The HERMES spectrograph on the 1.2 m Mercator telescope observes a star of visual…

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

The HERMES spectrograph on the 1.2 m Mercator telescope observes a star of visual magnitude

V=18

through its

2.5

arcsecond fibre, in one detector pixel near

550\ \mathrm{nm}

. The dark sky has a surface brightness of

21.8

magnitudes per square arcsecond in

V

. Take a

V=0

source to deliver

1.0\times10^3\ \mathrm{photons\,s^{-1}\,cm^{-2}\,\text{\AA}^{-1}}

at 550 nm, the collecting area as

1.13\times10^4\ \mathrm{cm^2}

, the pixel bandwidth as

0.029\ \text{\AA}

, and the factors after the top of the telescope as atmosphere

0.887

, telescope

0.60

and spectrograph

0.28

. The same budget, with the fibre's capture fraction

0.987

included, gives

3.3

electrons per second per pixel for a

V=10.4

star. ((a)) Find the sky-side etendue of the fibre,

\Xi = A_{\mathrm{tel}}\Omega

, in

\mathrm{cm^2\,arcsec^2}

, where

\Omega

is the solid angle the fibre subtends on the sky. ((b)) Decide which form of the throughput budget applies to the sky, and find the sky's detected rate in electrons per second per pixel. ((c)) Find the star's detected rate in electrons per second per pixel. ((d)) For a one-hour exposure with dark current

0.001

electrons per second and read noise

2

electrons, find the star's signal-to-noise ratio with the sky counted as background.

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