The above formulation in principle applies to any planet with a Dungey cycle-type stellar wind–magnetosphere interaction, and we thus consider here the appropriate parameters for exoplanets orbiting at arbitrary distances, with an emphasis on close-orbiting hot Jupiters. As discussed above, whereas for the RBL the radio powers are computed as functions of incident kinetic or Poynting flux, in our model, the powers are principally functions of the motional electric field of the solar wind, the dynamic pressure of the stellar wind and the Pedersen conductance of the ionosphere, all of which are dependent on further stellar and planetary parameters as described below. We examine results for both a solar-like stellar wind, and that representative of a young Sun-like star with high-mass-loss rate and magnetic field strength relative to the Sun. Considering first the Sun-like stellar wind, the relevant parameters are shown in Fig. 4 versus radial distance d normalized by the solar radius Rs [we truncate the inner radial distance of the plot at 2 $\mathrm{{\rm {\it R}}_s}$, being the canonical location of the heliospheric magnetic field ‘source surface’ (Owens & Forsyth 2013)]. Absolute distances in au are shown on the top axis for information, although we recognize that in reality the conversion from $\mathrm{{\rm {\it R}}_s}$ to au depends on the individual star. Specifically, Fig. 4(a) shows with the solid line the incident velocity of the solar wind on the magnetosphere vm, which is a function both of the stellar wind speed and the planet's orbital speed. For simplicity, we employ Parker's isothermal solution for the stellar wind speed vsw (Parker 1958), which is fully parametrized by the sound speed cs, and which, as shown by Cranmer (2004) has the closed-form solution

(18)

\begin{equation}
{v_{{\rm sw}}}^2= \left\lbrace \begin{array}{@{}l@{\quad }l@{}}-v_{\rm c}^2W_0[-D(d)] &amp; {\rm if } d \le d_{\rm c} ,\\
-v_{\rm c}^2W_{-1}[-D(d)] &amp; {\rm if } d \ge d_{\rm c} ,\\
\end{array}\right.
\end{equation}

where W0 and W−1 are branches of the Lambert W function, dc is the critical distance at which vsw passes through the sound speed cs, given by

(19)

\begin{equation}
d_{\rm c}=\frac{GM_{\rm s}}{2c_{\rm s}^2},
\end{equation}

where Ms = 1.9891 × 1030 kg is the solar mass and D(d) is given by

(20)

\begin{equation}
D(d)=\left(\frac{d}{d_{\rm c}}\right)^{-4}\exp \left[4 \left(1-\frac{d_{\rm c}}{d}\right)-1\right].
\end{equation}

For the Sun-like wind, we employ a sound speed cs = 130 km s−1 (which, for a Sun-like average particle mass of 1.92 × 10−27 kg corresponds to a temperature of ∼1.18 MK, though note for the present Sun calculation we actually make no assumptions in this regard), yielding a velocity at 1 au of ∼480 km s−1, consistent with observations, and ∼50–200 km s−1 in the hot Jupiter region of 3–10 Rs (indicated by the grey region). The dotted line indicates the Keplerian speed of a planet in a circular orbit vorb, and the solid line is the sum in quadrature of the two, giving the resultant incident stellar wind speed vm. Note that the two speeds are comparable in the inner region associated with hot Jupiters, and although this will modify the orientation of the magnetosphere with respect to the radial vector, it will not significantly alter the magnetospheric dynamics. We further show with the loosely dotted and dot–dashed lines the (constant) sound speed cs and the Alfvén speed vA given by equation (5). It is evident that the interaction is everywhere supersonic (modestly so in the hot Jupiter region, with a Mach number of ∼2) but becomes sub-Alfvénic inside of ∼15 $\mathrm{{\rm {\it R}}_S}$, such that Alfvén wings will form along the IMF field lines, as discussed by Saur et al. (2013), effectively shielding the stellar wind motional electric field and is related to KR saturation of the convection potential.
Citation: Parker 1958
Citation function: Uses; Uses