Lab 05: The Grating Equation and HIRES’s Resolving Power

Recompute Keck HIRES’s resolving power from its published groove density and blaze angle, and derive its velocity-resolution element.

Apparatus and Setup

This lab applies the grating equation (Lecture 09) and the resolving-power relations (Lecture 10) to Keck's HIRES echelle spectrograph, using its published groove density, blaze angle, and resolving power.

Materials and Data

QuantityValue
HIRES published resolving powerR ≈ 67,000
HIRES groove density, blaze angle52.67 grooves/mm, 70°
Representative illuminated beam width20 cm
HIRES published radial-velocity precision, 1 AU detection limit1.0 m/s, 0.2 M_Jup

Procedure

  1. Using R = 2W tan(θ_B)/λ with the representative beam width and HIRES's blaze angle, compute the predicted resolving power at V band, and compare it (percent difference) to the published R≈67,000.
  2. Using Δλ = λ/R at V band, compute HIRES's minimum resolvable wavelength interval, in picometers.
  3. Using Δv/c = 1/R, compute the velocity-resolution element corresponding to HIRES's resolving power, in km/s, and compare it to HIRES's published 1.0 m/s radial-velocity precision.
  4. Explain, using the distinction between resolving a single spectral line and centroiding many lines simultaneously, why HIRES's actual radial-velocity precision is roughly four orders of magnitude finer than the single-line velocity-resolution element computed in step 3.

Analysis and Uncertainty

The beam-width-derived resolving power in step 1 is not expected to exactly match the published value, because the representative beam width used is illustrative rather than HIRES's exact as-built optical prescription; state explicitly why this is an acceptable limitation for demonstrating the correct scaling relation but would not be acceptable for precision instrument design.

Deliverables

Assessment Criteria

References and Provenance