While there have been considerable advances in detection methods since 1969, the calculations and discussion remain valid, illustrating how filters integrate into a measurement system. We have therefore left it unchanged.
The challenge of detecting faint astronomical objects is compounded by the light of the night sky. This light consists primarily of starlight scattered by dust in the atmosphere and interstellar space (including sunlight), along with emissions from the upper atmosphere. It is mostly continuous in spectral nature, although emission lines are also present.
The sky light causes overall fogging of photographic plates, the most common detectors used for this work, although image tubes have seen increasing use in recent years. Maximum contrast between the photographic image of a star (or other object) and the sky background is achieved when the sky fog is just discernible on the plate. The exposure time is selected to reach this optimal fogging level, as efficiency falls off rapidly outside of this range.
The limit of detection of a faint object occurs when its image is just discernible against the background. This limit and its variation with system parameters have been studied extensively, particularly by Baum. A simplified account of the analysis is provided by Bowen, whose notation and methods we follow here.
The signal from an object consists of discrete photons arriving at a constant mean rate but randomly spaced. For faint signals, the photon arrivals follow a Poisson distribution. In this distribution, the standard deviation of the number of photons \(N\) arriving in a constant time is \(\sqrt{N}\).
Notation and Parameters
– \(D\): Telescope aperture diameter.
– \(f\): Telescope focal length.
– \(t\): Observation time.
– \(\beta\): Diameter of the object’s image.
– \(n\): Number of photons from the object received per unit area of the telescope aperture per second.
– \(s\): Number of background photons received per unit area of the telescope aperture per unit solid angle of the sky per second.
– \(p\): Limit of linear resolution of the photographic emulsion.
– \(q\): Quantum efficiency of the system, encompassing the photographic emulsion and optical system transmission.
– \(m\): Number of photons per unit area of the photographic plate to produce the correct background fog.
Bowen defines the faintness of an object as proportional to \(1/n\). The fractional error \(B\) in measuring \(N\) is given by:
\[
B = \frac{\sqrt{N}}{N} = \frac{1}{\sqrt{N}}
\]
The number of photons recorded from the object and an equal area of sky in time \(t\) is:
\[
D^2ntq + \beta^2sD^2tq
\]
The standard deviation for these measurements is:
\[
\sqrt{D^2ntq + \beta^2sD^2tq}
\]
Thus, the fractional error becomes:
\[
B = \frac{\sqrt{D^2ntq + \beta^2sD^2tq}}{D^2ntq}
\]
For very faint objects (\(n \ll \beta^2 s\)):
\[
B = \frac{\beta}{\sqrt{sD^2ntq}}
\]
The limiting faintness is:
\[
\frac{1}{n} = \frac{1}{\beta \sqrt{sD^2tq}}
\]
Bowen suggests \(B_1 = 0.2\) as the highest possible value for detectability. For photographic detectors, exposure time \(t_0\) must yield the correct background fog:
\[
t_0 = \frac{m f^2}{s D^2 q}
\]
Substituting \(t_0\) into the limiting faintness equation:
\[
\frac{1}{n} = \frac{1}{\beta} \sqrt{\frac{m f^2}{s D^2 q}}
\]
If the plate resolution is limiting (\(f\) small), \(\beta\) is replaced by \(p/f\), yielding:
\[
\frac{1}{n} = \sqrt{\frac{m f}{p s}}
\]
Influence of Telescope Parameters
For photographic detection, focal length \(f\) appears more important than aperture \(D\), as the longest exposure time \(t_m\) is limited to one night. The maximum allowable focal length \(f_m\) is:
\[
f_m = \sqrt{\frac{t_m s q}{D^2}}
\]
Substituting \(f_m\) into the equations:
1. For large \(\beta\):
\[
\frac{1}{n} = \frac{1}{\beta} \sqrt{\frac{t_m q}{s}}
\]
2. For small \(\beta\):
\[
\frac{1}{n} = \sqrt{\frac{t_m q}{p s}}
\]
These results confirm the advantage of larger telescopes.
Role of Filters
Filters can improve the ratio \(n/s\) if the spectral distribution of the object differs from the sky background. Assuming reductions \(n \to xn\) and \(s \to ys\):
\[
\frac{1}{n} = \frac{x}{\beta} \sqrt{\frac{t_m q}{y s}}
\]
A gain in detectability occurs if \(x > \sqrt{y}\). This enhancement applies primarily to objects with line spectra, such as hydrogen emission nebulae, observed using interference filters centered on \(H_\alpha\) (656.3 nm). The setup, shown in Figure 15.4, integrates the filter near the telescope’s prime focus.
Conclusion
For hydrogen emission nebulae, narrowband interference filters significantly enhance contrast by reducing sky background photons while maintaining object signal strength. This technique has revolutionized faint object detection.
