  We present millimeter-wave observations of several molecular ions in the disk
around the pre-main-sequence star DM Tau and use these to investigate the
ionization fraction in different regions of the disk. New Submillimeter Array
(SMA) observations of H2D+ J=1_10 - 1_11, N2H+ J=4-3 and CO J=3-2 are
presented. H2D+ and N2H+ are not detected and using the CO 3-2 disk size the
observations result in an upper limit of <0.47 K km s-1 for both lines, a
factor of 2.5 below previous single-dish H2D+ observations. Assuming LTE, a
disk midplane temperature of 10-20 K and estimates of the H2D+ o/p ratio, the
observed limit corresponds to NH2D+ < 4 - 21 \times 1012 cm-2. We adopt a
parametric model for the disk structure from the literature and use new IRAM 30
meter telescope observations of the H13CO+ J=3-2 line and previously published
SMA observations of the N2H+ J=3-2, HCO+ J=3-2 and DCO+ J=3-2 lines to
constrain the ionization fraction, xi, in three temperature regions in the disk
where theoretical considerations suggest different ions should dominate: (1) a
warm, upper layer with T>20 K where CO is in the gas-phase and HCO+ is most
abundant, where we estimate xi \simeq 4 \times 10-10, (2) a cooler molecular
layer with T = 16-20 K where N2H+ and DCO+ abundances are predicted to peak,
with xi \simeq 3\times10-11, and (3) the cold, dense midplane with T<16 K where
H3+ and its deuterated isotopologues are the main carriers of positive charge,
with xi < 3\times10-10. While there are considerable uncertainties, these
estimates are consistent with a decreasing ionization fraction into the deeper,
colder, and denser disk layers. Stronger constraints on the ionization fraction
in the disk midplane will require not only substantially more sensitive
observations of the H2D+ 1_10 - 1_11 line, but also robust determinations of
the o/p ratio, observations of D2H+ and stronger constraints on where N2 is
present in the gas phase.
