ASTR 340 Problem Set 05: The Grating Equation and a Second-Order Spectrograph Design

Problem set for lecture unit 5.

Problems

1. Diffraction angles for a fine-pitch grating 20 points

A spectrograph uses a 1200 grooves/mm grating at normal incidence. Compute the first-order diffraction angle for 400 nm light and for 700 nm light.

2. Angular dispersion 15 points

Using your Problem 1 results, estimate the average angular dispersion dθ/dλ (in degrees per nm) across this 400-700 nm range, and compare it to the exact derivative relation dθ/dλ=m/(d cosθ_m) evaluated at 550 nm.

3. Resolving power requirement 15 points

A science case requires resolving two spectral lines separated by 0.01 nm at 700 nm. Compute the minimum resolving power R required, and confirm it against 70000.

4. Illuminated groove count 20 points

Using R=mN, compute the minimum number of illuminated grooves N (at m=1) required to reach the Problem 3 resolving power, and state whether a 10 cm wide grating ruled at 1200 grooves/mm has enough illuminated grooves to meet this requirement.

Due and Scope

Submit a complete derivation with equations, units, labeled quantities, and a short narrative interpretation for each problem. Show every step; a correct final number without a visible derivation receives partial credit at most, and quoting a lecture number without adapting it to this problem's specific inputs receives no credit for that step.

OpenStax Companion Reading

OpenStax Astronomy 2e (local reference copy: references/openstax-astronomy-2e-extracted.txt), Chapter 6 (astronomical instruments), descriptive only; this course exceeds that introductory treatment throughout.

References and Data Sources