Selection of Homework Questions
Topic 9: Gas & Dust
(1) Interstellar radiation field (ISRF)
- Derive an expression for the radiation field, J, (erg s^{-1} cm^{-2} sr^{-1}) coming from a direction in which the luminosity density varies
with distance, r, from you as _{L}(r).
- Recover Olber's paradox by considering an infinite (static) uniform distribution of stars/galaxies.
- What's the intergalactic radiation field in erg
s^{-1} cm^{-2} sr^{-1} for a static Universe with radius ~ 4 Gpc
(the current horizon distance) and uniform galaxy luminosity density ~10^{8}
L_{} Mpc^{-3} (given in Topic 4.3
)
- What's the interstellar radiation field near the sun (a) in the galactic plane, and (b) towards the galactic poles (i.e. our night sky brightness inside and out of the Milky Way). Assume the sun is at the mid-plane, which has a luminosity density of 0.07 L_{} pc^{-3} and exponential scale height ~100 pc. Assume the
view is transparent away from the plane, but suffers 1.6 mag/kpc of attenuation in the plane.
- A galaxy has surface brightness µ_{V} mag/ss. Re-express this in units of erg s^{-1} cm^{-2} sr^{-1} (the V band zero point is 3.13×10^{-6} erg s^{-1}cm^{-2} at V = 0^{m}).
- Since surface brightness is independent of distance, the ISRF inside a galaxy is roughly half the galaxy's measured surface brightness (i.e. µ + 0.75 mags/ss; there is half ahead of you, and half behind).
Hence estimate the ISRF at the center of M87 (µ_{V} ~ 17 mag/ss) and M32 (µ_{V} ~ 11 mag/ss). Compare the hemispherical sky brightness in starlight in these
galactic nuclei to a moonlit night on earth (take the moon's albedo to be ~2%).
(2) Equilibrium Dust Temperatures and Spectra:
In the notes, we considered the simplified case of equilibrium dust temperature. Let's
return to that topic with a bit more precision.
- Let's start simple. Write the equation for the Eddington equilibrium temperature,
T_{Edd}, for a dust grain distance r from a star with temperature
T_{} and bolometric luminosity
L_{}. For this approach, we assume both star and
dust act like black bodies, so that Q_{abs} = Q_{em} = 1 at all
. What is T_{Edd} for dust at r = 0.1 pc from a BOV star
with T_{eff} = 30,000K and L_{bol} = 5.2 × 10^{4}
L_{}? How does T_{Edd} depend on r ?
- Now let's get more realistic. Consider a spherical silicate grain of radius a = 3 µm
and refractive index m = 1.5 - 0.05i, which obeys the Mie relation:
Q_{abs}() = Q_{em} () =
-4 X Im[ (m^{2}-1)/(m^{2}+2) ] for X < X_{c} and Q = 1 for X > X_{c};
where X = (2a/), and X_{c} is defined when Q reaches 1. What wavelength, _{c}, corresponds to X_{c}, beyond which
the grain exhibits reduced efficiency: Q()
^{-1} ? Compare _{c} with the Wein peak wavelength
for black bodies at T_{eff} and T_{Edd} (from above). How will this affect
Q_{abs} and Q_{em} and hence T_{d}?
- Now rewrite your original equilibrium equation, allowing for wavelength dependent
Q_{abs}() and Q_{em}(). Solve this equation for the above case to find the equilibrium temperature, T_{d}. You will need to use a numerical
integrator (qromb in Numerical Recipes works fine, or use another. Don't forget to check
your routine works properly!). You can either iterate
numerically to find T_{d} or try a few values of T to find when
heating = cooling. What is the "greenhouse factor", ie T_{d}/T_{Edd}, in this case?
- Redo the calculation for a smaller dust grain: a = 0.03µm and
explain why T_{d} has changed. Redo the calculation for the original
(a = 3 µm) grain, but now placed at r = 3 pc from the star. Has
T_{d} followed the r-dependence you derived in part 1 above, and if not why?
Redo the calculation to find the "sublimation distance", T_{d} 1500K for the small grains.
- For the larger grain at 0.1pc, plot its emitted spectrum (in F_{}) and compare it to a black body spectrum, B_{}, of
the same temperature. If you simply took the peak wavelength to derive
a dust temperature using the Wein relation, what temperature do you get? Not only are
the grains hotter than the simple Eddington value, but their peak emission suggests they
are even hotter than they actually are.
- Now consider a population of silicate grains with
size distribution dN/da a^{-3.5} between
0.03 µm (30nm) and 0.3µm (none outside this range). In this population,
which grains (small
or large) have most of the mass? Which have most of the area? Use numerical
methods to plot the spectrum generated from the population at 0.1 pc. Use the wavelength
of the peak
to derive a simple single Wein temperature and overplot a black body of this temperature.
Is this
single temparature closer to the temperature of the larger grains or the smaller ones?
- Finally, consider a uniform dust distrubution which extends from 1pc down to the
larger of the two sublimation radii. Divide the region into 10 radial shells and sum their
spectra to find the integrated IR spectrum from the region. Overplot a single black
body spectrum set to the Wein temperature derived from the peak. Does all this
averaging cause the spectrum
to deviate significantly from a black body?
(3) Small Grain Cooling Times
- Consider a small grain containing N=150 atoms of carbon, packed with density 1
gm cm^{-3}, which is struck by a single UV photon of energy 10eV. What's the
grain's temperature immediately after the photon is absorbed?
- Assuming the grain has refractive index m = 1.5 - 0.2i, estimate Q_{em} at
~ 10 µm. Estimate how long it takes the grain to
cool down.
(4) Extinction & Reddening
- Use the equations given in Cardelli , Clayton, & Mathis (1989, ApJ 345 245 e-link
) to
write a subroutine for A_{}/A_{V} and use this to generate a plots of
A_{}/A_{V}
vs 1/ for for R_{V} = 2.0, 3.1, 5.0 over the range
100 nm to 3 µm.
Describe, briefly, the various regions and their physical origin. Keep this subroutine for
your future career's, it is likely you will need it sometime.
- The spectrum of a galaxy nucleus has H flux of 1.2 × 10^{-14} erg s^{-1} cm^{-2} at observed wavelength 498nm, with
other lines at relative strength to H of :
H = 4.2; [OIII]5007 = 12.0;
[OII]3727 = 2.4; [OI]6300 = 0.20.
- Assuming R_{V} = 3.1 and an unreddened Balmer ratio of H/H = 2.86, what is A_{V}?
- Using this value of A_{V}, correct all the relative line strengths as well as the H flux.
- Calculate the uncorrected and corrected luminosity of H
- What is the "reddening vector" for A_{V} = 1 in a diagram of Log([OIII]/[OII]) (x-axis) vs Log(H/[OI]) (y-axis) (i.e. the x, y offset resulting from applying
1 magnitude of A_{V} to any plotted point in this diagram).
- From B&T, find M_{V}, (B-V)_{o} and (V-K)_{o} for a B0V main sequence star. From spectra, you identify an embedded star as type, B0V, and from photometry you find B=19.67, V=16.99.
- What is the reddening, E(B-V) ?
- Adopting the standard extinction law, for which R_{V} = 3.1, what is A_{V}?
- What is the unreddened apparent magnitude, V_{o}.
- What is the distance modulus, m_{V} - M_{V}, and hence distance to the star?
- If you also measure its K magnitude to be 7.25, is this consistent with the
standard reddening law given in B&T, and if not, should
R_{V} be larger or smaller? Which way does this affect your distance estimate?
(5) Dust to Gas Ratios & H columns :
(6) IR color-color diagrams :
Construct an IR color-color diagram with x/y axes Log(S_{12}/S_{25}) vs Log(S_{60}/S_{100}), where S_{12} etc are measures of f_{} at 12 µm etc (e.g. in Janskys)
- Plot the locus of a single temperature black body, from 20K to 500K, labelling the temperatures at 100, 200,.... 500K. Obviously, you will need to evaluate the Planck function B_{}(T); make sure you evaluate B_{} rather than B_{} since our axes are using f_{}
- Evaluate the locus of the sum of two black body components, one at 50K and one
at 250K with integrated fluxes in a ratio a/b; i.e. a/50^{4}×B_{}(50K) + b/250^{4}×B_{}(250K). Vary the relative contributions of the two, marking the locations where a/b = 0.0, 0.2, 1.0, 5.0, 100.0
- Plot spectra for the cases a/b = 0.2 & 5.0 in the range 5 µm to 500 µm,
showing the two components as well as their sum.
- Plot the locus of power-law spectra, f_{}
^{}, marking values of at -2, -1, 0, 1, 2.
- Using NED, find the IRAS fluxes for the following galaxies and plot them on
the IR color-color plot. M87 (giant elliptical); M101 (early type spiral); NGC 5548 (Seyfert 1);
Markarian 3 (Seyfert 2); Arp 220 (ULIG); M82 (starburst). Comment on their differing
locations.
(7) Dust heating: collisions vs radiation :
Let's revisit the claim made in the notes that dust in the ISM is heated more by radiation than particle collisions.
- Use kinetic theory to derive, or quote, the number of particles of mean speed
<v> striking unit area per second. For a gas of temperature T and number density n,
what
is the energy flux, F_{th}, impinging on unit area? Express this in terms of
the gas thermal energy density (i.e. pressure). Which component of the gas contributes
most to the collisional heating and why?
- For a radiation field, what is the relation between energy density, U_{ph} and flux F_{ph}?
- hence, show that the ratio of collisional to radiation heating of dust in
the ISM is F_{th}/F_{ph} = 3/8 × <v>/c × U_{th}/U_{rad}. Because U_{ph} U_{th} throughout much of the ISM, then radiation easily dominates the heating.
- Clearly, for thermal heating to compete with radiation, we need thermal energy densities larger than radiation energy densities by a factor of roughly c/<v>. What is
c/<v> for electrons at temperature T? Are there any environments you can think of
where this condition is satisfied?