UNIVERSITY OF CINCINNATI
Date:______
I, ______, hereby submit this work as part of the requirements for the degree of: in:
It is entitled:
This work and its defense approved by:
Chair: ______
Dust Grain Growth and Disk Evolution of a Set of Young
Stellar Objects
A dissertation submitted to the
Division of Research and Advanced Studies
of the University of Cincinnati
in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy (Ph.D.)
in the department of Physics
of the McMicken College of Arts and Sciences
2008
by
William Joseph Carpenter
B.S. The Ohio State University, 1998
M.S. Miami University of Ohio, 2001
Committee Chair: Prof. Michael Sitko, Ph.D.
Abstract
This work investigates the observed properties of a sample of young stellar systems using two computer-modeling codes, DUSTY and the Whitney TTSRE codes, to fit the spectra of many objects. The parameters of these computer models are used to attempt to answer several basic questions related to circumstellar disk evolution. Silicate band strengths and millimeter spectral indexes are related to grain growth and are found to be reasonably correlated for the group of models. Circumstellar disk self-shadowing can effect the spectral shape of far infrared data and the relation between the two are studied showing a possible correlation, however other factors can effect the shape of the far infrared data and so a firm correlation can not be confirmed. Six objects out of the nearly 50 stars fit with DUSTY are fit with the Whitney code. Chronological age estimates are compared with model parameters but no correlation could be found between either silicate band strength or millimeter spectral index and the objects’ ages. This indicates that the evolutionary ages and chronological ages of this set of objects are not closely related. For two stars, HD 31648 and HD 163296, time dependent data exists showing differences in the near infrared region. To fit the differing data, the inner radius of the model disk needed to move outwards and the geometrical thickness of the model disk needed to increase. This provides a possible disk evolution scenario to explain the differing spectra.
iii iv Acknowledgements
I express my deep gratitude to Prof. Sitko for all of his patient assistance in the completion of this project. I also thank him for giving me the opportunity to visit telescopes both in Arizona and Hawaii- even if it was up on a mountain and not down by the beach. I thank Moshe Elitzur, Barb Whitney, and all those legions of people that came before me and wrote the codes used in this project so I didn’t have to.
Finally, I super thank my wonderful wife, Amelia, for having such patience with me during these years and for supporting me so much! I also want to thank the two new
Carpenters, Ali and Jimmy, for being so awfully cute and also for not demanding TOO much of my time and allowing me to finish!
v Table of Contents
1. Introduction 11
2. Background Science 14
2.1. Young Star Formation 14
2.2. Circumstellar Material 16
2.3. Light Interaction with Dust 21
2.4. SED Modeling Codes 30
2.5. Nature and Use of the DUSTY Code 30
2.6. Nature and Use of the Whitney Code 40
3. DUSTY Fits and Results 54
3.1. AA Tau 63
3.2. AB Aur 64
3.3. BF Ori 66
3.4. BFS60 67
3.5. CO Ori 68
3.6. CQ Tau 70
3.7. CW Tau 72
3.8. CY Tau 73
3.9. DG Tau 75
3.10. DL Tau 77
3.11. DM Tau 78
3.12. GI Tau 80
3.13. GK Tau 81
1 3.14. GM Aur 83
3.15. GW Ori 85
3.16. HD 31648 86
3.17. HD 35187 88
3.18. HD 37806 89
3.19. HD 45677 90
3.20. HD 50138 93
3.21. HD 58647 94
3.22. HD 98800 96
3.23. HD 100546 98
3.24. HD 135344 (SAO 206462) 100
3.25. HD 141569 102
3.26. HD 163296 104
3.27. HD 169142 106
3.28. HD 190073 108
3.29. HD 250550 109
3.30. Hen 3-600 110
3.31. HK Ori 112
3.32. HL Tau 114
3.33. HP Tau 116
3.34. HP Tau G2 117
3.35. HP Tau G3 118
3.36. HR 4796 120
2 3.37. LkCa 15 121
3.38. LkCa 21 123
3.39. NV Ori 124
3.40. RR Tau 125
3.41. RW Aur 127
3.42. RY Ori 129
3.43. RY Tau 130
3.44. SU Aur 131
3.45. TW Hya 133
3.46. UX Ori 134
3.47. UY Aur 136
3.48. V590 Mon 137
3.49. V594 Cas 138
3.50. XZ Tau 140
3.51. Grain Growth and Settling Correlation 141
3.52. Disk Inner Wall and Self Shadowing Correlation Study 143
4. Whitney Code Fits and Results 150
4.1. HD 31648 155
4.2. HD 163296 162
4.3. Inner Disk Analysis 169
4.4. HD 100546 174
4.5. HD 135344 (SAO 206462) 179
4.6. HD 169142 183
3 4.7. AB Aur 190
4.8. Discussion 194
5. Conclusions 198
6. References 202
7. Appendix A 215
4 List of Figures
2-1 Sample SED and drawing of a Class II object 15
2-2 Basic flared disk model 18
2-3 More complex disk model with hole and ‘puffed up’ inner rim 19
2-4 Example close up SED of the NIR spectral region of HD 163296 20
2-5 Cartoon showing two suggested sources of excess NIR emission 20
2-6 Cartoon of stellar radiation incident on a grain and resulting emission 22
2-7 Wavelength dependence of Qabs 24
2-8 Wavelength dependence of Qabs for several grain sizes 25
2-9 Emission spectra of a 1000K blackbody and three grain types 25
2-10 Temperature vs. stellar distance for several grain types 27
2-11 Temperature vs. stellar distance for several grain sizes 27
2-12 DUSTY SED internal coded DL-Silicates and C# Mie code DL-Silicates 30
2-13 The DUSTY code model 31
2-14 An example of DUSTY halo disk reheating 33
2-15 The SED of HD 31648 as an example of limited grain growth 37
2-16 The SED of XZ Tau as an example of more advanced grain growth 38
2-17 An example SED of a ‘toy’ model for an early stage stellar system 39
2-18 The SED of HD 58647 40
2-19 Density plots of inner disk region and halo envelope 41
2-20 The SED of HD 135344 42
2-21 Cutaways diagrams showing cavity wall shape 43
2-22 Flattened envelope density structure and inner region disk structure 44
5 2-23 Schematic of the four dust file regions 45
2-24 Image file showing a face on object 48
2-25 Example of the same model with different photon counts 49
2-26 Whitney code produced model of Type 0 SED 50
2-27 Whitney code produced model of Type II SED 51
2-28 Density plot of a Whitney model disk with flaring parameter B=1.12 52
2-29 Density plot of a Whitney model disk with flaring parameter B=0.86 52
3-1 Comparison of pyroxene60 with olivine 56
3-2 Comparison of adding graphite and increasing the grain size 57
3-3 Comparison of replacing DL-Silicates with pyroxene 58
3-4 The SED and DUSTY model of AA Tau 63
3-5 The SED and DUSTY Model of AB Aur 64
3-6 The SED and DUSTY model of BF Ori 66
3-7 The SED and DUSTY model of bfs60 67
3-8 The SED and DUSTY model of CO Ori 68
3-9 The SED and DUSTY model of CQ Tau 70
3-10 The SED and DUSTY model of CW Tau 72
3-11 The SED and DUSTY model of CY Tau 73
3-12 The SED and DUSTY model of DG Tau 75
3-13 The SED and DUSTY model of DL Tau 77
3-14 The SED and DUSTY model of DM Tau 78
3-15 The SED and DUSTY model of GI Tau 80
3-16 The SED and DUSTY model of GK Tau 81
6 3-17 The SED and DUSTY model of GM Aur 83
3-18 The SED and DUSTY model of GW Ori 85
3-19 The SED and DUSTY model of HD 31648 86
3-20 The SED and DUSTY model of HD 35187 88
3-21 The SED and DUSTY model of HD 37806 89
3-22 The 1980 SED and DUSTY model of HD 45677 90
3-23 The 1992 SED and DUSTY model of HD 45677 91
3-24 The SED and DUSTY model of HD 50138 93
3-25 The SED and DUSTY model of HD 58647 consisting of a disk and 94
blackbody
3-26 The SED and DUSTY model of HD 58647 consisting of a large grain 95
halo
3-27 The SED and DUSTY model of HD 98800 96
3-28 The SED and DUSTY model of HD 100546 98
3-29 The SED and DUSTY model of HD 135344 100
3-30 The SED and DUSTY model of HD 141569 102
3-31 The SED and DUSTY model of HD 163296 104
3-32 The SED and DUSTY model of HD 169142 106
3-33 The SED and DUSTY model of HD 190073 108
3-34 The SED and DUSTY model of HD 250550 109
3-35 The SED and DUSTY model of Hen 3-600 110
3-36 The SED and DUSTY model of HK Ori 112
3-37 The SED and DUSTY model of HL Tau 114
7 3-38 The SED and DUSTY model of HP Tau 116
3-39 The SED and DUSTY model of HP Tau G2 117
3-40 The SED and DUSTY model of HP Tau G3 118
3-41 The SED and DUSTY model of HR 4796 120
3-42 The SED and DUSTY model of LickCa 15 121
3-43 The SED and DUSTY model of LickCa 21 123
3-44 The SED and DUSTY model of NV Ori 124
3-45 The SED and DUSTY model of RR Tau 125
3-46 The SED and DUSTY model of RW Aur 127
3-47 The SED and DUSTY model of RY Ori 129
3-48 The SED and DUSTY model of RY Tau 130
3-49 The SED and DUSTY model of SU Aur 131
3-50 The SED and DUSTY model of TW Hya 133
3-51 The SED and DUSTY model of UX Ori 134
3-52 The SED and DUSTY model of UY Aur 136
3-53 The SED and DUSTY model of V590 Mon 137
3-54 The SED and DUSTY model of V594 Cas 138
3-55 The SED and DUSTY model of XZ Tau 140
3-56 Band/continuum ratio and millimeter spectral index correlation 142
3-57 12µm/25µm correlation plot showing all selected objects 145
3-58 12µm/25µm correlation plot with “questionable” objects removed 145
3-59 12µm/60µm correlation plot showing all selected objects 146
3-60 12µm/60µm correlation plot with “questionable” objects removed 146
8 3-61 25µm/60µm correlation plot showing all selected objects 147
3-62 25µm/60µm correlation plot with “questionable” objects removed 147
4-1 The 1996 SED and Whitney model of HD 31648 155
4-2 The 2004 SED and Whitney model of HD 31648 155
4-3 Model Comparison of HD 31648 from 1996 and 2004 156
4-4 Surface brightness plot of HD 31648 158
4-5 Temperature and density cross-sections of HD 31648 160
4-6 The HD 31648 Whitney dust file model locations 161
4-7 The 2005 SED and Whitney model of HD 163296 162
4-8 The 2002 SED and Whitney model of HD 163296 162
4-9 The SED and Whitney model of HD 163296 including wall component 163
4-10 SED Comparison of HD 163296 from 2005 and 2002 164
4-11 Surface brightness plot of HD 163296 166
4-12 Temperature and density cross-sections of HD 163296 167
4-13 Temperature density plot of the HD 163296 model w/wo envelope 168
4-14 The HD 163296 Whitney dust file model locations 169
4-15 Close-up comparison of HD 31648 from 1996 and 2004 170
4-16 Close-up comparison of HD 163296 from 2005, 2002, and 2005+Wall 171
4-17 Diagram of a model of inner disk structure changes 172
4-18 The SED and Whitney model of HD 100546 174
4-19 Surface brightness plot of HD 100546 176
4-20 Temperature and density cross-sections of HD 100546 177
4-21 The HD 100546 Whitney dust file model locations 178
9 4-22 The SED and Whitney model of HD 135344 (SAO 206462) 179
4-23 Surface brightness plot of HD 135344 (SAO 206462) 181
4-24 Temperature and density cross-sections of HD 135344 (SAO 206246) 182
4-25 The SED and Whitney model of HD 169142 plus added DUSTY halo 183
component
4-26 Close-up of the SED and Whitney model of HD 169142 plus added 184
DUSTY halo component
4-27 The SED and Whitney model of HD 169142 plus added DUSTY wafer 184
disk component
4-28 Close-up of the SED and Whitney model of HD 169142 plus added 185
DUSTY wafer disk component
4-29 Diagram of physical representation of the added DUSTY disk 187
4-30 Surface brightness plot of HD 169142 188
4-31 Temperature and density cross-sections of HD 169142 189
4-32 The SED and Whitney model of AB Aur 190
4-33 Temperature and density cross-sections of AB Aur 192
4-34 The AB Aur Whitney dust file model locations 193
4-35 Plot of silicate band strength/continuum ratio vs. age 195
4-36 Plot of millimeter spectral shape vs. age 196
10 Chapter 1: Introduction
Since the dawn of time, people have wondered about how everything around us
came to be. More specifically, the question of the origin of planetary systems, including
our own, is of great interest. These systems evolve from a cloud of many tiny grains to a
‘disk’ of only a few very large grains (planets). The rise of infrared (IR) spectroscopy and increase in computing power has allowed better data collection and more efficient and sophisticated computer modeling. Details of first stage planet formation theory can now be better understood.
Since the timescale of planet formation is on the order of millions of years, it is not possible to observe the evolution of individual objects. By studying a large group of somewhat similar objects, it is possible to study the relationships between them to discover correlations between various properties of the object spectra.
This work investigates the observed properties of a sample of these young systems using two computer-modeling codes to fit the spectra of several objects. These are the DUSTY code, a simple model that can be used on a large number of objects quickly, and the Whitney TTSRE code, a more complex and realistic model that runs much slower. Therefore, nearly 50 objects are modeled with the DUSTY code. Out of those, six are selected to be modeled as well with the more complex and time consuming
Whitney code.
The parameters of these computer models are used to attempt to answer several basic questions related to circumstellar disk evolution. The standard model of early stage planet formation states that as a system ages, dust grains will grow in size and settle into a disk. The strength and shape of silicate grain features and the millimeter data spectral
11 slope in circumstellar spectra relate to grain growth and should be correlated. Part of this
investigation will be to find if these properties are correlated in a selection of pre main
sequence stars.
As disks settle and evolve, it is thought that the inner wall of the disk will shadow
some part of the rest of the disk. A shadowed disk should produce a differently shaped
spectrum than a non-shadowed disk. Therefore, another part of this study will be to
discover correlations, if any, that exist between the shadowing and shape of the far
infrared region of the spectra.
A subset of six objects will be modeled with the Whitney code. The relation
between silicate band strength and millimeter spectral slope will also be studied as well
as a very basic look at if any correlation can be seen between the development age and
chronological age for these six objects. Also, something that has never been done before,
time dependent models are constructed for two objects, HD 31648 and HD 163296. Data
exist for these objects from different epochs that show spectral differences. The different parameters of the model disks necessary to fit the different spectra will provide a glimpse at one possible scenario to explain what happened in the systems to create the different spectra.
Through answering these questions, more can be learned about the process of disk evolution and how some of the systems’ spectral properties relate to each other in this regard.
Chapter 2, the background science of circumstellar disk formation and evolution is briefly discussed as well as the basic science of the interaction of electromagnetic
12 radiation and interstellar dust grains. A general overview of the nature and use of two computer modeling codes, DUSTY and Whitney TTSRE, is also included.
Chapter 3 discusses the results of DUSTY fits for nearly 50 objects. The correlation between silicate feature band strength and millimeter spectral index as well as the correlation between stellar blackbody flux blockage and mid-infrared spectral shape are considered and analyzed.
Chapter 4 discusses the results of Whitney TTSRE fits of six objects and includes analysis of the correlation between silicate feature band strength, millimeter spectral index, and estimated ages of the six objects. Also, the nature of disk changes necessary to explain time-dependent spectra data for two objects is analyzed and discussed.
Finally, the last small chapter sums up what was found in the previous two chapters and also includes citations to the various instruments and sources of much of the data used in this work.
13 Chapter 2: Background
This chapter briefly examines the state of the science behind dusty disk formation and electromagnetic interactions with matter. Following is a short discussion of the
DUSTY and Whitney code models, inputs, and outputs.
Young Star Formation
The interstellar medium consists of mostly hydrogen gas with a small amount of helium and even smaller amounts of ‘metals’ such as oxygen and carbon. Local densities and temperatures of the gas clouds will vary over location. If a portion of the gas is cold and dense, the self-gravitational energy of the clump will be higher than the thermal kinetic energy and so the clump will begin to collapse. A local high density region will form in the collapsing gas. This point could be at the center, or for sufficiently large gas clouds, several high-density points may form thus ending in the birth of several stars.
Once a high enough pressure is reached, nuclear fusion of deuterium begins. This fusion releases a large amount of energy, which acts as radiation pressure against the gravitational pressure of the material still falling towards the now burning core. At first, the radiation pressure is insufficient to halt the contraction of the gas cloud. As more material falls and increases the pressure and temperature at the core, the radiation pressure increases to eventually equal the gravitational pressure and to create a stable star that then lives out its life on the main sequence of the HR diagram (Stahler et al. 1980).
For a real cloud of gas, this process is much more complicated than the simple sphere. A real cloud is turbulent. Very quickly at the beginning of its collapse, the
14 angular momentum of the cloud will establish a rotation axis. Material along the axis will find it much easier to fall into the protostar than material around the equator. This will form a disk of material orbiting around the star. This disk, whose interactions with the star are very complex and are being studied today (Boss 2007), will eventually collapse itself and form planetesimals that may eventually form planets.
When actually observing these objects, as the stars are first collapsing, they are still shrouded in gas and dust and so cannot be seen directly. However, as will be explained below, the surrounding dust absorbs the starlight and reemits it in the infrared.
These sources are called Class I sources and their flux is dominated by infrared (hereafter
IR) radiation (Wilking 1989). As the circumstellar material collapses onto the star and a disk takes shape, the star is more visible and so the flux is a ‘normal’ blackbody spectrum with an IR excess giving rise to a two peak appearance to the spectral energy distribution
(hereafter SED). The amount of excess
Figure 2-1: Sample SED and drawing of a Class II object (Wilking 1989).
depends on how much dust is in the envelope surrounding the star and how large the disk of material is. These are called Class II sources. There is also the Class III source which
15 is a near-naked star with just a tiny IR excess indicating that most of the dust and disk has
been accreted or dissipated. Class II sources are of interest for this study.
A subset of Class II sources is the Herbig Ae/Be (HAEBE) star. These stars are
roughly defined by three properties. These are: A or B spectral type with emission lines,
IR excess from hot or cold circumstellar material, and of luminosity class III through V
(Waters et al. 1998). They also tend to have masses of 2 – 10 Msun.
A second subset of Class II sources is the T Tauri star. Named for the first one
discovered in the constellation of Taurus, these are variable stars that generally are F, G,
or K spectral type. T Tauris are very young stars and are still surrounded by circumstellar
dust. These stars tend to have masses of 0.5 – 3 Msun (Bertout 1989).
Circumstellar Material
The IR excess in the above-mentioned object SED’s is the result of emission from
irradiated circumstellar dust (Dullemond 2001). The temperature of a piece of
circumstellar material is basically determined by the distance of the piece from its star.
Other smaller factors such as the albedo, or surface reflectance, of the grain and radiation
from the rest of the circumstellar material affect the temperature as well. The basic
1 temperature relation to distance for a blackbody is roughly 1 and is found from the r 2 energy equilibrium relation
4 Lstar 4σTblackbody = 2 4πdblackbody
16 where Lstar is luminosity of the star and dblackbody is the distance between the star and the blackbody. Since the dust can only exist at temperatures less than ~1500°K, this determines the minimum distance that dust could be orbiting the star. From Wien’s law, this temperature results in maximum intensity of emission of around 2.0 µm. This wavelength and longer (resulting from cooler grains) is in the IR and produces the IR excess seen in the SED’s.
The dust emission reveals the chemical make up of the dust through vibrational transition photons, the specific wavelengths showing the signature of specific minerals.
Individual grains consist of a large number of minerals and are mixtures of several different types. Circumstellar dust spectra have indicated the presence of silicates, ices, and other materials. Many of the objects exhibit amorphous (disordered lattice structure) silicates, whereas a few show evidence of crystalline silicates such as HD 100546
(Malfait et al. 1998). Several ideas have been proposed for how this process occurred.
The dusty material, as it accretes towards the protostar, could get annealed as it nears the star. Though most material will simply accrete onto the star, some of it, including the annealed grains, is ejected out along the polar jets while a smaller amount could be thrown back out above the circumstellar disk where it can fall back into the disk material, or the so called disk wind (Hill et al. 2001). Another idea has material annealing in the cool, high density stellar outflows of red giant stars (Waters 1999), however, crystalline material is essentially lacking in interstellar material from which young disk systems have formed (Kemper et al. 2004). Also shock waves from the collapsing protostar could anneal the material through friction as the front passes by (Harker and Desch 2002).
17 It is clear that most HAEBE stars’ IR excess comes from disks (Dullemond 2002), although some stars could also have envelopes of material as well. The disk is heated by direct starlight grazing the surface and heating the grains. Internal viscous forces also heat disks because material towards the inner radial edge of the disk will be orbiting its parent star at a higher velocity than material towards the outer radial edge. The basic model of a disk consists of a flared disk of optically thick material extending out to some radius. The model disk is made passive meaning there are no internal viscous forces because modeling the internal viscous forces is quite difficult. Therefore, the heating of the model disk grains comes from the direct starlight. The grains would then radiate
Figure 2-2: Basic flared disk model (Chiang 2004)
half of their energy into space (which we can see) and half deeper into the disk (Chiang
2004). In the simplest models of a passive disk such as this, the disk material reaches the surface of the star. However, this model would produce too much flux in the near-IR region (NIR) (1-5 microns). This is because the grains in this model unrealistically extend to the surface of the star where the temperature is above the grains’ evaporation temperature. By introducing a hole (meaning the disk simply does not extend to touch the star), the model could fit the SED’s in this region (Dullemond et al. 2001).
18 Since the disk has some physical thickness, if there is a hole, the inner edge of the hole will be under direct starlight. This inner wall will be heated to a higher temperature
Figure 2-3: More complex disk model with hole and ‘puffed up’ inner rim (Dullemond et al. 2001).
than if it were shadowed by material closer to the star. This means that the inner wall will
‘puff-up’ and become thicker than the region of disk behind it. To complicate things, this puffed up wall will block and shadow some of the disk behind it from direct starlight.
This part of the disk can then only be irradiated by emission from other parts of the disk or by emission or scattering from a thin dusty envelope if one is present. The disk only flares up again in the region that is out of the shadow. Therefore, most of the SED emission between 1 µm and 5 µm from the disk comes from the puffed up rim and the illuminated edge (Dullemond et al. 2001).
Many of these types of objects show variability in their SED’s - especially in the near infrared (1 - 5µm) region. As this is the region of the SED most closely associated
19
Figure 2-4: Example close up SED of the NIR spectral region of HD 163296.
with the inner disk region, the variability indicates that something is changing in the inner disk. Many ideas have been suggested to explain the changes. Two of them are illustrated in the image below (Vinkovic and Jurkic 2007). The increased NIR emission could be the result of a
Figure 2-5: Cartoon showing two suggested sources of excess NIR emission (Vinkovic and Jurkic 2007).
20
change in the “puffiness” of the inner disk edge. If the disk for some reason were to puff up more for a period of time, more energy would be emitted in the NIR region. This could possibly be caused by a local reduction in gas inwards of the inner disk edge allowing more starlight to reach to the disk thus puffing it up further. Also, if a piece of the disk were to ‘break off’ to follow the magnetic field lines and if that piece were to pass through the line of sight to the observer, it would create more NIR flux during that time.
Light Interactions with Dust
The interaction of a photon with a dust grain is the central mechanism that determines the resulting SED. Dust grains receive mostly visual and UV light from their
2900µm parent star. Due to Wien’s Law, T = , where λmax is the wavelength of λmax maximum intensity, and the fact that most dust grains vaporize above ~1500˚K (as mentioned above), dust grains will radiate at ~2µm and longer wavelengths. These wavelengths are in the infrared. The following diagram illustrates the process of converting the visible and ultraviolet photons into IR photons.
Wien’s Law describes a blackbody spectrum. However, real dust grains, especially smaller ones, do not behave like blackbodies. Real grains tend to emit energy at certain wavelengths better than others. This is because a real small grain is made up of atoms in a vibrating structure and the various vibrational modes emit at specific
21 wavelengths. This efficiency is called a cross section. When the size of the grain is factored out, they are called efficiency factors, frequently designated in the literature as
Figure 2-6: Cartoon of stellar radiation incident on a grain and resulting emission
Q’s. Each type of radiative interaction, absorption, emission, and scattering has its own efficiency and is dependent on wavelength. To determine the spectral response of a grain, the efficiency factors can simply be multiplied with an appropriate blackbody.
Another way to describe the optical properties (and is usually what can be measured in a lab) of a grain is by listing its optical properties. These are listed in the form of a complex refractive index, or m(λ) = n + ik, where n is the real part of the complex index of refraction and k is the imaginary part that acts as an extinction coefficient. These n’s and k’s have been measured for many substances (Jager et al. 1994,
22 Mutschke et al. 1995, Chihara et al. 2002, Koike et al. 2003) and are used in the DUSTY and Whitney TTSRE software to help determine dust interaction effects.
The complex refractive index is just the beginning to determine the response of a grain from incident photons. Efficiency factors are needed. Calculating efficiency factors is not straight forward and approximations must be used. One of the simplest is the use of
Mie Scattering (Bohren and Huffman 1983). In Mie Scattering, dust grains are modeled
2πa as a uniform sphere with size parameter x, defined as x = . Then using the size of the λ sphere, and the n and k for a specific wavelength photon, the efficiencies, polarization and scattering of the particle can be calculated. The problem is solved through boundary value equations and summing an infinite series of spherical Bessel functions. The equations for the Q’s, keeping only terms up to x5, are for extinction (ext) and scattering
(sca)
8πa 6nk Q ≈ 2 2 2 2 2 ext λ (n − k + 2) + 4n k
2 2 8 4 m −1 Q ≈ x 2 sca 3 m + 2 and the relationship between Q’s is
Qext = Qabs + Qsca where Qabs is the absorption efficiency. Some approximations to Mie Scattering can be used under certain grain size and wavelength regimes. In the limit of small x (a small grain and large wavelength) Qsca approaches zero. This results in the well known
∝ λ−4 , which means shorter wavelengths are scattered much more readily than Qsca,λ longer wavelengths. This is called Rayleigh Scattering and is why the sky is blue.
23 When the grains are large, x→∞. The n’s and k’s effects are lessened and the particle begins to act like a blackbody. Qsca and Qabs both then tend to 1 resulting in Qext
≈ 2.
As an interesting exercise, the Q’s can be calculated for a single size and type of grain and plotted. Plugging in n’s and k’s into the above Q equations along with the selected grain size results in a plot of Q vs. wavelength.
Figure 2-7: Wavelength dependence of Qabs for a single grain of pyroxene (MgSiO3) of
0.005µm in size. Multiplying the Q’s with a blackbody of temperature T, in this case
1000K, gives the resulting flux spectra.
24
Figure 2-8: Wavelength dependence of Qabs for a single grain of Draine and Lee Silicate for three different grain sizes, 0.005µm, 1.0µm, and 3.0µm.
Figure 2-9: Emission spectra of a 1000K blackbody and three grain types. Pyroxene 0 Fe is (MgSiO3) and Pryoxene 50 Fe is (Mg0.5Fe0.5SiO3). DL-Silicates and DL-Graphite are from Draine and Lee (1984)
25 As can be seen from the above plots, each example grain absorbs and emits more efficiently at certain wavelengths. However, the wavelength dependency lessens and disappears as the grain grows in size eventually reaching the blackbody result of Qabs = 1.
When a grain is placed at some distance, r, from a star the grain would, in general, be capable of having a different temperature than an equivalent blackbody at the same distance. The following plots illustrate the temperature vs. radius response of a few different grain types, all of 0.005µm. For larger grains, the grain’s spectral response approaches that of a blackbody which can be seen in Figure 2-11 where the radial temperature dependence for the largest grain size nearly matches the blackbody curve. I constructed these plots through a MS-Excel spreadsheet by integrating the flux a grain received from a solar like star (~5800K) and determining the grain temperature required to produce an equal integrated flux from grain emission. Once the process was repeated for several different orbital radii, the resulting power laws could be determined.
26
Figure 2-10: Log/Log plot of temperature vs. stellar distance for a blackbody, DL silicate and graphite, MgSiO3 and Mg0.5Fe0.5SiO3 pyroxene grains of radius 0.005µm illuminated by solar luminosity star.
Figure 2-11: Log/Log plot of temperature vs. stellar distance for a blackbody and DL silicate grains of radius 0.005µm, 1.0µm, and 3.0µm illuminated by solar luminosity star.
Larger grains tend towards blackbody responses.
27
Power laws were used to fit each of the three example lines. The 0% Fe pyroxene
1 grain has a temperature relationship of T ∝ while a 50% Fe pyroxene grain has a r −0.75
1 temperature relationship of T ∝ . The Draine and Lee (DL) “dirty” silicates have a r −0.73
1 1 relationship of T ∝ and DL-graphite T ∝ . A blackbody, which absorbs all r −0.59 r −0.39 energy and emits all energy at an efficiency of one, has a temperature relation of
1 T ∝ . Pyroxene and DL silicates, have temperature relationships that drop off faster r −0.5 than a blackbody, meaning that in general a silicate grain at a radial distance r larger than a few AU will be colder than an equivalent blackbody grain. However, other factors such as grain size (larger grains can be closer to the star than smaller grains) and the exact silicate grain composition effect the temperature of a grain. Therefore the following discussion on radial location of grains does not apply in all cases.
The Figures 2-7 and 2-8 show that small silicate and pyroxene grains are very efficient IR emitters and poor UV and visual absorbers, although in pyroxene, the efficiencies also depend on the ratio of Mg to Fe in their composition, which can be seen in the plot in the higher UV absorption of the 50% Fe pyroxene over the 0% Fe pyroxene.
High Fe concentrations have little effect on the slope of the temperature-radial power law, but serve to move the line to the right, meaning that the grains are hotter than a blackbody at the same radius. This is easy to understand as the 50% Fe pyroxene absorbs more starlight but has the same IR emission efficiency as a 0% Fe pyroxene grain.
Therefore, pyroxene grains with more Fe are hotter.
28 Since most of the stellar flux is UV and visual, in general the silicate grain absorbs less of the stellar flux than an equivalently size blackbody type grain would.
Also, since the silicate grain is efficient at emitting in the IR, the grain finds it easier to shed its energy. Therefore, a silicate grain of temperature T will be closer to its star than a blackbody of the same temperature. However, one further complication, which helps to explain the crossing of the DL-silicate and blackbody curves in the above plot, is that the actual radial location of a grain determines its spectral shape. This means that at certain temperatures, a grain may be attempting to shed its energy at an inefficient λmax, which is determined from Wien’s Law. This could force the grain to actually be farther from the star than a blackbody of the same temperature. Only at cooler temperatures, the grain’s
λmax would allow more efficient emission and bring the grain closer to the star than a blackbody of the same temperature.
Graphite, however, is efficient at absorbing and emitting in the UV and visual and not as efficient in the IR. These types of grains will have the opposite effect. Graphite grains will find it easy to absorb much of the stellar light incident upon it but then will have greater difficulty emitting energy in the IR. Therefore, graphite grains will be hotter than a blackbody grain at an equivalent radius from the star. Consequently, in general, a small graphite grain would have to be father from its star than a blackbody at the same temperature T.
The example of simple sphere Mie Scattering Fortran code in the Appendix A of
Bohren and Huffman (1983) was translated into the C# programming language to learn the scattering code and also to create a code that could calculate Q’s for a given set of n’s and k’s. The C# code is attached at the end in Appendix A. The code is capable of
29 calculating Q’s for a single grain type of a user-defined range of grain sizes. Running a similar grain type and size range with DUSTY’s internal functions produces nearly identical SED’s.
Figure 2-12: Plot of DUSTY internal coded DL-Silicates (Red line) and personal C# Mie code DL-Silicates (Blue line). The blue line has been slightly shifted down for clarity.
SED Modeling Codes
Calculating SED’s requires computer codes. Many are in existence, but the two codes used to model SED’s in this work are the University of Kentucky’s DUSTY code
(Ivezic 1999) and Space Science Institute’s Barb Whitney TTSRE code. (Whitney et al.
2003). A brief description of each code’s model and parameters follows.
Nature and Use of the DUSTY Code
The DUSTY code is an analytic FORTRAN code that directly calculates an output SED from the interaction of an input SED and circumstellar material. The DUSTY code comes in two versions, a full version and a version without the disk model. The
30 runtime for a single model calculation depends on the nature of the input parameters, but most models run in 15-30 seconds. These models are scalable, meaning that the inputs and outputs are normalized and can be fit to data simply by multiplying the resulting fluxes by some factor.
The DUSTY code model consists of a spherically symmetric nonrotating shell, or halo, of dust centered on the parent star. The inner and outer boundaries of this shell are user defined. An optically thick, but geometrically thin disk is embedded inside the halo.
Its outer boundary is user-defined while its inner boundary, in the original release of the code, is set to the surface of the star. Later modifications to the code allowed the inner boundary of the disk to also be a user-defined parameter. The halo dust density varies over radius according to user-defined input.
RHalo
RDisk RDisk
RHalo
Figure 2-13: The DUSTY code model of a spherical halo and an embedded wafer disk.
The diagram shows the inner and outer radii of each structure.
31
DUSTY can take input grain parameters and dust input files which then uses Mie
Scattering theory to calculate the scattering and absorption efficiencies of the dust mixture. Based on the halo density gradient and outer radius parameters, a radial grid is set up to divide the dust into “boxes”. The boxes are placed logarithmically so that there are more grid cells closer to the star where the number density of the dust (grains per unit volume) is greater. For each cell, an electromagnetic propagation matrix is calculated from the dust efficiencies. The temperature of each cell is calculated initially from the stellar temperature and then iteratively from radiation from the other cells. A blackbody curve of the appropriate temperature is multiplied by the previously calculated dust extinction efficiencies to determine the contribution to the SED from that cell. All the contributions are added up to result in the final SED that is outputted in the *.stb file.
The disk is heated by the star and halo emission, but the halo is heated only by the stellar radiation. The effect of this can be seen in below plot. The portion of the disk inside the halo’s inner radius is not reheated, but the rest of the disk is, creating the excess emission. The disk model is a straightforward blackbody, but the reheating can create non-blackbody curved disk SED’s.
32
Figure 2-14: An example of DUSTY halo disk reheating.
Since the wafer disk was modeled in the code to reach down to the surface of the star, it was necessary to modify the code to allow the presence of a “hole”. This is because the evaporation temperature of dust, roughly 1500K, would be reached at some distance away from the star. As an initial fix, the code was essentially executed twice, once with the full disk and then again with the disk’s outer edge set to the same temperature as the inner edge of the halo. Subtracting the second disk from the first would create a brute force hole. Later, the code was modified by the author with some assistance from Dejan Vinkovic so that the wafer’s inner edge was actually located at the same user-defined temperature used for the halo component thus allowing a single run to determine the SED instead of two. The two components have the same inner temperature, but will have different inner radii because of the different natures of how the components are heated. Since the disk is radiated by the star at such a shallow angle, a piece of disk will be much cooler than a piece of halo at the same radius.
33 To simulate the presence of a puffed up inner rim of a disk, a single temperature blackbody curve shown below was added during the fitting process. This blackbody could be multiplied
2hc 2 / λ5 Bλ (T ) = hc e λkt −1 by some power of λ that crudely models the emissivity of smaller grain dust. The temperature and λ-parameter are then set to produce the best fit of the combined blackbody and DUSTY model with the observed data SED.
The input parameters for the halo are as follows. The user defines the inner edge of the halo by setting the temperature of the hottest material. Normally, this would be the evaporation temperature of the material, but a lower temperature (representing a larger inner hole) can be set as well. The outer edge of the halo is defined by a user set number of inner halo radii. Thus changing the inner radius would change the outer radius even without the latter input being changed.
The density of the halo is determined by setting the optical depth τ of the halo dust. This optical depth is normally set at 550nm wavelength, though that too can be changed. The optical depths at the other model wavelengths are determined by interpolating the scattering and absorption cross-sections of the grain mixture on to the wavelength grid using the input τ and wavelength fiducially. The density distribution can be set to a power law with a user-defined exponent. The user can also define several different power laws depending on the location’s radius allowing for very complex density vs. radius curves.
34 The halo dust consists of a consistent mixture of user-defined dust. The dust grain sizes are determined by a power law where the user can set the exponent and both the lower and upper size limits. For all of the DUSTY models in this work, the power law exponent is –3.5, which is the standard parameter used in Mathis et al. (1977). The user also inputs the types of dust grains and their relative weights. A few dust types are built into the DUSTY code, including Drain and Lee interstellar silicate and graphite. Other dust types can be entered through input files of listing the dust’s n’s and k’s. Once this dust mixture is calculated, it makes up the entirety of the halo. The code does not allow different parts of the halo (such as inner regions vs. outer regions for example) to contain different dust mixtures. For most of these models in this study, a dust mixture of roughly
50/50 silicates and graphite was used. The types of silicates and graphite varied between a few different chemical compositions and also amorphous and crystalline varieties.
“Age” of the dust was simulated by setting the grain size larger to roughly model grain growth.
The input star is defined by a Kurucz model (1993- found at http://www.stsci.edu/hst/observatory/cdbs/k93models.html) of the appropriate temperature and surface gravity. Kurucz models describe the SED of the star itself and is the input radiation that the code uses to then calculate the resulting SED from the circumstellar dust structure.
The disk input is limited to its outer radius, which can be entered either in units of stellar radii, or by the temperature of the outer edge. In the modified code, the inner radius is calculated to be the location of the same temperature as that of the inner edge of the halo.
35 The code outputs a file listing the SED, which is then plotted on a λFλ vs. λ log- log scale. This file also lists the fraction of flux resulting from scattering, attenuation, and dust emission. The output spectrum must then be reddened to model the effect of the interstellar dust that the resulting light must travel through to reach our telescopes. This was done with a standard interstellar extinction curve from Whittet (2003), which is synthesis of a number of earlier sources. The disk flux is output into a separate file. The disk flux is added to the halo/stellar flux file by hand (in an MS-Excel spreadsheet) through the following equation:
f total = ρf disk + (1− ρ) f sph
Where:
2x cosi ρ = 1+ 2x cosi
where i is the disk inclination. And:
L x = disk Lsphere
Using the fact that Ltotal = Ldisk + Lsphere and then rearranging
Ldisk L x = total L 1− disk Ltotal
L The quantity disk is outputted in the DUSTY disk file and is used to Ltotal calculate the inclination angle from any given ρ. Likewise, ρ can be calculated from a given disk inclination angle. If the inclination angle of the disk is known, for example from observation, then ρ is calculated and set in the model fitting process.
36 The DUSTY model is not a very realistic one. The dust surrounding a star would not likely be distributed into a perfect sphere and also would not be stationary. But as a simple model that takes a very short time to run, it is a potentially useful tool for determining gross characteristics about the composition of the dust and estimates of grain growth.
Figure 2-15: The SED of HD 31648 as an example of limited grain growth. The combined line is the disk and halo (including halo extinction diminished star spectrum) spectral components added together. ISM Extinction shows the extinction effect of the interstellar medium on the combined spectrum.
37
Figure 2-16: The SED of XZ Tau as an example of more advanced grain growth.
In the above examples, the two SED’s illustrate two different stages of grain growth. The first, HD 31648, displays a sharp 10µm silicate feature, indicative of small silicate grains, and a millimeter region mostly dominated by the same small grains in the halo component. The second example, XZ Tau, shows a more rounded silicate feature (large silicate grains) and a millimeter region dominated by the large grains of the disk.
The SED’s (examples shown above and below) are plotted using IDL and display several pieces of information. The input star is displayed as a green dotted line. The orange dashed line is the DUSTY output star/envelope SED. The blue dot-dashed line is the SED of the disk and the red line is the model SED that is the result of the combination of the envelope and disk lines. The purple line in some plots shows the effect of interstellar reddening. The routine adds the disk and envelope components in the method described above. The entire set of plots is then multiplied by a scaling factor to fit with observational data.
38 The DUSTY model is a reasonably close approximation for very young systems, where the dust is still mostly contained in a thick halo. This is easily modeled with a large optical depth. The DUSTY model is also very good for modeling much more evolved systems where the material has settled into a thin ‘wafer’ disk, with ‘bowling-ball’ grains that act as blackbody emitters. The SED’s of systems ‘in between’ can be modeled, but potential problems can arise. An example of this is AB Aur and is discussed in the AB
Aur’s entry in the DUSTY results chapter.
Figure 2-17: An example SED of a ‘toy’ model for an early stage stellar system.
39
Figure 2-18: The SED of HD 58647, and example of a late stage evolved system.
Nature and Use of the Whitney Monte Carlo Radiative Transfer Code
The Whitney TTSRE code uses a Monte Carlo technique to model the effect of a circumstellar environment on an input stellar SED. This brute force method allows for more complex geometries than in the DUSTY code, but also takes dramatically longer to run.
The Whitney code model is a Chiang and Goldreich (CG 1997) flared disk with a rotating spherical envelope. Cavities can be carved out of the envelope and material of different makeup and density can be placed in the cavity to model an outflow if desired.
The sample density plots below illustrate some of the geometrical qualities of the model and how the disk is embedded into an example envelope. The figure and others like it were created with IDL plotting routines bundled with the Whitney source code download.
I heavily modified several of the routines to be able to produce specific types of plots.
40
Figure 2-19: Sample density plots depicting detail of inner disk region (left) and the relationship between the disk and the halo envelope (right) Colors indicate mass density of material. Blue is lowest density and yellow is highest.
For some of the objects modeled with this code, a blackbody was added to fit the near IR region of the SED. This is necessary when there is a flux deficit in the 10 – 20µm region creating an SED with two stellar flux excess ‘humps’, one in the NIR and the second in the far infrared to the millimeter. The Whitney code cannot fit such SED’s, an extreme example is HD 135344 (SAO 206462) displayed below. The Whitney model is used to fit the outer ‘hump’ while the blackbody is added to model the NIR ‘hump’.
41
Figure 2-20: The SED of HD 135344, an example of an SED with an additional blackbody component.
The disk is characterized by an inner and outer radius. The disk’s thickness is determined by a flaring parameter and an inner edge half-thickness. The disk height is calculated by:
Β ⎛ r ⎞ ⎜ ⎟ h = h0 ⎜ ⎟ ⎝ R* ⎠ where R* is the radius of the star, h0 the disk height at the stellar radius, and B is the flaring parameter that determines whether the disk is ‘flared’ (concave upward), ‘flat’
(actually conical), or ‘anti-flared’ (convex upward).
The envelope’s parameters include an inner and outer radius, which can be set independently of the disk’s inner and outer radii, and the opening angle of the cavity. The wall of the cavity can either be streamlined, meaning a straight line, or a user-defined polynomial shape.
42
Figure 2-21: Cutaways diagrams showing cavity wall shape- streamline (left) and polynomial (right).
The exact shape and end points of the curve are input parameters that describe the cavity.
The envelope’s physical shape is spherical, but its density profile is a rotationally flattened “bubble”. The density gradient is fixed at r-1.5, but the user can change the mass
43
Figure 2-22: Side view of sample system displaying flattened envelope density structure
(left) and inner region disk structure (right). The left image’s envelope density gradient has been plotted finer to see the density change. The disk portion of that image is bright white. The black region is empty space inside the model’s envelope inner edge.
of the envelope. As can be seen in the left image, the different color bands above and below the bright white area are regions of constant density in the envelope and are not spherical shells. The right image displays the same model as the left image, but with lower density resolution to see the disk’s density structure. The disk structure follows an overall flared shape with the individual gradients forming radial teardrop shapes. The equations describing these density profiles are shown farther below.
To determine the grain properties, the code accepts a grain file consisting of the grain scattering and extinction coefficients, kappa in units of area/mass, and information about the grain mixture’s forward scattering phase function, called g, and polarization parameters. The grain files must be obtained and/or calculated from other sources, such
44 as a Mie code. Parameters such as grain composition and size are embedded inside the grain files. The Whitney code has no direct way to change these parameters.
The model circumstellar geometry is split into three regions, the disk, the envelope, and the outflow cavity. The disk is then further split into two regions,
Figure 2-23: Schematic of the four dust file regions, disk midplane (yellow), disk atmosphere (blue), envelope (orange), and outflow region (red).
a midplane, and an atmosphere, with a user defined density threshold that determines the transition from one into the other. This threshold is usually set to a Hydrogen density of
8 -3 10 H2/cm . Any location in the disk that is of high density is considered midplane while the rest is atmosphere. A single dust file can be assigned to each of these four regions.
Three dust files could be used, “small” grains from 0.005 to 0.25 microns (KMH 1994),
“medium” grains up to 20 microns (Cotera et al. 2001), and “large” grains up to 1000 microns (Wood et al. 2002).
45 The code sets up a 2-dimension grid in radial and altitude angle space. Dust densities are calculated for all grid cells first for the disk and then for the envelope. The disk density is determined from:
A 2 ⎛ R ⎞ R ⎧ 1 ⎛ z ⎞ ⎫ ⎜ * ⎟⎛ * ⎞ ⎪ ⎪ ρ = ρ o 1− ⎜ ⎟ exp⎨− ⎜ ⎟ ⎬ ⎜ r ⎟ r 2 ⎜ h(r) ⎟ ⎝ ⎠⎝ ⎠ ⎩⎪ ⎝ ⎠ ⎭⎪ where A is the exponent in the density power law, r is the radial coordinate, h(r) is scale height of the disk at radius r, and z is the height coordinate measured from the midplane of the disk. The envelope density is determined from this equation:
−0.5 −1.5 −0.5 −1 M& ⎛ GM ⎞ ⎛ r ⎞ ⎛ µ ⎞ ⎛ µ 2µ 2 R ⎞ ρ = env ⎜ * ⎟ ⎜ ⎟ ⎜1+ ⎟ ⎜ + o C ⎟ ⎜ 3 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 4π ⎝ RC ⎠ ⎝ RC ⎠ ⎝ µo ⎠ ⎝ µo r ⎠
where RC is the centrifugal radius of the orbiting dust meaning that inside this radius, the dust is falling in towards the disk whereas outside the radius, it is able to be supported in orbit around the star. Also, µ is the cosine of the altitude angle and µo is the cosine of the angle defining the boundary of the cavity outflow (Whitney et al. 2003).
Once the densities are known, the code compares the disk and envelope densities at each grid location. That location is then set to the higher density and the appropriate dust file is assigned. The grid cells have different spatial sizes with areas of higher density containing more cells than areas of lower density. The spatial sized decrease with decreasing distance to the star. The optical properties, such as emissivity and optical depth, are calculated for each grid location. The model, being azimuthally symmetric, is then rotated around and placed into a three dimensional gird.
46 Once the grid is complete, photons are then propagated through, one at a time.
Each photon starts from the parent star emitting in a random direction. The photon is followed until it has traveled through a single optical depth of material. Then the photon is randomly scattered or absorbed at the location inside that grid cell (not necessarily the center). The heating effect of the photon is collected and then the scattered or re-emitted photon is propagated along its next path until it travels another single optical depth. This process is repeated for a single photon until it either leaves the system entirely, or is reabsorbed into the star. Once all photons have been propagated, the final wavelengths are tallied and normalized to create the resulting SED.
The output for the code is a single file that lists the final SED at 250 wavelengths over ten different disk inclinations. The output SED must be scaled and interstellar extinction added before any fit can be completed. IDL routines are used to complete this step. The KMH dust file, modeling interstellar dust, is used for interstellar reddening.
Similar files splitting the photons into the position of their last interaction (star, disk, envelope) are created as well allowing a plot showing the contribution of each geometry component to the total SED. However, the disk SED is the result of both disk emission from the star and emission from envelope reheating. If the SED of the unreheated disk is required, it is necessary to run a second time with the envelope mass set to zero. The disk also slightly reheats the envelope in this code, but the effect is far smaller than that of the envelope on the disk.
As an option, an image of the system at a particular inclination angle can be created for a number of wavelength filters. The code bins photons of the correct
47 wavelengths and builds up an image based on the last physical location of interaction for that photon. The grid for the image files is a simple 149 by 149 of uniform square cells.
Figure 2-24: Example Image file showing a face on object at a filter wavelength of
1.61µm.
The runtime for the code is roughly five minutes for one million photons on a 3
GHz desktop processor, which is sufficient to tell if a fit is good “enough” shortward of millimeter wavelengths. For smoother SED’s, usually 100 million is needed which takes upwards of a day. For very smooth fits of millimeter data, usually a billion photons need to be run which can take almost a week. Creating images doubles or triples the runtime and good images typically need a billion photons or more.
48
Figure 2-25: Example of the same model with different photon counts. The 1 million model has been shifted down by a factor of ten for clarity.
The image files can be used to determine the model’s predicted radial surface brightness for the disk of the system. Once the model is completed with sufficient photons, a simple plot of the intensity vs. radius will determine the surface brightness.
Integrating over progressively larger annuli allows slightly shorter runs as the photons over the entire disk surface are counted instead of just along one radial path.
In the process of fitting, it was discovered that the code did not allow the outer edge of the envelope to be inside the outer edge of the disk. This did not allow for creating models such as that from Vinkovic et al. (2006). The code was then modified by the author to allow such models by removing the restriction and setting envelope grid locations outside the outer radius to zero density.
49 The Whitney model, being more realistic than the DUSTY model, is useful in more applications. The Whitney model in principle could be used for all stages of disk modeling, starting from class 0 by setting a high mass envelope and a low mass disk down to class II sources with little or no envelope and a flattened, convex upward disk.
Figure 2-26: Example plot of Whitney code produced model of Type 0 SED. Three inclinations are shown, 0 (disk seen face-on; uppermost, red curve), 30 (middle, blue curve), and 90 (disk seen equator-on; lowermost, green curve) degrees. The dotted curve is the SED of the star, based on the Kurucz models.
50
Figure 2-27: Example plot of Whitney code produced model of Type II SED. Three inclinations are shown, 0 (disk seen face-on; uppermost, red curve), 30 (middle, blue curve), and 90 (disk seen equator-on; lowermost, green curve) degrees. The dotted curve is the SED of the star, based on the Kurucz models.
The model is most useful for class I objects where the disk has undergone some settling.
The ability to set the thickness and flaring of the disk allows some modeling of the amount of settling the disk has undergone. The following figures illustrate some example flarings.
51
Figure 2-28: Density plot of a Whitney model disk with flaring parameter B=1.32. Such a disk is very slightly flared – concave upward.
Figure 2-29: Density plot of a Whitney model disk with flaring parameter B=0.86. Such a disk is ‘anti-flared’ – convex upward.
52
Attempting to anti-flare the disk into a wafer increases the runtime dramatically making very thin wafer disks, disks approaching the DUSTY type disk, impractical to model.
The DUSTY code is used to model gross properties of 50+ objects. Some of the objects are then used in a correlation study to see what sorts of relationships may exist between added blackbody starlight blockages and far-IR observed data ratios. The
Whitney code is used to produce more detailed models of a small subset of the DUSTY objects. Two of these objects, HD 31648 and HD 163296, are modeled in different epochs to find how the Whitney model has to change to fit the two different states of their
SED’s.
53 Chapter 3: DUSTY Fits and Results
The results of 50+ DUSTY models follow. The data used for the modeling of each object will be listed with each star. Each object will include a short discussion of its model properties and any information on why such properties were chosen.
The UV data were taken from the IUE (International Ultraviolet Explorer) orbiting telescope that operated from 1978 to 1996. The data come from either the IUE archive (http://archive.stsci.edu/iue/), or the IUE Atlas of Pre-Main-Sequence stars
(Valenti et al. 2000). Some of the data from the J (1.25µm), H (1.65µm), and K (2.2µm) bands were taken from the ground based 2MASS (2 Micron All Sky Survey) (Skrutskie et al. 2006), which ran from 1997 to 2001. Most of the mid infrared data was taken from the Aerospace Corporation's Broadband Array Spectrograph system (BASS). IRAS
(Infrared Astronomical Satellite) was the source of the 12, 25, 60, and 100µm data points.
IRAS ran for 10 months in 1983 and catalogued several hundred thousand infrared sources (http://irsa.ipac.caltech.edu/IRASdocs/iras.html). The Spitzer Space Telescope is in heliocentric orbit and is in operation today. The IRS (Infrared Spectrograph) on board provided 5 - 37µm data for a few objects (Houck et al. 2004).
The remaining photometric data, the U (0.36µm), B (0.44µm), V (0.55µm), R
(0.7µm), and I (0.9µm) bands, come from several sources including de Winter et al.
(2001), though the data was taken in many different years and will be stated with brackets
“[ ]”. Other UBVRI data came from the EXPORT project (Oudmaijer et al. 2001). Other sources of data will be explained within each object.
A discussion of the data and modeling for each star follows. A full list of input parameters and results is listed in a table. The input stellar spectrum for each object was
54 chosen based first off of the SIMBAD (http://simbad.u-strasbg.fr/simbad/) stellar type reference or from the literature. The actual stellar spectrum was then selected based on the goodness of fit in the UV and visual regions combined with the estimate of the interstellar extinction. For most of the objects, the Kurucz models
(http://kurucz.harvard.edu/grids.html) exist for temperatures of every 250K with a minimum of 3500K. Also for most of the objects, the selected Kurucz model did not differ by more then 500K with the “accepted” spectral type. The Kurucz models are also defined by the model star’s surface gravity, or log g in units of cm/s2. For comparison, the Sun has a log g of 4.4. The Kurucz model’s temperature and surface gravity in terms of log g will be defined for each object.
Each model uses the built in DUSTY dust grain properties or one or more of several dust grain files provided by the Jena Group (Jäger 1994 and others). The built in
DUSTY grain types are Draine and Lee generic “Dirty” (i.e. absorptive) interstellar silicate and graphite, called DL-Sil and DL-grph (Draine and Lee 1984). For the Jena
Group files, there are five pyroxene files. The chemical formula is Mg(x)Fe(1-x)SiO3 where x is the mass fraction of magnesium. These fractions are 60%, 70%, 80%, 95%, and
100%. A “glassy” silicate (MgSiO3) is sometimes used (called cosil) as well as two crystalline silicates, a (Jäger et al. 1994) crystalline silicate (crsil) and crystalline
Ca2Al2SiO7 (cryst). Also, a model of olivine was used in a few models. Each fit was started with roughly 50%/50% Draine and Lee silicates and graphite. Additional components were added and relative weights changed when necessary for a good fit, which was determined by minimizing a χ2 fit of the available infrared and millimeter data.
55 The following examples illustrate some of the effects in the Mid-IR region of changing the material mixtures or grain sizes. The three following plots were all produced using a T = 10000K input star and a DUSTY halo of τ550nm = 0.5 and radial density ∝ r-2.0. The base grain size is 0.005 – 0.25µm. Different grain sizes are noted in the plot captions.
Figure 3-1: Comparison of pyroxene60 with olivine. Adding olivine to a small grain mixture results in a sharp spectral feature at 11µm.
56
Figure 3-2: Comparison of the effects of adding graphite and increasing the grain size of the 60/40 Sil/Grp mixture. Both increased grain size and increased graphite abundance result in a similarly reduced 10µm silicate feature and cannot be easily differentiated. The width of the feature is not changed by either operation.
57
Figure 3-3: Comparison of the effect of replacing DL-Silicates with pyroxene. The pyroxene60 has a sharper and narrower 10µm feature showing the difference between
“astronomical silicates” of Draine and Lee (1984) and the pyroxenes of Jäger (1994).
Interestingly, the DUSTY halo model is mathematically equivalent to the disk atmosphere component of the Whitney model (Vinkovic et al. 2003) described in the previous chapter. This means that the DUSTY model halo could also physically represent the upper layers of the Whitney style disk instead of a less realistic spherical envelope.
The inner and outer radial parameters of the DUSTY halo can therefore be considered the inner and outer radial parameters of the atmosphere of a Whitney model type disk.
Several of the objects described below need a massive cold halo in their models so that the SED could fit the millimeter data. This is because the DUSTY wafer disk is made up of “bowling ball” sized grains (grain size >> wavelength) and cannot fit the
58 steeper millimeter slopes of those objects. Since the DUSTY model is not capable of modeling a flared disk, which would fit the millimeter data, it is necessary to use a large enough halo to contain enough mass to produce the necessary fluxes in the millimeter regions. This is a failing of the DUSTY model as it requires unrealistically large halos to fit objects that are known to be much smaller, as will be noted in the individual entries.
Fortunately, this issue does not affect modeling of the inner disk regions, which is the focus of the correlation study at the end of this chapter.
The following tables list many of the model parameters of each of the 50+ objects modeled with DUSTY. The first table lists the input stellar temperature and the inner and outer radii of the halo and disk, if one is present in the model, in units of AU. The second table lists the grain model minimum and maximum grain radii and the relative grain type mixtures. The third table lists parameters of blackbodies added to some of the individual models including temperature and stellar light blockage. Table 4 lists the DUSTY models with improbable outer envelope radii and the observed data if known.
59 Table 1: DUSTY Model Star Temperature, Interstellar Ext. (Av), and Halo/Disk Sizes.
Object Kurucz Temp Av Inner Env AU Outer Env AU Inner Disk AU Outer Disk AU AA Tau 3750 0.1 0.73 73 0.03 11.3 AB Aur 9500 0.0 0.43 7344 BF Ori 8000 0.3 0.35 35 bfs60 13000 2.9 0.62 372 CO Ori 5750 1.9 0.15 145 CQ Tau 6500 0.65 3.96 23734 0.14 13.3 CW Tau 4250 4.4 0.13 59 0.01 23.2 CY Tau 3750 0.4 0.23 320 0.01 28.2 DG Tau 4000 0.73 0.08 79 0.02 7.9 DL Tau 4250 0.0 0.12 115 0.01 32.5 DM Tau Disk 4250 0.9 1.44 432 0.06 3.4 GI Tau 4250 1.0 0.09 434 GK Tau 4500 1.0 0.09 30660 GM Aur 4250 0.5 2.88 23072 GW Ori 4750 0.0 0.16 785 HD 100546 10500 0.05 13.72 27432 0.76 306.7 HD 135344 5500/7500 0.31 25.34 8482 HD 141569 10500 0.4 2.30 5751 HD 163296 9250 0.3 0.27 12006 0.06 358.8 HD 169142 5750/8750 0.6 7.38 74 0.53 9.1 HD 190073 9250 0.25 0.39 581 HD 250550 13000 0.8 0.61 1214 HD 31648 9250 0.25 0.29 28520 0.05 119.6 HD 35178 9000 0.85 1.03 103 0.16 21.6 HD 37806 10000 0.2 0.29 73 HD 45677 1980 21000 1.0 1.56 156 HD 45677 1992 21000 1.0 1.36 158 HD 50138 14000 0.75 0.60 121 HD 58467 10500 0.3 0.22 22 0.08 0.6 HD 98800 4500 0.6 1.76 353 0.09 9.1 Hens3-600 5000 1.5 0.77 177 HK Ori 8500 0.4 0.42 83 HL Tau 4000 0.7 0.16 3548 HP Tau 4000 2.0 0.30 305 0.01 5.3 HP Tau G2 6000 1.4 8.68 87 HP Tau G3 4000 1.2 3.83 38 HR 4796 9500 0.0 36.96 739 LkCa15 4250 4.0 2.78 278 0.06 17.8 LkCa21 3500 2.5 2.52 252 NV Ori 6250 1.05 1.01 1010 RR Tau 8750 1.25 0.22 1080 RW Aur 5000 0.3 0.21 103 0.02 7.2 RY Ori 6500 2.5 0.37 367 RY Tau 6000 1.5 0.35 348 0.02 16.4 SU Aur 6000 1.0 0.93 93 0.06 6.0 TW Hya 4000 0.7 1.02 1024 0.03 29.4 UX Ori 8750 1.0 0.74 1104 0.07 78.2 UY Aur 3750 0.65 0.15 145 0.01 9.5 v590mon 12000 0.65 0.34 101 v594cas 9750 2.1 0.26 1300 XZ Tau 5750 0.25 0.12 59 0.03 29.7
60
Table 2: DUSTY Model Dust Composition and Grain Sizes
Object Amin Amax Sil-DL Grp-DL Pyroxene Olivine Crystal Glassy AA Tau 0.8 9 15% 40% 35% 10% AB Aur 0.01 2.5 30% 50% 20% BF Ori 0.005 10 9% 46% 35% 10% bfs60 0.009 4 20% 48% 32% CO Ori 0.8 10 50% 50% CQ Tau 0.07 4 55% 33% 12% CW Tau 0.1 5 30% 59% 11% CY Tau 0.3 8 40% 60% DG Tau 1.1 10 17% 51% 32% DL Tau 2.4 10 40% 60% DM Tau Disk 4.2 10 40% 60% GI Tau 0.5 2.8 54% 46% GK Tau 0.5 2.8 54% 46% GM Aur 0.005 0.008 24% 40% 36% GW Ori 0.01 1.25 32% 41% 27% HD 100546 0.06 10 20% 50% 30% HD 135344 1 10 50% 50% HD 141569 2 10 60% 40% HD 163296 0.7 10 47% 24% 5% 24% HD 169142 10 11 50% 50% HD 190073 0.03 10 50% 50% HD 250550 0.1 10 45% 40% 15% HD 31648 0.15 10 45% 33% 22% HD 35178 0.9 10 50% 14% 3% 33% HD 37806 0.85 10 45% 17% 15% 23% HD 45677 1980 1.2 10 60% 17% 23% HD 45677 1992 2.2 10 60% 17% 23% HD 50138 1.6 10 61% 21% 18% HD 58467 5.8 10 53% 47% HD 98800 4.2 10 40% 60% Hens3-600 0.05 4 40% 60% HK Ori 0.01 10 20% 61% 18% 1% HL Tau 0.5 10 16% 49% 35% HP Tau 0.3 1.7 14% 46% 40% HP Tau G2 0.1 10 40% 60% HP Tau G3 0.01 10 40% 60% HR 4796 0.1 10 50% 50% LkCa15 0.25 1.25 10% 50% 40% LkCa21 0.005 0.25 50% 50% NV Ori 0.005 0.25 49% 51% RR Tau 3 10 56% 44% RW Aur 0.01 7 50% 37% 13% RY Ori 0.03 6 50% 50% RY Tau 0.03 6 2% 50% 42% 6% SU Aur 0.2 6 50% 47% 3% TW Hya 0.005 2 47% 50% 3% UX Ori 0.005 2 34% 50% 13% 3% UY Aur 0.5 2 43% 57% v590mon 0.35 2 11% 42% 47% v594cas 0.35 7 50% 50% XZ Tau 1 25 50% 50%
61
Table 3: DUSTY Model Blackbody Temperatures
Object BB Temp Lambda Block Object BB Temp Lambda Block AA Tau 950 -0.5 12% HD 45677 1992 AB Aur HD 50138 BF Ori HD 58467 1000 -2 0% bfs60 HD 98800 CO Ori Hens3-600 CQ Tau 1000 -2 14% HK Ori CW Tau HL Tau 750 -2 9% CY Tau 1100 -1 12% HP Tau 950 -1.5 6% DG Tau 1500 -0.5 7% HP Tau G2 DL Tau HP Tau G3 DM Tau HR 4796 1000 -2 0.36% GI Tau LkCa15 GK Tau LkCa21 GM Aur NV Ori GW Ori 750 -2 1.0% RR Tau HD 100546 1300 -1 7% RW Aur 950 -2 9% HD 135344 900 -1.5 25% RY Ori HD 141569 RY Tau HD 163296 1100 -2 9% SU Aur 900 -2 8% HD 169142 1200 0 4% TW Hya HD 190073 UX Ori 950 -2 3% HD 250550 1250 -0.8 11% UY Aur 1100 -2 14% HD 31648 v590mon HD 35178 1000 -2 5% v594cas 1200 -2 12% HD 37806 XZ Tau HD 45677 1980
Table 4: Probable Failed DUSTY Models
Object Modeled Outer Env AU Observed Outer AU (if known) CQ Tau 23734 200AU (Dent et al. 2005) GK Tau 30000 <2500 ( Vrba et al. 1986) GM Aur 23072 ? HD 31648 28520 450 (Grady et al. in prep) HD 100546 27432 100's (Grady et al. 2005) HD 135344 8482 ~200 (Grady et al. 2005) HD 141569 5751 600+ (Mouillet et al. 2001) HD 163296 12006 ~450 (Grady et al. 2000) HL Tau 3548 ? HR 4796 740 77 (Schneider et al. 1999)
62
AA Tau
Figure 3-4: The SED and DUSTY model of AA Tau.
AA Tau is a K7 (Andrews and Williams 2005) star in the Taurus region and is
140pc away (Kessler-Silacci et al. 2005). The data used for this object include 2MASS,
IRAS, Spitzer IRS, and the IUE Atlas. The millimeter data is from Andrews and
Williams (2005) and the UBVRI data is from Rydgren and Vrba (1983).
The input stellar spectrum for this model had a temperature of 3750K and log g of
1.0. This object displays a small silicate feature, which is indicative of a larger grain size distribution. The grains, a mixture of 15% DL-silicates, 40% DL-graphite, 35% pyroxene80 and 10% crsil, have a grain size distribution of 0.8 - 9µm. The addition of the crystalline material and the pyroxene was necessary to obtain the correct shapes of the
10µm and 20µm features. A 950K blackbody multiplied by λ-0.5 was added to fit the 2µm
63 - 5µm region. This blackbody blocks off 12% of the star’s radiation resulting in a half angle height of ~11˚.
The halo extends from 0.7 AU to about 70 AU and has a τv of 0.33. The slope of the mm-region indicates large grains and is fit by the wafer disk and not by the halo as can be seen from the difference between the halo’s (orange line) and mm data’s slopes.
This means the system has undergone some grain settling and growth.
The disk extends from 0.02 AU to 11.33 AU and has a best fit when the inclination angle is 77˚. The disk displays reheating, caused by the halo, from ~11µm onwards. This disk is then the cause of most of the mid to far IR flux.
AB Aur
Figure 3-5: The SED and DUSTY Model of AB Aur.
64 AB Aur is a A0 star in Auriga at a distance of 140pc (Acke and van der Ancker
2004). IRAS, 2MASS, and IUE Atlas data are included in this spectrum. A BASS spectrum was taken on February 9th, 1998. UBVRI data is from de Winter et al. [1993] while the millimeter data comes from Mannings (1994), Mannings and Sargent (1997), and Malfait et al. (1998). The Short Wave Spectrometer (SWS) on board the Infrared
Space Observatory (ISO) provided the continuum data from ~3 - 45µm. Finally, UV to visual data (0.33 – 0.58µm) was obtained from the Low Resolution SpectroPhotometer of the Pine Bluff Observatory (http://www.sal.wisc.edu/PBO/).
The input stellar spectrum has a T of 9500K and a log g of 4.0. The silicate feature for this star is larger and indicates small grains meaning that this system is in the early stages of evolution. The model contains 30% DL-silicates, 50% DL-graphite and
20% glassy silicates with the inclusion of the glassy material to narrow the 10µm silicate feature model and to sharpen its peak. The dust grain size distribution is 0.01 - 2.5µm.
The entire data set can be fit with just a halo model. This halo extends from 0.4AU to over 7000AU, which is quite large, but at least is consistent with extended envelope emission being seen to at least 1000AU (Corder et al. 2005).
The disk model in this plot is not being added to the halo. It is placed there for comparison and to show that the mm-slope of the disk does not match that of the data.
This is to show that a DUSTY disk model fails to fit the data for this object. A DUSTY disk model does not contain the small grains needed to produce the observed flux slope.
65 BF Ori
Figure 3-6: The SED and DUSTY model of BF Ori.
BF Ori is a A2IV star in Orion that is 510pc away (Acke and van der Ancker
2004). IRAS, 2MASS, and IUE Archive data sets SWP 15920, SWP 17863 LW 12255, and LWR 14105 are included in the SED. Two BASS spectra, from December 13th, 1994 and October 17th, 1996 are plotted as well as UBVRIJHK data from the Export group.
The input stellar spectrum has a T of 8000K and a log g of 2.0. This object displays a large silicate feature indicating small grains. The model consists of a halo extending from 0.35AU to 35AU and is made up of 9% DL-silicates, 46% DL-graphite,
35% pyroxene60 and 10% olivine. The addition of the pyroxene and olivine was necessary to narrow and sharpen the 10µm silicate feature peak with the exact type of pyroxene decided by the exact location of the peak. The dust grain size distribution is
0.005 – 10.0µm. The 60µm and 100µm points are upper limits and so this object is not
66 used in the 25µm/60µm correlation study. Since there is no data longward of 100µm, very little more can be said about this object.
BFS60
Figure 3-7: The SED and DUSTY model of bfs60.
BFS60 is a B9 star as listed in the SIMBAD database. IRAS, 2MASS, and a
BASS spectrum from January 8th, 2003 are plotted along with UBVRIJHK data from
Nickel and Staude (1984).
This object takes its reference name, BFS60, from the nebula it is embedded in.
Its actual names are GSC 05379-00359 (Hubble Guide Star Catalogue) and IRAS 06475-
67 0735. Because of this nebula, the object shows contamination beyond 60µm and is not used in the 25µm/60µm correlation study.
The input spectrum has a temperature of 13000K and a log g of 5.0. The model is a halo that extends from 0.62 AU to 370 AU, though the outer edge is not very well defined because of the nebulosity surrounding the object. The silicate feature is produced by grains ranging from 0.009 - 4.0µm in size and is consistent with a system that has undergone little grain growth. The material mixture is 20% DL-silicates, 48% DL- graphite, and 32% pyroxene80. The pyroxene80 was added to narrow the 10µm peak.
CO Ori
Figure 3-8: The SED and DUSTY model of CO Ori.
68 CO Ori is a F7 type star in Orion (Vink et al. 2005). IRAS, 2MASS, and IUE
Atlas data are plotted in the SED. A BASS spectrum from December 24th, 1999 is also plotted. The UBVRIJHK data is from the Export group and was taken over several nights in October 1998 and January 1999. As can then be seen, the variability is over a period of days.
The input spectrum has a temperature of 5750K and a log g of 5.0. The visual and near-infrared regions of CO Ori are highly variable. The model was chosen to fit the middle of the variability range and is modeled with a halo extending from 0.1 AU to 145
AU. The lack of a significant silicate feature indicates grain growth. The models grain size distribution is 0.8 - 10µm and is made up of 50% DL graphites and 50% DL silicates. No additional dust types were needed in the model. The lack of mm data does not allow much more to be said about this object or properties of the outer envelope. The
60µm IRAS point is an upper limit and therefore removes this object from both 60µm correlation studies.
69 CQ Tau
Figure 3-9: The SED and DUSTY model of CQ Tau.
CQ Tau is a F5IV type star in Taurus and is 99pc away (Grady et al. 2005). IRAS,
2MASS, and IUE Atlas data are plotted on this SED. The BASS spectrum was taken on
October 14th, 1996. The millimeter data comes from Mannings and Sargent (1997) and the UBVRIJHK data is from the Export group and was taken on several nights in October
1998. The 7mm point comes from Testi, et al. (2001).
The input spectrum has a temperature of 6500K and a log g of 5.0. CQ Tau is modeled with a halo that extends from 4 AU to nearly 23000 AU. The halo consists of
55% DL-graphite and a mixture of 33% pyroxene70 and 12% olivine and fits the silicate feature with a grain size distribution ranging from 0.07 - 4.0µm. The pyroxene and olivine were added to better fit the narrower shape of the 10µm silicate feature. The height of the feature indicates the presence of the smaller grains included in the model.
70 The slope of the mm data indicates that small grains are required and as can be seen in the plot, a wafer DUSTY disk has the wrong slope for a good fit. Therefore a very extended halo is needed in this model to fit the mm points properly though its slope is still not quite right. The model disk extends from 0.1 AU to only 13 AU and fits best at a nearly edge on inclination angle. It provides the small amount of flux needed in the millimeter region. The two components, then combined, provide a good fit. A blackbody of 1000K multiplied by λ-2 completes the model. The blackbody blocks off 14% of the star’s light which is a half height angle of 13˚. When comparing the enormous halo size to observational evidence, material has been seen around CQ Tau out to only 200AU
(Dent et al. 2005), which is substantially less then the 23,000AU in the model. This object is a good example of where the DUSTY model fails. If the model disk contained smaller grains as well, then the millimeter data could be fit by the disk alone and the model halo could be much closer to what is observed.
71 CW Tau
Figure 3-10: The SED and DUSTY model of CW Tau.
CW Tau is a K2 star in Taurus (Andrews and Williams 2005). IRAS and 2MASS data are plotted along with a BASS spectrum that was taken on February 1st, 2001. The millimeter data are from Beckwith et al. (1990), Beckwith and Sargent (1991), and
Andrews and Williams (2005).
The input spectrum has a temperature of 4250K and a log g of 4.0. The halo for the model of CW Tau extends from 0.10 AU to 90 AU and consists of a 30% DL- silicates, 59% DL-graphite, and 11% pyroxene80. The small amount of pyroxene was needed to help produce the 10µm silicate feature shape in the data. The small silicate feature indicates grain growth and is modeled by a grain distribution of 0.1 - 5µm. The
72 disk extends from 0.007 AU to 20 AU and best fits at nearly edge on. Some halo reheating is evident in the disk beyond 10µm. Due to the steep positive slope of the JHK data, this model required a very large interstellar extinction, or Av = 4.4. A blackbody was needed for the extra emission from the inner edge of the disk. The blackbody has a temperature of 1000K and is multiplied by λ-1. It blocks 26% of the star’s light indicating a half angle height of 23˚.
The disk is necessary as its slope better fits the mm data. The halo millimeter slope is simply too steep. There is some ambiguity in that the 450µm point is an upper limit, yet is lower then what the slope of the most of the rest of the mm data would imply.
CY Tau
Figure 3-11: The SED and DUSTY model of CY Tau.
73
CY Tau is in Taurus and is a M1 star (Sargent et al. 2006). This SED consists of data from 2MASS and the IRS on the Spitzer Space Telescope. The millimeter data are from Andrews and Williams (2005) and UBVRI data is from Rydgren and Vrba (1983).
The input spectrum has a temperature of 3750K and a log g of 5.0. A disk that extends from 0.01 AU to 28 AU dominates this model and fits best when almost face on.
The halo extends from 0.2 AU to 320 AU and has a low optical depth of 0.15. The grains in the halo are a 40% DL-silicates and 60% DL-graphite mix and have a grain distribution of 0.3 - 8.0µm which agrees with the grain growth indicated by the very small silicate feature. The shape of the 10µm silicate feature required more graphite for a good fit, but did not require the addition of other materials. The slope of the mm data requires the shallower sloped disk, which is made up of very large grains (blackbodies) instead of the steeper sloped halo. A blackbody was needed for the extra emission from the inner edge of the disk. The blackbody has a temperature of 1100K and is multiplied by λ-1. It blocks 12% of the star’s light indicating a half angle height of 11˚.
The lack of IRAS data for this object means it is not included in any of the
IRAS/Blackbody correlation studies.
74 DG Tau
Figure 3-12: The SED and DUSTY model of DG Tau.
DG Tau is a K7 star (Andrews and Williams 2005) in Taurus. IRAS and 2MASS data are included in this SED. The UV data comes from the IUE Atlas. Three BASS spectra are plotted together, taken on January 31st, 2001, January 17th, 2002, and on
January 10th, 2003. The millimeter data come from multiple sources, Weintraub et al.
(1989), Adams et al. (1990), Beckwith and Sargent (1991), and lastly Mannings and
Emerson (1994).
The input spectrum has a temperature of 4000K and a log g of 5.0. This object is highly variable and as such, not much can be determined from modeling. The variability in the BASS data shows that in 2001 the silicate band was actually in absorption, while in
75 most of the other data sets it was in emission, or was flat. The 2002 data was modeled for the above plot. The millimeter data slope and the small silicate feature indicate grain growth and this is matched in the model by the need for a disk to fit the millimeter data.
The grain size distribution runs from 1.1 - 10µm meaning there are few small grains. An optically thick halo extends from 0.8 AU to 80 AU and is made up of 17% DL-silicates,
51% DL-graphite, and 32% pyroxene80. The pyroxene was required in the model to narrow the shape of the 10µm silicate feature. The disk extends from 0.1 AU to 8 AU and is inclined near edge on. The disk also experiences reheating longwards of 3µm. The model halo is optically thick enough to absorb some of the disk flux creating the spectral features seen in the disk. The optically thick halo was necessary to produce the flux to fit the IRAS points and also to fit the shape of the JHK data. The optically thick nature of the halo would block off most of the stellar photospheric emission. Since the actual star can be seen, this model is very poor.
A blackbody of 1500K multiplied by λ-0.5 is added to fit the near infrared region.
This blackbody blocks 7% of the star’s light and has a half height angle of 6˚. This indicates the presence of some sort of larger puffed up inner disk rim. This object is also located in a region of extended emission in the IRAS data and therefore is not included in any of the IRAS/blackbody correlation studies.
76 DL Tau
Figure 3-13: The SED and DUSTY model of DL Tau.
DL Tau is a K7 type star in Taurus (Furlan et al. 2005). IRAS, 2MASS, and IUE
Atlas data are plotted in this SED. The BASS spectrum was taken on February 1st, 2001.
The millimeter data comes from Beckwith et al. (1990) and Mannings and Emerson
(1994).
The input spectrum has a temperature of 4000K and a log g of 2.0. This object is another object along with DG Tau that shows evidence of grain growth from the small silicate feature and the blackbody like slope of the millimeter data. The model consists of a wafer DUSTY disk extending from 0.01 AU to 33 AU. The halo extends from 0.1 AU to 115 AU and is made up of 40% DL-silicates and 60% DL-graphite grains. The 10µm silicate feature required the addition of no further materials. The grain distribution is 2.4 -
10µm and is consistent with significant grain growth which the nearly non-existent 10µm silicate feature illustrates in the above figure. This combined with the domination of the
77 wafer disk towards the millimeter is evidence for a highly evolved and very flattened disk system.
DM Tau
Figure 3-14: The SED and DUSTY model of DM Tau.
DM Tau is a M1 type star in Taurus (Bergin et al. 2004). IRAS and 2MASS data are plotted on this SED. The BASS spectrum was taken on four nights, January 15th and
16th, 2001, January 8th, 2003, and February 20th, 2003. The millimeter data come from
Beckwith et al. (1990) and Beckwith and Sargent (1991).
The input spectrum has a temperature of 4250K and a log g of 5.0. This object has been modeled by a disk extending from 0.05 AU to 35 AU and a halo extending from 1.5
AU to 430 AU. The disk shows reheating from longward of 20µm. The halo consists of a
78 mixture of 40% DL-silicates and 60% DL-graphite and has a grain distribution of 4.2 -
10µm microns. No other materials were needed to produce a fit. Much like the two previous objects, the small silicate feature and shallow millimeter data slope indicate strong grain growth, disk evolution and flattening. Also, the near-photospheric flux levels out to around 10µm show evidence of a cleared out inner disk region. The cleared out region indicates that the IRAS data ratios could be suppressed due to an actual lack of inner material instead of possible disk shadowing. Therefore this object has been left out of the correlation studies.
The model of Bergin et al. (2004) consists of a blackbody “wall” located at the inner radius of a disk. The emission from this disk is ignored as they are modeling the inner region only. An optically thin inner disk is added extending from the “wall” inward to the dust destruction radius. This inner disk contains smaller grains and gives rise to the silicate features in their models. They also found it necessary to add an inner blackbody of 1400K to fit the near infrared. The DUSTY model is able to fit the data of this object with only two components and does not need an inner blackbody. A halo is used instead of the inner disk. The results are somewhat comparable, the Bergin inner wall has a temperature of 117K while the DUSTY model inner edge is 300K. The Bergin model has a grain size of 0.1 – 2.0µm, which is smaller than the 4.2 – 10.0µm range the DUSTY model uses. The smaller grains could result in a ‘hotter’ SED and so would have to be modeled cooler to result in a similar SED to the DUSTY model. Their puffy inner rim heights were small, which approaches DUSTY’s wafer thin disk. They also estimate optically thin dust in the inner disk, which is matched by the optically thin (optical depth
0.3) halo in the DUSTY model.
79
GI Tau
Figure 3-15: The SED and DUSTY model of GI Tau.
GI Tau is a K6 type star in Taurus (Watson et al. 2007). This star is very near GK
Tau. The data for GI Tau consists of an IRAS and 2MASS photometry as well as an IUE
Atlas spectrum. The BASS spectrum was taken on December 24th, 1999. The millimeter point is from Beckwith et al. (1990).
The IRAS flux for this object includes the flux from GK Tau also. The two stars are very close to each other and IRAS was not able to resolve them apart. The IRAS flux was split apart such that the 12 µm point would match closely with the BASS data. This was found to be when 45% of the flux was assigned to GI Tau and the other 55% to GK
Tau. Therefore, the IRAS data are multiplied by 0.45 for the plot of this object.
80 The input spectrum has a temperature of 4250K and a log g of 5.0. The model for this object consists of a halo extending from 0.1 AU to 430 AU and is made up of nearly
54% DL-silicates and 46% DL-graphite. The slight increase in the silicates allowed a better fit of the 10µm silicate feature width and strength. The grain distribution is 0.5 -
2.8µm, which is “small” grains, to produce the prominent silicate feature in the SED. The
100µm IRAS point and the 1.3mm point are both upper limits and therefore not much can be said about the outer disk regions of this object.
GK Tau
Figure 3-16: The SED and DUSTY model of GK Tau.
GK Tau is a M0 type star in Taurus (Watson et al. 2007). This star is very near GI
Tau. The data for GK Tau consists of an IRAS and 2MASS photometry as well as an IUE
81 Atlas spectrum. The BASS spectrum was taken on December 24th, 1999. The millimeter point is from Beckwith et al. (1990).
The IRAS flux for this object also includes the flux from GI Tau. The flux was divided between the two objects such that the IRAS 12 µm point would match closely with the BASS data. 55% of the flux was assigned to GK Tau meaning that the IRAS data was multiplied by 0.55 before plotting.
The input spectrum has a temperature of 4500K and a log g of 5.0. This model is similar to the GI Tau model. It has the same grain and material composition but because the 1.3mm point for this object is not an upper limit, it became necessary to extend the radius of the halo by a significant amount to be able to fit that point. The halo for this model extends out to over 30000 AU to produce enough flux to fit the data. Here is another case of the Dusty model failing since the two stars, GI Tau and GK Tau, are roughly 13” apart (Vrba 1986) at a distance of 140pc (Kenyon and Hartmann 1995). This corresponds to a separation distance of ~1800 AU, which would put GI Tau inside GK
Tau’s DUSTY halo.
82 GM Aur
Figure 3-17: The SED and DUSTY model of GM Aur.
GM Aur is a K3 type star in Auriga (Bergin et al. 2004). The data for GM Aur consists of an IRAS and 2MASS photometry as well as an IUE Atlas spectrum. The
BASS spectrum was taken on January 8th, 2003. The millimeter data are from several sources, Weintraub et al. (1989), Adams et al. (1990), Beckwith et al. (1990), Beckwith and Sargent (1991), and lastly Koerner et al. (1993).
The input spectrum has a temperature of 4250K and a log g of 5.0. This objects is modeled with a halo extending from 2 AU to 23000 AU and is composed of 24% DL- silicates, 40% DL-graphite, and 36% pyroxene95. The addition of the pyroxene helped to fit the sharp and narrow 10µm silicate feature. The grain distribution is 0.005 - 0.008µm which is quite consistent with the small grains indicated by the large silicate feature in the
SED. Like DM Tau above, this system shows some clearing out of an inner region from the near photospheric flux out to 10µm. As with AB Aur, a disk cannot fit the millimeter
83 data because the millimeter data slope is too steep for a blackbody disk. Small grains must be present in the outer regions to produce such a slope and a DUSTY disk does not contain such grains. The enormous size of the halo is necessary to provide sufficient cold grains to fit the millimeter data and is very unlikely that the actual dimensions of the DM
Tau system is that large.
The near photospheric flux levels out to around 8µm show evidence of a cleared out inner disk region. The cleared out region indicates that the IRAS data ratios could be suppressed due to an actual lack of inner material instead of possible disk shadowing.
Therefore this object has been left out of the correlation studies.
DM Tau shows grain growth along with a cleared out region whereas interestingly this star only shows the inner region and indeed shows very little grain growth. The
Bergin model of this system is similar to the DM Tau model as well. The Bergin wall for this object has a temperature of 124K compared with the 300K the DUSTY models uses.
Like DM Tau, this difference could be explained by the smaller grain sizes that the
Bergin model uses. Also, the DUSTY model for this object can fit the SED by using only a single component, instead of the three components Bergin uses. The Bergin model does not consider the outer regions of the system and so no comparisons can be made.
84 GW Ori
Figure 3-18: The SED and DUSTY model of GW Ori
.
GW Ori is a K5 type star in Orion (Watson et al. 2007). The data for GW Ori consists of an IRAS and 2MASS photometry as well as an IUE Atlas spectrum. The
BASS spectrum was taken on January 16th, 2002. The BASS and 2MASS spectra were taken during different times when the object was in different flux states.
The input spectrum has a temperature of 4750K and a log g of 4.0. This object is modeled with a halo that extends from 0.15 AU to 800 AU and is composed of 32% DL- silicates, 41% DL-graphite, and 27% glassy silicates. The addition of the glassy silicates helped to provide the sharp and narrow 10µm silicate feature. The grain size distribution is 0.01 - 1.25µm which is consistent for producing the large silicate feature in the SED.
There is no millimeter data for this object and so little can be said of the nature of any disk that may be present. The JHK data was not able to be fit and does not match well
85 with the BASS or IRAS data. This object is known to have variable flux by a factor of two to three.
HD 31648
Figure 3-19: The SED and DUSTY model of HD 31648.
HD 31648, also know as MWC 480, is a A5 type star that is ~140pc away (Grady et al. 2005). The data for HD 31648 consists of IRAS and 2MASS photometry as well as an IUE Atlas spectrum. The BASS spectrum plotted was taken on October 14th, 1996.
The UBVRI and JHK data come from the Export group. LRSP data is also plotted. The millimeter data come from Mannings et al. (1997) and Piétu et al. (2006).
The input spectrum has a temperature of 9250K and a log g of 4.0. This model consists of a halo extending from 0.3 AU to 28500 AU and consists of 45% DL-graphite,
33% pyroxene100, and 22% crystalline silicates. The crystalline silicate help to fit the small peak near 11µm while the pyroxene narrowed and sharpened the 10µm silicate
86 feature. The grain size distribution is 0.15 - 10.0µm. A wafer disk extending from 0.05
AU to 120 AU provides a small amount of extra emission that when combined with the halo, match the slope of the millimeter data.
The slope of the millimeter data is slightly steeper than that of a blackbody and so small grains are needed, which cannot be provided by a DUSTY disk. The halo has to have the large radius to be able to provide the emission necessary to fit the millimeter data. Compared to the TTSRE model, and considering that the halo of HD 31648 is seen to only extend a few hundred AU from the star (Grady et al, in preparation), it is evidence that the disk of HD 31648 cannot be modeled as a simple wafer. This means the system has not undergone sufficient grain growth and disk settling for the disk to be a wafer.
This model is an example of where the DUSTY model fails despite the very nice fit of the SED. Future modifications to the DUSTY code disk model allowing a non-blackbody disk would give the DUSTY code better ability to fit such objects.
The Whitney model (SED in Whitney results section) of this object is a flared disk reaching to 250AU and a small halo shell that extends to only 10AU. The grain sizes are roughly similar between the two models. The biggest difference is that the Whitney model does not extend to thousands of AU’s. This is because the far IR spectra region can be fit with a flared disk which is much more massive and capable of generating the flux necessary in a smaller space. Also, explained above, the millimeter data slope is consistent with a flared disk, and not a wafer blackbody disk, thus requiring the DUSTY model to use a massive envelope to produce the correct slope and flux. The observational imagining is more consistent with the Whitney model component dimensions.
87 HD 35187
Figure 3-20: The SED and DUSTY model of HD 35187.
HD 35187 is an A2 type star and is 150pc away (Dent 2005). The data for HD
35187 include IRAS and 2MASS photometry as well as an IUE archive spectrum. The
BASS spectrum was taken on October 17th, 1996. UBVRIJHK data comes from Sylvester et al. (1996) and the millimeter point comes from Walker and Butner (1995).
The input spectrum has a temperature of 9000K and a log g of 4.0. The model for
HD 35187 consists of a halo extending from 1 AU to 100 AU and a disk that extends from 0.16 AU to 22 AU. The disk dominates at millimeter wavelengths. The grain size distribution of the halo is 0.9 - 10.0µm and is consistent with the smaller size of the silicate feature and lack of small grains evident in the millimeter data. The halo dust consists of roughly 50% DL-graphite, 14% pyroxene95, 33% glassy silicates, and 3% olivine. The glassy silicates and olivine were added to help produce the small 11µm bump while the pyroxene narrowed the 10µm silicate feature slightly. This feature is
88 similarly shaped to the one shown in HD 31648 although it is wider, hence the need for less pyroxene dust. This object is similar to DL Tau and CW Tau in that all three objects show smaller silicate bands and blackbody type slopes in their millimeter data. Some amount of grain growth and disk settling is evident here.
HD 37806
Figure 3-21: The SED and DUSTY model of HD 37806.
HD 37806 is a B9 type star (Oudmeijer and Drew 1999). The data for HD 37806 consists of IRAS and 2MASS photometry as well as an IUE Atlas spectrum. The BASS spectrum was taken on October 14th, 1996. The UBVRIJHK data all comes from de
Winter et al. [1980, 1981, and 1992].
The input spectrum has a temperature of 10000K and a log g of 4.0. The model of
HD 37806 consists of a halo that extends from 0.3 AU to 72 AU and has a grain distribution of 0.85 - 10.0µm. The halo dust is made up of 45% DL-graphite, 17%
89 pyroxene80, 23% crystalline silicates, and 15% olivine. The crystalline silicates and olivine form the 11µm bump while the pyroxene slightly narrows the entire silicate feature. The silicate feature is smaller than in HD 31648 and indicates more grain growth in this system than in that one. There is no millimeter data and so nothing can be said about the presence or properties of a wafer disk. The 60µm IRAS point is an upper limit and therefore removes this object from both 60µm correlation studies.
HD 45677
Figure 3-22: The 1980 SED and DUSTY model of HD 45677.
90
Figure 3-23: The 1992 SED and DUSTY model of HD 45677.
HD 45677 is a B2 type star and is ~1000pc away (Monnier et al. 2006). The data for HD 45677 consists of IRAS photometry and low resolution spectrophotometer
(LRSP- Pine Bluff Observatory) spectrum (Sitko 1981). The BASS spectrum was taken on January 8th, 2003. IUE Archive data of the appropriate time frame (1980 and 1992) is plotted on each SED as well. UBVRI data is from de Winter et al. [1980] for the 1980 plot and [1992 and 1994] for the 1992 plot. The JHK data is from de Winter et al. [1980] for the 1980 plot. 1980 IR data comes from Sitko (1980) and the 1992 IR data comes from Sitko (1994).
The input spectrum has a temperature of 21000K and a log g of 5.0. The model for HD 45677 consists of an envelope extending from 1.5 AU to 150 AU. The halo for the model consists of 60% DL-graphite, 17% pyroxene100, and 23% glassy silicates. The pyroxene narrowed the feature while the glassy silicates helped to shape the silicate feature to match the data. The model has a grain size distribution of 1.2 - 10.0µm for the
1980 fit and 2.2 - 10.0µm for the 1992 fit. Though there is no millimeter data, the slope
91 of the 60µm and 100µm IRAS points indicate the presence of small grains. The slope is too steep for a wafer disk. To complete the fit of the 1980 data, a percentage of star light was subtracted off the resulting fit by simply multiplying the stellar flux by 0.37 and subtracting before interstellar extinction was applied. This allowed the UV portion of the model to shift downward to fit the lower UV 1980 data. This method is crude and other effects such as reducing the flux interacting with the circumstellar dust and the resulting emission from the “blocking” material have been ignored. However, this model adjustment does suggest the idea that the circumstellar material is somehow “clumpy” and one of these optically thick, blackbody clumps was in the line of sight in 1980 and then was not in 1992.
The model of Schulte-Ladbeck et al. (1992, hereafter SL) is a dust halo with a thicker inhomogeneous dusty torus/disk. Though the DUSTY model is only a halo, it is consistent in that the SL model contains a non-wafer disk and is also clumpy. Though the
SL model does not indicate its radius, references to other such models of the circumstellar dust (Coyne and Vrba 1976) show that it is probably smaller than 100 AU, which is slightly smaller, though consistent with the DUSTY model radii of 150 AU.
There has been recent evidence to suggest that this system may actually be a binary star system without a dusty disk system (Miroshnichenko 2007) and is not similar to many of the other HAEBE’s in this survey. Therefore this object is considered a possible rejection from the correlation study.
92 HD 50138
Figure 3-24: The SED and DUSTY model of HD 50138.
HD 50138 is a B6 type star (Oudmeijer and Drew 1999). The data for HD 50138 consists of IRAS and 2MASS photometry as well as LRSP and IUE Archive spectra. Six
BASS spectra taken from 1994 through 2003 are plotted as well. UBVRI and JHK data comes from de Winter et al. [1980, 1981, and 1992].
The input spectrum has a temperature of 14000K and a log g of 5.0. The model for HD 50138 is very similar to HD 45677 and is a halo that extends from 0.6 AU to 120
AU. This halo consists of 60% DL-graphite, 21% pyroxene100, and 18% olivine, a mixture similar to HD 45677’s model. The model has a grain size distribution of 1.6 -
10.0µm. The 60µm and 100µm data points have a slope that is steeper than that of a blackbody wafer disk and so are only fit by a halo. Like HD 45677, HD 50138 has a smaller silicate feature that indicates grain growth. There is no millimeter data for either object and very little can be said about disk flattening beyond the slope of the outer IRAS
93 data points ruling out a wafer disk. Since a DUSTY disk is a model of the most advanced dust settling and disk flattening, it is not surprising that these two objects can undergo advanced grain growth, but be modeled by diskless halos.
There has been recent evidence to suggest that this system may actually be a binary star system without a dusty disk system (Miroshnichenko 2007) and is not similar to many of the other HAEBE’s in this survey. Therefore this object is considered a possible rejection from the correlation study.
HD 58647
Figure 3-25: The SED and DUSTY model of HD 58647 consisting of a disk and
blackbody.
94
Figure 3-26: The SED and DUSTY model of HD 58647 consisting of a large grain halo.
HD 58647 is a B9 type star (Oudmeijer et al. 2001). The data for HD 58647 consists of IRAS and 2MASS photometry. The BASS spectrum was taken on December
14th, 1994. UBVRI data comes from the Export group and was taken in 1999. The UV data is from the IUE archives.
The input spectrum has a temperature of 10500K and a log g of 5.0. The model of
HD 58647 consists of a disk ranging from 0.07 AU to 0.6 AU and is modeled at a nearly face on inclination angle to fit the correct flux levels in the SED. The halo is modeled with an optical depth of nearly zero thus creating a model of only a wafer disk. There is no silicate feature in this object, which is consistent with a blackbody wafer disk. The slope of the 25µm and 60µm IRAS data points is also nearly consistent with a blackbody disk. A blackbody of 1000K is multiplied by λ-2 to provide the excess near infrared emission that the wafer disk could not. The lambda factor does indicate that there are some small grains remaining at the inner edge of the disk located in a wall of some kind.
95 A second model could also fit the SED, that of a halo ranging from 0.26 – 216
AU with nearly 50/50 DL-sil and DL-grp. The grains range from 5.8 – 10.0µm in size.
Both models fit the data equally well and neither can be ruled out.
This system has undergone advanced grain growth and disk settling as evidenced by the two models’ parameters. Because of this system’s advanced evolution, it is considered a possible rejection from the correlation study.
HD 98800
Figure 3-27: The SED and DUSTY model of HD 98800.
HD 98800 is pair of binary stars with the hottest object a type K5 or K4 star based on its estimated temperature or ~4500K (Koerner et al. 2000). The data for HD 98800 consists of IRAS and 2MASS photometry. The BASS spectrum was taken on February
96 8th, 1998. UBVRIJHK and millimeter data is from Sylvester et al. (1996). An infrared spectrum from the PHT (PHT40 mode) on the ISO is also plotted.
The input spectrum has a temperature of 4500K and a log g of 5.0. The model for
HD 98800 consists of a halo extending from 1.7 AU to 350 AU and is made up of 40%
DL-silicates and 60% DL-graphite. The grain size distribution is 4.2 - 10.0µm leading to the very low 10µm silicate strength. The disk extends from 0.09 AU to 9 AU and experiences reheating longward of 10µm from the halo. The almost nonexistent silicate feature is consistent with the large grain size. The slope of the two millimeter data points with smaller error bars are fit by a wafer disk, though the slope of the 800µm point and the 1.3mm point, the point with the larger error bars, would be fit better by the halo.
Photospheric flux out to 7µm is evidence of inner clearing of material and is consistent with the model halo’s inner radius. The cleared-out region indicates that the IRAS data ratios could be suppressed due to an actual lack of inner material instead of possible disk shadowing. Therefore this object has been left out of the correlation studies.
This object is actually two sets of binary stars (Koerner et al. 2000). This complicates possible fitting in that the models assume a single stellar source located at the middle of the system and that the source’s radius is small compared to the circumsteller material. When there are more than one radiation source, if they are clustered close together in relation to the orbiting dust, then the central source issue can be avoided, however, each star shines separately on the surrounding dust. When one object is much brighter than the others, it is possible to take the brightest star as the radiation source and ignore the others. When they are more similar, it is possible to
97 simply average their separate Kurucz models in an appropriate fashion to represent the total stellar flux shining on the circumstellar dust.
HD 100546
Figure 3-28: The SED and DUSTY model of HD 100546.
HD 100546 is a B9.5 star and is ~103pc away (Grady et al. 2005). The data for
HD 100546 consists of IRAS photometry, IUE Archive data in the UV, and spectra from the Short Wave and Long Wave Spectrometers on the ISO. JHKLM data comes from
Malfait et al. (1998) and UBVRI data, taken in 1992, comes from de Winter et al. [1992 and 1994]. Millimeter data is from Wilner et al. (2003) and Henning et al. (1994). Also,
ISO data from Sloan et al. (2003) is included.
The input spectrum has a temperature of 10500K and a log g of 5.0. The model for HD 100546 consists of a halo that extends from 14 AU to 27000 AU and is made up of 50% DL-graphite, 20% DL-silicates and 30% crystalline silicates. The crystalline
98 material was necessary to fit the detailed structure from 8µm to 30µm. The dust grain size distribution is 0.06 - 10.0µm. A wafer disk extends from 0.7 AU to 300 AU. A blackbody of 1300K multiplied by λ-1 blocks off 7% of the stellar light, which gives a half height angle of 6˚. The slope of the millimeter data points is slightly steeper than a blackbody disk and so smaller grains than blackbody grains are needed to fit the data and is the reason for the enormous halo. Crystalline material gives rise to the detailed silicate features in the data from 10µm to 30µm. The halo and the disk have relatively large inner radii, but there is still emission inward of that. The blackbody models a “wall” located inside the wafer and models what is thought to be a puffed up inner rim that the DUSTY model cannot calculate.
The 27000 AU halo outer radius was necessary to provide enough cold grains to fit the millimeter data. This is a case of the DUSTY model failing since this system’s circumstellar material is observed to extend on the order of 100 AU from the star (Grady et al. 2005).
The Whitney model for this object is a flared disk extending from 20 – 100 AU and contains large grains. A blackbody of similar parameters models the NIR spectra region. The DUSTY wafer disk actually extends out further than the Whitney flared disk.
The DUSTY model, much like HD 31648, contains an envelope with a massive outer radius, which is necessary to provide the flux with the necessary slope in the millimeter region. The Whitney model is able to fit this region with just the flared disk.
99 HD 135344 (SAO 206462)
Figure 3-29: The SED and DUSTY model of HD 135344.
HD 135344, also known as SAO 206462, is a F4 type star 140pc away (Grady et al. 2005). The data for HD 135344 includes IRAS and 2MASS photometry as well as an
IRS spectrum from Spitzer Space Telescope. The BASS spectrum was taken on July 9th,
2007. UBVRIJHKL’M comes from Sylvester et al. (1996). The millimeter data is from
Sylvester et al. (1994) and Walker and Butner (1995). Also, data (Beerman, L, private communication) from the SpeX instrument (Rayner et al. 2003) is plotted. The data just shortward of 100µm is from the Multiband Imaging Photometer for Spitzer (MIPS) courtesy of P.S. Smith from University of Arizona. UV data is from Grady (private communication).
100 The input spectrum for this object consists of a 50/50 mixture of two input stellar spectra, 5500K and 6250K both with log g’s of 4.5. This mixture was necessary because of the high rotation speed of the central star. The model of HD 135344 has a somewhat similar overall spectral shape as that of HD 100546. This model consists of only a halo extending from 25 AU to 8500 AU and is made up of 50% DL-silicates and 50% DL- graphite. No additional grain types were needed. The grain size distribution is 1 - 10µm.
The lack of a silicate feature is explained both by the larger grain size and the large inner radius such that there is little silicate material close enough to the central star to be radiated enough to emit at those wavelengths. A blackbody of 900K multiplied by λ-1.5 models the excess near infrared emission and blocks off 25% percent of the starlight.
This is a half height angle of 22.5˚. The large radial size of the model halo is necessary to provide enough cold grains to produce the millimeter flux. This is a failure of the
DUSTY model as the observed size of this object is ~200AU (Grady et al. 2005).
The slope of the millimeter data is too steep for a blackbody wafer disk and is fit by the halo. The Whitney model for this object is a flared disk extending ~60 – 155 AU and is made up of large grains. A blackbody of similar parameters is used to fit the NIR region. No envelope is used in the Whitney model. The flared disk of the Whitney model is able to fit the millimeter data where the wafer DUSTY model could not. Because of this, the DUSTY model needed a massive thousands AU halo to fit the millimeter flux whereas the Whitney mode could use a flared disk of much more reasonable size consistent with observational imaging.
101
HD 141569
Figure 3-30: The SED and DUSTY model of HD 141569.
HD 141569 is an A type star ~100pc away (Zuckerman 2001). The data for HD
141569 consists of IRAS and 2MASS photometry. The BASS spectrum was taken on
May 12th, 1998. The UV data is from the IUE Archive and the UBVRI data is from de
Winter et al. [1992]. The millimeter data comes from Walker and Butner (1995).
The input spectrum has a temperature of 10500K and a log g of 5.0. The model for HD 141569 is a halo that extends from 2 AU to 5570 AU and consists of 60% DL- silicates and 40% DL-graphite. The excess silicates helped to slightly better fit the 10µm region. The dust grain size distribution is 2 - 10.0µm. This object shows grain growth and has a cleared out inner area as evidenced by the presence of only photospheric flux out to
~8µm. Combined with the lower ratio of infrared emission to stellar emission, this object shows many signs of advanced evolution of the grains in size. The peak near 8µm is due
102 to emission by organic material (usually thought to be polycyclic aromatic hydrocarbons
– PAHs), which cannot be modeled by DUSTY and is ignored. Observations of this object point to circumstellar material extending at least 600AU (Mouillet et al. 2001), but it is likely that the DUSTY model is overestimating the size of the circumstellar dust. The
DUSTY model required the large halo size to provide enough cold grains to produce the millimeter flux. The model of Merin et al. (2004) uses a non-blackbody disk that extends from 0.24 – 428 AU, which is far smaller than the DUSTY model halo. This object is a good example of the need for non-blackbody disk model capability in modeling codes.
DUSTY’s disk, since it does not contain non-blackbody grains, can not fit the SED.
The lack of excess emission above stellar flux shortward of 8µm indicates a cleared out inner disk. The cleared out region indicates that the IRAS data ratios could be suppressed due to an actual lack of inner material instead of possible disk shadowing.
Therefore this object has been left out of the correlation studies.
103 HD 163296
Figure 3-31: The SED and DUSTY model of HD 163296.
HD 163296 is an A3 type star that is 122pc away (Grady et al. 2005). The data for
HD 163296 consists of IRAS, 2MASS, and LRSP data. The BASS spectrum was taken in
2005. The UV data is from the IUE Atlas and the millimeter data is from Mannings
(1994), Henning et al. (1994) and Henning et al. (1998). The UBVRI data is from de
Winter et al. [1997]. The bright green data in the visual and infrared is a spectrum taken at Lick Observatory in 2005.
The input spectrum has a temperature of 9250K and a log g of 5.0. The model for
HD 163296 consists of a halo extending from 0.27 AU to 12000 AU and is made up of
47% DL-graphite, 24% pyroxene95, 24% crystalline silicates, and 5% olivine. The crystalline silicates and the small amount of olivine help to fit the small 11µm bump while the pyroxene narrows the 10µm silicate feature. The grain size distribution is 0.7 -
10.0µm. A disk extending from 0.06 AU to 360 AU when combined with the halo
104 allowed a fit of the slope of the millimeter data. The slope of the blackbody disk is too shallow while the slope of the halo is too steep to fit alone. The small grains necessary to produce the 10µm silicate feature also force the halo millimeter slope steeper and so cannot fit the millimeter data. A blackbody of 1100K multiplied by λ-2 blocks 9% of the stellar light and has a half height angle of 8˚.
The size of the silicate feature and the slightly shallower millimeter data slope both indicate a moderate amount of grain growth and dust settling. This system is very similar to HD 31648 but appears to be slightly more evolved in grain growth as evidenced by the slightly smaller silicate feature and shallower millimeter data slope.
The large outer radius of the model halo is necessary to contain sufficient cold grains to produce the millimeter flux. This is a case where the DUSTY model fails because observations have shown that material only extends 450AU from the star (Grady et al. 2000).
The Whitney model for this object is a flared disk extending to 450AU with an attached small envelope out to only 10AU. The Whitney model does not require an extra blackbody, as the inner region of a flared disk model is sufficient to produce the NIR flux. The grain sizes are similar between the two models. As with HD 31648, the SED of
HD 163296 has a millimeter data slope that is steeper than a blackbody and therefore cannot be fit by a wafer blackbody disk. Because of this, the DUSTY model required an envelope contribution which required the enormous thousands AU halo size. The
Whitney model, capable of having smaller grains inside the disk, could fit this SED with a massive disk instead. The sizes of the Whitney model components are more consistent with the observational imaging.
105
HD 169142
Figure 3-32: The SED and DUSTY model of HD 169142.
HD 169142 is an A5 type star 145pc away (Dent et al. 2005). The data for HD
169142 consists of IRAS and 2MASS photometry as well as UV data from the IUE
Archives. The millimeter data comes from Sylvester et al. (1996) and the UBVRI data comes from van der Veen et al. (1989). Also, spectra from the PHT on the ISO and SST
IRS data are also plotted.
The structure in the data from 7 - 10µm is similar to the same structure in RR Tau and is thought to arise from polycyclic amorphous hydrocarbons (reference), or PAH’s.
106 There were no dust input files for PAH’s available and so DUSTY could not be used to model that feature.
In a similar fashion to HD 135344, the input spectrum for this object consists of a
50/50 mixture of 5750K and 8750K spectra, both with log g’s of 5.0. The model for HD
169142 consists of a halo extending from 7 AU to 73 AU and is made up of 50% DL- silicates and 50% DL-graphite. No further grain types were needed. A wafer blackbody disk extends from 0.5 – 9 AU. The dust grain size distribution is 3.5 - 10.0µm and is consistent with the lack of a 10.0µm silicate feature. A blackbody of 1200K blocks off
4% of the stellar light and has a half height angle of 4˚. The slope of the millimeter data matches with a blackbody wafer disk and cannot be fit by a halo alone. Combined with the lack of a 10µm silicate feature, this object has undergone advanced grain growth and disk settling.
The Whitney model for this object is a flared disk reaching from ~44-230AU that is made up of very large grains providing a blackbody type slope for the millimeter data.
The model has no envelope, but does add a blackbody of similar temperature as the
DUSTY model’s. The large grain size of the flared disk in the Whitney model is consistent with the ‘bowling balls’ of the DUSTY wafer disk. However, because of the low incidence angle of starlight on the wafer disk, the wafer disk dimensions are much smaller than those of the flared disk model. The Whitney model for this object also includes elements of the DUSTY model itself (Grady et al. 2007). These shall be explained fully in the Whitney results section.
107 HD 190073
Figure 3-33: The SED and DUSTY model of HD 190073.
HD 190073 is a A2 type star 400pc away (Dent et al. 2005). The data for HD
190073 consists of IRAS and 2MASS photometry as well as a LRSP spectrum. The
BASS spectrum was taken on October 14th, 1996. The UV data comes from the IUE
Archive. The UBVRI data comes from the Export group.
The input spectrum has a temperature of 9250K and a log g of 4.5. The model for
HD 190073 consists of a halo extending from 0.3 AU to 580 AU and is made up of 50%
DL-silicates and 50% DL-graphite. The dust grain size distribution is 0.03 - 10.0µm. This system is similar to HD 37806 and has similar halo parameters as well. The silicate feature is of moderate size and is indicative of some grain growth, though the model shows very little. There is no millimeter data and so little can be said about the presence or properties of a wafer disk. Miroshnichenko (2007) has suggested that this star may be similar to HD 45677 and HD 50138 in not being a true pre-main sequence object.
108
HD 250550
Figure 3-34: The SED and DUSTY model of HD 250550.
HD 250550 is a B7 Herbig star that is 700pc away (Feunte 2002). The data for
HD 250550 consists of IRAS and 2MASS photometry. The two BASS spectra were taken on January 31st, 2001 and January 9th, 2003 showing little variability between those two epochs. The UV data comes from the IUE Atlas. The UBVRI data comes from Herbst &
Shevchenko (1999) where we have used the mean values of the magnitudes (inspection of the on-line data table indicates that the brightness was stable to within 10% over a time span of 9 years). The JHK data is from de Winter et al. [1981]. The millimeter data comes from Henning et al. (1998).
109 The input spectrum has a temperature of 13000K and a log g of 4.5. The model of
HD 250550 consists of a halo extending from 0.6 AU to 1200 AU and is made up of 45%
DL-graphite, 40% pyroxene70, and 15% crystalline silicates. The pyroxene dust narrows the 10µm silicate feature while the crystalline silicates help to shape the feature and provide the small bump at 11µm. The dust grain size distribution is 0.1 - 10.0µm. A blackbody of 1250K multiplied by λ-0.8 blocks off 11% of the stellar light and has a half height angle of 10˚.
The silicate band is of medium size and is consistent with the modeled grain sizes.
There is only a single millimeter upper limit point and so nothing can be said about the presence or properties of a wafer disk. The halo is sufficient to fit the known data for the outer regions.
Hen 3-600
Figure 3-35: The SED and DUSTY model of Hen 3-600.
110
Hen 3-600 is a M4 type star that is 50pc away (Kessler-Silacci et al. 2005). The data for Hen 3-600 consists of IRAS and 2MASS photometry as well as an IUE Archive spectrum. The BASS spectrum was taken on January 16th, 2001.
The input spectrum has a temperature of 5000K and a log g of 2.0. The model of
Hen 3-600 is a halo that extends from 0.76 AU to 176 AU and is made up of a 40% DL- silicates and 60% DL-graphite. The dust grain size distribution is 0.05 - 4µm leading to the smaller 10µm silicate feature of the model, matching the observed SED indicating some grain growth. There is no millimeter data and nothing can be said about the presence or properties of a wafer disk. Since this is a triplet star system, the same assumptions that went into the model of HD 98800 also apply to this object.
In the Zuckerman (2001) model of this system, the “hot” dust detected by IRAS is orbiting around two of the stars in the Hen 3-600 system, while a second mass of cooler dust detected by the James Clerk Maxwell Telescope’s Submillimeter Common-User
Bolometer Array (SCUBA) is orbiting the entire system of three stars. From this model, the third star is actually outside the “hot” dust and is 70 AU from the other two stars. This would make modeling this system nearly impossible because the third star would be embedded inside the DUSTY halo.
The Sargent model (2006) of this object is a flared disk that orbits Hen 3-600A.
The outer radius of the Sargent disk models is on the order of a few hundred AU.
However, the wavelength region used to obtain a fit is limited and is only 8-14µm. The size of the disk is comparable to the extent of the DUSTY halo. In the DUSTY model,
111 only the halo component is capable of producing the silicate feature while the DUSTY disk, not containing small grains, cannot.
The lack of excess emission above stellar flux shortward of 10µm indicates a cleared out inner disk. IRAS data ratios would be unhelpful in determining amounts of shadowing or blackbody blockages. Therefore Hen 3-600 is separated out from the other objects in the correlation studies
HK Ori
Figure 3-36: The SED and DUSTY model of HK Ori.
HK Ori is an A4 Herbig star in Orion 450pc away (Feunte et al. 2002). The data for HK Ori consists of IRAS and 2MASS photometry as well as an IUE Archive spectrum. The two BASS spectra were taken on October 17th, 1996 and December 24th,
1998. UBVRI and JHK data come from de Winter et al. [1992 and 1980 respectively]
112 with other JHK data from the Export group. The millimeter point is from Henning et al.
(1998).
The input spectrum has a temperature of 8500K and a log g of 5.0. The model for
HK Ori is a halo extending from 0.4 AU to 80 AU and is made up of 20% DL-silicates,
61% DL-graphite, 18% pyroxene80, and 1% crystalline silicates. The small amount of crystalline material and the pyroxene helped to narrow and shape the silicate feature. The dust grain size distribution is 0.01 - 10µm leading to a slightly reduced model 10µm silicate feature matching the observed SED indicating some grain growth. The IRAS
60µm data point is an upper limit and because of its low relative flux makes it unlikely that a blackbody type disk is present in this system. The silicate band strength is variable and sometimes is very weak. This indicates grain growth, but the lack of a large grain disk does not. The 60µm IRAS point is an upper limit and therefore removes this object from both 60µm correlation studies.
113 HL Tau
Figure 3-37: The SED and DUSTY model of HL Tau.
HL Tau is a K7 T Tauri star in Taurus (Andrews and Williams 2005). The data for HL Tau consists of IRAS and 2MASS photometry. The BASS spectrum was taken on
January 10th, 2003. The continuum data from 2 - 40µm is from the SWS camera on ISO.
UBVRI data is from Rydgren and Vrba (1983). The millimeter data is from several sources, Weintraub et al. (1989), Adams et al. (1990), Beckwith et al. (1990), Beckwith and Sargent (1991) and Mundy et al. (1996).
The input spectrum has a temperature of 4000K and a log g of 5.0. The model for
HL Tau is a halo extending from 0.16 AU to 3500 AU and is made up of a 16% DL- silicates, 49% DL-graphite, and 35% pyroxene80 dust. The pyroxene was added to help shape the small silicate feature. The dust grain size distribution is 0.5 - 10µm. The optical depth of this halo is quite large (τ550nm = 8.0) and as such, the small silicate feature appears in absorption instead of emission. A blackbody of 750K multiplied by λ-2 blocks
114 9% of the stellar light and has a half height angle of 8˚. The slope of the millimeter data cannot be fit with a blackbody wafer disk because the DUSTY disk does not contain small grains. Therefore, a large halo containing enough small grains to produce the millimeter flux was necessary. Though this halo is not as large as some of the other failures, such as HD 31648, the DUSTY model is still likely overestimating the radial extend of the circumstellar material. The ratio of the infrared flux to the stellar flux indicates that this is a young system as the dust surrounding the star is still thick and little processed. This is supported by the lack of a large disk in the model. However, the small silicate feature and minimum grain size do indicate advanced grain growth. Extended emission is known to surround this object and it is possible that some of the quite optically thick halo may be explained by extraneous material. The data exhibits a water ice absorption band around 3µm, which is near the lower wavelength limit of the BASS instrument. The DUSTY model does not attempt to fit that feature.
115 HP Tau
Figure 3-38: The SED and DUSTY model of HP Tau.
HP Tau is a K3 T Tauri star in Taurus (Andrews and Williams 2005). The data for
HP Tau consists of IRAS and 2MASS photometry. The BASS spectrum was taken on
January 10th, 2003. The millimeter data is from various sources, Weintraub et al. (1989),
Adams et al. (1990), Beckwith et al. (1990), Duvert et al. (2000), and Andrews and
Williams (2005).
The input spectrum has a temperature of 4000K and a log g of 5.0. The model for
HP Tau consists of a halo extending from 0.3 AU to 340 AU and is made up of 14% DL- silicates, 46% DL-graphite, and 40% pyroxene100. The pyroxene helps to narrow the silicate feature. The grain size distribution is 0.3 - 1.7 µm. A wafer disk extends from
0.01 AU to 5 AU and experiences reheating due to the halo longwards of 10µm. A blackbody of 950K multiplied by λ-1.5 blocks 6% of the stellar light and has a half height angle of 5˚. The millimeter data is not quite as clear as to what the “correct” slope is, but
116 when picking the points with smaller error bars, the slope is nearly that of a large grain wafer disk.
The silicate feature is of moderate size and is more rounded than in younger systems. This is consistent with the millimeter data slope indicating that some grain growth and dust settling has occurred.
HP Tau G2
Figure 3-39: The SED and DUSTY model of HP Tau G2.
HP Tau G2 is a G0 type star (Strom et al. 1989). The data for HP Tau G2 consist of 2MASS photometry and a BASS spectrum taken on February 1st, 2002. The millimeter point is from Osterloh and Beckwith (1995).
117 The input spectrum has a temperature of 5750K and a log g of 5.0. The model for
HP Tau G2 is a halo that extends from 8 AU to 80 AU and consists of 40% DL-silicates and 60% DL-graphite. The grain size distribution is 0.1 - 10.0µm though the data around
10µm is not sufficient to say much of anything about the grain sizes or possible growth.
The lack of non-photospheric emission shortward of 10µm indicates an inner region cleared out of material. Only a single upper limit point exists at 1.3mm and so only the inner halo radius of this object is relatively constrained. The lack of data especially longward of 10µm does not allow much more to be said for this object.
There is no IRAS data of this object and so HP Tau G2 is not included in the correlation studies.
HP Tau G3
Figure 3-40: The SED and DUSTY model of HP Tau G3.
118
HP Tau G3 is a K7 type star (Strom et al. 1989). The data for HP Tau G3 consist of 2MASS photometry and a BASS spectrum taken on January 16th, 2002. The millimeter point is from Osterloh and Beckwith (1995).
The input spectrum has a temperature of 5500K and a log g of 5.0. The model for
HP Tau G3 is a halo extending from 3.8 AU to 38 AU and consists of 40% DL-silicates and 60% DL-graphite. The grain size distribution is 0.01 - 10.0µm. Only a single upper limit 1.3mm data point constrains the outer regions of the system, therefore the inner halo radius is the only parameter relatively constrained. This object is similar to HP Tau G2 and has a similar model except there appears to be slightly more of a 10µm silicate feature for HP Tau G3 meaning a smaller grain size for this model. However, the quality and amount of data is insufficient to make any sort of definitive judgments about grain growth or grain size constraints. The stellar sources are nearly the same for both objects.
The inner halo radius for HP Tau G3 is somewhat smaller than for HP Tau G2. There is no IRAS data for this object and so it is not included in the correlation studies.
119 HR 4796
Figure 3-41: The SED and DUSTY model of HR 4796.
HR 4796 is an A0 Herbig star 76pc away (Kessler-Silacci et al. 2005). The data for HR 4796 consist of IRAS and 2MASS photometry. The BASS spectrum was taken on
May 12th, 1998. The BVRIJHK data and millimeter point are from Jura et al. (1995).
The input spectrum has a temperature of 9500K and a log g of 5.0. The model for
HR 4796 consists of a halo extending from 37 AU to 740 AU and is made up of 50% DL- silicates and 50% DL-graphite grains. No other grain types were needed. The grain size distribution is 0.1 - 10.0µm. A blackbody of 1000K multiplied by λ-2 blocks off 0.4% of the stellar light making a half height angle of only 0.4˚. The small blackbody contribution is necessary to provide the small amount of excess emission in the 2 - 5µm region.
The lack of a silicate feature indicates grain growth. The very large nearly cleared out inner region extending to 37AU indicates a very evolved system. The slope of the
100µm IRAS point and the millimeter upper limit data point is too steep for a DUSTY
120 large grained wafer disk. Observations of this object indicate that the surrounding material is in a disk ‘ring’ that extends from 60 – 77 AU (Schneider et al. 1999).
Therefore, this DUSTY model can be considered a failure as it greatly overestimates the outer radius of the circumstellar material. The lack of sufficient millimeter data does not allow anything more to be said about the possible presence or properties of a wafer disk other than that such a disk is not excluded by the 0.8mm data point upper limit.
The 12µm IRAS point is an upper limit, therefore this object is not included in either 12µm correlation studies.
Lk Ca 15
Figure 3-42: The SED and DUSTY model of LickCa 15.
121 LickCa15 is a K5 T Tauri star (Bergin et al. 2004) that is 140pc away (Kessler-
Silacci et al. 2005). The data for LickCa15 consists of IRAS and 2MASS photometry.
The BASS spectrum was taken on four nights, January 15th, 2001, January 16th, 2002,
January 9th, 2003, and February 20th, 2003. The millimeter points are from Osterloh and
Beckwith (1995) and Andrews and Williams (2005).
The input spectrum has a temperature of 4250K and a log g of 4.0. The model for
LickCa 15 consists of a halo extending from 2.7AU to 300AU and consists of 10% DL- silicates, 50% DL-graphite, and 40% pyroxene80. The addition of the pyroxene narrowed the silicate feature of the model and allowed a better fit. The grain size distribution is
0.25 - 1.5µm. A wafer disk extends from 0.06AU to 17AU and experiences reheating longward of 30µm. The millimeter data slope is consistent with a large grain wafer disk.
The somewhat rounded 10µm silicate feature indicates a modest amount of grain growth.
The model of Bergin for this object consists of an inner disk wall blackbody of
177K which matches somewhat with the inner halo boundary temperature of 300K for the DUSTY model. As with the other Bergin modeled objects, this temperature difference may be the result of the smaller grain size in the Bergin model. Much like the model for
DM Tau, the Bergin model also includes an inner blackbody of 1400K to explain the excess near-infrared emission that the other components of their models cannot. DUSTY is able to model the system without the need for an extra blackbody. Since their models do not consider the SED longward of ~20 µm, no comparisons can be made for the outer regions of the system. The Bergin model disk has an inner radius of 3AU, which agrees closely with the DUSTY model halo inner radius of 2.7AU.
122 The near photospheric flux levels out to around 8µm show evidence of a cleared out inner disk region. The cleared out region indicates that the IRAS data ratios could be suppressed due to an actual lack of inner material instead of possible disk shadowing.
Therefore LickCa15 has been left out of the correlation studies.
Lk Ca 21
Figure 3-43: The SED and DUSTY model of LickCa 21.
LkCa21 is a M3 T Tauri star (Andrews and Williams 2005). The data for
LickCa21 consists of 2MASS photometry and a BASS spectrum that was taken on
January 17th, 2002. The millimeter data is from Osterloh and Beckwith (1995) and
Andrews and Williams (2005).
The input spectrum has a temperature of 3500K and a log g of 4.0. The model for
LickCa 21 consists of a halo extending from 2.5AU to 250AU, however, the outer edge
123 of the halo is not very well constrained since there are only a few upper limit data points in the millimeter region. The halo dust is composed of 50% DL-graphite and 50% pyroxene80. The use of pyroxene narrowed the silicate feature. The grain size distribution is 0.005 - 0.25µm. The prominent size of the 10µm silicate feature indicates very little grain growth has occurred in this system. The lack of good millimeter data does not allow anything to be said about the presence or properties of a wafer disk.
The lack of IRAS data removes this object from any of the correlation studies.
NV Ori
Figure 3-44: The SED and DUSTY model of NV Ori.
NV Ori is a F6 type star (Blondel and Tjin a Djie 2006). The data for NV Ori consists of 2MASS photometry and an IUE Atlas spectrum. The BASS spectrum was taken on December 24th, 1998. UBVRI data comes from the Export group.
124 The input spectrum had a temperature of 6250K and a log g of 4.0. The model for
NV Ori consists of a halo extending from 1AU to 1000AU. However, the outer radius is unconstrained as there is no data longwards of 20µm. The halo dust is composed of 49%
DL-silicates and 50% DL-graphite and has a dust grain size distribution of 0.005 -
0.25µm. The very prominent 10µm silicate feature indicates very little grain growth has occurred. Since there is no data longwards of 20µm, nothing can be said about the presence or properties of a wafer disk.
The lack of IRAS data removes this object from any correlation studies.
RR Tau
Figure 3-45: The SED and DUSTY model of RR Tau.
RR Tau is an A3-5 Herbig star 160pc away (Acke and von der Ancker 2004). The data for RR Tau consists of IRAS and 2MASS photometry with two BASS spectra taken
125 on October 15th, 1996 and January 15th, 2002. The UV data is from the IUE Atlas and the
UBVRI and some JHK data are from the Export group.
The input spectrum has a temperature of 8750K and a log g of 5.0. The model for
RR Tau consists of a halo extending from 0.2AU to 1080AU and is composed of 56%
DL-silicates and 44% DL-graphite. The dust grain size distribution is 3 - 10µm. The
IRAS points may be contaminated by extended emission and so the outer edge of the halo model is not too constrained. The complex features in the SED around 10µm are thought to result from PAH’s. The lack of a silicate feature indicates grain growth, which is consistent with the model’s large grain sizes. The lack of millimeter data makes it impossible to comment on the possible presence or properties of a wafer disk.
Due to the possible extended emission contamination of the IRAS data, this object is not included in the correlation studies.
126 RW Aur
Figure 3-46: The SED and DUSTY model of RW Aur.
RW Aur is a binary system made up of K1 and K5 T Tauri stars (Furlan et al.
2005). The data for RW Aur consists of IRAS and 2MASS photometry. The two BASS spectra were taken on October 15th, 1996 and February 8th, 1998. The UV data comes from the IUE Atlas and the UBVRI data is from Herbst and Shevchenko (1999). The
Herbst data shows that this object displays variability. The higher flux data is from 1983 and the lower flux from 1986. The millimeter data is from Osterloh and Beckwith (1995) and Andrews and Williams (2005). The flux of the Osterloh and Beckwith 1.3mm point listed is ten times too large and is a typo. The correct flux has been plotted in the figure above.
127 The input spectrum has a temperature of 5000K and a log g of 5.0. The model of
RW Aur consists of a halo extending from 0.2AU to 100AU and is composed of 50%
DL-silicates, 37% DL-graphite, and 13% pyroxene95. The pyroxene allowed the narrowing of the silicate feature. The dust grain size distribution is 0.01 - 7µm. A wafer disk extends from 0.02AU to 7.2AU and experiences reheating longward of 8µm. A blackbody of temperature 950K multiplied by λ-2 blocks 9% of the stellar light and has a half height angle of 8˚.
The 10µm silicate feature shows a rounded shape and is weak indicating grain growth. The slope of the millimeter data is consistent with a large grain disk. This is evidence that RW Aur has undergone grain growth and dust settling and is a more evolved system than others. The small minimum grain size in the model is necessary to allow a fit of the silicate feature when added to the big grain disk. This would indicate that any small grains still remaining in the system are well within the halo and have not settled into a disk.
128 RY Ori
Figure 3-47: The SED and DUSTY model of RY Ori.
RY Ori is a F6 type star (Mora et al. 2001). The data for RY Ori consists of IRAS and 2MASS photometry. The BASS spectrum was taken on December 24th, 1999. The
UBVRI data is from de Winter et al. [1992] while the JHK data is also from de Winter et al. [1986]. Both UBVRI and JHK data also come from the Export group.
The input spectrum has a temperature of 6500K and a log g of 5.0. The model for
RY Ori consists of a halo extending from 0.4AU to 370AU and is composed of 50% DL- graphite and 50% pyroxene95. The use of pyroxene provided the narrower silicate feature in the model. The grain size distribution is 0.03 - 6µm. The large 10µm silicate feature indicates that little grain growth has occurred. The model’s grain size distribution is consistent with this. There is no millimeter data and so nothing can be said about the presence or properties of a large grain disk. The 60µm IRAS point is an upper limit and therefore removes this object from both 60µm correlation studies.
129
RY Tau
Figure 3-48: The SED and DUSTY model of RY Tau.
RY Tau is a G5 star in Taurus (Muzerolle et al. 2003). The data of RY Tau consists of IRAS and 2MASS photometry as well as an IUE Atlas spectrum. The BASS spectrum was taken on October 14th, 1996. The UBVRI data is from the Export group.
The millimeter data is from several sources, Beckwith et al. (1990), Beckwith and
Sargent (1991), Mannings and Emerson (1994), and Andrews and Williams (2005).
The input spectrum has a temperature of 6000K and a log g of 5.0. The model of
RY Tau consists of a halo extending from 0.3AU to 340 AU and is composed of 2% DL- silicates, 50% DL-graphite, and 48% pyroxene95. The addition of pyroxene allowed the
130 narrowing of the silicate feature. The dust grain size distribution is 0.03 - 6µm. A disk extending from 0.02AU to 17AU experiences reheating longward of 8µm.
The slope of the millimeter data is consistent with a large grain wafer disk.
However, the prominent and sharp 10µm silicate feature is consistent with little grain growth. This is an example of a system that does not fit with the model of grain growth occurring with dust settling. As expected, RY Tau stands out on a plot of silicate feature band strength vs. millimeter spectral index shown after the individual star entries.
SU Aur
Figure 3-49: The SED and DUSTY model of SU Aur.
SU Aur is a G2 type star in Auriga (Muzerolle et al. 2003). The data for SU Aur consists of IRAS and 2MASS photometry as well as an IUE Atlas spectrum. The BASS spectrum was taken on October 17th, 1996. The UBVRI data is from Herbst and
131 Shevchenko (1999) and is the mean of data taken on October 14th and 20th, 1996. The millimeter data is from Weintraub et al. (1989), Beckwith et al. (1990), and Andrews and
Williams (2005).
The input spectrum has a temperature of 6000K and a log g of 5.0. The model for
SU Aur consists of a halo extending from 1AU to 90AU and is composed of 50% DL- graphite, 47% pyroxene95, and 3% olivine. The pyroxene narrowed the silicate feature while the olivine provided fit the tiny bump at 11µm. The grain size distribution is 0.2 -
6µm. A wafer disk extends from 0.06AU to 6AU and experiences reheating longward of
10µm. A blackbody of temperature 900K multiplied by λ-2 blocks 8% of the stellar light and has a half height angle of 7˚. The slope of the millimeter data is consistent with that of a blackbody large grain disk. The 10µm silicate is somewhat rounded and slightly weaker indicating some grain growth and is consistent with the dust settling evidenced from the wafer disk.
Extended emission surrounds this object and contaminates the IRAS data to some extent. This causes the data points to show more flux than coming from just SU Aur itself. Because of this, this object is not included in the correlation studies.
132 TW Hya
Figure 3-50: The SED and DUSTY model of TW Hya.
TW Hya is a K7 type T Tauri star in the Hydae Association (Sargent et al. 2006).
The data for TW Hya consists of IRAS and 2MASS photometry as well as an IUE Atlas spectrum. The BASS spectrum was taken on March 21st, 1998. The UBVRIJHK data comes from Rucinski & Krautter (1983). The millimeter data is from Weintraub et al.
(1989) and Wilner et al. (2003).
The input spectrum has a temperature of 4000K and a log g of 5.0. The model for
TW Hya consists of a halo extending from 1.0AU to 1000 AU and is composed of 47%
DL-silicates, 50% DL-graphite, and 3% olivine. The small amount of olivine was necessary to fit the tiny bump at 11µm. The dust grain size distribution is 0.005 - 2µm. A
133 wafer large grain disk extends from 0.03AU to 30.0AU and experiences reheating longward of 20µm. The 10µm silicate feature is prominent and sharp and is consistent with small grains. However, photosperic flux shortward of 8µm indicates a cleared out inner region as does the domination of a large grain disk at millimeter wavelengths. The disk does not fit the flux of the 1.0mm and 1.3mm points though it does fit the slope.
The cleared out inner region indicates that the IRAS data ratios could be suppressed due to an actual lack of inner material instead of possible disk shadowing.
Therefore TW Hya has been left out of the correlation studies. Interestingly, a very recent paper (Setiawan et al 2008) claims the detection of a planet in this system making this the first observational link between a planet and a disk.
UX Ori
Figure 3-51: The SED and DUSTY model of UX Ori.
134
UX Ori is an A4 type Herbig star 430pc away (Dent et al. 2005). The data for UX
Ori consists of IRAS and 2MASS photometry as well as an IUE Archive spectrum. The
BASS spectrum was taken on December 14th, 1994. The UBVRI come from de Winter et al. [1980, 1988, and 1992] while JHK data also came from de Winter et al. [1980, 1981, and 1986]. Other UBVRIJHK data came from the Export group. Millimeter data comes from Natta et al. (1999). Also, the PHT infrared spectrometer on the ISO provides some data in the near infrared.
The input spectrum has a temperature of 8750K and a log g of 5.0. The model for
UX Ori consists of a halo extending from 0.7AU to 1100AU and is composed of 34%
DL-silicates, 50% DL-graphite, 13% pyroxene100, and 3% olivine. The olivine added the bump at 11µm while the pyroxene narrowed the silicate feature. The grain size distribution is 0.005 - 2.0µm. A large grain wafer disk extends from 0.07AU to 72AU and experiences some reheating from the halo longward of 8µm. A blackbody of temperature 950K multiplied by λ-2 blocks 3% of the stellar light and has a half height angle of 3˚. The 10µm silicate feature is quite prominent and sharp and is evidence of small grains. However, a large grain disk dominates at millimeter wavelengths. This object is another example where grain growth and dust settling are not linked.
The object is highly variable in the UV through near infrared.
135 UY Aur
Figure 3-52: The SED and DUSTY model of UY Aur.
UY Aur is a binary system consisting of M0 and M2.5 T Tauri stars (Furlan et al.
2005). The data for UY Aur consists of IRAS and 2MASS photometry as well as an IUE
Atlas spectrum. The BASS spectrum was taken on January 9th, 2003. UBVRI data comes from Rydgren and Vrba (1983) while the millimeter data is from Beckwith et al. (1990),
Osterloh and Beckwith (1995), and Andrews and Williams (2005).
The input spectrum has a temperature of 3750K and a log g of 5.0. The model of
UY Aur consists of a halo extending from 0.15 AU to 1450 AU and is composed of 43%
DL-silicates and 57% DL-graphites. The grain size distribution is 0.5 - 2.0µm, reducing the model 10µm silicate feature and matching the observed SED. A wafer large grain disk extends from 0.01 AU to 9.5 AU and experiences reheating longward of 8µm. A blackbody of temperature 1100K multiplied by λ-2 blocks 14% of the stellar light and has a half height angle of 13˚. The slope of the large grain wafer disk is consistent with that
136 of the millimeter data. The 10µm silicate feature is weak and rounded indicating grain growth. Together, the weak silicate feature and the fit of a DUSTY wafer disk is evidence that this object has undergone advanced evolution although it is still shrouded in a thick halo of dust with an optical depth of τ550nm = 1.78. The optically thick halo is necessary to contain sufficient material for the cool star used in the model to generate enough thermal emission from the dust to fit the IR region. This higher optical depth also produces a good fit of most of the UV and visual region.
V590 Mon
Figure 3-53: The SED and DUSTY model of V590 Mon.
V590mon is a B7 type Herbig star (Hernandez et al. 2004). The data of v590mon consists of IRAS and 2MASS photometry as well as an IUE Atlas spectrum. The BASS
137 spectrum was taken December 14th, 1994. The UBVRIJHK data comes from de Winter et al. [1997] while one set of UBVRI data is from Mendoza and Gomez (1980) and Rydgren and Vraa (1987). The millimeter data is from Henning et al. (1998).
The input spectrum has a temperature of 12000K and a log g of 5.0. The model of v590mon consists of a halo extending from 0.3AU to 100AU and is composed of 11%
DL-silicates, 42% DL-graphite, and 47% pyroxene100. The addition of the pyroxene narrowed the silicate feature fit. The dust grain size distribution is 0.35 - 2.0 µm. The roundness of the 10µm silicate feature indicates grain growth, which is consistent with the model’s minimum grain size. Lack of good millimeter data makes it impossible to say anything about the presence or properties of a large grain disk. The 60µm IRAS point is an upper limit and therefore removes this object from both 60µm correlation studies.
V594 Cas
138 Figure 3-54: The SED and DUSTY model of V594 Cas.
V594cas is thought to be a B8 type Herbig star (Finkenzeller 1985). The data for v594cas consists of IRAS and 2MASS photometry an IUE Atlas spectrum. The BASS spectrum was taken on October 14th, 1996. The millimeter point is from Henning et al.
(1998).
The input spectrum has a temperature of 10750K and a log g of 5.0. The model of v594cas consists of a halo extending from 0.89AU to 890AU and is composed of 50%
DL-silicates and 50% DL-graphite. No additional dust grain types were needed. The grain size distribution is 0.35 - 7µm. A blackbody of temperature 1200K multiplied by λ-2 blocks 12% of the stellar light and has a half height angle of 11˚. The weakness and roundness of the 10µm silicate feature indicates grain growth and is matched by the model minimum grain size. The lack of good millimeter data prohibits saying anything about the presence or properties of a large grain disk.
139 XZ Tau
Figure 3-55: The SED and DUSTY model of XZ Tau.
XZ Tau is a M3 T Tauri star (Watson 2007). The data for XZ Tau consists of
2MASS photometry with a BASS spectrum taken on January 10th, 2003. The millimeter points are from Osterloh and Beckwith (1995) and Andrews and Williams (2005).
The input spectrum has a temperature of 5750K and a g of 5.0. The model for XZ
Tau consists of a halo extending from 0.1AU to 60AU and is composed of 50% DL- silicates and 50% DL-graphite. No additional dust grain types were needed. The grain size distribution is 1.0 – 25µm. A large grain disk extends from 0.03AU to 30 AU. The
10µm silicate feature is quite weak and round and is indicative of grain growth, which is matched by the model’s minimum grain size. The large grain wafer disk dominates at millimeter wavelengths and is consistent with the grain growth shown by the small
140 silicate feature. However, more millimeter data would be needed to confirm the fit of a large grain disk.
This object is an extreme example of the grain growth/dust settling model. As there is no IRAS data, this object is not used in the correlation studies.
Grain Growth and Settling Correlation
In several of the objects discussed above, grain growth and millimeter SED data slope are mentioned together. For many of the objects, grain growth, evidenced by the presence of a rounded and smaller 10µm silicate feature, and disk settling, evidenced by a shallower millimeter data slope, occurred together in the same object. It is interesting to show this graphically. The following plot shows the relationship between the 10µm silicate band/continuum ratio and the spectral index of the millimeter region for several objects. The band/continuum ratio is the ratio of the 10µm silicate strength with the underlying spectral continuum. The spectral index of the millimeter data was found by finding the power law that fit the data. Many times this could simply be done graphically.
Objects were rejected if there was no millimeter data or if there were known contamination issues with the millimeter data or the 10µm spectral region. Other objects were rejected if the band/continuum ratio could not be calculated or if only an upper limit on the ratio could be found.
Most of the objects are clustered along the fit line, showing that grain growth and disk settling are correlated with each other to some extent. However, a few objects, LkCa
15, RY Tau, UX Ori, and to some extent GM Aur do not. These stars show large silicate features combined with low spectra indexes indicative of evolved, settled disks. In
141 general, a star probably would move downwards and to the right along the fit line on the graph as it evolved. However, it is possible that grain growth and disk settling do not have to occur at the same rate. One method to explain the placement of the ‘outlier’ stars on the plot could be if disk material is gravitationally scattered by protoplanetary objects forming in the disk. This would remove disk material, thus making the disk appeared more ‘settled’ while not significantly changing the average dust grain size and the silicate band strength. Also, protoplanets would move material out of the disk and into the surrounding halo. This movement of material could cause collisions and increase the amount of small grains in the system. Thusly, a larger silicate band would appear in the
SED even though a large grain wafer disk dominates the millimeter region. Much further study is needed to begin to answer these sorts of questions.
Figure 3-56: The correlation of the 10µm silicate band/continuum ratio and spectral index of the millimeter SED region for selected objects.
142 Disk Inner Wall and Self Shadowing Correlation Study
The addition of blackbodies to some of the fits attempts to model a possible puffed up inner disk rim. This puffed up inner rim would shadow part of the disk behind it. It is interesting to see if there is any sort of correlation between the amount of starlight the blackbody blocks and flux ratios from the IRAS data. For example, most of the flux for the 12µm IRAS point (and to lesser extents, the 25µm and 60µm points) would come from the region of the disk that would be shadowed in these types of systems. Therefore, this would imply anticorrelation between the percentage of starlight blocked and the flux ratio of 12µm/25µm as an example.
Not all of the objects are included in the study. A few, mentioned within their entries either have no IRAS spectra or have contaminated IRAS spectra, usually by nearby nebulae emission, and are not included. A few other objects, ones with large inner gaps such as HD 98800, are also not included as the IRAS ratios for these objects could have low ratios because inner disk material was simply not there as opposed to only being shadowed. These objects are DM Tau, GM Aur, HD 98800, HD 141569, Hen 3-
600, LkCa15, and TW Hya.
Also, four objects, HD 190073, HD 45677, HD 50138, and HD 58647 all are
“questionable” in that their IRAS ratios may be deeply affected by factors other than shadowing such as advanced disk settling, and small thick halos. Observing the SED’s and noting that the IRAS data has a significantly different slope from most of the other objects shows that these objects may not be quite the same as most of the other objects.
Therefore, each correlation plot has been completed twice, one with the four questionable
143 objects included and one without. Differences between the two will be noted in the discussion that follows.
Each plot is constructed from the log of the appropriate IRAS flux ratio vs. the log of the % of unobscured starlight. Linear fits with weighted errors for both the log of the ratios and log of % unobscured starlight are completed. The errors on the ratios were calculated by propagating the errors on the IRAS measurements themselves. The errors on the blackbody blockages were estimated at 1% of the blockage value. The slope of the linear fits is listed on each plot.
144
Figure 3-57: 12µm/25µm correlation plot showing all selected objects.
Figure 3-58: 12µm/25µm correlation plot with “questionable” objects removed.
145
Figure 3-59: 12µm/60µm correlation plot showing all selected objects.
Figure 3-60: 12µm/60µm correlation plot with “questionable” objects removed.
146
Figure 3-61: 25µm/60µm correlation plot showing all selected objects.
Figure 3-62: 25µm/60µm correlation plot with “questionable” objects removed.
147
Out of the three pairs of plots, only the 12µm/25µm pair of plots shows little significant difference in the linear fit after removing the four questionable objects. The other two pairs show a large change in the slope fit. This is because the IRAS ratios, the
12µm/60µm and 25µm/60µm ratios, of the four objects are larger than that of most of the other objects and thus skew the fit upwards. However, the change in the fits after removing the questionable objects seems to pivot more around the middle cluster of stars.
HD 135344 and CW Tau do not have enough weight to pin the fit in place at the far left end. In all three sets of plots, the general shape is the same.
Unfortunately, though a higher amount of shadowing indicates smaller ratios, a smaller ratio does not always indicate a high amount of shadowing. This would mean that mechanisms other than shadowing could produce smaller ratios, such as planet formation cleaning out material from some portion of the disk or extra material in the outer regions increasing the flux at higher wavelengths.
The Whitney code model for HD 163296 did not require an added blackbody whereas the DUSTY model did. This means that the wafer disk model is too geometrically thin to explain the NIR flux and it is likely that the disk for this object is flared instead of a wafer.
For most of the objects that have addition blackbodies in their models, the blackbody could represent a puffed up inner disk wall or a ‘chunk’ of material that has broken off and happens to be in the line of sight at the moment of the observation. It could also be a combination of the two. In a system with a puffed up inner wall, the disk will experience shadowing in its mid-IR region thus creating a flux deficit. In a system
148 with a line of sight chunk, this disk would not experience any significant shadowing from the chunk, though the system may still be shadowed if it also has a wall. To complicate things further, a system may have a mid-IR flux deficit because it actually has little material in that region to emit, the material cleared out possibly by a forming planet. HD
135344 (SAO 206462) may be an example. Thus, systems with a wall and a shadowed disk would have similar SED’s as a system with a line of sight chunk and a cleared out mid disk region. Finally, it is entirely possible that BOTH geometries could be happening at once in the same system as could possibly be happening in HD 135344.
Methods of separating these various situations are mostly observational in nature and would require much more time related data than presently exists. A few objects, such as HD 163296 and HD 31648 have enough NIR data to see changes in that spectral region (discussed more fully in the Whitney code results section), but there is not the equivalent IRAS data at the same times to determining the change in the mid-IR to see any correlations. Imaging could also be an aid if an actual gap in the disk could be observed, though the gaps that would produce the above mentioned deficits would have to be on the order of 10 AU from its star. Imaging disks that close to its parent star is quite problematic because of the overwhelming brightness of the star itself.
Unfortunately, little more can be said about the detailed structure of the disks from these models.
149 Chapter 4: Whitney Code Fits and Results
The general picture of disk evolution today is one where grain growth and setting into the disk with increased disk age/evolution will produce a set of correlated changes in disk properties. With increased grain size, the spectral energy distribution at millimeter and sub-millimeter wavelengths will become less steep as the grains behave more and more like blackbodies without any steepening due to wavelength-dependent emissivity.
Increased grain sizes will also reduce the strength of the silicate bands, which become weak when the mean grain sizes exceed the wavelength of grain emission.
Dust settling will manifest itself in a change in disk thickness and flaring. With increased settling, the disk will become thinner, and depending on the details of the accretion, the disks may evolve from being flared concave upward, as is needed to explain the SEDs of Class I sources, to ‘flat’ (no flaring – conical) to flared convex upward (‘anti-flared’).
In this chapter we investigate the relationships among these parameters for 6 objects. AB Aur is a ‘classical’ Herbig star still embedded in nebulosity, and very young.
HD 31648 and HD 163296 are Herbig stars without nebulosity, presumably older, yet associated with known star-forming regions (Taurus-Auriga T association, the the Sco-
Cen OB association). HD 100546, HD 135344 (SAO 206462), and HD 169142 are not associated with any known star-forming regions. Their ages are not known precisely, due to the difficulty of age determination for young A-F stars, but are presumably older than the first three stars listed. HD 31648 and HD 163296 were chosen because of their data variability while the others were chosen for comparison because they are mostly older than the above two objects.
150 The data used to construct the observational SED for each object is the same as that used for the DUSTY models. The origins of the data are repeated in this section for completeness. Input Kurucz models were chosen to be the same temperature and surface gravity as those used in the DUSTY models.
Each model uses one or more of the three pre-calculated dust input files discussed in the Whitney code description section of the background chapter. For convenience, the grain files will be listed here again. They are, “small” grains from 0.005 to 0.25 microns
(Kim et al. 1994), “medium” grains up to 20 microns (Cotera et al. 2001), and “large” grains up to 1000 microns (Wood et al. 2002).
The process of fitting each model followed a trial and error system until a reasonably good fit was found. No χ2 fit was attempted as the run time for individual final smooth fits could last days.
The fits fall into two types. HD 31648 and HD 163296 are fit with small-inner radius disks, small envelopes, and no blackbodies, such as those used in some of the
DUSTY models. HD 100546, SAO 206462, and HD 169142 are fit with large inner- radius disks, no envelopes, and NIR blackbodies. In addition, a Whitney model of AB
Aur is discussed for comparison between more evolved disk systems, such as the five objects above, and less evolved disk/large envelope systems such as AB Aur.
Under each entry, the SED fits are shown along with a short discussion of the model’s parameters and observation SED characteristics that helped to determine the model parameters. Additional plots and short discussions follow, including plots of the temperature and density distributions of the objects, radial surface brightness plots and dust input file location plots. The temperature, density and dust file plots are made with
151 pre-generated IDL routines using polar coordinate output files from the Whitney code itself. The surface brightness plots are made taking a Whitney code x-y image file output and inputing it into a self-constructed MS-Excel sheet. The spreadsheet performs the necessary calculations, such as integrating around radially increasing annuli, to find the surface brightness vs, radius values. The Excel sheet also finds the best power law fit to the surface brightness. The values and fits are inputed into an IDL routine to produce the final plots.
Follow is a table listing the important model parameters for each of the six objects. HD 31648 and HD 163296 have two models each to model the two different flux states that their data displays (discussed in their relevant sections), some parameters contain two values separated by a ‘/’. The number before the slash refers to the lower flux model while the number to the right refers to the higher flux model.
Table 4-1: Whitney Model Parameter List for HD 31648, HD 100546, and HD 135344.
Parameter (units) HD 31648 HD 100546 HD 135344 Kurucz input file t8250g4 kt10500g5 5500g4.5/6250g4.5 (50/50) Disk midplane dust file medium large large disk dust file medium large large envelope dust file small star radius (Solar radii) 1.7 1.5 1.9 star temp (K) 8200 10500 6000 star mass 2 3.5 1.36 disk accretion rate (solmass/yr) 4.00E-09 7.90E-08 8.00E-08 disk mass (solar masses) 0.8 0.006 0.009 disk outer radius (AU) 250 100 155 disk inner radius (stellar radii) 25.04/31.3 2652 6720 disk inner radius (sublimation radii) 0.8/1 51 420 disk inner radius (AU) 0.20/0.25 18.30 58.73 disk height at Rstar (stellar radii) 0.14/0.21 0.071 27 A- disk density exponent r^(-A) 1.6 2.06 1.31 B- disk height exponent r^(B) 0.6 1.06 0.31
152 Disk scale height (Stellar radii) 0.97/1.66 302.17 414.81 Disk scale height (AU) 0.007/0.013 2.08 3.63 envelope max radius (AU) 10 envelope min radius (Stellar radii) 31.3 envelope min radius (sub radii) 1 envelope min radius (AU) 0.24 env mass infall rate (solar mass/yr) 7e-7/1.2e-6 inner cavity wall opening angle 10 sublimation radius (stellar radii) 31.3 52 16 accretion luminosity % 0.020% 0.004% 0.001% % flux hitting envelope+disk 39%/53% 41% 18.0% % flux scattered 13%/17% 15% 6.0% Mag Co-rotation radius (stellar radii) 15.0 Stellar Hotpot Fraction 0.0002 Hotspot Lum fraction 1.1% Envelope Mass (solar masses) 4e-8/6e-8 d (parsecs) 140 110 140 Av 0.05 0 0.31 Blackbody Temp 1500 1075 BB lambda power -0.7 -0.7 BB scaling factor 2.00E-16 5.00E-16 BB blockage 14% 23% Dusty Disk Add- Inner Edge (AU) Dusty Disk outer edge (AU) Ldisk/Lstar Dusty Halo Add- Inner Edge (AU) Dusty Halo outer edge (AU) Dusty Halo radial density dropoff Lhalo/Lstar Surface Brightness (r^) -3.1 / 1.14µm -3.05 / 1.14µm -9.66 / 1.14µm
Table 4-2: Whitney Model Parameter List for HD 163296, HD 169142, and AB Aur.
Parameter (units) HD 163296 HD 169142 AB Aur Kurucz input file kt8750g35 t4750g5/7750g5 (50/50) t9750g35 Disk midplane dust file large large medium disk dust file small large small envelope dust file small small star radius (Solar radii) 1.7 3 2.4 star temp (K) 8750 6300 9750 star mass 2 1.5 2.45 disk accretion rate (solmass/yr) 8.00E-08 7.00E-10 1.00E-09 disk mass (solar masses) 0.4 0.015 0.2 disk outer radius (AU) 450 230 450 disk inner radius (stellar radii) 37.17/44.25 3186 44.50 disk inner radius (sublimation radii) 1.05/1.25 180 1.00
153 disk inner radius (AU) 0.29/0.35 43.97 0.50 disk height at Rstar (stellar radii) 0.043/0.054 0.03 0.05 A- disk density exponent r^(-A) 1.99 2.065 2.06 B- disk height exponent r^(B) 0.99 1.065 1.06 Disk scale height (Stellar radii) 1.54/2.30 161.46 2.79 Disk scale height (AU) 0.012/0.018 2.2300 0.03 envelope max radius (AU) 10 460.00 envelope min radius (Stellar radii) 106.2 890.00 envelope min radius (sub radii) 3 20.00 envelope min radius (AU) 0.83 10.03 env mass infall rate (solar mass/yr) 7.00E-07 1.70E-07 inner cavity wall opening angle 20 30 sublimation radius (stellar radii) 35.4 17.7 44.5 accretion luminosity % 0.020% 0.013% 0.001% % flux hitting envelope+disk 35%/38% 19% 50% % flux scattered 10%/11% 6.3% 13% Mag Co-rotation radius (stellar radii) 3.00 Stellar Hotpot Fraction 0.02 Hotspot Lum fraction 11.0% Envelope Mass (solar masses) 2.60E-08 d (parsecs) 122 155 144 Av 0.1 0.15 0.3 Blackbody Temp 1500 BB lambda power 0 BB scaling factor 7.50E-10 BB blockage 5% Dusty Disk Add- Inner Edge (AU) 0.15 Dusty Disk outer edge (AU) 40 Ldisk/Lstar 0.005 Dusty Halo Add- Inner Edge (AU) Dusty Halo outer edge (AU) Dusty Halo radial density dropoff Lhalo/Lstar Surface Brightness (r^) -3.1 / 1.14µm -3 / 1.14µm
154 HD 31648
Figure 4-1: The 1996 SED and Whitney model of HD 31648 showing 0˚, 40˚, and 90˚ disk inclinations.
Figure 4-2: The 2004 SED and Whitney model of HD 31648 showing 0˚, 40˚, and 90˚ disk inclinations.
155
Figure 4-3: Comparison of the 40˚ disk inclination fits of HD 31648 from 1996 (solid line) and 2004 (dashed line).
HD 31648, also known as MWC 480, is an A5 type star that is ~140pc away
(Grady et al. 2005). The data for HD 31648 consists of IRAS, 2MASS, and IUE Atlas spectra. The BASS spectrum plotted was taken on October 14th, 1996. The UBVRI and
JHK data come from the Export group. LRSP data is also plotted. The millimeter data come from Mannings et al. (1997) and Piétu et al. (2006). The star has age estimates from
2.5 Myr old (van den Ancker 1998) to 8.0 Myr old (Simon et al. 2001). Blondel and Djin
A Djie (2006) also estimate the age to be ~6 Myr old. The input spectrum has a temperature of 9250K and a log g of 4.0.
156 The Whitney model for HD 31648 consists of a convex upwardly flared disk, flaring parameter B = 0.6. The disk consists of medium grains and extends from 0.2 –
250AU. The disk mass is 0.8 solar masses. The model also contains an envelope “shell” massing 4x10-8 Solar masses that extends from 0.25 – 10AU and is made up of small grains. The shell has an opening angle of 10˚ from vertical. The best model disk inclination angle is 40˚. Pietu at al. (2006), from millimeter observations, finds a disk extending to ~190 AU and inclined at 36˚.
To obtain a reasonable fit of the 10µm silicate feature, the slope of the IRAS data, and the slope of the millimeter data simultaneously, it was necessary to use a medium grain anti-flared disk with an attached small halo. Large grains produced too shallow of a slope in the millimeter and so only medium grains could be placed in both the disk midplane and atmosphere. The anti-flared nature of the disk and medium grains indicate some amount of grain growth and settling. The addition of the envelope was necessary to provide sufficient flux in the 10µm region to fit the silicate feature. The small envelope’s emission and scattering reheated the disk, mostly in the far IR and millimeter regions.
The reheating helped to determine the exact flaring parameter necessary for a good fit to the SED because disks with larger flaring produced too much flux in the far IR and millimeter.
The 1996 and 2004 BASS data show differences. To fit these differences, the disk scale height and inner radius and envelope mass were modified from the 1996 model parameters to determine the necessary inputs to fit the 2004 data. The inner radius of the disk model was pushed out to 0.25AU from 0.20AU and its scale height increased to
0.007 from 0.013AU. Also, the envelope mass was increased from 4x10-8 Solar masses to
157 6x10-8 solar masses. Thus the inner edge of the disk model moved farther outward as the disk also grew in height. The 10µm silicate feature flux increase was greater than the model flux increase due to the disk parameter differences. Therefore it was necessary to increase the envelope mass to produce the extra flux. However more time dependent data, especially observations during another ‘high’ period, across a larger portion of the spectrum would be needed to help determine if the disk grew in thickness or a piece of it broke off, or both. More discussion on the inner disk changes follows the entry for HD
163296.
Attempts to fit the SED with a disk alone failed to fit the three spectral regions,
NIR, 10µm silicate, and far IR, at the same time and would only usually be able to fit one or two of the three regions.
Figure 4-4: Surface brightness plot of HD 31648.
158 The surface brightness of this HD 31648 model has a radial dependence of r-3.1.
This does not match the latest observations of ~r-5.0 (C.A. Grady, private communication). Attempts were made to fit the steeper surface brightness dependence, however it was not possible to find a model that both fit the SED and produced such a steep surface brightness. Though a steeper surface brightness was modeled for another object, r-9.6 for SAO 206462, that model consisted of an anti-flared disk with a large inner radius and no envelope. That sort of geometry allows steep surface brightness dependences because of the self-shadowing nature of the anti-flared disk. The large inner hole also opens up the flaring parameter space that makes fine-tuning of such steep surface brightness dependences possible. When an envelope is added to the model, the envelope’s reheating has the effect of flattening out the surface brightness dependence of the disk making it impossible for the Whitney model to produce a surface brightness steeper than ~r-3.4 when the model contains an envelope. In the case of HD 31648, the steepest surface brightness possible from a simultaneous SED fit was the surface brightness plotted above, with a dependence of r-3.1.
The combination of spectral shapes of the NIR and 10µm silicate features and the necessity of an envelope component in the model determined the necessary flaring parameter and envelope mass. Thusly the surface brightness could not be fit and thus the
Whitney flared disk model is insufficient to explain the apparently more complex circumstellar dust system of HD 31648.
159
Figure 4-5: Temperature and density cross-sections of HD 31648 at three different radial views. Color indicates temperature or density and progresses from lowest to highest: blue, red, orange, and yellow. Both the radial and vertical distance axes are in units of
AU.
From the plot above, the anti-flared disk contains the highest density of material, especially towards its midplane. The envelope does not have a high density of material and has roughly ten orders of magnitude less dense than the disk midplane. However the envelope contains much hotter material, ranging from ~500-700K, than most of the disk,
<100K for much of its material. This is because the envelope receives the full brunt of the starlight. The star only shines directly on the thin inner wall of the model disk and the
160 exposed disk surfaces. The rest of the disk is illuminated by emission from other sources such as the envelope or the surfaces of the disk and is consequently cooler.
Figure 4-6: The HD 31648 Whitney dust file model locations in the circumstellar geometry. The yellow and blue regions contain ‘medium’ grains while the orange region consists of ‘small’ grains.
161 HD 163296
Figure 4-7: The 2005 SED and Whitney model of HD 163296 showing 0˚, 50˚, and 90˚ disk inclinations.
Figure 4-8: The 2002 SED and Whitney model of HD 163296 showing 0˚, 50˚, and 90˚ disk inclinations.
162
Figure 4-9: The SED and Whitney model of HD 163296 showing 0˚, 50˚, and 90˚ disk inclinations. This SED is constructed by adding an additional ‘wall’ component to the
2005 model.
163
Figure 4-10: Comparison of the 50˚ disk inclination fits of HD 163296 from 2005 (solid line) and 2002 (dashed line).
HD 163296 is an A3 type star that is 122pc away (Grady et al. 2005). The data for
HD 163296 consists of IRAS, 2MASS, and LRSP data. The BASS spectrum was taken in
2005. The UV data is from the IUE Atlas and the millimeter data is from Mannings
(1994), Henning et al. (1994) and Henning et al. (1998). The UBVRI data is from de
Winter et al. [1997]. The bright blue and pink data in the visual and infrared is a spectrum taken at Lick Observatory in 2002 and 2005. The input spectrum has a temperature of
9250K and a log g of 5.0. The star’s age is estimated to be from 4.0 Myr (van Boekel et al. 2004) to 8.0 Myr old (van den Ancker et al. 1998).
The Whitney model for the observations of HD 163296 in 2005 consists of a marginally convex upwardly flared disk, flaring parameter B = 0.99, extending from 0.3
164 – 450 AU. The disk has a half height thickness at its inner edge of 0.012 AU. The disk mass is 0.4 solar masses. An envelop of mass 2.6x10-8 solar masses extends from 0.8 – 10
AU. The shell has an opening angle of 20˚ from vertical. The disk midplane contains large grains while the rest of the disk and the envelope contain small grains. The slope of the millimeter data requires the large sized grains in the midplane. The size and shape of the 10µm silicate feature requires the small grains to be placed in the envelope and the atmosphere of the disk. The best model inclination is 50˚. Grady et al. (2000) observes a disk extending to ~450 AU and is inclined at ~50˚.
In 2002, HD 163296 exhibited excess NIR flux compared to that of 2005. Two methods were used to model the increased flux. First the Whitney 2005 model parameters were changed until the 2002 SED was fit. The model changed by increasing the half height of the disk inner edge to 0.018 AU from 0.012 AU and increasing the disk inner radius to 0.35 AU from 0.30 AU. Since the 10µm silicate feature showed little extra flux over the 2005 data, the disk parameter changes were sufficient to model the differences between the two data sets and unlike for HD 31648, the envelope mass did not need to be increased. Thusly, the inner edge of the model disk moved out slightly while the disk itself grew in thickness.
The second method involved subtracting the 2005 flux from the 2002 flux and fitting the difference of the two spectra with an emission spectrum calculated from Mie theory. The details on the exact grain types, sizes and temperatures are described in Sitko et al. (ApJ, in press).
Further discussion on the inner disk changes follows this section.
165
Figure 4-11: Surface brightness plot of HD 163296.
The observed surface brightness radial dependence of the model of HD 163296 is r-3.1 and is close to the observed relation of r-3.0 observed beyond 2.2 arcsec = 270 AU
(Wisniewski et al. 2007). However, the observed surface brightness for HD 163296 varies over the radial distance whereas the Whitney model’s does not. This means that the actual structure of HD 163296’s circumstellar material is more complex than that of the Whitney model.
166
Figure 4-12: Temperature and density cross-sections of HD 163296 at three different radial views. Color indicates temperature or density and progresses from lowest to highest: blue, red, orange, and yellow. Both the radial and vertical distance axes are in units of AU.
The temperature and density profiles for HD 163296 are shown above. The disk’s density is highest at the inner edge and in the disk’s midplane. The density drops off in the characteristic teardrop shape when going to large radius. Much like HD 31648, HD
163296’s envelope density is rough ten orders of magnitude less dense than that of the disk midplane. However, the envelope, because it receives much more direct starlight, is much hotter than the disk. The envelope is ~500-700K while most of the disk is <100K.
The model disk is hottest directly at its inner edge and also along its surface. The envelope, through its own thermal emission, helps to heat the disk and is one of the
167
Figure 4-13: Temperature density plot of the HD 163296 model with and without the model envelope. Radial and vertical scales are in units of AU.
factors determining the exact flaring parameter of the disk when fitting the SED. As can be seen in the above plot, when the envelope is removed from the model, the disk is correspondingly cooler. Most of the internal disk without the reheating of the envelope is
<50K whereas the envelope reheating increased it to <100K. A similar effect occurs in
HD 31648.
The following plot shows the locations of the dust file regions in the HD 163296 model. The yellow region depicts the disk midplane while the blue is the disk atmosphere. Orange depicts the envelope.
168
Figure 4-14: The HD 163296 Whitney dust file model locations in the circumstellar geometry. The yellow region consists of large grains and the blue and orange regions are made up of small grains.
Inner Disk Analysis
In both HD 31648 and HD 163296, observations show changes in the NIR and the
10µm silicate feature. HD 31648 showed increased fluxes in these regions in 2004 over its 1996 levels. HD 163296 showed increased flux in 2002 over its 2005 levels. The
SED’s of both epochs for both objects were fit. Using the lower flux levels (1996 for HD
31648 and 2005 for HD 163296) as a baseline, fitting the increased flux levels required
169 that the Whitney disk inner edge move farther out and an increase in the disk’s scale height parameter. Also, in HD 31648, the mass of the envelope shell was increased in the model to fit the SED. The specific details of the model changes are explained in each objects’ entry.
Figure 4-15: Close-up comparison in the NIR of the 30˚ disk inclination fits of HD
31648 from 1996 (solid line) and 2004 (dashed line).
170
Figure 4-16: Close-up comparison in the NIR of the 50˚ disk inclination fits of HD
163296 from 2005 (solid line), 2002 (dashed line), and 2005+Wall (red dashed line). The
Wall component is plotted as the dark solid line towards the bottom of the plot.
The above plots show close ups of the spectral region related to the inner disk changes. In HD 31648, the combination of increased inner edge radius, disk height, and envelope mass produces the dashed line that fits the higher flux data. In HD 163296, the combination of an increase in the disk inner edge radius and height results in the blue dashed line fitting the higher flux data. A second line fitting the high flux data, the red dashed line, is the result of adding the lower solid black line, called the wall, to the 2005 fit, which is the solid blue line fitting the low flux data. The wall component could represent some portion of the disk that has broken off and is in the process of being ejected (Vinkovic and Jurkic 2007). The particulars of the wall component have been mentioned above in the HD 163296 entry and are detailed in Sitko et al. (ApJ, in press).
171 These changes in the model fits indicate that the inner edge of the disk moved outward as the disk slightly increased in thickness. A physical explanation for this could be the result of a lowering of the optical depth of gas inside the inner edge of the disk
(which the Whitney model does not model) thus allowing more starlight to shine on the inner edge of the disk. This increased stellar flux would heat up the disk evaporating a portion of the inner disk, resulting in a larger inner edge radius, and would also puff up the disk at the new inner edge. The following diagram shows an example of this.
Figure 4-17: Diagram of a model of inner disk structure changes due to increased starlight illumination.
The red section represents the gas inside the inner radius of the dusty dusk while the green is the disk itself. In the second diagram, the vertical scale of the red region has changed allowing more starlight to strike the disk itself. The blue portion represents the piece of disk that has evaporated due to the increased heat of the extra starlight. The disk itself has also increased in thickness due to the extra heating.
172 The Whitney model is only able to approximately describe this change because increasing the model’s disk height actually increases the disk thickness along its entire radial extent, whereas the actual disk would only increase its thickness close to the star.
This has been depicted in the diagram by overlaying a lighter colored thickened disk, representing the ‘real’ disk change, over the top of the dark green thickened disk that represents the Whitney disk change.
The 10µm silicate feature flux in HD 31648 increased more than the disk changes alone produced, the small envelope mass in the model had to increase as well. This may indicate that both processes are occurring in this object. The some portion of the disk evaporated and expanded in height while other pieces broke off and are in the process of ejection.
173 HD 100546
Figure 4-18: The SED and Whitney model of HD 100546 showing 0˚, 10˚, and 90˚ disk inclinations.
HD 100546 is a B9.5 star and is ~103pc away (Grady et al. 2005). The data for
HD 100546 consists of IRAS photometry, IUE Archive data in the UV, and spectra from the Short Wave and Long Wave Spectrometers on the ISO. JHKLM data comes from
Malfait et al. (1998) and UBVRI data, taken in 1992, comes from de Winter et al. [1992 and 1994]. Millimeter data is from Wilner et al. (2003) and Henning et al. (1994). Also,
ISO data from Sloan et al. (2003) is included. HD 100546’s age is thought to be greater than 10.0 Myr old (Van Boekel et al. 2004 and van den Ancker 1998). The input spectrum has a temperature of 10500K and a log g of 5.0.
174 The Whitney model for HD 100546 consists of a flared disk of mass 0.006 solar masses with flaring parameter B = 1.06, extending from 18-100 AU. The disk’s half scale height at its inner edge is 2 AU. This height is much larger than the model disk scale height of HD 31648 and HD 163296 because the inner edge of the model disk for HD
100546 starts much farther out than in the previous two objects, 18 AU as oppose to 0.3
AU. Large sized grains are used throughout the disk and are necessary to fit the sub- millimeter and millimeter data. The model does not contain an envelope. A blackbody of
1500˚K and multiplied by λ-0.7 is added to fit the 1-8µm region. The blackbody blocks off
14% of the star’s light, resulting in a half angle height of 13˚. The lack of a 10µm silicate feature and the shallow slope into the millimeter region is indicative of grain growth and is supported by the use of large grains in the model. The inclination angle of the model disk is 10˚. Grady et al. (2001) observes a disk ranging from ~200 – 350 AU depending on the wavelength measured. Also, the observed disk has an inclination of ~35˚.
The Whitney model did not contain dust grain input files capable of fitting the complex crystalline features longward of 15µm. For the above plot, a DUSTY model, using the same material composition as the HD 100546 DUSTY model, is added to a modified Whitney model of HD 100546. Decreasing the disk scale height by a factor of four modified the Whitney model such that adding in the DUSTY halo model’s crystalline features resulted in a good fit. Since the DUSTY halo is mathematically equivalent to the atmospheric portion of a Whitney disk, this modification could be thought of as a brute force method of adding materials to a Whitney model.
175
Figure 4-19: Surface brightness plot of HD 100546.
The surface brightness radial dependence of the disk portion of the HD 100546 model is r-3.05 which matches well with the observed r-3.1 (C. Grady, private comm.). The observed surface brightness is measured beyond ~0.3 arcsec (30 AU) and is matches with that portion of the model surface brightness. Since the model matches both the observed surface brightness and SED, it is good evidence that the disk structure in that radial region, corresponding to ~50+ AU, is a slightly concave upward flared (flaring parameter
B=1.06) Whitney type disk.
176
Figure 4-20: Temperature and density cross-sections of HD 100546 at three different radial views. Color indicates temperature or density and progresses from lowest to highest: blue, red, orange, and yellow. Both the radial and vertical distance axes are in units of AU.
The temperature and density plots show the expected type of response for a flared disk. The inner edge of the disk is the hottest along with the surfaces where the temperatures range around 200K. Portions of the disk deeper inside are cooler. The density of the disk shows the characteristic teardrop shape of higher density at the midplane and towards the inner edge.
The plot below displays the placement of the dust files. Both the yellow midplane and blue disk atmosphere regions contain large grains. This is interesting in that the
177 crystalline features in the SED require small grains. The crystalline small grains and the fat disk shape indicate this system has undergone little grain growth or disk settling, which contradicts the existence of the large grains needed to fit the millimeter data.
Therefore, this system is not following the standard model of coupled grain growth and dust settling. Something more complex is happening in this system.
Figure 4-21: The HD 100546 Whitney dust file model locations in the circumstellar geometry. Both the yellow and blue regions consist of large grains.
178
HD 135344 (SAO 206462)
Figure 4-22: The SED and Whitney model of HD 135344 (SAO 206462) showing 0˚,
10˚, and 90˚ disk inclinations.
HD 135344, also known as SAO 206462, is a F4 type star 140pc away
(Grady et al. 2005). The data for HD 135344 includes IRAS and 2MASS photometry as well as an IRS spectrum from Spitzer Space Telescope. The BASS spectrum was taken on July 9th, 2007. UBVRIJHKL’M comes from Sylvester et al. (1996). The millimeter
179 data is from Sylvester et al. (1994) and Walker and Butner (1995). Also, data (L.
Beerman, private communication) from the SpeX instrument (Rayner 2003) is plotted.
The data just shortward of 100µm is from the Multiband Imaging Photometer for Spitzer
(MIPS) courtesy of P.S. Smith from University of Arizona. UV data is from Grady
(private communication). This object’s age is estimated at 8.0 Myr old (van Boekel et al.
2004). The input spectrum for this object consists of a 50/50 mixture of two input stellar spectra, 5500K and 6250K both with log g’s of 4.5.
The Whitney model of HD 135344 consists of a convex upward flared disk, with flaring parameter B = 0.31, extending from 58-155 AU and has a mass of 0.009 solar masses. The disk has a half scale height at the inner radius of 3.6 AU. Like HD 100546 above, the large inner radius for this model disk results in a large inner edge disk height.
Large sized grains are used throughout the entire disk. The model does not contain an envelope. A blackbody of 1075˚K and multiplied by λ-0.7 is added to fit the 1-8µm region.
The blackbody blocks off 23% of the star’s light, resulting in a half angle height of 21˚.
The lack of a 10µm silicate feature and the shallow slope of the millimeter data indicates that grain growth and disk settling has occurred. The use of large grains in the model and a very flattened convex upwardly flared disk agree with the observations. The disk model inclination is 10˚. Dent et al. (2006) observed an inclination angle of 11˚ for this object.
The observed disk radii is not as constrained, one estimate being ~210 AU (Doucet et al.
2006).
180
Figure 4-23: Surface brightness plot of HD 135344 (SAO 206462).
The surface brightness radial dependence of the anti-flared disk of HD 135344 is r-9.66 which matches with the observed dependence of r-9.6. (C.A. Grady, private communication)
181
Figure 4-24: Temperature and density cross-sections of HD 135344 (SAO 206246) at three different radial views. Color indicates temperature or density and progresses from lowest to highest: blue, red, orange, and yellow. Both the radial and vertical distance axes are in units of AU.
The temperature plot of the disk of HD 135344 shows the normal temperature distribution for an anti-flared disk. The inner edge and surfaces have the highest temperature of the circumstellar material, ~50K. The interior of the disk is cooler. The saw tooth shape of the upper and lower edges of the disk are a result of the flatness and geometrical thickness of the disk and the grid size of the outer disk regions of the
Whitney code plotting routine. The density plot shows the normal density distribution of
182 this type of disk. The inner edge and midplane of the disk contain the highest density of material.
The disk density is such that the midplane density is never reached. Therefore, the entire model disk uses the disk atmosphere dust file and no dust file plot is shown.
HD 169142
Figure 4-25: The SED and Whitney model of HD 169142 showing 0˚, 30˚, and 90˚ disk inclinations plus added DUSTY halo component. Dotted lines are the Whitney model fits without the additional DUSTY component.
183
Figure 4-26: Close-up of the SED and Whitney model of HD 169142 showing 0˚, 30˚, and 90˚ disk inclinations plus added DUSTY halo component. The dashed lines indicate the model SED’s without the added DUSTY Halo component. Dotted lines are the
Whitney model fits without the additional DUSTY component.
Figure 4-27: The SED Whitney model of HD 169142 showing 0˚, 30˚, and 90˚ disk inclinations plus added DUSTY wafer disk component. Dotted lines are the Whitney model fits without the additional DUSTY component.
184
Figure 4-28: Close-up of the SED and Whitney model of HD 169142 showing 0˚, 30˚, and 90˚ disk inclinations plus added DUSTY wafer disk component. The dashed lines indicate the model SED’s without the added DUSTY wafer component. The green dashed line along the 40 AU wafer line shows the SED of the modeled DUSTY disk when face on. Dotted lines are the Whitney model fits without the additional DUSTY component. The dashed line just above the 40 AU wafer shows the flux of the wafer at 0˚ inclination.
HD 169142 is a A5 type star 145pc away (Dent et al. 2005). The data for HD
169142 consists of IRAS and 2MASS spectra as well as UV data from the IUE Archives.
The millimeter data comes from Sylvester et al. (1996) and the UBVRI data comes from van der Veen et al. (1989). Also, spectra from the PHT on the ISO and SST IRS data are
185 also plotted. In a similar fashion to SAO 206462, the input spectrum for this object consists of a 50/50 mixture of 5750K and 8750K spectra, both with log g’s of 5.0. The star is thought to have an age ranging from 1 Myr old (Grady et al. 2007) to 8.0 Myr old
(van Boekel et al. 2004).
The Whitney model of HD 169142 consists of a marginally flared disk, flaring parameter B = 1.065, that extends from 44 – 230 AU and mass 0.015 solar masses. The disk consists of large grains. There is no envelope in this model. A blackbody of 1500˚K is added to help fit the NIR region and models the inner disk wall located at the dust sublimation radius. The blackbody blocks off 5% of the starlight, resulting in a half angle height of 5˚. PAH features are observed in the 7 - 10µm region and cannot be modeled by the Whitney code. The PAH spectrum of RR Tau, also used in the DUSTY model of that object, is added to the model SED to fit the observed SED. The disk model inclination angle is 10˚. The observed inclination of the disk is < 30˚ while the disk radius is ~200
AU (Grady et al. 2007).
Elements of the DUSTY model were added to attempt further refinements to fitting the SED of HD 169142, especially around the 10µm region (Grady et al. 2007).
The added elements would have to have negligible reheating or outer disk shadowing effects on the Whitney model flared disk. Two models were made, one using a DUSTY
Halo and one using a DUSTY wafer disk. The following diagram illustrates the 2 different models and the three geometries possible from the two models.
186
Figure 4-29: Diagram of physical representation of the added DUSTY disk (a) and halo
(b) components to the Whitney model. (Grady et al. 2007)
One possible model to explain the excess 10µm region emission is an optically thin spherical halo. This model halo component is a DUSTY halo that extends from 0.15
– 25 AU and has a flux of ~1% that of the star. The Halo has a radial density distribution of r-2. With an outer radius of 25 AU, the DUSTY halo is entirely within the radius of the inner edge of the Whitney disk at 44 AU. This halo could represent an optically thin cloud of grains, smaller than those in the flared disk, containing a shadowed grain disk within. Since DUSTY halos are mathematically equivalent to the atmosphere of a flared disk, the DUSTY halo could also model a small grain atmosphere of a possible shadowed inner disk. This conjectured inner disk has negligible flux and little effect on the SED because it is shadowed by the structure modeled by the blackbody component, but would be inferred to be there because of the atmosphere. However, the actual physical existence of the inferred inner disk in either the halo or atmospheric case cannot be proved.
187 The DUSTY wafer component extends from 0.15 – 40 AU and has a flux of 0.5% that of the star. This wafer disk models an evolved inner disk that is made up of very large grains.
In both models, the SED plots above show the effects of the added DUSTY components on the Whitney models plus blackbodies.
Figure 4-30: Surface brightness plot of HD 169142.
The surface brightness radial dependence of the Whitney model of HD 169142 is r-3.0. This agrees with the observational surface brightness of r-3.0. (Grady et al. 2007) The
188 combination of good fits of both the SED and surface brightness lends evidence that the
50+ AU region of the disk is likely to be very similar to a Whitney type disk.
Figure 4-31: Temperature and density cross-sections of HD 169142 at three different radial views. Color indicates temperature or density and progresses from lowest to highest: blue, red, orange, and yellow. Both the radial and vertical distance axes are in units of AU.
The temperature and density plots of the Whitney disk component of HD 169142 show the characteristic profiles for a flared disk. The highest temperatures are on the inner edge of the disk and along its surface and are ~200K. The temperature is lower farther inside the disk. The density of the disk is highest at the inner edge and along the
189 midplane. The density at the midplane inner edge is roughly 10,000 times that of the outer radial edges of the disk.
AB Aur
Figure 4-32: The SED and Whitney model. of AB Aur showing 0˚, 40˚, and 90˚ disk inclinations.
AB Aur is a A0 star in Auriga at a distance of 140pc (Acke and van der Ancker
2004). IRAS, 2MASS, and IUE Atlas data are included in this spectrum. A BASS spectrum was taken on February 9th, 1998. UBVRI data is from de Winter et al. [1993] while the millimeter data comes from Mannings (1994), Mannings and Sargent (1997), and Malfait et al. (1998). The Short Wave Spectrometer (SWS) on board the Infrared
190 Space Observatory (ISO) provided the continuum data from ~3 - 45µm. Finally, UV to visual data (0.33 – 0.58µm) was obtained from the Low Resolution SpectroPhotometer of the Pine Bluff Observatory (http://www.sal.wisc.edu/PBO/). The input stellar spectrum has a T of 9500K and a log g of 4.0. The star is estimated to be 2 Myr old (van Boekel et al. 2004).
The Whitney model for AB Aur consists of a flared disk, flaring parameter B =
1.06, that extends from 0.44 – 450 AU. The half scale height of the disk at its inner edge is 0.03 AU. The disk mass is 0.2 solar masses. The model also consists of an envelope that extends from 9 – 1000 AU and has an opening angle of 30° from the vertical. The envelope has a mass of 7x10-5 solar masses. The disk midplane consists of medium grains while the rest of the system consist of small grains. The disk model inclination angle is
40˚. Corder et al. (2005) observed a disk extending to ~615 AU with an inclination of
~21˚.
AB Aur’s SED is displayed to compare a less evolved system to the more evolved systems shown above. The model for AB Aur consists of a concave upward flared disk
(B=1.06) embedded in a large envelope. The use of smaller grains than those used in the other objects matches the pronounced 10µm silicate feature and steeper millimeter slope.
Comparing to the other objects, the five discussed above use larger grains for most of their geometry and have little or no envelopes. Those systems show more grain growth and disk settling than AB Aur.
The model (using the same code) of Robitaille et al. (2007) did not attempt to fit the BASS data and does not fit quite as well as the model for AB Aur in this work.
191
Figure 4-33: Temperature and density cross-sections of AB Aur at three different radial views. Color indicates temperature or density and progresses from lowest to highest: blue, red, orange, and yellow. Both the radial and vertical distance axes are in units of
AU.
The temperature and density profiles for AB Aur are shown above. The disk’s density is highest at the inner edge and in the disk’s midplane. The density drops off in the characteristic teardrop shape when going to large radius. Much like HD 31648, HD
163296’s envelope density is rough ten orders of magnitude less dense than that of the disk midplane. The envelope of AB Aur has the same relative density to its disk but it is
192 physically much larger and so therefore more massive than the small envelopes of HD
31648 and HD 163296.
The envelope, because it receives much more direct starlight, is much hotter than the disk. The envelope is ~500-700K while most of the disk is <100K. The model disk is hottest directly at its inner edge and also along its surface.
Below is a plot depicting the locations of the dustfiles used in the model of AB
Aur. Small grains are used in the orange and blue regions while the tiny sliver of yellow at the inner edge of the disk midplane consists of medium grains.
Figure 4-34: The AB Aur Whitney dust file model locations in the circumstellar geometry. The yellow region consists of medium grains and the blue and orange regions are made up of small grains.
193 Discussion
The results of the six models can be compared to see how well these six objects fit into the standard model of grain growth and dust settling. As a star grows older, its circumstellar grains would grow in size as well as settle towards the midplane and the disk. These changes would show in the SED’s by a softening and weakening of the 10µm silicate feature as well as flattening of the millimeter spectral slope. Also, the disk itself would flatten from concavely flared upward to convexly flared upward.
Though the ages for the six objects are not very well known, their relative ages can be reasonably presumed and split into three groups. AB Aur is the youngest, HD
31648 and HD 163296 are slightly older, while the last three, HD 100564, HD 135344
(SAO 206462), and HD 169142 are thought to be the oldest of the studied group.
Therefore, it would be expected that AB Aur would have the smallest grains, largest and sharpest 10µm silicate feature, and steepest millimeter slope. The last three objects would have the largest grains, weakest 10µm silicate feature, and shallow millimeter slope. HD
31648 and HD 163296’s parameters would fall somewhere in between. The following qualitative table displays the relevant model parameters for the six objects. The objects are listed by increasing age group. There is no attempt at age ordering within groups.
Grain size refers to the main type of grains used in the model for that object.
194 Table 4-3: Table of Whitney model parameter/age comparisons. The three object age groups have been color-coded. Red for the youngest group, blue for slightly older, and green for oldest. The age of HD 169142 has been averaged between the Grady et al.
(2007) and van Boekel et al. (2004) values. The age of HD 31648 has been averaged between the van den Ancker (1998), Blondel and Djin A Djie (2006), and Simon et al.
(2001) values.
Object Grain size Sil. Band Str MM Slope Log Age Disk Flaring/Thickness AB Aur small 2.21 -4.30 6.3 1.06/fat HD 31648 medium 2.01 -3.66 6.7 0.6/thin HD 163296 large/small 2.11 -2.99 6.8 0.99/medium HD 100546 large 2.04 -3.66 >7.0 1.06/fat HD 135344 large 0 -4.15 6.9 0.3/thin HD 169142 large 0 -3.13 6.5 1.06/medium
Figure 4-35: Plot of silicate band strength/continuum ratio vs. age of the six studied objects. Age Error on HD 100546 has been estimated to be 0.3 dex, similar to the errors of the other objects.
195
Figure 4-36: Plot of millimeter spectral shape vs. age of the six studied objects. Age
Error on HD 100546 has been estimated to be 0.3 dex, similar to the errors of the other objects.
As can be seen in the table and the plots, there is a suggestive relationship of increased grain growth and disk flattening with time. But due to the small sample size and the large error estimates on the ages, it is simply impossible to say much about any correlations. The objects are too scattered, for example HD 100546 is the oldest object in the group, yet has significant silicate band strength. HD 169142 has no discernable silicate feature, but its age places it as one of the younger objects. However, HD
196 169142’s age was found by averaging the ages in the literature and some estimates place its age closer to 10 Myr- which would place that object more in line with the others.
Age correlations are also unclear in the spectral index plot. The object with the flattest mm-index is HD 163296 but it is ‘middle-aged’. The models also do not show a nice, tidy progression from fat concavely flared disks to thin convexly flared wafer-like disks. The model disk of AB Aur is fat and flared concavely, as expected for a younger object, but so is the disk model of HD 100546, which is in the oldest group. HD 100546 also does not have the flattest mm-spectral index as would be expected for the oldest object in the list. Also, the model disk for HD 31648 is very flat despite its silicate band strength and somewhat steeper millimeter slope. Interestingly, a similar correlation study done with a much larger sample found little correlation either (Kessler-Silacci et al.
2006). The non-correlations indicate that disk system evolutionary age has little to do with chronological age.
For these six objects, the disk flaring parameter and thickness do not correlate well with the grain size or millimeter spectral slope. The stars that do not follow the pattern may have something more complex happening within their systems than that allowed by the standard circumstellar evolution models. It is possible that, for these systems, grain growth and disk settling are not as coupled as in the more standard systems.
Further work is needed to construct more complex models.
197 Chapter 5: Conclusions
The goals of this work were to investigate the nature of the dusty disks surrounding a sample of pre-main sequence stars. We were interested in investigating the evolution of these structures as a function of age. Here “age” might be chronological, or degree of disk development. These may or may not be closely related to each other.
Two modeling environments were used to study these systems: DUSTY, which is relatively simple, but can be used on a large number of objects in a relatively short amount of time, and the Whitney code, which is more complex and realistic, but is computationally intensive.
As these systems evolve, it is expected that grain growth and settling would occur. These should produce a silicate band of diminishing strength, since the band strength is a function of the grain size. As the grains grow larger we expect a flattening in the spectral index at millimeter wavelengths. Small grains tend to have wavelength- dependent emissivities that produce emission spectra that are steeper than that of a blackbody, while larger grains will have emissivities independent of wavelength and emit more like blackbodies. Hence grain growth will tend to flatten the SEDs at long wavelengths.
Roughly half of the DUSTY fit objects could be used to find correlations between the silicate feature band strength and the millimeter spectral index. The study found that there is a reasonably clear correlation showing that the silicate feature band strength weakens as the millimeter spectral slope flattens, as expected in our general picture of these disk systems. A handful of objects, notably LkCa 15, RY Tau, and UX Ori, fell outside the clustering of the other objects.
198 In general, the majority of objects have SEDs consistent with an inner disk wall and an outer disk, which may be partially shadowed by the inner disk region. The
DUSTY model is capable of describing the effects of a thin disk plus spherical halo, although the latter is mathematically equivalent to a “disk atmosphere”. We approximated the emission from the inner disk wall with a blackbody (necessary to fit many of the SEDs) and searched for a correlation between the strength of this feature and the shape of the SED at longer wavelengths. It was expected that if these disks were partly shadowed by an inner disk wall of some height, then the strength of the emission produced by this region should be correlated to the shape of the SED produced by the disk further out.
The subset of objects used to find correlations between disk inner wall height and the spectral shape of the IRAS data did not produce results as clear as the band strength/millimeter index plot. These correlation plots did indicate general trends of increased stellar flux blockage going with lower 12µm/25µm, 12µm/60µm, and
25µm/60µm IRAS data ratios. However, these same three IRAS data ratios can be lower for reasons other than blocked stellar light, such as an actual deficit of disk material.
Methods of telling these cases apart involve much more observations and techniques that are beyond the scope of this work.
We selected a small set of six objects to study in greater detail using the Whitney radiative transfer code. Here, we attempted to derive detailed information about the structures of the disks, based upon their observed SEDs, while constraining the models with other sources of information. These latter included the observed sizes and surface brightness distributions of the disks, observed disk inclinations to the line of sight to the
199 observer from imaging and millimeter interferometry, mass accretion rates determined form UV and X-ray observations, etc. In addition, for two of the objects (HD 31648 and
HD 163296) we attempted to investigate the variability of these sources within the framework of the disk modeling, something that had never been done before. Much of this work was done as part of large (by astronomy standards) projects (Grady et al. 2007,
Sitko et al. 2008). We also used the model parameters for these six objects to investigate possible correlations among grain sizes and chronological age.
We found that the silicate band strength and millimeter spectral index of the six
Whitney fit objects were not strongly correlated with the object’s ages. The sample size, only 6 objects, was limited, and the ages of the objects are only poorly known.
Due to the relative wealth of observational data on HD 31648 and HD 163296 over the past decade or so, they were the focus of time-dependence of the inneer disk structure. HD 31648 and HD 163296 showed increased NIR flux in 2004 and 2002 respectively. To fit the increased flux of those periods, the Whitney disk’s inner edge moved further out and grew geometrically thicker. This provides one option to explain what happened to these systems during the increased flux period. However, other physical explanations are possible and more observations are needed.
In some of the objects here, such as HD 169142 and HD 135344 (SAO 206462) it was clear that these systems required hot material close to the star, often near the sublimation temperature, and a much colder thick disk further away. For example, in HD
169142, the inner material radiated at such a high temperature that it must be within 1
AU from the star, while the outer disk begins at 43 AU. It is not yet known how much material might be between these two distances, but it is expected that gaps in the disk
200 consistent with these sizes can be opened up through the gravitational influence of planets forming in these regions.
Through the use of SED fits and the correlation studies, a little more of the details of planet formation have been unveiled.
This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and
Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.
The IRS was a collaborative venture between Cornell University and Ball Aerospace
Corporation funded by NASA through the Jet Propulsion Laboratory and Ames Research
Center.
This research has made use of the SIMBAD database, operated at CDS, Strasbourg,
France
Some/all of the data presented in this paper were obtained from the Multimission Archive at the Space Telescope Science Institute (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.
Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NAG5-7584 and by other grants and contracts.
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214 Appendix A
Simple Sphere Mie Scattering C# code.
// first attempt at bhmie code in c# // created on 2/27/2005 at 2:10 PM using System; using System.IO; using System.Text; public class complex { public double re; public double im;
public complex(double re, double im) { this.re = re; this.im = im; } public complex() { this.re = 0.0; this.im = 0.0; } static public complex operator +(complex arg1, complex arg2) { return(new complex(arg1.re + arg2.re, arg1.im + arg2.im)); } static public complex operator +(complex arg1, double fac) { return(new complex(arg1.re + fac, arg1.im)); } static public complex operator +(double fac, complex arg1) { return(new complex(arg1.re + fac, arg1.im)); } static public complex operator -(double fac, complex arg1) { return(new complex(fac - arg1.re, arg1.im)); } static public complex operator -(complex arg1, double fac) { return(new complex(arg1.re - fac, arg1.im)); } static public complex operator -(complex arg1, complex arg2) { return(new complex(arg1.re - arg2.re, arg1.im - arg2.im)); } static public complex operator *(complex arg1, complex arg2) { return(new complex(arg1.re*arg2.re - arg1.im*arg2.im, arg1.re*arg2.im + arg1.im*arg2.re));
215 } static public complex operator *(complex arg1, double fac) { return(new complex(fac*arg1.re, fac*arg1.im)); } static public complex operator *(double fac, complex arg1) { return(new complex(fac*arg1.re, fac*arg1.im)); } static public complex operator /(complex arg1, complex arg2) { double c1, c2, d; d = arg2.re*arg2.re + arg2.im*arg2.im; if(d == 0) return (new complex(0,0)); c1 = arg1.re*arg2.re + arg1.im*arg2.im; c2 = arg1.im*arg2.re - arg1.re*arg2.im; return (new complex(c1/d,c2/d)); } static public complex operator /(complex arg1, double fac) { return(new complex(arg1.re/fac,arg1.im/fac)); } static public complex operator /(double fac,complex arg1) { double d; d = arg1.re*arg1.re + arg1.im*arg1.im; if (d == 0) return(new complex(0,0)); return(new complex(fac*arg1.re/d,-1*fac*arg1.im/d)); }
public double cabs() { return (Math.Sqrt(re * re + im * im)); } public void ccjg() { im = -1*im; return; } public complex conj() { return(new complex(re,-1*im)); } public double real() { return re; } public double imag() { return im; } public void display() { Console.WriteLine("({0},{1})",re,im); return;
216 } }
class callbh { public static void Main() { complex refrel = new complex(); complex [] s1 = new complex[200]; complex [] s2 = new complex[200]; complex shold = new complex(); double refmed; double[] refre = new double[10000]; double[] refim = new double[10000]; double[] rad = new double[50]; double[] nsda = new double[50]; double[] wavel = new double[10000]; double[] csca = new double[10000]; double[] cext = new double[10000]; double[] cabs = new double[10000]; double x; int nang; double dang; double qsca = 0.0; double qext = 0.0; double qback = 0.0; double s11nor; int nan; int j; int aj; int ghf; int counter; int sizes; int scount; double amin; double amax; double q; double s11; double s12; double s33; double s34; double pol; double ang; double qabs; double pw1; double fac; double dcabs; double dcext; double dcsca; string buffer; StringBuilder w = new StringBuilder(); StringBuilder r = new StringBuilder(); StringBuilder i = new StringBuilder();
217
amin = 0.005; amax = 0.25; q = 3.5; sizes = 50; for(j=0;j<200;j++) { s1[j] = new complex(); s2[j] = new complex(); } for(j=0;j<10000;j++) { csca[j] = 0.0; cext[j] = 0.0; cabs[j] = 0.0; } Console.WriteLine("Sphere Scattering Program"); StreamReader myFile = File.OpenText("sil-dlee.nk"); StreamWriter outFile = File.CreateText("sil-dlee.qab"); outFile.WriteLine("------"); outFile.WriteLine("Wave Qext Qabs"); outFile.WriteLine("------"); StreamWriter outFile2 = File.CreateText("sil-dlee.cas"); outFile2.WriteLine("------"); outFile2.WriteLine("Wave Cabs Csca"); outFile2.WriteLine("------"); // FileStream myBile = new FileStream("miebi.dat",FileMode.CreateNew); // BinaryWriter bwFile = new BinaryWriter(myBile); // myFile.WriteLine("Sphere Scattering Program"); // bwFile.Write("Sphere Scattering Program"); for(ghf=0;ghf<7;ghf++) buffer = myFile.ReadLine(); counter = 0; while ((buffer = myFile.ReadLine()) != null) { // reading a file has to be fine tuned to specific file format // Console.WriteLine(buffer); w.Append(buffer); r.Append(buffer); i.Append(buffer); w.Remove(15,30); r.Remove(0,15); r.Remove(15,15); i.Remove(0,30); wavel[counter] = Convert.ToDouble(w.ToString()); refre[counter] = Convert.ToDouble(r.ToString()); refim[counter] = Convert.ToDouble(i.ToString()); // Console.WriteLine("w {0}, r {1}, i {2}",w,r,i); // Console.WriteLine("w {0}, r {1}, i {2}",wavel[counter],refre[counter],refim[counter]); w.Remove(0,w.Length); r.Remove(0,r.Length); i.Remove(0,i.Length);
218 counter++; } myFile.Close(); pw1 = 1.0/(sizes - 1); fac = Math.Pow(amax/amin,pw1); // Console.WriteLine("{0}, {1}, {2}",sizes,pw1,fac); for(scount=0;scount for(ghf=0;ghf 219 dcext = qext*nsda[scount]*3.14159265*Math.Pow(rad[scount],3.0)*Math.Log(fac); dcsca = qsca*nsda[scount]*3.14159265*Math.Pow(rad[scount],3.0)*Math.Log(fac); Console.WriteLine("dCabs = {0}, dCext = {1}, dCsca = {2}",dcabs,dcext,dcsca); cabs[ghf] = cabs[ghf] + dcabs; cext[ghf] = cext[ghf] + dcext; csca[ghf] = csca[ghf] + dcsca; // Console.WriteLine("Angle s11 pol s33 s34"); // myFile.WriteLine("Qsca = {0} Qext = {1} Qback = {2}",qsca,qext,qback); // myFile.WriteLine("Qabs = {0}",qabs); // myFile.WriteLine("Angle s11 pol s33 s34"); // bwFile.Write("Qsca = {0} Qext = {1} Qback = {2}",qsca,qext,qback); // bwFile.Write("Qabs = {0}",qabs); // bwFile.Write("Angle s11 pol s33 s34"); s11nor = 0.5*(s1[1].cabs()*s1[1].cabs()+s2_1].cabs()*s2[1].cabs()); nan = 2*nang-1; // Console.WriteLine("nan {0}",nan); for (j=1;j<=nan;j++) { aj = j; s11 = 0.5*s2[j].cabs()*s2[j].cabs(); s11 = s11 + 0.5*s1[j].cabs()*s1[j].cabs(); s12 = 0.5*s2[j].cabs()*s2[j].cabs(); s12 = s12 - 0.5*s1[j].cabs()*s1[j].cabs(); pol = -s12/s11; shold = s2[j] * s1[j].conj(); s33 = shold.real(); s33 = s33/s11; s34 = shold.imag(); s34 = s34/s11; s11 = s11/s11nor; ang = dang*(aj-1)*57.2958; // Console.WriteLine("{0} {1} {2} {3} {4}",ang,s11,pol,s33,s34); // myFile.WriteLine("{0} {1} {2} {3} {4}",ang,s11,pol,s33,s34); // bwFile.Write(ang,s11,pol,s33,s34); } } } for(ghf=0;ghf 220 outFile.Close(); outFile2.Close(); // bwFile.Close(); } public static void bhmie(double x, complex refrel, int nang, ref complex[] s1, ref complex[] s2, ref double qsca, ref double qext, ref double qback) { double[] amu = new double[100]; double[] theta = new double[100]; double[] pi = new double[100]; double[] tau = new double[100]; double[] pi0 = new double[100]; double[] pi1 = new double[100]; complex[] d = new complex[3000]; complex y = new complex(); complex xi = new complex(); complex xi0 = new complex(); complex xi1 = new complex(); complex an = new complex(); complex bn = new complex(); // complex ctest = new complex(); double psi0; double psi1; double psi; double dn; double dx; double dang; double xstop; double nstop; int nmx; int nn; double chi0; double chi1; int j; int n; double p; double t; double apsi0; double apsi1; double apsi; double fn; double rn; double chi; int jj; double ymod; // if(s1 == null) // s1 = new complex[200]; // if(s2 == null) // s2 = new complex[200]; for(j=0;j<3000;j++) 221 d[j] = new complex(); dx = x; y = x * refrel; // Console.WriteLine("dx {0}",dx); // Console.Write("y "); // y.display(); xstop = x + 4*Math.Pow(x,.3333333)+2.0; nstop = (int)xstop; ymod = y.cabs(); // Console.WriteLine("xstop {0}, ymod {1}",xstop,ymod); nmx = (int) Math.Max(xstop,ymod)+15; dang = 1.570796327/(float)(nang-1); for (j=1;j<=nang;j++) { theta[j] = (j-1)*dang; amu[j] = Math.Cos(theta[j]); // Console.WriteLine("j {0}, theta[j] {1}, amu[j] {2}",j,theta[j],amu[j]); } // Console.WriteLine("nmx {0}",nmx); d[nmx].re = 0.0; d[nmx].im = 0.0; nn = nmx - 1; // Console.WriteLine("nn {0}",nn); for(n = 1; n <= nn; n++) { rn = nmx - n + 1; d[nmx-n] = (rn/y)-(1/(d[nmx-n+1]+rn/y)); // Console.Write("n {0}- rn {1}, d ",n,rn); // d[nmx-n].display(); } for(j=1;j<=nang;j++) { pi0[j] = 0.0; pi1[j] = 1.0; } nn = 2*nang-1; for(j=1;j<=nn;j++) { s1[j].re = 0.0; s1[j].im = 0.0; s2[j].re = 0.0; s2[j].im = 0.0; } psi0 = Math.Cos(dx); psi1 = Math.Sin(dx); chi0 = -1*Math.Sin(x); chi1 = Math.Cos(x); apsi0 = psi0; apsi1 = psi1; // Console.WriteLine("psi0 {0}, psi1 {1}",psi0,psi1); // Console.WriteLine("chi0 {0}, chi1 {1}",chi0,chi1); xi0.re = apsi0; xi0.im = -1*chi0; xi1.re = apsi1; xi1.im = -1*chi1; qsca = 0.0; 222 n = 1; do { dn = n; rn = n; fn = (2*rn+1)/(rn*(rn+1)); psi = (2*dn-1)*psi1/dx-psi0; apsi = psi; chi = (2*rn-1)*chi1/x - chi0; // Console.WriteLine("rn {0}, fn {1}, psi {2}",rn,fn,psi); // Console.WriteLine("chi {0}",chi); xi.re = apsi; xi.im = -1*chi; // Console.WriteLine("apsi {0}, apsi1 {1}",apsi,apsi1); // d[n].display(); // xi.display(); // xi1.display(); // d[n].display(); // Console.WriteLine("n {0}, rn/x {1}, apsi1 {2}",n,rn/x,apsi1); an = (d[n]/refrel+rn/x)*apsi-apsi1; /* ct_st = d[n]/refrel; ctest.display(); ctest = ctest + rn/x; ctest.display(); ctest = ctest * apsi; ctest.display(); ctest = ctest - apsi1; ctest.display(); an.display(); */ an = an/((d[n]/refrel+rn/x)*xi - xi1); bn = (refrel*d[n]+rn/x)*apsi - apsi1; // bn.display(); bn = bn/((refrel*d[n]+rn/x)*xi - xi1); // an.display(); // bn.display(); qsca = qsca+(2*rn+1)*(an.cabs()*an.cabs()+bn.cabs()*bn.cabs()); // Console.WriteLine("qsca {0}",qsca); for(j=1;j<=nang;j++) { jj = 2*nang-j; pi[j] = pi1[j]; tau[j] = rn*amu[j]*pi[j]-(rn+1)*pi0[j]; p = Math.Pow(-1,n-1); s1[j] = s1[j]+fn*(an*pi[j]+bn*tau[j]); t = Math.Pow(-1,n); s2[j] = s2[j]+fn*(an*tau[j]+bn*pi[j]); // Console.WriteLine("amu[j] {0}",amu[j]); // Console.WriteLine("pi {0}, tau {1}, pi0[j] {2}, pi1[j] {3}",pi[j],tau[j],pi0[j],pi1[j]); if(j != jj) { s1[jj] = s1[jj]+fn*(an*pi[j]*p+bn*tau[j]*t); 223 s2[jj] = s2[jj]+fn*(an*tau[j]*t+bn*pi[j]*p); } // Console.WriteLine("s1 and s2"); // s1[j].display(); // s2[j].display(); } // Console.ReadLine(); psi0 = psi1; psi1 = psi; apsi1 = psi1; chi0 = chi1; chi1 = chi; xi1.re = apsi1; xi1.im = -chi1; n = n+1; rn = n; for(j = 1; j<=nang;j++) { pi1[j]=((2*rn-1)/(rn-1))*amu[j]*pi[j]; pi1[j] = pi1[j]-rn*pi0[j]/(rn-1); pi0[j] = pi[j]; } }while(n-1-nstop < 0); qsca = (2/(x*x))*qsca; qext = (4/(x*x))*s1[1].real(); qback = (4/(x*x))*s1[2*nang-1].cabs()*s1[2*nang-1].cabs(); return; } } 224