  M dwarf stars are excellent candidates around which to search for exoplanets,
including temperate, Earth-sized planets. To evaluate the photochemistry of the
planetary atmosphere, it is essential to characterize the UV spectral energy
distribution of the planet's host star. This wavelength regime is important
because molecules in the planetary atmosphere such as oxygen and ozone have
highly wavelength dependent absorption cross sections that peak in the UV
(900-3200 $\r{A}$). We seek to provide a broadly applicable method of
estimating the UV emission of an M dwarf, without direct UV data, by
identifying a relationship between non-contemporaneous optical and UV
observations. Our work uses the largest sample of M dwarf star far- and near-UV
observations yet assembled. We evaluate three commonly-observed optical
chromospheric activity indices -- H$\alpha$ equivalent widths and log$_{10}$
L$_{H\alpha}$/L$_{bol}$, and the Mount Wilson Ca II H&K S and R$'_{HK}$ indices
-- using optical spectra from the HARPS, UVES, and HIRES archives and new HIRES
spectra. Archival and new Hubble Space Telescope COS and STIS spectra are used
to measure line fluxes for the brightest chromospheric and transition region
emission lines between 1200-2800 $\r{A}$. Our results show a correlation
between UV emission line luminosity normalized to the stellar bolometric
luminosity and Ca II R$'_{HK}$ with standard deviations of 0.31-0.61 dex
(factors of $\sim$2-4) about the best-fit lines. We also find correlations
between normalized UV line luminosity and H$\alpha$ log$_{10}$
L$_{H\alpha}$/L$_{bol}$ and the S index. These relationships allow one to
estimate the average UV emission from M0 to M9 dwarfs when UV data are not
available.
