# Apparent celestial speed

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Image: The angular displacement (θ = ωt) from the apparent motion of a star in a given amount of time (t) is proportional to the cosine of its declination. In this long-exposure image from the Terran Northern Hemisphere, stars appear to be circling the North Celestial Pole. [1]

Apparent celestial speed is the apparent angular speed (ω) of an astronomical object due to the relative motions between the astronomical object and the observer. In the simplest approximation between an extrasolar object and an observer on the surface of the planet Terra, this is one revolution per day or fifteen degrees per hour (1 rev/d = 15 deg/h) for an extrasolar object positioned on the Celestial Equator of Terra and zero (0) for an extrasolar object positioned at a Celestial Pole of Terra. An extrasolar star close to a celestial pole of a rotating planet (such as Polaris/Alpha Ursae Minoris [α UMi] when viewed from Terra in 2021) is known as a polestar since it exhibits very little apparent speed. Apparent sidereal speed (the apparent celestial speed of an extrasolar star) as viewed from the Terran surface is equal to the apparent celestial speed of an extrasolar star at the Terran Celestial Equator (ω0) multiplied by the cosine of the star's declination (δ).

• ω = ω0cos(δ)

For the purposes of observational (particularly photographic) astronomy, it is better to overestimate than to underestimate the apparent angular speeds of celestial objects. As such, ω0 (on Terra) can then be approximated using the following periods: 0.99726968 days (d) for one rotation of Terra (Pr), 365.256363004 days for one orbit of Terra (Po), 25771.57534 years (a) for one axial precession of Terra (Pa), and 235 megayears (Ma) for one orbit of Sol (P). [2] [3] [4] The last period of 235 megayears should only be used for extragalactic objects, but it can be safely excluded from calculations since the uncertainty in the rotation period of Terra (which is constantly changing) is greater than the contribution to apparent angular speed due to the orbit of the Solar System around the Milky Way Galaxy.

I recommend to use the value of 7.312025662628254 × 10−5 radians per second (equal to 15.08213556575813 degrees per hour [deg/h] or 0.2513689260959688 degrees per minute [deg/min]) for computations with ω0 (given here to sixteen [16] figures, excluding the extragalactic term). Note that one degree per hour is equal to one arcminute per minute (arcmin/min) or one arcsecond per second (arcsec/s). The (Terran) declination of Polaris is 89.3527 degrees (on `2021-04-27`), the cosine of which is 0.0112973, meaning that the apparent angular speed of Polaris is 88.5169 times slower than (or about one percent [1%] of) the apparent angular speed of a star on the Terran Celestial Equator (which is about 15.082136°/h [degrees per hour], giving the apparent angular speed of Polaris as about 0.170387°/h). [5] The declination of the zenith is equal to the latitude (φ).

• 15.082136°/h = ω[0°] = apparent sidereal speed at Celestial Equator
• 11.739484°/h = ω[38.888411°] = apparent sidereal speed at zenith in Washington, District of Columbia, United States of America (USA) [6]
• 11.612390°/h = ω[39.651198°] = apparent sidereal speed at zenith in Cumberland, Allegany, Maryland, USA [7]
• 0.170387°/h = ω[89.3527°] = apparent sidereal speed of Polaris on `2021-04-27`
• 0.116137°/h = ω[89.5588°] = apparent sidereal speed of SAO (Smithsonian Astrophysical Observatory) 75 on `2021-04-27`
• 0°/h = ω[90°] = apparent sidereal speed at Celestial North Pole

Objects within the Solar System can have an apparent angular speed greater than that of the sidereal (or extrasolar) apparent angular speed for an object of the same declination, so for a margin of error, it is best to use the equatorial apparent sidereal speed (ω0) for most Solar System objects. The further away the object is from the observer, the closer that it's apparent motion should appear to be sidereal. However, some satellites and NEOs (near-earth objects) can have apparent angular speeds greater than the equatorial sidereal rate.

## calculations

• 1 rev ≡ 2π rad
• 1 deg ≡ (1/360) rev
• 1 arcmin ≡ (1/60) deg
• 1 arcsec ≡ (1/60) arcmin
• 1 min ≡ 60 s
• 1 h ≡ 60 min ≡ 3600 s
• 1 d ≡ 24 h ≡ 86400 s
• 1 a ≡ 365.25 d ≡ 3.15576 × 107 s
• Pr = 0.99726968 d = 86164.100 s
• Po = 365.256363004 d = 3.15581497635 × 107 s
• Pa = 25771.57534 a = 8.132890659 × 1011 s
• P = 235 Ma = 7.42 × 1015 s
• ωr = 7.2921150 × 10−5 rad/s
• ωo = 1.99098659277 × 10−7 rad/s
• ωa = 7.725648322 × 10−12 rad/s
• ω = 8.47 × 10−16 rad/s
• ω0[1] ≈ 1.0027378 rev/d ≈ 7.2921150 × 10−5 rad/s ≈ ωr
• ω0[2] ≈ 1.0054756 rev/d ≈ 7.3120249 × 10−5 rad/s ≈ ωr + ωo
• ω0[3] ≈ 1.0054757 rev/d ≈ 7.3120257 × 10−5 rad/s ≈ ωr + ωo + ωa ≈ 7.31202566263 × 10−5 rad/s
• ω0[4] ≈ 1.0054757 rev/d ≈ 7.3120257 × 10−5 rad/s ≈ ωr + ωo + ωa + ω ≈ 7.31202566271 × 10−5 rad/s

### Maxima

```k: 10^3 /* kilo */ \$
pi: %pi /* Archimedean constant */ \$
rad: 1 /* radian */ \$
s: 1 /* second */ \$

min: 60*s /* minute */ \$
M: k^2 /* mega */ \$
pi2: 2*pi /* circle constant */ \$

h: 60*min /* hour */ \$
rev: pi2*rad /* revolution */ \$

d: 24*h /* day */ \$
deg: rev/360 /* degree */ \$

a: 365.25*d /* year */ \$
arcmin: deg/60 /* arcminute */ \$
Po: 365.256363004*d /* Terran orbital period */ \$
Pr: 0.99726968*d /* Terran rotational period */ \$

arcsec: arcmin/60 */ arcsecond */ \$
Cy: 100*a /* century */ \$
omegao: rev/Po /* Terran orbital angular speed */ \$
omegar: rev/Pr /* Terran rotational angular speed */ \$
PSol: 235*M*a /* Solar orbital period */ \$

omegaa: 5028.796195*arcsec/Cy /* Terran axial precession angular speed */ \$
omegaSol: rev/PSol /* Solar orbital angular speed */ \$

omega0: omegar+omegao+omegaa /* apparent celestial speed of extrasolar object at Terran Celestial Equator */ \$
Pa: rev/omegaa /* Terran axial precession period */ \$

omega0Gal: omega0+omegaSol /* apparent celestial speed of extragalactic object at Terran Celestial Equator */ \$```

## references

1. `wikipedia:star trails`
2. `wikipedia:Planet Terra`
3. `wikipedia:Axial precession#Values`
4. `wikipedia:Star Sol#Orbit in Milky Way`
5. Stellarium
6. `https://www.googlemaps.com/`
7. `Nicole Sharp's location`

## keywords

astronomy, motion, Polaris, polestar, rotation, speed