# Spectrographs

This problem set is designed for astronomy to help the student, teacher, and researcher understand the inner workings of a spectrograph.

## Spectrography A ray trace through a prism with apex angle α is shown. Regions 0, 1, and 2 have indices of refraction $n_{0}$ , $n_{1}$ , and $n_{2}$ , and primed angles $\theta '$ indicate the ray angles after refraction. Credit: NathanHagen.

Def. a machine for recording spectra, producing spectrograms is called a spectrograph.

Def. a visual representation of the spectrum of a celestial body's radiation is called a spectrogram.

A prism is a transparent optical element with flat, polished surfaces that refract light [over a range of wavelengths]. At least two of the flat surfaces must have an angle [α] between them. The exact angles between the surfaces depend on the application. The traditional geometrical shape is that of a triangular prism with a triangular base and rectangular sides, and in colloquial use "prism" usually refers to this type.

Ray angle deviation and dispersion through a prism can be determined by tracing a sample ray through the element and using Snell's law at each interface. For the prism shown at right, the indicated angles are given by

{\begin{aligned}\theta '_{0}&=\,{\text{arcsin}}{\Big (}{\frac {n_{0}}{n_{1}}}\,\sin \theta _{0}{\Big )}\\\theta _{1}&=\alpha -\theta '_{0}\\\theta '_{1}&=\,{\text{arcsin}}{\Big (}{\frac {n_{1}}{n_{2}}}\,\sin \theta _{1}{\Big )}\\\theta _{2}&=\theta '_{1}-\alpha \end{aligned}} .

For a prism in air $n_{0}=n_{2}\simeq 1$ . Defining $n=n_{1}$ , the deviation angle $\delta$ is given by

$\delta =\theta _{0}+\theta _{2}=\theta _{0}+{\text{arcsin}}{\Big (}n\,\sin {\Big [}\alpha -{\text{arcsin}}{\Big (}{\frac {1}{n}}\,\sin \theta _{0}{\Big )}{\Big ]}{\Big )}-\alpha$ If the angle of incidence $\theta _{0}$ and prism apex angle $\alpha$ are both small, $\sin \theta \approx \theta$ and ${\text{arcsin}}x\approx x$ if the angles are expressed in radians. This allows the nonlinear equation in the deviation angle $\delta$ to be approximated by

$\delta \approx \theta _{0}-\alpha +{\Big (}n\,{\Big [}{\Big (}\alpha -{\frac {1}{n}}\,\theta _{0}{\Big )}{\Big ]}{\Big )}=\theta _{0}-\alpha +n\alpha -\theta _{0}=(n-1)\alpha \ .$ The deviation angle depends on wavelength through n, so for a thin prism the deviation angle varies with wavelength according to

$\delta (\lambda )\approx [n(\lambda )-1]\alpha$ .

## Problem 1

The image at the top of this resource appears to have a deviation angle of 45°. The detector may be about 4 cm from the prism. Using a refractive index n = 1.732 and a representative wavelength for the optical colors calculate the width of each wavelength channel and the total detector width to capture the incoming photons. Use an apex angle of 35°.

## Problem 2

Let the deviation angle be 30°, the detector distance be half a meter with the same refractive index and apex angle of 37.5°.

For the optical colors calculate the width of each wavelength channel and the total detector width to capture the incoming photons.

## Problem 3

Using the configuration of Problem 2 and assuming a prism for X-rays and gamma rays existed, calculate the width of each wavelength channel, for five representative wavelengths of each, and the total detector width to capture the incoming photons.

## Problem 4

Using the configuration of Problem 1 and representative wavelengths for each of the infrared bands described in infrared astronomy, calculate the width of each wavelength channel and the total detector width to capture the incoming photons.

## Problem 5

Using each configuration of the problems above and representative wavelengths for submillimeter, microwave, and radio waves, calculate the width of each wavelength channel and the total detector width to capture the incoming photons.

## Hypotheses

1. Amateur astronomers may be able to build or buy a spectrograph.