Context. The boundary conditions of the heliosphere are set by the ionization, density and composition of inflowing interstellar matter. Aims. Constraining the properties of the Local Interstellar Cloud (LIC) at the heliosphere requires radiative transfer ionization models. Methods. We model the background interstellar radiation field using observed stellar FUV and EUV emission and the diffuse soft X-ray background. We also model the emission from the boundary between the LIC and the hot Local Bubble plasma, assuming that the cloud is evaporating because of thermal conduction. We create a grid of models covering a plausible range of LIC and Local Bubble properties, and use the modeled radiation field as input to radiative transfer/thermal equilibrium calculations using the Cloudy code. Data from in situ observations of He 0, pickup ions and anomalous cosmic rays in the heliosphere, and absorption line measurements towards ɛ CMa were used to constrain the input parameters. Results. A restricted range of assumed LIC H i column densities and Local Bubble plasma temperatures produce models that match all the observational constraints. The relative weakness of the constraints on N(H i) and Th contrast with the narrow limits predicted for the H 0 and electron density in the LIC at the Sun, n(H 0) = 0.19 − 0.20 cm −3, and ne = 0.07 ± 0.01 cm −3. Derived abundances are mostly typical for low density gas, with sub-solar Mg, Si and Fe, possibly subsolar O and N, and S about solar; however C is supersolar. Conclusions. The interstellar gas at the Sun is warm, low density, and partially ionized, with n(H) ≈ 0.23 − 0.27 cm −3, T ≈ 6300 K,
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