2017年IOAA理论第10题-GOTO望远镜
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英文原题
(T10)GOTO [25 marks]
The Gravitational-Wave Optical Transient Observer (GOTO) aims to carry out searches of optical counterparts of any Gravitational Wave (GW) sources within an hour of their detection by the LIGO and VIRGO experiments. The survey needs to cover a big area on the sky in a short time to search all possible regions constrained by the GW experiments before the optical burst signal, if any, fades away. The GOTO telescope array is composed of 4 identical reflective telescopes, each with 40-cm diameter aperture and f-ratio of 2.5, working together to image large regions of the sky. For simplicity, we assume that the telescopes’ fields-of-view (FoV) do not overlap with one another.
a) Calculate the projected angular size per mm at the focal plane, i.e. plate scale, of each telescope. [6]
b) If the zero-point magnitude (i.e. the magnitude at which the count rate detected by the detector is 1 count per second) of the telescope system is 18.5 mag, calculate the minimum time needed to reach 21 mag at Signal-to-Noise Ratio (SNR) = 5 for a point source. We first assume that the noise is dominated by both the Read-Out Noise (RON) at 10 counts/pixel and the CCD dark (thermal) noise (DN) rate of 1 count/pix/minute. The CCDs used with the GOTO have a 6-micron pixel size and gain (conversion factor between photo-electron and data count) of 1. The typical seeing at the observatory site is around 1.0 arcsec. [8]
The Signal-to-Noise Ratio is defined as
\[ \text{SNR} \equiv \frac{\text{Total Source Count}}{\sqrt{\Sigma_{i}{ } \text{Noise}_{I}^{2}}}= \frac{\text{Total Source Count}}{\sqrt{\sigma_{\text{RON}}^{2}+\sigma_{\text{DN}}^{2}+\ldots}} \text{ , }\]
\[\sigma_{\text{RON}} =\sqrt{N_{\text{pix}}\cdot \text{RON}^2}\text{ , } \sigma_{\text{DN}} =\sqrt{N_{\text{pix}}\cdot \text{DN}\cdot t}\text{ , }\]
where t is the exposure time.
c) Normally when the exposure time is long and the source count is high then Poisson noise from the source is also significant. Determine the relation between SNR and exposure time in the case that the noise is dominated by Poisson noise of the source. Recalculate the minimum exposure time required to reach 21 mag with SNR = 5 from part b) if Poisson noise is also taken into consideration. The Poisson noise (standard deviation) of the source is given by $$\sigma_\text{source}=\sqrt{\text{Source Count}}$$ . In reality, there is also the sky background which can be important source of Poisson noise. For our purpose here, please ignore any sky background in the calculation. [6]
d) The typical localisation uncertainty of the GW detector is about 100 square-degrees and we would like to cover the entire possible location of any candidate within an hour after the GW is detected. Estimate the minimum side length of the square CCD needed for each telescope in terms of the number of pixels. You may assume that the time taken for the CCD read-out and the pointing change are negligible. [5]
