UBV filters

Stefan-­Boltzmann Law• The flux (fl) of a hotter object is greater at all wavelengths than a cooler object, and

f = fλ0

∞

∫ dλ ∝T 4

Abundances and Line StrengthsQ:Why do we see the trends we do in spectral sequences?Specifically, why does H line strength peak at spectral type A0 (Teff ≈ 10000 K)?Also, why do we say the Sun is mostly made of Hydrogen when the lines of Calcium are stronger than those of Hydrogen?

To answer these questions, we have to delve into statistical mechanics, to answer the following questions:

Q: What determines the strength of an absorption line?A: The relative number of atoms in the lower energy state of the transition.

Q: What determines the number of atoms in a given energy state?

A: Excitation and Ionization

Hydrogen Energy Levels

Excitation Ionization

Maxwell-­Boltzmann Velocity Distribution

The energy difference between the n=1 and n=2 states of Hydrogen is 10.2 eV. At what temperature are there equal numbers of atoms in those two states?

(hint: kT = 1 eV for T = 12000 K)

55,000 K

85,000 K

120,000 K

170,000 K

A

B

C

D

Calcium has an ionization energy about half as much as hydrogen’s ionization energy: about 6.5 eV versus 13.6 eV (recall that kT = 1 eV, for T = 12000 K)In an equal mixture of these gases (at the same temperature and pressure) how does the ionization fraction of calcium compare to that of hydrogen?

NCaII/NCaI is less than half of NHII/NHI

NCaII/NCaI is about half of NHII/NHI

NCaII/NCaI is about twice of NHII/NHI

NCaII/NCaI is more than twice NHII/NHI

A

B

C

D

Boltzmann-­Saha for Balmer Lines

Line strengths v. temperature

Key takeaways from stellar spectroscopy

• At high T, n, the populations of energy levels and ionization state of an atom are determined by collisions

• The line strength of a transition is proportional to the population of the initial lower (or upper) state for absorption (or emission) lines

• The population of a given energy state is determined by the balance between excitation (or de-­excitation) and ionization

• At low temperature (T à 0), the populations are dominated by excitation;; at high temperature (T à ∞) by ionization

• To determine abundances of an element, it is important to correct for the fact that a single line represents only a fraction of the total abundance

• For example, in the Sun, only 1 in 200,000,000 Hydrogen atoms are in the n=2 state

N2

N= (g2 g1)e−ΔE12 kT

1+ A(kT )3 2

Ne

e−χ0 kT⎡

⎣⎢

⎤

⎦⎥

(A = const.)