# Derivation of the orbital parameters of a exoplanet discovered by direct imaging

I wonder which orbital elements of a exoplanet can be found only applying direct imaging. I know that the angular separation combined with parallax can give us the semi-major axis, but what about the other elements? Is it possible? And what about the mass or the radius of the planet (given known mass and radius of the star)?

All of the orbital elements can be derived by direct imaging, so long as you observe long enough to see a fair fraction of the orbit.

Kepler's third law then gives the total mass of the system and the relative orbits may give the mass ratio and hence individual masses (though this would be quite uncertain if the mass ratio was large).

Absolute masses could also come from dynamical measurements (i.e. velocities), but you can get these if the system is at a known distance. Again, because the star wouldn't move very much, this would likely lead to a highly uncertain planetary mass.

The planet mass might instead be estimated from its luminosity, an estimate of its age, and a cooling model. The radius can be estimated from its temperature and luminosity.

• Strictly speaking, direct imaging gives you the orbital elements with a degeneracy corresponding to mirroring the orbit in the sky-plane, so you end up with two possible values of the inclination corresponding to these mirror images. To resolve that degeneracy you need something sensitive to motion in the radial direction, e.g. RV. – antispinwards Jan 4 at 21:45
• Yeah I mean only with direct imaging, not using radial velocities nor astrometry or spectroscopy – Carlos Vázquez Monzón Jan 4 at 22:23
• @antispinwards True. Though the degeneracy would not affect any mass estimate. – Rob Jeffries Jan 5 at 0:07
• @uhoh that we're talking about exoplanets? – Rob Jeffries Jan 5 at 9:20
• Ah! yes that's what I missed. – uhoh Jan 5 at 11:47