Gas has been detected in many exoplanetary systems ($>$10 Myr), thought to be released in the destruction of volatile-rich planetesimals orbiting in exo-Kuiper belts. In this letter, we aim to explore whether gas is also expected in the Kuiper belt (KB) in our Solar System. To quantify the gas release in our Solar System, we use models for gas release that have been applied to extrasolar planetary systems, as well as a physical model that accounts for gas released due to the progressive internal warming of large planetesimals. We find that only bodies larger than about 4 km can still contain CO ice after 4.6 Gyr of evolution. This finding may provide a clue as to why Jupiter-family comets, thought to originate in the Kuiper belt, are deficient in CO compared to Oort-clouds comets. We predict that gas is still produced in the KB right now at a rate of $2 times 10^{-8}$ M$_oplus$/Myr for CO and orders of magnitude more when the Sun was younger. Once released, the gas is quickly pushed out by the Solar wind. Therefore, we predict a gas wind in our Solar System starting at the KB location and extending far beyond with regards to the heliosphere with a current total CO mass of $sim 2 times 10^{-12}$ M$_oplus$. We also predict the existence of a slightly more massive atomic gas wind made of carbon and oxygen (neutral and ionized) with a mass of $sim 10^{-11}$ M$_oplus$. We predict that gas is currently present in our Solar System beyond the Kuiper belt and that although it cannot be detected with current instrumentation, it could be observed in the future with an in situ mission using an instrument similar to Alice on New Horizons with larger detectors. Our model of gas release due to slow heating may also work for exoplanetary systems and provide the first real physical mechanism for the gas observations.
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