  We present the results of an 80 ks Chandra ACIS-S observation of the double
pulsar system J0737-3039. Furthermore, we report on spectral, spatial and
timing analysis of the combined X-ray observations performed so far for this
system. Fitting a total of ~1100 photons, we show that the X-ray spectrum of
the J0737-3039 system is very soft, and not satisfactorily modeled by a simple
blackbody or an atmospheric model. However, it is not possible yet to
discriminate between a predominantly non-thermal and a predominantly thermal
origin for the X-ray emission. Adopting a simple power-law emission model, the
photon index (Gamma=3.7) and the implied conversion efficiency of the
rotational energy of PSR J0737-3039A into X-ray emission (4.1x10^-4, for a
distance to the source of 500 pc) are compatible with the X-ray photons being
emitted in the magnetosphere of PSR J0737-3039A. This hypothesis is also
supported by the absence of detectable X-ray orbital modulation (up to ~20%) or
any X-ray nebular emission and it is in agreement with the high (~75%) X-ray
pulsed fraction of PSR J0737-3039A. A two blackbody or a Comptonized blackbody
model also reproduce the data, and the upper limit to the value of the hydrogen
column density N_H <10^20 cm^-2, is in better agreement (with respect to the
power-law model) with the Galactic N_H in that direction and at that distance.
For the two blackbody model the implied emission radii and temperatures are
also compatible with those seen in other recycled pulsars, calling for the bulk
of the X-ray photons being originated from heated regions at the surface of
pulsar A. On the other hand, in the Comptonized blackbody model, the electron
temperature seems to be significantly smaller than in other similar objects.
