Atmospheric Refraction: Geometric and Apparent Solar Altitude

Why can the Sun appear higher than its geometric position?

A useful explanation needs more than a single “about half a degree” number. It must say which altitude is the input, what units the correction uses and whether a sunrise convention has already included refraction. This guide calculates a named approximation and separates it from actual weather-dependent observations.

Keep three quantities in separate columns

Geometric altitude is the angle used before this atmospheric correction. Apparent altitude is the corrected angle. Refraction is their difference. If h = 1° and R is about 0.3634°, the apparent result is about 1.3634°, not 0.3634° and not 0.6366°.

The term apparent can mean different things in different astronomy products, because coordinate pipelines can include several corrections. Here its meaning is explicit: the only difference being demonstrated is the declared atmospheric R(h). When reading another service, inspect its definition rather than assuming the field name guarantees the same calculation.

The Hong Kong Observatory explains the physical direction of the effect through bending of sunlight in the atmosphere: the observed image is raised relative to the unrefracted location. Its sunrise discussion concerns the upper edge of the solar disk. The centre-altitude worksheet below is a different, carefully labelled quantity.

The declared model covers geometric altitude 0°–45°

We use two branches of NOAA's published solar-position refraction approximation. The exercises deliberately restrict h to 0°–45°. They do not extrapolate the near-horizon formula through every negative altitude, a zenith singularity or unusual atmospheric layers.

For 0° ≤ h ≤ 5°, the correction in degrees is R = (1735 − 518.2h + 103.4h² − 12.79h³ + 0.711h⁴) / 3600. Here h is a decimal-degree number. The polynomial's output before division is in arcseconds.

For 5° < h ≤ 45°, put T = tan(h), with the angle converted correctly for the trigonometric function. Then R = (58.1/T − 0.07/T³ + 0.000086/T⁵) / 3600. A programming library expecting radians must receive h × π/180, not the degree value directly.

The exact 5° input uses the polynomial branch, following the published calculator's branch condition. The two fitted expressions need not join with mathematically perfect smoothness. A small feature at a piecewise-fit boundary is a property of the approximation, not a sudden physical layer that every ray encounters at exactly 5°.

Eleven altitude calculations

At h = 0°, all polynomial terms containing h vanish. The correction is 1735/3600 = 0.481944° after rounding, or 28.9167 arcminutes. The model therefore places the apparent centre above the geometric horizon when the geometric centre itself is at 0°.

At 1°, the polynomial numerator is 1735 − 518.2 + 103.4 − 12.79 + 0.711 = 1308.121 arcseconds. Dividing by 3600 gives approximately 0.363367°, and adding it to the input gives 1.363367°.

At 45°, tan h = 1, so the second expression becomes (58.1 − 0.07 + 0.000086)/3600, approximately 0.016119°. That is less than one arcminute, much smaller than the near-horizon correction in this fit. The table illustrates an altitude-dependent effect rather than a constant offset.

Six printed decimals make the arithmetic reproducible; they do not claim six-decimal observational accuracy. Real atmospheric conditions have not been supplied. The displayed precision belongs to a mathematical exercise with fixed coefficients.

Arcminutes are not minutes on a clock

One angular degree equals 60 arcminutes or 3600 arcseconds. Thus 30 arcminutes is 0.5°. These are angle units. They do not become 30 minutes of time simply because both words contain minutes.

The table's R in arcminutes is 60 times R in degrees. To recover degrees, divide by 60. For the h = 0° row, 28.9167 arcminutes is approximately 0.481944°, not 28.9167°.

Likewise, 0.5° is decimal notation, whereas 0°30′ is sexagesimal notation for the same angle. Writing 0.30° would mean 18 arcminutes. A clean report prints the unit with the number and keeps any later time conversion in a separately labelled calculation.

Recovering geometric altitude is an inverse problem

If the input is already apparent altitude a, the unknown h satisfies a = h + R(h). Simply subtracting R(a) uses the apparent angle where the formula expects the geometric angle. That shortcut generally does not reproduce the original forward relation.

The examples solve the equation numerically by bracketing h between 0° and 45° and repeatedly narrowing the interval. The selected targets lie within the model's attainable range and away from a fit-boundary ambiguity. Substituting each solved h back into h + R(h) reconstructs the target.

Not every apparent input belongs to this restricted worksheet. Its smallest geometric input is 0°, whose apparent result is already about 0.481944°. An apparent value below that cannot be inverted within the stated 0°–45° geometric range. It would require a broader, explicitly specified model, rather than silently accepting a negative solution.

Different corrections at the two limbs compress the vertical diameter

For a simple illustration, declare a geometric vertical disk diameter d = 0.5°. If the centre is h, the lower and upper geometric limb altitudes are h − 0.25° and h + 0.25°. Apply the correction to each limb separately.

The apparent vertical diameter is dapp = d + R(h + d/2) − R(h − d/2). In these cases the lower limb receives a larger upward correction, so the vertical separation shrinks. Translating the entire disk upward by the centre correction would miss that difference.

All listed limb inputs remain inside the model's 0°–45° range. The chosen 0.5° disk is a teaching input, not a date-specific measurement of the Sun. The last column compares vertical diameters using 100 × (d − dapp)/d; its denominator is the declared geometric vertical diameter.

This calculation is one-dimensional. It does not model the horizontal outline, magnification in every direction, a mirage or the full path of rays through a measured atmosphere. Vertical compression alone should not be reported as the same percentage change in total disk area.

See the correction and the differential effect

The upper curve answers how far a single direction is raised. The lower curve answers how differently two nearby directions are raised. Those are related but distinct questions. A large shared upward shift does not by itself determine the change in diameter; the difference between the limb corrections does.

Read each axis before comparing panels. The upper values are arcminutes, while the lower values are percentages of a declared diameter. The graph is a reproducible model calculation, not a photographed solar profile or a reconstruction of today's atmosphere.

Why −0.833° is not a universal correction to add

The familiar sunrise centre altitude near −0.833° combines two ideas: a solar semidiameter and an approximate horizon-refraction allowance. The Hong Kong Observatory describes an illustrative total of about 50 arcminutes, comprising roughly 16 arcminutes of solar radius and 34 arcminutes of refraction under average conditions.

That is an event convention for the upper limb at the horizon. It is not a rule to add 0.833° to every geometric centre altitude. It is also not the same input as our h = 0° row. In one case the centre is below the horizon when the upper limb appears; in the other the geometric centre has reached 0°.

If a published sunrise time already includes the conventional correction, adding it again double counts the effect. First establish whether a value is a geometric angle, an apparent angle or an event time calculated under an apparent-limb convention. The sunrise and twilight guide helps separate those definitions.

The same angular shift can imply different time offsets

A local linear estimate is Δt ≈ R / |dh/dt|, where the vertical altitude rate is expressed in degrees per unit time. A shallow crossing has a smaller vertical rate, so the same angular correction can correspond to a larger time displacement.

The rates are hypothetical and R is deliberately held at 0.5° for this comparison. No row is a predicted sunrise correction for a named place. In particular, the Sun's hour-angle rate is not automatically its vertical altitude rate. Latitude, declination and hour angle affect the path relative to the horizon.

The approximation becomes unreliable if the vertical rate changes appreciably over the estimated interval. Near a tangent touch, dividing by a rate close to zero is a warning that a simple linear estimate is inadequate. The polar-light article explains why a threshold may have no crossing at all.

A six-step comparison of two altitude reports

1. Confirm the same location, instant, body and reference horizon.
2. Identify whether each altitude is geometric or already corrected.
3. Record the model, its input range and the correction's units.
4. Apply R to the correct input angle, or solve the inverse relation if needed.
5. Keep solar radius, horizon obstruction and event conventions separate.
6. Compare the reconstructed result before rounding and document the remaining assumptions.

For example, two fields differing by approximately R(h) may be describing the same geometry before and after atmospheric correction. Two fields differing by a full sunrise convention may be answering different questions. The size of the discrepancy alone does not identify its cause.

For location-based event information, visit the sunrise and sunset page. This guide does not certify the refraction model used by every external application or every polar-coordinate response.

Model limits and source notes

NOAA's calculation notes explain that atmospheric conditions vary and that observed events can differ from calculated ones, especially when the solar path meets the horizon shallowly. This worksheet has no temperature profile, pressure observation, terrain silhouette or observer-height correction. It cannot predict an unusual refraction event from its fixed coefficients.

The coefficients and branch boundary were checked against both NOAA's explanatory page and its published calculator source. That source adds the returned correction to the geometric elevation. NOAA now states that the calculator is no longer actively supported or maintained; the references here document the method rather than endorse it for a critical schedule.

The altitude, inverse, limb and timing examples are original calculations with explicit assumptions. They explain the distinction between geometry and an atmospheric approximation. They do not replace an observed horizon, a date-dependent ephemeris or a measurement of atmospheric structure.

Frequently asked questions