The mechanical resolution is determined by the minimum angle ΔN by which the viewing telescope can turn. For the Phasmatron spectroscope's angle measuring devices, this is 10^-3 degrees (a millionth of a degree). Converting to radians, ΔN=1.74532*10^-5 rad. Next we need dE/dλ so we can approximate Δλ.

We know that dE/dn=2/cos{arcsin(n_D/2)}. This for n_D=1.72803 gives dE/dn=3.9724624 rad^[1], and dn/dλ=1.2702*10^-5/A^[1], => dE/dλ=(dE/dn)*(dn/dλ)=5.04582*10^-5 rad/A. Therefore we can approximate and use ΔE/Δλ=5.04582*10^-5 rad/A, and since ΔE almost equals ΔN, Δλ=ΔN/5.04582*10^-5 rad/A. For ΔN=1.74532*10^-5 rad, this gives:

Δλ_mechanical=0.3458942A^[1]. Compare this with Δλ_optical=0.3866148A^[1][2].

Suppose we wanted to calculate instead the mechanical resolution in the area of the blue Mercury line (4358.35A) Then:

dn/dλ=(1.76197-1.74805)/|4358.35-4799.9107|=3.1524545*10^-5/A. dE/dn=2/cos{arcsin(n_4358.35/2)}=4.22704 rad. dE/dλ=4.22704 rad*3.1524545*10^-5/A=1.3325551*10^-4 rad/A. Δλ_mechanical=1.74532*10^-5 rad/1.3325551*10^-4 rad/A=0.131A.

Compare this with Δλ_optical=0.1152104A from Lord Rayleigh's formula Δλ=λ/{B*dn/dλ}, for dn/dλ=3.1524545*10^-5/A, B=12*10^-8A, and λ=4358.35A.

We observe that as a function of wavelength, the mechanical resolution varies roughly in the same way that the optical resolution does.

Notes

1. In the area of the sodium D lines.
2. The true mechanical resolution can be calculated using the program in section Measuring Wavelengths. If we input N=60°, M=59.097° and on another run input N=60° and M=59.098°, we can subtract the two wavelength values found and get thus the Δλ. This way we will get that the true mechanical resolution in the area of sodium lines is 0.3564453A, which is very close to the value of Δλ_mechanical found above.

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