To determine the required exposure time ($t$) for a specific Signal-to-Noise Ratio ($SNR$) in a CCD, the standard SNR formula is rearranged into a quadratic equation, as noise sources depend differently on time (photon/dark noise are $\propto t$, read noise is constant). [1, 2, 3]  
The inversion result is typically given by the formula: $t = \frac{SNR^2 (C_{sky} + C_{dark}) + \sqrt{SNR^4 (C_{sky} + C_{dark})^2 + 4 SNR^2 C_{source} N_{pix} R_{N}^2}}{2 C_{source}^2}$ 
(Note: This formula assumes the source signal ) 
1. The Standard CCD SNR Equation 
The signal-to-noise ratio is given by: $SNR = \frac{C_{s}t}{\sqrt{C_{s}t + N_{pix}(C_{sky}t + C_{dark}t + R_{N}^2)}}$ 

Where: 

• : Exposure time (seconds) 
• : Signal count rate from source (e⁻/pixel/sec) 
• : Sky background rate (e⁻/pixel/sec) 
• : Dark current rate (e⁻/pixel/sec) 
• : Read noise (e⁻ rms/pixel) 
• : Number of pixels in the aperture [4, 5, 6, 7]  

2. Inverted Equation for Exposure Time ($t$) 
To solve for $t$, rearrange the equation into a quadratic form ($At^2 + Bt + C = 0$). The solution for $t$ that yields the required $SNR$ is: [1, 8]  
$t = \frac{SNR^2 \left(C_{s} + N_{pix}(C_{sky} + C_{dark})\right) + \sqrt{SNR^4 \left(C_{s} + N_{pix}(C_{sky} + C_{dark})\right)^2 + 4 SNR^2 C_{s}^2 N_{pix} R_{N}^2}}{2 C_{s}^2}$ 

3. Simplified Scenarios 
Depending on the dominant noise source, the equation simplifies: 

• Background/Dark Current Limited ():$t \approx \frac{SNR^2 (C_{sky} + C_{dark})}{C_{s}^2}$ 
• Source Photon Noise Limited ():$t \approx \frac{SNR^2}{C_{s}}$ 
• Read Noise Limited ():$t \approx \frac{SNR \cdot R_{N}\sqrt{N_{pix}}}{C_{s}}$ 

4. Input Parameters Needed 
To calculate $t$, you must know: 

1. Target SNR (e.g., 10, 100) 
2. Source Rate (): Detected electron rate from target. 
3. Background Rate (): Sum of sky and thermal noise. 
4. Read Noise (): Per-pixel camera read noise. 
5. : Size of the aperture or ROI. [9, 10, 11, 12, 13]  

AI responses may include mistakes.

[1] https://www.stsci.edu/instruments/wfpc2/Wfpc2_hand5/ch6_exposuretime7.html
[2] https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0101810
[3] https://stacks.cdc.gov/view/cdc/193834/cdc_193834_DS1.pdf
[4] https://hst-docs.stsci.edu/stisihb/chapter-6-exposure-time-calculations/6-4-computing-exposure-times
[5] https://scientificimaging.com/knowledge-base/signal-and-noise-quantitative-explanation/
[6] https://www.astropy.org/ccd-reduction-and-photometry-guide/v/dev/notebooks/03-01-Dark-current-The-ideal-case.html
[7] https://academic.oup.com/mnras/article/509/4/6111/6442285
[8] http://www2.lowell.edu/rsch/LMI/ETCMethod.pdf
[9] https://www.allaboutcircuits.com/technical-articles/quantum-efficiency-and-snr-ccd-image-sensors/
[10] https://pmc.ncbi.nlm.nih.gov/articles/PMC8272111/
[11] https://www.reddit.com/r/astrophotography/comments/2hx4v7/issues_with_biasoffset_master_frame_in_dss/
[12] https://www.frontiersin.org/journals/astronomy-and-space-sciences/articles/10.3389/fspas.2022.871163/full
[13] https://irsa.ipac.caltech.edu/data/SPITZER/SWIRE/SWIRE_EN1_columns.html

