Neutron stars do not shine like "normal" stars do: isolated neutron stars are visible only because 1) they are born very hot in the SN event which generates them or 2) they slowly lose their rotational energy through radiation processes connected with the superstrong magnetic field co-rotating with the star. Both such thermal and non-thermal processes contribute to neutron star luminosities in X-rays, especially in the .1-to-10 kev energy region. It is precisely for this two-decade range that the combination of the ESA XMM/Newton optics and its EPIC focal plane detector was optimised. We present two case studies of isolated neutron stars observed with XMM-Newton. The two objects are Geminga and 1E1207.4-5209,both were previously known to be intrinsically unique, and the XMM-Newton observations have enriched substantially their phenomenology. Geminga was discovered in 1973 by NASA's SAS-2 satellite as an unidentified, rather ill-defined gamma-ray source in the galactic anticentre. The object is not visible in radio, very faint in optical, easily seen in X-rays and has by far the majority of its energy output in > 50 MeV gamma-rays(see Bignami and Caraveo , 1996 Ann.Rev. Astron. Astrophys. 34,331). An XMM-Newton 100 ksec exposure revealed an extended x-ray feature trailing Geminga with a symmetric structure, consisting of two tails, ~2 arcmin long, or ~0.1 pc at 160 pc. The tails are aligned and symmetric with respect to the pulsar proper motion vector. No diffuse emission is detected in the region between the tails, and their shape gives the impression of the projected view, in the plane of the sky, of an empty paraboloid or cone of X-ray emission. We interpret such elongated structure as a velocity driven bow-shock originated by the highly supersonic motion of Geminga in the interstellar medium. It is the first time that a similar faint structure is seen in X-ray and it allows to estimate the inclination of the motion of Geminga with respect to the plane of the sky, the energy of the electrons accelerated by Geminga and the density of the local interstellar medium ( Caraveo et al, 2003 Science, 301,1345).1E1207.4-5209 is a radio quiet isolated neutron star at the center of a well defined supernova remnant. The discovery of deep spectral features in its X-ray spectrum made 1E1207.4-5209 a unique case, since no other INS has shown significant features in its X-ray continuum. On August 2002, XMM-Newton performed a 260 ksec observation which yielded 360,000 photons from 1E1207.4-5209.Analysis of this long observation, while confirming the two phase-dependent absorption lines at 0.7 and 1.4 keV, unveiled a statistically significant third line at ~2.1 keV, as well as a possible fourth feature at 2.8 keV. The nearly 1:2:3:4 ratio of the line centroids, as well as the phase variation, naturally following the pulsar B-field rotation, strongly suggest that such lines are due to cyclotron resonance scattering (Bignami et al.2003 Nature, 423,725; De Luca et al, 2004 Astro Astrophys, 418,625). If the 0.7 keV line is the fundamental cyclotron frequency, then B ~6 1010(1+z) ~8 1010 G for electrons, assuming a "standard" 20% gravitational redshift. For protons, the field should be a factor mp/me higher, i.e. close to 2 1014 G, i.e. a magnetar type magnetic field. Neither of these values agrees with the B field inferred from the 1E1207.4-5209 timing parameters, which give 2-3 1012G under the rotating dipole hypothesis.
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