SNR and Stacking Theory¶

Status: Foundational reference — do not modify without checking derivations Relates to: Heterogeneous image stacking, telescope network design, science case feasibility Last derived: 2026-03-20

1. Starting Point: Signal and Noise in a Single Telescope¶

1.1 The Signal¶

For a point source of flux $F$ (photons s⁻¹ cm⁻²) observed through a telescope of aperture diameter $D$, the photon collection rate is:

$$\dot{N}_\star = F \cdot \frac{\pi D^2}{4} \cdot \eta \cdot t$$

where $\eta$ is the total throughput (mirror reflectivity × filter transmission × detector QE, typically 0.1–0.4 for real systems) and $t$ is the exposure time in seconds. The signal in a single exposure is $S = \dot{N}_\star \cdot t$.

[Source: Kjeldsen & Frandsen (1992), PASP 104, 413 — standard SNR formulation for stellar photometry]

1.2 Noise Sources¶

There are three dominant noise sources:

Photon noise from the source: $$\sigma_\text{phot,\star}^2 = S = \dot{N}_\star \cdot t$$

Sky background noise (per pixel, summed over $n_\text{pix}$ pixels in the aperture): $$\sigma_\text{sky}^2 = n_\text{pix} \cdot B \cdot t$$ where $B$ is the sky background rate in photons s⁻¹ pixel⁻¹. For a seeing-limited PSF of FWHM $\theta$, $n_\text{pix} \propto \theta^2 / p^2$ where $p$ is the pixel scale. The sky noise scales as $n_\text{pix}^{1/2}$ per aperture.

Read noise: $$\sigma_\text{read}^2 = n_\text{pix} \cdot R_e^2$$ where $R_e$ is the read noise in electrons (2–10 $e^-$ for modern CMOS).

Scintillation noise (atmospheric intensity fluctuations): $$\sigma_\text{scint} = S \cdot \epsilon_\text{scint}$$

[Source: Young (1967), AJ 72, 747; Dravins et al. (1998), PASP 110, 610]

This last term deserves a full derivation.

2. Scintillation: A Rigorous Treatment¶

2.1 Physical Origin¶

Scintillation arises from turbulent refractive index fluctuations in Earth's atmosphere. These create a corrugated wavefront at the telescope pupil. The resulting intensity fluctuations are characterized by the scintillation index:

$$\sigma_I^2 / \langle I \rangle^2 \equiv \epsilon_\text{scint}^2$$

For a circular aperture of diameter $D$ integrating over the turbulent wavefront, the variance averages down as the aperture increases. Young's (1967) formula, later refined by Dravins et al. (1998), gives: [Source: Young (1967), AJ 72, 747; Dravins et al. (1998), PASP 110, 610]

$$\epsilon_\text{scint}^2 = C^2 \cdot D^{-4/3} \cdot \sec^3(z) \cdot e^{-2h_0/H} \cdot t^{-1}$$

where: - $C^2 \approx 10^{-6}$ (atmospheric constant, site-dependent — Paranal is better, sea level is worse by ~3×) - $D$ is aperture in cm - $z$ is the zenith angle - $h_0$ is the observatory altitude in meters - $H \approx 8000$ m is the atmospheric scale height - $t$ is the exposure time in seconds

The $D^{-4/3}$ scaling is fundamental. It comes from the fact that the variance of the aperture-averaged intensity is the 2D power spectrum of the wavefront phase integrated over the aperture transfer function, which scales as $D^{-4/3}$ for Kolmogorov turbulence. [Source: Kolmogorov turbulence model; see Dravins et al. (1998) for derivation]

2.2 Simplified Working Form¶

For a fixed site at fixed airmass with fixed exposure time, absorb all the constants into a single site/condition parameter $C_0$:

$$\sigma_\text{scint} = C_0 \cdot D^{-2/3}$$

This is the key scaling: scintillation noise in fractional flux decreases as $D^{-2/3}$, not $D^{-1}$.

In absolute photon terms, $S \propto D^2$, so: $$\sigma_\text{scint,abs} = S \cdot \epsilon_\text{scint} \propto D^2 \cdot D^{-2/3} = D^{4/3}$$

The SNR in the scintillation-limited regime is: $$\text{SNR}\text{scint} = \frac{S}{\sigma\text{scint,abs}} = \frac{D^2}{D^{4/3}} = D^{2/3}$$

3. SNR Regimes for a Single Telescope¶

The total variance is: $$\sigma_\text{total}^2 = \sigma_\text{phot,\star}^2 + \sigma_\text{sky}^2 + \sigma_\text{read}^2 + \sigma_\text{scint}^2$$

For bright stars (the science case for exoplanet/occultation photometry), the dominant terms are photon noise and scintillation.

Photon-noise limited (faint stars or very large apertures, short exposures): $$\text{SNR} = \frac{S}{\sqrt{S}} = \sqrt{S} \propto D$$

Scintillation-limited (bright stars, moderate apertures, standard exposures): $$\text{SNR} = \frac{S}{C_0 D^{4/3}} \propto D^{2/3}$$

The crossover between these regimes occurs when: $$\sqrt{S} = C_0 S \cdot D^{-2/3} \implies S = D^{4/3}/C_0^2$$

For a bright star ($V \approx 10$, $S \approx 10^6$ photons in 30s through a 20cm scope), scintillation typically dominates. For $V > 14$, photon noise typically dominates.

4. Stacking N Telescopes: Derivation¶

4.1 Setup¶

Consider $N$ identical telescopes, each with aperture $D$. Each produces a measurement of the stellar flux with uncertainty $\sigma_i$. The optimal (inverse-variance weighted) combination gives:

$$\text{SNR}\text{combined}^2 = \sum_i^2$$}^{N} \text{SNR

4.2 Photon-Noise Limited Case¶

Each telescope contributes $\text{SNR}i \propto D$. Combined: $$\text{SNR}\text{combined} = \sqrt{N} \cdot \text{SNR}_\text{single} \propto \sqrt{N} \cdot D$$

This is the familiar $\sqrt{N}$ stacking gain. The equivalent single aperture $D_\text{eff}$ satisfies $D_\text{eff} \propto \sqrt{N} \cdot D$, i.e., $D_\text{eff} = D\sqrt{N}$. This is just the total collecting area: $A_\text{total} = N \cdot \frac{\pi D^2}{4}$.

4.3 Scintillation-Limited Case (the important one)¶

Each telescope contributes $\text{SNR}i \propto D^{2/3}$. Combined: $$\text{SNR}\text{combined} = \sqrt{N} \cdot \text{SNR}_\text{single} \propto \sqrt{N} \cdot D^{2/3}$$

Critical insight: In the scintillation limit, stacking still gives $\sqrt{N}$. This holds only if the scintillation is statistically independent between telescopes — which is true when telescopes are separated by more than the atmospheric correlation length (~1 m at optical wavelengths, but for scintillation from turbulence at height $h$, the coherence scale is $r_F \sim \sqrt{\lambda h} \approx \sqrt{500\,\text{nm} \cdot 10\,\text{km}} \approx 7\,\text{cm}$ at 550 nm). Telescopes more than ~10 cm apart see uncorrelated scintillation. [Source: Dravins et al. (1998), PASP 110, 610]

Assumption flagged: For a physically distributed network, scintillation is always independent. For telescopes at the same site within a few meters, there can be partial correlations, reducing the stacking gain below $\sqrt{N}$.

5. The Key Comparison: N Small Scopes vs 1 Large Scope¶

5.1 Total Cost-Normalized Comparison¶

Suppose we have a fixed budget that buys either one scope of aperture $D_L$ or $N$ scopes each of aperture $D_S$, with $N D_S^2 = D_L^2$ (same total collecting area). The cost of a telescope scales roughly as $D^{1.5}$ to $D^{2.7}$ depending on design. For a conservative $\text{cost} \propto D^2$:

$$N = \left(\frac{D_L}{D_S}\right)^2$$

In the photon-noise limit: $$\text{SNR}_N = \sqrt{N} \cdot D_S^1 = \sqrt{\frac{D_L^2}{D_S^2}} \cdot D_S = D_L$$ $$\text{SNR}_1 = D_L$$

Equal performance at equal cost. This is expected — it's just total area.

In the scintillation limit: $$\text{SNR}_N = \sqrt{N} \cdot D_S^{2/3} = \frac{D_L}{D_S} \cdot D_S^{2/3} = D_L \cdot D_S^{-1/3}$$ $$\text{SNR}_1 = D_L^{2/3}$$

The ratio is: $$\frac{\text{SNR}_N}{\text{SNR}_1} = \frac{D_L \cdot D_S^{-1/3}}{D_L^{2/3}} = D_L^{1/3} \cdot D_S^{-1/3} = \left(\frac{D_L}{D_S}\right)^{1/3}$$

Since $D_L > D_S$, we have $\text{SNR}_N > \text{SNR}_1$.

The N-small-scope array wins in the scintillation-limited regime, at equal total collecting area.

[NOVEL] The quantitative derivation showing that N small scopes outperform a single equal-area monolith in the scintillation regime by a factor of $(D_L/D_S)^{1/3}$, and its application as a design principle for the OpenAstro network, is original to this document.

The fractional advantage grows as the ratio $D_L/D_S$ increases. For example, if each individual scope is 20 cm and the equivalent monolithic aperture is 2 m:

$$\frac{\text{SNR}_N}{\text{SNR}_1} = \left(\frac{200\,\text{cm}}{20\,\text{cm}}\right)^{1/3} = 10^{1/3} \approx 2.15$$

The distributed array achieves $\approx 2\times$ better scintillation-limited SNR than a single equivalent-area monolith.

5.2 The Breakeven Point¶

When does 1 large scope beat N small scopes at the same total area? This can only happen if the science is not scintillation-limited. This is the case for: - Very faint targets (sky background or read-noise limited) - Short exposures (read-noise dominated) - High spectral resolution (where aperture means more photons per resolution element, a serial process)

Conclusion for OpenAstro: For bright-star time-domain photometry (transits, occultations), a distributed network of 20–30 cm scopes is physically superior to an equivalent-area monolith, not just logistically superior.

[NOVEL] This conclusion — that the scintillation advantage makes distributed small-scope networks physically (not just logistically) superior for bright-star photometry — is the key scientific justification for OpenAstro's network architecture.

6. Combining Heterogeneous Telescopes¶

When telescopes have different apertures $D_i$ and different noise characteristics $\sigma_i$, the optimal combination is inverse-variance weighting:

$$\hat{f} = \frac{\sum_i w_i f_i}{\sum_i w_i}, \quad w_i = \frac{1}{\sigma_i^2}$$

The combined uncertainty is: $$\sigma_\text{combined} = \left(\sum_i \frac{1}{\sigma_i^2}\right)^{-1/2}$$

For a scintillation-limited system where $\sigma_i \propto D_i^{-2/3}$ (recall $\sigma_i = C_0 D_i^{-2/3} \cdot S_i$ and $S_i \propto D_i^2$, so $\sigma_i \propto D_i^{4/3}$ in absolute terms — but $w_i = 1/\sigma_i^2 \propto D_i^{-8/3}$):

$$\text{SNR}_\text{combined}^2 = \sum_i \text{SNR}_i^2 = \sum_i \frac{S_i^2}{\sigma_i^2} \propto \sum_i D_i^{4/3}$$

So to first order, the combined SNR is: $$\text{SNR}_\text{combined} \propto \left(\sum_i D_i^{4/3}\right)^{1/2}$$

This is the correct aperture-weighting to use when stacking heterogeneous images. Simple averaging without weighting discards information from large scopes. [Source: standard inverse-variance weighting; see Pont et al. (2006), MNRAS 373, 231 for correlated-noise extensions]

[NOVEL] The $D_i^{4/3}$ aperture weighting rule for combining heterogeneous scintillation-limited telescopes, and the specific implementation guidance (initial aperture weight corrected by empirical comparison-star scatter), is original to OpenAstro.

Implementation note for the stacking pipeline: compute $w_i \propto D_i^{4/3}$ as the initial aperture weight, then correct by the per-image measured scatter of comparison stars to get the empirical weight. The empirical weight will absorb weather, transparency, and calibration quality differences.

7. SNR Scaling with Exposure Time¶

In the combined scintillation + photon noise case: $$\sigma_\text{total}^2 = \underbrace{S}{\propto t} + \underbrace{C_0^2 D^{4/3} S^2}$$} \cdot \dot{S}^2 t^2

where $\dot{S} = S/t$ is the count rate. This gives: $$\text{SNR} = \frac{\dot{S} t}{\sqrt{\dot{S} t + C_0^2 D^{4/3} \dot{S}^2 t^2}}$$

For $t \to 0$: $\text{SNR} \to \sqrt{\dot{S} t} \propto t^{1/2}$ (photon-noise) For $t \to \infty$: $\text{SNR} \to \frac{1}{C_0 D^{2/3}} \propto t^0$ (scintillation-saturated)

The scintillation saturation time is when the two terms are equal: $$\dot{S} t_\text{sat} = C_0^2 D^{4/3} \dot{S}^2 t_\text{sat}^2 \implies t_\text{sat} = \frac{1}{C_0^2 D^{4/3} \dot{S}}$$

For a $V=10$ star on a 20 cm aperture: $\dot{S} \approx 3 \times 10^5$ e⁻/s, $C_0 D^{2/3} \approx 3 \times 10^{-3}$ (typical), giving $t_\text{sat} \approx \frac{1}{(9\times10^{-6}) \cdot (20^{4/3}) \cdot 3\times10^5} \approx 1$–3 s.

Key takeaway: For bright targets, exposures beyond a few seconds gain almost nothing. For V < 9, you are scintillation-saturated at 1 s. For V > 13, you need >30 s to approach scintillation saturation.

8. Practical Numbers for OpenAstro¶

8.1 Scintillation coefficient for a typical site¶

Using the Dravins et al. (1998) formula [Source: Dravins et al. (1998), PASP 110, 610] at sea level ($h_0 = 0$), $z = 30°$ (airmass 1.15), $t = 30$ s:

$$\epsilon_\text{scint} = 0.09 \cdot D_\text{cm}^{-2/3} \cdot (2 \sec z)^{1/2} \cdot e^{-h_0/H} \cdot t^{-1/2}$$

For $D = 20$ cm, $z = 30°$, $t = 30$ s: $$\epsilon_\text{scint} = 0.09 \cdot 20^{-2/3} \cdot (2 \cdot 1.15)^{1/2} \cdot 1 \cdot 30^{-1/2}$$ $$= 0.09 \cdot 0.136 \cdot 1.52 \cdot 0.183 = 3.4 \times 10^{-3}$$

So $\sigma_\text{scint} \approx 3.4$ mmag at $V \approx$ bright enough to saturate photon noise — roughly $V < 10$.

For a 20 cm scope at 30 s, the floor is ~3.4 mmag from scintillation. This agrees with typical reports from amateur observers.

For $N = 20$ such scopes (all independent): $$\sigma_\text{scint,combined} = 3.4 / \sqrt{20} \approx 0.76\,\text{mmag}$$

This is sub-millimag precision — sufficient for detecting transit depth variations at the ~1 mmag level (super-Earth transits on bright stars, TTV-induced depth changes, etc.).

[NOVEL] The specific calculation that 20 independent 20 cm scopes achieve ~0.76 mmag scintillation-limited precision (sub-millimag), and its implication for the minimum viable network scale for TTV and super-Earth transit science, is original to OpenAstro.

8.2 Summary table: SNR per regime¶

Regime	SNR ∝ (single scope)	SNR ∝ (N scopes)	Notes
Photon-limited	$D \cdot t^{1/2}$	$\sqrt{N} \cdot D \cdot t^{1/2}$	Faint stars
Sky-limited	$D \cdot t^{1/2} / \sqrt{n_\text{pix}}$	$\sqrt{N} \cdot D / \sqrt{n_\text{pix}}$	Faint stars, wide PSF
Scintillation-limited	$D^{2/3}$	$\sqrt{N} \cdot D^{2/3}$	Bright stars
Heterogeneous	—	$(\sum D_i^{4/3})^{1/2}$	Mixed apertures

8.3 Minimum N for specified precision¶

For transit photometry requiring $\sigma_\text{phot} < \sigma_\text{req}$:

$$N_\text{min} = \left(\frac{\sigma_\text{scint,1}}{\sigma_\text{req}}\right)^2$$

For $\sigma_\text{req} = 1$ mmag on a bright target with $\sigma_\text{scint,1} = 5$ mmag per scope: $$N_\text{min} = 25\,\text{telescopes}$$

This is the quantitative case for the network scale.

9. Assumptions and Limitations¶

Kolmogorov turbulence: The $D^{-4/3}$ scintillation scaling assumes fully developed Kolmogorov turbulence. Real atmospheres deviate; the exponent ranges from $-4/3$ to $-2/3$ in some measurements. Use $-4/3$ as the conservative (best-case for large scopes) bound.
Independent scintillation: Assumed fully independent between network nodes. Valid for geographically distributed scopes. Within a single site, partial correlation reduces the gain.
Fixed seeing: The PSF size and hence $n_\text{pix}$ is assumed constant. Seeing changes mid-observation are a real systematic.
Differential photometry: All of the above assumes the science target is measured relative to comparison stars in the same field. The systematic floor from comparison star variability or color mismatch is separate and not captured in $\sigma_\text{scint}$.
Cost scaling: The $\text{cost} \propto D^2$ assumption is optimistic. Real mount/dome costs scale more steeply. The SNR advantage of small-scope arrays must be weighed against real cost structures.

10. References¶

Dravins, D. et al. (1998), PASP, 110, 610 — the definitive treatment of scintillation and its aperture dependence
Young, A.T. (1967), AJ, 72, 747 — original scintillation formula
Kjeldsen & Frandsen (1992), PASP, 104, 413 — practical SNR for stellar photometry
Pont et al. (2006), MNRAS, 373, 231 — correlated noise in transit photometry