Astronomy Principles and Practice Fourth Edition.pdf

242 The optics of telescope collectors
If the stellar spectrum is relatively flat over the detected wavelength interval and if it is assumed
that the photon energy is constant over this interval, then we may write
N T = π 4 D2 t λ hc λλ (16.5)
where λ is the spectral interval for the measurements.
The arrival of photons at the telescope is a statistical process. When the arriving flux is low,
fluctuations are clearly seen in any recorded signal as a result and any measurements are said to suffer
from photon shot noise. If no other sources of noise are present, the uncertainty of any measurement is
given by √ N T . Hence, any record and its ‘error’ may be expressed as N T ± √ N T . In this circumstance,
the signal-to-noise (S/N) ratio of the observation is given by
N T
√
NT
= √ N T ∝
√
D 2 t −→ D √ t. (16.6)
Although the light gathering power of a telescope depends on its collection area and hence on D 2 ,
equation (16.6) shows that the signal-to-noise ratio of basic brightness measurements only increases
according to D. It may also be noted that the S/N ratio improves according to the square root of
the observational time illustrating a law of diminishing returns according to the time spent making
any measurement. As will be seen later, in the real situation, the signal strengths depend on the
photon detection rate, after taking into account the transmittance of the telescope and its subsidiary
instrumentation and the efficiency of the detector. However, these additional factors do not alter the
conclusions coming from equation (16.6).
In passing, it may also be mentioned that under some circumstances of detection of faint objects,
the effectiveness of a telescope may not be proportional to D 2 . For example, when faint stars are
being detected photoelectrically against a background night sky which is also emitting light (no sky is
perfectly black), measurements of the night sky brightness need to be made also so that this background
signal can be subtracted and allowed for. In this circumstance, where observations are required of the
star plus sky background and then sky background alone, the effectiveness of the telescope is then only
proportional to D.
Example 16.1. At 6300 Å (630 nm) the flux from a source is 10 −18 Wm −2 Å −1 . Determine the
photon rate passing through a telescope aperture with D = 2·2 m over a wavelength interval of 100 Å.
Calculate the best possible S/N ratio of a measurement with an integration time of 30 s.
Using equation (16.5)
N T = π 630 × 10 −9
4 (2·2)2 × 30 ×
6·63 × 10 −34 × 3 × 10 8 × 10−18 × 100
this assuming that the flux exhibits no spectral variation over the wavelength interval of the
measurements and that the photon energy is constant over this interval. By performing the arithmetic,
N T ≈ 7463.
If the measurements are limited purely by photon shot noise, the S/N ratio is simply √ N T which
equals 86·4. In other words, the accuracy is of the order of 1 part in 86·4 or just better than 1%.
16.3.2 Brightness of an extended object
The brightness of any extended object is defined in the same way as surface brightness (see
section 15.5.1) as the flow of energy it provides through a unit area normal to the radiation per unit