  We determine stellar parameters for the M dwarf GJ 436 that hosts a
Neptune-mass planet. We employ primarily spectral modeling at low and high
resolution, examining the agreement between model and observed optical spectra
of five comparison stars of type, M0-M3. Modeling high resolution optical
spectra suffers from uncertainties in TiO transitions, affecting the predicted
strengths of both atomic and molecular lines in M dwarfs. The determination of
Teff, gravity, and metallicity from optical spectra remains at ~10%. As
molecules provide opacity both in lines and as an effective continuum,
determing molecular transition parameters remains a challenge facing models
such as the PHOENIX series, best verified with high resolution and
spectrophotometric spectra. Our analysis of GJ 436 yields an effective
temperature of Teff = 3350 +/- 300 K and a mass of 0.44 Msun. New Doppler
measurements for GJ 436 with a precision of 3 m/s taken during 6 years improve
the Keplerian model of the planet, giving a minimum mass, M sin i = 0.0713 Mjup
= 22.6 Mearth, period, P = 2.6439 d, and e = 0.16 +/- 0.02. The noncircular
orbit contrasts with the tidally circularized orbits of all close-in
exoplanets, implying either ongoing pumping of eccentricity by a more distant
companion, or a higher Q value for this low-mass planet. The velocities indeed
reveal a long term trend, indicating a possible distant companion.
