Astronomy

Photosphere Thickness and Pressure vs stellar type

Photosphere Thickness and Pressure vs stellar type

Where can I find information on how the thickness and pressure of a stellar photosphere is expected to vary with star type (temperature). I, as an amateur, have attempted a "simple" analysis of spectral line profiles to estimate these properties and whilst it gives good results for the Sun I can find no data on other stars with which to test my analysis. My work can be found on my website and I am particularly interested in values for the blue component of Albireo and for Altair.


Photosphere

V. Martínez Pillet , in COSPAR Colloquia Series , 2002

INTRODUCTION

The photosphere is the portion of the atmosphere where we are able to measure with some confidence the magnetic field that pervades the Sun outer layers. Our understanding of the structuring of the magnetic field in the photosphere ( Solanki, 1999 ) has provided the background to explain some of the energetic processes that take place in higher layers. It is of interest to see the large number of works that have appeared in recent years where a relation between interacting bipoles and energy release events is presented. They are described to some extend in this work. The flux losses (cancellation) associated with these interacting bipoles is believe to play a major role in filament formation, eruption and, in general, CME initiation. We study several examples where this mechanism is seen to operate and discuss the alternatives to explain it.


Solar and Stellar Variability

2.2.1.3 Zeeman Doppler Imaging

In addition to these global observations of integrated stellar signals, new surveys aimed at obtaining information about the geometry of the magnetic field have been undertaken. They rely on the Zeeman effect in spectral lines, which is the only direct source of information about the strengths and topologies of stellar magnetic fields . ZDI, first introduced by Semel (1989) and Donati et al. (1989) , is a powerful tomographic technique to map the large-scale distribution of stellar magnetic fields. It often uses a set of circular polarization profiles collected over one or several rotations and converts these profiles into a magnetic map of the stellar photosphere. The BCool Collaboration 4 has undertaken a ZDI survey of 170 cool stars to determine how the field geometry and properties depend on fundamental stellar parameters magnetic fields were detected on 67 of them ( Marsden et al., 2014 ).

Although the circular polarization signal in individual lines is within the observational noise multiline techniques (least square deconvolution) Donati et al. (1997) , or multiline singular value decomposition, Carroll et al. (2012) allow to detect a polarization signature. However, most of the time the linear polarization signal remains small and hardly detectable.

Tomographic inversion of the signal uses of decomposition of the magnetic structures on spherical harmonics. Only the large-scale, i.e., low-order terms with l < 6 may be recovered. A test of the method is presented in Kochukhov et al. (2017) , in which simulated Sun-as-a-star spectropolarimetric observations are reconstructed from SDO/HMI solar magnetograms and photospheric images for one solar Carrington rotation. Then they are analyzed with the ZDI techniques to recover the large-scale magnetic map. Comparison of the maps of radial magnetic field given by the ZDI techniques and the original HMI data after filtering its high-order harmonic components shows qualitatively similar configurations. A quantitative analysis of the longitudinal magnetic flux in the detailed solar magnetograms shows that 95% of the flux is already contained in the low-harmonic modes with l < 6. However, less than 1% of the total magnetic energy is present in the large-scale reconstruction. The magnetic energy content at a small scale has a major role that is not recovered by ZDI based on circular polarization profiles and the mean magnetic strength is underestimated.

Another difficulty of ZDI inversions is that the temperature distribution caused by spots on the stellar surface must be disentangled from the magnetic field distribution, so the magnetic reconstruction is improved if both I and V profiles are used in the inversion procedure. The average magnetic strength is also better recovered when the information on the transverse component contained in the linear polarization is available.


Luminosity classification is based upon the widths of the absorption lines in the star's spectrum.

  • Lines get broader as the pressure increases.
  • Big stars are puffier, which means the pressure in their atmosphere is lower.
  • Larger stars have narrower absorption lines (lower gravity, lower pressure).
  • Larger stars are brighter at the same temperature (larger R, larger L).

This gives us a way to assign a relative Luminosity to stars based upon their spectral line properties!


II. COMPARISON OF TWO CANONICAL SOURCE TYPES

    • Structures
    • Spectra: Continua
    • Spectra: "Lines" (fine structure)
    • Information carried by their spectra

      NB: photospheres (10 17 particles/cm 3 in the Sun) are DENSE by the standards of AGN emitting regions

      Supermassive black hole (M

    3 x 10 13 M8 cm) at center of an accretion disk with r 10 16 cm.

      Primary component is UVOIR radiation from photosphere.

      Thermal source emergent spectrum Planck function

      Primary component is very broad-band, nonthermal radiation extending from radio to XR.

      Complex, narrow, absorption lines

      Produced by transitions in those atoms, ions, and molecules which are prevalent at characteristic Te and pressure.

      Continuum slope & structure: Temperature

      Fit full energy distribution with combinations of single generation models to determine star formation history & abundances

      Slope & structure of continuum related to energy distribution of electrons, importance of Compton scattering, accretion disk structure, dust emission/absorption, etc.

      Less definitive interpretation than stellar continua because of multiple components, complex generation mechanisms, absence of near-TEQ.

    where f is the flux, B is Planck function and is the angular area (or upper limit) of the source. Is T "unphysically" high?

      . although of different species than in stars: e.g. O, N, He, S, Fe (uncommon). but not Ca, Mg (unless UV access), etc.

      Properties of stars: D. F. Gray (1992) "Observation & Analysis of Stellar Photospheres"

    Last modified December 2020 by rwo

    Text copyright © 2000-2020 Robert W. O'Connell. All rights reserved. These notes are intended for the private, noncommercial use of students enrolled in Astronomy 511 at the University of Virginia.


    KINETIC THEORY

    George B. Arfken , . Joseph Priest , in University Physics , 1984

    25.1 The Atomic Model of Matter

    Calculate the mass in kilograms of (a) one atom of 12 C (b) one atom of hydrogen.

    The size characteristic of the neutron is 10 −15 m. The mass of the neutron is 1.67 × 10 −27 kg. A star composed entirely of closely packed neutrons has a mass of 2 × 10 30 kg (the mass of our sun). Estimate the radius of such a neutron star.

    (a) Estimate your mass and mass density and then use the values to compute your volume. (b) Use your computed volume and knowledge of the characteristic atomic size to estimate the number of atoms in your body. (c) Devise a method for estimating the number of atoms in your body that does not require knowledge of the characteristic atomic size. Compare the result with that of part (b).

    A laboratory meter rod has a width of 2.5 cm and a thickness of 0.70 cm. Assuming that the characteristic dimension of a molecule is 3 Å, estimate the number of molecules in the rod.


    S type stars: how can they be gaseous ?

    Stars like S Cas which are S class stars (yes that online astronomy textbook is very useful !) has a surface temperature of 'only' 1800 K which is about the melting point of steel ald below the melting point of many rock minerals, and these do occur on these stars. S type stars have TiO spectral bands and ZrO as well.

    These compounds have melting points even above 1800 K, so does that occur as dust in the stellar atmosphere ? Or is the surface (partially) liquid or even solid ? Solid appears to be more realistic, as the low pressure does not allow liquid state of these compounds.

    #2 spereira

    #3 jrschmidt2

    I'm not an astronomer, but coming this as a physical chemist (my area), I would guess that the temperatures are high enough and the concentrations of these minerals is low enough that they are likely primarily in the vapor as small clusters/ ion pairs. At those temperatures, the vapor pressure of the minerals would be very high (increases with temperature -- just like heating a glass of water). Given that they are (likely?) quite dilute in the atmosphere, that would make the vapor vs. solid the most thermodynamically favored phase.

    #4 KBHornblower

    If I am not mistaken, these substances can and do precipitate as fine dust in the rarified outer parts of these stars, much as water vapor in the stratosphere can precipitate into ice crystals which remain suspended as cirrus clouds.

    #5 ZetaOrionis

    Stars like S Cas which are S class stars (yes that online astronomy textbook is very useful !) has a surface temperature of 'only' 1800 K which is about the melting point of steel ald below the melting point of many rock minerals, and these do occur on these stars. S type stars have TiO spectral bands and ZrO as well.

    These compounds have melting points even above 1800 K, so does that occur as dust in the stellar atmosphere ? Or is the surface (partially) liquid or even solid ? Solid appears to be more realistic, as the low pressure does not allow liquid state of these compounds.

    Remember that the listed temperature of a star ussually refers to the temperature of its "surface" (photosphere): deeper within the star, it's much hotter. Many of the elements that make up most rocks and minerals such as iron, calcium, etc. are fused in the cores of only more massive stars than red giants. However S-type stars can fuse carbon in their core, and while that carbon may be in plasma form deep within the star, some of that carbon can actually make it to the surface and can get cool enough to condense, into solid dust grains which collect in the atmosphere giving the star a deep crimson color, hence the name "carbon stars." While it itself is much more massive than most carbon stars, it is believed that the 2019-2020 dimming of Betelgeuse: a red supergiant star the constellation of Orion, may have been due to such as dust cloud of elements being expelled from the atmosphere of the star


    'Shell Stars', Beta Mon A & 28 Tau

    Here’s another interesting spectral comparison of two Be type stars. Beta Monocerotis A and 28 Taurus (Pleione) are fast rotating B stars, both with circumstellar decretion disks. Due to inclination angle of the decretion disk to our line of sight, disk matter substantially blocks our view of either star’s photosphere. For this reason, both stars are designated ‘Be shell stars.’ Typical of the shell star phase are the presence of metal lines such as iron (Fe) and magnesium (Mg) in the spectrum, and, radiometric continuum profiles analogous of the atmospheres of cooler super giant stars. In my two rectified spectra below, the radiometric differences are not apparent, but the lines for Fe and Mg are quite prominent. Helium absorption lines (He I) are also present in both spectra suggesting that the photospheres are not entirely masked by the decretion disk. Particularly interesting is the contrast of metal lines in each spectrum. In 28 Tau’s spectrum, Fe and Mg lines are absorption troughs, much as you would expect to find in spectra of a stellar photosphere, while the same lines in Beta Mon A are emission peaks. From my reading, this suggests that more of the inside diameter of the decretion disk is visible where photospheric radiation is sufficient to excite the metal atoms, resulting in emission peaks in the spectrum.

    More than one author of papers I have been reading suggests that no two ‘shell stars’ are alike……can’t argue with that. Spectroscopy rocks!

    Here are the full spectra for both stars:

    Edited by old_frankland, 03 March 2017 - 12:27 PM.

    #2 Organic Astrochemist

    I hate that name "shell stars" because it obfuscates far more than it elucidates. Most of us have an image of a shell as a roughly radially symmetric covering, but it is more often the radially asymmetric,disc-like nature of shell stars, seen at low inclination angles, that explains their spectra.

    This is an excellent overview of Be and shell stars showing the effect of viewing angle relative to the decretion disk: Shelyak

    See the illustration by Slettebak, 1988. They don't say, but I bet the emission shown is H-beta. Notice how the high inclination view A looks a lot more like your spectrum of Beta Mon and the edge-on view C looks more like your spectrum of Pleione.

    One fact complicating spectral interpretation is that the spectra are not spatially resolved so elements from different physical locations in a system are all superimposed on top of each other. To understand the spectrum we have to try to separate the components of a spectrum by their physical location as well as by their chemical origin.

    I really liked your earlier post of zeta Tau because you could see the broad emission of He I at 5048 from the photosphere (due to fast rotation). You could also see that the absorption of He I at 5016 had both a broad component from the photosphere and also a narrow component due to absorption near the disk along the line of sight. In contrast, for Pleione, which must be nearly edge on, the photospheric absorption of He I at 5048 is not visible and the very sharp absorption of He I at 5016 likely only results from absorption near the disk along the line of sight.

    In the present case, I think that Beta Mon has the higher inclination angle, which allows one to see both the broad photospheric absorptions and also the emission peaks. Let's assume that the chemical composition of the decretion disk is roughly the same at all distances from the star. In the case of Beta Mon, photons from the excited metals (and H-beta and even the continuum), which are produced in a location near the star can travel more directly towards us without being absorbed, (high inclination angle, away from the disk). However, the photons from the excited metals (and H-beta and the continuum) in Pleione that are produced near the star, have to travel down the length of the disk, near the disk (low inclination angle) which contains those metals and hydrogen which can then absorb photons at precisely those wavelengths where we see emission in Beta Mon, hence the absorptions observed in Pleione.

    The emission occurs in the disk near the UV radiation of the star and the sharp absorption occurs in the disk further from the star. The absorption in the photosphere is broad due to fast rotation of the star whereas absorption in the disk is sharp because of slower rotation.

    Thanks for the spectra and the opportunity to think and talk about spectroscopy.

    Edited by Organic Astrochemist, 03 March 2017 - 08:52 PM.

    #3 old_frankland

    Great spectra.

    I hate that name "shell stars" because it obfuscates far more than it elucidates. Most of us have an image of a shell as a roughly radially symmetric covering, but it is more often the radially asymmetric,disc-like nature of shell stars, seen at low inclination angles, that explains their spectra.

    This is an excellent overview of Be and shell stars showing the effect of viewing angle relative to the decretion disk: Shelyak

    See the illustration by Slettebak, 1988. They don't say, but I bet the emission shown is H-beta. Notice how the high inclination view A looks a lot more like your spectrum of Beta Mon and the edge-on view C looks more like your spectrum of Pleione.

    One fact complicating spectral interpretation is that the spectra are not spatially resolved so elements from different physical locations in a system are all superimposed on top of each other. To understand the spectrum we have to try to separate the components of a spectrum by their physical location as well as by their chemical origin.

    I really liked your earlier post of zeta Tau because you could see the broad emission of He I at 5048 from the photosphere (due to fast rotation). You could also see that the absorption of He I at 5016 had both a broad component from the photosphere and also a narrow component due to absorption near the disk along the line of sight. In contrast, for Pleione, which must be nearly edge on, the photospheric absorption of He I at 5048 is not visible and the very sharp absorption of He I at 5016 likely only results from absorption near the disk along the line of sight.

    In the present case, I think that Beta Mon has the higher inclination angle, which allows one to see both the broad photospheric absorptions and also the emission peaks. Let's assume that the chemical composition of the decretion disk is roughly the same at all distances from the star. In the case of Beta Mon, photons from the excited metals (and H-beta and even the continuum), which are produced in a location near the star can travel more directly towards us without being absorbed, (high inclination angle, away from the disk). However, the photons from the excited metals (and H-beta and the continuum) in Pleione that are produced near the star, have to travel down the length of the disk, near the disk (low inclination angle) which contains those metals and hydrogen which can then absorb photons at precisely those wavelengths where we see emission in Beta Mon, hence the absorptions observed in Pleione.

    The emission occurs in the disk near the UV radiation of the star and the sharp absorption occurs in the disk further from the star. The absorption in the photosphere is broad due to fast rotation of the star whereas absorption in the disk is sharp because of slower rotation.

    Does that make sense?

    Thanks for the spectra and the opportunity to think and talk about spectroscopy.

    Thanks for the input. Lots of great points made. and, yeah, the term 'shell star' has long out lived its usefulness. The more papers I look through, the more obvious it becomes that decretion disks are considerably complex, not the simple 'Saturn' type disk illustrated on the Shelyak site (which as you point out is still very instructive). A couple interesting variables that more than one author has used in their modeling is the divergence of disk thickness and perturbation patterns in the disk. Even the suggestion of asymmetrical and symmetrical disk lobes. As you say, the challenge is sorting out the physical locations of the elemental components.

    An interesting variation is a paper by Anne Cowley and Elain Gugula, University of Michigan, in which they suggest the presence of a close binary companion to Beta Mon A (not associated with Beta Mon B or C). A more massive, close orbiting B type star that distorts the decretion disk of Beta Mon A. Certainly seems to address some of the emission variations over Beta Mon A's 12.5 year cycle.

    Cyclic Variations of the Be Star Beta1 Monocerotis

    <<snip>> In the present case, I think that Beta Mon has the higher inclination angle, which allows one to see both the broad photospheric absorptions and also the emission peaks. Let's assume that the chemical composition of the decretion disk is roughly the same at all distances from the star. In the case of Beta Mon, photons from the excited metals (and H-beta and even the continuum), which are produced in a location near the star can travel more directly towards us without being absorbed, (high inclination angle, away from the disk). However, the photons from the excited metals (and H-beta and the continuum) in Pleione that are produced near the star, have to travel down the length of the disk, near the disk (low inclination angle) which contains those metals and hydrogen which can then absorb photons at precisely those wavelengths where we see emission in Beta Mon, hence the absorptions observed in Pleione. <<snip>>

    Very clearly explained! Thank you.

    <<snip>> I really liked your earlier post of zeta Tau because you could see the broad emission of He I at 5048 from the photosphere (due to fast rotation). You could also see that the absorption of He I at 5016 had both a broad component from the photosphere and also a narrow component due to absorption near the disk along the line of sight. In contrast, for Pleione, which must be nearly edge on, the photospheric absorption of He I at 5048 is not visible and the very sharp absorption of He I at 5016 likely only results from absorption near the disk along the line of sight. <<snip>>

    This is excellent. I need to take a closer look at those spectra to better sort out your points.

    Thank you for the very instructive commentary. . so much to learn, but totally cool that I can see, and begin to understand so much of this with my modest equipment from my own backyard!


    Equations of stellar structure

    The simplest commonly used model of stellar structure is the spherically symmetric quasi-static model, which assumes that a star is in a steady state and that it is spherically symmetric. It contains four basic first-order differential equations: two represent how matter and pressure vary with radius two represent how temperature and luminosity vary with radius. [4]

    In forming the stellar structure equations (exploiting the assumed spherical symmetry), one considers the matter density , temperature , total pressure (matter plus radiation) , luminosity , and energy generation rate per unit mass in a spherical shell of a thickness at a distance from the center of the star. The star is assumed to be in local thermodynamic equilibrium (LTE) so the temperature is identical for matter and photons. Although LTE does not strictly hold because the temperature a given shell "sees" below itself is always hotter than the temperature above, this approximation is normally excellent because the photon mean free path, , is much smaller than the length over which the temperature varies considerably, i. e. .

    First is a statement of hydrostatic equilibrium: the outward force due to the pressure gradient within the star is exactly balanced by the inward force due to gravity.

    ,

    where is the cumulative mass inside the shell at and G is the gravitational constant. The cumulative mass increases with radius according to the mass continuity equation:

    Integrating the mass continuity equation from the star center () to the radius of the star () yields the total mass of the star.

    Considering the energy leaving the spherical shell yields the energy equation:

    ,

    where is the luminosity produced in the form of neutrinos (which usually escape the star without interacting with ordinary matter) per unit mass. Outside the core of the star, where nuclear reactions occur, no energy is generated, so the luminosity is constant.

    The energy transport equation takes differing forms depending upon the mode of energy transport. For conductive luminosity transport (appropriate for a white dwarf), the energy equation is

    In the case of radiative energy transport, appropriate for the inner portion of a solar mass main sequence star and the outer envelope of a massive main sequence star,

    where is the opacity of the matter, is the Stefan-Boltzmann constant, and the Boltzmann constant is set to one.

    The case of convective luminosity transport (appropriate for non-radiative portions of main sequence stars and all of giants and low mass stars) does not have a known rigorous mathematical formulation, and involves turbulence in the gas. Convective energy transport is usually modeled using mixing length theory. This treats the gas in the star as containing discrete elements which roughly retain the temperature, density, and pressure of their surroundings but move through the star as far as a characteristic length, called the mixing length. [5] For a monatomic ideal gas, when the convection is adiabatic, meaning that the convective gas bubbles don't exchange heat with their surroundings, mixing length theory yields

    where is the adiabatic index, the ratio of specific heats in the gas. (For a fully ionized ideal gas, .) When the convection is not adiabatic, the true temperature gradient is not given by this equation. For example, in the Sun the convection at the base of the convection zone, near the core, is adiabatic but that near the surface is not. The mixing length theory contains two free parameters which must be set to make the model fit observations, so it is a phenomelogical theory rather than a rigorous mathematical formulation. [6]

    Also required are the equations of state, relating the pressure, opacity and energy generation rate to other local variables appropriate for the material, such as temperature, density, chemical composition, etc. Relevant equations of state for pressure may have to include the perfect gas law, radiation pressure, pressure due to degenerate electrons, etc. Opacity cannot be expressed exactly by a single formula. It is calculated for various compositions at specific densities and temperatures and presented in tabular form. [7] Stellar structure codes (meaning computer programs calculating the model's variables) either interpolate in a density-temperature grid to obtain the opacity needed, or use a fitting function based on the tabulated values. A similar situation occurs for accurate calculations of the pressure equation of state. Finally, the nuclear energy generation rate is computed from particle physics experiments, using reaction networks to compute reaction rates for each individual reaction step and equilibrium abundances for each isotope in the gas. [6] [8]

    Combined with a set of boundary conditions, a solution of these equations completely describes the behavior of the star. Typical boundary conditions set the values of the observable parameters appropriately at the surface () and center () of the star: , meaning the pressure at the surface of the star is zero , there is no mass inside the center of the star, as required if the mass density remains finite , the total mass of the star is the star's mass and , the temperature at the surface is the effective temperature of the star.

    Although nowadays stellar evolution models describes the main features of color magnitude diagrams, important improvements have to be made in order to remove uncertainties which are linked to the limited knowledge of transport phenomena. The most difficult challenge remains the numerical treatment of turbulence. Some research teams are developing simplified modelling of turbulence in 3D calculations.


    Measuring the Stars

    Most stars are too distant, or too small, to measure the size directly Combining Luminosity with Temperature gives Stellar Size The total luminosity is the area of the star times its surface temperature to the fourth power (Blackbody theory) OR -> Luminosity

    Area x Temp 4 or -> Luminosity

    Radius 2 x Temp 4 and this implies Radius

    • Stars above the main sequence are giants
      • The red giants are giant stars with surface temperatures making them red (about 3000-4000 K)
      • The white dwarf are very hot, small stars
      • loosely packed, groups of younger stars
      • tightly packed, groups of the oldest stars known
        • older than 10,000,000,000 years

        These lecture notes were developed for Astronomy 122 by Professor James Brau, who holds the copyright. They are made available for personal use by students of the course and may not be distributed or reproduced for commercial purposes without my express written consent.