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📊 Deep Sky Imaging Optimizer – Math & Logic

This document explains the equations and reasoning behind the imaging optimizer.

1. Signal-to-Noise Ratio (SNR)

For stacked images, the raw SNR is estimated as:

$$ SNR = \sqrt{N \cdot t} $$

Where:

  • $N$ = number of light frames
  • $t$ = exposure time per frame (in seconds)

👉 Interpretation:

  • Doubling the number of frames improves SNR by $\sqrt{2}$
  • Doubling the exposure time per frame also improves SNR by $\sqrt{2}$

2. Normalized SNR

We scale the raw SNR against a maximum possible SNR ($SNR_{max}$) to get a percentage value:

$$ SNR_{\text{norm}} = \frac{SNR}{SNR_{max}} \times 100 $$

Where:

  • $SNR_{max}$ is an arbitrary "perfect" SNR reference (e.g. 1000 in code)

3. Required Time for Target SNR

To compute how much total integration time is needed to reach a target normalized SNR:

$$ T_{req} = \frac{SNR_{target}^2}{t} $$

Where:

  • $SNR_{target}$ = desired raw SNR (derived from normalized percentage)
  • $t$ = exposure time per frame

This determines how long you must expose in total to meet the SNR goal.

4. Exposure Time Recommendation

Exposure time is based on sky brightness (Bortle scale) and whether you use autoguiding:

  • With guiding:

$$ t = \begin{cases} 240 & \text{if } Bortle \leq 3 \\ 180 & \text{if } Bortle \leq 5 \\ 90 & \text{if } Bortle \leq 7 \\ 45 & \text{otherwise} \end{cases} $$

  • Without guiding:

$$ t = \begin{cases} 120 & \text{if } Bortle \leq 3 \\ 90 & \text{if } Bortle \leq 5 \\ 60 & \text{if } Bortle \leq 7 \\ 30 & \text{otherwise} \end{cases} $$

5. Calibration Frame Scaling

Calibration frames are balanced against the number of light frames:

  • Bias Frames:

$$ N_{bias} = \min(\max(\lfloor 0.2 \cdot N_{light} \rfloor, 10), 50) $$

  • Dark Frames:

$$ N_{dark} = \min(\max(\lfloor 0.2 \cdot N_{light} \rfloor, 5), 30) $$

  • Flat Frames:

$$ N_{flat} = \min(\max(\lfloor 0.15 \cdot N_{light} \rfloor, 5), 20) $$

This ensures you don’t take too few or too many calibration frames.

6. Calibration Time Estimate

Total time spent on calibration:

$$ T_{calib} = N_{bias} \cdot 0.005 + N_{flat} \cdot \max(0.1, 0.1t) + N_{dark} \cdot t $$

Where:

  • $0.005$ = 5 ms per bias
  • Flats ≈ 10% of light exposure (min 0.1 s)
  • Darks = same exposure as light frames

7. Channel Balancing

For LRGB (or SHO narrowband) imaging, integration is divided by weights:

$$ T_{channel} = w_i \cdot T_{total} $$

Where:

  • $w_i$ = weight for each channel (e.g., 0.5 for Luminance, 0.166 for RGB)
  • $T_{total}$ = total planned integration time

8. Grand Total Time

Finally, the total session time (lights + calibration) across all channels:

$$ T_{grand} = \sum_{i=1}^{channels} \big( T_{light,i} + T_{calib,i} \big) $$

✨ With these equations, the program estimates:

  • Optimal exposure time
  • Number of light frames
  • Required calibration frames
  • Session time per channel and grand total