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Giovanni Covone » 10.Ionized systems in the ISM: Warm ionized medium and …

The diffuse “warm ionized medium”

The diffuse “warm ionized medium” (WIM): about 90% of ionized hydrogen (HII), in a warm and diffuse phase.

Visible trough its H- $\alpha$ emission in our Galaxy.

Kinetic temperature: about $10^4$ K.
Mass: 30 \% of the ISM total mass.
Density: about $0.1 {\rm cm}^{-3}$ .
Ionization fraction: $\frac{n_e}{n_H} \sim 0.7$ .

Source of heat: ionizing photons from hot massive stars.

Unsolved problem: a homogeneous galactic HI distribution should absorbe the ionizing UV flux.

Solution: H I clouds cannot be spread evenly throughout the galaxy; large portions of the galaxy must have been cleared of H I.

Observational tools for the WIM

Techniques used to detect the Warm Ionized Medium:

1) low frequency radio observations,

2) the dispersion of radio pulses of pulsars

3) hydrogen recombination emission.

Pulsar pulses

Electromagnetic waves from pulsars interact with the free electrons in the WIM.

Consequence: time dispersion of the pulse.

The plasma frequency $\nu_p$ is given by:

$\nu_p = \, \left( \frac{n_e e^2}{\pi m_e} \right)^{1/2} \simeq 9 \times 10^3 n_e^{1/2} {\rm s}^{-1} .$

Comparison with pulsar pulse.

Image of single pulse from a pulsar, in arbitrary units. Credit: C2H2 & HI CA$Hed Online Education.

Observational tools for the WIM (cont.)

Index of refraction of the ISM: $n_{\rm ref} = \sqrt{1 - \nu_p^2 / \nu^2} \, .$

Difference between group velocity and phase velocity $c$ : $\frac{v_{\rm group}}{c} = \sqrt{1 - \nu_p^2 / \nu^2} \, .$

At high frequency in the pulse ( $\nu \gg \nu_p$ ), the term ( $\nu_p^2/\nu^2$ ) is negligiable. Time for a pulse of frequency $\nu$ to arrive from a pulsar at a distance $d$ : $t = \int_0^d \frac{d l} {v_{\rm group}}$

For the non-negligiable case, we can Taylor expand the term $\frac{1}{v_{\rm group}}$ : $\frac{1}{v_{\rm group}} \simeq \frac{1}{c} + \frac{nu_p^2}{2 c^2}$

The time delay between the phase and group velocities is:
$\Delta t = t - \frac{d}{c}$ , which reads:

$\Delta t \, = \, (e^2 / 2 \pi m_e c) \, \int_0^d n_e dl / \nu^2 \, .$

Define the dispersion measure $D = \int_0^d n_e d l$ .
The change in time delay with frequency is: $\frac{d \Delta t}{d v} = \frac{e^2 D}{\pi m_e c \nu^3} \, .$

Application: the Crab Nebula

The dispersion measure used to determine the mean electron density in the ISM.

Application: determine electron density to the Crab pulsar.
Observed delay between pulses at frequencies 100 MHz and 400 MHz: 23.5 s.

First, we obtain the electron column density to the Crab pulsar:

$\int_0^D n_{\rm e} \, {\rm d}s = 1.86 \times 10^{24} \, {\rm m}^{-2} \, .$

Distance of the Crab Nebula: $D \sim 6000$ light years

The average electron density:
$= \frac{1}{D} \, \int_0^D n_{\rm e} {\rm d}s = 3 \times 10^4 \, {\rm electrons/m}^3 \, .$

Observational tools for the WIM (cont.)

H-alpha Emission from the WIM

Weak H-alpha emission from the WIM.

Intensity of emission from $H_{\alpha}$ is given by:

$I_{\alpha} \, = \, \left( \frac{\alpha_{32} h \nu_{\alpha}}{4 \pi} \right) \int n_e n_p dl \simeq 8.8 \times 10^{-8} \, {\rm EM} \, {\rm erg \, cm}^{-2} \, {\rm s}^{-1} \, {\rm sr}^{-1} \, ,$
where $\alpha_{32} = 1.2 \times 10^{-13} {\rm cm}^3 \, {\rm s}^{-1}$ is the effective recombination coefficient, at $temperature T = 10^4 \,$ K.

The integral part of the equation above is called the emission measure, designated EM.
The emission measure is important for calculating line strengths in WIM.

We assume: $n_p = n_e$ .

Optical depth, $\tau = \int \kappa(\nu) dl$ .

Using the opacity for ionized hydrogen, the optical depth is: $\int n^2 d l \, .$
Constant source temperature: then only the density changes along the line of sight.

$\tau (\nu)$ is then proportional to $EM = \int n^2 dl =D \, .$

Observational tools for the WIM (cont.)

EM is the amount of emission and absorption along the line of site.

Particles approach each other along the line of site.

Recombination line strengths are proportional to the emission measure.

Measuring both the emission measure and the temperature of the region can be determined.

In the Milky Way: 90% of the ionized gas is in fully ionized, diffuse form.

Volume and spatial distribution.

ISM around a hot star

What happens to the nearby ISM under the action of their strong UV radiation?

A blackbody with $T \sim 3.5 \times 10^4 \, {\rm K}$ emits about 1/3 of its radiation in photons
with energy higher than 13.6 eV

Remember: $h \, \nu_I \sim 13.6$ eV is the ionization threshold of the hydrogen atom.

These photons can ionize a substantial fraction of the surrounding ISM.

O stars in Rosette Nebula

Credit: NASA's Spitzer Space Telescope.

Some exercises

Exercise 1: Compute the rate of Lyman continuum photons of an O5 star (assume a blackbody spectrum).
Mass: $M=50$ solar masses;
Effective temperature $T \sim 4 \times 10^4 {\rm K}$ ,
Radius $R=5 \times 10^{11}$ cm.

Exercise 2: Compute the same quantity for the Sun.

Exercise 3: The peak emission measure in M42 is ${\rm EM+\, = 5 \times 10^6 \, {\rm cm}^{-6} \, {\rm pc} \, .$ Approximate the nebula as a sphere with radius 0.3 pc. Compute the rate of H recombinations in the nebula. Assume a gas temperature $T = 10^4 \, {\rm K}$ , and assume singly ionized He with He/H=0.1.

Exercise 4: The mass distribution of GMCs in the Galaxy is given by
$\frac{{\rm d} N_{\rm GMC}} {{\rm d} ln \, M_{\rm GMC}} \sim N_u \left( \frac{ M_{\rm GMC} }{M_u} \right)^{-\alpha}$
which holds for $10^3 \, M_{\odot} < M_{\rm GMC} < M_u$ , with $M_u \sim 6 \times 10^6 M_{\odot} \, ,$ $N_u \sim 63$ , and $\alpha \sim 0.6$ (a) Calculate the total mass in GMCs in the Galaxy. (b) Calculate the number of GMCs in the Galaxy with $M > 10^6 M_{\odot} \, .$

Solution: UV radiation from an O star

Lyman continuum photons: energy larger than 13.6 eV. Compute the corresponding frequency $\nu_0$ .

The number of photons emitted in over all solid angles per unit time per unit area at the surface of the star:

${\rm d} q_v = \pi \, \frac{B_{\nu} (T_*)}{h \nu} \, .$

The number of photons emitted by a star (per unit time) :

$Q_{\rm star} = 4 \pi R^2_* \times \int_{\nu_0}^{\infty} \frac{\pi B_{\nu} (T_*) }{h \nu} d \nu \, .$
In the range of interest, we have:
$\frac{h \nu}{k T_*} \geq \frac{h \nu_0}{k T_*} \, = \,<br /> \frac{13.6 {\rm eV}}{8.6 \times 10^{-5} {\rm eV K}^{-1} \, 3.8 \times 10^4 {\rm K}} \, = 4.2$

Apptoximation of the black body curve with the the Weins approximation:
$B_{\nu } \simeq \frac{2 h \nu}{c^2} {\rm e}^{(- \frac{h \nu}{k T})} \, .$

Change of variables and integration by parts:
$Q_{\rm star} \, = \, \frac{8 \pi R^2_*}{c^2} \left( \frac{k T_*}{h} \right)^3 \,<br /> \int_{(\frac{h \nu_0)}{k T_*}}^{\infty}<br /> \left( \frac{h \nu}{k T_*} \right)^2 \, {\rm exp} \left( \frac{h \nu}{k T_*} \right) {\rm d} \left(\frac{h \nu}{k T_*}\right) \, .$

Solution: UV radiation from an O star (cont.)

The definite integral yields the result:

$\left[ -x^2 e^x - 2 x e^{-x} - 2 e^{-x} \right]_{\frac{h \nu_0}{k T_*}}^\infty$

The results is:
$Q_{\rm star} \sim 4.5 \times 10^{48}$ photons per second.

For a Sun-like star (spectral type G3, effective temperature $T = 5800 K$ ):
the quantity $(\frac{h \nu_0}{k T_*})$ is much larger:
$\left( \frac{h \nu_0}{k T_*} \right) \simeq 27 \, .$

Fraction of ionizing: 25% for an O5 star; only $1.5 \times 10^{-11}$ for a G3 star.

How many ionizing UV photons does the Sun emit?

Answer only depends on temperature and radius of the star.
The production rate for a Sun-like star is smaller by the following quantity:

$\left( \frac{R_{\rm G3}}{R_{\rm O}} \right)^2 \, \left( \frac{T_{\rm G3}}{T_{\rm O}} \right)^4 \times<br /> 1.6 \times 10^{11} \times 4 \simeq 2 \times 10^{-16} \, .$

The Stromgren sphere

Definition of the Stromgren Sphere.

Within Stromgren Sphere: ionization is maintained by absorption of ionizing photons radiated by a central hot star.

In a steady state, hydrogen recombination is balanced by photoionization.

$Q_0$ : rate of emission of hydrogen-ionizing photons.

Radiative recombination: $H^+ + e^- \rightarrow H + h \nu$

Photoionization: $H + h\nu \rightarrow H^+ + e^-$ .

Equating the rates of photoionization and radiative recombination, gives the steady state condition for ionization balance:

$Q_0 \, = \, \frac{4 \pi}{3} \, R_S \, \alpha_B \, n(H^+) \, n_e \, ,$
where we used the recombination coefficient $\alpha_B$ .
We assume that the hydrogen density be $n(H) = n(H^+) .$