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In the August 2021 preprint by WeiJia Sun et al. Exploring the stellar rotation of early-type stars in the LAMOST Medium-Resolution Survey. II. Statistics there is histogram plot with the caption:

v sin i distributions of (blue) normal stars, (cyan) binary stars, (green) CP stars, (orange) periodic variables, and (red) cluster members.

Which velocity is $v$ is in the parameter of a binary's $v \sin i$?

v sin i distributions of (blue) normal stars, (cyan) binary stars, (green) CP stars, (orange) periodic variables, and (red) cluster members.

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    $\begingroup$ Could you provide where exactly this formula appears? $\endgroup$
    – Heopps
    Sep 5, 2022 at 17:50
  • $\begingroup$ For instance, here researchgate.net/figure/… . $\endgroup$
    – Elena Greg
    Sep 5, 2022 at 18:07
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    $\begingroup$ @ElenaGreg I've added your link back into your question. When replying to a comment don't forget to use the (at)replies function so that they receive a notification that you've responded. If for example you start your comment by typing @Heo you should see an autocomplete function with their ID and you can just select it.(screenshot of example) For more see How do comment @replies work? (found in FAQ) $\endgroup$
    – uhoh
    Sep 5, 2022 at 22:48
  • $\begingroup$ @Heopps info added. $\endgroup$
    – uhoh
    Sep 5, 2022 at 22:48

2 Answers 2

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That said, a common velocity observed in binary systems (or also generally multiple systems and exoplanet systems) is the radial velocity of the component(s) around the barycenter. That's the velocity projecton onto the plane directed towards the observer as derived from spectroscopic data (periodic blue shift and red shift). Yet we do not know the angle of inclination $i$ of the binary system with respect to this plane. Thus the "real" radial velocity $v$ in the observed system compared to the observed radial velocity $v_r$ equates as $v = v_r / \sin i$ or in other words we (only) know that the projection of the radial velocity of the observed component around their barycenter is $v\sin i$. Usually the astrometric motion within the image plane (thus perpendicular to the line of observation) is too small to be measurable.

We DO know the inclination reasonably well for all those binary or multiple or planetary systems where we observe periodic brightness changes (eclipsing binaries) as this means that the rotational axis is (nearly) perpendicular to our line of sight and one component passes in front of the other regularily.

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    $\begingroup$ I've moved the OP's comment back into their question, it has a specific example now. $\endgroup$
    – uhoh
    Sep 5, 2022 at 22:50
  • $\begingroup$ $v \sin i$ isn't connected with the radial velocity. $\endgroup$
    – ProfRob
    Jan 11, 2023 at 0:53
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$v \sin i$ is the parameterised amount of rotational broadening that is required to match the spectrum being considered. $v$ is the equatorial velocity of a star and $i$ is the (generally unknown) angle between the line of sight and the spin-axis of the star.

When you have a binary system then it is possible that either one or both of the stars contributes significantly to the spectrum. These are labelled as SB1 or SB2 systems respectively.

Most binary systems are SB1s. That is because even a small mass difference between the components leads to a large luminosity difference. In these cases, the $v \sin i$ value is the projected equatorial velocity of the luminous component.

In the case of an SB2 then it is quite possible that the $v\sin i$ value is meaningless, or at least requires careful interpretation, because the lines of both stars are present in the spectrum and may be displaced in velocity with respect to each other.

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