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Maryam Saberi

Publications and source records attributed to Maryam Saberi.

At least 19 recordsLinked to original sources

The unique capabilities of HST for stellar physics: Probing Atmospheric Structure, Chromospheres, and Mass Loss of Evolved Stars

Evolved stars are among the primary sources of chemical enrichment and dust production in galaxies. During the giant phases, stars return a substantial fraction of their mass to the interstellar medium (ISM) through stellar winds, enriching galaxies with newly synthesized elements and dust. However, the atmospheric structure and physical processes that initiate mass loss remain poorly constrained observationally. Understanding the origin, structure, and evolution of stellar chromospheres remains a long-standing problem in stellar astrophysics. While the mechanisms responsible for chromospheric heating and atmospheric dynamics are not fully understood even in the Sun, they become more complex in evolved stars due to pulsation, shocks, convection, extended atmospheres, and possible magnetic activity. Determining the thermal, density, and velocity structure of these extended atmospheres is therefore essential for understanding atmospheric heating, the onset of mass loss, and the late stages of stellar evolution. High-resolution NUV and FUV spectroscopy (R ~ 30,000-100,000) provided by HST/STIS occupies a unique observational parameter space that cannot be replaced by existing facilities. HST/STIS therefore remains essential for understanding the atmospheric physics and mass-loss processes of evolved stars. We highlight the need to preserve and prioritize high-resolution NUV and FUV spectroscopic capabilities with HST. Such programs would provide essential benchmarks for stellar atmosphere modeling, complement ongoing ALMA and optical observations, and help define future UV-optical capabilities for the Habitable Worlds Observatory (HWO).

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HST observations of chromospheric UV lines in the AGB star R Leo

The role of stellar chromospheres in the chemistry, mass loss, and evolution of cool evolved stars remains poorly understood. We present high-resolution ultraviolet spectra of the nearby Mira-type AGB star R~Leo obtained with STIS on board the \textit{Hubble Space Telescope}. We focus on two strong chromospheric diagnostics, the Mg\,\textsc{ii} h \& k resonance lines and the C\,\textsc{ii}] 2325\,Å multiplet, and model their formation with the NLTE radiative-transfer code RH using phase-dependent hydrodynamic atmospheric structures that include pulsation-driven shocks. The observed C\,\textsc{ii}] multiplet ratios imply an electron density of order $10^9\,\mathrm{cm^{-3}}$. The RH contribution functions show that the C\,\textsc{ii}] emission is strongly localized in a compact shock-heated shell at the first temperature peak of the model atmosphere, near $R\simeq1.8~R_{\rm phot}$, where $T\sim10^4$\,K and the local electron density is a few $10^8\,\mathrm{cm^{-3}}$. In contrast, the Mg\,\textsc{ii} h \& k lines probe a more extended region ($R< 16~R_{\rm phot}$), with the line cores forming at greater radial distances than the wings. The Mg\,\textsc{ii} h line is reproduced more reliably than the k line, which is more strongly affected by circumstellar and interstellar absorption. As an additional low-opacity kinematic check, the semi-forbidden Al\,\textsc{ii}] $\lambda2669$ line shows a projected stellar-rest-frame blueshift of $\sim6~{\rm km~s^{-1}}$, consistent with shock-related motions expected in Mira atmospheres. Overall, the compact C\,\textsc{ii}] formation region, the extended Mg\,\textsc{ii} contribution functions, and the strong phase dependence of the synthetic Mg\,\textsc{ii} profiles support a picture in which pulsation-driven shocks shape the ultraviolet chromospheric emission lines in Mira variable stars.

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Millimeter-Wavelength Observations of the Active Sun: Unveiling the Origins of Space Weather

Societal dependence on space-based services demands major advances in predicting the impacts of eruptive solar events. Millimeter-wavelength observations offer uniquely direct access to the time-dependent physical conditions in the atmospheric layers of the Sun where these events originate. A facility capable of full-disk, high-cadence, multi-frequency imaging would provide a transformative view of the Sun and its influence on the heliosphere. AtLAST is ideally suited to deliver this capability, and to establish a European leadership role in advancing the scientific foundations that will enable reliable, operational space-weather forecasting for the first time.

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Probing sulfur chemistry in oxygen-rich AGB stars with ALMA

Sulfur and its isotopic ratios play a crucial role in understanding astrophysical environments, providing insights into nucleosynthesis, ISM processes, star formation, planetary evolution, and galactic chemistry. We investigate the distribution of sulfur bearing species $\rm{SO_2}$, $\rm{^{34}SO_2}$, SO, and $\rm{^{34}SO}$ towards five oxygen rich Asymptotic Giant Branch (AGB) stars ($o$ Ceti, R Dor, W Hya, R Leo, and EP Aqr), along with their excitation temperatures, column densities, and isotopic ratios. Using ALMA Band 6,7,8 data and CASSIS, we detect these species and estimate excitation temperature and column density via the rotational diagram and MCMC methods under LTE. Line imaging of various transitions is used to infer spatial distributions. The excitation temperatures of $\rm{SO_2}$ range from $\sim$200-600 K with column densities of $\rm{1-7\times10^{16}\ cm^{-2}}$, while $\rm{^{34}SO_2}$ shows comparable or slightly lower values and about an order of magnitude lower column densities. The $\rm{^{32}S/^{34}S}$ ratios for R Dor and W Hya are near solar, slightly higher for $o$ Ceti, and lower for EP Aqr and R Leo. Most detected lines exhibit centralized emission: high excitation $\rm{SO_2}$ traces compact hot gas in inner CSEs, whereas low-excitation lines trace more extended structures. Morphological differences, irregular emission in $o$ Ceti, circular in R Leo and W Hya, clumpy in R Dor, and unresolved in EP Aqr may arise from variations in physical conditions, multiplicity, outflows, rotation, desorption processes, UV or cosmic ray effects, or observational resolution. Overall, the centralized SO and $\rm{SO_2}$ emissions support previous findings for low mass-loss rate AGB stars, and the $\rm{^{32}S/^{34}S}$ ratios likely reflect natal cloud composition, with deviations linked to metallicity or excitation conditions.

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The SOFIA Massive (SOMA) Star Formation Q-band Follow-up. II. Hydrogen Recombination Lines Toward High-Mass Protostars

Hydrogen recombination lines (HRLs) are valuable diagnostics of the physical conditions in ionized regions around high-mass stars. Understanding their broadening mechanisms and intensity trends can provide insights into the densities, temperatures, and kinematics of HII regions. We investigate the properties of ionized gas around massive protostars by analyzing hydrogen recombination lines (H-alpha and H-beta) in the Q-band. Observations were conducted using the Yebes 40m radio telescope in the Q-band (30.5~50 GHz) toward six high-mass protostars selected from the SOMA Survey (G45.12+0.13, G45.47+0.05, G28.20-0.05, G35.20-0.74, G19.08-0.29, and G31.28+0.06). The line profiles were analyzed to assess broadening mechanisms, from which electron densities and temperatures were derived. We compared our results with Q-band data from the TianMa 65m Radio Telescope (TMRT) and ALMA Band 1 Science Verification observations of Orion KL. A total of eight H-alpha (n = 51 to 58) and ten H-beta (n = 64 to 73) lines were detected toward G45.12+0.13, G45.47+0.05, and G28.20-0.05, with non-detections in the other sources. Electron densities of ~1-5$\times$10$^6$ cm$^{-3}$ and temperatures of 8000-10000 K were derived. Orion KL shows one order of magnitude lower electron density, but a similar temperature. Notably, G45.12 and G28.20 show increasing intensity with frequency for both H-alpha and H-beta, in contrast to the decreasing trend in Orion KL. The observed line widths indicate contributions from both thermal and dynamical broadening, suggesting high-temperature ionized gas affected by turbulence, outflows, rotation, or stellar winds. Pressure broadening may also play a minor role. The contrasting intensity trends likely reflect differences in local physical conditions or radiative transfer effects, warranting further study through higher-resolution observations and modeling.

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Deciphering Galactic Halos: A Detailed Review of Star Formation in NGC 5128 (Cen A)

NGC 5128 (Centaurus A), the closest giant elliptical galaxy outside the Local Group to the Milky Way, is one of the brightest extragalactic radio sources. It is distinguished by a prominent dust lane and powerful jets, driven by a supermassive black hole at its core. Using previously identified long-period variable (LPV) stars from the literature, this study aims to reconstruct the star formation history (SFH) of two distinct regions in the halo of NGC 5128. These regions reveal remarkably similar SFHs, despite being located about 28 kpc apart on opposite sides of the galaxy's center. In Field 1, star formation rates (SFRs) show notable increases at approximately 800 Myr and 3.8 Gyr ago. Field 2 exhibits similar peaks at these times, along with an additional rise around 6.3 Gyr ago. The increase in SFR around 800 Myr ago is consistent with earlier research suggesting a merger event. Since no LPV catalog exists for the central region of NGC 5128, we focused our investigation on its outer regions, which has provided new insights into the complex evolutionary history of this cornerstone galaxy. The SFH traced by LPVs supports a scenario in which multiple events of nuclear activity have triggered episodic, jet-induced star formation.

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The Sun at millimeter wavelengths V. Magnetohydrodynamic waves in a fibrillar structure

Magnetohydrodynamic (MHD) waves, playing a crucial role in transporting energy through the solar atmosphere, manifest in various chromospheric structures. Here, we investigated MHD waves in a long-lasting dark fibril using high-temporal-resolution (2~s cadence) Atacama Large Millimeter/submillimeter Array (ALMA) observations in Band 6 (centered at 1.25~mm). We detected oscillations in brightness temperature, horizontal displacement, and width at multiple locations along the fibril, with median periods and standard deviations of $240\pm114$~s, $225\pm102$~s, and $272\pm118$~s, respectively. Wavelet analysis revealed a combination of standing and propagating waves, suggesting the presence of both MHD kink and sausage modes. Less dominant than standing waves, oppositely propagating waves exhibit phase speeds (median and standard deviation of distributions) of $74\pm204$~km/s, $52\pm197$~km/s, and $28\pm254$~km/s for the three observables, respectively. This work demonstrates ALMA's capability to effectively sample dynamic fibrillar structures, despite previous doubts, and provides valuable insights into wave dynamics in the upper chromosphere.

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Atacama Large Aperture Submillimeter Telescope (AtLAST) Science: Solar and stellar observations

Observations at (sub-)millimeter wavelengths offer a complementary perspective on our Sun and other stars, offering significant insights into both the thermal and magnetic composition of their chromospheres. Despite the fundamental progress in (sub-)millimeter observations of the Sun, some important aspects require diagnostic capabilities that are not offered by existing observatories. In particular, simultaneous observations of the radiation continuum across an extended frequency range would facilitate the mapping of different layers and thus ultimately the 3D structure of the solar atmosphere. Mapping large regions on the Sun or even the whole solar disk at a very high temporal cadence would be crucial for systematically detecting and following the temporal evolution of flares, while synoptic observations, i.e., daily maps, over periods of years would provide an unprecedented view of the solar activity cycle in this wavelength regime. As our Sun is a fundamental reference for studying the atmospheres of active main sequence stars, observing the Sun and other stars with the same instrument would unlock the enormous diagnostic potential for understanding stellar activity and its impact on exoplanets. The Atacama Large Aperture Submillimeter Telescope (AtLAST), a single-dish telescope with 50\,m aperture proposed to be built in the Atacama desert in Chile, would be able to provide these observational capabilities. Equipped with a large number of detector elements for probing the radiation continuum across a wide frequency range, AtLAST would address a wide range of scientific topics including the thermal structure and heating of the solar chromosphere, flares and prominences, and the solar activity cycle. In this white paper, the key science cases and their technical requirements for AtLAST are discussed.

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AtLAST Science Overview Report

Submillimeter and millimeter wavelengths provide a unique view of the Universe, from the gas and dust that fills and surrounds galaxies to the chromosphere of our own Sun. Current single-dish facilities have presented a tantalising view of the brightest (sub-)mm sources, and interferometers have provided the exquisite resolution necessary to analyse the details in small fields, but there are still many open questions that cannot be answered with current facilities. In this report we summarise the science that is guiding the design of the Atacama Large Aperture Submillimeter Telescope (AtLAST). We demonstrate how tranformational advances in topics including star formation in high redshift galaxies, the diffuse circumgalactic medium, Galactic ecology, cometary compositions and solar flares motivate the need for a 50m, single-dish telescope with a 1-2 degree field of view and a new generation of highly multiplexed continuum and spectral cameras. AtLAST will have the resolution to drastically lower the confusion limit compared to current single-dish facilities, whilst also being able to rapidly map large areas of the sky and detect extended, diffuse structures. Its high sensitivity and large field of view will open up the field of submillimeter transient science by increasing the probability of serendipitous detections. Finally, the science cases listed here motivate the need for a highly flexible operations model capable of short observations of individual targets, large surveys, monitoring programmes, target of opportunity observations and coordinated observations with other observatories. AtLAST aims to be a sustainable, upgradeable, multipurpose facility that will deliver orders of magnitude increases in sensitivity and mapping speeds over current and planned submillimeter observatories.

astro-ph.IM

The Complex Star Formation History of the Halo of NGC 5128 (Cen A)

NGC 5128 (Cen A) is the nearest giant elliptical galaxy and one of the brightest extragalactic radio sources in the sky, boasting a prominent dust lane and jets emanating from its nuclear supermassive black hole. In this paper, we construct the star formation history (SFH) of two small fields in the halo of NGC 5128: a northeastern field (Field 1) at a projected distance of $\sim 18.8$ kpc from the center, and a southern field (Field 2) $\sim 9.9$ kpc from the center. Our method is based on identifying long period variable (LPV) stars that trace their sibling stellar population and hence historical star formation due to their high luminosity and strong variability; we identified 395 LPVs in Field 1 and 671 LPVs in Field 2. Even though the two fields are $\sim 28$ kpc apart on opposite sides from the center, they show similar SFHs. In Field 1, star formation rates (SFRs) increased significantly around $t\sim 800$ Myr and $t\sim 3.8$ Gyr; and in Field 2, SFRs increased considerably around $t\sim 800$ Myr, $t\sim 3.8$ Gyr, and $t\sim 6.3$ Gyr, where $t$ is look--back time. The increase in SFR $\sim 800$ Myr ago agrees with previous suggestions that the galaxy experienced a merger around that time. The SFH reconstructed from LPVs supports a scenario in which multiple episodes of nuclear activity lead to episodic jet-induced star formation. While there is no catalog of LPVs for the central part of NGC 5128, applying our method to the outer regions (for the first time in a galaxy outside the Local Group) has enabled us to put constraints on the complex evolution of this cornerstone galaxy.

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The formation and evolution of Andromeda IX

Local Group (LG), the nearest and most complete galactic environment, provides valuable information on the formation and evolution of the Universe. Studying galaxies of different sizes, morphologies, and ages can provide this information. For this purpose, we chose the And\,IX dSph galaxy, which is one of the observational targets of the Isaac Newton Telescope (INT) survey. A total of 50 long-period variables (LPVs) were found in And\,IX in two filters, Sloan $i'$ and Harris $V$ at a half-light radius of 2.5 arcmin. The And\,IX's star formation history (SFH) was constructed with a maximum star formation rate (SFR) of about $0.00082\pm0.00031$ M$_\odot$ yr$^{-1}$, using LPVs as a tracer. The total mass return rate of LPVs was calculated based on the spectral energy distribution (SED) of about $2.4\times10^{-4}$ M$_\odot$ yr$^{-1}$. The distance modulus of $24.56_{-0.15}^{+0.05}$ mag was estimated based on the tip of the red giant branch (TRGB).

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The Isaac Newton Telescope Monitoring Survey of Local Group Dwarf Galaxies. VI. The Star Formation History and Dust Production in Andromeda IX

We present a photometric study of the resolved stellar populations in And IX, the closest satellite to the M31, a metal-poor and low-mass dwarf spheroidal galaxy. We estimate a distance modulus of $24.56_{-0.15}^{+0.05}$ mag based on the tip of the red giant branch (TRGB). By probing the variability of asymptotic giant branch stars (AGB), we study the star formation history of And IX. We identified 50 long period variables (LPVs) in And IX using the Isaac Newton Telescope (INT) in two filters, Sloan $i'$ and Harris $V$. In this study, we selected LPVs within two half-light radii with amplitudes in the range of 0.2-2.20 mag. It is found that the peak of star formation reaches $\sim$ $8.2\pm3.1\times10^{-4}$ M_sun yr$^{-1}$ at $\approx 6$ Gyr ago. Our findings suggest an outside-in galaxy formation scenario for And IX with a quenching occurring $3.65_{-1.52}^{+0.13}$ Gyr ago with the SFR in the order of $2.0\times10^{-4}$ M_sun yr$^{-1}$ at redshift < $0.5$. We calculate the total stellar mass by integrating the star formation rate (SFR) within two half-light radii $\sim$ $3.0\times10^5$ M_sun. By employing the spectral energy distribution (SED) fitting for observed LPVs in And IX, we evaluate the mass-loss rate in the range of $10^{-7}$ $\leq$ $\dot{M}$ $\leq$ $10^{-5}$ M_sun yr$^{-1}$. Finally, we show that the total mass deposition to the interstellar medium (ISM) is $\sim$ $2.4\times10^{-4}$ M_sun yr$^{-1}$ from the C- and O-rich type of dust-enshrouded LPVs. The ratio of the total mass returned to the ISM by LPVs to the total stellar mass is $\sim 8.0\times10^{-10}$ yr$^{-1}$, and so at this rate, it would take $\sim$ 1 Gyr to reproduce this galaxy

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The Isaac Newton Telescope monitoring survey of Local Group dwarf galaxies--V. The star formation history of Sagittarius dwarf irregular galaxy derived from long period variable stars

We conducted an optical monitoring survey of the Sagittarius dwarf irregular galaxy (SagDIG) during the period of June 2016 -- October 2017, using the 2.5-m Isaac Newton Telescope (INT) at La Palama. Our goal was to identify Long Period Variable stars (LPVs), namely asymptotic giant branch stars (AGBs) and red supergiant stars (RSGs), to obtain the Star Formation History (SFH) of isolated, metal-poor SagDIG. For our purpose, we used a method that relies on evaluating the relation between luminosity and the birth mass of these most evolved stars. We found $27$ LPV candidates within two half-light radii of SagDIG. $10$ LPV candidates were in common with previous studies, including one very dusty AGB (x-AGB). By adopting the metallicity $Z = 0.0002$ for older population and $Z=0.0004$ for younger ages, we estimated that the star formation rate changes from $0.0005\pm0.0002$ M$_{\odot}$yr$^{-1}$kpc$^{-2}$ ($13$ Gyr ago) to $0.0021 \pm 0.0010$ M$_{\odot}$yr$^{-1}$kpc$^{-2}$ ($0.06$ Gyr ago). Like many dwarf irregular galaxies, SagDIG has had continuous star formation activity across its lifetime, though with different rates, and experiences an enhancement of star formation since $z \simeq 1$. We also evaluated the total stellar mass within two half-light radii of SagDIG for three choices of metallicities. For metallicity $Z = 0.0002$ and $Z=0.0004$ we estimated the stellar mass M$_*$ = ($5.4 \pm 2.3$) $\times$ $10^ 6$ and ($3.0 \pm 1.3$) $\times$ $10^ 6$ M$_{\odot}$, respectively. Additionally, we determined a distance modulus $μ$ = $25.27\pm0.05$ mag, using the tip of the red giant branch (TRGB).

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Star Formation History of Two Fields in the Halo of NGC5128

NGC5128 galaxy is a giant elliptical galaxy located in the Centaurus group of galaxies at 3.8 Mpc. We aim to study the star formation history (SFH) of two different fields of the galaxy. The northeastern field (Field 1) is located at a distance of 18.8 kpc, while the southern field (Field 2) is at 9.9 kpc. We use a photometric method that is based on identifying long period variable (LPV) stars and asymptotic giant branch (AGB) stars, as they are strong tracers of star formation and galaxy evolution due to their luminosity and variability; 395 LPVs in Field 1 and 671 LPVs in Field 2 have been identified. These two fields present similar SFHs, although the SF rate of Field 2 is more enhanced. We find that the galaxy has three major star formation episodes t $\sim$ 800 Myr ago, t $\sim$ 3.2 Gyr ago, and t $\sim$ 10 Gyr ago, where t is look-back time. The rate of star formation at $\sim$ 800 Myr ago agrees with previous studies suggesting that the galaxy experienced a merger around that time. Furthermore, NGC5128 has experienced a lower star formation rate in its recent history which could have been driven by jet-induction star formation and multiple outbursts of AGN activity in this galaxy, as well as a minor merger around 400 Myr ago.

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EMISSA -- Exploring Millimetre Indicators of Solar-Stellar Activity II. Towards a robust indicator of stellar activity

An activity indicator, which can provide a robust quantitative mapping between the stellar activity and the physical properties of its atmosphere, is important in exploring the physics of activity across spectral types. But the common activity indicators show large variability in their values which makes defining a robust quantitative scale difficult. Millimetre (mm) wavelengths probe the different atmospheric layers within the stellar chromosphere providing a tomographic view of the atmospheric dynamics. The project aims to define a robust mm-based activity indicator for the cool main-sequence stars ($\mathrm{T_{eff}} \sim$ 5000 - 7000 K). We derive the mm-brightness temperature ($\mathrm{T_B(ν)}$) spectral indices ($\mathrm{α_{mm}}$) for cool stars including the Sun using archival data in the 30 - 1000 GHz range. The derived values for $\mathrm{α_{mm}}$ are explored as a function of various physical parameters and empirical power-law functions were derived. $\mathrm{α_{mm}}$ estimates were also compared with other activity indicators. Despite the estimation errors, $\mathrm{α_{mm}}$ values could well distinguish the cool stars, unlike common activity indicators. The low estimation errors on the derived trends of $\mathrm{α_{mm}}$ versus physical parameters suggest that $\mathrm{α_{mm}}$ could be a robust activity indicator. $\mathrm{α_{mm}}$, which is linked to chromospheric thermal stratification and activity in cool stars can well distinguish and physically characterise the stars more robustly than common activity indicators. We emphasise the need for multi-frequency data across the mm-band for stars, with a range of physical parameters and gathered at multiple epochs during activity cycles. This will help explore $\mathrm{α_{mm}}$ in a statistically robust manner and study the emergence of chromospheric heating on the main-sequence.

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Tracing the role of AGB stars in the Galactic Fluorine budget

The cosmic origin of fluorine is still under debate. Asymptotic giant branch (AGB) stars are among the few suggested candidates to efficiently synthesis F in our Galaxy, however their relative contribution is not clear. In this paper, we briefly review the theoretical studies from stellar yield models of the F synthesis and chemical equilibrium models of the F-containing molecules in the outflow around AGB stars. Previous detections of the F-bearing species towards AGB and post-AGB stars are also highlighted. We suggest that high-resolution ALMA observations of the AlF, one of the two main carriers of F in the outflow of AGB stars, can provide a reliable tracer of the F-budget in AGB stars. This will be helpful to quantify the role of AGB stars in the Galactic F budget.

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EMISSA -- Exploring Millimeter Indicators of Solar-Stellar Activity I. The Initial mm-cm Main Sequence Star Sample

Due to their wide wavelength coverage across the millimetre to centimetre (mm - cm) range and their increased sensitivity, modern interferometric arrays facilitate observations of the thermal and non-thermal emission from different stellar atmospheric layers. We study the spectral energy distribution ($S_{obs}(ν)$) of main sequence stars using archival mm - cm data with the aim to study their atmospheric stratification as a function of stellar type. The main-sequence stars with significant detection in mm bands were identified in the ALMA Science Archive. These data were complemented with spectral flux data in the Ultra violet to centimetre range as compiled from various catalogues and observatory archives. We compare the resultant $S_{obs}(ν)$ of each star with a photospheric emission model ($S_{mod}(ν)$) calculated with the PHOENIX code. The departures of $S_{obs}(ν)$ from the model are quantified in terms of a "spectral flux excess" parameter ($ΔS/S_{mod} (ν)$) for every star in the sample. The initial sample consists of 12 stars across a range of spectral type from A1 to M3.5 and the Sun-as-a-star as reference. The stars with $T_{eff} = 4000 - 7000\,K$ (F - M type) showed a systematically higher $ΔS/S_{mod}$ in the mm - cm range, with the values rising with decreasing $ν$. The steepness of this rise is higher for cooler stars, though the fully convective 3000 K star in the sample deviated from this trend. For the A-type stars, $ΔS/S_{mod} \sim 0$ within errors. The high $ΔS/S_{mod}$ in cool stars points to the presence of hotter upper atmospheric layers, i.e. a chromosphere and corona, like for the Sun. The mm - cm $ΔS/S_{mod}$ spectrum offers a way to estimate the efficiency of the heating mechanisms across various atmospheric layers and thereby to understand their structure and activity. We emphasise the need for more mm - cm data.

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Reflections on the photodissociation of CO in circumstellar envelopes

Carbon monoxide (CO) is the most abundant molecule after molecular hydrogen and is important for the chemistry in circumstellar envelopes around evolved stars. When modelling the strength and shape of molecular lines, the size of the CO envelope is an input parameter and influences the derived mass-loss rates. In particular the low-J transition CO lines are sensitive to the CO photodissociation radius. Recently, new CO photodissociation radii have been published using different formalisms that differ considerably. One set of calculations is based on an escape-probability formalisms that uses numerical approximations derived in the early-eighties. The accuracy of these approximations is investigated and it is shown that they are less accurate than claimed. Improved formalism are derived. Nevertheless, the changes in CO envelope size are small to moderate, less than 2\% for models with $10^{-7}< \dot{M}< 10^{-4}$ \msolyr\ and at most 7\% for model with $\dot{M} = 10^{-8}$ \msolyr.

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