= Declination
{title2=$\delta$}
{wiki}
<Declination> $\delta$ is the signed <angle> from the reference <celestial equator> to a direction, measured along its <hour circle>: $-90^\circ\leq\delta\leq+90^\circ$. Positive <declination> means celestial north, rather than necessarily north of the observer's <zenith>. A direction with $\delta=0$ lies on the <celestial equator>; $\delta=+90^\circ$ is the north reference <celestial pole>. <Declination> is an <angle>, not a height or a measure of <Euclidean distance>. https://aa.usno.navy.mil/faq/asa_glossary[US Naval Observatory: Astronomical Almanac glossary, declination and culmination]{ref}.
The connection to an observation follows from a <dot product>. In the elementary spherical-<Earth>, unrefracted model, an observer at <geographical latitude> $\phi$ has <zenith> direction $\mathbf z=(\cos\phi\cos\Theta,\cos\phi\sin\Theta,\sin\phi)$ in the same reference axes. Since $\mathbf n\cdot\mathbf z=\sin a$ for <altitude> $a$, the <cosine addition formula> gives
$$
\sin a=\sin\phi\sin\delta+\cos\phi\cos\delta\cos H.
$$
At upper <meridian transit>, $H=0$, so $a_{\rm upper}=90^\circ-|\phi-\delta|$. For the synthetic site $\phi=52^\circ$ and synthetic direction $\delta=20^\circ$, this yields $a_{\rm upper}=58^\circ$. A direction with $\delta=70^\circ$ instead reaches $72^\circ$, north of the <zenith>; positive <declination> does not imply a crossing on the southern half of the <celestial meridian>. A negative computed upper <altitude> means the direction never rises above the geometric <astronomical horizon> at that site.
For an angular readout, define signed zenith distance $z_s$ on the upper <celestial meridian>, positive toward the south and negative toward the north. Then $\delta=\phi-z_s$. A synthetic southward readout $z_s=32^\circ$ at $\phi=52^\circ$ recovers $\delta=20^\circ$. An unsigned distance $|z_s|$ alone would leave two possibilities, $\phi\pm|z_s|$, when both are in range. A calibrated <optical telescope> must therefore record which side of the <zenith> it points toward. In this linear relation, a $1'$ error in the assumed <geographical latitude> produces a $1'$ error in inferred <declination>; averaging repeated readings cannot remove that shared offset. Near the <astronomical horizon>, uncorrected <atmospheric refraction> also invalidates the geometric readout.
The <unit vector> construction explains a subtle feature of sky maps. At fixed <declination>, changing <right ascension> traces a parallel whose radius on the unit <sphere> is $\cos\delta$. Its arc <length> is $\cos\delta\,|d\alpha|$, with <angles> in <radians>; at $\delta=60^\circ$, a $15^\circ$ <right ascension> increment spans a $7.5^\circ$ arc along that parallel. This arc is not generally the shortest <great circle> separation. At either <celestial pole>, all <right ascension> values represent the same direction: the <coordinate singularity> belongs to the coordinate grid. The <unit vector> remains regular there.
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