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Is it possible to determine orbit orientation (in space) of an exoplanet without direct imaging or astrometry (for example with transits and rv observations only)? I have read somewhere that orbits of exoplanets in TRAPPIST-1 system have negligible mutual inclination (below one degree) but I cannot see a way how to determine that from observations.

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    $\begingroup$ Planets around TRAPPIST-1 were discovered by the transit method, which is why their orbit inclinations are known. $\endgroup$
    – James K
    Jan 7 at 14:20
  • $\begingroup$ I see but how you can determine their orbits are coplanar? $\endgroup$
    – Leos Ondra
    Jan 7 at 15:11
  • $\begingroup$ If all the planets have very different longitude of ascending node, but the same inclination =90, they might not be co-planar but this is dynamically unstable, doesn't fit with notions of solar system formation, and is statistically unlikely $\endgroup$
    – James K
    Jan 7 at 15:15
  • $\begingroup$ Ok, I understand. I assume that for direct determination of longitude of ascending node we need imaging oř astrometry? $\endgroup$
    – Leos Ondra
    Jan 7 at 15:47

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Yesish.

The transit method joined with radial velocity measurements can give quite tight constraints on the inclination of the orbits with respect to our line of sight as well as their eccentricity and orientation of the semi major axis through the rv data, amplitude and shape and phase offset, relative to the transit. Alone the fact that the planet and the star align in our line-of-sight during a transit combined with typical sizes of orbits indicates that the inclination of the plane has to be such that we look (nearly) edge-on onto the orbital plane, thus very close to 0° wrt our line-of-sight. One of the focal points of its elliptical orbit has to be near or in the usually (much-)more-massive star.

The direct imaging and radial velocity method is sensitive to close-in, massive planets. Conversely, direct imaging and astrometry are sensitive to massive, far-out planets.

Further, but beyond this question, through the transit also the relative size of the planet compared to its host star are known by assessing the amount of light blocked by the planet during transit. Through spectroscopy of the star we can determine its spectral class and its mass, and then the mass of the planet by the amplitude of its rv variations on the star,and thus finally also the planets density.

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