# How spectrographs that measure radial velocities manage to translate variations in the stars' spectrum lines into the orbital speed of the star?

Like ESPRESSO or CARMENES, for example. I just don't understand the process or the way these spectrographs manage to obtain the line-of-sight velocity of the "wobble" of the star (due to the presence of another companion such as an exoplanet) just by analyzing variations on spectral lines.

Then further, how do they go from a line-of-sight velocity to get the orbital speed of the star?

• Yeah that's a way to detect that something is moving, but how to measure the speed? Oct 18, 2019 at 0:26
• Doppler shift - change in wavelength of a line compared to the 'at-rest' (not moving) wavelength of the line, gives velocity ("speed") Oct 18, 2019 at 1:06
• $v \sim \Delta \lambda/c \lambda_0$, where $\lambda_0$ is the wavelength at rest and $\Delta \lambda$ is the measured wavelength shift. Oct 18, 2019 at 7:42
• CVM, your suspicions are correct; that only tells you there is something there. Despite the upvotes on the comment answers using the simple formula suggested doesn't take into account eccentricity and inclination which would leave too many unknown variables and add too large an error on the velocity calculated in such a manner.
– Rob
Oct 19, 2019 at 21:40
• @ProfRob, I'm not the only one who disagrees with you: en.wikipedia.org/wiki/… physics.stackexchange.com/a/664843/170832 nor is this the first time you've been unhelpful.
– Rob
Dec 11, 2023 at 3:31

There are many ways to measure a star's motion, radial velocity (wobble) can be measured using doppler spectroscopy. The first exoplanet discovered by this method was 51 Pegasi b by Michel Mayor and Didier Queloz, who discovered the planet on December 1995.

A drawback of this method is that it can only detect the movement of a star towards or away from the Earth. If the orbital plane of the planet is "face on" when observed from the Earth the wobble of the star will be perpendicular to an observer's line of vision, and no spectrum shift will be detected.

In most cases a distant planet's orbital plane is neither "edge-on" nor "face-on" when observed from the Earth. Most likely it is tilted at some angle to the line of sight, which is usually unknown. This means that a spectrograph would not detect the full movement of the star, but only that component of its wobble that moves it towards the Earth or away from it.

The ESPRESSO instrument uses two 90x90 mm CCD detectors, one red and one blue sensitive. The detectors view the light after it is reflected off of an echelle grating, which is optimized for use at high incidence angles and therefore in high diffraction orders. Higher diffraction orders allow for increased dispersion (spacing) of spectral features at the detector, enabling increased differentiation of these features.

The equations are relatively simple. The observed Doppler velocity is $$K=V_{\mathrm {star} }\sin(i)$$, where $$i$$ is the inclination of the planet's orbit to the line perpendicular to the line-of-sight.

Reference:

"A Jupiter-mass companion to a solar-type star", Nature volume 378, pages 355–359 (1995), by Michel Mayor and Didier Queloz

An improvement to Mayor and Queloz's equations is offered in:

"The Rossiter-McLaughlin effect and analytic radial velocity curves for transiting extrasolar planetary systems" (Mar 25 2005), by Yasuhiro Ohta, Atsushi Taruya, and Yasushi Suto

"7. Conclusions and discussion
We have discussed a methodology to estimate the stellar spin angular velocity and its direction angle with respect to the planetary orbit for transiting extrasolar planetary systems using the RM effect previously known in eclipsing binary stars (Rossiter 1924; McLaughlin 1924; Kopal 1990). In particular we have derived analytic expressions of the radial velocity anomaly, $$\Delta_{vs}$$, which are sufficiently accurate for extrasolar planetary systems. If the stellar limb darkening is neglected, the expression is exact. We have extended the result to the case with limb darkening and obtained approximate but accurate analytic formulae. For a typical value of $$\gamma = R_p / R_s \sim 0.1$$, the formulae reduce to a simple form (eqs. [40], [43], [44], [45], [48], and [49]):

$$\Delta v_s = \Omega s X_p \sin I_s \frac{\gamma^2 \{1-\varepsilon(1-W_2) \} }{ 1-\gamma^2-\varepsilon \{ \frac{1}{3}-\gamma^2 \} } \tag{56}$$

during the complete transit phase and (the following is scrollable):

$$\small { \Delta v_s = \Omega_s X_p \sin I_s \frac { (1- \varepsilon) \{ - z_0 \zeta + \gamma^2 \cos^{-1} (\zeta / \gamma) \} + \frac { \varepsilon }{ 1 + \eta_p } W_4 }{ \pi (1- \frac {1}{3} \varepsilon ) - (1 - \varepsilon) \{ \sin^{-1} z_0 - (1 - \eta_p) z_0 + \gamma^2 \cos^{-1} (\zeta / \gamma) \} } \tag {57} }$$

during the egress/ingress phases, where

\begin{align} W_2 & = \frac { (R^2_s - X^2_p - Z^2_p)^{1/2} }{ R_s }, \tag {58} \\ W_4 & = \frac {\pi}{2} \, \gamma^{3/2} (2 - \gamma)^{1/2} \, (\gamma - \zeta) \; x_c \frac {g(x_c ; \eta_p , \gamma) }{ g(1- \gamma ; - \gamma , \gamma) }, \tag {59} \end{align}

where $$g(x; a, b)$$ is defined in equation (A17). The definition and the meaning of the variables in the above expressions are summarized in Table 1.

The numerical accuracy of the above formulae was checked using a specific example of the transiting extrasolar planetary system, HD 209458, and we found that they are accurate within a few percent. Our analytic formulae for the radial velocity anomaly are useful in several ways. One can estimate the planetary parameters much more efficiently and easily, since one does not have to resort to computationally demanding numerical modeling. Furthermore, the resulting uncertainties of the fitted parameters and their correlations are easily evaluated.

...

Table 1. List of notation
Variables    Definition     Meaning

Orbital Parameters
$$m_p$$        Sec.2        Planet mass
$$m_s$$        Sec.2        Stellar mass
$$a$$          Fig.1       Semimajor axis
$$e$$          Fig.1        Eccentricity of planetary orbit
$$\varpi$$         Fig.1       Negative longitude of the line of sight
$$i$$           Fig.2        Inclination between normal direction of orbital plane and y-axis
$$r_p$$         Eq.[1]       Distance between star and planet (see Fig.1)
$$f$$          Eq.[2]        True anomaly (see Fig.1)
$$E$$         Eq.[2]       Eccentric anomaly
$$n$$          Eq.[3]       Mean motion
$$M$$        Eq.[4]       Mean anomaly

Internal Parameters of Star and Planet
$$Is$$        Fig.2         Inclination between stellar spin axis and y-axis
$$λ$$          Fig.3         Angle between $$z$$-axis and normal vector $$\hat{n}_p$$ on $$(x, z)$$-plane
$$Ω_s$$         Eq.[12]       Annular velocity of star (see Fig.2)
$$R_s$$         Sec.4      Stellar radius
$$R_p$$         Sec.4      Planet radius
$$\varepsilon$$         Eq.[38]       Limb darkening parameter
$$V$$           Sec.6      Stellar surface velocity, $$R_sΩ_s$$

Mathematical Notation
$$X_p$$         Sec.4      Position of the planet
$$γ$$         Eq.[25]       Ratio of planet radius to stellar radius, $$Rp/Rs$$
$$η_p$$          Eq.[28]       See Fig.6
$$x_0$$          Eq.[33]       See Fig.6

• In the picture, where the spectrograph is shown in section, there are a lot of blocks, BCA, RCA, BTM, RTM, CM, DC, EG, etc. In this regard, I have a question. Is a spectrograph a static device with the position of the blocks fixed once and for all, or does it also need regulation and control (for example, the position of mirrors, intermediate optical calculations)?
– dtn
Dec 12, 2023 at 5:35

Your initial question was rather unclear, but now I see that you might be interested in how line-of-sight velocities are used to fully constrain the orbit and orbital speed of the star.

First, to answer what I thought was the original question. The accurately measured positions of known spectral lines is turned into a line-of-sight velocity, by comparing those line wavelengths with the same lines in a source that isn't moving. For example, a laboratory source.

This gives an accurate line-of-sight (or "radial") velocity using the Doppler shift $$V_r = c \frac{\Delta \lambda}{\lambda_0}\ ,$$ where $$\lambda_0$$ is the wavelength of the line in a source at rest and $$\Delta \lambda$$ is the observed change in wavelength, which could be positive (a redshift) or negative (a blue shift).

More precision is obtained by a cross-correlation of the observed spectrum with template spectra,which then uses information from many spectral features at the same time to work out a Doppler shift and $$V_r$$ relative to the template.

Then, to go from a series of $$V_r$$ measurements to a characterisation of the orbit is an inference problem. You design a model of an elliptical orbit that will depend on the masses of the components, their separation (semi-major axis), their orbital eccentricity, the average velocity of the system along the line-of-sight to the Earth, and two angles that specify the orientation of the orbit in space (since in general, one does not have any spatial information about the orbit). From the orbital model one can then predict what values a series of radial velocity measurements will give. An iterative, Bayesian inference procedure (usually Markov Chain Monte Carlo, MCMC, these days) will then yield the "best fit" orbital parameters.

Unfortunately, radial velocities alone cannot unambiguously yield the system orientation. This leads to an ambiguity in the inferred orbit. The problem is usually couched by making one of the two orbital angles the "inclination" of the system, $$i$$. When $$i= 0$$ degrees, the system is viewed with it's orbital axis in the line of sight and there would be no racial velocity variations at all. When $$i=90$$ degrees, the orbital axis is at right angles to the line-of-sight, and the radial velocity variations are maximised.

Thus the orbital speed can never be fully determined using spectral line measurements alone, only that component along the line of sight. This leads to an ambiguity in deduced masses, such that the determined mass is actually $$M \sin i$$ and is hence a lower limit to the true mass $$M$$.