I know by observing the dimness of a star it is possible to calculate an exoplanet's distance from the star and its mass by how much the star wobbles. However, is there any way to accurately determine if said planet has any magnetic field through observation alone directly?


There are three approaches with which people have looked for this, and not yet been too successful:

  • A transiting planet planet with a strong magnetic dipole and/or very strong host star winds might produce a visible signal when the magnetospheric bow-shock passes in front of the star as well. The idea is that at the bow-shock the streaming hydrogen would be heated up, ionize and produce a detectable ultraviolet photometry signal upon recombination at a characteristic distance (the stand-off distance) from the planet.
    HD189733b was for a time the first planet to have measured its magnetic field in that way, however the discovery was retracted because the ultraviolet opacities that the group used to determine the bow-shock position were wrong.
    It turns out there is no signal from the bow-shock.

  • The second idea is to look at a hot Jupiter that has an evaporating atmosphere. The shape of the ionized hydrogen tail in the magnetosphere will be influenced by the strength of the planetary dipole. But then, the amount of hydrogen that is in the planet's shadow is also a function of the magnetic field strength.
    Hydrogen that comes into the planet's shadow will recombine with magnetospheric electrons due to the lack of direct irradiation from the star and produce a signal in the Lyman-$\alpha$ recombination line.
    Going further, only the particles that stream away from the planet, and towards us will be part of this process. Thus the blue-shifted side of the Lyman-$\alpha$ line can inform us about the strength of the magnetic field.
    So far the theory. The practice is that our last remaining UV-spectrometer in space that could perform such a thing (mounted on the Hubble) is even only barely capable of observing this process. But it has been done for the planet HD209458b The data quality is lousy, and the interpretation that has been done in Kislyakova et al. 2015 has more parameter than datapoints and is therefore questionable.
    But if you choose to believe the authors, then this planet has a magnetic moment of $0.1$ Jupiter magnetic moments.

  • The third possibility that could yield real results are radio observations with the LOFAR array. LOFAR observes at the edge of observable frequencies from Earth (at very low frequencies the ionosphere interferes with radio observations) at the for the moment highest achievable spatial resolution. The idea here is to look for analogues of Jupiters decametric radiation, which is simply synchrotron radiation produced when ionized particles start gyrating around a magnetic field line.
    This already has been tried at higher frequencies / different observatories with the planets around $\epsilon$ Andromedae and $\tau$ Bootes, as yet to no avail.


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