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Ionization & the Saha Eq.
The ionization state of a gas in LTE can be expressed in a fashion similar to the Boltzmann Eq. ,
N K +1 2U K +1 2πme kT χ = ( ) exp[ K ], NK ne U K h2 kT
1 3 m < v 2 >= kT 2 2 <v2>: mean-square particle speed. Root-mean-square (r.m.s) particle speed:
vrms ≡ < v 2 > = 3kT / m
most probable speed: mean speed:
2
§3. 2. Dark & Light Matter
Most of matter invisible.
3
Evidence for dark matter
Galaxies rotate much faster in their outer regions. extended dark matter surrounding galaxies. Dark matter is needed to bound galaxy clusters. Gravitational lensing Mtotal >>Mvisible 2 independent pieces of evidence for DM in GCs.
E Nn gn = exp[ ], N 1 g1 kT
Where n is the principle quantum number, Nn is the number of atoms in which electrons are in the nth energy level (e.g., N1 is the number of atoms with electrons in the ground state), gn is the statistical weight (e.g., for hydrogen, gn=2n2), and ⊿E is the energy difference between state n and the ground state.
8
§3. 4. The Gaseous Universe
H & He Metallic-like liquid H at the center of Jupiter Temperature T Density ne State of ionization Spitzer (1978): for T<8x104K, particle encounters are almost always elastic. thermal timescale ~ hours – years astrophysical gases are in thermal equilibrium. H (n=2)
vmp = 2kT / m
8kT 8 RT = πm πM
10
<v> =
the ideal gas
An ideal gas is a gas that obeys the ideal gas law (particle pressure):
Pp = nkT =
where ≡
ρ m H
kT
<m> 1 = mH mH N
where UK+1 & UK are the partition functions of the (K+1)th & Kth ionization states, respectively, ne is the electron density, me the electron mass, χK is the energy required to remove an electron from the ground state to the Kth ionization state. e.g., hydrogen only has one electron to be removed, & can only exist in the singly ionized or neutral states, the Saha Eq. reduces to,
=
1 2 X + 3Y / 4 + Z / 2
(ionized )
+ electron degeneracy
P = Pp + Prad
11
Statistical equilibrium, (Local) Thermal Equilibrium=(L)TE
collision excitation/de-excitation absorption excitation/ionization emission de-excitation The populations of energy levels are determined by including all processes that both populate & de-populate any given level. In a steady state, the transition rate into any level equals the rate out – statistical equilibrium. Equations of statistical equilibrium are set up for each level and involve the density of the particles, the energy density of the radiation field, and coefficients describing collisional, radiative, and spontaneous transition probabilities. The coefficients may themselves be functions of other quantities, such as quantum mechanical parameters or temperature. Extremely complex simplification If a gas is in TE, the energy in the radiation field is in equilibrium with the kinetic energy of the particles. LTE: a gas has TE properties, but only locally.
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Boltzmann Equation
For LTE, the equations of statistical equilibrium are much simplified and the population of states is given by the Boltzmann Equation:
(%) LightWeighted Fractional Contribution
40 30 20 10 0 30 20 10 0 80 60 40 20 0 3 2.5 2 1.5 1 0.5 0 log10(t) (Gyr) 0.5 1 1.5 2
IMF = Initial Mass Function: The admixture of stars of different masses when first formed.
M<8Msun planetary nebula WD
Zhou+06
M>8Msun SNe II elements heavier than Fe Typical kinetic energy ~ 1051 erg
7

star formation history & IMF
60 50
Lu, Zhou, Wang et al. 2006
1
§3. 1. The Big Bang
Part 4. Cosmology
Not an event occurred sometime somewhere. Spacetime came into being with the Big Bang.
Redshifts of galaxies The age of the Universe vs. oldest stars: 12.7 – 13.2 Gyr CMB = Cosmic Microwave Background Abundances of light elements nucleosynthesis in the first moments
How to know T, ne, and H+/H0 ?
9
Kinetic temperature & MaxwellMaxwell-Boltzmann velocity distribution
Gas, energy changed between particles via elastic collisions Statistical mechanics Maxwell-Boltzmann velocity distribution kinetic temperature
X = 0.77, Y = 0.23, Z = trace
Stellar evolution & ISM enrichment Part II. Stars
P-P chain (M<1.5Msun) He: the only product