Calculating TimeΒΆ

The Exposure Time is calculated based on the user input SNR, instrument configuration and input source spectrum. For spectroscopic modes, this value is calculated at a particular wavelength which is specified by the user in Section 2 of the ETC forms.

Please make sure the wavelength entered in this section is covered by the selected mode.

With this information, the count rate terms (in photons/sec) are retrieved from pysynphot. Each photon count rate term is depicted below with an r subscript.

Taking formula (ii) from the previous page and expanding the count terms, we get the following:

\mathrm{SNR} = \frac{S_{r}t}{\sqrt{S_{r}t + (\mathrm{Sky}_{r}t) + \mathrm{DC}_{r}t + \mathrm{RN}^{2} + (\mathrm{Thermal}_{r}t) }}

This can be easily changed to:

S_{r}^2 \cdot t^{2} - \mathrm{SNR}^{2} \cdot [ S_r + (\mathrm{Sky}_r) + \mathrm{DC}_r + (\mathrm{Thermal}_r) + ] \cdot t - \mathrm{SNR}^{2} \cdot \mathrm{RN}^{2} = 0

Then solving for t gives us:

t = \frac{b + \sqrt{b^2 + 4ac}}{2a}

Where

a = S_{r}^{2}
b = (S_{r} + \mathrm{BG}_{r}) \cdot \mathrm{SNR}^{2}
c = \mathrm{RN}^{2} \cdot \mathrm{SNR}^{2}
\mathrm{BG}_{r} = (\mathrm{ES}_{r}) + (\mathrm{ZL}_{r}) + (\mathrm{GL}_{r}) + \mathrm{DC}_{r} + (\mathrm{Thermal}_{r})

S_{r} = Source count rate
\mathrm{SNR} = Signal To Noise ratio
\mathrm{ES}_{r} = Earthshine count rate (optional field)
\mathrm{ZL}_{r} = Zodiacal Light count rate (optional field)
\mathrm{GL}_{r} = Geocoronal Emission Line(s) count rate (optional field)
\mathrm{DC}_{r} = Dark Current count rate
\mathrm{Thermal}_{r} = Thermal background count rate (optional field)
\mathrm{RN} = Read Noise

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