Context. This is the second paper in a series of papers showing the results of extrasolar planet population synthesis calculations using our extended core accretion model. In the companion paper (PaperI), we presented in detail the methods we use. In subsequent papers, we shall discuss the effect of the host star's mass on the planetary population and the influence of various properties of protoplanetary disks. Aims. In this second paper, we focus on planets orbiting solar-like stars. The goal is to use the main characteristics of the actually observed extrasolar planet population to derive in a statistical manner constraints on the planet formation models. Methods. Drawing initial conditions for our models at random from probability distributions derived as closely as possible from observations, we synthesize a number of planetary populations. By applying an observational detection bias appropriate for radial velocity surveys, we identify the potentially detectable synthetic planets. The properties of these planets are compared in quantitative statistical tests with the properties of a carefully selected sub-population of actually observed extrasolar planets. Results. We use a two dimensional Kolmogorov-Smirnov test to compare the mass-distance distributions of synthetic and observed planets, as well as the one dimensional version of the test to compare the ,the semimajor axis and the [Fe/H]distribution. We find that while many combinations of parameters lead to unacceptable distributions, a number of models can account to a reasonable degree of statistical significance for most of the properties of the observed sample. We concurrently account for many other observed features, e.g. the ``metallicity effect''. This gives us confidence that our model captures several essential features of giant planet formation. In addition, the fact that many parameter combinations could be rejected indicates that planet population synthesis is indeed a promising approach to constrain formation models. Our simulations allow us also to extract a number of properties of the underlying exoplanet population that are not yet directly detectable. For example, we have derived the planetary initial mass function (PIMF) and have been led to conclude that the planets detected so far represent only the tip of the iceberg(9%) of all the existing planets. The PIMF can also be used to predict how the detectable extrasolar planet population will change as the instrumental precision of radial velocity surveys improves from 10m/s to 1m/s, or even to an extreme precision of 0.1m/s. Key words: planetary systems - planetary systems: formation - planetary systems: protoplanetary disks - planets and satellites: formation - solar system: formation
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