# G. G. Bennett — The Calculation of Astronomical Refraction in Marine Navigation **Citation:** Bennett, G. G. "The Calculation of Astronomical Refraction in Marine Navigation." *The Journal of Navigation* 35, no. 2 (1982): 255–259. --- ## Overview Bennett's paper addresses a practical problem: Garfinkel's (1967) refraction algorithm — the standard adopted by the British and American Nautical Almanac Offices — is too complex for handheld calculators. The paper develops simple closed-form formulas that reproduce Garfinkel's output as a function of three measurable real-world variables: 1. **Observed altitude angle** ($h$, degrees above horizon) 2. **Air temperature** ($T$, °C) 3. **Atmospheric pressure** ($P$, mb) Bennett explicitly states he is **not assessing whether Garfinkel itself represents physical reality** — he is curve-fitting to an accepted computational standard. --- ## How the Real-World Variables Enter ### Altitude Angle ($h$) — The Primary Variable The relationship between refraction and altitude is highly nonlinear: - **Above 15°**: Refraction is small and well-approximated by the classical one-parameter form $R_M = K \cot h$, where $K \approx 1$. Deviations between computed and actual refraction are expected to be small. - **Near 0° (horizon)**: Refraction balloons to ~34 arcminutes and $\cot h$ diverges, so the simple formula breaks down completely. The entire challenge of the paper is bridging these two regimes with a single formula. ### Temperature and Pressure — Multiplicative Corrections Temperature and pressure modify the air's density and thus its refractive index. Bennett notes that **refraction is more sensitive to temperature than pressure**: a 1°C change in temperature causes approximately the same change in refraction as a 4 mb change in pressure. These variables enter as multiplicative corrections applied to the mean (standard-conditions) refraction. --- ## The Curve-Fitting Strategy Bennett's approach is **empirical curve fitting against Garfinkel's algorithm output**, not derivation from first principles. ### Step 1: Physics-Motivated Functional Form Start with $R_M = \cot(h + \text{something})$ — the cotangent captures the dominant geometric behavior of refraction through the atmosphere. ### Step 2: Fit the Low-Altitude Correction Bennett's best single formula (**Formula G**): $R_M = \cot\!\left(h + \frac{7.31}{h + 4.4}\right) \quad [\text{arcminutes}]$ The rational term $\frac{7.31}{h+4.4}$ shifts the cotangent argument just enough to: - Prevent divergence as $h \to 0$ - Match Garfinkel across the full 0–90° range **Max error: 0.07' (at 12° altitude)** — negligible for navigation. ### Step 3: Sinusoidal Residual Correction **Formula H** refines Formula G with an iterative correction on the residuals: $R_M = R'_M - 0.06 \sin(14.7\, R'_M + 13)$ where $R'_M$ is Formula G. This trims residuals to **0.015' max error** over the entire 0–90° range. ### Step 4: Temperature and Pressure Correction Rather than the crude Bowditch formula $\frac{510}{460+t} \cdot \frac{p}{29.83}$, Bennett fits: $R = \frac{P - 80}{930}\left(\frac{1}{1 + 8 \times 10^{-5}(R_M + 39)(T - 10)}\right) R_M$ This introduces a **coupling term** between refraction magnitude and temperature — $(R_M + 39)(T - 10)$ — capturing how temperature sensitivity itself varies with altitude. | | Bennett | Bowditch | |---|---|---| | Max error vs Garfinkel ($T$: -20 to 40°C, $P$: 970–1050 mb) | **0.2'** | **3.5'** | --- ## Comparison of Formulae Reviewed Bennett evaluates eight formulae (A–H) against Garfinkel. Key findings: | Formula | Source | Max Error (low alt) | Max Error (high alt) | Full Range? | |---|---|---|---|---| | A | HP 65 Nav Pac | $\infty$ (0°) | 0.1' | No (piecewise) | | B | Sinclair | 0.02' | 0.05' | No (piecewise, break at 15°) | | C | Bennett (simple) | 1.8' (0–3°) | 0.1' | Yes (single formula) | | D | HP 67/97 Nav Pac | 1.1' (0–4°) | 0.1' | Yes | | E | Almanac for Computers | 0.8' (0–0.3°) | 0.1' | Yes (complex) | | F | Lewis (Chebyshev) | 0.04' | 0.04' | Yes (10 coefficients) | | **G** | **Bennett (recommended)** | **0.07'** | **0.07'** | **Yes (single formula)** | | **H** | **Bennett (refined)** | **0.015'** | **0.015'** | **Yes (single formula)** | All formulae are accurate above 15°. The key differentiator is low-altitude performance and simplicity. --- ## Connection to Other Reviews - [[1944_Garfinkel_Review]] — Bennett curve-fits Garfinkel's algorithm output. The entire paper is a practical simplification of Garfinkel. - [[1939_Barrell_Sears_Review]] — The refractivity values encoded in Garfinkel's output (and thus in Bennett's fit) descend from B&S (1939). - [[1966_Edlen_Review]] — The B&S → Edlén → Garfinkel chain means Bennett's constants implicitly encode Edlén-era refractivity values. - [[1996_Ciddor_Review]] — If Garfinkel's refractivity were updated to Ciddor, Bennett's fitted constants would change slightly. - [[1941_Willis_Review]] — Willis provides the tropopause constant ($x_1 = 0.778$) that Garfinkel uses, which propagates into the tables Bennett fits. --- ## What This Means for Refraction Curve-Fitting ### This Is Empirical Fitting, Not Physics Derivation - The constants (7.31, 4.4, 0.06, 14.7, 13, 80, 930, $8 \times 10^{-5}$, 39) are **determined by minimizing residuals** against Garfinkel's tabulated output — not derived from atmospheric physics equations. - The functional forms (cotangent with rational offset, sinusoidal residual correction, coupled $T$/$P$ multiplier) are **chosen for mathematical convenience** while being flexible enough to capture the true curve shape. ### The Chain of Validation ``` Physical atmosphere → Garfinkel's algorithm (ray tracing through polytropic atmosphere) → Bennett's formula (curve fit to Garfinkel) → Inputs: h, T, P → Output: refraction in arcminutes (≤0.2' error) ``` The refraction values are "correct" in the sense that they match the Nautical Almanac tables to within fractions of an arcminute — well below the observational uncertainty of a marine sextant (~0.5–1'). ### Key Takeaway Bennett demonstrates that with the right choice of functional form and just three real-world measurables ($h$, $T$, $P$), you can reproduce the full complexity of atmospheric refraction to better than 0.2 arcminutes across all practical conditions. The paper is a practical demonstration that refraction is a **deterministic, predictable function** of observable variables — not a free parameter or a fudge factor. --- ## References (from paper) 1. Garfinkel, B. (1967). "Astronomical refraction in a polytropic atmosphere." *The Astronomical Journal* 72, No. 2. 2. Hewlett Packard (1974). *HP 65 Navigation Pac*. 3. Sinclair, A. T. "Data for astro-navigation in 1978 for use with small calculators." NAO Technical Note No. 44. 4. Hewlett Packard (1978). *HP67/HP97 Navigation Pac 1*. 5. *Almanac for Computers, 1981*. Nautical Almanac Office, USNO. 6. Bowditch, N. *American Practical Navigator* vol. 1 (1977); vol. 2 (1975). DMA Hydrographic Center.