Radial velocity

Question

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

Answer 1

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.