# Radial velocity stability to spectrum shift calculation

I am engineer new to Astronomy and am trying to understand spec of an spectrograph (RV stability = 2m/s). There is a note in document saying "RV shift of 2 m/s is equivalent to a shift of the spectrum on the detector of about 30 nm". The measurement range is 380nm-850nm.

My understanding so far is as follow: Stars moving away/towards earth causes shift in known spectrum lines towards red or blue depending on direction of movement. The magnitude of shift can be derived from Doppler effect as:

$$\Delta_\lambda = \lambda_0 \frac{V_r}{c}$$

where $$\Delta_\lambda$$ is the shift in wavelength, $$\lambda_0$$ is the rest wavelength, and $$V_r$$ is the radial velocity

or $$V_r = c\frac{\Delta_\lambda }{\lambda_{0}}$$

RV stability would be difference between radial velocity of consecutive measurements.

Taking an example here:

$$V_{r1} = \frac{\Delta\lambda_1 c}{\lambda_0}$$ $$V_{r2} = \frac{\Delta\lambda_2 c}{\lambda_0}$$

where $$V_{r1}$$ and $$V_{r2}$$ are the two radial measurements, and $$\Delta\lambda_1$$ and $$\Delta\lambda_2$$ are the two wavelength shifts.

$$\Delta V_{r} = \frac{(\Delta\lambda_1 - \Delta\lambda_2 )*c}{\lambda_0}$$

where $$\Delta V_{r}$$ is Radial velocity stability between measurements.

$$\Delta_s = \frac{\Delta V_r \lambda_0}{c}$$

Where $$\Delta_s$$ is $$\Delta\lambda_1- \Delta\lambda_2$$ = shift on ccd due to measurement

Plugging numbers from note.

Shift of spectrum on detector = $$\frac{2 \cdot 380nm}{3\cdot 10^8} \neq 30nm$$

I am definitely doing something wrong but do not know what exactly.

• @fasterthanlight Thanks a lot for making it legible. Mar 24, 2022 at 22:19

Most CCDs have pixels of size $$\sim 15$$ $$\mu$$m, so each such pixel would cover 0.00126 nm of wavelength, which is indeed, very high resolution.
• My understanding is that the $\Delta_s$ calculated above is the physical shift of spectral lines on detector, which is way off from the document's value. Mar 24, 2022 at 23:10