<p><span class="math-container">$v \sin i$</span> is the parameterised amount of rotational broadening that is required to match the spectrum being considered. <span class="math-container">$v$</span> is the equatorial velocity of a star and <span class="math-container">$i$</span> is the (generally unknown) angle between the line of sight and the spin-axis of the star.</p>
<p>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.</p>
<p>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 <span class="math-container">$v \sin i$</span> value is the projected equatorial velocity of the luminous component.</p>
<p>In the case of an SB2 then it is quite possible that the <span class="math-container">$v\sin i$</span> 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.</p>

CC BY-SA 4.0
2022-09-21T21:25:54.793Z
2023-01-11T00:52:37.777Z
2023-01-11T00:52:37.777Z
ProfRob
<p>That said, a common velocity observed in binary systems (or also generally multiple systems and exoplanet systems) is the <a href="https://en.wikipedia.org/wiki/Radial_velocity" rel="nofollow noreferrer">radial velocity</a> 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 <span class="math-container">$i$</span> of the binary system with respect to this plane. Thus the &quot;real&quot; radial velocity <span class="math-container">$v$</span> in the observed system compared to the observed radial velocity <span class="math-container">$v_r$</span> equates as <span class="math-container">$v = v_r / \sin i$</span> or in other words we (only) know that the projection of the radial velocity of the observed component around their barycenter is <span class="math-container">$v\sin i$</span>. Usually the astrometric motion within the image plane (thus perpendicular to the line of observation) is too small to be measurable.</p>
<p>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.</p>

CC BY-SA 4.0
2022-09-05T22:50:45.587Z
I've moved the OP's comment back into their question, it has a specific example now.
uhoh
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2023-01-11T00:53:54.813Z
$v \sin i$ isn't connected with the radial velocity.
ProfRob
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2022-09-05T18:12:08.300Z
2022-09-06T22:58:30.390Z
2022-09-06T22:58:30.390Z
planetmaker
<p>In the August 2021 preprint by WeiJia Sun et al. <a href="https://www.researchgate.net/publication/353678217_Exploring_the_stellar_rotation_of_early-type_stars_in_the_LAMOST_Medium-Resolution_Survey_II_Statistics" rel="nofollow noreferrer">Exploring the stellar rotation of early-type stars in the LAMOST Medium-Resolution Survey. II. Statistics</a> there is histogram plot with the caption:</p>
<blockquote>
<p>v sin i distributions of (blue) normal stars, (cyan) binary stars, (green) CP stars, (orange) periodic variables, and (red) cluster members.</p>
</blockquote>
<p>Which velocity is <span class="math-container">$v$</span> is in the parameter of a binary's <span class="math-container">$v \sin i$</span>?</p>
<p><a href="https://i.sstatic.net/HfE69.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/HfE69m.png" alt="v sin i distributions of (blue) normal stars, (cyan) binary stars, (green) CP stars, (orange) periodic variables, and (red) cluster members." /></a></p>

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2022-09-05T17:50:31.897Z
Could you provide where exactly this formula appears?
Heopps
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2022-09-05T18:07:17.807Z
For instance, here https://www.researchgate.net/figure/v-sin-i-distributions-of-blue-normal-stars-cyan-binary-stars-green-CP-stars_fig1_353678217 .
Elena Greg
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2022-09-05T22:48:31.713Z
@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](https://i.sstatic.net/VSkk1.png) of example)  For more see [How do comment @replies work?](https://meta.stackexchange.com/q/43019/303080) (found in [FAQ](https://meta.stackexchange.com/q/7931/303080))
uhoh
CC BY-SA 4.0
2022-09-05T22:48:53.620Z
@Heopps info added.
uhoh
<observational-astronomy><rotation><binary-star><eclipsing-star-systems>
What is the v sin i parameter?
Elena Greg
# What is the v sin i parameter?

(Asked by: username_0 on 2022-09-05)

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](https://www.researchgate.net/publication/353678217_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.](https://i.sstatic.net/HfE69m.png)](https://i.sstatic.net/HfE69.png)

**Tags:** observational-astronomy, rotation, binary-star, eclipsing-star-systems

> **Comments:**
> **username_1**: Could you provide where exactly this formula appears?  
> **username_0**: For instance, here https://www.researchgate.net/figure/v-sin-i-distributions-of-blue-normal-stars-cyan-binary-stars-green-CP-stars_fig1_353678217 .  
> **username_2**: @username_0 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](https://i.sstatic.net/VSkk1.png) of example)  For more see [How do comment @replies work?](https://meta.stackexchange.com/q/43019/303080) (found in [FAQ](https://meta.stackexchange.com/q/7931/303080))  
> **username_2**: @username_1 info added.  

---

## 2 answers

### Answer by username_4 on 2022-09-05. Score: 5 | Accepted answer

That said, a common velocity observed in binary systems (or also generally multiple systems and exoplanet systems) is the [radial velocity](https://en.wikipedia.org/wiki/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.

> **Comments:**
> **username_2**: I've moved the OP's comment back into their question, it has a specific example now.  
> **username_3**: $v \sin i$ isn't connected with the radial velocity.  

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### Answer by username_3 on 2022-09-21. Score: 1

$v \sin i$ 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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