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ASTR 551, Majewski [SPRING 2008]. Lecture Notes

ASTR 551 (Majewski) Lecture Notes

Extinction by Dust

Reference: Read Binney & Merrifield Section 3.7.

An important complication affecting distance estimates within the Milky Way is the diminution of starlight by absorption/scattering of dust.

Extinction and Optical Depth


Then the extinction causes I λo to drop:

dI λ = -I λ n(x) kλ dx


Integrating, we have:


I λ = I λo eλ


is the total optical depth between observer and source, and

N(x) = column density of absorbing grains (cm-2) = total absorptivity along line of sight.

In terms of magnitudes:

A λ = -2.5 log(I λ / I λo) =

= -2.5 log(eλ )

= -2.5 log(e) ln(eλ )

= -2.5 (0.4) τλ

= 1.086 τλ

Therefore, absorption in magnitudes is numerically equivalent to optical depth along line of sight.

Since N(x) is a constant to any star, and A λ = 1.086 τλ = k λ N(x), any variation in A λ reflects k λ.

Interstellar Reddening in the UBV System

ISM dust makes objects redder (like Sun and Moon on horizon on hazy, dusty, polluted days).

Recall λ-1 dependence from Mie scattering (scattering radius large compared to wavelength) -- found by Trumpler -- allows us to determine net interstellar absorption: A(λi ), which is defined by the amount of absorption in magnitudes at λi :

The observed and intrinsic color indices are defined by:

The observed color index is related to the intrinsic color index as:

So, for example, if (B - V)0, (U - B)0 are the intrinsic, true colors of a source, (B - V), (U - B) are observed. Then:

  1. Colors are always written λi - λj where λi < λj (so red = larger color index). Then:

  2. Because A(λi ) > A(λj ) (Trumpler). Then: E(λi - λj ) = Eij > 0 (generally)

We determine the relation between Eij (λ) by observation.

Total Extinction

Thus far, we have learned how to correct colors, i.e. relative extinction, A(λi ) - A(λj ).

What about correcting magnitudes for total extinction, A(λi )?


It turns out that we can define a relation that ties the selective and relative extinctions:

As seen above, Av is roughly proportional to wavelength dependence of extinction cross-section at V.

The Extinction Law

The extinction law is the function Aλ.

Tables of E(X-V) / E(B-V) exist for "standard extinction law" -- e.g., Binney & Merrifield Table 3.21.

Band E(X-V) / E(B-V)
U 1.72
B 1.00
V 0.00
R -0.78
I -1.60

Note this recovers what we found before:

Because it is more intuitive to use absolute absorption, more recently it has become fashionable to plot:


Band E(X-V) / E(B-V) AX / AV
U 1.72 1.55
B 1.00 1.32
V 0.00 1.00
R -0.78 0.75
I -1.60 0.48


Because it is (sometimes) easier to use absolute extinction rather than relative extinction, it has become fashionable to plot:

The 2175 Å UV peak is probably due to tiny graphite grains ~ 50 atoms in size.

In Binney & Merrifield, Figure 3.17, the log of both axes of the above plot is shown (to stress the IR part of the extinction curve).

Visible (left) and Near-Infrared View of the Galactic Center
Visible image courtesy of Howard McCallon. The infrared image is from the 2 Micron All Sky Survey (2MASS). Images and caption from

Cluster Method for Determining RV

Variations in RV

Why is the universal shape so unexpected?

Dust-to-Gas Ratio

  • A related way in which Nature is somewhat kind -- E(B-V) is found to be directly proportional to the column density of Hydrogen in all its forms (H, H2, H+):

    N (Htot) = 5.8 x 1025 E(B-V) / [(mag)(m2)]

    Contours of the Galactic n(H) column density at |b| < 40 deg in units of 1020 cm-2 (Dickey & Lockman 1990). From
  • Because we also have E(B-V)AV ∝ kλ N(X) (where kλ is the cross section of each grain and N(X) is the column density of grain)

    we find, then, that N(X) / N(Htot) ~ constant.

  • We call:

    the dust / gas ratio.

    Its constancy suggests that a fixed number and size distribution of dust grains is associated with given Htot mass.

    Note the close correspondence of dust and gas (in this case CO) in the new view of the dust distribution near the Galactic plane by UVa grad student David Nidever & SRM.
  • A typical volume density for gas near the Sun is n H = 106 m-3 = 1 cm-3, so, along distance d, find:

  • Then:

    This is for "standard" lines of sight through same density gas, i.e. in disk, b ~ 00

  • Note: it has been found that dust/gas ratio is lower in fast-moving clouds compared to slow-moving clouds (Spitzer, 1982). Fast-moving clouds may be more recently shocked, which destroys grains.

    Measuring Reddening to (Hot) Gaseous Sources

    e.g. H II regions, AGN

    Extinction Maps

    Several ways to map the extinction with position in the galaxy.

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    All material copyright © 2003,2006,2008 Steven R. Majewski. All rights reserved. These notes are intended for the private, noncommercial use of students enrolled in Astronomy 551 at the University of Virginia.