# Extinction, the elephant in the room that hinders optical Galactic
observations

J. Maíz Apellániz1
1Centro de Astrobiología CSIC-INTA  campus ESAC.
Camino bajo del castillo s/n $28\,692$ Villanueva de la Cañada  Madrid  Spain

###### Abstract

Extinction is the elephant in the room that almost everyone tries to avoid
when analyzing optical/IR data: astronomers tend to find a quick fix for it
that the referee will accept, but that does not mean such a solution is
correct or even optimal. In this contribution I address three important issues
related to extinction that are commonly ignored and present current and future
solutions for them: [1] Extinction produces non-linear photometric effects,
[2] the extinction law changes between sightlines, and [3] not all families of
extinction laws have the same accuracy.

###### keywords:

Dust, extinction — Galaxy: structure — Methods: data analysis — Methods:
observational — Stars: early-type

##  1 Introduction

Interstellar extinction has been studied for almost a century and for a novice
in the topic it may be surprising how much was already known about it at the
early stages of that period. ? describes extinction phenomena in terms of
monochromatic (for single lines), selective (color excess) and general
absorption (or proper extinction if one includes scattering), and as the
existence of obscuration effects (dark areas of the sky devoid of visible
stars). ? and ? noticed that the extinction law towards Orion is peculiar,
implying that a uniform extinction law does not exist for the whole sky. ?
produced the first attempt to compute an extinction law as a function of
wavelength.

The 1950s brought the advent of medium-scale (hundreds of stars) photoelectric
$UBV$ surveys of OB stars and, from them, the use of the $Q$ parameter to
measure their extinctions and estimate spectral types based on the behavior of
the extinction law and the fact that the ratio of total to selective
extinction $R_{V}\equiv A_{V}/E(B-V)$ is close to 3.0 for many sightlines
(??). At around the same time, ?? recognized the existence of photometric
bandwidth effects in the analysis of the photometry of those stars, leading to
non-linear extinction trajectories in color-magnitude and color-color diagrams
and, eventually, to the dependence of $R_{V}$ not only on the type of dust but
also on its amount as well as on the SED of the star. The following two
decades saw a number of such optical extinction studies of OB stars using
mostly the Johnson and Strömgren photometric systems.

The extension to other wavelength regimes started in the 1970s and took off
with the launch of IUE in 1978 and the opening of moderately large ultraviolet
samples and with the subsequent pioneering work of ? in the infrared. Those
led to the seminal work of ?, the first family of UV-optical-IR extinction
laws dependent on a parameter, the ratio of total to selective extinction, to
determine both the amount and type of dust along the sightline.

The most recent development in extinction studies has taken place in the last
quarter of a century with the appearance of large-scale photometric surveys,
with sample increases of several orders of magnitude and the development of
dedicated pipelines for the uniform processing of the data. In the NIR the
most important step was the publication of 2MASS in 2003 (?) and in the
optical a number of surveys became available in the period such as SDSS (?)
and IPHAS (?). The final step was the publication of the different Gaia data
releases (?), with the availability of astrometry, (spectro)photometry,
spectroscopy, and variability for $\sim 10^{9}$ stars. The combination of
those surveys have brought us to a new era, both in terms of data volume and
in the types of stars used to derive extinction, which are now of late type in
most studies.

Why do I call extinction the elephant in the room? Because it is an obvious
phenomenon in astronomy but it is undesired and most researchers want to just
correct for it in a simple and quick way. The object of many papers is to
obtain magnitudes and colors of astronomical objects and extinction is just a
nuisance that gets in the way. Unfortunately, some of the simple and quick
solutions ignore the complexities of the problem and lead to biased results.
In this contribution I address the three main reasons why this happens:

  * 1.

Extinction is a non-linear effect in magnitude or color. In general, doubling
the amount of dust does not double $A_{V}$ or $E(B-V)$ and the same amount and
type of dust does not produce the same $A_{V}$ or of $E(B-V)$ in two stars
with different SEDs. One needs to use monochromatic definitions for the amount
and type of dust, so neither $E(B-V)$ nor $R_{V}$ are appropriate. One should
use their monochromatic equivalents, e.g. $E(4405-5495)$ and $R_{5495}$.

  * 2.

The extinction law changes between sightlines. $R_{5495}$ is not the same
everywhere. Many times it is within the canonical range of 3.0-3.2 but in a
significant number of occasions it is outside that range (mostly $>3.2$).
$R_{5495}$ is not even constant within a small region, especially if it
includes diverse ISM environments such as H ii regions. Furthermore, some
measurements of $R_{5495}$ use invalid or outdated approximations.

  * 3.

No existing family of extinction laws accurately covers the UV-optical-IR
range. The two most commonly used, ? and ? can have significant photometric
residuals already for stars with $\mbox{$E(4405-5495)$}\sim 1.0$. ? works well
up to $\mbox{$E(4405-5495)$}\sim 2.0$ and does a reasonable job even around
$\mbox{$E(4405-5495)$}\sim 3.0$ but does not extend to the UV and has the
incorrect behavior in the IR.

In the next three sections I address each of those reasons and provide recipes
when available. I end up analyzing prospects for the future.

##  2 Non-linearity of extinction

Extinction as a function of wavelength is defined as the ratio of the observed
(extinguished) to the unextinguished spectral energy distributions (SEDs)
expressed in magnitudes:

$A(\lambda)=-2.5\log_{10}\left(\frac{F_{\lambda}(\lambda)}{F_{\lambda,0}(\lambda)}\right).$
(1)

The extinction in a given band such as $V$ is the difference between the
observed and the unextinguished magnitudes, which for a photon-counting
detector such as a CCD can be expressed as:

$A_{V}\equiv V-V_{0}=-2.5\log_{10}\left(\frac{\int
P_{V}(\lambda)F_{\lambda}(\lambda)\lambda\,d\lambda}{\int
P_{V}(\lambda)F_{\lambda,0}(\lambda)\lambda\,d\lambda}\right),$ (2)

where $P_{V}(\lambda)$ is the sensitivity curve for the $V$ band. In an
equivalent manner, for the $B$ band we have:

$A_{B}\equiv B-B_{0}=-2.5\log_{10}\left(\frac{\int
P_{B}(\lambda)F_{\lambda}(\lambda)\lambda\,d\lambda}{\int
P_{B}(\lambda)F_{\lambda,0}(\lambda)\lambda\,d\lambda}\right).$ (3)

To compute the values in Eqns 2 and 3 we need to substitute $F_{\lambda}$
using Eqn. 1 and define $A(\lambda)$. The traditional (band-integrated) $B-V$
color excess is defined as the difference in extinction between the $B$ and
$V$ bands:

$E(B-V)\equiv A_{B}-A_{V}=(B-V)-(B-V)_{0}$ (4)

and the (band-integrated) ratio of total to selective extinction is defined as
the ratio of Eqns. 2 and 4:

$R_{V}\equiv\frac{A_{V}}{E(B-V)}.$ (5)

Figure 1: (left) $E(B-V)-\mbox{$E(4405-5495)$}$ and (right) $R_{V}$ as a
function of $E(4405-5495)$ for a ? extinction law with $\mbox{$R_{5495}$}=3.1$
and three main-sequence stars with different $T_{\rm eff}$.
$E(B-V)\approx\mbox{$E(4405-5495)$}$ and $R_{V}\approx\mbox{$R_{5495}$}$ only
for hot stars with low extinctions.

The extinction law is simply the extinction normalized by some measurement of
the amount of dust. This can be, for example, the value of $A(\lambda)$ at a
given wavelength or the difference between $A(\lambda)$ at two different
wavelengths. What should not be used as a normalization is a band-integrated
extinction (such as those in Eqns. 2 or 3) or color excess (such as that in
Eqn. 4). Why? Because band-integrated quantities depend in a complex way on
the SEDs. In other words, $A_{V}$ and $E(B-V)$ are not direct measurements of
the amount of dust, as the same amount of dust on the sightline of a red and a
blue star yield different values of $A_{V}$ or $E(B-V)$. To show that,
following ? I use as the amount of dust $E(4405-5495)$, the monochromatic
equivalent of $E(B-V)$, and apply the family of extinction laws of ?111The
result does not change substantially for other families, see ? for the
equivalent plots using ?. to different SEDs from the library of ?. Results are
shown in Fig. 1, where it is clearly seen that the commonly used
approximations of $E(B-V)\approx\mbox{$E(4405-5495)$}$ and
$R_{V}\approx\mbox{$R_{5495}$}$ are only valid for hot stars with low
extinction. Indeed, the reason why in ? the wavelengths of 4405 Å and 5495 Å
were chosen as representative of $B$ and $V$, respectively, was to allow for
those approximations to be valid in that regime and, in that way, let the
classic extinction measurements of OB stars remain approximately valid.

Figure 2: Throughputs for the Gaia EDR3 $G$+$G_{\rm BP}$+$G_{\rm RP}$
photometric system (Weiler et al. in prep.).

As I have previously mentioned, bandwidth effects in photometry were known
already in the 1950s. However, there is no mention of them in ?, where
$E(B-V)$ is used as the amount of extinction and $R_{V}$ as the type of
extinction. ? mentions the difference between monochromatic and band-
integrated quantities but he still uses $E(B-V)$ and $R_{V}$ as
parameters222The latter is called simply $R$ but its definition is that of
$R_{V}$. Also, due to the different definitions of central wavelengths, if $R$
is considered as the monochromatic definition for the extinction law, then it
is not exactly $R_{5495}$.. For their samples the difference is small, as both
papers use OB stars with low extinction (see below). Still, the formulae in
those papers are inconsistent and need to be adapted to monochromatic
quantities when implemented to allow them to be applied to SEDs of arbitrary
temperatures and extinctions.

Despite claims to the contrary, bandwidth effects are always present. If they
have been ignored in the past it may have been because samples were more
restricted in temperature and extinction and because past photometric
calibrations were not as accurate as current ones (see ??? for examples).
Having said that, non-linearity starts becoming noticeable at lower
extinctions for broader filters than for narrower ones (e.g. Johnson vs.
Strömgren) and for wavelength regions where the extinction law is steeper.
Those conditions make non-linearity a stronger effect for the Gaia $G$+$G_{\rm
BP}$+$G_{\rm RP}$ system than for other optical photometric systems, as the
$G$ band is very broad (Fig. 2) and the photometric callibration of Gaia is
better than that of any optical ground-based system (?).

Figure 3: Gaia EDR3 Carina OB1 CMD adapted from ?. The orange lines show
extinction tracks for MS stars of different $T_{\rm eff}$ (indicated in kK at
their left ends) and $R_{5495}$ (also labelled). For the 52.5 kK extinction
track with $\mbox{$R_{5495}$}=4.5$ I have also drawn a blue straight line next
to it to show the curvature effect.

Gaia data processing correctly differentiates between monochromatic and band-
integrated quantities. For example, Apsis (?) provides one monochromatic
extinction ($A_{0}$, with the wavelength being 5414 Å), three band-integrated
extinctions ($A_{G}$, $A_{\rm BP}$, and $A_{\rm RP}$), and one band-integrated
color excess or reddening [$E(\mbox{$G_{\rm BP}$}-\mbox{$G_{\rm RP}$})$], each
one adequately identified. Non-linearity appears both in Gaia CMDs and color-
color diagrams. A sample CMD is shown in Fig. 3, adapted from the analysis of
Carina OB1 by ?. The extinction tracks are shown in orange and computed for
different $T_{\rm eff}$ and $R_{5495}$ values using the family of ?. As the
vertical scale comprises 13 magnitudes, the curvature is not apparent at first
but plotting a straight line next to one of them (in blue) shows that the
effect amounts to several tenths of a magnitude over the extinction range
plotted in this case (most stars that appear in the plot are blue MS stars
that have been displaced towards the lower right corner by the strong
differential extinction in the region). A sample color-color diagram is shown
in Fig. 4, as it appeared in ?. The stellar locus is highly curved due to
extinction and the different values of $T_{\rm eff}$, with both effects being
relatively similar due to the Gaia filter design, and follows the result of
the throughputs in that paper (labelled as MAW) with $\mbox{$R_{5495}$}=3.0$
quite well (the extension towards the upper left part of the diagram is caused
mostly by stars with contaminated photometry due to crowding or nebulosity).

Figure 4: Gaia DR2 color-color diagram from ?. The lines show the extinction
tracks for red-clump stars with different assumptions about $R_{5495}$ and the
filter throughputs (the purpose of that paper was to identify those), with the
associated symbols plotted at intervals of $E(4405-5495)$ of 0.1 mag.

Figure 5: Band-integrated $E(J-H)$ color excess as a function of the
monochromatic amount of extinction $A_{1}$ from ?. Two effective $T_{\rm eff}$
(4 kK and 40 kK) and three values of the NIR slope $\alpha$ (1.5, 2.0, and
2.5) are used. Symbols are used for the actual values and lines for the
extrapolation from the low-extinction regime.

In the NIR, extinction is smaller than in the optical for the same dust column
but, at the same time, we can reach higher extinction values and non-linear
effects appear sooner or later. As discussed in ?, if non-linearity is ignored
in IR studies, extinction measurement can be compromised. Some papers, such as
??? correctly deal with it but others do not. I show this in Fig. 5, which is
taken from the Appendix in ?. As we increase $A_{1}$, the monochromatic
extinction at 1 $\mu$m, the color excess $E(J-H)$ clearly deviates from a non-
linear behavior. The effect is stronger for higher values of $\alpha$, the
exponent in the assumed power-law NIR extinction. As we will see later, the
current preferred value is $\alpha\sim 2.2$, so the effect is easy to notice
already at relative lower values of $A_{1}$.

As a summary of this section, extinction has a clear non-linear behavior in
CMDs and color-color diagrams. The corollary is that one should not apply the
commonly used approximation of defining a central wavelength for a filter and
extinguish the photometry assuming that only the extinction law at that point
is relevant. Extinction should be calculated using integrals such as Eqns. 2
and 3. As that may be computationally too expensive when dealing with large
samples and fitting procedures, one alternative is to compute a grid for the
desired values of the extinction and interpolate with splines within the grid.
This is the approach used by CHORIZOS (?).

##  3 Variability of the extinction law

###  3.1 Introduction

As I mentioned in the introduction, the variability of the extinction law in
the optical/IR has been known since the 1930s but it was not until the 1980s
that the much stronger variability in the UV was discovered with IUE. ?
produced an average UV extinction law from OAO-2, Copernicus, and TD-1 data
but subsequent work revealed variability among Galactic sightlines (???) and
between those and LMC (?????) and SMC (?????) sightlines.

The studies of extinction of the 1980s culminated in the work of ?, who, as
previously mentioned, produced the first $R_{5495}$-dependent family of
extinction laws. That family has a uniform behavior in the IR (based on ?) and
a highly variable dependence on $R_{5495}$ as a single parameter in the
optical and UV. Another commonly used family of extinction laws, that of ?,
appeared a decade later and has a similar behavior to the ? one: non-variable
in the IR and dependent on an $R$-like parameter in the optical and UV. In the
next section I analyze their similarities and differences in more detail. I
also address extinction-law variability in the IR later on. For the time
being, I just point out that, given the range of extinctions covered by the
previously mentioned two families of extinction laws, such IR extinction
variability would have been very difficult to measure with the data available
in the 1980s and 1990s.

###  3.2 Methodology

Before addressing the variability of the extinction law, I need to discuss how
the extinction law and extinction in general are measured. In principle,
$A(\lambda)$ is determined by Eqn. 1, so what we need is the observed SED
determined from spectrophotometry, divide it by the unextinguished SED, obtain
the logarithm, and multiply the result by $-2.5$. As expected, that is more
easily said than done for two reasons: [1] We do not know a priori the
unextinguished SED and [2] in most cases we do not have spectrophotometric
data but only photometry. I address those two issues starting with the first
one.

The unextinguished SED. The traditional method used, for example, in most of
the 1980s work with IUE data, is called the pair method. One obtains the
spectrum of an identical star with no extinction and uses it as a reference or
standard for the unextinguished SED. The three problems associated with the
pair method are: obtaining the ratio of the distances needed to normalize the
fluxes, determining that the two stars are indeed identical, and finding the
reference star itself. Of those, the third one is the most difficult one: OB
stars are concentrated in the plane of the Milky Way and are usually at
considerably distances, so extinction is almost never negligible (?). Due to
those problems, in the last two decades most works have shifted towards using
synthetic SEDs as a reference instead of observed ones (??). For this
alternative method we still have the first problem (flux normalization due to
distance) and the last two are substituted by another two: making sure that
the grid of synthetic SEDs is spectrophotometrically accurate and determining
the parameters of the star ($T_{\rm eff}$ being the most important one but
also $\log g$ and metallicity in some cases) to select the specific SED in the
grid. I discuss these issues below.

Photometry instead of spectrophotometry. Analyses of UV extinction using IUE
data in the 1980s had access to spectrophotometry in the ultraviolet but even
those complemented the optical and IR ranges with photometry. As most current
works use large-scale photometric optical/IR surveys as their data source for
extinction determinations, photometry has become the most popular method to
determine extinctions. Photometry has the advantage of large sample sizes but
introduces problems with calibration for both the analyzed stars and for the
standard system (see below) and only samples the extinction law at a coarse
wavelength grid which then needs to be interpolated or modelled.

There are actually two related problems we are dealing with. One is
determining the family of extinction laws that applies to an astrophysical
regime (such as a given galaxy or environment) and another one calculating the
extinction for a set of stars. In this section I start with the second
problem, which assumes we already know the family of extinction laws, and in
the next section I will address the more complex problem of determining the
family of extinction laws. To determine the extinction parameters (amount and
type) we need to fit the observed photometry to the synthetic photometry
derived from the SED models. Using $\chi^{2}$ minimization, the problem can be
expressed with these three equations:

$\chi_{\rm red}^{2}=\frac{1}{N-{\rm
DoF}}=\sum_{i=1}^{N}\frac{(\fcolorbox{red}{white}{$m_{i,{\rm obs}}$}-m_{i,{\rm
syn}})^{2}}{\fcolorbox{red}{white}{$\sigma_{i,{\rm
obs}}^2$}}\fcolorbox{violet}{white}{$\sim 1$}$ (6)

$m_{i,{\rm
syn}}=-2.5\log_{10}\left(\frac{\int\fcolorbox{red}{white}{$P_i(\lambda)$}\,F_{\lambda}(\lambda)\,\lambda\,d\lambda}{\int\fcolorbox{red}{white}{$P_i(\lambda)$}\,\fcolorbox{OliveGreen}{white}{$F_{\lambda,r}(\lambda)$}\,\lambda\,d\lambda}\right)+\fcolorbox{OliveGreen}{white}{${\rm
ZP}_{i,r}$}$ (7)

$F_{\lambda}(\lambda)=\fcolorbox{blue}{white}{$F_{\lambda,0}(\lambda)$}\,e^{-0.4\fcolorbox{orange}{white}{\scriptsize$A(\lambda)$}}$
(8)

with the following definitions:

  * •

$\chi_{\rm red}^{2}$: Goodness of fit (reduced chi square).

  * •

$N$: Number of filters/passbands.

  * •

DoF: Degrees of freedom of fit (number of parameters).

  * •

$m_{i,{\rm obs}}$: Observed magnitude for filter/passband $i$.

  * •

$m_{i,{\rm syn}}$: Synthetic magnitude for filter/passband $i$.

  * •

$\sigma_{i,{\rm obs}}$: Observed magnitude uncertainty for filter/passband
$i$.

  * •

$P_{i}(\lambda)$: Sensitivity curve/throughput for filter/passband $i$.

  * •

$F_{\lambda}(\lambda)$: Observed SED.

  * •

$F_{\lambda,r}(\lambda)$: Reference system SED ($r$ = Vega, ABmag, STmag).

  * •

ZPi,r: Photometric zero point for filter $i$ in the reference system $r$.

  * •

$F_{\lambda,0}(\lambda)$: Unextinguished SED.

  * •

$A(\lambda)$: Extinction as a function of $\lambda$ in mag.

The logical procedure is to generate an extinguished grid of SEDs using Eqn. 8
as a function of intrinsic (such as $T_{\rm eff}$ and $\log g$ or luminosity
class) and extrinsic (such as amount and type of dust) parameters, including
distance if it is not independently determined. The extinguished SED grid is
transformed into an extinguished magnitude grid using Eqn. 7 and the
parameters and their uncertainties are obtained through the fitting process
that uses Eqn. 6. This is the procedure used in CHORIZOS (?) and in other
equivalent codes. However, the devil is in the details and there are five
different aspects that have to be considered. I describe them here, each
color-coded in Eqns. 6 to 8 above.

Photometric calibration. There are three related aspects of photometric
calibration to be considered: the absence of biases in the observed
magnitudes, the proper estimation of the photometric uncertainties, and the
correct characterization of the sensitivity curves. As already mentioned, old-
style ground-based systems such as Johnson and Strömgren have problems
inherent with the small samples analyzed by each author and with the presence
of inconsistencies among different authors, leading to relatively large
realistic uncertainties (??). Gaia photometry, on the other hand, has the
opposite problem: the large sample and consistent reduction leads to
calibrations that are so accurate that small systematic effects can be
detected, thus leading to improvements in the calibration itself (?). In this
respect, it is important to distinguish that there are two types of
photometric uncertainties. Internal uncertainties are those that do not make a
comparison with an external system and, as such, are appropriate for
variability studies such as in ?. However, if we want to compare our observed
magnitudes with synthetic ones, we also have to include the uncertainties in
the knowledge of the photometric system (external uncertainties). The easiest
thing to do is to use a calculated minimum uncertainty or to add a systematic
uncertainty in quadrature to the observed one (?). If this is not done,
chances are that we will have problems in our final check (see below). A
similar issue can appear if our source is intrinsically variable and our
photometry is obtained in different epochs, which is why a check for source
variability is convenient (?).

Standard calibration. Photometric systems are calibrated with a reference SED
that can be an astrophysical source such as Vega (Vega system) or a simple
function such as in the ABmag or STmag systems (see e.g. ?). In the case of an
astrophysical source, it needs to be determined independently (???). In all
cases, the photometric zero point (in some references, called the differential
zero point) needs to be calculated. It is a quantity that is supposed to be
close to zero but seldom is actually zero due to the way photometric surveys
are calibrated (see ?? and references therein).

Model SEDs. Nowadays there are a number of choices for SED grids for stellar
atmospheres of different characteristics. However, before using them one needs
to read the fine print, as some are optimized to accurately describe stellar
lines but not so much the absolute fluxes or have a detailed analysis in just
some wavelength regions. One way to test them is to compare the derived
synthetic photometry derived with the observed magnitudes of low-extinction
stars. As an example, CHORIZOS uses a tailored SED grid built from different
sources (?). For hot stars, the SEDs are mostly from TLUSTY (??) and
accurately represent the optical colors of O and B stars. However, the NIR
synthetic colors derived from TLUSTY are incorrect, so for long wavelengths
the ? grid is used instead. Also, one should always check whether the grid
includes the type of system we are observing: stars with emission lines or IR
excesses and binary systems with components of very different $T_{\rm eff}$
will usually need to be analyzed with an SED grid with additional parameters.

Extinction. The main issue here is extinguishing the model SEDs (Eqn. 8)
before integrating in wavelength (Eqn. 7) to correctly account for non-linear
photometric effects, as discussed in the previous section. Furthermore, some
papers use simple linear approximations to determine the amount and type of
extinction from color-color relations: those determinations of $E(B-V)$ and
$R_{V}$ should be distrusted in general (and, once again, one should use
monochromatic quantities for the amount and type of extinction, not band-
integrated ones).

Final check. The last step to perform is to verify the goodness of the fit.
Fitting a functional form to a data set and obtaining a result does not mean
that the result makes sense: a straight line can be fit to data extracted from
a parabola to derive a slope and an intercept but that does not mean that
those two quantities accurately describe the data. Unfortunately, this step is
omitted in many papers and one is left to wonder whether the results make
sense or not. Going over the four previous aspects, if the fit is not as good
as it should it could be due to a number of reasons: [1] The observed
magnitudes may have systematic effects, their random uncertainties may be
underestimated, or the sensitivity curve may not be properly characterized.
[2] The photometric zero points may be incorrect. [3] The model SEDs may not
properly represent the object. [4] The extinction law may be incorrect. Any of
these reasons may apply in different situations. Sometimes the problems will
be in the calibration, in others we may have an interesting astrophysical
case, such a star with a previously undetected IR excess. One way to
understand what is going on is to analyze the fit residuals, as those may
point out to e.g. systematic effects as a function of the amount of
extinction, a telltale sign of issues with the extinction law (see ? for an
example).

###  3.3 Results

Figure 6: $R_{5495}$ as a function of $E(4405-5495)$ for the sample of 120 OBA
stars in ?. The color coding indicates whether the data appeared in ? or not.
Green lines correspond (from left to right) to the values of $A_{5495}$ of 3,
6, and 9 mag.

In ? we used the method described above to derive $E(4405-5495)$ and
$R_{5495}$ for 562 Galactic O stars. In a subsequent paper (?), we selected a
subsample of those and added some OB stars with notoriously high
$E(4405-5495)$ or $R_{5495}$ to obtain a golden sample of 120 OBA stars with
measurements not only of those extinction properties but also of interstellar
K i and C2. We also detected a very broad interstellar absorption band of
unknown origin centered around 7700 Å that should be incorporated into future
famiies of extinction laws.

The results for the golden sample are shown in Fig. 6 and are derived from a
combination of NIR (mostly 2MASS) and optical photometry (Gaia, Tycho-2,
Johnson, and Strömgren). In general, it is difficult to determine $R_{5495}$
from optical photometry alone, as the information encoded at longer
wavelengths is needed to determine $A_{5495}$. The most important result from
Fig. 6 is that there is a large scatter in the Galactic $R_{5495}$ values:
from a floor around 2.6 to values higher than 6.0. A similar range (3.1-6.7)
was found for the 30 Dor sample in ? using HST photometry. The distributions
in $R_{5495}$ and $E(4405-5495)$ are not independent: the scatter in
$R_{5495}$ decreases as $E(4405-5495)$ increases and, in the limit of high
extinction, most $R_{5495}$ values concentrate in the 2.8-3.2 range.
Therefore, the canonical values of 3.0-3.2 are still valid, but only as an
approximation in that circumstance.

Figure 7: Extinction measurements for O stars in two H ii regions: (left) the
Carina Nebula and (right) M 16/NGC 6611. The two images are from ESO press
releases 1250 ($Bgr$H$\alpha$) and 0926 ($BVR$), respectively. North is
towards the top and east towards the left and the approximate physical scale
is indicated. Only a section of each H ii region is shown to better visualize
the dust lanes.

Figure 8: Extinction measurements for Galactic O stars plotted on top of the
full-sky H$\alpha$ image of ?. Different colors are used for five ranges of
$R_{5495}$ with the size of the symbol increasing with $E(4405-5495)$. The
background image is showed in a logarithmic intensity scale and in Galactic
coordinates with a Cartesian projection centered on the Galactic anticenter.

It is important to understand the uncertainties associated with both
$E(4405-5495)$ and $R_{5495}$. Parameter uncertainties in CHORIZOS are
calculated by properly sampling the multi-parameter space around the minimum
$\chi^{2}$ location, thus allowing for a correct characterization of their
correlations (?). In general, $E(4405-5495)$ can be determined reasonably well
(uncertainties of $\sim 0.02$ mag) at most extinctions as long as there is
good-quality photometry. This is easy to understand as the intrinsic $(B-V)$
of OB stars $(B-V)_{0}$ is well determined and, to a first approximation,
$\mbox{$E(4405-5495)$}\sim E(B-V)$ (but see Fig. 1) and from Eqn. 4 the
uncertainties arise only from those in the measured (Johnson) $B-V$, which are
of the order of 0.02 mag (see Table 9 in ? and remember the previous point
about realistic external photometric uncertainties)333This is just a
simplified version of what the code does and assumes that Johnson $BV$
photometry is available. In reality, all used magnitudes contribute in some
degree to the detrmination of $E(4405-5495)$.. The uncertainties on
$R_{5495}$, on the other hand, are much larger for small values of
$E(4405-5495)$ than for large ones. The explanation is that $R_{5495}$ is a
measurement of the type of extinction and in the limit of low extinction
information about it is limited. Interestingly, the sampling of $\chi^{2}$ in
the multi-parameter space shows that in most cases the uncertainties in
$E(4405-5495)$ and $R_{5495}$ are anticorrelated: this is seen in Fig. 6 by
the elongation of the ellipse uncertainties along the lines of constant
$A_{5495}$. Therefore, the relative uncertainty of $A_{5495}$ can be lower
than those of both $E(4405-5495)$ and $R_{5495}$, something that is impossible
in the absence of an anticorrelation. In other words, a method that properly
measures extinction combining optical and NIR photometry can arrive at a more
precise determination of $A_{5495}$ (and hence, of $A_{V}$) than of either
$E(4405-5495)$ or $R_{5495}$.

Where does the variability in the extinction law originate? Already ? pointed
out that the high values of $R_{5495}$ in Orion are likely due to a larger
than average grain size for the dust particles, a view that has remained since
then. In ? we reviewed the literature and presented results showing how H ii
regions have large internal variability in $R_{5495}$ and $E(4405-5495)$, with
regions with intense nebular emission typically having high $R_{5495}$ and low
$E(4405-5495)$ and regions close to or on top of dust lanes having low
$R_{5495}$ and high $E(4405-5495)$ (Fig. 7). In addition, a map of the
$R_{5495}$ measurements in ? over the whole sky (Fig. 8) shows that high-
latitude stars, which are relatively nearby and for which the Local Bubble
contribution should be larger than average, tend to have high values of
$R_{5495}$. This led us in ? to propose a scenario in which there are three
different ISM regimes depending on the ambient UV radiation level:

  * •

Regions with intense UV radiation have large values of $R_{5495}$ ($>4$). This
includes H ii regions but also cavities filled with hot gas (e.g. the Local
Bubble) where UV radiation can travel for long distances.

  * •

Regions with low intensity of UV radiation have small values of $R_{5495}$
($<3$). These regions are those with significant column densities of CO or
those that are easily detected as dust lanes in the optical.

  * •

Regions with intermediate levels of UV radiation have values of $R_{5495}$
between 3 and 4. These regions represent a typical warm to cold ISM (excluding
H ii regions, cavities, and molecular clouds) that fills the majority of the
volume in the Galactic disk.

Therefore, high levels of UV radiation correspond to large average grain sizes
(high $R_{5495}$). The likely explanation is that ISM regions with large
values of $R_{5495}$ are produced by the selective destruction of smaller dust
grains with respect to large grains. Such a selective destruction could be
caused by thermal sputtering (?) or heating by EUV radiation (??).

###  3.4 A comparison with Gaia DR3 results

Figure 9: Comparison between the CHORIZOS results of ? and the Gaia XP results
of ?. The black/red lines are the input XP spectra using two choices of basis
functions (to showcase the uncertainty in flux), the blue line is the SED
derived from the CHORIZOS fit, and the green line the SED derived from the ?
XP fit (displaced to fit as much of the XP spectra as possible). The names of
the stars and their CHORIZOS-derived values of $E(4405-5495)$ and $R_{5495}$
are given in each panel, with the three stars selected to showcase different
values of $R_{5495}$.

To see the importance of including $R_{5495}$ in an analysis of extinction,
here I compare the results from ? with the Gaia DR3 results of ?. That paper
presents the Gaia-DPAC analysis using the Astrophysical parameters inference
system (Apsis). With a variety of information from different Gaia sources,
they derive stellar and extinction parameters for a large sample of stars.
Here I concentrate on the results from the ESP-HS module, the Extended Stellar
Parametrizer for Hot Stars, as that is the module which is more appropriate
(and favorable) to use against the OBA-star analysis with CHORIZOS in ?.

The ESP-HS module uses a library of hot-star atmospheres to derive $T_{\rm
eff}$, $\log g$, $A_{0}$ (the equivalent to $A_{5495}$, see above), and $d$
based on the parallax and the XP spectra (plus the RVS spectra for some
stars). It fixes the equivalent of $R_{5495}$ to 3.1 using the family of
extinction laws of ?. As a comparison, the analysis of ? is based on
photometric data (including Gaia and 2MASS) and fits $E(4405-5495)$,
$R_{5495}$, and $\log d$, fixing $T_{\rm eff}$ and the luminosity class (a
proxy for $\log g$) from the spectral types, and using the family of
extinction laws of ?. I compare the results from the two procedures for three
stars of different $R_{5495}$ in Fig. 9.

The predicted CHORIZOS SED is a reasonable fit for the XP spectra of the three
stars despite having been obtained prior to Gaia DR3. There are some small
differences, both at small and large wavelength scales, some of which may be
due to small deficiencies in the ? family of extinction laws and some due to
systematic biases in the XP spectra. Among the first, I note that for V747 Cep
there is a hint of the 7700 Å bump and for V662 Car there is a clear
detection. Among the second, many of the wiggles seen are likely artifacts due
to the extraction procedure as a series of basis functions (?).

As for the ? results, the fit is reasonably good for V747 Cep, which has an
$R_{5495}$ of 3.135$\pm$0.078 that is compatible with the assumed value of
3.1. On the other hand, the fit is bad for either V662 Car and $\theta^{1}$
Ori Ca,Cb, which have values of $R_{5495}$ significantly different from the
assumed 3.1. The discrepancy is a natural consequence of the CHORIZOS fit
being a good one, as there is no stellar SED of any $T_{\rm eff}$ with
$\mbox{$R_{5495}$}=3.1$ that resembles that of an O star with values of
$\mbox{$R_{5495}$}>4.0$ and a minimum amount of extinction. In other words,
the ? results are inconsistent with their own input data because they did not
include the real solution among the set of possible ones, an example of
“fitting a straight line to a parabola” (see above).

The conclusion of this analysis is that the parameters derived by assuming
that $R_{5495}$ is 3.1 everywhere are bound to fail in the cases where
$R_{5495}$ is significantly different and those cases, at least for OBA stars,
are relatively common (Fig. 6). As an example, the ESP-HS analysis of V662 Car
derives a value of $A_{0}=3.639\pm 0.042$ mag. The equivalent $A_{5495}$
calculated from CHORIZOS is $5.206\pm 0.038$ mag, significantly higher.
Another lesson is that NIR data need to be added to optical
(spectro)photometry to correctly fit both the type and amount of extinction: ?
could not include 2MASS photometry in their analysis due to the DPAC policy of
excluding non-Gaia data.

##  4 Families of extinction laws

###  4.1 History

As already mentioned, ? or CCM published the first family of extinction laws
with a parameter describing the type of extincion ($R_{V}$ though it should
have been a monochromatic quantity such as $R_{5495}$) to add to the amount of
extinction [$E(B-V)$ though it should have been a monochromatic quantity such
as $E(4405-5495)$]. It was based on $UBVRIJHKL$ photometry for the optical+IR
part and IUE spectrophotometry for the UV part, thus covering the range from
1250 Å to 3.5 $\mu$m. As with most 1980s extinction papers, CCM is a UV-
centric work, as the access to that range provided by IUE was the novelty at
that time and that was where most of the interesting information was
available. As it turned out, most papers that ended up using the family of
extinction laws eventually applied it to the optical+IR ranges instead of to
the ultraviolet, especially after the first decade444As proof of the
importance of the paper, it is currently the 11th most cited paper on ADS,
with a citation count approaching $10\,000$ at the time of the writing of this
contribution..

The CCM family is based on 29 low-extinction Galactic sightlines, of which
only three have listed values with $E(B-V)>1.0$ mag and even for those they
are only 1.09 mag, 1.11 mag, and 1.22 mag. The listed covered range in $R_{V}$
is 2.6-5.3. The reasons for those limitations are easy to understand: as an
UV-centric study, objects with higher reddenings would have been mostly
inaccessible to IUE, and at that time there were few objects with known high
values of $R_{5495}$. For the three wavelength ranges the following functional
forms were used:

  * •

UV: An adaptation of the ? functional form (see below).

  * •

Optical: A 7th degree polinomial in $x\equiv 1/\lambda$ to interpolate between
the central wavelengths of the Johnson photometric bands.

  * •

IR: A power law $A(\lambda)\propto\lambda^{-\alpha}$ with a fixed value of
$\alpha=1.61$ from ?.

A decade later, ? or F99 produced another family of extinction laws. His
sample was larger than that of CCM but with a similar sampling555Some of the
details of the analysis are hard to follow in that paper, as the sample is not
presented in detail but is a modification of that of ?, their $R_{V}$ values
are not given, and the notation confuses monochromatic and band-integrated
quantities. in terms of $E(B-V)$, as the paper is also UV-centric. The main
modification with respect to CCM is the use of splines instead of a seventh
degree polynomial to interpolate between filter central wavelengths and, in
that way, avoid unphysical oscillations in wavelength (Fig. 10). The spline
interpolation is extended into the IR and, as a result, the functional form
does not follow a power law there. Instead, the slope is flatter at
wavelengths longer than the $H$ band and steeper at shorter wavelengths (Fig.
11).

Figure 10: Comparison between the STIS spectrophotometry for two 30 Dor stars
in the MA14 sample and the fits using the three families of extinction laws
discussed in this contribution. Note how the use of splines in F99 and MA14
allow them to follow the approximate shape of the spectrophotometry while the
seventh degree polynomial used by CCM introduces unphysical oscillations in
wavelength.

A third family of extinction laws was derived by ? or MA14 using a sample of
85 OB stars in 30 Dor observed with HST/WFC3 $UBVI$ optical and ground-based
$JHK$ photometry. The advantages of HST/WFC3 photometry are its better
calibration and spatial resolution, which leads to a reduction in the
systematic effects present in Johnson photometry and allows for the detection
of possible companions. The covered range in $E(4405-5495)$ was smaller than
those of CCM or F99 (an effect offset by the better photometric calibration)
but with a better sampling in $R_{5495}$. As opposed to the two previous
families, the TLUSTY models used for CHORIZOS were the source for
unextinguished SEDs instead of real standard stars.

Figure 11: Comparison between the CCM, F99, and MA14 families of extinction
laws for two values of $R_{5495}$, 3.0 and 5.1. The left panel shows the
extinction laws normalized to the value at 5495 Å. The right panel shows the
ratio of the CCM and F99 laws to the MA14 one of the same $R_{5495}$. Note
that the CCM and MA14 laws are identical in the IR for a given $R_{5495}$.

The idea behind MA14 was to combine the best of both worlds from CCM and F99:
the proven behavior of the first as a function of $R_{5495}$ with the use of
splines in the optical from the second to eliminate the unphysical
oscillations in wavelength. To that purpose, MA14 started with the CCM family,
maintained the IR behavior of ?, and substituted the optical part with a
spline equivalent. Then, the photometric residuals were studied to discover
that the CCM laws were underestimating extinction in the $U$ band (the same is
true for F99) and corrected for that effect (Fig. 12). The shortcomings of the
family are that it does not include the UV; that the IR slope is assumed and
not determined (something difficult to do for the range of extinctions of the
sample); and that, being derived from 30 Dor data, its validity for Galactic
sightlines was unproven at the time of the writing of the paper (but see
below).

Figure 12: 30 Dor star observed with HST/STIS spectrophotometry and their
corresponding SED fits from the photometry in MA14 using the CCM and the new
families in that paper. Both fits are very good in the optical but in the $U$
band the new laws are significantly better, leading to the possibility that
some of the differences in the UV slope arise not from a difference in
extinction laws between the Milky Way and 30 Dor but due to a bad stitching
between the optical and UV regimes in CCM. I have not removed the effects of
narrow ISM lines in the NUV, the mismatch in the profiles of stellar lines in
the optical, or the noise in the FUV and around 1 $\mu$m.

There are other published families of extinction laws but I restrict the
analysis to these three, as the first two are the ones most often used in the
literature and the third one is used for the comparison in the next
subsection. In Fig. 11 I plot a comparison between the three families, which
is a graphical summary of some of the points I have made in this subsection.

  * •

The CCM and MA14 families follow a power law with $\alpha=1.61$ and are
identical in the NIR, so they are shown as superimposed straight lines in the
left panel and a value of 1.0 in the right one for that range. The F99
families, on the other hand are not power laws in the NIR and they predict
less extinction there for the same amount of optical extinction. The
differences between CCM/MA14 and F99 in the NIR increase with $R_{5495}$.

  * •

The differences between the three families are smaller in the optical than
those between CCM/MA14 and F99 in the NIR.

  * •

The CCM family has an oscillatory behavior in the optical range due to the use
of a seventh degree polynomial in $1/\lambda$ for interpolation.

  * •

Both the CCM and F99 families predict a lower extinction in the $U$ band for
the same amount of extinction in the $V$ band.

###  4.2 The optical/NIR

Figure 13: Reduced $\chi^{2}$ as a function of $E(4405-5495)$ using the CCM
(top left), F99 (top right), and MA14 (bottom left) families of extinction
laws for the CHORIZOS runs of the Galactic O-star sample of ?. (bottom right)
Reduced $\chi^{2}$ histograms for the first three plots and expected
distribution.

Having presented the characteristics of the three families of extinction laws,
the outstanding question is: which one of the three is a better approximation
to reality? The answer was provided by ?, whose work on the distribution of
$E(4405-5495)$ vs. $R_{5495}$ for 562 O-type Galactic stars was presented in
the previous section. That paper also processed their Johnson+2MASS+Gaia
photometry (with Tycho-2 and/or Strömgren included when available) through
CHORIZOS using the three families. The resulting reduced $\chi^{2}$
distributions are shown in Fig. 13

  * •

As seen in the bottom right panel, the best results are clearly obtained with
the MA14 familiy. The observed reduced $\chi^{2}$ distribution follows the
expected one closely, with no target having a value over 4. On the other hand,
both the CCM and F99 families have extended tails into high reduced $\chi^{2}$
values that indicate that they cannot provide accurate fits for some of the
stars in the sample, with four of the five the worst outliers coming from F99.
The family of extinction laws of ? provides the best description of Galactic
optical extinction for $\mbox{$E(4405-5495)$}<3.0$ available to date.

  * •

For CCM, the worst results are those where Strömgren photometry is included.
This is expected given the seventh degree polynomial used for that family. The
tight correlation shown between the reduced $\chi^{2}$ and $E(4405-5495)$ in
the top left panel for those targets is a telltale sign that the effect is
caused by extinction. Other than that, CCM provides a reasonable behavior in
the sampled range.

  * •

For F99, the worst results are those for high values of $R_{5495}$, marked
with stars in the top right panel. However, there is also a significant trend
of increasing reduced $\chi^{2}$ with $E(4405-5495)$ for targets with near-
canonical values of $R_{5495}$.

  * •

I point out once more that the three families of extinction laws were
developed using stars that are esentially in the leftmost one-third of the
$E(4405-5495)$ range of the CCM/F99/MA14 panels of Fig. 13. For stars in the
remaining two-thirds we are extrapolating the behavior from there.

The analysis of ? was extended by ? to include more stars, especially with
large values of extinction (reaching $\mbox{$E(4405-5495)$}=3.5$). They found
that even with the MA14 family, some stars have reduced $\chi^{2}$ values
larger than 4. This means that the extrapolation from low extinctions is
already failing and that an improved family is needed. Below I address how new
data could be used to achieve that.

###  4.3 The UV

Figure 14: F99 UV extinction for $R_{5495}$ of 3.1 and 5.0. The
$\mbox{$R_{5495}$}=3.1$ case has been decomposed into its three components
(linear, Drude profile, and far-UV rise).

? parameterized UV extinction as a function of $x\equiv 1/\lambda$ (in
$\mu{\rm m}^{-1}$) as the sum of three components: a linear term, a Drude
function (for the 2175 Å bump) and a far-UV rise. The formula in that paper
is:

$\frac{A(\lambda)-A_{V}}{E(B-V)}=c_{1}+c_{2}x+c_{3}D(x;\gamma,x_{0})+c_{4}F(x),$
(9)

$D(x;\gamma,x_{0})=\frac{x^{2}}{(x^{2}-x_{0}^{2})^{2}+x^{2}\gamma^{2}},$ (10)

$F(x)=\left\\{\begin{array}[]{ll}0.5392(x-5.9)^{2}+0.05644(x-5.9)^{3}&{\rm
for}\;\;x\geq 5.9\\\ 0&{\rm for}\;\;x<5.9\end{array}\right..$ (11)

As I have previously mentioned, such formulae should not be expressed in terms
of band-integrated quantities [$A_{V}$ or $E(B-V)$] and in a consistent
implementation they should be substituted by their monochromatic equivalents.
In Fig. 14 I represent the UV extinction as implemented by F99 using those
formulae but renormalized to 1.0 at the beginning of the UV range and based on
$R_{5495}$. CCM used a similar but slightly different parameterization in the
UV but the overall characteristics are the same. Both CCM and F99 used a
number of sightliness to derive relationships that express the six parameters
in Eqns. 9-11 ($c_{1}$, $c_{2}$, $c_{3}$, $c_{4}$, $x_{0}$, and $\gamma$) as a
function of $R_{5495}$ (or its equivalent). When combined with the optical/IR
results, the final outcome in each paper is a single-parameter family that
covers the three wavelength ranges.

The main effect of increasing $R_{5495}$ in either the CCM or F99 families is
to make the UV slope flatter (for extreme $R_{5495}$ values, the linear
component can actually be negative). For the LMC and SMC (see references at
the beginning of the previous section) the most important effect is the
reduction in $c_{3}$, the intensity of the Drude profile.

As opposed to the results in CCM and F99, the analysis of ? found a very
different result: with the exception of few cases with extreme values of
$R_{5495}$, the UV and IR portions of the Galactic extinction curves are not
correlated with each other. In other words, it is not possible to derive a
single-parameter (e.g. $R_{5495}$) family of extinction laws for the three
wavelength ranges, as UV and IR extinctions behave independently. In a
subsequent paper, ? attempted to derive a two-parameter family of extinction
laws but, as shown in Appendix E of ?, the analysis was flawed because they
inadvertantly made one parameter a function of the other one. In a recent
paper involving those authors (?) they derive a new family of extinction laws
based on a single parameter for the UV/optical/IR ranges and only reference ?
in passing.

###  4.4 The IR

Figure 15: IR extinction laws normalized to 2.144 $\mu$m from ? and ?. As a
comparison, two power laws with $\alpha$ of 2.20 and 1.61 are also shown. The
location of the $JHK$ bands and the identification of the main absorption
features are printed.

As already mentioned, ? approximated the NIR extinction law with a power law:

$A(\lambda)=A_{1}\lambda^{-\alpha},$ (12)

where $A_{1}$ is the extinction at 1 $\mu$m, $\alpha$ is the power-law
exponent, and $\lambda$ is expressed in $\mu$m. ? obtained $\alpha=1.61$,
which is the value that CCM and MA14 used. On the other hand, most results of
the last 15 years (????????) have obtained significantly higher values of
$\alpha$ between 2.11 and 2.47, with most in the range $2.2\pm 0.1$. One of
the reasons for the discrepancy with the ? result is that the latter
inadvertantly included Cyg OB2-12 in their sample, a B-type variable
hypergiant with a strong wind and a significant IR excess (??) which flattens
the derived extinction law if not corrected for. Also, most recent papers show
little sightline-dependence variability either with direction or amount of
extinction (?????) and a progressive flattening of the slope as we move from
the $JH$ bands to the $K$ band and beyond (???), indicating that IR extinction
cannot be approximated by a power law over the whole range (?).

There are no known strong absorption features in the $JHK$ bands (or NIR) but
the situation changes in the MIR, with significant absorption features arising
from H2O, CO, CO2, aliphates, and silicates (???), see Fig. 15. The flattening
detected by some authors between the $JH$ and $HK$ bands becomes more
significant in the 3-8 $\mu$m region (once the absorption features are
removed) with values of $\alpha\sim 1.5$ or even lower.

One apparently discrepant voice is that of ? who measure $\alpha=1.7$ using
relatively low-extinction OB stars. The difference is not due to the nature of
the targets (as OB stars inhabit, in principle, different typical environments
than late-type giants, see ? for the effects on extinction), as ? measured a
value of $2.13\pm 0.08$ also using OB stars (albeit more extinguished).
Instead, I suspect the difference is caused by a combination of two effects:

  * •

The low extinction of their targets (compared to most other similar studies)
makes their results more subject to systematic errors. Indeed, the values they
obtain for $\alpha$ vary between 1.36 and 2.20, a range much larger than what
is seen in other studies. Also, their Fig. 5 reveals that the power-law fit is
not a good approximation for the data of some of their wavelength regions and
stars.

  * •

Their wavelength range is large, covering 0.8-5.5 $\mu$m for some of their
stars, as opposed to the narrower $JHK$ range of most studies above. We have
already seen that the extinction law flattens for wavelengths longer than the
$K$ band and the same is true in the 0.8-1.0 $\mu$m region (Fig. 11).
Therefore, an average $\alpha$ in the 0.8-5.5 $\mu$m range is likely to be
lower than its equivalent for the $JHK$ bands.

##  5 The future

Extinction makes stars look faint but the future for extinction studies looks
bright. The most significant improvement will be spectrophotometry
substituting the role of photometry, thus dissipating the doubts over the
functional form of the extinction law and allowing for the detection of
possible structures at intermediate wavelength scales, such as the already
known 7700 Å bump. For the case of the Magellanic Clouds (and, later on, for
other galaxies) there is plenty of room to significantly increase the number
of studied sightlines. Regarding spectrophotometry, in the optical Gaia XP is
likely to play the major role thanks to its large sample size and dynamic
range and despite its poor spectral resolution. That may be complemented with
HST/STIS data for a few sightlines, especially if the mission does not end
soon. In the IR there is still ISO and Spitzer archival data to be fully
analyzed but new data may come from ground-based observations and from JWST,
though in the latter case I am somewhat pessimistic that panels will assign
much time to bright objects (as many extinguished stars are in the IR). In the
UV HST is likely to still be the main contributor in the short term with
projects such as ULLYSES (?) until new missions are launched.

Another line that should be explored is the improvement in calibrations
provided by Gaia (spectro)photometry applied to ground-based photometric
surveys. That will allow detailed extinction studies to complement Gaia with
narrow and intermediate bands. An example of that type of study is GALANTE (?)

Extinction should be understood as the effect of ISM material on astronomical
light sources, independently of whether the material is “generic” dust
extinguishing the continuum or is in lines/bands assigned to a known or
suspected material. This is already done in the NIR (Fig. 15) and in the UV
with the 2175 Å band (long suspected to originate in graphite). Therefore, the
extinction law should eventually incorporate atomic and molecular lines, which
would introduce further complications. First, is that different environments
in terms of composition, depletion, and ionization state will contribute to
different degrees to extinction, thus necessitating multi-parameter families.
Second, some lines like the optical Na i and Ca ii doublets easily saturate
near the Galactic plane and their measured EWs depend not only on the column
density but on the temperature and kinematics along the sightline. However,
even approximately incorporating them into an extinction line would provide us
information on the conditions associated with different components of the ISM,
such as the excess Ca ii absorption associated with high-velocity clouds (??).

The final components that should be added to the extinction law are DIBs, ISM
bands discovered a century ago (??) but of still uncertain nature save a few
adscribed to fullerenes (?). It is known that there are different types of
sightlines (??), named $\sigma$ and $\zeta$ after the prototypes $\sigma$ Sco
and $\zeta$ Oph, that sample different regimes of the ISM based on the
exposure to UV radiation. The DIB EWs are well correlated with $E(4405-5495)$
and the sightline type is weakly correlated with $R_{5495}$ (?), so there is
hope that a multi-parameter extinction law could incorporate a significant
part of the DIB behavior.

## Acknowledgements

I would like to thank the LOC for organizing a nice meeting in such a
beautiful location. I acknowledge support from the Spanish Government
Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de
Investigación ($10.130\,39$/ $501\,100\,011\,033$), and FEDER/UE through grant
PGC2018-0$95\,049$-B-C22 and grant PID2022-$136\,640$NB-C22 from the Consejo
Superior de Investigaciones Científicas (CSIC) through grant 2022-AEP 005.

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