<p>The relative position between the components of the binary moves on a circular orbit with velocity <span class="math-container">$v_{\mathrm{circ}}=\sqrt{G(m_1+m_2)/r}$</span>, where <span class="math-container">$r=1459$</span>AU is the distance, <span class="math-container">$G$</span> is Newton's constant of gravity and <span class="math-container">$m_{1,2}$</span> are the masses of the components.</p>
<p>Thus, the radial velocity semiamplitude of the secondary is
<span class="math-container">\begin{align}
v_{\mathrm{rad},2} &amp;= \frac{m_{1}\sin i}{m_1+m_2} v_{\mathrm{circ}} =  m_{2,1}\sin i \sqrt{\frac{G}{r(m_1+m_2)}},
\end{align}</span>
where <span class="math-container">$i$</span> is the inclination of the binary orbit's angular momentum vector w.r.t. the line of sight (i.e. <span class="math-container">$i=90^\circ$</span> if the line of sight lies in the orbital plane). For the primary, the amplitude is smaller by a factor <span class="math-container">$m_2/m_1$</span>.</p>
<p>If <span class="math-container">$m_1\gg m_2$</span> (as is the case here, where the uncertainty in <span class="math-container">$m_1$</span> presumably exceeds <span class="math-container">$m_2$</span>), the above simplifies to
<span class="math-container">\begin{align}
\frac{v_{\mathrm{rad},2}}{\sin i} &amp;= \sqrt{\frac{Gm_1}{r}} = 2\pi \sqrt{\frac{0.76}{1459}} \frac{\mathrm{AU}}{\mathrm{yr}} \approx 0.143 \frac{\mathrm{AU}}{\mathrm{yr}} \approx 0.678 \frac{\mathrm{km}}{\mathrm{s}}, \\[0.5ex]
\frac{v_{\mathrm{rad},1}}{\sin i} &amp;= \frac{m_2}{m_1} \frac{v_{\mathrm{rad},2}}{\sin i} \approx 0.639 \frac{\mathrm{m}}{\mathrm{s}},
\end{align}</span>
where I used <span class="math-container">$m_1=0.76\,$</span>M<span class="math-container">$_\odot$</span> and <span class="math-container">$G=(2\pi)^2\,$</span>AU<span class="math-container">$^3$</span>/yr<span class="math-container">$^2$</span>M<span class="math-container">$_\odot$</span>.</p>
<p>Note that <span class="math-container">$v_{\mathrm{rad},2}$</span> can presumably not be measured, since the brown dwarf is too faint and very close to the primary, while measuring stellar radial velocity amplitudes smaller than 1 m/s is one of the main methods for inferring the presence and/or mass of secondaries in the planet to brown-dwarf range.</p>

CC BY-SA 4.0
2023-01-02T17:30:30.460Z
From my understanding it is actually the radial velocity semiamplitude of the second object (ε indi B) that I need to find. This is based on how the question starts with the introduktion of the dwarf. Also all previous questions were about ε indi A.
The mass of the primary star is 0.76 solar masses. and thanks for the help!
C H
CC BY-SA 4.0
2022-12-30T13:30:44.477Z
2023-01-10T09:11:24.087Z
2023-01-10T09:11:24.087Z
Walter
<p>Is there a way to calculate the radial velocity without wavelength? I can only find two formulas so far. One of them needs wavelengths and the other one needs the velocity of the star, but I have none of these. I have distances, parallax, magnitudes, luminosities, astrometric signal, radii, effective temperature, <span class="math-container">$T_{eff}$</span> and masses.</p>
<p>It's about a star with a &quot;double star&quot; of two dwarf stars in orbit. (It is assumed that they can be considered as total mass)</p>
<p>It's assumed that the orbit is circular.</p>
<p>Edit:
<br> It's an exam question from last year's exam:
<br>
A star ε indi A has a &quot;double star&quot; of brown dwarfs in orbit. Assume that the brown dwarfs can be considered as one mass of 75 Jupiter masses. The distance between the brown dwarfs (ε indi B) and ε indi A is 1459 AU.
<br>
Calculate the maximum astrometric signal and the maximum radial velocity semiamplitude under the assumption that the orbit is circular.</p>
<p>I've already calculated the maximum astrometric signal. It's just the last part that I can't figure out to do with the information I have.</p>
<p>The formulas I'm talking about are:
<br> <span class="math-container">$v_r=c\frac{\Delta \lambda}{\lambda}$</span>
<br> and
<br> <span class="math-container">$v_r = v_\text{star} \sin(\frac{2 \pi}{P} t)$</span></p>

CC BY-SA 4.0
2022-12-28T17:44:31.063Z
I can only guess what you are asking for but I doubt you can estimate the radial velocity of a star (and if it is a binary, then there are two stars, each with a varying radial velocity) without knowing something about its motion...
ProfRob
CC BY-SA 4.0
2022-12-29T14:45:43.160Z
I've added the full question to my post to clarify things
C H
CC BY-SA 4.0
2023-01-03T18:40:37.670Z
So I finally got an answer that makes sense. Thanks a lot for the help! However, I need to explain why the two brown dwarfs, despite the magnitude of the signal, are not detected in some of these ways. I have absolutely no idea what ways/methods they are referring to (I can try and ask my instructor if this needs to be clarified). But is there any other explanation than the fact that brown dwarfs are extremely faint and relatively cold (thus their spectrum is in the infrared making them difficult to observe because of Earth's atmosphere)?
C H
<orbital-mechanics><astrometry><radial-velocity>
Radial velocity
C H
# Radial velocity

(Asked by: username_0 on 2022-12-28)

Is there a way to calculate the radial velocity without wavelength? I can only find two formulas so far. One of them needs wavelengths and the other one needs the velocity of the star, but I have none of these. I have distances, parallax, magnitudes, luminosities, astrometric signal, radii, effective temperature, $T\_{eff}$ and masses.

It's about a star with a "double star" of two dwarf stars in orbit. (It is assumed that they can be considered as total mass)

It's assumed that the orbit is circular.

Edit:
  
 It's an exam question from last year's exam:
  
A star ε indi A has a "double star" of brown dwarfs in orbit. Assume that the brown dwarfs can be considered as one mass of 75 Jupiter masses. The distance between the brown dwarfs (ε indi B) and ε indi A is 1459 AU.
  
Calculate the maximum astrometric signal and the maximum radial velocity semiamplitude under the assumption that the orbit is circular.

I've already calculated the maximum astrometric signal. It's just the last part that I can't figure out to do with the information I have.

The formulas I'm talking about are:
  
 $v\_r=c\frac{\Delta \lambda}{\lambda}$
  
 and
  
 $v\_r = v\_\text{star} \sin(\frac{2 \pi}{P} t)$

**Tags:** orbital-mechanics, astrometry, radial-velocity

> **Comments:**
> **username_1**: I can only guess what you are asking for but I doubt you can estimate the radial velocity of a star (and if it is a binary, then there are two stars, each with a varying radial velocity) without knowing something about its motion...  
> **username_0**: I've added the full question to my post to clarify things  
> **username_0**: So I finally got an answer that makes sense. Thanks a lot for the help! However, I need to explain why the two brown dwarfs, despite the magnitude of the signal, are not detected in some of these ways. I have absolutely no idea what ways/methods they are referring to (I can try and ask my instructor if this needs to be clarified). But is there any other explanation than the fact that brown dwarfs are extremely faint and relatively cold (thus their spectrum is in the infrared making them difficult to observe because of Earth's atmosphere)?  

---

## 1 answers

### Answer by username_2 on 2022-12-30. Score: 3

The relative position between the components of the binary moves on a circular orbit with velocity $v\_{\mathrm{circ}}=\sqrt{G(m\_1+m\_2)/r}$, where $r=1459$AU is the distance, $G$ is Newton's constant of gravity and $m\_{1,2}$ are the masses of the components.

Thus, the radial velocity semiamplitude of the secondary is
\begin{align}
v\_{\mathrm{rad},2} &= \frac{m\_{1}\sin i}{m\_1+m\_2} v\_{\mathrm{circ}} = m\_{2,1}\sin i \sqrt{\frac{G}{r(m\_1+m\_2)}},
\end{align}
where $i$ is the inclination of the binary orbit's angular momentum vector w.r.t. the line of sight (i.e. $i=90^\circ$ if the line of sight lies in the orbital plane). For the primary, the amplitude is smaller by a factor $m\_2/m\_1$.

If $m\_1\gg m\_2$ (as is the case here, where the uncertainty in $m\_1$ presumably exceeds $m\_2$), the above simplifies to
\begin{align}
\frac{v\_{\mathrm{rad},2}}{\sin i} &= \sqrt{\frac{Gm\_1}{r}} = 2\pi \sqrt{\frac{0.76}{1459}} \frac{\mathrm{AU}}{\mathrm{yr}} \approx 0.143 \frac{\mathrm{AU}}{\mathrm{yr}} \approx 0.678 \frac{\mathrm{km}}{\mathrm{s}}, \\[0.5ex]
\frac{v\_{\mathrm{rad},1}}{\sin i} &= \frac{m\_2}{m\_1} \frac{v\_{\mathrm{rad},2}}{\sin i} \approx 0.639 \frac{\mathrm{m}}{\mathrm{s}},
\end{align}
where I used $m\_1=0.76\,$M$\_\odot$ and $G=(2\pi)^2\,$AU$^3$/yr$^2$M$\_\odot$.

Note that $v\_{\mathrm{rad},2}$ can presumably not be measured, since the brown dwarf is too faint and very close to the primary, while measuring stellar radial velocity amplitudes smaller than 1 m/s is one of the main methods for inferring the presence and/or mass of secondaries in the planet to brown-dwarf range.

> **Comments:**
> **username_0**: From my understanding it is actually the radial velocity semiamplitude of the second object (ε indi B) that I need to find. This is based on how the question starts with the introduktion of the dwarf. Also all previous questions were about ε indi A. The mass of the primary star is 0.76 solar masses. and thanks for the help!  

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