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Searching for exoplanets by measuring the change of a stars brightness when an exoplanet transits the star's disk also has the potential to detect exo-sunspots. Have they ever been detected?

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Can we see evidence of sunspots on other stars?

Strictly speaking, no. Since there is one and only one Sun in the entire universe, there is one and only one star in the universe that has sunspots. That's rather pedantic. Going beyond that, the answer is almost certainly an emphatic yes. The generic term is starspots rather than sunspots, and they have been observed on other stars. For a very nice review, see the open source, peer reviewed article on starspots by K.G. Strassmeier.

The key issue with observing starspots on other stars is that even the largest sunspot observed on our Sun would be far too small / far too dim to resolve with even the largest of telescopes. But some stars apparently do have starspots much larger and much cooler than the largest / coolest sunspot on our Sun. Other than being larger and cooler, these very large starspots on other stars exhibit behaviors analogous to the sunspots observed on our Sun.

Starspots the size of the sunspots observed on our Sun almost certainly exist, but they currently can't be detected given the limitations of current technology.

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    $\begingroup$ Thanks for the edit, @MikeG. $\endgroup$ – David Hammen Sep 13 at 23:27
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    $\begingroup$ I add a note mostly for OP. Measuring the dimming of a star and resolving what causes the dimming are two different things. In principle one might measure a fainter star without seeing a starspot. As for the transit of exoplanet. The same is in analyzing an exoplanet atmosphere. Changes are always easy detectable. $\endgroup$ – Alchimista Sep 14 at 8:58
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The evidence for cool spots on stars, analogous to sunspots is extensive and overwhelming. Everything from the modulation of the measured light from stars as the spots rotate, to spectroscopic indicators of cool regions on the surface, to Doppler imaging that provides maps of the spot distribution on the surfaces of fast-rotating stars.

However, to answer your specific question, evidence for spots is seen as small "wobbles" in the transit signal from an exoplanet as it covers and uncovers individual spots or spot groups. An example of the literature on the topic is Mancini et al. (2016). The light curve data from Kepler and now TESS has made this a growing field of research.

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Starspots can be detected indirectly, as seen here. From a talk I heard about this some years ago, the basic idea, or at least one of them, is to observe a sharp spectral line very precisely over time, measuring the exact "shape" of the line -- how much of the light was shifted to slightly higher or lower frequencies -- as it varied over time. A starspot slightly reduces the amount of light at higher frequencies when its on the part of the star rotating towards us, and slightly reduces the amount at lower frequencies when it's on the part rotating away. This signal is very faint, but has a precisely predictable structure and is periodic, so on rapidly rotating stars it can be detected among the noise with a lot of care.

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    $\begingroup$ That blogpost actually links to [Mapping starspots and magnetic fields on cool stars ](star-www.st-and.ac.uk/~acc4/coolpages/imaging.html) which seems to be a pretty cool technique, and probably deserves some specific elaboration. $\endgroup$ – uhoh Sep 16 at 8:51

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