We present radio observations of the dM5.5e fare star Proxima Centauri
at 20, 13, 6, and 3.5 cm. The star was only detected during an impuls
ive, highly circularly polarized, and apparently narrow-band flare at
20 cm, similar to those seen on other dMe flare stars. This flare was
detected in just similar to 1.7 hr of observing time spanning an simil
ar to 12 hr period, suggesting that Proxima Cen may be a prolific prod
ucer of coherent bursts at 20 cm. On the other hand, despite similar t
o 30 hr of observing time at both 6 and 3.5 cm over a 4 yr period, the
star was not detected as either a flaring or a quasi-steady (quiescen
t) source at these wavelengths. We place an upper limit of similar to
2 x 10(11) ergs Hz(-1) s(-1) on its radio luminosity at 6 and 3.5 cm,
the lowest detection threshold yet reached for a star other than the S
un. This upper limit is approximately equal to the radio luminosity of
the active Sun (i.e., at or close to the peak on its activity cycle)
outside of flares. Our results place important constraints on the fill
ing factor of similar to 500-1000 G magnetic loops containing X-ray-em
itting plasma on Proxima Cen. Because such loops should be optically t
hick to gyroresonance emission at centimeter wavelengths, their fillin
g factor can be inferred directly from the measured stellar radio flux
density. Our radio results imply that loops at temperatures similar t
o 2 x 10(7) K, representative of the hot stellar X-ray component, have
a filling factor of similar to 13%. Loops at temperatures similar to
3 x 10(6) K, similar in temperature to the nonflaring solar active reg
ion corona, have a filling factor of less than or equal to 88%. Our re
sults are compatible with present empirical relationships for the magn
etic field parameters of late-type dwarf stars as applied to Proxima C
en. Based on its measured rotation period (P-rot approximate to 41 day
s) and the ratio of its soft X-ray to bolometric luminosity (L(x)/L(bo
l) x 2.4 x 10(-4)), these relationships predict that Proxima Cen shoul
d be about an order of magnitude below the saturation limit in magneti
c activity, where the entire surface of stars is thought to be covered
by kilogauss X-ray loops. We compare our results with the contrasting
case of UV Ceti, a dM5.5e flare star that according to the same empir
ical relationships should be approximately as magnetically active as P
roxima Cen. UV Ceti, however, displays quiescent radio emission with a
Luminosity that is more than an order of magnitude higher than the up
per limit placed on Proxima Cen. Our radio observations place an upper
limit of similar to 7 x 10(-12) M. yr(-1) on the mass-loss rate by an
y stellar wind (assumed to have a velocity of 300 km s(-1)) from Proxi
ma Cen. This upper limit is almost 2 orders of magnitude lower than th
at inferred by Mullan et al. from millimeter wavelength observations o
f other dMe flare stars. We show that the high mass-loss rate inferred
by Mullan et al. is untenable if our present understanding of the cen
timeter wavelength radio emission of dMe flare stars is correct.