The corrections — SEXTANT

The spine

A reading is not a position

A sextant measures one angle, once, from a moving deck. Between that angle and a place on the Earth stand five named corrections, each with a sign, a table and a reason. A navigator who skips one is not slightly wrong; they are somewhere else.

Index error

The instrument does not read zero when it is pointed at zero. Every sextant has some, it is constant, and you measure it by sighting the horizon against itself before you start.

Here: The token's own scale and the pool's rounding: how many decimals the equity reports, and what the quote does at the last one. Constant, knowable in advance, and the only correction on this list that is free to remove.

Dip

The visible horizon is below true horizontal, by an angle that grows with the square root of your height above the sea. It exists only because of where the observer stands.

Here: Everything between you and the price that is a property of your position rather than the market: the fee, the spread you cross, the block you happen to land in.

Refraction

The atmosphere lifts every body above where it is. Zero overhead, and larger than the Sun at the horizon. It depends on the APPARENT altitude — the thing it is correcting — so it cannot be looked up from anything the observer can see directly.

Here: The gap between the standing answer and the market while it stands. It is not a function of the answer's age. It is a function of where in the interval you are, and the interval ends at the next print, which has not happened yet.

Semidiameter

You measure the edge of the Sun, because you can see an edge and you cannot see a centre. The half-width has to be added back.

Here: You transact at an edge — a bid or an ask — and the number everyone quotes is the middle. The half-spread has to be added back, with its sign, and the sign depends on which way you are going.

Parallax

You are on the surface of the Earth and the tables are computed for its centre. Negligible for a star, nine arcseconds for the Sun, a degree for the Moon: it grows as the body gets closer.

Here: The price is quoted for a size you are not trading. Negligible for a small order and the whole answer for a large one, and it grows as you get closer to the depth available.

The one that will not sit still

Refraction

Four of the five are constants or lookups. The third is the one this whole site is named for.

Refraction against apparent altitude, in arcminutes. Solid: the traced integral in js/refraction.js. Dashed: Bennett (1982), a published closed form. They agree to 0.13 arcminutes above five degrees and part company at the horizon, where the answer depends on the lowest few metres of air.

Light entering the atmosphere is bent downward, so every body is seen higher than it is. The lift is nothing overhead and 33.8 arcminutes at the horizon — larger than the Sun, which is 32.0 arcminutes across.

The consequence is on the front page of this site, moving: at the moment the Sun's upper limb appears to touch the sea, its centre is 49.8 arcminutes below the horizon and the entire disc has already set.

Horizon refraction traced at 7 surface temperature gradients, all of them ordinary. Highlighted: the standard atmosphere, the one the tables assume.

And here is the whole site in one chart. Refraction at the horizon is not a constant — across ordinary surface conditions the traced value runs from 29.1 to 52.2 arcminutes, a factor of 1.79. The almanac prints 34.5′, because the air a navigator is standing in is not a thing they can measure from the deck. That is a reading standing in for a condition, and it is exactly what a mark is.

The Sun's outline at five true altitudes, computed from the same integral as the hero. Dashed: a circle of the true angular size, for comparison.

The model against things it did not compute

Nineteen checks, and four of them are controls

A model that agrees with itself is not evidence. Every check compares the traced integral against something derived a different way: a published closed form, an analytic limit, an invariant the integrator never uses, or a control that is required to fail.

`node tools/verify-model.mjs`, run by every build.
check	got	want
refractivity of standard air at 633 nm	276.517ppm	276.5 ± 0.3	ok
plane-parallel limit, 45 to 70 degrees	0.250%	0 ± 0.6	ok
agrees with Bennett (1982) above 5 degrees	0.127'	0 ± 0.15	ok
control — and disagrees at the horizon	0.721'	0.72 ± 0.5	ok
refraction falls monotonically from horizon to zenith	1.000	1	ok
refraction at the zenith	0.000'	0 ± 0.001	ok
control — CONTROL — uniform steps in height give a wrong answer	14492.909'	must differ	ok
quadrupling the step count moves the horizon answer by	0.007'	0 ± 0.01	ok
traced dip against the 1.758 sqrt(h) rule	0.171%	0 ± 1.5	ok
true depression of the Sun's centre at apparent sunset	49.751'	50 ± 2	ok
the whole disc is below the horizon when its top edge appears to touch	49.751'	31.9888	ok
vertical flattening of the disc at the horizon	0.844	0.83 ± 0.09	ok
control — CONTROL — the blue end is lifted more than the red	0.688'	must differ	ok
inversion at which a horizontal ray curves as fast as the Earth	129K/km	115 ± 25	ok
a trace through a steeper inversion reports a duct	1.000	1	ok
lookup table against the integral, off-knot	0.005'	0 ± 0.02	ok
and its inverse round-trips	0.000'	0 ± 0.01	ok
control — CONTROL — horizon refraction across ordinary surface lapse rates	23.071'	must differ	ok
Bouguer invariant closes at the top of the atmosphere	0.000'	0 ± 0.01	ok