Metallicity Evolution

1. Metallicity:
   1. In astro jargon everything that is NOT hydrogen (H, Z=1) or helium (He, Z=2) is a metal---just accept it.

      Note:

      1. cosmic composition (meaning inside modern galaxies: fiducial values by mass fraction: 0.73 H, 0.25 He-4, ∼< 0.02 metals).
      2. primordial cosmic composition (which is also almost cosmic present intergalactic medium composition: fiducial values by mass fraction: 0.75 H, 0.25 He-4, 0.001 D, 0.0001 He-3, 10**(-9) Li-7).

   2. The term metallicity is the amount of metals.

      The amount of metals is expressed in several ways. Two common ways:

      1. metallicity mass fraction Z (which is NOT atomic number Z).
      2. [Fe/H] = log[(N_Fe/N_H)/(N_Fe/N_H)_☉]: The logarithmic ratio of iron (Fe, Z=26) to hydrogen (H, Z=1) relative the solar composition of the same ratio.

      Metallicity is usually determined by modeling stellar spectra.

      Examples of metallicity measurements are given below in Image 1.

      

   3. Image 1 Caption: The absolute V magnitude M_V (center wavelength 0.551 μm, full width half maximum 0.088 μm) versus metallicity [Fe/H] = log[(N_Fe/N_H)/(N_Fe/N_H)_☉]: for Local Group galaxies. Except for the Milky Way (MW), the points are all for dwarf galaxies it is thought. The only identified galaxies are the Milky Way (MW) and M32 (NGC 221) (a dwarf elliptical satellite galaxy of the Andromeda Galaxy (M31, NGC 224)).

      Note absolute V magnitude M_V (center wavelength 0.551 μm, full width half maximum 0.088 μm) is the logarithmic integrated luminosity proxy through the V band passpand. It is roughly the intrinsic brightness to the human eye of an astronomical object seen at 10 parsecs (pc).

      The magnitude scale is a wrong-way logarithmic scale: upward is brighter and 5 magnitudes is a factor of 100 in brightness.

      The correletation between metallicity [Fe/H] and absolute V magnitude M_V is probably just as follows. Metallicity increased with cosmic time until ∼ 5. The galaxies that galaxy quenched (i.e., stopped star formation) before then stopped increasing in metallicity in some proportion to the cosmic time of quenching. The longer a quenched galaxy has been quenched, the dimmer it tends to be since more stars have evolved into dim white dwarfs.

      So the basic trend seen in Image 1 is understandable: the more metal poor, the dimmer. However, for quantitative understanding, a detailed analysis must be done.

2. Gas Inflows and Outflows:
   1. Most galaxies whether unquenched galaxies or quenched galaxies have inflows and outflows of gas. The inflows are largely have the primordial cosmic composition (maybe sometimes somewhat enriched by metals from stellar nucleosynthesis and supernova nucleosynthesis from past outflows from galaxies).

      Inflows and outflows of gas are illustrated in the figure below (local link / general link: gas_inflow_outflow.html).

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   2. In unquenched galaxies, star formation occurs and stellar winds from stars (mostly post-main-sequence stars) and supernovae enrich interstellar medium (ISM) in metals. Recall in astro-jargon, metals are everything which is NOT hydrogen (He) and helium (He). Some deuterium (D, H-2) is counted as a metal too.

   3. In quenched galaxies the gas stays too hot for new star formation. For fullish explication of galaxy quenching, see file galaxy_quenching.html.

      But we can say here that galaxy mergers often cause starbursts by strongly compressing the colliding interstellar medium (ISM) which induces a high star formation rate (SFR). After a starburst has exhausted the ISM, the SFR falls steeply and will usually go to near zero permanently if the merged galaxy mass (including dark matter mass) exceeds the golden mass 10**12 M_☉ after order 1--3 Gyr???? (see Dekel et al. (2019)). The golden mass 10**12 M_☉ galaxy quenching rule is NOT a law of nature, but rather a conspiracy of nature: i.e., various physical effects combine to effect it.

      Note quenched galaxies (which are usually elliptical galaxies) are often referred to as red. But this actually means that their spectrum in the visible band (fiducial range 0.4--0.7 μm) is stronger in red light than unquenched galaxies. The psychophysical response to their spectrum is actually white light or yellowish white in true color.

   4. The inflows to galaxies are caused by the attraction of the gravitational force of galaxies which is dominated by the gravitational force of the dark matter halos. The baryonic matter (stars and interstellar medium (ISM)) contributes ∼< 1/6 of the gravitational force.

   5. The outflows from galaxies are mostly ejecta from supernova explosions and relativistic bipolar jets from active galactic nuclei (AGNs) or weaker jets from inactive galactic nuclei. Note whether strong jets or weak jets, the jets are powered by the magnetic fields in the accretion disks around central supermassive black holes (SMBHs) which probably all large galaxies have (see Wikipedia: Supermassive black hole). In quenched galaxies (which are mostly elliptical galaxies), there can be convective blobs of hot gas that rise from accretion disks around central supermassive black holes (SMBHs) that contribute to net outflow????.

   6. In most galaxies the inflow and outflow are probably about equal most of the time. In starburst galaxies, the outflow dominates as shown in the figure above (local link / general link: gas_inflow_outflow.html). The increased outflow is caused by the much higher rate of supernova explosions.

      The star formation rate (SFR) in starburst galaxies probably varies tremendously, but the typically it is thought that their rate is ∼ 30 times that of ordinary unquenched galaxies like the Milky Way (see Wikipedia: Starburst galaxy), and so the supernova rate is probably typically ∼ 30 times that of ordinary unquenched galaxies.

   

3. Metallicity Evolution and Saturation:
   1. The metallicity of galaxies does evolve with cosmic time: i.e., the time since the Big Bang (a time equal to age of the observable universe = 13.797(23) Gyr (Planck 2018)).

      However, even for unquenched galaxies (i.e., those with at least moderate star formation to cosmic present), metallicity tended to saturate or plateau at solar metallicity or a bit above after cosmic time ∼ 5 Gyr.

   2. Image 2 Caption: The cosmichemical evolution of metallicity illustrated by the proxy of simulated oxygen (O) abundance evolution calculated for a simple model (David Weinberg 2016, On the Deuterium-to-Hydrogen Ratio of the Interstellar Medium, p. 3).

   3. The horizontal axis is cosmic time from the Big Bang in gigayears (Gyr).

   4. The vertical axis is abundance ratio for oxygen (O).

      Abundance ratio is the logarithmic ratio of a metallicity proxy to hydrogen both in number of atoms per unit volume divided by the same for the Sun. The generic formula is

            [A/H] = log[ ( N_A/N_H )/( N_A/N_H )_☉ ] ,

      where A is the element (usually oxygen (O) or iron (Fe)), N_A is number of atoms per unit volume, A = H is hydrogen, and ☉ is the Sun symbol.

   5. The curves shown are for various sets of free parameters of the model used in the calculations.

      The fiducial curve is the green one that saturates at the solar metallicity which is zero for [O/H].

   6. Qualitatively, the real metallicity of galaxies should evolve as in the Image 2.

   7. To understand cosmichemical evolution, first examine Table: Cosmic Composition below which shows fiducial values for the (modern) cosmic composition and the primordial cosmic composition.

   8. Big Bang nucleosynthesis generated the primordial cosmic composition.

      The subsequent production of metals was in stars and supernovae which ejected the metals into the interstellar medium (ISM). The stars did this via strong stellar winds mainly in their post-main-sequence phase.

      The ISM enriched in metals, then contributes to ISM would strictly increase in metallicity with cosmic time as new generations of stars contributed to its metallicity. The increasingly enriched ISM would then cause new generations of stars to also be enriched in metals

      But this is NOT the case.

   9. The case is that supernovae and/or strong activity from the central supermassive black holes of galaxies eject matter processed through stars into the intergalactic medium.

      The ejected matter is largely replaced by inflows from the intergalactic medium (IGM of mostly primordial matter from the Big Bang: i.e., matter with the primordial cosmic composition which is nearly pure hydrogen, helium, and a small about of lithium.

   10. So at first the ISM becomes enriched in metals, but in a few gigayears after the Big Bang, the ISM and the stars that form out of it reach a near-equilibrium metallicity. This near-equilibrium is of order 0.01 to 0.04 ???? metals by mass fraction.

   11. So Carl Sagan (1934--1996) was only partially right when he said We Are Made Of Star Stuff (Carl Sagan). Our hydrogen (a fair amount in the human body) and helium (very little in the human body) is Big Bang stuff.

   12. Can we prove the near-equilibrium metallicity? Yes, but this is a digression beyond the scope of an intro astronomy course.

       Assume a 1-zone model of galaxy of fixed size with average density ρ. As a simplifying assumption, we assume that no matter is permanently locked up into compact remnants (white dwarfs, neutron stars, black holes), brown dwarfs, planets, or smaller astro-bodies. We can easily dispense with this assumption if want to.

       A differential equation determining the change in ρ with time t is

       (dρ/dt) = (dρ/dt)_inflow - κρ ,

       where (dρ/dt) is rate of change in ρ, (dρ/dt)_inflow is a constant inflow rate, κ = 1/τ is the rate of outflow constant, and τ is time parameter that is the inverse of κ. The solution of the differential equation via an integrating factor is

       ρ = (dρ/dt)_inflow*τ*[1-exp(-t/τ)] + ρ_0*exp(-t/τ)     in general;

       ρ = (dρ/dt)_inflow*t + ρ_0                                            for t << τ;

       ρ = ρ_∞ = (dρ/dt)_inflow*τ                                           for t → ∞,

       where ρ_0 is the initial density at time zero whatever that is for the galaxy. We see that ρ saturates at ρ_∞ = (dρ/dt)_inflow*τ. Note that as the rate of outflow constant decreases (i.e., τ increases), the saturation density increases: i.e., τ ↑ ρ_∞ ↑.

       Now metallicity times density Zρ obeys the differential equation

       [d(ρZ)/dt] = Z_IGM(dρ/dt)_inflow - κ(Zρ) + γρ ,

       where Z_IGM is the constant metallicity of the intergalactic medium (IGM) and γ is the rate of creation of metallicity constant which has units of inverse time. For the saturation solution, we set [d(ρZ)/dt] = 0 and ρ = ρ_∞, and obtain

       Z = γτ + Z_IGM .

       If γτ = 0, then the metallicity is just that of the intergalactic medium (IGM). Note that as (γτ) ↑ Z ↑.

   13. The near-equilibrium of metallicity will probably last many gigayears into the future.

   14. Eventually, the intergalactic medium itself might become sufficiently enriched with metals that inflow from it is enriched in metals. In this case, metallicity of the ISM and stars would probably begin to strictly increase.

   15. However, inflow from the intergalactic medium itself will gradually decrease as the intergalactic medium is used up or ceases to inflow due to the universal expansion. This will eventually bring star formation to an end.

   16. Star formation, in fact, has already been decreasing for gigayears as we know because the overall emission from stars has been decreasing since ∼ 10 Gyr ago when it peaked in what is called the cosmic noon (z≅2, cosmic time ∼ 4 Gyr) (Driver et al. 2015). We know this factoid because we can see the earlier observable universe---whenever you look out, you look back.

       Note the age of the observable universe = 13.797(23) Gyr (Planck 2018) (see Planck 2018: Age of the observable universe = 13.797(23) Gyr) according to the Λ-CDM model which is our current standard model of cosmology (SMC, Λ-CDM model) The Λ-CDM model. might be revised or replaced sometime in the 2020s. See Tensions of the Λ-CDM Model Circa 2020s, for a discussion of current problems with the Λ-CDM model.

   17. To be a bit more precise about the end of star formation according to the standard model of cosmology (SMC, Λ-CDM model) wildly extrapolated, at cosmic time of order 10**14 years (about 10000 times the current cosmic time), all star formation will have ended and all the long-lived M stars will have left the main sequence???. This is the end of the Stelliferous Era (i.e., the star-making era).

   18. For a further discussion of the end of Stelliferous Era and later eras of the observable universe according to the Λ-CDM model wildly extrapolated, see The Fate of the Universe According to the Λ-CDM Model.

Images:

1. Credit/Permission: © User:Vallastro, 2022 / CC BY-SA 4.0.
   
2. Credit/Permission: © David H. Weinberg 2016 / None.
   Image link: On the Deuterium-to-Hydrogen Ratio of the Interstellar Medium, David H. Weinberg, 2016, p. 3, Figure 1.
   Placeholder image alien_click_to_see_image.html.