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Telescope Resolution Observation Time Estimator

The Telescope Resolution Observation Time Estimator solves for the exposure time needed to reach a target signal-to-noise ratio with a specified source rate, background rate, read noise, and number of exposures. It is useful when an optical resolution goal is already defined and the next question is how much integration time is needed to obtain statistically useful data at that resolution.

The estimator uses a standard photon-counting noise model and reports both per-exposure and total integration time. It is most useful for preliminary planning; observatory-specific exposure-time calculators should be used when detailed throughput, sky brightness, detector behavior, atmospheric transmission, or saturation limits matter.

Calculator inputs

e⁻/s
e⁻/s
e⁻ rms
exp
SNR
Result
—
Required total integration time
Exposure time each—
Total source electrons—
Solved SNR—

1. Enter the detected source rate
Use an electron rate after telescope throughput and detector efficiency are applied.

2. Enter background and read noise
Use rates and noise values for the same extraction aperture as the source.

3. Choose the number of exposures
Split the total integration into the planned number of detector reads.

4. Set the target SNR
Enter the combined signal-to-noise ratio required for the observation.

5. Review the solved time
Check both total and per-exposure time, then verify saturation, cosmic-ray, visibility, and scheduling limits separately.

For N exposures of duration t: SNR = (S × N × t) / sqrt((S + B) × N × t + RN² × N)
Solve for total time T = N × t using: S²T² − SNR²(S+B)T − SNR²RN²N = 0
Use the positive quadratic root for T

S is source electron rate, B is background electron rate, RN is effective read noise per exposure, N is number of exposures, and T is total integration time. The solution assumes independent photon and read noise and constant rates throughout the observation.

What the result means

This is the total integration time needed for the positive solution of the entered source, background, read-noise, exposure-count, and target-SNR model.

The result should be checked against instrument saturation limits, overheads, target visibility, and observatory-specific performance models.

Given

  • Source = 25 e⁻/s
  • Background = 40 e⁻/s
  • Read noise = 5 e⁻
  • 8 exposures
  • Target SNR = 50

Calculation
Solve 25²T² − 50²(25+40)T − 50²×5²×8 = 0. The positive root gives T ≈ 263.0 s. Per exposure = 263.0/8 = 32.9 s.

Result
Required total integration ≈ 263 s (4.38 min).

The target SNR is reached in the simplified model with about 33 seconds per exposure across eight exposures.

Why can more exposures require slightly more total time?

More detector reads add read-noise variance. When read noise matters, splitting the same observation into many short exposures can reduce efficiency.

What happens in a background-dominated observation?

When background shot noise is much larger than read noise, the required time is driven mainly by source and background rates. The exact number of exposures then matters less, provided saturation and other limits are satisfied.

Does this estimator account for telescope resolution directly?

No. It assumes the source and background rates already correspond to the chosen optics, aperture, sampling, and extraction region. Use a resolution calculator separately to verify the image-quality requirement.

Can I use photon rates instead of electron rates?

Only if detector quantum efficiency is effectively included so that the rates represent detected electrons. Otherwise convert expected photons to detected electrons first.

Why should I still check a facility exposure-time calculator?

Real instruments have wavelength-dependent throughput, detector nonlinearity, saturation, sky emission, aperture losses, and other effects that this compact analytical model does not include.