  Recent low-frequency radio observations suggest that some nearby M dwarfs
could be interacting magnetically with undetected close-in planets, powering
the emission via the electron cyclotron maser (ECM) instability. Confirmation
of such a scenario could reveal the presence of close-in planets around M
dwarfs, which are typically difficult to detect via other methods. ECM emission
is beamed, and is generally only visible for brief windows depending on the
underlying system geometry. Due to this, detection may be favoured at certain
orbital phases, or from systems with specific geometric configurations. In this
work, we develop a geometric model to explore these two ideas. Our model
produces the visibility of the induced emission as a function of time, based on
a set of key parameters that characterise magnetic star-planet interactions.
Utilising our model, we find that the orbital phases where emission appears are
highly dependent on the underlying parameters, and does not generally appear at
the quadrature points in the orbit as is seen for the Jupiter-Io interaction.
Then using non-informative priors on the system geometry, we show that
untargeted radio surveys are biased towards detecting emission from systems
with planets in near face-on orbits. While transiting exoplanets are still
likely to be detectable, they are less likely to be seen than those in near
face-on orbits. Our forward model serves to be a powerful tool for both
interpreting and appropriately scheduling radio observations of exoplanetary
systems, as well as inverting the system geometry from observations.
