F4 - breaks

image: H Cegla, lecture notes

H has a well known spectrum defined by the starting and ending levels (quantum number $n$ ) of the radiation emitted

$n$	series	symbol	range
1	Lyman	Ly	UV
2	Balmer	Ba/H	visible
3	Paschen	Pa	near-IR
4	Brackett	Br	IR
5	Pfund	Pf	IR
6	Humphrys	Hu	IR

the transitions are indicated by Greek letter
the closest (lowest E) with $α$ , $β$ for the second, $γ \dots$
eg: $H_{α}$ for $n = 2 \leftrightarrow n = 3$ , Ly $_{α}$ for $n = 2 \leftrightarrow 1$
the Rydberg formula allows the calculation of the vacuum wavelength of photons corresponding to any transition from energy level $n_{1}$ to $n_{2} :$

\frac{1}{λ_{v a c}} = R_{H} (\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}})

where, $R_{H} = 1.1 \times 10^{7}$ m $^{- 1}$

the energy levels become progressive closer for growing $n$ , asymptotically approaching the continuum level, which is $13.6$ eV for H
any incoming photons with energy higher than the difference between the ground state and the continuum can ionise the atom (bound-free transition)
the spectra are a balance between neutral and ionised states
the difference between the lowest energy of any series and the continuum energy is called the series limit

image: NASA

every time a series limit is approached, the spectrum shows a sharp drop in flux, called a break
- Lyman break at $912$ angstorm
- Balmer break at $3646$ angstorm
any photon with energy higher than the limit (shortre $λ$ ) can ionise the atom
they are therefore removed from the line of sight (absorbed or scattered) producing a drop in the measure flux
in order to actually see the Ly break, there has to be enough photons to start with
this is only possible in galaxies with recent episodes of star formation (abundant blue stars, hence UV/blue dominated spectrum)
the existence of Ly/Ba breaks is crucial to measure the redshift for far-away galaxies with brightness too low to see individual spectral lines
due to the Doppler effect: $λ_{o b s} = λ_{r e s t} (z + 1)$
the Ly/Ba breaks thus occur in the visible/IR for far away galaxies
at the extreme end, both can be observed in the IR (eg. $z = 7$ in image above)

H in star-forming regions

image: H Cegla, lecture notes

the density and temperature of H gas are considerably higher in the immediate surroundings of star-forming regions than in the rest of the ISM
carefully consider the balance between neutral and ionised H to understand the spectrum of star-forming regions
the balance is a product of:
- temperature: determines the population of energy levels
- density: drives collisions and transfers energy to atoms
- EM radiation: photons carry energy and are absorbed/emitted
stellar cores are young, hot, luminous precursors of stars, emitting mostly in UV
moving radially outward from stellar cores, all of the above decrease
this creates a gradient in the ionisation state of the surrounding H gas

HII region

to find the size of a HII region, the ionisation rate can be computed as:

N_{i o n} = N_{U V} = \frac{ϵ L}{13.6 eV}

where, $ϵ$ is the luminosity fraction emitted above the Lyman limit

for type O stars, $ϵ \approx 1$
the ionisation process is balanced by recombination, happening at a rate:

N_{r e c} = V_{H I I} n_{p} n_{e} α_{2}

where, $V_{H I I}$ is the volume of the HII region, $n_{e}, n_{p}$ are electron and proton number densities, $α_{2}$ is the recombination constant

assuming a spherical shell of radius $r$ , and imposing a balance between the ionisation and the recombination rates:

r = {[\frac{3 N_{i o n}}{4 π n_{p} n_{e} α_{2}}]}^{1 / 3}

for any O5 star, the rate of ionisation = the rate of photons $\approx 5 \times 10^{49}$ s $^{- 1}$
typical-star forming regions have $n_{e} \approx n_{p} \approx 10^{9}$ m $^{- 3}$ and $α_{2} = 3 \times 10^{1} 0^{19}$ m $^{3}$ s $^{- 1}$
this results in a typical size of $1$ pc for a HII region (similar to the distance between the sun and Proxima Cen.)

HI and radiation

HI absorbs radiation very efficiently
considering a distance $d x$ travelled by radiation, the change in intensity:

\frac{d I_{λ}}{d x} = - κ_{λ} I_{λ} + η_{λ}

where, $κ$ is the absorption coefficient, and $η$ is the emissivity, with the $λ$ subscript signifying the dependence of all quantities on the wavelength

the source function:

S_{λ} = \frac{η_{λ}}{κ_{λ}}

the optical depth:

τ_{λ} = - \int κ_{λ} d x ⟹ d τ = - κ d x

thus, the change in intensity equation can be rewritten as:

\frac{d I_{λ}}{d τ} = I_{λ} - S_{λ}

$τ$ increases along the line of sight moving from the observer to the object, opposite to the natural coordinate $x$

\int_{I_{0}}^{I} \frac{1}{I S} d I = \int_{τ}^{0} d τ ∴ I (λ) = I_{0} e^{- τ} + S (1 - e^{- τ})

the first term is Bear-Lambert law
thus, the measured intensity depends on the material composition through $S_{λ}$ , and the total absorption through $τ$
considering two limiting cases
optically thick medium $(τ ≫ 1) : e^{- τ} \approx 0 ⟹ I \approx S$
so there is no information about the original path of the radiation
only the 'surface' of the material is visible
similar to how only the first treeline is visible when looking at a thick forest
optically thin medium $(τ ≪ 1) : e^{- τ} \approx 1 - τ ⟹ I \approx I_{0} + (S - I_{0}) τ$
so the intensity is proportional to the total absorption along the line of sight