  Aims: The aim of this paper is to demonstrate that millimeter wave data can
be used to distinguish between various atmospheric models of sunspots, whose
temperature structure in the upper photosphere and chromosphere has been the
source of some controversy. Methods: We use observations of the temperature
contrast (relative to the quiet Sun) above a sunspot umbra at 3.5 mm obtained
with the Berkeley-Illinois-Maryland Array (BIMA), complemented by submm
observations from Lindsey & Kopp (1995) and 2 cm observations with the Very
Large Array. These are compared with the umbral contrast calculated from
various atmospheric models of sunspots. Results: Current mm and submm
observational data suggest that the brightness observed at these wavelengths is
low compared to the most widely used sunspot models. These data impose strong
constraints on the temperature and density stratifications of the sunspot
umbral atmosphere, in particular on the location and depth of the temperature
minimum and the location of the transition region. Conclusions: A successful
model that is in agreement with millimeter umbral brightness should have an
extended and deep temperature minimum (below 3000 K). Better spatial resolution
as well as better wavelength coverage are needed for a more complete
determination of the chromospheric temperature stratification above sunspot
umbrae.
