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Is there any way through which we could determine the ROTATION period of an exoplanet based on the data we observed?(from transit)

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Do you mean the time it takes for the exoplanet to rotate around its axis? The transit method is used to detect the presence of an exoplanet as it passes in front of the star (as seen from Earth). After several transits you can get the orbital period, i. e. the time it takes for the planet to orbit around the star.

There are methods to measure the rotation period of astronomical objects (exoplanets included). You can use the Doppler effect to measure the change of wavelengths of the planet's light as it rotates. This technique has been used to measure the rotation period of exoplanet Beta Pictoris b.

Suppose the planet (as seen from our point of view) rotates from right to left. If so the right-hand edge of the planet's disc will be moving towards us, while the left-hand edge will be moving away from us. Therefore, the wavelength of the light coming from the right will be slightly shorter (i. e. it will be blue-shifted) than that coming from the left (which will be red-shifted). The greater the difference in wavelengths, the faster the planet is rotating. The method cited in the article above seems to be a bit more complicated, but that's the basics of it.

Alternatively, if you can image the planet with enough resolution, you might be able to spot different terrains or patterns on its surface, and by measuring the time it takes for those patterns to appear again in the same apparent position you might be able to calculate the rotation period. But this is probably unfeasible for most exoplanets, at least with the instruments we have right now.

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    $\begingroup$ The easiest targets, super-Jupiters around faint stars, are already having their (atmospheric) rotation measured by direct imaging, e.g. 2M1207b. But Venus' atmosphere rotates what, about 60 times(?) faster than its solid ground. A problem for doppler measurements as well. $\endgroup$ – LocalFluff Jan 6 '17 at 14:27
  • $\begingroup$ OK , thank you. I was searching for an example and I didn't find any.( like Beta Pictoris b) $\endgroup$ – Andrei Cosmin Jan 6 '17 at 15:00

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