The Astronomy Module, Stellar Evolution - Lecture Notes | PHYS 2018, Study notes of Physics

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Physics 2018: Great Ideas in Science:

The Astronomy Module

Stellar Evolution Lecture Notes

Dr. Donald G. Luttermoser East Tennessee State University

Edition 1.

Abstract

These class notes are designed for use of the instructor and students of the course Physics 2018: Great Ideas in Science. This edition was last modified for the Fall 2007 semester.

c) Population class which is measure of the metalicity – the abundance of elements heavier than helium.

4. All three of these classification schemes above can be deduced from a star’s spectrum. A spectrum is a plot of the star’s energy flux as a function of wavelength (or frequency, or energy). Note that the energy flux is essential the brightness of the spectrum at a particular wavelength.
5. If we assume that star’s shine like a blackbody radiator, it is easy to show from the Stefan-Boltzmann radiation law that a star’s luminosity, L, is related to the star’s radius, R, and average photospheric (i.e., ‘surface’) temperature, the so-called effective temperature, Teff, via

L = 4π σ R^2 T (^) eff^4 , (II-1)

or in terms of solar ( ) values

L L

( R

R

) 2   Teff Teff( )

 

4 , (II-2)

where the effective temperature of the Sun is Teff( ) = 5770 K.

B. Spectral Classification.

1. The first large-scale classification of stellar spectra was under- taken by Mrs. W.P. Fleming, Antonia Maury, and Annie Jump Cannon in the 1920’s at Harvard College Observatory and be- came known as the Henry Draper (HD) catalog. a) Over 400,000 stars were classified.

b) Stars were grouped by hydrogen (Balmer) line strengths with designations A-S.

c) Classes C, D, E, H, I, J, L, P, and Q were dropped for one reason or another or merged into other classes (see Jaschek and Jaschek 1987, The Classification of Stars, Cambridge Press).

d) The R and N stellar classifications corresponded to car- bon stars and now have been merged into one classifica- tion designated C (although not the same as the original C stars, which were spectroscopic binaries). Many peo- ple however still use the R and N classification (including me) to describe carbon stars. R stars are the hotter of the two and correspond to the oxygen-rich K stars in temper- ature. The N-type carbon stars correspond to the coolest oxygen-rich M stars in temperature. Carbon stars dif- fer from oxygen-rich stars in that there visual spectrum is dominated by carbon molecule (i.e., C 2 , CN, and CH) absorption bands.

e) S stars are another special class similar in temperature to late K and M stars. The S star’s spectrum is dominated by LaO, VO, and ZrO molecular bands.

2. Groups were rearranged from the hottest (called early-type stars ) to the coolest (called late-type stars ) and 10 subdivisions for each group introduced. Each “spectral type” is determined by various line strengths and line ratios. A brief overview is displayed in Table II-1. A more complete description can be found in Jaschek and Jaschek (1987) and Kaler (1989, Stars and Their Spectra, Cambridge Press).
3. Later it was found by Saha, that the sequence of spectral types from hottest to coolest stars should follow: O B A F G K M (R N S)

a) Classes R, N, and S are special as described above.

b) The weakness of the H lines in O stars is due to most of the hydrogen being completely ionized.

c) The weakness of the H lines in M stars is due to their cool atmospheres, most of the electrons are in the ground state, with virtually none in the 2nd level where the Balmer lines arise.

C. Luminosity Classification.

1. Absorption lines that appear in a star’s spectrum arise in the outer layers of a star’s atmosphere. Spectral lines are broadened (i.e., made thicker = stronger) via a variety of processes. One of these processes is particle collisions. a) Since collisional rates are a function of density and density depends on the surface gravity of a star; bigger, lower gravity stars will tend to have sharper lines than high gravity stars for a given spectral type.

b) A broader line “for the same spectral type” generally im- plies higher gravity.

2. As we just saw, luminosity depends both on the radius and tem- perature of a star.
3. Table II-2 displays the luminosity classification system.
4. The Morgan-Keenan (M-K) classification of a star is simply the spectral type along with the luminosity class and defines a star’s location on an H-R diagram (i.e., the Sun is a G2 V star, α Boo a K1 III star, see below).

Table II–2: Luminosity Classifications Luminosity Absolute Visual Mag Class Type B0 F0 M Ia Luminous supergiants –6.7 –8.2 –7. Ib Less luminous supergiants –6.1 –4.7 –4. II Bright giants –5.4 –2.3 –2. III Normal giants –5.0 1.2 –0. IV Subgiants –4.7 2. V Main sequence (dwarfs) –4.1 2.6 9. sd (VI) Subdwarfs (pop II dwarfs) 10. wd (VII) White dwarfs 10.2 12.

D. Stellar Populations.

1. Chemical abundances are determined from line strengths. Typi- cally, an elemental abundance is tabulated in the form of [X/H],

[X/H] ≡ log[n(X)/n(H)]? − log[n(X)/n(H)] , (II-3)

where X is the element in question. a) To make matters more confusing, elemental abundance is sometimes scaled to the hydrogen abundance, with the logarithm of the hydrogen abundance normalized to 12.00. For instance, the solar abundance for sodium in this sys- tem is 6.33. This means that the actual abundance, α = n(X)/n(H), is 2. 14 × 10 −^6 (log α = 6. 33 − 12 .00 = − 5 .67).

b) Confusing things even further, the abundance is tabulated with respect to the total number density and not the hy- drogen density. One can adjust from one to the other by realizing that n(H)/n(total) = 0.908 in the Sun. Except for the peculiar hydrogen deficient stars, it is always as- sumed that stellar hydrogen abundance is equivalent to the solar value.

c) Population III stars: i) Zero metal abundance (Z = 0).

ii) They no longer exist in the Galaxy.

iii) These were the first stars to form out of the pri- mordial baryons formed during the Big Bang.

3. Besides representing metalicity with the “Z” index (i.e., mass fraction of metals to all particles), metalicity also is defined by a star’s Fe abundance with respect to the Sun following Eq. (II-3):

[Fe/H] ≡ log 10

  n(Fe) n(H)

  (^) − log 10

  n(Fe) n(H)

  (^). (II-5)

a) The Sun’s Fe abundance is about log 10

( (^) n(Fe) n(H)

) = 10−^5.

b) If a star has solar metalicity, then [Fe/H] = 0.0.

c) Population I stars range from − 0. 5 < [Fe/H] < +0.5.

d) Population II stars have [Fe/H] < − 0 .8.

e) The few stars that have − 0. 8 < [Fe/H] < − 0 .5 are typ- ically called old disk population stars, though they are typically grouped with the Population I stars.

f) Population III stars will have [Fe/H] = 0 if they are ever observed in the youngest galaxies at the farthest reaches of the Universe.

Figure II–1: The Hertzsprung-Russell Diagram.

E. The Hertzsprung-Russell Diagram

1. In the early 1900s, 2 astronomers found peculiar groupings of stars when they plotted their absolute magnitude (a star’s bright- ness at 10 parsecs distance) to their (B − V ) (blue - visual mag- nitudes) colors (Hertzsprung) or spectral type (Russell) =⇒ the Hertzsprung-Russell (HR) Diagram (see Figure II-1). a) Main sequence stars (∼90% of all stars are of this type).

b) Giant stars (most of them red in color).

c) Supergiant stars (very luminous, hence large).

d) White dwarf stars (very faint, hence small).

2. When plotting the HR diagram for a star cluster, the stars are vir- tually at the same distance, hence differences in observed bright- ness corresponds to actual luminosity differences (and not dis-

upper right.

b) Theoretical HR diagrams plot luminosity (L/L ) versus effective temperature (Teff) of the star. i) Eddington, through the process of radiative diffu- sion, showed that L ∝ M^4 on the main sequence, or L L

( M

M

) 4

. (II-6)

ii) Since the time that a star spends on the main sequence depends on the amount of fuel the star has (M) divided by the rate at which it burns that fuel (L), the main sequence lifetime of a star is tlife ∝ M/L or M−^3 , hence

tlife =

(M

M

) 3 × 1010 years. (II-7)

iii) The y-axis goes from low (lower portion) to high luminosity (upper portion).

iv) The x-axis goes from hotter temperatures (left side) to lower temperatures (right side).

F. Stellar Nurseries.

1. Stars form from the dust and gas that lie between the stars =⇒ the interstellar medium.
2. Stellar nurseries are found within giant molecular clouds (GMC) as shown in Figure II-2. The nearest GMC to the solar system is the Orion complex.

Hot Gas [dark shaded area] (0.1 atoms/cm^3 ) (T = 10 6 K)

Giant Molecular Clouds (10 - 1000 atoms/cm^3 ) (T = 10 - 50 K)

Figure II–2: Structure of the Interstellar Medium.

G. The Jeans’ Length and Jeans’ Mass.

1. J.H. Jeans first analyzed the gravitational instability of gas clouds in 1902 =⇒ the minimum size of a cloud which will collapse under its own gravity is called the Jeans’ Length.
2. It is nothing more than the conservation of energy. In an in- terstellar gas cloud, there are two energy sources: the thermal energy of the gas, Eth, and the gravitational energy of the clouds gravitational field, Eg. For a static cloud, energy conservation gives: Eth + Eg = 0. (II-8)

a) The thermal energy is just the pressure multiplied by the volume of the material:

Eth = P · V = ρ kB T μ mH

π R^3. (II-9)

c) Since this free-fall time depends only on density, all parts of the cloud will collapse at the same as long as the cloud has uniform density =⇒ homologous collapse.

6. A GMC will not necessarily collapse as a single unit, typically just a portion of it will collapse based upon the triggering mechanism (see below). a) The density of the collapsing cloud will increase by orders of magnitude during the free-fall time.

b) Though we have developed the collapse criteria under the assumption of constant density, in reality, there will be pockets of inhomogeneities in the cloud.

c) As a result, sections of the cloud will independently sat- isfy the Jeans’ mass limit and begin to collapse locally, producing smaller features within the original cloud.

d) This cascading collapse could lead to the formation of large numbers of smaller objects.

H. Triggers of Star Formation (SF).

1. Observations. a) In the Milky Way Galaxy, SF occurs in GMCs.

b) OB stars form in associations at edges of GMCs.

c) OB association ionizes the surrounding gas producing an H II region.

d) Lower mass (T Tauri-type) stars form throughout the vol- ume of the GMC.

e) SF defines the optical spiral arms of the Milky Way.

2. What is the trigger? Any process that can cause a stable (M < MJ ) cloudlet to become unstable (M > MJ ). a) Agglomeration: Component cloudlets of GMC’s collide and sometime coallace until M > MJ.

b) Shock Wave Compression: A shock can be the trigger =⇒ it acts like a snow plow causing ρ to increase, and as a result, MJ drops (see Figure II-3). i) Spiral Density Wave: As Milky Way Galaxy rotates, its two spiral arms can compress a GMC, which then leads to star formation.

ii) Ionization Front: O & B stars form very quickly once cloud collapse has started (see below). These produce H II regions from their strong ionizing UV flux, which initially expand outward away from the OB association. This ionization front heats the gas causing a shock to form. The shock can compress the gas such that M > MJ , which once again, leads to star formation.

iii) Supernova Shocks: O & B stars evolve very quickly on the main sequence and die explosively as supernovae. The shock sent out by such a su- pernova can excite further star formation.

I. The Free-Fall Stage of Stellar Birth.

1. As a portion of a GMC begins to contract, cloud complexes with masses greater than ∼ 50 M become unstable and fragment into smaller cloudlets (see Figure II-4). Each little cloudlet continues to collapse as described above.

Collapsing Cloudlets

Figure II–4: Collapsing cloud fragmentation to smaller collapsing “cloudlets.”

axis

Cloudlet

Figure II–5: Cloudlet contraction and spin as a result of internal eddies.

axis

Bulge

Disk

Figure II–6: Formation of protostar and protoplanetary disk.

i) In the outer protoplanetary disk, ice crystals con- dense out of the gas along with some dust grains.

ii) In the inner protoplanetary disk, it is too hot for ice crystals to form, only dust condenses out.

iii) The dust (and ice) begin to conglomerate to- gether in a process known as advection (similar to building a snowman), building bigger and bigger particles. This processed continued until boulder to mountain sized objects existed =⇒ the planetesi- mals.

iv) Planetesimal is the name given to big rocks in orbit about a “protostar.” Due to their small size (D < 1000 km), they are not spherical in shape.

• The “rocky” planetesimals are now called aster- oids.