IAL 20: Star Basics II

Don't Panic

Sections

  1. Modeling Stars (Reading Only)
  2. The Average Atmosphere Temperature of Stars (Omit)
  3. Spectral Types (Reading Only)
  4. The Hertzsprung-Russell (HR) Diagram
  5. Luminosity Classes
  6. Stellar Mass
  7. Binaries and Physical Star Groups
  8. Population I, II, and III Stars

  1. Modeling Stars (Reading Only)

    2. The modeling of stars in general is very complex.

    In this section, we outline the simplest approach to star modeling in order to understand how it is done and how models are fitted to observations. The fit verifies the models insofar as they are realistic: i.e., contain the right physical structures and the right physics theories.

    Verified models then allow us to know things about star which CANNOT be known by direct NOR indirect observations (which are both discussed in section Introduction).

    As preview/review of some of the topics of to come in IALs on stars, see the Pleiades open star cluster in the figure below/above (local link / general link: pleiades.html).

    In the sections following this one, we explicate further the ingredients of     star modeling: i.e., sections Star Shape and Structure, Stellar Composition, Luminosity, Flux, Photometry, Distance and Stellar Parallax, Luminosity Determination and Spectroscopic Parallax.

    In the subsections below, we cover star modeling itself:

    1. The Very Basic Qualitative Model of a Star:

      The very basic qualitative model of a star is a large self-gravitating sphere of hot gas.

      Going beyond this to quantitative star modeling necessarily includes the relevant physics theories: hydrostatics, hydrodynamics, nuclear physics, quantum mechanics, radiative transfer, statistical mechanics, thermodynamics, and whatever other physics theory is needed.

    2. The Stellar Structure Model and the Atmosphere Model:

      Going beyond the very basic qualitative star model actually requires two quantitative models: the (interior) stellar structure model and the stellar atmosphere model.

      The stellar structure model is entirely about what we CANNOT observe since it is all beneath the opaque photosphere---except for luminosity.

      The stellar atmosphere model (which is from just below the photosphere outward) is about lots that we do NOT see too, but we do get direct observations of photometry and spectroscopy From the solar atmosphere we observe a lot more, but that is a special case since we are so close.

      The reason for needing two models is that the scales of stellar structure modeling and stellar atmosphere modeling are so different that doing them in one model is vastly impractical. The two kinds of models can be fitted together: outer boundary conditions of the stellar structure model are the inner boundary conditions of the stellar atmosphere model and vice versa. The two kinds of models are connected as we discuss below subsection Stellar Atmosphere Models.???? (Well we will one day when I write that up explicitly.???)

      Since we are discussing very simple modeling, we ignore the complications of stellar rotation, stellar magnetic fields, and star-star interactions which occur close binaries. Close binaries show stellar evolution NEVER seen for single stars.

    3. Stellar Structure Models:

      One can develop a simple stellar structure model with only mass and composition as free parameters.

      The range of possible masses and compositions is suggested in List of Solar Units below (local link / general link: solar_units.html):

        EOF

      Note that stellar composition is in general complex.

      Zero-age main sequence (ZAMS) stars just require the cosmic composition metallicity Z as a free parameter.

      But as star ages from zero-age on the main sequence (i.e., when it first starts hydrogen burning in its core), the core gradually enriches in helium-4 (He-4). In the post-main-sequence phase, there is helium burning in the core and then nuclear burning of heavier elements. We will NOT consider the details in IAL.

    4. An Example Stellar Structure Model:

      What a stellar structure model gives you is "runs" of quantities (i.e., their distribution with radius coordinate).

      The figure below (local link / general link: sun_model_interior.html) is an example of the "runs" for a stellar structure model of the Sun with a brief discussion of how stellar structure models are calculated.

      We discuss stellar structure models further in IAL 22: The Main Sequence Life of Stars.

    5. Stellar Structure Models and Observables:

      Neither of the prime free parameters of stellar structure models (i.e., stellar mass and stellar composition) are direct observables in general.

      So they CANNOT be set by direct observation.

      In fact, the only synthetic direct observable calculable from simple stellar structure models is luminosity (i.e., energy output per unit time: i.e., power)---which is typically given in units of solar luminosities L_☉ = 3.828*10**26 W.

      But luminosity is only a direct observable when you know the distance to the star (see sections Distance and Stellar Parallax and Luminosity Determination and Spectroscopic Parallax), can effectively integrate observed flux over all wavelengths (see section Luminosity, Flux, Photometry), and can account for extinction.

      Now observed luminosity and stellar mass (if we had them) are enough (if we can assume composition) to constrain simple stellar structure models, and so tell us what stars of thoses masses and luminosities are like.

      But in general we do NOT have stellar mass and often NOT luminosity.

      The upshot of the above is that in order verify our understanding of stars based on stellar structure models we need another kind of model from which we can calculate sufficient sythetic observables to fit to actual observables to verify and constrain our stellar structure models. That model is the stellar atmosphere model which we discuss below in the subsections Stellar Atmosphere Models and Spectral Types and Stellar Models.

    6. Stellar Atmosphere Models:

      What we directly observe for stars are photometry (broad wavelength band measurements of flux: see section Luminosity, Flux, Photometry below), spectroscopy (narrow wavelength band measurements of flux), and, for sufficiently near stars, distance.

      Spectroscopy gives more detailed information than photometry, but is harder to obtain to the same level of accuracy and for distant stars NOT obtainable at all.

      With sufficient photometry and spectroscopy we can model the stellar atmosphere: i.e., create a model of the stellar atmosphere.

      Adjusting the free parameters of the model to fit the photometry and spectroscopy gives us values for those free parameters. The values are as good as the photometry, spectroscopy, and modeling allow.

      For simple stellar atmosphere modeling, the free parameters (which are determined by fits to observed photometry and spectroscopy) are usually:
      1. Composition. Assuming the cosmic composition (which is NOT always valid), composition effectively means metallicity Z.
      2. Gravitational field 
          g =  GM/R**2  ,
        where gravitational constant G = 6.67430(15)*10**(-11) (MKS units), M is stellar mass and R is stellar radius (i.e., photospheric radius). But note the modeling only gives g, NOT M and R separately.
      3. Effective temperature 
          T_eff = (F/σ)**(1/4) = [(L/(4πR**2))/σ]**(1/4) ,
        F is flux, σ is the Stefan-Boltzmann constant σ = (5.670374 19 ...)*10**(-8) W/*m**2/K**4 (exact) (see NIST: Fundamental Physical Constants --- Complete Listing 2018 CODATA adjustment)), L is luminosity, and R is stellar radius (i.e., photospheric radius). Note effective temperature is the temperature the star would have if it radiated like an exact blackbody radiator of radius R. Stars do NOT radiate like exact blackbody radiators. Nevertheless, effective temperature is a good characteristic or sort-of average temperature for their photosphere.

      From the above, we can obtain three fitted free parameters: stellar atmosphere (or nearly equivalently metallicity Z), gravitational field g, and effective temperature.

      Alas, the there are 3 unknowns M, R, and L for 2 equations: the ones for gravitational field g, and effective temperature.

      So we cannot solve for M, R, and L separately without more information.

      If we had any 2 of M, R, and L, and some estimate of core composition, then stellar structure model could be fitted and we would understand the star insofar as simple modeling allows.

      However, as discussed in section Stellar Structure Models and Observables we usually do NOT have stellar mass or luminosity. And stellar radius is known to accuracy/precision only for the Sun. A few red supergiants can be barely resolved and have their radii determined: e.g., Betelgeuse with R = 887(203) R_☉ ≅ 4 AU (see solar radius R_☉ = 6.957*10**5 km = 109.1 R_eq_⊕ = 4.650*10**(-3) AU; Dolan et al. 2016).

      So are we stuck?

      No. Distances (and therefore luminosities) can be obtained for relatively nearby stars by stellar parallax (see section Distance and Stellar Parallax below). Distances by stellar parallax were originally only obtainable to very nearby stars within a few parsecs in the 19th century, (see Wikipedia: Stellar parallax: 19th and 20th centuries), but with advancing technology, smaller stellar parallaxes, and so greater distances have been obtained progressively.

      As of 2018, the Gaia spacecraft (mission 2013--2025?) has provided us with accurate/precise stellar parallax to distances up to 8 kpc (see Wikipedia: Gaia spacecraft: Objectives)) which is about the distance to center of the Milky Way.????

      Also as mentioned above in section Stellar Structure Models and Observables, stellar mass is a direct observable for some binary star systems.

      The upshot is that we can nowadays fit stellar models (both stellar structure models and stellar atmosphere models) to vastly many stars and thereby understand them and know their parameters insofar as our modeling is sufficiently realistic.

      We can go well beyond the simple modeling described here and include stellar rotation, stellar magnetic fields, and star-star interactions which occur close binaries.

    7. Spectral Types and Stellar Models:

      As previewed in IAL 8: The Sun and as discussed at greater length in IAL 20: Star Basics II, stars can be classified by spectral type.

      The full spectral type classification (which includes its luminosity class) very full characterizes a star. The spectral type classification is just empirical: it is a direct observable.

      Now all stars of the same full spectral type classification are very much alike and so all have the same stellar model (both stellar structure model stellar atmosphere model) insofar as they are alike.

      So we do NOT have to model every star we want to understand. We just have to model stars of all spectral types.

      And this has been done.

      Now there remain imperfections in spectral type classification and modeling of stars, but there is continually work to reduce those imperfections.

    8. Beyond the Main Sequence:

      The current status is that we understand main-sequence stars in their bulk properties and evolution very well.

      Pre-main-sequence stars and post-main-sequence stars are more difficult to model, and so are less well understood, especially quantitatively.

      A general reason is that pre-main-sequence stars and post-main-sequence stars evolve more rapidly than main-sequence stars, and that just makes them harder to model.

      Also in the case of pre-main-sequence stars, they are embedded in star forming regions which are opaque in the visible band (fiducial range 0.4--0.7 μm = 4000--7000 Å) and this makes them harder to undersand observationally.

      And in the case of post-main-sequence stars, they are subject to explosive events (core helium flashes, thermal pulses (AKA helium shell flashes), and for stars > ∼ 8 M_☉, supernova explosions) which are hard to model because they are so complex.

      See the figure below (local link / general link: stellar_evolution_overview.html) for an overview of the stellar evolution of a star of less than ∼ 8 M_☉.

  2. The Average Atmosphere Temperature of Stars (Omit)

    3. SECTION UNDER RECONSTRUCTION

    The photosphere of a star is a layer from which of order half the photons can escape to infinity. It is often called the surface of a star for convenience. In fact, stars have no sharp surface: they just extend outward morphing into stellar winds.

    The photosphere itself is NOT a sharp surface, but has a thickness defined in some way and the whole stellar atmosphere extends from a bit below the photosphere to well above: in fact, it is the thing that morphs into stellar winds.

    For a cartoon of the solar atmosphere which an example of stellar atmospheres, see the figure below (local link / general link: sun_outer_layers_cartoon.html).

    For characterizing     stellar atmospheres, we would like to have an average atmosphere temperature.

    There are actually 4 kinds of average atmosphere temperature that can be mentioned.

    If stars radiated exactly like a blackbody radiators from sharply defined surface (which would necessarily be a sharply defined photosphere), then the four kinds of the average atmosphere temperature would be exactly the blackbody radiator temperature. Since stars are only blackbody radiators to 1st order, the four kinds of average atmosphere temperature will NOT be equal in general, but they will usually be rather close to equal in some sense.

    We explicate the four kinds of average atmosphere temperature in the subsections below.

    1. Stellar Atmosphere Model Inner Boundary Temperature:

      A stellar atmosphere model inner boundary temperature is a basic parameter of stellar atmosphere and is obtained by fitting the stellar atmosphere model to observations of the star or star type you are modeling.

      So the model inner boundary temperature is a model-dependent result and NOT a direct NOR an indirect observable.

      We can explicate stellar atmosphere modeling a bit.

      The simplest model of a stellar atmosphere is specified as follows:

      1. A plane-parallel slab with only radial dependence for quantities: e.g., temperature, density, pressure, etc.: See the example model of the Sun's atmosphere in figure below (local link / general link: sun_atmosphere_model.html).
      2. Inner and outer boundaries: The inner boundary is deep the star where the matter and electromagnetic radiation (EMR) are in mutual thermodynamic equilibrium and the EMR is necessarily blackbody radiation. The inner boundary does radiate like blackbody radiator except for a well-understood small angular dependence of the emitted radiation beams. The outer boundary is set far enough out for you to capture all the stellar atmosphere effects of interest.
      3. A set of free parameters: The free parameters are set by fitting the model to observations (in particular, spectroscopy photometry, and luminosity) of the star or star type you are modeling. The fitted free parameters are themselves main model results and are useful in understanding stellar atmospheres in shorthand. They are also useful in setting the outer boundary conditions for stellar-structure modeling of the interior of stars.

        The most basic free parameters are:
        1. The inner boundary temperature or, alternatively, an effective temperature (which we describe below).
        2. A gravitational field strength g. The gravitational field itself depends on the mass and radius which can be taken as alternative free parameters. In the simplest model (which is what we are considering), the model is geometrically thin enough that a single value is adequate for the gravitational field and the radius of the stellar atmospheres.
        3. Composition which is usually solar composition with the abundance of metals (i.e., the metallicity Z) being often the single adjustable value.

      The atmosphere model gives you among other things a temperature PROFILE for the stellar atmospheres that varies to 1st order only with radius.

      Of course, atmosphere model fitted free parameters, temperature PROFILE, and other model results are only accurate insofar as your model fits the observations AND is sufficiently realistic to capture actual stellar atmosphere behavior.

      More complex models can be developed for greater realism which means they give more accurate model results. But more complex models are harder to calculate.

      An example of an stellar atmosphere model is the model of the Sun's atmosphere shown in the figure below (local link / general link: sun_atmosphere_model.html).

      UNDER RECONSTRUCTION BELOW

    2. The Effective Temperature:

      The effective temperature is explicated in the figure below (local link / general link: solar_spectrum_graph.html).

    3. The Wien's-Law Temperature:

      If we apply Wien's law to a stellar spectrum, we obtain a characteristic or sort of average photospheric temperature which we call the Wien's-law temperature.

      Wien's law is described in the figure below (local link / general link: wien_law.html).

      For example, Sun's flux peak is at ∼ 0.475 μ (Solar Spectral Irradiance Handbook of Chemistry and Physics 82nd Edition Boca Raton, FL: CRC Press 2001.), and so by Wien's law we get T_W = (2897.7729 μ*K)/(0.475 μ) = 6100 K.

    4. Color Temperature:

      A general definition of color temperature is:

      In astronomy, we usually use a more specific definition replacing "color comparable to" by "B-V color index equal to" (Wikipedia: Color temperature: Color temperature in astronomy).

      The B-V color index is the difference of the B and V magnitudes which is roughly speaking the ratio of yellow to blue light.

      The UBVRI passband system is illustrated in the figure below (local link / general link: photometry_ubvri.html).

    5. Spectroscopy:

      Because of the problem of extinction and other reasons, it usually better to determine temperature from spectroscopy and stellar atmosphere modeling rather than photometry if one has spectroscopy available.

        Question: Spectroscopy is:

        1. the study of spectroscopes.
        2. the study of flux transmitted by broad passband filters.
        3. the study of spectra.
        4. a set of spectral observations.











        Answer 3 and 4 are right, but answer 3 is the primary definition.

        Answer 4 is a secondary definition. One often says something like "the spectroscopy of star X shows ...".

      Spectroscopy is done using spectroscopes. A spectroscope is illustrated in the figure below (local link / general link: spectroscope.html).

      The process of determining the       photospheric temperature is to model the stellar atmosphere and adjust the model atmosphere to get to fit to an observed spectrum.

      The accuracy of the fit depends on the physical realism of the model.

      A crude model that gives a good fit is NOT usually very accurate.

      A very realistic model that gives a good fit should give a high accuracy photospheric temperature as well as all the other details of the stellar atmosphere.

      But highly realistic stellar atmosphere models are demanding to create.

      However, whole online catalogues of advanced stellar atmosphere models have been created and are always being improved.

      Nowadays, one's computer searches the catalogues and finds a match to your observations. So it's all become rather easy.

      However, even advanced models are NOT perfect.

      A detailed model of the Sun's atmosphere is illustrated in the figure below.

      Atmosphere models for the Sun are particularly demanding if one wants to understand all of the Sun's atmosphere.

      We just know so much about the Sun's atmosphere that a complete model incorporating all we know is challenging to build.

  3. Spectral Types (Reading Only)

    4. Empirically, stars are classified by spectral types.

    We explicate in the following subsections.

    1. Spectra and Spectral Types:

      A spectrum is taken by measuring fluxes in very narrow wavelength bands.

      A spectrum therefore gives information about the spectral lines which are usually (but NOT always) very narrow in wavelength for stars.

      The analysis of a stellar spectrum gives one the photospheric temperature of a star as discussed above in the section The Surface or Photosphere Temperature of Stars as well as lots of other information.

      This analysis in general takes modeling of the stellar atmosphere, but stars fall into standard classes as determined by spectra and other means.

      Once a stellar class has been modeled accurately, then the modeling has been done once-for-all---well once-for-all until better modeling is done.

      One obtains spectra by dispersing (i.e., spreading out) the light of different wavelengths with a prism or diffraction grating.

    2. Spectral Lines:

      In order to understand stellar spectra, a key determinant of the strength of the spectral lines is obviously the stellar composition which is approximately solar composition, except for variations in the total metalliticity. See Table: Gross Solar and Primordial Cosmic Compositions by Mass Fraction below (local link / general link: solar_composition_metallicity.html).

          EOF

      Because stars are mostly hydrogen, we expect to see atomic hydrogen lines: in the visible these would be the Balmer lines. For the Balmer lines, see the figure below (local link / general link: line_spectrum_hydrogen_balmer.html).

      The       Balmer lines are further explicated in the atomic hydrogen (H I) Grotrian diagram in figure below (local link / general link: grotrian_01_00_H_I.html).

      We also can expect       helium lines, metal lines, and, in cooler stars, molecular lines (lines of bound systems of atoms). Although metals and molecules are only traces in stars, they have many lines, some of them very strong.

        Typically above about 4000 K, molecules CANNOT exist because the collisions of the particles are sufficiently energetic to prevent formation or to break molecules apart (FK-423).

      The Balmer lines and helium and some metal lines are shown in spectra in the figure below.

    3. Where Did the Spectral Types Come From?

      In the late 19th century and early 20th century, the spectral types were fixed just empirically (i.e., based on observed characteristics alone) before modern spectral analysis was invented.

      Each spectral type was designated by a capital letter.

      Originally, the spectral types went AB...P and represented decreasing strength of Balmer lines (FK-422; CK-286).

      But this ordering turned out NOT to be very physically significant: it was NOT a temperature ordering.

        The strength of Balmer lines is NOT monotonic with temperature: i.e., they don't just get stronger or weaker as temperature increases.

        At low temperatures, the lines are weak because the energy state they depend on is NOT much EXCITED: i.e., NOT many atoms are in that state.

        At high temperatures, hydrogen tends to be ionized: i.e., it has lost its only electron and become a bare proton. Bare protons are very simple objects and have no atomic transitions and NO lines.

        The Balmer lines tend to be strongest for temperatures of order 9000 K (FK-424).

      Rather than rename the spectral types---that would have been the easy way---the types were re-ordered and a few were dropped (FK-422; CK-286).

      The standard modern spectral types are OBAFGKM: these are in order of decreasing photospheric temperature .

      The OBAFGKM stellar classification can be remembered by the mnemonic "O be a fine girl/guy kiss me."

        Sometimes the only sensible thing to say.

      There are other less common spectral types.

      The spectral types are divided into subtypes: each type has a ten subtypes: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 in order of decreasing temperature within the type. Thus, for example, one has stars of spectral types: 

      
        O0, O1, ...  , O9

        G0, G1, G2, G3, ...  , G9 

        M0, M1, M2, M3, M5, M6, ..., M9 .

      We can look at examples of star spectra of various spectral types in the figure below (local link / general link: star_spectra.html).

      And       "you must remember this ...," the Sun is a G2 star.

      Thus, it's a middle of the pack star---but NOT an "average star": spectral types are too diverse for average of all stars to be a useful concept.

      Still we use the Sun as convenient standard and use solar mass M_☉, solar luminosity L_☉, and solar radius R_☉ as natural units for, respectively, stellar mass, luminosity, and stellar radius.

      For the solar units in detail, see the insert solar_units.html below (local link / general link: solar_units.html).

        EOF

      We can look at       Table: Spectral Types to see the characteristics of the main spectral types. 

        Table:  Spectral Types

        
Spectral   Color         Photospheric        Spectral lines       Examples
 Type                    Temperature
                            (K)

        
 O          blue-violet  30,000--50,000      Ionized atoms,       Mintaka (δ Ori)
                                               especially helium   

 B          blue-white   11,000--30,000      Neutral helium,      Spica (α Vir)
                                               some hydrogen       

 A          white          7500--11,000      Strong hydrogen,     Sirius (α CMa)
                                               some ionized   
                                               metals

 F          yellow-white     5900--7500      Hydrogen, ionized    Canopus (α Car)
                                               metals (e.g.,     
                                               calcium, iron)

 G          yellow           5200--5900      Neutral and ionized  Sun, Capella (α Aur)
                                               metals especially   
                                               ionized calcium    

 K          orange           3900-5200       Neutral metals       Aldebaran (α Tau)
                                                                   
 M          red-orange       2500--3900      Strong titanium      Betelgeuse (α Ori)
                                               oxide, some         
                                               neutral calcium  

 L          red              1300--2500      Neutral potassium,   Teide 1
                                               rubidium, cesium,
                                               metal hydrides

 T          red              below 1300      Strong neutral       Gliese 229B
                                               potassium, some
                                               water (H_2O)
        References: FK-425; CK-286, except that it does NOT have the brown dwarf types LT.

        Notes: The OBAFGKM spectral types can be remembered by the mnemonic "O be a fine girl/guy kiss me." (Sometimes the only sensible thing to say.)

        The LT types are brown dwarfs which are NOT stars. Brown dwarfs have masses in the range 13--75 M_Jupiter. The upper limit is 0.08 M_☉. They do NOT burn ordinary hydrogen. They briefly burn deuterium (the stable heavier isotopes of hydrogen) and, if above 60 M_Jupiter, lithium. Most of the their electromagnetic radiation emission comes at the expense of the gravitational potential energy lost in contraction. See CK-306 and FK-424.

        The OBAFGKMLT types can be remembered by another mnemonic, but I can't remember what it is (FK-424).

  4. The Hertzsprung-Russell (HR) Diagram

    5. Now we have considered two basic INTRINSIC stellar quantities: luminosity and photospheric temperature.

    In understanding and modeling stars, it would be interesting to know if these parameters were related.

    The plot of luminosity versus photospheric temperature (or nearly equivalently spectral types) for stars is called a Hertzsprung-Russell (HR) diagram.

    We will look at several different HR diagrams below each of which highlights different features of HR diagams.

    1. A Cartoon HR Diagram:

      A cartoon of an HR diagram is given in the figure below (local link / general link: star_hr_named_stars_cartoon.html) that illustrates the general features of HR diagrams.

    2. A More Elaborate HR Diagram:

      Now let's look just below to a more elaborate representative HR diagram---for the sake of redundancy. See the figure below (local link / general link: star_hr_named_stars.html).

    3. The Main Groupings of Stars on the HR Diagram:

      As we have seen, the HR diagram is NOT an uncorrelated scatter diagram.

      But it's NOT altogether simple either.

      There are various groupings of stars in luminosity-temperature/spectral-type space: i.e., on the HR diagram.

      Below we just list the main groupings that everyone should keep in mind when discussing stars and the HR diagram:

      1. Main sequence: a smooth band of increasing luminosity with temperature.

        Here we are just being redundat with the discussion above for completeness in our list.

        In the solar neighborhood and similar stellar neighborhoods---but NOT all stellar neighborhoods---about 90 % of stars lie on the main sequence (FK-428).

        The Sun is G2 star.

        As mentioned above, main sequence stars are those undergoing hydrogen burning (i.e., nuclear burning of hydrogen to helium in their cores).

        Typically a star spends about 90 % ????? of its nuclear-burning life on the main sequence---which explains why main sequence stars are the most abundant stars undergoing nuclear burning.

      2. Giants and Supergiants: These are very luminous post-main-sequence stars.

        They are NOT undergoing hydrogen burning in their cores: they are burning hydrogen and, perhaps, other elements in concentric shells about a core which is perhaps burning some heavier element than those in the burning shells.

        They can have either hot or cool surfaces.

        The cool ones emit primarily red light, and are called red giants or red supergiants.

        In the solar neighborhood and similar stellar neighborhoods---but NOT all stellar neighborhoods---about 1 % of stars are giants or supergiants (FK-429).

        But because they are so luminous, they are much more conspicuous than their numbers indicate.

        Many of the best known naked-eye stars are giants or supergiants: e.g., Betelgeuse (a red supergiants) and Deneb (a blue-white supergiants).

      3. White Dwarfs (WDs): These are the compact remnants.

        They are what is left when the all nuclear burning has stopped and a lot of mass has been ejected by stellar winds and explosions.

        They are or are NOT stars depending on what you mean when you are discussing stars.

        Though typically their masses are about 0.5 M_☉, they are of order Earth-size: they are compact and super-dense.

        They shine by residual heat or heat from continuing contraction.

        They just cool off forever.

        In the solar neighborhood and similar stellar neighborhoods---but NOT all stellar neighborhoods---about 9 % of stars are white dwarfs (FK-429).

        Actually, because they can be very dim, there might be more white dwarfs around than we notice.

      We will discuss the kinds of stars further in subsequent IALs:

    4. An HR diagram for a Large Sample of Stars:

      An HR diagram for a large sample of stars is shown in the figure below (local link / general link: star_hr_large_sample.html).

    5. Stellar Radii on an HR Diagram:

      How one obtains stellar radii approximately and contour lines of constant radius are explicated in the figure below (local link / general link: hr_radius.html).

      Note that relatively close, very-large-radius       stars can be resolved with special techniques or instruments: one still CANNOT see a lot of detail, but at least the finite size of the star in the image is NOT just the diffraction pattern of a point light source.

      In the figure below (local link / general link: betelgeuse.html) is a resolved image of Betelgeuse imaged by the Hubble Space Telescope (HST).

    6. An HR Diagram Applet:

      One can gain more insight into the HR diagram by exploring the applet in the figure below (local link / general link: naap_hr_diagram.html).

  5. Luminosity Classes

    6. Spectral types essentially classify stars by photospheric temperature.

    There is also a luminosity class classification which is illustrated in the Hertzsprung-Russell (HR) diagram in the figure below (local link / general link: star_hr_lum.html).

    Note the luminosity class are NOT determined by luminosity: they are determined by being bands on the HR diagram.

    Note also that very often, we just conflate the expressions spectral type and luminosity class as spectral type.

    For examples of full stellar classification, we can look back at Table: Stars of Highest Apparent Brightness.

    And "you must remember this ...," the Sun is a G2 V star.

  6. Stellar Mass

    7. Stellar mass is the most basic controlling parameter of stars. It controls most of their behavior over their lifetimes.

    Composition, rotation, and having a close binary companion are also important, but distinctly secondary.

    There are other parameters, not controlling, that are important for star behavior at any point in time and in understanding and modeling it: most notably the aforementioned luminosity, photospheric temperature, and photospheric radius.

    1. Determining Stellar Mass:

      In fact, for an isolated star there is no observational way of determining its stellar mass.

      But orbital parameters of gravitationally bound pairs of stars (binaries) allow mass to be determined using Newtonian physics and some information about the inclination angle of the binary.

      The masses of all the main-sequence spectral types can be determined by examples in binaries: we expect the mass of a main-sequence spectral type to be an approximately fixed value usually.

      Note the mass of a post-main-sequence spectral type probably has a range of values.

      There is also a mass-luminosity relation for the main sequence. A cartoon of the mass-luminosity relation is shown in the figure below (local link / general link: mass_luminosity.html).

    2. The Main-Sequence Rule Redux:

      Recall stellar mass also enters the main-sequence rule which we reiterate in the figure below (local link / general link: hr_mass.html) which shows the main-sequence star stellar masses on a cartoon of a Hertzsprung-Russell (HR) diagram.

    3. Main-Sequence Lifetimes are Functions of Stellar Mass:

      Main-sequence lifetimes are functions of stellar mass.

      The functional behavior is illustrated in the figure below (local link / general link: star_lifetimes.html).

    4. The Initial Mass Function (IMF):

      The initial mass function (IMF) is the frequency of stars as a function stellar mass at the time of their nearly simultaneous birth (hence the word "initial") in star formation regions.

      The IMF decreases at least for stars more massive than about 1 M_☉ and probably for stars (which all have stellar mass >∼ 0.08 M_☉ which is the lower limit on stellar mass). The initial mass function (IMF) is explicated in the figure below (local link / general link: initial_mass_function.html) showing the initial mass function (IMF).

  7. Binaries and Physical Star Groups

    8. Many stars are relatively isolated---they are NOT gravitationally bound to or interacting strongly with any other gravitational source smaller than a galaxy or a large fraction thereof such as a spiral arm.

    To reiterate, many stars are relatively isolated.

    But many stars are NOT relatively isolated.

    Let's consider these physical star groups---starting with binaries and then larger physical groupings of stars: e.g., gravitationally bound or gravitationally interacting groups. For the smaller groupings (i.e., those below the scale of galaxies and NOT including stellar association), see the figure below (local link / general link: binary_cluster.html).

    1. Binaries:

      Binaries (i.e., gravitationally bound pairs of stars) are a whole massive subject in themselves---but we will just give the short story.

      The stars in a binary orbit their mutual center of mass in elliptical orbits as shown in the figure below.













      The figure below (local link / general link: orbit_elliptical_equal_mass.html) shows an animation of a binary orbit.

      In a       binary, usually the brighter star is called the primary and the other star, the secondary. Sometimes, primary means most massive and secondary least massive. Usually, the two meanings for primary and secondary give the same stars, but NOT always.

      We will just mention why binaries are important:

      1. They are numerous.

        In the solar neighborhood---which may be representative of observable universe as a whole---about 2/3 of all stars are in binaries.

        Thus, binaries are as common as singles (i.e., single stars).

        Why are they numerous? Some feature of star formation since virtually all binary pair stars form as binaries at the same time.

        Binary systems from gravitational capture are very rare and cosmically insignificant.

      2. Stellar masses and other stellar parameters impossible/difficult to obtain for singles can be obtained in some cases by analyzing the observations of binaries using Newtonian physics and various pieces of information.

        Mass determinations are particularly important since they allow the masses of all spectral types to fixed.

        Mass is a basic parameter in understanding and modeling stars.

        This importance is an importance to our understanding NOT an importance to universe.

      3. Close binaries (i.e., binaries where the binary companions orbit relatively closely to each other) often interact: e.g., they can heat each other or exchange mass during certain phases of stellar evolution.

        Thus, binaries can show behaviors singles never can.

        These behaviors are sometimes cosmically important: e.g., some kinds of supernovae happen only in or nearly only in close binaries.

        For a close binary (which is also an eclipsing binary), see the figure below (local link / general link: star_binary_eclipsing.html).

    2. Multiple Star Systems:

      Multiple star systems are systems of 3 or more gravitationally bound stars with complex orbits. The 3-star systems are naturally called triple star systems.

      Multiple star systems are much rarer ??? than binary star systems and become rarer with increasing multiplicity. ???

    3. Open Clusters:

      Open clusters are loosely-bound, irregularly-shaped groups of stars consisting of order 100 to 1000 stars with a size scales of order 4 to 20 pc (HI-392--393).

      The stars formed at about the same time (i.e., same time to within a few million or tens of millions of years: HI-338) in a star formation region.

      In the Milky Way, open clusters are only in the Galactic disk.

      Both internal and external gravitational perturbations tend to break up open clusters and they probably only survive for a few hundred million years.

      Probably the best known example of an open cluster is the Pleiades in Taurus. The Pleiades are well known in many cultures because they form a distinct group on the sky.

      In Europe, they have also been called the Seven Sisters and their Japanese name is Subaru like the car and the telescope. Geoffrey Chaucer (c. 1343--1400) alludes to them in his Chauntecleer tale.

      Usually 6 Pleiades at least can be seen with the naked-eye; 9 can be seen under good conditions; 14 were claimed visible by Johannes Kepler (1571--1630). Telescopically, the cluster has about 1000 known members, but this number does NOT include unresolved binaries.

      See the Pleiades in the figure below (local link / general link: pleiades.html).

      To find the       Pleiades, one can use the constellations as SKYMARKS

      First, locate Orion and Sirius (the brightest star in the sky) off to the lower left of Orion (south-east on the sky). A line from Sirius though the belt of Orion and then through the bright orangy Aldebaran (the eye of Taurus leads pretty much to the Pleiades---a distinct close little group of six or more naked-eye stars---there are at least 1000 stars altogether in the cluster recall.

      The method is illustrated for northern constellations in the winter sky map in the figure below (local link / general link: sky_map_winter.html).

    4. Stellar Associations:

      Stellar associations are structures of a few to a few hundred stars and span of order 10 to 100 pc (HI-393,395).

      They are generally gravitationally UNBOUND though gravitationally interacting and moving together (FK-456). The kinetic energy of the stars and gravitational perturbations will break them up within a few 10s of millions of years.

      The stars in an stellar association formed at about the same time (i.e., same time to within a few million or tens of millions of years: HI-338) in a star formation region.

      In the Milky Way, they are only in the Galactic disk.

      The most discussed kind of stellar associations are the OB associations which are stellar associations containing of order 10--100 hot (and therefore bluish), young OB stars.

    5. Globular Clusters:

      Globular clusters are compact, dense, spherical, gravitationally-bound systems of stars (HI-395).

      They can have from of order 20,000 to several million stars and their central concentrations have diameters of order to 5 to 25 pc.

      Near the center of a globular cluster, there could be 10,000 stars per cubic parsec: the night sky would be 10 times brighter than the full Moon.

      In the Milky Way, there are about 150 globular clusters and they are spread around in the Galactic halo.

      Globular clusters mostly consist of Population II stars with ages >∼ 12.5 Gyr???? (FK-638), and they so are very old (Wikipedia: Globular cluster: Formation).

      Most stars in globular clusters formed at the same time over a span of ∼ 30 Myr ????. However, most globular clusters have younger populations of stars. Possibly an encounters and mergers ??? with giant molecular clouds (GMCs) triggered renewed bursts of star formation (Wikipedia: Globular cluster: Formation).

      The age determinations for globular clusters is an important lower limit on the age of the observable universe (see Wikipedia: Age of the universe = 13.799(21) Gyr).

      Why did globular clusters form only in the relatively young observable universe before cosmic noon (z≅2, cosmic time ∼ 4 Gyr)? Well, globular cluster formation is still NOT well known. It may be that the low metallicity of that era favored their formation whereas in later times open clusters were favored.

      For an example globular cluster, see the figure below.

    6. Galaxies and Larger Groupings:

      Galaxies, galaxy clusters, galaxy superclusters, and the large-scale structure of the universe: These are all large groupings of stars that we will take up in:

  8. Population I, II, and III Stars

    9. There is another classification scheme for stars called stellar populations which divides stars into populations I, II, and III stars based on their age or, nearly equivalently, their metalliticity (i.e., abundance of metals) (HI-413--414):

    1. Population I Stars:

      These are relatively young and metal-rich stars:

      The Sun is a Population I star.

      Population I stars are found mainly in the galactic disks and galactic bulges, but much NOT in galactic halos of spiral galaxies.

      They are NOT much found in elliptical galaxies.

      We will discuss disks, bulges, and halos of galaxies later in IAL 27: The Milky Way and IAL 28: Galaxies, but as a preview we discuss galaxies a bit in the figure below (local link / general link: galaxy_sombrero.html).

    2. Population II Stars:

      These are relatively old and metal-poor stars:

      In spiral galaxies, Population II stars are found mainly in the galactic halos (most notably globular clusters in galactic halos) and galactic bulges with only a small fraction in galactic disks.

      In elliptical galaxies, most of the stars are Population II stars (see Characteristics of Galaxies).

    3. Population III Stars:

      Population III stars are as-yet-unobserved nearly zero-metallicity stars formed in the first age of star formation after the Big Bang: they were the first generation of stars. Their only metal is lithium-7 (Li-7) (the only metal produced in Big Bang nucleosynthesis) at mass fraction ∼ 10**(-9).

      They should all be ∼ 13.5 Gyr old.

      As aforesaid, they have never been observed.

      In the modern universe, they must either be very rare or nonexistent.

      It is believed be that the Population III stars were all massive stars and were NOT long-lived since near zero-metallicity led to the star formation of massive stars.

      On the other hand, maybe some small, long-lived Population III stars are around and just have NOT been recognized as such because their surfaces have accreted some metals from the interstellar medium (ISM) (see reference ????). Thus, they do NOT look like Population III stars.

      On the third hand, remember the farther you look out, the further back in time you see. So potentially very short-lived Population III stars can be seen. In fact, a science goal of the James Webb Space Telescope (JWST, 2021--2031) is to discover Population III stars in the early universe (see Wikipedia: Stellar population: Population III stars), but such discoveries may be beyond its reach (see Rydberg et al. 2013)). However, recall the Population III stars are supermassive compared to modern stars, and so are much brighter than modern stars. Even if the JWST CANNOT detect individual ones, it may be able to detect star clusters of Population III stars.

    4. Summary Tables for Cosmic Composition and Stellar Population Composition:

      The insert below (local link / general link: cosmic_composition_table.html) shows Table: Cosmic Composition which summarizes cosmic composition and Table: Stellar Population Metallicity for the Milky Way which summarizes stellar populations.

    5. Why Is There Varying Metalliticity with Star Age?

      Why is there varying metalliticity with star age?

      As we will discuss in subsequent lectures, stars synthesize metals in their nuclear burning lifetimes and eject those into interstellar medium (ISM) either through stellar winds (mostly post-main-sequence stellar winds) or as supernovae.

      Out of the ISM, new generations of stars form as is discussed in IAL 21: Star Formation.

      Thus, every new generation of stars has higher metalliticity (higher abundance of metals) on average than the generation before---if the galaxies were like closed-boxes.

      Actually, they are NOT like closed-boxes. There is inflow of intergalactic gas which has nearly primordial composition and outflow of interstellar medium (ISM).

      The upshot is that the metalliticity of stars has saturated at ∼ 1--4 % ???? (see Table: Stellar Population Metallicity for the Milky Way: also shown above: local link / general link: cosmic_composition_table.html) for probably billions of years into the future (see David Weinberg 2016, "On the Deuterium-to-Hydrogen Ratio of the Interstellar Medium", p. 3, but this NOT the best reference---but where is that mythical beast).

      Because stars have vastly varying lifetimes, some old, metal-poor stars persist on and overlap with younger, metal-richer stars.

      Thus, there are very old, very metal-poor stars today: e.g., stars in globular clusters have calculated ages of about 12.5 Gyr (FK-638) as we mentioned above.

      Old stars are necessarily small stars. Recall the more massive a star is the faster it runs through all of the stages of its nuclear burning lifetime. Population II stars are probably mostly less massive than the Sun. Therefore, their colors must be mostly red or yellow. Massive hot main-sequence stars (OB stars) are blue or blue-white and go through their lifetimes quickly.