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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
$v_r = v_\text{star} \sin(\frac{2 \pi}{P} t)$

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    $\begingroup$ 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... $\endgroup$
    – ProfRob
    Commented Dec 28, 2022 at 17:44
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    $\begingroup$ Those formulas are used when you already have observed the binary star system by Spectroscopy imaging so they Use the Doppler effect for the calculation thus you need the wavelength/$λ$. The formula you're talking about is probably $Δλ/λ=v/c$. I'm not sure which other formula you're talking about This [formula][1] (referred by Wikipedia) most probably will help: $V=DR/DT$. It is expressed in terms of arc lengths. [1]: en.wikipedia.org/wiki/Radial_velocity $\endgroup$
    – user47732
    Commented Dec 28, 2022 at 18:31
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    $\begingroup$ I've added the full question to my post to clarify things $\endgroup$
    – C H
    Commented Dec 29, 2022 at 14:45
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    $\begingroup$ 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)? $\endgroup$
    – C H
    Commented Jan 3, 2023 at 18:40

1 Answer 1

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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.

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    $\begingroup$ 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! $\endgroup$
    – C H
    Commented Jan 2, 2023 at 17:30

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