An observationally based wind model contemporaneous with the radio detections in $τ$ Boötis

TL;DR

This work presents a contemporaneous magnetic map of τ Boötis A and a 3D Alfvén-wave–driven wind model to evaluate the space environment at the hot Jupiter τ Boötis Ab during LOFAR detections. By combining an unresolved Bode’s-law approach with a resolved planet-centered MHD model, the authors quantify kinetic and magnetic wind power and the auroral Poynting flux that could power exoplanetary radio emissions. They find that, under best-case conditions, the magnetic-energy channel can approach the observed bursty signal only if the stellar surface magnetic field is scaled by ~10; kinetic-energy transfer is insufficient to explain the detection in all scenarios. The resolved model reveals that sub-Alfvénic wind regimes reduce energy transfer and that emission likely originates from regions along downstream field lines with favorable plasma conditions, underscoring the importance of contemporaneous magnetism measurements and wind modeling for star–planet interaction studies.

Abstract

Recent low-frequency array (LOFAR) radio signal detections bearing from the $τ$ Boötis system have been cautiously attributed to auroral emissions from the hot Jupiter $τ$ Boötis Ab. The auroral emissions are believed to be excited by interaction between the exoplanet and the winds of its host star. Since stellar winds respond to stellar surface magnetism, three-dimensional stellar wind modelling, able to account for the star's contemporaneous magnetic field geometry, can aid the interpretation of radio detections. For the first time, we present spectropolarimetric observations of $τ$ Boötis A from the same epoch as the LOFAR detections. We derive a contemporaneous large-scale magnetic map of $τ$ Boötis A, which shows a poloidally dominated field with mean strength 1.6 G. From our magnetic map, we create a three-dimensional numerical wind model and characterise the wind properties around $τ$ Boötis Ab. To compute the wind power dissipated in $τ$ Boötis Ab's magnetosphere, we apply two approaches: A) the solar system-based empirical relation called Bode's law; and B) a resolved numerical model of the planetary magnetosphere. When consistently applying best-case assumptions, we redict radio flux densities around 50 mJy and 0.68 mJy respectively. Our values are much too small to be consistent with the reported observation of $890_{-500}^{+690}$ mJy; a stellar surface magnetic field scaling $\gtrsim 10$ is required to reproduce the observed signal strength. As $τ$ Boötis A has a rapid magnetic cycle, we speculate that wind variations cased by variation in stellar magnetism may explain the lack of detections from follow-up observations. Our work emphasises the importance of contemporaneous observations of stellar magnetism and observational signatures of star-planet interaction.

Figures (11)

• Figure 1: Time and orbital phase of the LOFAR observations of 2021AA...645A..59T and the spectropolarimetric observations presented in this work. For each observation, the orbital configuration can be read of the $x$-axis and the date can be read of the $y$-axis. The orbital configuration is based on the model of 2012Natur.486..502B. The planetary configuration is shown as a grey line. The phase is zero at inferior conjunction, which is when $\tau$ Boötis Ab is at its closest distance to the observer. The times of spectropolarimetric observations are indicated with blue dots (these times are also given in Table \ref{['tab:obs']}). Out of the the LOFAR observations made by 2021AA...645A..59T, shown as coloured line segments, the 'bursty' L569131 signal is contemporaneous with the magnetic observations while the 'slowly varying' L570725 signal was observed about two weeks later. The bursty signal was observed near quadrature while the slowly varying signal was observed near the inferior conjunction of $\tau$ Boötis Ab.
• Figure 2: Radial surface magnetic field of $\tau$ Boötis A based on spectropolarimetric observations conducted contemporaneously with the detected LOFAR radio signal. The rotation axis of $\tau$ Boötis A is inclined by about 40° to the line of sight, hence stellar latitudes below -40° are minimised by the ZDI regularisation. The radial field strength ranges from -3.385.85 and the mean value $\langle|B_r|\rangle = \qty{1.21}{\gauss}$. The location of the minimum and maximum values are indicated by a white cross and a white circle. The subplanetary point on the stellar surface is indicated by a black dot. The dashed square around the subplanetary position indicates the longitude and latitude range that will be used for computing quantities at the position of the planet. The phase of the spectropolarimetric observations are indicated by the short tick marks near the longitude axis. Note that the full vector magnetogram is shown in Fig. \ref{['fig:vector-magnetogram']}.
• Figure 3: Alfvén characteristic flow lines $\boldsymbol{u} \pm \boldsymbol{u}_\textsc{a}$ that intersect the planetary magnetosphere, represented by a sphere of radius $R_\text{m}$. The purple and blue curves correspond to flow lines that originate at the the planetary magnetosphere, while the yellow and grey curves correspond to flow lines that terminate at the planetary magnetosphere. Alfvén wave energy thus flows away from the planet along the blue/purple curves, and towards the planet along the yellow/grey curves. A graticule (white) has been added to help visualise the orbital plane and its inclination to the observer. The equatorial plane is coloured by the smallest value of the radial component of $\boldsymbol{u} \pm \boldsymbol{u}_\textsc{a}$. The planetary orbit is indicated as a black ellipse. The intersection of the Alfvén surface and the planetary orbital plane is shown as a green curve.
• Figure 4: Radio flux density variations as a function of planetary position. For the radio-kinetic (top) and radio-magnetic (bottom) mechanisms, the variations shown are computed on a spherical shell of radius equal to the orbital distance of $\tau$ Boötis Ab. In each panel, the values have been normalised to the computed radio flux density at the estimated planetary position. The stellar longitude and latitude values indicate the position of the substellar point. Wind speeds are calculated in the planetary frame. In every point the planet is assumed to be travelling in the direction of increasing longitude (this is used for the calculation of $v$). The black circle indicates the position of the planet, and the black dashed square indicates the area over which the averages in Table \ref{['tab:results']} are calculated.
• Figure 5: Plane $xz$ cross sections of the magnetosphere of $\tau$ Boötis Ab in our resolved three-dimensional model of the planetary magnetosphere (method B). The background colours show the plasma number density $\text{H}^{+}/\text{m}^3$ according to the panel's top colourbar. Velocity vectors and are indicated by arrows coloured by magnitude according to the right colourbar. Black and magenta lines depict magnetic field line projections into the $xz$-plane. The magnetic dipole moment and rotation vector of the planet are parallel to $z$. The solid dark violet line indicates the equality between the electron plasma frequency and gyrofrequency, $f_\text{pe}/f_\text{ce}=1$. Radio emission originating in a region with $f_\text{pe}/f_\text{ce}<1$, i.e., inside the closed violet curve, may escape the planet's vicinity due to the gyrofrequency exceeding the local plasma frequency. Alfvén wings develop in the subalfvénic cases B10 and B100. A bow shock forms in the superalfvénic scenario SA10. In the transalfvénic scenario B1 neither Alfvén wings nor a bow shock form; the discontinuity visible at $-7.5\,R_\textsc{p}$ is the upstream magnetopause.