Echelle Spectrograph Lab: Grating Orders, Stellar Absorption Lines and Radial Velocity

Setting up the lab bench...

Presets

Live readouts

Graphs

Inspected order: observed spectrum and the blaze envelope

Resolving power R against slit width

Selected line: true profile, instrument-broadened profile and fitted centre

Numerical checks

running...

How to use

Pick a star preset and look at the stack of stripes: every stripe is one grating order, blue at the bottom; dark gaps are absorption lines.
Drag the inspected wavelength: the highlighted stripe and the first graph show that order with its blaze envelope.
Narrow the slit or enlarge the telescope beam: R rises and the Na doublet (0.6 nm apart) separates. Make the slit wider than the seeing and watch the throughput fall.
Change the radial velocity and the line selection; the third graph shows the fitted centre against the rest wavelength.
Keys: Space pauses the noise refresh, R resets.

The 3D page shows the spectrograph bench. The rays follow the real grating angles: the echelle is tilted to its blaze angle, the fan of colours leaving the cross-disperser lands on the CCD, and the monitor shows the echellogram produced by the same model as the 2D page.

The science

An echelle spectrograph uses a coarse grating at a steep blaze angle and a very high diffraction order m, so that light of many wavelengths is sent into nearly the same direction. A second dispersing element (a prism or low-density grating, the cross-disperser) separates the overlapping orders sideways, and the CCD records a stack of spectral stripes: the echellogram. A star therefore gives a long spectrum, about 400-800 nm, in a single exposure.

m λ = d ( sin α + sin β ), Littrow blaze (α = β = θ_B): m λ_B = 2 d sin θ_B

With R2 gratings tan θ_B = 2 (θ_B = 63.4 degrees), 31.6 grooves/mm, 2 d sin θ_B = 56.6 μm, so order m = 103 is blazed at 550 nm. Neighbouring orders overlap in wavelength by the free spectral range and the blaze function sets the efficiency across an order:

FSR = λ_B / m, η(λ) = sinc²[ π m ( λ / λ_B - 1 ) ]

Resolving power is limited by whichever is worst: the grating (R = m N, N the number of illuminated grooves), the slit (R_slit = 2 tan θ_B W / (χ D_T), W the collimated beam diameter, χ the slit angle on the sky, D_T the telescope diameter) and the detector sampling (about two pixels per resolution element). Seeing larger than the slit simply loses light.

1/R² = 1/R_slit² + 1/R_grating², dλ/dx = λ / (2 f tan θ_B) (linear dispersion)

A radial velocity v shifts every line by Δλ/λ = v/c (relativistic: √((1+v/c)/(1-v/c)) - 1). The page fits the line centre by a depth-weighted centroid on pixel samples with photon noise. The classical photon-limited estimate is about σ_v ≈ (c/R) / (SNR √n_pix) times a factor near 0.6.

Honest simplifications. The stellar spectrum is a blackbody with 25 Gaussian absorption lines at real wavelengths and invented depths scaled by simple temperature rules; there is no model atmosphere, no stellar rotation, no telluric lines, no scattered light, no order tilt or optical aberrations, no CCD gaps and no calibration lamp. The echellogram is drawn at screen resolution, so narrow lines are shown wider than on a real detector. The instrument profile is a Gaussian of constant R, applied in three wavelength bands. It is a teaching model, not a data-reduction pipeline.