The stellar wind cycles and planetary radio emission of the τ Boo system

τBoo is an intriguing planet-host star that is believed to undergo magnetic cycles similar to the Sun, but with a duration that is about one order of magnitude smaller than that of the solar cycle. With the use of observationally derived surface magnetic field maps, we simulate the magnetic stellar wind of τBoo by means of three-dimensional magnetohydrodynamics numerical simulations. As the properties of the stellar wind depend on the particular characteristics of the stellar magnetic field, we show that the wind varies during the observed epochs of the cycle. Although the mass-loss rates we find (~2.7 × 10^-12M_⊙yr^-1) vary less than 3per cent during the observed epochs of the cycle, our derived angular-momentum-loss rates vary from 1.1 to 2.2 × 10³²erg. The spin-down times associated with magnetic braking range between 39 and 78Gyr. We also compute the emission measure from the (quiescent) closed corona and show that it remains approximately constant through these epochs at a value of ~10^50.6cm^-3. This suggests that a magnetic cycle of τBoo may not be detected by X-ray observations. We further investigate the interaction between the stellar wind and the planet by estimating radio emission from the hot Jupiter that orbits at 0.0462au from τBoo. By adopting reasonable hypotheses, we show that, for a planet with a magnetic field similar to Jupiter (~14G at the pole), the radio flux is estimated to be about 0.5-1mJy, occurring at a frequency of 34MHz. If the planet is less magnetized (field strengths roughly smaller than 4G), detection of radio emission from the ground is unfeasible due to the Earth's ionospheric cut-off. According to our estimates, if the planet is more magnetized than that and provided the emission beam crosses the observer line-of-sight, detection of radio emission from τBoob is only possible by ground-based instruments with a noise level of ≲1mJy, operating at low frequencies.