A fibre-fed echelle spectrograph for a telescope must reach a resolving power with a…

PHYSICS · Multi-part · Fermi University Physics · Optics & Photonics · Geometrical optics and imaging · Geometrical optics and imaging

What an Instrument Can See, Ch Radiance, etendue and a throughput budget that is spent

Problem

A fibre-fed echelle spectrograph for a

4.0\ \mathrm{m}

telescope must reach a resolving power

R = 60\,000

with a fibre that subtends

1.5

arcseconds on the sky. Its echelle has

\tan\theta_{\mathrm{bl}} = 2.0

. Assume the fibre carries

D_{\mathrm{tel}}\phi

to the collimator without growth. Take the seeing as Gaussian with a full width at half maximum of

1.0

arcsecond, so that a centred fibre of angular diameter

d

, in arcseconds, catches a fraction

1-2^{-d^2}

of a star's light. ((a)) Find the collimated beam diameter

D_{\mathrm{coll}}

that reaches

R = 60\,000

with the unsliced fibre. ((b)) The beam can be at most

200\ \mathrm{mm}

across. An image slicer that cuts the fibre image into

N

equal strips and stacks them along the slit divides the width the collimator sees by

N

. Find the smallest

N

that reaches

R = 60\,000

. ((c)) Instead of slicing, the fibre could be narrowed, with the

200\ \mathrm{mm}

beam. Find the angular diameter that reaches

R = 60\,000

, and the fractions of a star's light caught by that fibre and by the

1.5

arcsecond fibre.