Dust emission modelling of AGB stars

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Dust emission modelling of AGB stars

Uppsala University

Emelie Siderud

28 August 2020

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1 Introduction

1.1 Mass loss of AGB stars

A star spends most of its life on the main sequence, burning hydrogen (H) to helium (He) in its core. For a star with an initial mass between about 0.5−8M, the evolution off the main sequence starts when hydrogen is depleted in the core and instead the H-burning move outward in a shell surrounding the inert He core. The outer layers of the star expand, and the star goes through a first red giant branch phase. Convection causes materials from the deep interior to be mixed with the material above which changes the observed surface abundance of the star. The central temperature and density increase in the core until He- burning begins. Inert carbon (C) and oxygen (O) are produced and analogue to the H-burning shell, the He-burning will eventually move outward to form a He-burning shell. Nuclear fusion in the stellar core is now over and the star will ascend a red giant branch for the second time, the asymptotic giant branch (AGB), which is powered by alternating H- and He-burning shells. This is the final phase for low- and intermediate-mass stars, before they turn into white dwarfs.

During the early AGB phase, the He-burning shell dominates the energy output. As the He-shell becomes thinner, the H-burning shell is reignited and becomes the dominating energy source of the star. From here on, a cyclic process of alternating intense He-shell burning follows, which leads to a phenomenon called He-shell flash, or thermal pulse. The H-burning shell dumps He ash onto the He layer below under a duration of 10⁴ years, depending on mass. The temperature at the base of the He-shell is increased until a He shell flash occurs.

The H-burning shell is pushed outward, causing it to cool and turn off until the He-shell diminishes, a period that last a few hundred years. The H-burning shell is finally recovered and the process repeats. During a flash episode, the convective zone in the outer envelope of the star will reach down to the region between the two shells, which leads to a dredge-up of carbon-rich materials from this region to the surface. The relative abundances of carbon and oxygen in the AGB star thus changes during its evolution due to the dredge-up of processed material. Initially, AGB stars are of spectral type M where all carbon is bound in CO molecules and the excess oxygen is available for formation of other molecules and dust. Increasing the carbon abundance leads to a C/O ratio equal to 1, where nearly all carbon and oxygen atoms are bound in CO (spectral type S).

Finally, as the carbon abundance increases, all oxygen atoms are bound in CO and carbon becomes available for formation of other molecules (spectral type C).

During the AGB phase, the effective temperatures reaches around 3000 K and below, while the luminosities reaches 10³ to 10⁴ L, which correspond to typical stellar radii of several hundred R. The surface layers are thus loosely bound as the resulting surface gravities are 4-5 orders of magnitude below that of a sun-like star. Observations have shown that AGB stars are losing mass at rates in the range of 10⁻⁸to 10⁻⁴Mper year. The most common and general

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view on the cause of mass loss in AGB stars is the pulsation-enhanced dust- driven outflow (PEDDRO), where stellar pulsation induce strong shock waves that push gas above the stellar surface where it is cooled and become dense enough for dust to form. The dust is then in turn pushed outward by radiation pressure, dragging the gas along, resulting in a dust-driven outflow of mass with wind velocities in the range of 5-30 km/s. The mass loss of AGB stars leads to the enrichment of the interstellar medium and is one of the most important sources of cosmic dust. [5] [4] [3]

1.2 Dust driven wind

AGB stars are long period variables with a typical pulsation period of about 100 – 1000 days. The cause of the regularity or amplitude of the pulsations is not clearly established. It is thought to be caused by radial pulsations originating in the convective stellar envelope. As a result of the pulsations, gas is elevated above the stellar surface through induced shock waves and reach distances from the star where dust particles can form. For dust particles to survive and grow, the grain temperature must be below the stability limit of the respective con- densate. To estimate the closest distance at which such grains can exist, R_c, the point where the grain temperature is equal to the condensation temperature, T_c, of the material must be found. Further assuming a Planckian radiation field cor- responding to the effective temperature T_∗, geometrically diluted with distance from the star, and approximating the grain absorption coefficient with a power law κabs∝ λ^−pin the relevant wavelength range where λ is the wavelength and p is a material-dependent constant, the condensation radius becomes

Rc

R_∗ ≈ 0.5 Tc

T_∗

−^4+p₂

. (1)

As the absorption coefficient depends on the wavelength (i.e. the value of p), different grain materials will react differently to the stellar environment. A grain material with a positive value of p will push the condensation distance further out as it is more efficient at absorbing than emitting radiation, i.e. it heats up as a result of interacting with the stellar radiation field. For a grain material with a negative value of p, the opposite is true, i.e. the condensation distance is closer to the star. Hence, how close to a star a grain type can survive depends on the combination of condensation temperature and the slope of the absorption coefficient of the material. In figure 1, the efficiency of different grain materials is plotted as a function of wavelength, where the efficiency is defined as the ratio between the radiative and the geometrical cross-section of an individual grain. The plot shows that for example silicates are more efficient absorbers at longer wavelengths while carbon and iron are more efficient absorbers at shorter wavelengths.

The high luminosity of AGB stars result in a radiative pressure that acts on the dust particles and causes them to accelerate away from the star. The outwards directed radiative acceleration is critically dependent on the total ra-

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Figure 1: Efficiency Q per grain radius a as a function of wavelength for a selection of dust species.[1]

diative cross section per mass of circumstellar material. By comparing it to the inwards directed gravitational acceleration, an estimation of the amount of opacity required to drive a stellar wind by radiative pressure can be made, i.e.

Γ = a_rad agrav

= hκiL∗

4πcGM_∗ (2)

where κ is the total cross section per unit mass, or flux mean opacity, M_∗ and L_∗ are the stellar mass and luminosity respectively, c is the speed of light, and G is the gravitational constant. When the two forces are equal in magnitude, i.e. Γ = 1, a critical value of the flux mean opacity is found, given by

κ_crit= 4πcGM_∗ L∗

. (3)

If Γ > 1, and thus hκi > κcrit, the matter is accelerated away from the star.

The expression for Γ either depends solely on stellar parameters (Eq. 3) or on hκi (Eq. 2), that is grain material dependent. For a dust species to be considered a potential wind driver it needs to fulfil two basic criteria. Firstly, its condensation distance must fall within the shock-levitated atmosphere, and secondly, its flux-integrated opacity must be comparable to, or exceed, κ_crit. In C-type AGB stars (C/O > 1), carbon dust is considered the most probable wind driver while in M-type AGB stars (C/O < 1), magnesium-iron-silicates are considered good candidates for driving the winds. [4] [3] [1]

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2 Radiative transfer through dust

Light from a star that travel through a medium will be affected by scattering, emission and absorption depending on the physical and chemical properties of the medium. This propagation of radiation is referred to as radiative transfer and can be described mathematically by knowing or estimating the parameters of the medium. For an AGB star that is surrounded by a circumstellar envelope (CSE) of gas and dust particles, the result from such computations can for example be used to determine the dust density distribution and mass-loss rate of the star.

2.1 Dust opacities

Dust opacities are much more capable of affecting radiative heat transfer than gas opacities. This is due to the strong continuum opacities of dust which cover large bands of the whole electromagnetic spectrum, with few wavelength windows by which one can peer inside, while gas opacities only cover a fraction of the spectrum. The emission retrieved from a star enshrouded in a dusty envelope is thus likely dominated by thermal emission from dust. The optical properties of a dust particle is expressed in terms of efficiency Q, and includes contributions from both absorption, Qabs, and scattering, Qsca. According to Mie theory, for a grain with radius a much smaller than the relevant wavelength, the efficiencies behave like Qabs∝ a and Qsca ∝ a⁴ (small particle limit).

If considering a dust particle as a transparent sphere, no incident light is absorbed, instead through refraction on the surface it is redirected into another direction. In this case the opacity κν is nearly entirely a scattering opacity.

If our dust particle is made up of graphite, then only a small fraction of the incident light is scattered and most is absorbed. Silicates are dominated by Si-O bonds and typically have two main opacities at 10 and 20 micrometer (see Figure 1). The absorption opacity in the optical and near-infrared is strongly dependent on the amount of iron that is present. If a silicate grain has little or no iron, the absorption opacity drops while the scattering opacity stays roughly the same. The opacities of carbon grains are simpler than those of silicates as there is no major dip in the near-infrared. This suggests that carbon dust is more likely to dominate in the near-infrared region.

2.2 Dust emission modelling

Radiative transfer calculations can be used to determine the temperature of the dust and to compute a spectral energy distribution (SED) of a dusty object.

The transfer of radiation through the dusty CSE assumes a model of a spherically symmetric, continuously expanding wind. The dust properties also have to be assumed, and the total mass-loss rate of the star can be derived from the dust density distribution assuming a gas-to-dust ratio and a wind velocity distribution.

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The most important parameter in the dust radiative transfer modelling is the dust optical depth

τ_λ= κ_λ Z re

r_i

ρ_d(r)dr (4)

where κλ is the dust opacity per unit mass, ρd is the dust mass density, the integration is made from the inner (ri) to the outer (re) radius of the CSE.

Figure 2 shows the emerging SED for three optical depth cases of carbon dust.

For tau = 0.01, the SED is dominated by radiation around 0.5 micrometer which represents the stellar radiation. The dust emission starts to appear at longer wavelengths which is referred to as the infrared excess as the flux is in excess of the expected radiation from a star without any dust around it. With increased optical depth, the dust extincts the stellar radiation which cause the stellar flux to drop and the infrared flux to increase, i.e. the SED moves to longer wavelengths. The color of the star becomes redder as the photosphere moves outward. At even larger optical depths, the millimeter and centimeter wavelength flux also increases, and at these wavelengths the CSE is optically thin. [2]

Figure 2: Dusty envelope example around an AGB star.

3 Observations

To make an image of a dusty cloud that emits thermal radiation, a process known as volume rendering or forward ray tracing can be performed. The formal transfer equation is integrated along a ray starting behind the cloud and

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ending up at the observer. For an “observer at infinity” approach, which is a good approximation when studying stars, the rays of the image will be parallel, and the pixel size must be specified in centimeters instead of an expression of angle. This means that the image can be computed without knowing the distance to the observer. The flux is found by integrating over the image and the result retrieved from several telescopes, operating at different wavelengths, can be used to make a single SED. [2]

In this project, observations from different telescopes are combined to make the SED for each carbon star in the Deathstar¹ sample. The SED data points are collected from literature and converted to flux densities in units of Jansky.

Figure 3 shows the wavelength coverage of currently available data (right image) compared to previously available data (middle image) that was used when the latest dust emission modelling for nearby Galactic stars were performed. Hence, this work will include contributions of cold dust to the SED as well as the newly collected data points. A well-sampled SED is essential to improve the reliability of dust estimates. A short description of some of the telescopes and instrumentation used in this work is given below.

Figure 3: The SED of a typical AGB star peaking around 1 micron.

3.1 AKARI

The main objective of AKARI was to perform an all-sky survey, like that of IRAS (Infrared Astronomical Satellite launched in 1983) but with better sensitivity, spatial resolution and wider wavelength coverage. AKARI was launched in February 2006 and had a lifetime of 550 days. Its 68.5 cm-diameter telescope covered more than 94% of the sky from a sun-synchronous polar orbit at 700 km altitude. It observed the universe in the wavelength range 2-180 microns. Two catalogues have been publicly released; AKARI-FIS, containing far-infrared (65, 90, 140, 160 µ m) fluxes of 427,071 sources, and AKARI-IRC, containing mid- infrared (9 and 18 µ m) fluxes of 870,973 sources. The AKARI maps improve those from IRAS by a factor of four or five.²

1https://www.astro.uu.se/deathstar/index.html

2http://www.isas.jaxa.jp/en/missions/spacecraft/past/akari.html

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3.2 Herschel

The Herschel Space Observatory was launched in May 2009 and placed in an orbit around the second Lagrangian point of the Sun–Earth system. At this location it was able to point outward into the Universe without interference from the Earth, Moon or Sun which all emit strong infrared radiation. Its primary mirror was 3.5 m in diameter which focused the light onto three scientific instruments. Observations from two of the instruments are used in this work: PACS (Photoconductor Array Camera and Spectrometer) and SPIRE (Spectral and Photometric Imaging REceiver), which are sensitive in the wavelength range 60-210 microns and 200-670 microns respectively. Herschel had a nominal mission lifetime of three years and provided observations in wavelengths previously unseen by infrared satellites and radio telescopes on ground.³

3.3 WISE

Wide-field Infrared Survey Explorer (WISE) is a mid-infrared-sensitive space telescope that performed an all-sky astronomical survey during its initial op- eration from December 2009 to February 2011. The WISE instrument is a four-channel imager (3.4, 4.6, 12 and 22 µm) that includes a 40-cm telescope.

The satellite was placed at an altitude of 525 km, and like AKARI it had a sun-synchronous polar orbit in order to look out at right angles to the Sun and always point away from Earth. A picture was taken every 11 seconds and covered an area of the sky three times larger than the full Moon. A total of 1.5 million images was retrieved, covering hundreds of millions of astronomical objects and more than 99 % of the sky.⁴

3.4 ALMA

Atacama Large Millimeter/submillimeter Array (ALMA) is located on the Cha- jnantor Plateau in the Chilean Andes and consist of 66 antennas: fifty-four 12- meter diameter antennas and twelve 7-meter diameter antennas. The location with an altitude of 5000 meters above sea level, as well as its dry climate and clear skies, allows ALMA to capture far infrared waves that is emitted from var- ious sources in the universe. All 66 antennas work together as an interferometer which enable them to act as one single giant radio telescope equal in size to the total array. For the Deathstar sample, ALMA observations were performed in bands 6 (230 GHz) and 7 (345 GHz) with a smaller part of the array (the Atacama Compact Array, ACA), thus providing data points at a previously unavailable wavelength range.⁵

3https://herschel.jpl.nasa.gov/mission.shtml

4http://wise.ssl.berkeley.edu/mission.html

5https://www.almaobservatory.org/en/about-alma-at-first-glance

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4 Results

The available data points for each star in the sample were combined to create the SED plots. To account for the variability of the stars, particularly at shorter wavelengths, an uncertainty was assumed as follows: 80% for wavelengths less than 1.25 micron, 50% between 1.25 and 2.5 micron, 40% between 2.5 and 3.75 micron, 30% between 3.75 and 5 micron, and 20% for wavelengths above 5 micron. To further constrain the contribution of the variability at shorter wavelenghts, data points below 0.8 micron were not included in the calculations for finding the best-fit model.

The publicly available code DUSTY⁶ was used to solve the dust radiative transfer. The CSE is assumed to be spherically symmetric with a continuously expanding wind. Dust properties of amorphous carbon grains from Rouleau and Martin [6] were applied, assuming a grain radius of 0.1 micron and density 2 g cm⁻³. Three parameters are adjustable; the dust optical depth at 10 µm, τ10, the dust temperature, Td, and the stellar temperature, T?. The input stellar spectra was retrieved from DARWIN starting models (Sara Bladh, priv.

comm.) for stellar temperatures between 2000-2400 K, and COMARCS models⁷ for stellar temperatures between 2500-3200 K. The optical depth was ranged from 0.01 to 5 µm in steps of 10%, and the dust temperature was varied in steps of 100 K from 500 to 1500 K. This resulted in a large grid of models which can be scaled to the luminosity and distance of any star. The distances used for our sample was retrieved from Gaia data release 2, and the luminosity was adjusted from [7] to fit the new distances.

The best fit model is found by minimizing

χ²=

N

X

i=1

(Fmod,λ− Fobs,λ)²

σ²_λ , (5)

where Fλ is the flux density and σλ is the uncertainty in the measured flux density at wavelength λ, and the summation is done over all N independent observations. The reduced χ²for the best fit model is given by

χ²_red= χ²_min

N − p, (6)

where p is the number of adjustable parameters (3 in our case). Table 1 lists the result from the SED modelling. SS Vir had one of the best fits and the plot can be seen in figure 4. Overall the result varies from relatively well fitted models (χ²_redclose to 1), see figure A.1, to poorly fitted data points (large χ²_red). Y Pav, V688 Mon and V1259 Ori could not be fitted with the Gaia distances due to that unreasonable high luminosity was needed to compensate for the increased distance. In general, the ALMA data points in the cold dust region did not fit the models, which cause larger χ²_redvalues. Further, it was not possible to find

6http://faculty.washington.edu/ivezic/dusty_web/

7http://stev.oapd.inaf.it/atm/lrspe.html

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Table 1: SED modelling

Source D L? T? τ10 Td χ²_red N

[pc] [L] [K] [K]

C-type semi-regular and irregular stars:

TW Oph 539 3000 2500 0.03 1500 1.78 18

NP Pup 443 3500 3000 0.01 1500 6.57 18

TW Hor 395 6700 3000 0.01 1500 15.03 20

T Ind 553 8800 2900 0.01 1500 19.00 16

RT Cap 404 5900 2900 0.01 1500 17.62 19

AQ Sgr 507 4000 2900 0.01 1400 1.35 13

U Hya 158 2000 3100 0.01 1500 6.40 11

W Ori 782 9000 2300 0.05 1500 1.84 21

V Aql 369 5000 2800 0.01 1500 2.20 19

Y Pav 360^a 3300 2900 0.01 1500 1.75 17

X Vel 582 5000 2100 0.01 1500 1.88 20

Y Hya 437 4200 2700 0.01 1500 1.04 18

SS Vir 639 5400 2000 0.02 1500 1.03 22

W CMa 509 4000 2900 0.01 1500 1.87 18

C-type Mira stars:

R Lep 400 5500 2400 0.03 1500 2.23 18

CZ Hya 2222 14000 2000 0.09 1500 1.21 21

R For 584 5800 2100 0.04 1500 3.73 21

R Vol 750 6800 2000 0.09 1500 1.61 17

RV Aqr 782 8000 2300 0.07 1500 5.44 19

V688 Mon 700^a 2000 2000 0.4 1500 22.68 16

V1259 Ori 1600^a 9300 2100 0.9 1500 30.73 29

aNot Gaia distances

a good fit for T Ind due to the profile of the data points, see figure B.1. In the region of about 100 micron, the data points deviate from the expected shape, which could be an indication of a detached shell.

5 Conclusions

Dust emission modelling of 21 nearby AGB carbon stars, all a part of the Death- star sample, have been performed and the following conclusions were made:

– When comparing our result with previous work ([7]), the main differences we have adopted are that we are using a stellar spectrum input instead of a blackbody, that we have used other distances (retrieved from Gaia release 2), and that we have more data points. We find that the sample of

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Figure 4: SED of SS Vir, the dashed line is the model input spectrum and the solid line is the best-fit model.

stars that have previously been modeled (about 10/21) are surrounded by less dust, i.e. higher Td and lower τ10. We also retrieve a slightly higher χ²_red value which could be related to the increased number of data points at longer wavelengths.

– The ALMA data points at 230 GHz (1300 µm) and 345 GHz (868 µm) were generally difficult to fit. One possible explanation could be that the optical properties of the dust at wavelengths larger than 300 µm (i.e. not covered by laboratory measurements in [6]) are different than what has been assumed.

– The stars in our sample cover a stellar temperature range from 2000 to 3100 K, with 11 sources ≥ 2500K and 10 sources < 2500K which clearly shows that models below 2500 K should not be ruled out when studying AGB stars.

– The SED of T Ind, which deviates from the expected line shape of the model, suggests that the star is a possible detached shell candidate which requires further investigation.

– The fact that the Gaia distances were unreasonable to use for Y Pav, V688 Mon and V1259 Ori is an indication that the retrieved distances might not be applicable to Mira stars, and that the period-luminosity relation is a more reliable method for measuring distances to these stars.

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References

[1] S. Bladh and S. H¨ofner. Exploring wind-driving dust species in cool luminous giants. I. Basic criteria and dynamical models of M-type AGB stars. A&A, 546:A76, 2012.

[2] C.P. Dullemond. Lecture notes - chapter 5: Radiative transfer in dusty media, July 2012.

[3] S. H¨ofner. Dust Formation and Winds around Evolved Stars: The Good, the Bad and the Ugly Cases. Astronomical Society of the Pacific Conference Series, 414, 2009.

[4] H. Olofsson and S. H¨ofner. Mass loss of stars on the asymptotic giant branch.

Astron Astrophys Rev, 26, 2018.

[5] Sofia Ramstedt. Molecules and dust around AGB stars. PhD thesis, Stock- holm University, Stockholm, 2009.

[6] F. Rouleau and P.G. Martin. Shape and Clustering Effects on the Optical Properties of Amorphous Carbon. Astrophysical Journal, 377, 1991.

[7] F. L. Sch¨oier, S. Ramstedt, H. Olofsson, M. Lindqvist, J.H.Bieging, and K.B. Marvel. The abundance of HCN in circumstellar envelopes of AGB stars of different chemical type. A&A, 550:A78, 2013.

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A SED best-fits

(a) (b)

(b) (c)

(d) (e)

Figure A.1: SED plots of the best-fits. The dashed line is the model input spectrum and the solid line is the best-fit model.

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B T Ind

Figure B.1: SED of T Ind, the dashed line is the model input spectrum and the solid line is the best-fit model which clearly deviates from the the data points.

Dust emission modelling of AGB stars