= Right ascension
{title2=$\alpha$}
{wiki}
<Right ascension> $\alpha$ is the eastward <angle> along the reference <celestial equator> from the <reference equinox> to a direction's <hour circle>, conventionally reduced to $0\leq\alpha<24^{\rm h}$. Its hours label <angles>: $24^{\rm h}=360^\circ$, $1^{\rm h}=15^\circ$, $1^{\rm m}=15'$ and $1^{\rm s}=15''$. Thus $6^{\rm h}30^{\rm m}=97.5^\circ=13\pi/24$ <radians>. A minute of <right ascension> differs from a minute of arc, and the actual arc on a fixed-<declination> parallel also contains the <cosine> factor described under <declination>.
The hours become useful through <sidereal time>. At local <sidereal time> $8^{\rm h}$, a synthetic direction of <right ascension> $6^{\rm h}30^{\rm m}$ has west-positive <hour angle> $1^{\rm h}30^{\rm m}=22.5^\circ$ and has passed its upper <meridian transit>. Conversely, at $5^{\rm h}30^{\rm m}$ it has <hour angle> $-1^{\rm h}$ and has yet to cross. Interpret these intervals in sidereal hours; ordinary civil hours are not interchangeable with them. The <right ascension> of the upper <celestial meridian> advances as the <Earth> turns, while the synthetic fixed target's <right ascension> stays unchanged.
An instrument makes this geometry tangible. DePauw University's description of McKim Observatory documents a meridian <optical telescope> with an angular circle, astronomical clocks and a chronograph. The circle measures zenith distance, and the crossing <time> supplies <right ascension>. Clock pulses and the observer's electrical key mark a rotating paper record, so crossing <time> is read relative to the clock marks. One clock kept <sidereal time> and another solar <time>. This apparatus separates two measured quantities: when a <star> crosses the <celestial meridian>, and at what <angle> from the <zenith>. https://www.depauw.edu/about/campus/mckim-observatory/observatory-instruments/[DePauw University: McKim's meridian telescope, angular circle, clocks and chronograph]{ref}.
A teaching reconstruction would fix an <optical telescope> to sweep the <celestial meridian>, align its central sighting line, and observe a recognizable reference <star> followed by a target. Record each central-line crossing against a clock or timestamped camera, and read the angular circle at upper <meridian transit>. A short sequence shows the image moving across the stationary sighting line; the central crossing supplies a repeatable event. In the ideal model, successive upper crossings of a fixed direction recur once per sidereal day, and directions with increasing <right ascension> cross later in the same cycle. These are predictions for the reconstruction, rather than reported McKim measurements. Tatum derives the equality between local <sidereal time> and <right ascension> at upper <meridian transit>. https://www.astro.uvic.ca/~tatum/celmechs/celm6.pdf[Jeremy Tatum: Section 6.3, timing upper meridian transit]{ref}.
Calibrate the clock's zero by comparing its reference crossing readout $C_r$ with the reference <star>'s predicted apparent <right ascension> $\alpha_r$ in the same date convention: the additive correction is $b=\alpha_r-C_r$, modulo $24^{\rm h}$. Then infer target <right ascension> $\alpha_t=C_t+b$ if the clock rate is correct between crossings. In a purely synthetic example, $\alpha_r=6^{\rm h}00^{\rm m}00^{\rm s}$ and $C_r=6^{\rm h}00^{\rm m}20^{\rm s}$ imply $b=-20^{\rm s}$. A target readout $C_t=6^{\rm h}30^{\rm m}20^{\rm s}$ therefore gives $\alpha_t=6^{\rm h}30^{\rm m}00^{\rm s}=97.5^\circ$. The $20^{\rm s}$ clock offset is $300''$ in <right ascension>, while its local east-west angular effect at <declination> $60^\circ$ is approximately $150''$. The reference and target in this calculation are invented directions, with no claim about any current <star>'s coordinates.
\Image[/equatorial-coordinates-synthetic-transit.png]
{title=Synthetic meridian observation: changing altitude, fixed right ascension, and a calibrated sidereal-clock readout}
Controls give the inference its meaning. Use independently established <geographical latitude> and an angular-circle zero for <declination>; check sighting-line alignment against several reference <stars>, and bracket the target in <time> with references to reveal clock drift. Record north or south of the <zenith>, distinguish upper from lower <meridian transit>, and account for <atmospheric refraction>. Convert the observing timestamp to local <sidereal time> using the site's east-positive longitude correction, rather than a time-zone offset; precise conversion uses the <Earth>'s measured rotation <time>. A reference <star> calibrates a zero only after its catalogue direction has been converted to the observation convention. Repeated crossings reduce random <measurement error>, while an incorrect clock rate, axis alignment or catalogue convention remains a systematic limitation. https://aa.usno.navy.mil/faq/GAST[US Naval Observatory: Local sidereal time, longitude convention and rotation-time input]{ref}. These controls are requirements inferred from the geometric and clock model, not a claimed historical observing log.
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