IAL 22: The Main Sequence Life of Stars

Don't Panic

Sections

  1. Being on the Main Sequence
  2. Nuclear Physics and Nuclear Fusion
  3. Nuclear Burning Stable on the Main Sequence
  4. Stellar Structure and Stellar Modeling
  5. Radiative Transfer and Convection
  6. Brown Dwarfs
  7. Evolution on the Main Sequence
  8. Preview of Post-Main-Sequence Evolution

  1. Being on the Main Sequence

    2. Observationally, being on the main sequence means that a star is located on that band on the HR diagram that has been called the main sequence. See the figure below (local link / general link: star_hr_named_stars.html).

    From a combination of observation and modeling, we now know a lot about     main sequence stars which are, in fact, the relatively easy stars to understand.

    Physically, being on the main sequence means that a star is stably fusing hydrogen nuclei to helium nuclei in its core and that its structure is very nearly in hydrostatic equilibrium (i.e., it is very nearly a fluid at rest).

    The heat energy released by the nuclear fusion compensates for the energy the star loses by radiating electromagnetic radiation (EMR) continuously into space.

    Main sequence stars are very nearly in STEADY STATE (i.e., an unchanging state) for long periods of time.

  2. Nuclear Physics and Nuclear Fusion

    3. This section is largely a recapitulation of IAL 8: The Sun: 5. Nuclear Fusion in the Sun, but some new aspects are presented.

    1. Nuclear Physics:

      Atoms consist of negatively charged electrons in a swarm about a nucleus made of positively charged protons and neutral neutrons.

      See the figure below (local link / general link: atom_nucleus_symbol.html).

      One really needs       quantum mechanics to understand atoms and nuclei in detail---that is well beyond the scope of these lectures.

      The number of protons determines the chemical species.

      Species with the same number of protons and different numbers of neutrons are isotopes of each other.

        Isotope is sort of a tricky word to define since it is a relationship word like brother: all men are brothers, but NOT all men are brothers to each other.

        All atoms are isotopes, but NOT all atoms are isotopes to each other.

      Isotopes of an atom are chemically nearly identical, but nuclei have somewhat different nuclear behavior: sometimes very different nuclear behavior.

        The electronic structure of atoms determines the chemistry. Isotopes of the same atom have the same number of protons, and thus their nuclei bind to electrons in almost identical manner. Thus the electronic structure and chemistry of isotopes is nearly identical.

        The chemical behavior CANNOT be quite identical because the mass of the isotopes is somewhat different because of the differing number of neutrons. The mass of an atom affects the reaction rates for chemical processes to some degree and even the electronic structure which has a small effect on the spectroscopy of the atom.

      The ordinary hydrogen nucleus just consists of a single proton. It is the simplest of all nuclei.

      A nucleus consisting of one proton and one neutron is a heavy hydrogen: a deuteron symbolized by D or H-2.

      Nuclei are held together against the electrostatic repulsion of the protons by the strong nuclear force.

      The strong nuclear force is a very strong force, but it is very short range. It acts only over a distance of 10**(-15) meters or less. This distance of 10**(-15) meters is 10**5 times smaller than an atom's size.

      Thus, the protons and neutrons in a nucleus are compacted close together within about 10**(-15) meters of their nearest neighbors and nuclei have a size scale of order 10**(-15) to 10**(-14) meters depending on the number of protons and neutrons.

    2. Hydrogen Burning in Main-Sequence Stars:

      Now H nuclei strongly repel by the electrostatic force because they are like-charged particles.

      In stars, only in the cores is it ordinarily sufficiently hot and dense that the electrostatic repulsion can be overcome and the H nuclei can collide closely enough that the strong nuclear force can bind them (i.e., fuse them).

        Astrophysicists, usually call fusion, nuclear burning because the fusion process is the nuclear analogue of chemical burning.

      The       deuteron is a reactive nucleus compared to ordinary hydrogen and it burns He-3 (two protons and one neutron in the nucleus) relatively quickly (HRW-1106).

      But the final product in stellar hydrogen burning is the very stable He-4 nucleus.

      There are several H-to-He-4 burning processes in stars. The figure below (local link / general link: stellar_nuclear_burning_processes.html) illustrates the two dominant ones: the proton-proton (PP) chain reaction and the CNO cycle.

      The dominant one in       stars less massive than about 1.5 M_☉ is the PP I branch (a special case of the PP chain reaction) (HI-343; Cl-369; HRW-1106).

      We will just look at the PPI chain in detail to illustrate a nuclear burning chain. See the figure below (local link / general link: nuclear_burning_ppi_chain.html).

      The dominant one in       stars more massive than about 1.5 M_☉ is the carbon cycle (HI-343; Cl-390).

      In the carbon cycle, the carbon nucleus acts as a catalyst: i.e., a reactant that facilitates the process without being destroyed in a net sense in the process. We will NOT go into details though.

        Hans Behte (1906--2005) discovered the carbon cycle in 1938. In the early 1930s, he left Germany because of the Nazis and in WWII he was the leader of the theory group at Los Alamos in the Manhatten Project. He was almost certainly the last of the great pre-war generation physicists.

      The net process in both H-to-He-4 burning processes is 
      
    4H + 2 electrons to  He-4 + 2 neutrinos + heat energy 

     (HRW-1107;
      Cl-390).
  
         The heat energy is in the form of kinetic energy 
         of the particles and photons.

         The neutrinos mostly just freely escape the
         Sun, 
         fly off into space, and have very little effect on
         the universe it seems.

    3. Stellar Lifetime and the Sun's Lifetime:

      Stellar lifetimes and main-sequence lifetime can be calculated. We consider these in general below in subsection Main-Sequence Lifetimes.

      Here let's just ask how long is the Sun's main sequence lifetime? For the answer, see the figure below (local link / general link: sun_lifetime_estimate.html).

  3. Nuclear Burning Stable on the Main Sequence

    4. Nuclear burning is stable on the main sequence.

    By STABLE we mean the nuclear burning neither just turns off suddenly nor goes into a thermonuclear runaway and blows up the star.

    The cartoon in the figure below (local link / general link: stability_mechanical.html) illustrates the STABILITY in general via a mechanics analogue.

    A practical example of stability is all buildings. When kicked they don't fall down---one hopes NOT even when they are kicked really hard.

    A RULER balanced on a finger is an example of an unstable system. Balance scales take advantage of instability: their instability allows a fine determination of mass.

    Hydrogen burning in the Sun and all main sequence stars is STABLE due to the process discussed in the figure below (local link / general link: sun_hydrogen_burning_stability.html).

    To recapitulate from     IAL 8: The Sun: Controlled Fusion and Fusion Power, down here on Earth we would like to have STABLE hydrogen burning or, as it is called, controlled fusion for fusion power.

    Controlled fusion and fusion power are explicated in the figure below (local link / general link: nuclear_fusion_deuteron_triton.html).

  4. Stellar Structure and Stellar Modeling

    5. A star's stellar structure (i.e., its interior structure) can be specified by giving the "RUNS" of its thermodynamic and other physical quantities:

    These quantities CANNOT be directly observed.

    They follow from stellar structure modeling using:

    1. The BASIC CONTROLLING STELLAR PARAMETERS: stellar mass and composition.

      Usually, somewhat less important controlling paramters are rotation and having a close binary companion.

    2. Known physics: convection, thermodynamics, hydrodynamics, hydrostatics, Newtonian physics, nuclear physics (especially nuclear burning rates which are functions of temperature, density, and composition), quantum mechanics, radiative transfer, statistical mechanics, thermodynamics, and whatever other physics theory is needed.

    3. The 4 equations of stellar structure which we explicate below in subsection The 4 Equations of Stellar Structure.

    4. Using numerical methods on the computer to solve simulataneously the 4 equations of stellar structure. Analytic solutions are NOT possible.

    Recall we discussed both stellar structure modeling and stellar atmosphere modeling in IAL 20: Star Basics II: Modeling Stars. Also recall, 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.

    Here we will focus a bit on stellar structure modeling and in particular on the 4 equations of stellar structure and recapitulate a bit.

    1. The 4 Equations of Stellar Structure:

      The 4 equations of stellar structure are explicated in the figure below (local link / general link: stellar_structure_4_equations.html).

      We will NOT detail of the       4 equations of stellar structure, but one of them that students should know of (because of its general importance) is the stellar hydrostatic equilibrium equation. This equation actually applies to all spherically-symmetric hydrostatic self-gravitating pressure-supported bodies. So it applies moons, planets, and stars. Corrections do have to be made for deviations from spherical symmetry, of course.

      We derive and discuss the stellar hydrostatic equilibrium equation in the figure below (local link / general link: hydrostatic_equilibrium.html).

        Question: The requirement of hydrostatic equilibrium means that pressure:

        1. increases with depth.
        2. stays constant with depth.
        3. decreases with depth.











        Answer 1 is right.

        This is true in stars, in the Earth's atmosphere, in the oceans, and in a bottle of water sitting at rest on table.

        The pressure must increase with depth to sustain the greater overlying mass with depth.

    2. The Kind of Pressure in Ordinary Stars:

      The pressure in ordinary stars (i.e., NOT white dwarfs, NOT neutron stars) is a combination of ordinary ideal gas pressure (provided by ions and electrons) and radiation pressure.

      The radiation pressure component increases with importance with stellar mass: for a star of 1 M_☉, it is almost negligible; for a star of 8 M_☉, it is about 10 % of the central pressure and a smaller fraction as one moves away from the center (Cl-162--164).

      The compact objects white dwarfs and neutron stars are held up by, respectively, degenerate electron gas pressure and degenerate neutron gas pressure. We discuss degeneracy pressure briefly in the figure below (local link / general link: gas_classical_quantum.html).

    3. An Example of a Stellar Structure Model:

      As mentioned above, the equations of stellar structure can ONLY be solved on a computer to determine star models.

      People have been doing so since the 1950s and nowadays the models can be quite sophisticated and accurate.

      Stellar structure models of main sequence stars are particularly easy to model since they are time-independent or STEADY STATE to good approximation. Good stellar structure models for all spectral types are available. As aforesaid, stellar structure models allow us to know many things we CANNOT know by direct or indirect observations.

      Pre- and post-main-sequence modeling is a lot tougher because the stars are evolving more rapidly and hydrostatic equilibrium is NOT always a good approximation.

      As an example, we present a cartoon of a stellar structure model of the Sun in the figure below (local link / general link: sun_model_interior.html).

  5. Radiative Transfer and Convection

    6. Well, we know heat energy is created in stars by nuclear burning in their cores and we know that it is radiated away. But how does the heat energy get out? The "how" is, of course, part of the modeling process discussed above: here we just go into qualitatively how the energy is transported.

    Recall a basic fact of thermodynamics: 2nd law of thermodynamics which requires entropy increase to a maximum for a closed system.

    Why this is so can be understood at the microscopic level, but we'll give it a pass.

    The interiors of stars are hot because of nuclear burning brought about by high temperature (initially due to stellar contraction and later due to nuclear burning itself) and high density due to gravity and the huge mass of stars.

    The stars actually have low entropy because they have so much compacted heat energy: i.e., highly ordered heat energy.

    But space has very high entropy because of its low density.

    The overall entropy of stars and space is increased by spreading out of the heat energy of the stars throughout space.

    The stars emit EMR to do this.

    In fact, according to our current understanding of cosmology, the stars will never succeed in heating space up.

    There may be a very cold thermodynamic equilibrium in of order 10**100 years, but that is a very speculative extrapolation of what we know today (HI-477). This future is discussed in IAL 30: Cosmology.

    1. Energy Transfer in Space and in Stars:

      In space, the spreading out energy is mainly transported by free-streaming radiative transfer of EMR.

      But how is it transported in stars. The heat energy transport in stars is mainly by two mechanisms: radiative transfer (but NOT free-streaming) and convection.

      In the interior of a star, radiative transfer is always occurring and the process is essentially a photon random walk process that we discussed in IAL 6: Light and Electromagnetic Radiation (EMR): Photon Propagation.

    2. Convection Redux:

      Convection is a universally important, macroscopic heat transfer process. It occurs in:

      1. almost stars including the Sun.
      2. the interior of the Earth. It is the driver of the plate tectonics of the crust as is discussed in IAL 11: The Earth.
      3. in the Earth's atmosphere.

        Question: What is a common, everyday, obvious example of convection?

        1. The freezing of water.
        2. The boiling of water.
        3. The churning of butter.











        Answer 2 is right.

        Boiling is actually the formation of bubbles of gas throughout a liquid and does NOT necessarily involve convection. But when water boils convection is usually also happening.

      Why and how does convection occur? An explication of convection is given in the figure below (local link / general link: convection.html).

      Convection happens whenever the TEMPERATURE GRADIENT becomes sufficiently steep and the insulation is a fluid---or at least sufficiently fluid-like as inside the Earth's mantle as is discussed in IAL 11: The Earth.

        Question: Why can't convection happen in truly rigid solids?

        1. The parts of the solid arn't free to move relative to each other.
        2. There are no heat flows through solids.
        3. Solids are perfect insulators.











        Answer 1 is right.

        Answer 3 is wrong, but if true, implies answer 2, but answer 2 does NOT imply answer 3 since there could be no heat flows in solids for other reasons. Just a little exercise in logic.

      Convection is a chaotic, turbulent process. Thus it is very hard to calculate its behavior in detail.

      The full calculation requires 3-dimensional hydrodynamics dynamics which is still difficult even with supercomputers. Frequently, one gets the wrong answer.

      Dealing with convection is one of the difficult and uncertain parts of astrophysics. For example, our understanding of stellar evolution is we think quite good, but uncertainty about convection is one of the weak links.

      The figure below (local link / general link: star_convection.html) illustrates convection in stars as a function of stellar mass.

      Convection happens in radial zones (or layers) in stars where for some reason the TEMPERATURE GRADIENT becomes too steep for stability against convection.

      Remember radiative transfer happens in convective zones too: it is just NOT the dominant heat transport mechanism there.

      In the Sun, the convective zone extends from about 0.71 R_☉ to the photosphere (Cox-342).

      In the Sun, we actually see the tops of the CONVECTION CELLS. They are the solar granules. See the figure below (local link / general link: sunspots_intro.html).

  6. Brown Dwarfs

    7. Brown dwarfs are substellar objects which we explicate in the figure below (local link / general link: brown_dwarf_comparison.html).

    UNDER RECONSTRUCTION BELOW

    Currently, there is no consensus on what the lower mass limit for brown dwarfs should be. But one popular choice is that a brown dwarf should have more than 13 M_Jupiter. Objects smaller than that are NOT planets, are sometimes called sub-brown dwarfs.

    At mass greater than 13 M_Jupiter, brown dwarfs can burn deuterons to helium; at mass greater than 60 M_Jupiter, they can burn lithium (3 protons, 4 neutrons) to helium by some process.

    During these burning phases, the brown dwarfs are fully convective and so they burn up all their deuterons and lithium in ???tens of millions of years????. (I need to check that time period).

    After their brief burning phase, brown dwarfs can only generate heat by contraction, and so cool off forever: ultimately they must become very cold, dim objects.

    People long expected brown dwarfs to exist---because there is no reason why substellar objects should NOT exist---but they were hard to find because they are very dim and radiate principally in the infrared as aforesaid.

    One of the first discovered brown dwarf was Gliese 229B in 1994. See the figure below (local link / general link: brown_dwarf_gliese_229b.html).

    Hundreds of brown dwarfs are known.

    Brown dwarfs and sub-brown dwarfs together may be about as numerous as stars in the universe based on current statistics.

    If this is so, then they do NOT contribute much mass to the universe and have little effect on cosmological evolution. Because they do NOT eject any material back into space as stars do, they are just quasi-eternal sinks for matter, and thus do NOT contribute to cosmic chemical evolution.

    In the solar neighborhood, one estimate puts the net mass of brown dwarf at about 10 % of the luminous mass (in stars, brown dwarfs, sub-brown dwarfs, gas, and dust).

    Of course, if brown dwarfs are much more numerous than stars, then they could make a significant contribution to the total mass of the universe.

  7. Evolution on the Main Sequence

    8. Stars do evolve on the main sequence.

    1. For Example, the Sun:

      For example of main-sequence stellar evolution, let us consider the main-sequence stellar evolution Sun as illustrated in the figure below.

      The decrease in hydrogen fuel in the core has the seemingly paradoxical result that stars will get more luminous over their main sequence life:

      1. Pressure support depends on the number of particles for ordinary gases. In fact, pressure is linearly proportional to the number of particles.

      2. When hydrogen is fused to helium, 4 particles are converted to 1.

      3. This causes a tendency to loss in pressure which in turn causes a tendency to contraction due the gravitational force on the Sun's own mass.

      4. But contraction increases the core density and increases the temperature essentially due to gravitational potential energy being converted into thermal energy.

      5. The higher density and temperature, the higher the pressure and thus collapse is resisted.

      6. But higher density and temperature tend to increase the collision rate of hydrogens and other isotopes in hydrogen burning chains, and thus the hydrogen burning rate.

      7. This tendency more than compensates for the decrease in fuel innermost core, and so there is a higher rate of energy production and the star gets brighter.

        Note there is lots of hydrogen in the outer parts of the core.

      8. The upshot is that main sequence stars burn brighter as they exhaust their fuel.

        Reference Se-246; FK-467.

      Again, the core of a star on the main sequence, thus gets denser and hotter.

      The extra energy tends to make the outer layers expand: so as the core gets denser, the star actually tends to increase in size.

      Stars larger than about 1 M_☉ get a little cooler on the surface during main sequence evolution. Since they are getting brighter too, they move upward and rightward on an HR diagram (CK-314).

      The Sun and smaller stars??? will actually get slightly hotter on the surface for awhile anyway (FK-467). They would move upward and/or to the left on the HR diagram. See the figure below (local link / general link: star_hr_named_stars_cartoon.html).

        These results for main-sequence stellar evolution are probably only understood in a computer modeling sense. This means we put the right physical ingredients in a computer model and we get the results. The results defy any simple explanation.

        A lack of simple explanation is often the case in computer calculations: we simply say we understand in a computer modeling sense.

      The brightening of Sun-size       main sequence stars is NOT a lot actually.

      The Sun is now about 30 % brighter than when initially on the main sequence about 4.6 Gyr ago.

      It will be about 30 % brighter than now in about 3.5 Gyr (WB-106; FK-493)

      This gradual brightening of the Sun is known only theoretically. But we think we understand main sequence stars well in their main behavior.

      So the brightening is about as certain a result as a purely theoretical result can be.

      The brightening is a pretty modest change for the Sun.

      But it probably means the doom of complex life on Earth within of order a gigayear or two by first eliminating the carbon dioxide from the atmosphere and then eliminating liquid water.

        Well one might imagine that life evolves in a way to overcome its current physical limitations, and endure or modify the brute physical changes that are coming. But that's more than physical science can know.

      The unhappy fate of life on Earth is discussed in IAL 11: The Earth.

      The brightening of the Sun is further explicated in the figure below (local link / general link: sun_evolution.html).

    2. Main-Sequence Lifetimes:

      The main sequence lifetime of a star begins when the star settles on the zero-age main sequence on the HR diagram???? (CK-304).

      It ends when the star's core hydrogen is sufficiently exhausted for the burning rate in the core to slow down or, more loosely, when the core hydrogen is exhausted (CK-311).

        Question: How does main sequence lifetime vary with star mass?

        1. It increases with mass.
        2. It stays constant with mass.
        3. It decreases with mass.











        Answer 3 is right.

      As one goes up in mass for main sequence stars, there is more hydrogen fuel, but the rate of burning increases even more rapidly and the main sequence lifetime becomes shorter.

      The lifetimes fall very quickly with mass in fact: 

        lifetime t ≅ constant/M**(2.5)  .

      This is a faster decrease than an inverse-square law decrease illustrated in the figure (local link / general link: function_behaviors_plot.html).

      A table of approximate       main-sequence lifetimes and a derivation of a main sequence lifetime formula are given in the figure below (local link / general link: star_lifetimes.html).

      According to the       Λ-CDM model (concordance model) of cosmology, the age of the observable universe = 13.797(23) Gyr (Planck 2018) since the Big Bang.

      This precise value for the universal age could be wrong if the Λ-CDM model is wrong---which is possible---but probably by less than 1 gigayear.

    3. Ages of Stars on the Main Sequence:

      We know the duration of main-sequence lifetimes from computer simulations to relatively high accuracy/precision nowadays.

      But how do we know the ages of observed main sequence stars?

      In general, we CANNOT. But we can if the main sequence stars are in star clusters.

      The method of determining the ages of main sequence stars in star clusters by means of Hertzsprung-Russell (HR) diagrams and turn-offs is explicated in the figure below (local link / general link: star_hr_turn_off.html).

    4. Stars with Stellar Mass Less Than 0.9 M_☉:

      If we look again at the table of approximate main-sequence lifetimes (see below), we that some low-mass stars will be on the main sequence for hundreds of gigayears.

      Post-main-sequence stars of that initially had masses less than about 0.9       M_☉ don't exist now and won't for some time to come.

      The fate of very low-mass stars (and also stars of mass up to about 8 M_☉) is to become white dwarfs whose nature is discussed in IAL 24: Compact Remnants: White Dwarfs and Neutron Stars.

      Here will just say white dwarfs are star-mass objects of Earth-size (and thus very dense) that are NOT burning any nuclear fuel and are just cooling off forever.

        White dwarfs exist, of course: there are just none that originated from stars of less than about 0.75 M_☉ when on the main sequence.

      Because we have never observed even snapshots of this transformation for low-mass stars and never will (on the ordinary human time-scale), we know less about it than what happens to larger-mass star: i.e., star from about 0.9 M_☉ and up.

      But it is is probably a comparatively gentle process??? particularly for red dwarfs which are main sequence stars in the mass range 0.08--0.4 M_☉ (CK-311).

      Red dwarfs are convective throughout and will convert almost all there hydrogen to helium.

      They will never burn helium unlike more massive stars, and so should just contract probably pretty slowly into white dwarfs.

      They probably will just cool in the post-main-sequence phase (CK-311), but since contraction causes heating they might heat up for awhile???? but I've NOT been able to find out for sure.

      IAL 23: Late Star Evolution and Star Death takes up the subject of the post-main-sequence evolution of stars of about solar mass and larger.

  8. Preview of Post-Main-Sequence Evolution

    9. As preview of the post-main-sequence evolution of stars (which is more fully explicated in IAL 23: The Post-Main-Sequence Life of Stars), see the two Hertzsprung-Russell (HR) diagrams in the two figures below (local link / general link: star_hr_post_main_sequence.html; local link / general link: sun_evolution_hr.html).