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Theoretical Elements

A little bit of trigonometry...

Some theoretical elements can allow us to better understand needs and constraints to consider for the realization of a spectrohéliographe:

• Angular dispersion of a reflective grating :

sin(α) = k . λ / (2 . p)

• Linear dispersion in the spectrum plane :

D = p . cos(α) / (k . f)

• Resolving power :

R = k . N

• Spectral resolution :

Dl = λ / R

with :
α = diffraction angle (in rd)
f = focal length of the camera lens (in mm)
k = order of spectrum
λ = wavelength (en nm)
N = total number of grooves of the grating
p = groove spacing of the grating = 1000000/number of grooves per mm (in nm)
R = resolving power of the grating

• example:
For a 1200 gr/mm grating enlightened perpendicularly to the surface, the wavelength of the Ha line is diffracted in the first order according to an angle of :

a = arcsin( 656.3 /(2 x 1000000/1200) ) = 0.4047 rd = 23° 11',4
With a 500 mm f.l. camera, the linear dispersion is :
d = 1000000/1200 . cos(0.4047) / (1 . 500) = 1.5 nm/mm

A 30 mm aside grating has, in 1st order, a resolving power :
R = 1 . 1200 . 30 = 36000

At this wavelength, one can resolve, in theory, two lines separated by :
656.3 / 36000 = 0.018 nm

... geometry ...

Here is a layout of spectroheliograph :

• Lens of telescope : Aperture D1 and focal length F1
• Collimator : Aperture D2 and focal lengthF2
• Camera lens : ApertureD3 and focal lengthF3
• Width of the entrance slit : S
• Width of the grating : C

The telescope gives a Sun image diametre = F1 . tg(alpha) with alpha = apparent angle of the Sun, about 1/2 degree.

focus plane of telescope, focus plane of collimator and entrance slit are merged.

The spectrograph gives a magnification of this image = F3/F2 . It is possible to magnify or to reduce the size of the solar image on the sensor. Width of the entrance slit image varying in the same ratio, it may be interesting to use long f.l. F1 and F2 with a rather large slit, and to reduce final image with F3 if the sensor is small. In the case of Littrow mount (collimator and camera are the same), magnification is obviously = 1.

The grating must be totally illuminated to obtain maximum resolving power. this implies to use a collimateur with aperture D2 = 1,414 C or greater. If you have only a 40 mm aperture collimator, a 1 pouce width grating is sufficient. This lens must also be illuminated in the same ratio (1.414 C) by the beam comming from the entrance slit of the spectro . We can see that, aroud the slit, the incident and emerging angles are equals (in yellow) and the F2/D2 ratio must be equal to F1/D1 (or smaller).

Camera lens must have the same aperture D3 than D2.

... and a spectroheliocomputer (use mm)

 Grating Télescope Collimator Camera spectrum image Dispersion (Ha) set set set set set set Width D1 D2 D3 heigth A/mm gr/mm F1 F2 F3 Order