# Difference between revisions of "Apparent celestial speed"

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, the angle between the star's position and the Terran Celestial Equator).

• 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). The last period of 235 megayears should only be used for extragalactic objects, but it can be safely excluded from any calculations since the uncertainty in the rotation period of Terra (which is constantly changing) is greater than any 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 declination of Polaris is 89.26411 degrees, the cosine of which is 0.01284333, meaning that the apparent angular speed of Polaris is 77.86142 times slower than (or about one percent [1%] of) the apparent angular speed of a star on the Celestial Equator (which is about 15.082136 degrees per hour, giving the apparent angular speed of Polaris as about 0.1937049 degrees per hour).

## calculations

### Maxima

pi: %pi /* Archimedean constant */ $rad: 1 /* radian */$
s: 1 /* second */ $min: 60*s /* minute */$
pi2: 2*pi /* circle constant */ $h: 60*min /* hour */$
rev: pi2*rad /* revolution */ $d: 24*h /* day */$
deg: rev/360 /* degree */ $a: 365.25*d /* year */$

Pr: 0.99726968*d /* Terran rotational period */ $Po: 365.256363004*d /* Terran orbital period */$
Pa: 25771.57534*a /* Terran axial precession period */ $PSol: 235*10^6*a /* Solar orbital period */$

omega0: (rev/Pr)+(rev/Po)+(rev/Pa) /* apparent celestial speed of extrasolar object at Terran Celestial Equator */ $omega0Gal: omega0+(rev/PSol) /* apparent celestial speed of extragalactic object at Terran Celestial Equator */$

### MathJax

• $\displaystyle{ 1~\mathrm{rev} \equiv 2\pi~\mathrm{rad} }$
• $\displaystyle{ 1~\mathrm{min} \equiv 60~\mathrm{s} }$
• $\displaystyle{ 1~\mathrm{h} \equiv 60~\mathrm{min} \equiv 3600~\mathrm{s} }$
• $\displaystyle{ 1~\mathrm{d} \equiv 24~\mathrm{h} \equiv 86400~\mathrm{s} }$
• $\displaystyle{ 1~\mathrm{a} \equiv 365.25~\mathrm{d} \equiv 3.15576\times10^7~\mathrm{s} }$
• $\displaystyle{ P_\mathrm{r} = 0.99726968~\mathrm{d} = 86164.100~\mathrm{s} }$
• $\displaystyle{ P_\mathrm{o} = 365.256363004~\mathrm{d} = 3.15581497635\times10^7~\mathrm{s} }$
• $\displaystyle{ P_\mathrm{a} = 25771.57534~\mathrm{a} = 8.132890659\times10^{11}~\mathrm{s} }$
• $\displaystyle{ P_☉ = 235~\mathrm{Ma} = 7.42\times10^{15}~\mathrm{s} }$
• $\displaystyle{ \omega_0 \approx 1.0027378~\mathrm{rev}/\mathrm{d} \approx \frac{1~\mathrm{rev}}{P_\mathrm{r}} \approx 7.2921150\times10^{-5}~\mathrm{rad}/\mathrm{s} }$
• $\displaystyle{ \omega_0 \approx 1.0054756~\mathrm{rev}/\mathrm{d} \approx \frac{1~\mathrm{rev}}{P_\mathrm{r}} + \frac{1~\mathrm{rev}}{P_\mathrm{o}} \approx 7.3120249\times10^{-5}~\mathrm{rad}/\mathrm{s} }$
• $\displaystyle{ \omega_0 \approx 1.0054757~\mathrm{rev}/\mathrm{d} \approx \frac{1~\mathrm{rev}}{P_\mathrm{r}} + \frac{1~\mathrm{rev}}{P_\mathrm{o}} + \frac{1~\mathrm{rev}}{P_\mathrm{a}} \approx 7.3120257\times10^{-5}~\mathrm{rad}/\mathrm{s} \approx 7.31202566263\times10^{-5}~\mathrm{rad}/\mathrm{s} }$
• $\displaystyle{ \omega_0 \approx 1.0054757~\mathrm{rev}/\mathrm{d} \approx \frac{1~\mathrm{rev}}{P_\mathrm{r}} + \frac{1~\mathrm{rev}}{P_\mathrm{o}} + \frac{1~\mathrm{rev}}{P_\mathrm{a}} + \frac{1~\mathrm{rev}}{P_☉} \approx 7.3120257\times10^{-5}~\mathrm{rad}/\mathrm{s} \approx 7.31202566271\times10^{-5}~\mathrm{rad}/\mathrm{s} }$

## references

1. wikipedia:star trails