Using Multiple Emission Line Ratios to Constrain the Slope of the Dust
  Attenuation Law
  We explore the possibility and practical limitations of using a three-line
approach to measure both the slope and normalization of the dust attenuation
law in individual galaxies. To do this, we focus on a sample of eleven galaxies
with existing ground-based Balmer H$\alpha$ and H$\beta$ measurements from slit
spectra, plus space-based grism constraints on Paschen-$\beta$. When accounting
for observational uncertainties, we show that one galaxy has a well-constrained
dust law slope and normalization in the range expected from theoretical
arguments; this galaxy therefore provides an example of what may be possible in
the future. However, most of the galaxies are best-fit by unusually steep or
shallow slopes. We then explore whether additional astrophysical effects or
observational biases could explain the elevated Paschen-$\beta$/H$\alpha$
ratios driving these results. We find that galaxies with high
Paschen-$\beta$/H$\alpha$ ratios may be explained by slightly sub-unity
covering fractions ($>$97%). Alternatively, differing slit losses for different
lines can have a large impact on the results, emphasizing the importance of
measuring all three lines with a consistent spectroscopic aperture. We conclude
that while the three-line approach to constraining the shape of the dust
attenuation law in individual galaxies is promising, deep observations and a
consistent observational strategy will be required to minimize observational
biases and to disentangle the astrophysically interesting effect of differing
covering fractions. The James Webb Space Telescope will provide more sensitive
measurements of Balmer and Paschen lines for galaxies at $z\approx0.3-2$,
enabling uniform constraints on the optical-infrared dust attenuation law and
its intrinsic variation.
