Summary of Materials science doctoral thesis: Rotational spectral line formation and radiative transfer in circumstellar envelopes

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The research in this thesis follows two directions: The first, using high performance computing currently available, we studied atomic and molecular emissions in astrophysical environment. The radiative transfer model was applied to explain the characteristics of the recombination line masers Hnα in the envelope of the star MWC 349A.

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Nội dung Text: Summary of Materials science doctoral thesis: Rotational spectral line formation and radiative transfer in circumstellar envelopes

MINISTRY OF EDUCATION VIETNAM ACADEMY AND TRAINING OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY -------------------- TRAN NGOC HUNG ROTATIONAL SPECTRAL LINE FORMATION AND RADIATIVE TRANSFER IN CIRCUMSTELLAR ENVELOPES Specialization: Optics Numerical code: 944 01 10 SUMMARY OF MATERIALS SCIENCE DOCTORAL THESIS HANOI, 2021
The thesis was completed at: Department of Physics, Graduate University of Science and Technology, VAST Center of Engineering Physics, Institute of Physics, VAST Supervisors: Assoc. Prof. Dr. Dinh Van Trung Assoc. Prof. Dr. Pham Hong Minh Reviewer 1: .............................................................................................................................. .............................................................................................................................. Reviewer 2: .............................................................................................................................. .............................................................................................................................. Reviewer 3: ............................................................................................................................. .............................................................................................................................. The thesis will be defended at: .............................................................................................................................. .............................................................................................................................. Time:..................................................................................................................... The thesis could be found at: • National Library of Vietnam • Library of Graduate University Science and Technology • Library of Institute of Physics
PREFACE The atomic and molecular emission in the radio region is the opportunity for us to explore the kinetic and physical properties of astrophysical environments. Strong and directional maser emissions allow high angular resolution up to milliseconds to study astro-objects at unprecedented details. In recent decades, advances in computational techniques have greatly assisted the analysis of astronomical data. Numerically modeling can accurately simulate the optical images or radio spectrum to help us understand the processes going on in the universe. Predictions from these numerical models serve as guidance for many studies. The research in this thesis follows two directions: The first, using high performance computing currently available, we studied atomic and molecular emissions in astrophysical environment. The radiative transfer model was applied to explain the characteristics of the recombination line masers Hnα in the envelope of the star MWC 349A. The seconds, we modeled a gaseous envelope of a binary system of stars by using numerically hydrodynamic calculation. The hydrodynamic simulation using the open-source PLUTO program models envelopes of asymptotic giant branches (AGBs) to study the morphology of spiral structures surrounding these stars. The thesis is presented in four chapters as follows: - Chapter 1. An overview of the astronomical environment and the theoretical basis in radio astronomy research. Two important basic theories presented are the radiative transfer theory and the hydrodynamic theory. - Chapter 2. Present the radiative transfer modeling process in a circumstellar and the work blockes in building simulation programs. Hydrodynamics in an envelope of a binary system has also been modeled by using the open-source program PLUTO. 1
- Chapter 3. Modeling the transmission of molecular radiation in a circumstellar medium and applying it to recombination line masers Hnα in the envelope of the star MWC 349A. - Chapter 4. Modeling hydrodynamics of binary systems for considering the influence of the orbital factors on the morphology of the envelope. This model is combined with the radiative transfer model to obtain images of molecular radiation that can be compared with the observed data. 2
Chapter 1. Overview circumstellar media, radiative transfer and structures of envelopes of evolved stars 1.1 Formation and development of stars Stars are formed from nebulae which are giant clouds of gases. If a gaseous cloud is massive enough that the gaseous pressure is not strong enough to balance gravity, it undergoes gravitational collapse. As a result of this collapse, new stars are formed in the core of the gaseous cloud. 1.2 Observations of circumstellar envelopes The advances in astronomical observations help us resolve many structures of circumstellar envelopes. The Hubble space telescope became a pioneer icon in the exploration of the universe by optical photography. Besides, many radio observatories around the world help collect a lot of important data. One of the famous observatories is the ALMA observatory which has had many important discoveries. 1.3 Atomic and molecular spectra observed in circumstellar envelopes 1.3.1 Hydrogen recombination lines Hydrogen gas around stars is ionized by UV radiation to create an ionized envelope known as the H II region. This ionization occurs concurrently with recombination process between protons and free electrons. The process will emit continuous bound-free radiation and produce H atoms at high-energy levels. Movement from high-energy levels to low-energy levels in the H II region is accompanied by recombination line radiation. 1.3.2 Molecular rotation spectrum The energy difference between two successive rotational energy levels is determined as follows: 3
(1.1) Where, J a rotational level with J = 0,1,2…; B0 the rotation constant; m the reduced mass; and r0 the effective bonding radius. Transitions from an upper rotation level to a lower rotation level are accompanied by photon emission resulting in a molecular rotation spectrum. 1.4 Emission and radiative transfer 1.4.1 The basic concepts Some basic concepts related to radiative transfer in astrophysical environments are intensity, emission coefficent, attenuation coefficient, Einstein coefficients, source function. 1.4.2 Pumping processes and equations of statistical equilibrium Interactions between radiation and molecules are absorption and emission processes that affect the radiation intensity and the molecular populations. In equilibrium, the pump rate on an energy level is equal to the loss rate of this energy level. (1.2) Where, P1 and P2 are the pump rates from other levels to 1 and 2 level; Γ1 and Γ2 are the loss rates of 1 and 2 level; A21 and B21 are Einstein coefficients for emissions; Jν is mean intensity. Pumping processes present in circumstellar envelopes: photoionization, recombination, collision… 1.5 Radiative transfer equation The radiative transfer equation shows that the change in the intensity of the light beam depends on the optical properties of the medium: 4
(1.3) The general solution of the equation is: (1.4) 1.5.1 Radiative transfer in media 1.5.1.1 LTE medium When the light enters a very thick optical medium, τν(D) >> 1, it will be completely absorbed. The contribution of spontaneous emissions is shown in the source function of the medium: (1.5) Where, hν = Eul the energy of the spontaneous photon. This function takes the form of the Planck function for black-body radiation. 1.5.1.2 Maser If the medium has an optical depth τν(D) < 0, the light beam will be amplified. The absorption coefficient of this medium is also negative. The medium has a negative optical depth if it satisfies the inverse population condition. The inverse processes of populations are called the atomic and molecular excitation in the medium. The radiative transfer equation in a two-level molecular medium has the form: (1.6) Where, Js is the saturation intensity. 5
In the case of Jν Js, the intensity increases linearly with the the transmission distance: (1.8) 1.5.2 Mechanism of spectral broadening 1.5.2.1 Natural broadening Natural broadening is caused by the intrinsic nature of the molecule. This profile has a spectral form of Lorentz: (1.9) 1.5.2.2 Collisional broadening In high-density environments molecules often collide with each other. This profile has a spectral form of Lorentz: (1.10) 1.5.2.3 Doppler broadening We know that atoms and molecules are always in thermal motion. Therefore, the frequency of the photon emited or absorbed by them in the associated frame will differ from the frequency observed by the observer. Each molecule will induce a Doppler shift causing a line broadening effect: 6
(1.11) After normalization, the Doppler broadening function describes the frequency distribution of the given form: (1.12) 1.5.2.3 Voigt broadening In regions of high density and high temperature gas, thermal motion and collisions are major causes of spectral broadening. The convolution of both kinds of broadening is described by the Voigt distribution function: (1.13) 1.5.3 Some radiative transfer models in astrophysics To study molecular emission, several programs focus on numerically solving the radiative transfer equation in molecular gases as MORELI, LIME… Some other programs simulate radiative transfer in a scattering medium as HYPERION, TORUS, RADMC3D… 1.6 The influence of the companion on the circumstellar structure 1.6.1 Modeling binary systems The Kepler equation for a binary star system can be written in the form: M = E − e.sinE (1.14) Where, e is the orbital eccentricity, M = Ωt is the mean anomaly, and E is the eccentric anomaly. 7
1.6.2 Euler equations The dynamics of gas in the circumstellar envelopes is described by the following Euler equations: (1.15) The gravitational influence of the companion star is shown through the source term S(U). Chapter 2. Modeling radiative transfer and hydrodynamics 2.1 A numerical method to solve the radiative transfer problem 2.1.1 Grid setup We use a nested three-dimensional Cartesian grid centered at the star. A coarse mesh indexed as level 1 covers the entire computational domain with the side length of 120 AU. This mesh is divided into 643 cells. The level 2 mesh is the same number of cells but finer in resolution (factor of 2) due to the fact that it only covers a half computational domain. 2.1.2 Ray tracing algorithm The ray tracing algorithm is used to find the serial cells through which the light ray passes and the corresponding distances. 2.1.3 Directions of rays To create beams of light to a point evenly distributed in directions in space, we use the Delaunay triangulation algorithm on the unit sphere. 8
2.1.4 Numerical solving the statistical equilibrium equations To determine the average intensity, we solve simultaneously the radiative transfer equation and the statistical equilibrium equations using Newton-Raphson’s iterative method. 2.2 Hydrodynamics model We simulate the circumstellar envelope as an uncompressible, non- cooling and non-selfgravitating gaseous environment. 2.2.1 Riemann problem In numerical analysis, Riemann problems appear in a natural way in finite volume methods for the solution of conservation law equations due to the discreteness of the grid. For that it is widely used in fluid dynamics and in magnetohydrodynamics simulations. In these fields, Riemann problems are calculated using Riemann solvers. 2.2.2 Godunov’s scheme Godunov's scheme is a conservative numerical scheme, suggested by S. K. Godunov in 1959, for solving partial differential equations. 2.2.3 Courant-Friedrichs-Lewy condition The convergence condition by Courant–Friedrichs–Lewy is a necessary condition for convergence while solving certain partial differential equations numerically. 2.2.4 Grid setup with PLUTO To help minimize computation time, we implement meshing according to the hierarchical grid as built in the radiative transfer problem. 2.2.5 Initial conditions 9
The initial positions of the two stars are chosen at the points closest to the mass center of the system. The initial gas density of the envelope of the AGB is 10−6 times that of the surface of the star. 2.2.6 Boundary conditions The inner boundary is defined at the spheric surface with the center AGB and the radius equal to 2. 2.2.7 Companion’s gravitation We can describe gravitation in both scalar and vector potential forms. It is the source term in the system of equations Euler. 2.2.8 Physical conditions Physical conditions are given to solve hydrodynamic equations in a certain mechanism. 2.3 Conclusion Chapter 2 presents discussions on the hydrodynamics and radiative transfer models: - A three-dimensional numerical model for molecular radiative transfer is developed for applications in simulating masers around young stars and AGBs. - The hydrodynamic model of a binary star system consisting of an AGB star and a companion star is built on the open-source PLUTO platform. 10
Chapter 3. Investigating hydrogen recombination line masers in the envelope of the star MWC 349A 3.1 Introduction MWC 349A is classified as a B[e] star located at a distance of 1.2 kpc. From optical emission line and radio continum imaging observations, it was inferred that the star is surrounded by an ionized envelope driven by wind from the massive central star. MWC 349A is the strongest hydrogen recombination line (HRL) source in the radio and NIR region. A well-known characteristic of the HRL spectra of the star is the double- peaked profile which is thought to originate from the Kepler rotation of the ionized disk formed by photoionization occuring in the inner neutral disk. The HRLs of the star MWC 349A has the potential to provide crucial information on kinematics and physical condition of the disk and wind. 3.2 Theoretical studies on hydrogen maser and the problems to be solved Several theoretical models have been constructed to explain the HRL observations. Mart´ın-Pintado et al. (2011) propose an axisymmetric double-cone model to interpret the high resolution observation of H30α. B´aez-Rubio et al. (2013) further developed the model and could explain a number of single disk and interferometric observation for n ≥ 30. However their model does not consider interaction between maser radiation and the atom, thus unable to account for the maser saturation. As such for n < 30 the predictions of the model are unrealistic in comparison to observations. One of the important features ignored in the model is saturation effect. Strelnitski et al. (1996b) argued that the Hnα lines are saturated with n ≤ 30, and the degree of saturation increases with decrease of quantum numbers n. They derived a simple estimation of the degree of saturation and presented phenomenology of the saturation effect affecting on adjacent emission lines. 3.3 Model of HRL maser in MWC 349A The geometry of envelope of MWC 349A follows closely the double- cone model and the disk wind model showed in Figure 3.1. 11
Figure 3.1. Geometric and kinametic configuration of MWC 349A: the left panel shows the 3D description, the right panel shows a slide plane containing the revolution axis. 3.3.1 The ionized wind The rotating and expanding ionized wind is launched from the thin Keplerian disk near the equatorial plane that has the minimal and maximal radius of Rdmin and Rdmax. 3.3.2 The ionized disk The ionized disk is contained in the double-cone with the revolution axis perpendicular to the equatorial plane of the neutral disk. 3.3.3 H II maser model Recombination processes, together with collisional excitation and ionization, occur frequently in H II regions. In the open and turbulent plasma environment, it is not appropriate to assume LTE. Goldreich & Kwan (1974) showed that the intensities of the observed RRLs were a consequence of departures from LTE. 3.3.4 Voigt broadening The line profile ϕν is the Voigt profile which is convolution of the Gaussian profile with Lorentz profile. 12
3.3.5 Gaunt factors There are two mechanisms for creating continuous spectra: first, protons and free electrons interact with each other and then separate (free-free interaction); second, after the interaction, protons and electrons recombine into H atoms (bound-free interaction). The contributions of these two kinds of emission are characterized by Gaunt factors. 3.3.6 Departure coefficients There are two sets of metrics used for recombination line calculation: one by Walmsley and the other by Storey and Hummer. 3.3.7 Saturation of HRL masers We present a full three-dimensional radiative transfer model for two- level maser and use that to numerically model the HRLs and saturation of maser emission in the ionized envelope of MWC 349A. 3.4 HRL masers 3.4.1 Profiles In Figure 3.2 we show the predicted line profiles of the H30α and H26α from our models 1 and 2 both lines exhibit the double-peak structures, the well-known characteristics of line emission arising from the rotation disk. Both the line profiles have the peak intensities consistent with observations. Figure 3.2. Compare the H26α and H30α maser spectra between observation (right image) and model (left image): dashed line for the absence of saturation effect; solid line for the case of saturation effect. 13
3.4.2 Channel maps of brightness temperature Figure 3.3 shows the channel maps of brightness temperature of maser emission for the two lines H26α. Figure 3.3. Channel maps of H26α intensity in MWC 349A. 3.4.3 Effects of the ionized wind To investigate how the wind affects this asymmetry, we also consider Model 2 in which the wind’s electron density is half that in Model 1 in order to reduce contribution of the wind in line emission. Therefore, our results emphasize the effects of the ionized wind on the HRLs which can be important for explaining the observations of higher frequency HRL masers. 14
Figure 3.4. H26α and H30α maser spectra in two ionization wind models: Model 1 (solid line) has twice the electron density in the wind as Model 2 (dotted line). 3.5 Saturation of HRL masers Our full three-dimensional radiative transfer model takes into account saturation effect neglected by the previous models and reveals for the first time the saturation property of recombination line masers. Figure 3.5 presents the saturation degree of H26α, H27α and H30α masers in a plane containing the symmetry axis. Figure 3.5. The saturation degree of recombination line masers H26α, H27α and H30α in a plane containing the symmetric axis. 3.6 Conclusion Using the new model we have investigated the saturation properties of two typical recombination line masers H26α, H27α and H30α in the envelope of MWC 349A. The radiative transfer calculations using the long characteristics method allow correct treatment of maser beaming effect. We find that H30α maser is unsaturated while H26α maser is 15
partially saturated. Spectral intensities predicted by the model are appropriate to recent observations. In addition, the velocity separation between maser peaks is predicted to increase marginally from the H30α line to the H26α line, more consistent with observations. The effects of the ionized wind on the spectral profiles and the spatial distribution of recombination line maser emission are also obtained from the results of our model. Specifically, the absorption due to ionize wind leads to asymmetries in the spectral profile and spatial distribution of stronger and partially saturated maser such as H26α. Our analysis could potentially provide useful constraints on characteristics and physical properties of the bipolar wind from MWC 349A. Chapter 4. Studying circumstellar structures of AGBs under the influence of the companion star 4.1 Hydrodynamic model for a binary star system Our simulations are carried in a three dimensional domain ∼4800 AU (astronomical unit) in size. We use 800 grid points in each direction to reach a spatial resolution of ∼1.3 AU near the primary star. The binary system includes the primary star with a mass of 2.2 times the solar mass (Ms) and the secondary star with a mass of 0.8 times the solar mass. The primary star has the mass loss rate of about 10-5 Ms/yr, typical for AGB stars with known spiral pattern such as CIT 6, CRL 3068 or CW Leo. The orbital period of the binary system is set at 325 years, well in the range of the expected orbital periods of envelopes with spiral pattern. The orbital period corresponds to the distance between binary components of about 68 AU. The wind velocity V’w is set at 15.7 km/s, also typical for the above systems such as CIT 6 (wind velocity of 18 km/s) or CW Leo (wind velocity of 14 km/s). The sound speed at the inner boundary is set to 2 km/s, which corresponds to a gas temperature of about 700 K, also a typical value for the inner region near the mass losing AGB star. 16
4.2 Isothermal and ideal gas In Fig. 4.1 we show the results of our simulations in the two cases of isothermal and adiabatic EOS and the binary system moving in a circular orbit. The spiral patterns show significant differences between the two cases (i) the width of the spiral arm in the isothermal case is much larger than that in the adiabatic case; (ii) the density contrast is noticeably higher in the adiabatic case. These differences are directly related to the expanding speeds of the inner and outer surface of the spiral pattern. Figure 4.1. Density distributions in the logarithmic scale of the gas in the orbital plane from the two cases of isothermal EOS (left panel) and adiabatic EOS (right panel) in circular orbits. The dashed curves are the Archimedean spiral matched to the density distribution. The red curves correspond to the inner side and the black curves correspond outer surface of the spiral pattern. 4.3 Circular and elliptic orbits In this section we compare the properties of the envelope under the influence of binary companion in circular and elliptical orbits. The adiabatic EOS is used in both cases and the elliptical orbit has an eccentricity of e=0.5. In Figures 4.2 and 4.3 we show the density distributions for the circular and elliptical orbit configuration on the orbital plane and in the plane containing the polar axis of the envelope. 17
Figure 4.2. Density distributions in the circular orbit case shown in logarithmic scale. Left panel shows gas density distribution in the orbital plane; right panel shows gas density in a the plane containing polar axis and perpendicular to the orbital plane. Figure 4.3. Same as Fig. 4.2 but for the elliptical orbit case. 4.4 Temperature profiles As the gas moves radially outward, it expands and cools adiabatically. In Figure 4.4 we show the gas temperature along two radial directions, one along the X axis and the other along the Y axis for the circular orbit case. The gas temperature in the inter-arm region in both directions decreases monotonically with radial distance as expected. However, across the spiral arm the gas is strongly compressed due to the collision between gas moving at different velocities, leading to elevated gas temperature. From Figure 4.4 we can estimate the jump in temperature due to shock 18