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:

\text{\sl SNR} = \frac{S_{r}t}{\sqrt{S_{r}t + (Sky_{r}t) + \text{\sl DC}_{r}t + RN^{2} + (Thermal_{r}t) }}

This can be easily changed to:

S_{r}^2 \cdot t^{2} - \text{\sl SNR}^{2} \cdot [ S_r + (Sky_r) + \text{\sl DC}_r + (Thermal_r) + ] \cdot t - \text{\sl SNR}^{2} \cdot \text{\sl 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} + \text{\sl BG}_{r}) \cdot \text{\sl SNR}^{2}
c = \text{\sl RN}^{2} \cdot \text{\sl SNR}^{2}
\text{\sl BG}_{r} =  (\text{\sl ES}_{r}) + (\text{\sl ZL}_{r}) + (\text{\sl GL}_{r}) + \text{\sl DC}_{r} + (Thermal_{r})

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

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