= Astronomical horizon
The <astronomical horizon> is the <great circle> where the horizontal <plane> through the observer intersects the <celestial sphere>. It is perpendicular to the local vertical, so every direction on the <astronomical horizon> has <altitude> $0^\circ$ and <zenith distance> $90^\circ$. In the local <horizontal coordinate system>, it consists of the <unit vectors> satisfying $\mathbf n\cdot\mathbf e_U=0$. This geometric reference remains defined when buildings or mountains hide it; a visible skyline is not a calibration of zero <altitude>.
Three boundaries must be distinguished. The <astronomical horizon> fixes the angular reference. Terrain supplies a site-dependent boundary with apparent <altitude> $t(\eta)$ that changes with <azimuth>. <Atmospheric refraction> changes the apparent direction of incoming <light>, and can also change the apparent distant boundary. A point source clears an opaque terrain boundary only if $\gamma_{\rm app}>t(\eta)$ when both are expressed in the same apparent <horizontal coordinate system>; this clearance condition does not guarantee visibility through clouds or bright sky. In the synthetic diagram, a source with geometric <altitude> $3^\circ$ and an imposed illustrative upward correction $R=0.2^\circ$ remains behind a $4^\circ$ boundary. The chosen $R$ is an algebraic illustration, not a fitted atmospheric prediction.
\Image[/horizontal-sky-coordinates-level-and-skyline.png]
{title=Synthetic calibration triangle and invented terrain boundary; angular zero, apparent altitude and geometric horizon are separate}
Even an unobstructed sea boundary need not coincide with the <astronomical horizon>. On a spherical <Earth> of radius $r$, an observer at height $q$ sees the unrefracted tangent to the surface at a downward <angle> $D$ satisfying $\cos D=r/(r+q)$. The right-angle tangent <triangle> supplies this relation; the visible boundary then has <altitude> $-D$. With illustrative $r=6371\,\mathrm{km}$ and $q=10\,\mathrm m$, $D\simeq0.1015^\circ$, before <atmospheric refraction>. Raising the eye therefore changes the visible boundary without redefining the horizontal <plane>.
The US Naval Observatory's rise/set definitions explicitly distinguish the geometric reference from apparent contact with a visible disk: the standard <Sun> calculation puts its centre $50'$ below the horizontal <plane>, combining an average $16'$ radius with $34'$ of <atmospheric refraction>. An angular minute is $1'=1/60^\circ$ in the <minute and second of arc> convention. This is an operational average under stated conditions, rather than a universal zero of <altitude>. The same source identifies weather, local topography and observer height as limits on predicted rise/set <time>. An observation log should therefore retain the level reference, site and eye height, terrain profile, observation <time>, and whether each reported <altitude> is geometric or apparent. https://aa.usno.navy.mil/faq/RST_defs[US Naval Observatory: Rise, set and twilight definitions, geometric horizon and observational limitations]{ref}.
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