU] 0.0 0.1 0.2 0.3 0.4 z / r logχ /n=-3.5 Tdust=20K -12 -10 -8 -6 -4 log ε (CO)
Figure 1.3: The CO abundance for our small (Rout=60 AU) and large (Rout=400 AU)
brown dwarf disk models. If the disk mass is kept constant, the CO ice line that begins in the mid-plane at around 1 AU in our small model almost disappears in the large model.
structure between the small and large brown dwarf disks in Chapter 2, given equal disk mass. Because the larger disks are much more optically thin, there is very little CO ice in the disk. A further survey by ALMA was published by Testi et al. (2016), finding disk masses for a sample of brown dwarf disks in Ophi-uchus, and while poorly-resolved, the data do suggest that the disks have sharp
outer radii at around R .25 AU. Earlier work has suggested that dynamical
interactions in a dense star-forming region may truncate the disks to small radii Bate (2009, 2012). Additionally, the disks have masses that as a fraction of stellar mass are similar to (but possibly somewhat lower than) their higher-mass T Tauri counterparts, with Mdisk/M∗ratios generally between 10−1and 10−3. Pascucci
et al. (2016) also attempt to find whether or not brown dwarf disks are generally less massive, but conclude that reliable gas mass measurements are necessary. This is indeed a crucial step: both Testi et al. (2016) and Pascucci et al. (2016) assume a gas-to-dust ratio of 100:1. There is no guarantee that the dust-to-gas ratio does not also change with spectral type, making it impossible to draw a reasonable conclusion.
Although it is clear that brown dwarf protoplanetary disks are common, there are currently no observations of an exoplanet (Mplanet < 0.013 M ) around a
non-binary brown dwarf. There may truly be a relative lack of planets around brown dwarfs, or this lack of observations may simply be a sensitivity issue. Mulders et al. (2015) find a stellar-mass dependent drop in the occurrence of planets found with Kepler, such that planets around M-type stars occur twice as frequently as around G type stars. However, they report that Kepler is only sensitive to stellar masses as low as 0.3 M , excluding the brown dwarf regime. If
100
10
1
0.1
0.01 Semi-Major Axis [Astronomical Units (A
U)] 0 1 2 3 4 5 20 15 10 5 0
Mass of Star [Solar Mass]
Planet Mass [J
upiter Mass]
exoplanets.org | 9/1/2014
Figure 1.4: The masses of confirmed exoplanets and their host stars, from the exoplan-ets.org database.
there truly were fewer planets around brown dwarfs than T Tauris, this situation would require a dramatic reversal of that trend.
1.6 planets and brown dwarfs
Observations of exoplanets around brown dwarfs are still few in number, and none have been found around a non-binary host – the crucial science question here is whether or not planets can form in a protoplanetary disk around a typical brown dwarf, and if they can, what types of planetary systems we are likely to find.
Figure 1.4 shows that most confirmed exoplanets orbit stars of around 1 M .
There is a significant degree of observational bias: although most microlensing discoveries have been of planets around M dwarf stars, most other discoveries (dominated by Kepler) are around generally warmer stars (Skowron et al. 2015). However, the rate of discovery is not equal to the occurrence rate. Mulders et al. (2015) find that after accounting for known observational biases, M type stars appear twice as likely to host planets as G type stars. The likely reason for the lack of brown dwarf discoveries is observational: we need higher sensitivities.5 5 Unfortunately, the recently-launched TESS exoplanet-finding mission is only searching for planets
14 introduction
The first report of planets around a brown dwarf is Bennett et al. (2008), who report by gravitational lensing a 3.3+4.9−1.6M⊕exoplanet around the 0.060+0.028−0.021M dwarf star MOA-2007-BLG-192L – the authors suspected the substellar nature of the host, but it could not be confirmed at the time. VLT observations later refined these determinations, placing the planet’s mass at 3.2+5.2
−1.8M⊕, the host
star’s mass at 0.084+0.015−0.012M (Kubas et al. 2012). Although this means that the host star is most likely stellar, the planet’s credible mass and close separation of 0.66+0.51−0.22 AU gives good confidence that it formed within a protoplanetary disk. Clearly, planets can form around a star that is sitting right above the brown dwarf threshold. So either there are planets around brown dwarfs waiting to be discovered, or there is a surprising and unknown reason as to why they cannot. By 2012, researchers had observed circumstellar disks around brown dwarfs, and also exoplanets around stars that are very close to the sub-stellar boundary. The objects 2MASS J1207334-393254 (Chauvin et al. 2004, 2005; Mohanty et al. 2007) and 2MASS J04414489+2301513 (Todorov et al. 2010) might appear promis-ing, but they have both high planet:star mass ratios of∼0.2 and 0.25 to 0.5, with orbital separations of 45 AU and 15 AU respectively. Lodato et al. (2005) thus suggest that the companions formed by the gravitational fragmentation mech-anism, as its mass ratio and separation are statistically consistent with higher mass binaries and such high disk masses are likely to be unstable under a core accretion or disk fragmentation model.
Following these discoveries, Han et al. (2013) used microlensing observations
to find a tightly separated (∼ 0.87 AU), 1.9±0.2 MJup planet orbiting a ∼
0.022 M brown dwarf (two separate and slightly different parameter solutions
are proposed in the paper, due to ecliptic degeneracy). The authors are still cautious about this detection: the planet is nearly 10% of the stellar mass, so its formation is still challenging to explain – if indeed it is a planet at all, following the semantic definition that it must have formed by accreting matter within a protoplanetary disk.6
Perhaps the closest discovery yet to a bona fide planet around a solitary brown dwarf star is the TRAPPIST-1 planetary system (Gillon et al. 2016). The host star has a mass of 0.080±0.009 M , which is right on the sub-stellar boundary. However, it is most likely not sub-stellar: absorption features at the 670.8 nm lithium line were not detected using high-resolution spectroscopy, and the star’s
estimated age is 7.6±2.2 Gyr (Burgasser & Mamajek 2017). A main-sequence
star of this age will be depleted in lithium, whereas a brown dwarf of this age is expected to have an enhanced lithium abundance (Basri 1998). Thus, both Gillon et al. (2016) and Burgasser & Mamajek (2017) conclude that TRAPPIST-1 is not a brown dwarf but a very low-mass, main sequence M dwarf.
Researchers have observed planets around brown dwarfs in a binary system, but not around a solitary brown dwarf: this is puzzling. In the binary system
6 By mass alone, the companion is officially defined as a brown dwarf. The official IAU definitions of “planet” and “brown dwarf” state nothing about their mechanisms of formation.
analyzed by Han et al. (2013), they argue that the extreme properties of the system suggest that the stellar companion formed like a regular planet. Bate (2012) finds through hydrodynamical simulation that close binary stars are more likely to have a near-equal mass ratio between components. These difficult conditions necessitate both further statistical results on the distribution of masses in low-mass multiple systems, and more theoretical and modelling work to better ascertain boundary conditions for each model of formation.
If brown dwarfs can indeed have disks massive enough to form terrestrial planets, and a 0.08 M star can host multiple planets, there is no obvious process which might prevent a planet forming around a brown dwarf. Brown dwarf
disks have been observed to have dust masses up to 6 M⊕ (Testi et al. 2016), so
substantial amounts of dust can exist in these disks. Although the gas mass of a brown dwarf disk has never been measured, assuming a 100:1 gas-to-dust ratio the total disk mass is thus 1.9 MJup, which should be a sufficient mass budget to
form terrestrial or Neptune-sized planets.7 The remaining puzzles are how stable
brown dwarf disks are, how frequently planets form, why planets might fail to form, and how the disk eventually dissipates.
1.7 dust structure, grain growth, and migration
The migration and evolution of dust is a significant factor in the evolution of a protoplanetary disk. We cannot yet fully explain why small dust grains manage to remain in the outer disk for several million years (Birnstiel et al. 2009). This is a problem: radial drift (Brauer et al. 2008; Birnstiel et al. 2010b) and grain coagulation (Dominik & Dullemond 2008) should rapidly deplete the population of small dust grains. Radial drift depletes the outer disk of dust grains, and is necessary for describing accretion. Grain coagulation allows rapid grain growth in the inner disk, around 1 AU.
Below, we briefly describe the mechanisms of dust grain growth and migration, following closely the formulae described in Brauer et al. (2008) and Birnstiel et al. (2010b).
The vertical mass density distribution of the gas is given by ρg(z, r) = √Σ(r)
2πHexp
−z2/2H2 , (1.1)
where H is the pressure scale height of the gas H = cs/Ωk, the isothermal
sound speed is cs = pkT/µ, and Ωk = pGM /r3 is the Keplerian rotation
frequency. k and G are the Boltzmann and gravitational constants, and µ is the
7 These masses have been measured with ALMA for disks in the ρ Ophiuchi cloud. The sub-mm observations are blind to any dust that may exist as decimetre-sized (or larger) objects, which may very well be present in a∼Myr old disk. In order to improve these mass estimates, we need a better understanding of how dust evolves in disks.
16 introduction
mean molecular weight, typically assumed to be 2.3 times the mass of a proton (representing a mixture of helium and molecular hydrogen).
The Stokes number describes the coupling of dust and gas, defined by Stk =Ωkackρs
sρgα
2q−1, (1.2)
where akand ρsare the radius and density of the dust particle, csis the isothermal
sound speed, and ρg is the gas density. α is the dimensionless turbulence
parameter, and q can range from q=1 (describing large, slow eddies) to q=0
(describing small, fast eddies).
The maximum radial drift velocity is defined by
vn= c 2 s 2Vk δ+7 4 , (1.3)
where δ is the power law index of the radial surface density distribution and Vk
is the velocity of Keplerian rotation, Vk =Ωkr. Finally, the radial drift of a dust
grain of mass mk is given by Weidenschilling (1977) and Whipple (1972):
vdust,k =− 2vn
Stk+St1k
. (1.4)
To give numerical context to these equations, the maximum drift velocity at 1 AU is 45 m s−1 (Brauer et al. 2008).
Given the rapidity of the predicted inwards radial drift, decimetre-sized par-ticles are then expected to rapidly drift inwards to the central star (Nakagawa et al. 1986). Due to dust settling, we do not expect to observe micron-sized dust particles in the surface layers of the disk. However, observations of mid-infrared silicate emission clearly indicate the presence of these dust particles (Kessler-Silacci et al. 2006). Thus, either settling does not sufficiently deplete this population or there is some mechanism which replenishes these grains.
In addition to radial drift, dust grain coagulation also depletes the population of small particles. These equations are complex and difficult to solve, in part due to the many orders of magnitude involved between the sizes of the smallest and largest grains. Full details are left to papers which describe dust coagulation and the algorithms used (Dullemond & Dominik 2005; Brauer et al. 2008). The simplified models in Chapter 4 use the growth rate of mono-disperse coagulation (Brauer et al. 2008; Birnstiel et al. 2012):
da
dt =
ρd
ρs∆u, (1.5)
where∆u is the approximate relative velocity between two grains in turbulent
In order to explain observations of sub-mm sized grains in the outer disk, there must be processes which counteract radial drift, and replenish the small grain population that grain coagulation rapidly depletes (Birnstiel et al. 2010a; Pinilla et al. 2013). In brown dwarf disks, it has been suggested that turbulence plays a role in counteracting radial drift and slowing the inwards migration of dust (Pinilla et al. 2013).
To counter grain growth, we introduce fragmentation, whereby collisions between larger aggregated particles can fragment them into smaller particles due to turbulence and (in non-turbulent disks) differences in the relative drift speed of dust grains (Brauer et al. 2008; Birnstiel et al. 2010b).
1.8 modelling approaches
Protoplanetary disk models can have greatly varying degrees of complexity. Simple models are easy to understand and fitting them to observational data is relatively straightforward. More complex models, such as 2D thermochemical models, allow for a more complex analysis but become increasingly difficult to fit to observations due to degeneracies and the large number of parameters.
Some of the simplest models are slab models (for example, those introduced by Drake & Ulrich 1980). They model a single column of gas, typically with a fixed gas temperature, and either a variable gas density through the column (a “1D” slab model), or a fixed gas density (a “0D” slab model). By parametrizing a 2D geometry for the slab, we can calculate not only line fluxes, but a model spectrum: we assume that the emission comes from an annulus under Keplerian rotation, and thus can calculate a double-peaked line profile. Other broadening effects such as pressure broadening can be applied based on the assumed gas temperature, and the line optical depths from a 1D slab model can be determined using escape probability methods.
For cases such as modelling CO ro-vibrational lines as done by Brittain et al. (2009) for the disk HD 100546, such slab models are sufficient because the properties such as the gas temperature, continuum optical depth, and volume density do not vary significantly over the line-emitting region (Hein-Bertelsen 2015). However, such a slab model encompasses no information about the spatial distribution of the line and might in many cases not allow an accurate measurement of simple parameters such as gas column densities.
In order further to study the mid-infrared line-emitting regions of protoplan-etary disks beyond our current knowledge, we need 2D disk models that can
encompass structure in both the radial and vertical directions.8 2D disk models
have been used for some years in studying the outer disks in the sub-mm, but only recently have they been applied to mid-infrared ro-vibrational lines in the
8 3D models are not yet feasible: they would take a long time to compute, and further code development is required.
18 introduction
inner few AU of disks (for example, see Bruderer et al. 2015). In some cases, simplifications such as using parametrized chemistry or setting Tgas=Tdustare
used. However, they are not sufficient to study every aspect of the disk.
Thermochemical disk models began with Kamp & Dullemond (2004), where the gas and dust temperatures are decoupled and a true chemical network is introduced (that is, a network of species, reactions, and reaction rates that can be run to self-consistently reach thermochemical equilibrium). This chemical network code was further developed by Peter Woitke with the addition of 2D radiative transfer, dust opacities, and an expanded network of chemical heating and cooling processes to create the ProDiMo code (Woitke et al. 2009). These models trade the ability to do quick Monte Carlo fitting of observational data for a much more complex model that turns the disk model into an astrochemical laboratory, where different species and chemical and radiative processes can be studied across a wide range of wavelengths (Kamp 2015). It is these most complex models with which we work in this thesis, using the thermochemical disk modelling code ProDiMo.
1.9 molecular spectroscopy
The spectral lines that we observe in disks come from simple molecules changing energy levels. A molecule can change from one rotational state to another, from one vibrational state to another, or it can simultaneously change both its rotational and vibrational states. This latter process is called a rotational-vibrational transition, or ro-rotational-vibrational for short. Typically, if we observe simple molecules in the sub-mm we are seeing rotational transitions. If we observe those same molecules in the mid-infrared, we often see ro-vibrational transitions. A more complete overview of these processes and their application to protoplanetary disks is given by Dionatos (2015), and here we quickly review the fundamental concepts.
The vibrational energy levels for a diatomic molecule are given by the formula for a harmonic oscillator, with energy
E= ¯hν
v+12
, (1.6)
where v is the vibrational quantum number and ν is the is the frequency of vibration that depends on the atomic masses. The rotational energies of a diatomic molecule are given by
E= h
2
8π2IJ(J+1) (1.7)
where I is the moment of inertia and J is the rotational quantum number. These
Jand v quantum numbers are used to define the transitions of individual lines.
dipole transition is where the rotational quantum number J increases or decreases by 1.
A set of energy transitions for a molecule is called a ladder, and the energy levels of each “step” on the ladder are separated by the rotational constant. For
CO, this constant is about 115 GHz. This means that the CO J=1→0 line has a
frequency of 115 GHz, the CO J=2→1 line has a frequency of 230 GHz, and
so on.
Molecules that have no permanent dipole moment may display much weaker quadrupole transitions, following the selection rule where the rotational quantum number goes as∆J =±2. Thus for H2, the lowest energy transition is J=2→0.
For species such as OH that have open electronic shell configurations, or for
polyatomic species such as H2O, the energy level formulae become more complex
due to the unpaired electrons and additional axes of inertia that these species have.
As opposed to rotational transitions, where not all transitions are permitted for every molecule, any vibrational transition is allowed. However, the strongest
transitions are generally where∆v= ±1. Additionally, vibrational transitions
can occur simultaneously with a change in the rotational state. These are the ro-vibrational transitions.
The ro-vibrational lines of each species are classified into branches: especially
the P-, Q-, and R-branches. A P-branch transition is where∆J =−1. Another
way of writing this is that the difference between the upper(Ju)and lower(Jl)
rotational states is Ju = Jl−1. A Q-branch transition is where∆J =0, and an
R-branch transition is where∆J=1.
The rotational constant for ro-vibrational energy levels can change. This is because the distance between each atom in the molecule is not fixed: it is affected by the molecule’s moment of inertia. This can give rise to so-called band-heads. The reason why band-heads appear in the R-branch is that the rotational constant tends to decrease with the vibrational energy level. In an R-branch line, the vibrational energy level decreases while the rotational energy level increases. This leads to the spectral lines “bunching up” around a particular wavelength. Such a
concentration of lines makes the band-heads of molecules such as CO, CO2, and
HCN very bright and easily detectable with a spectrograph. Such band-heads are visible in Fig. 1.5.
20 introduction
1.10 infrared spectra
Since the launch of the Spitzer space telescope, we have been able to observe the mid-infrared spectra of protoplanetary disks. Generally, the brightest species to be observed in the mid-infrared are CO2, C2H2, H2O, and HCN.
In the past, we have most often used zero-dimensional slab models to derive molecular abundances and gas temperatures for each species in the disk. For example, Salyk et al. (2011) and Pontoppidan et al. (2010) use LTE slab
mod-els to model the C2H2, HCN, and CO2 emission lines. Notably, they find an
anti-correlation between line detections and the mid-infrared SED slope. Their interpretation is that line detections seem more likely in disks where small dust grains have begun to settle into the mid-plane. The reasoning is straightforward: if there is less dust in the upper layers of the disk, the line-emitting regions will be less optically thick and thus the line fluxes will increase. This is supported in studies by Antonellini et al. (2015, 2017) using thermochemical disk models.
However, analyzing Spitzer spectra is not without its difficulties. The signal-to-noise ratios are relatively low, and the spectral resolution of R=600 is insufficient to disentangle line blends. The result is that it is very difficult to distinguish between the dust continuum and the H2O spectrum: Fig. 1.5 illustrates this point.
If the noise levels are comparable to the water line strength, it becomes difficult to place the continuum and to de-blend the spectrum.
Another weakness of past models is in the use of slab models. Although simple and quick to compute, they rely on assuming a single gas and dust temperature and a line-emitting area scaled to match the observed flux. It is likely that the profile of the gas temperature in the line-emitting region (and along our line of sight) plays an important role in interpreting observations of mid-infrared spectra.
The work in this thesis aims to improve our understanding by calculating the spectra of 2D disk models, and understanding how the spectra and line-emitting regions respond to changes in the mode parameters.
1.11 disk models
Our approach to modelling protoplanetary disks is to further our understanding with thermochemical models that include both gas and dust self-consistently.
In Chapters 2 and 3, our models assume a parametrized gas and dust structure, and calculate the radiative transfer of the disk. Afterwards, the disk chemistry is calculated. For these models we use the thermochemical disk modelling code ProDiMo (Woitke et al. 2009; Kamp et al. 2010; Aresu et al. 2011), following the so-called “DIANA standard” approach that aims to standardize the parameters and techniques used in order to calculate disk models (Woitke et al. 2016; Kamp et al. 2017).
13.5 14 14.5 15 15.5 16 16.5 17 17.5 wavelength (micron) 0 0.002 0.004 0.006 0.008 0.01 0.012 Flux (Jy) C2H2 HCN H2O CO2 OH NH3 all species CO2 bandhead HCN bandhead C2H2 bandhead
Figure 1.5: The mid-infrared spectra of our standard T Tauri disk model from chapter
4, computed with FLiTs and convolved to a spectral resolution of R = 600. Because
each individual spectrum has been arbitrarily shifted for visibility purposes, the black, horizontal dashed line indicates the continuum level for each species.
1.11.1 ProDiMo
The ProDiMo models begin by creating the 2D distribution of gas and dust. Although it is possible to create a disk structure that is in hydrostatic equilibrium, the DIANA standard model uses a parametrized disk structure (Woitke et al. 2016). The gas and dust are distributed according to a power-law surface density distribution, and then scaled vertically to a given scale height:
ρ(r, z)∝ exp −z2 2Hg(r)2 where Hg(r) =H0 rr 0 β . (1.8)
ρ(r, z)is the gas mass density using cylindrical coordinates, H0is the scale height
of the gas at radius r0, and β is the flaring exponent. By vertically integrating
ρ(r, z), we obtain the gas surface densityΣ(r)that is used to fix the constant of proportionality in Eq. (1.8).
The dust is then concentrated towards the mid-plane through a dust settling mechanism (Dubrulle et al. 1995), but the gas structure remains fixed.
Following the disk structure calculations is the 2D dust continuum radiative transfer, solving the equation
dIν
dτν
22 introduction
while assuming coherent isotropic scattering Sν=
κabsν Bν(Td) +κscaν Jν
κextν (1.10)
where Iνis the spectral intensity, Jν= 4π1
R
IνdΩ is the mean intensity, Sνis the
source function, Bνis the Planck function, and κabsν , κscaν , and κextν =κabsν +κscaν
are the dust absorption, scattering, and extinction coefficients (Woitke et al. 2009). For every grid point in the disk, the ray-based radiative transfer traces a path backwards along the direction of photon propagation, solving these equations of radiative transfer. Given an input stellar spectrum, the radiative transfer provides the dust temperature and spectral intensity (from 90 nm to 10 mm) for every grid point in the disk model.
After the dust radiative transfer has been fixed, the chemistry and gas thermal balance routine can be run. The gas surface densities and scale heights remain fixed in our models, but the chemistry is allowed to run to chemical and radia-tive equilibrium. The large, state-of-the-art DIANA standard chemical network produces abundances for 235 species, and level populations for a subset of these (Kamp et al. 2017). The reaction rate databases that can be used are UMIST2006 (Woodall et al. 2007), UMIST2012 (McElroy et al. 2013), KIDA2011 (Wakelam et al. 2012), and the OSU (Ohio State University) network from Eric Herbst. In addition to the reactions (up to three-body) and rate constants from these databases, we also include the adsorption and desorption of gas onto dust grains. Full details of the chemical networks and reaction rates are described in Kamp et al. (2017). In this thesis, we adopt the UMIST2012 database. These calculations are done one grid point at a time. However, if necessary, global iterations can be computed. For example, when solving a model for hydrostatic equilibrium (rather than having a parameterized disk structure), we begin a new global iteration starting with the new dust structure.
Following the completion of the thermochemical part of the code, we can finally compute the line radiative transfer. The level populations of many species are calculated in the model using an escape probability method (Woitke et al. 2009), and thus for any of these species and transitions we can compute the profile individual spectral lines. These methods are detailed in Woitke et al. (2011).
The result of these steps is a full ProDiMo disk model, where the 2D abundances and reaction rates of 235 simple species can be studied, as well as observable parameters such as the SED and individual spectral lines.
Although protoplanetary disks evolve on time-scales that are relatively short by astrophysical standards, we model only static disks (the radiative transfer and opacities are fixed at the start of the model). The main reason is that coupling hydrodynamics with thermochemistry is a computationally expensive task which will likely require new code that can use Open MPI on a modern computer cluster. A partial exception to this is in chapter 4, where we use a dust evolution code to model the dust distribution in a T Tauri disk at several discrete time-steps,
but this is not a self-consistent implementation. The coupling of these codes does not result in global iterations, and the dust evolution code knows nothing about the thermochemical component: the output from the dust evolution code is simply fed downstream to create separate ProDiMo models with different dust structures.
1.11.2 FLiTs
The line radiative transfer module of ProDiMo is not suitable for producing mid-infrared spectra: we need a tool that can compute spectral lines much more quickly, and one that can treat the physical and spectral overlapping of lines in a self-consistent manner. FLiTs is able to self-consistently compute these blended lines, and each individual spectral line takes only about∼0.2 s to compute.
FLiTs is a new tool (written by Michiel Min and described by Woitke et al. 2018) which can quickly calculate the spectrum of a disk in the mid-infrared. It accepts a completed ProDiMo model as an input, and the infrared spectrum can be calculated for every species that has level population data.
FLiTs uses a ray-tracing algorithm that is optimized for both speed and accuracy. We solve the monochromatic formal solution of radiative transfer to construct an image of the entire disk at each wavelength point, and this usually is a relatively slow procedure: the line profile must be sampled at a very high spectral resolution. Although FLiTs can produce a very high-resolution spectrum, it can also produce accurate line fluxes when sampling at a reduced spectral resolution. This is done by taking the exact wavelength used by the ray-tracer randomly from the wavelength bin concerned. The effect of this is that we integrate over the strictly defined width of each wavelength bin. Similarly, the positions of each ray used by the ray-tracer are also randomized. These techniques avoid the appearance of aliasing effects, while producing accurate integrated line fluxes
with a relatively coarse (for example, 5 km s−1) velocity resolution. However,
with such a coarse resolution the individual line profiles are poorly resolved. The line radiative transfer in FLiTs follows the same approach as ProDiMo, as described by Woitke et al. (2009). The line radiative transfer equation is given by
dIν
ds =κextν (Sν−Iν), (1.11)
where Sνis the source function, Iνis the spectral intensity, κextν is the extinction
coefficient, and s is the distance along a ray.
Although in principle FLiTs can be used for any wavelength region, we use FLiTs in Chapters 3 and 4 in order to study the infrared spectra of T Tauri disk models. Figure 1.5 shows FLiTs spectra for a “standard” T Tauri model. One of the unique features of FLiTs is the self-consistent computation of line blends, even across different species. Thus, FLiTs can either produce the spectrum for a single species, or the spectrum of multiple species. In Chapter 3 we show that this is of
24 introduction
minor importance for the ro-vibrational water lines, where para- and ortho- lines can interfere with each other. The superposition of separately-computed lines may thus slightly over-estimate the flux of the lines when computed self-consistently. One consequence of this approach is that a multi-species spectrum cannot be separated into its component species. However, in practice, we find that the molecular lines of different species do not interact with each other, so it is feasible to calculate the spectrum for each species individually.
1.11.3 Model results
The results from ProDiMo and FLiTs allow almost every observable feature of a protoplanetary disk to be studied in a model. The greatest difficulty in both fitting observations and understanding the models is finding a unique solution: because there are so many features and parameters in the models, there is a great deal of degeneracy in the results. An approach that aims to study only a few aspects at once is most feasible: this is why the “DIANA standard” model has been introduced (Woitke et al. 2016). For example, if we wish to study the effects on the mid-infrared spectra of several parameters such as the UV flux and disk flaring, we can create a consistent approach by only changing one parameter at once.
Figure 1.6 shows an example of the results that we can obtain a ProDiMo model. The 2D disk models include basic information such as the gas and dust temperature structures, opacities and optical depths, the dust distribution, abundances and reaction rates for up to 235 species, and level populations for a subset of these species. Thus, any single ProDiMo model can be used not only to study the inner few AU of a disk in the infrared, but also to study the outer parts of the disk that emit in the far-infrared and sub-mm regimes.9
1.12 this thesis
The question that we ultimately desire to answer is how to get from a molecular cloud to a planetary system. Although we know most of the answers in fairly broad strokes, we are often missing many finer details. For example, we have not yet observed a planetary companion to a non-binary brown dwarf. Given that we
know of multiple exoplanet systems around stars that lie just above the 0.08 M
upper limit for a brown dwarf, this raises the question: is the lack of observations simply a sensitivity issue, or does some process related to the sub-stellar nature of the brown dwarf reduce the likelihood of planets forming? Such a dramatic process would likely change the chemistry of the disk enough to leave a chemical
9 In doing this, it is necessary to use a very high resolution in order to ensure that the entire disk is adequately resolved, for example a 240×180 model. To concentrate additional grid points in the inner disk, increase theNinnerparameter, to perhapsNinner=80.
101 100 101 102 r [au] 0.0 0.1 0.2 0.3 0.4 z/r 1.0 10.0 0.0 1.0 5.0 7.5 10.0 12.5 15.0 log n<H > [cm 3] 101 100 101 102 r [au] 0.0 0.1 0.2 0.3 0.4 z/r 0.0 0.8 1.6 2.4 3.2 4.0 log Tgas (K) 101 100 101 102 r [au] 0.0 0.1 0.2 0.3 0.4 z/r -12.0 -10.0 -8.0 -6.0 -4.0 log ( H2 O) /( H)
Figure 1.6: Some results from our standard T Tauri model in chapter 3. Upper left: the total hydrogen number density. The black contours are the AV=1 and AV=10 vertical optical depth contours at 20 µm, and the red contours are the radial optical depth contours for AV,rad =0.01 and AV,rad=1. Upper right: the gas temperature with gas temperature contour lines. Bottom: abundance of H2O gas. Note the hole in H2O abundance from around 1 AU to 30 AU: this is where water ice resides.
26 introduction
signature that we could observe. The first science question which we address is to what extent brown dwarf disks can be modelled like a scaled-down T Tauri, and whether or not the chemistry of these models is fundamentally different.
The second science question addresses the infrared spectra of disks, in partic-ular T Tauri disks. Upcoming infrared observatories will give us significantly improved infrared spectra. With enough sensitivity and resolution to resolve the emission bands of several species in the near- and mid-infrared – and even individual spectral lines – we can learn much about where these species exist in the disk, their temperatures, and their abundances. In order to reach such an in-depth understanding, we need also to incorporate similar levels of complexity into our disk models.
The final science question is that of dust evolution: how does it affect the inner few AU of protoplanetary disks, where terrestrial planets are thought to form? To address this question, we combine dust evolution models with our thermochemical ProDiMo models in order to simulate the effect of dust evolution on the disk chemistry and analyze the infrared spectra.
This thesis makes some progress towards answering these three science ques-tions. Concerning protoplanetary disk models, we wish to unify our models such that a single modelling technique is able to describe disks around brown dwarf, T Tauri, and Herbig disks, such that simply by changing a few parameters we can change a typical Herbig model into a typical brown dwarf model. So far, there are many observational differences between Herbig and T Tauri disks that we do not yet fully understand. For example, why do we not see species such as
H2O and HCN in the infrared spectra of Herbig disks? These disks are bright
and easy to observe, yet the only species that we regularly detect is CO.
The regime between T Tauri and brown dwarf disks is equally important, but less explored. The aim of Chapter 2 is to probe the differences between T Tauri and brown dwarf disks. The question which we address here is whether or not a brown dwarf disk can be treated simply as a T Tauri disk but scaled down in mass and radius, or if there are significant additional processes that need to be incorporated. We use ProDiMo models in order to understand how we might understand the structure of a brown dwarf disk, and to find any differences that might arise between a brown dwarf and a T Tauri disk. Using our model, we reproduce the SED and sub-mm CO line flux of the brown dwarf disk ρ Oph 102 observed by Ricci et al. (2012). Our conclusion is that it is feasible to model brown dwarf disks using the same techniques as used to model their more massive counterparts, and that overall the disks look similar to a scaled-down T Tauri disk.
Chapter 3 investigates the mid-infrared spectra of T Tauri disks. The infrared spectra have huge potential for analysis because the lines of each species trace different layers in the disk. With the help of thermochemical modelling we can try to understand the abundances of each species, their formation and destruction mechanisms, and their locations in the disk. We use the codes ProDiMo and
FLiTs to calculate the spectra of a series of disk models, analyzing the effects that parameters such as the UV flux and the degree of flaring can have on the spectra (these parameters are generally quite uncertain, and heating processes have a significant effect on the disk chemistry). We analyze the line fluxes, the sizes of the line-emitting regions, and the gas and dust temperatures from which the lines are emitted. We find that these properties can change significantly between models, and suggest that using 2D models that can incorporate gradients in the gas temperature is key to understanding these spectra in the future. We also find that increasing the gas-to-dust ratio of our models to 1000 increases the line fluxes better to match the fluxes we observe in T Tauri disks. Using and understanding more advanced models is a necessary step towards understanding all of the wonderful data that new observatories such as JWST will give us.
Although there is a lot of complex information buried in the spectral lines of disks, the dust is equally important. Small dust grains are the main carrier of opacity in the disk, and larger dust grains eventually come together to form terrestrial planets. This means that the mass ratio between the total gas content and the mass of small dust grains can strongly affect the fluxes of mid-infrared spectral lines. However, the mechanisms of dust evolution are still poorly under-stood. Chapter 4 analyses the effect of dust evolution on the mid-infrared spectra of a typical T Tauri disk. For the first time, we couple dust evolution with 2D thermochemical models to create a simplified model of an evolving disk from a thermochemical perspective. We find a strong relationship between the age of our disk model and the mid-infrared line fluxes. We find that the dust distribution can have a profound effect on the inner few AU of a disk, and find that the
line flux increases with age for every species except C2H2. We conclude that
because each species emits from different radii and heights in the disk, the flux ratios of mid-infrared lines may hold the key to understanding the distribution of dust in the inner disk. A natural consequence of this dust evolution is that the disk becomes more optically thin over time, which explains why we need to substantially increase the gas-to-dust ratio of our disk models in order to match observed infrared line fluxes.
Although there is still much yet to understand, we hope that this work lays a valuable foundation for the future: towards understanding the differences in disks across spectral types and down to the sub-stellar brown dwarf regime, how to model and analyze their infrared spectra in detail, and how to couple hydrodynamical and thermochemical disk models together.
