NavMuseum

Interlude Right ascension, declination and sidereal time

The sky has a grid of its own

Earth’s latitude and longitude, projected outward, become the coordinates every star chart uses. The difference is that this grid turns with the stars, not with you.

10 min read · 5 interactive figures

Open any star atlas and you’ll find a grid printed over the stars, with numbers like 18h 36m 56s and +38° 47′ beside Vega. Those two numbers are its address. They don’t change from night to night or from place to place, which is remarkable, because Vega itself is in a different part of your sky every hour. This interlude explains how that works: what the grid is, why one coordinate is measured in hours, and how a star’s fixed address turns into “look up there, now.”

You’ll need it for everything that follows. Chapter 1 measured one star, the pole. Navigators measure dozens, and to do that they need a map of the sky and a way to lay it over the horizon.

Two grids

There are two natural ways to say where a star is.

The first belongs to you. Its altitude is its angle above your horizon, from 0° at the horizon to 90° straight overhead. Its azimuth is its compass direction, measured clockwise from true north, so east is 90° and south is 180°. This grid is fixed to your horizon, and the stars wheel through it: a star’s altitude and azimuth change every minute and differ from town to town.

The second belongs to the stars. Take Earth’s grid of latitude and longitude and imagine it projected outward from Earth’s centre onto the sky. Earth’s equator becomes the celestial equator, Earth’s poles become the celestial poles, and every star gets a latitude and a longitude on the sky. Those are called declination and right ascension. Because the stars are fixed relative to one another, and this grid is pinned to them rather than to Earth, a star keeps the same coordinates while Earth turns underneath.

Figure 1 The star sphere from outside, with Earth turning inside it. Drag to turn the view; press Spin Earth. Your latitude on Earth (cyan ring) projects straight out to the same declination on the sky, and the point above your head (your zenith) slides along it as Earth turns. The right panel unrolls the same sky into a flat chart; click a star there to select it.#

Press Spin Earth and watch what moves. The stars and their grid stay put. Earth turns inside them, carrying you around once a day, and the point directly above your head, your zenith, traces a circle on the sky. Notice that the circle never changes height: it always runs along the declination equal to your latitude. That is the Chapter 1 result again, seen from outside.

Declination: latitude on the sky

Declination, written δ, is how far a star is north or south of the celestial equator, in degrees: +90° at the north celestial pole, 0° on the equator, −90° at the south pole. Polaris is at about +89.4°, Vega at +38.8°, Sirius at −16.7°, and the Southern Cross around −60°[1].

Declination is the coordinate that ties a star to a latitude on Earth. A star of declination δ passes directly overhead, once a day, for everyone at latitude δ. Vega crosses the zenith above Washington, Lisbon and Beijing; Sirius above northern Madagascar and Bolivia. That simple fact is why navigators in many traditions used “zenith stars” to recognize latitude, as Chapter 2 describes.

Declination also decides whether a star ever sets. From latitude φ in the north, any star with δ greater than 90° − φ never dips below your horizon: it is circumpolar. At 40° N, that’s everything north of +50°, including the Big Dipper. Anything south of −50° never rises at all, which is why northern observers never see the Southern Cross.

Right ascension: longitude with a clock in it

Right ascension, written α or RA, is the sky’s longitude. It goes all the way round the celestial equator, increasing eastward, but with two differences from longitude on Earth.

First, its zero point isn’t a city. RA starts at the March equinox, the point on the sky where the Sun crosses the celestial equator heading north, around 20 March each year[2]. It is traditionally called the First Point of Aries and marked ♈. In the figures it’s the brass line at 0h.

Second, it is measured in hours rather than degrees. The full circle is 24h, so

1h = 15°    1m = 15′    1s = 15″(i.1)

Hours are not a quirk. The sky turns once a day, so hours of right ascension are also hours of time: a star at RA 7h crosses your meridian, and stands highest, one hour after a star at RA 6h, wherever you are. Right ascension is the sky’s timetable.

The name is older than the clockwork. To an observer on the equator, every star rises straight up out of the horizon. Astronomers called that the “right” (upright) sphere, and a star’s right ascension was the arc of the equator that rises together with it there.

On a printed chart, RA increases to the left. That surprises everyone the first time. Lie on your back with your head to the north and your feet to the south, and east is on your left; a star chart is drawn from that point of view, from inside the sky looking out.

Sidereal time: the clock that joins the two grids

Here is the key to putting a star map over your horizon. At any moment, one line of right ascension runs straight through your zenith, north–south: it lies on your meridian. Which line that is depends on the time and on your longitude, and it changes by 1h of RA per hour of (sidereal) time. That RA is called your local sidereal time, LST. It’s a clock that reads the sky, and observatories keep one on the wall.

Once you know LST, every star’s position in your sky follows. Its hour angle, how far it has turned west past your meridian, is

H = LST − RA(i.2)

and its altitude h comes from the navigational triangle, using your latitude φ and its declination δ[3]:

sin h = sin φ sin δ + cos φ cos δ cos H(i.3)

A companion formula gives the azimuth. That’s the whole recipe: latitude tilts the grid, sidereal time turns it.

Figure 2 Your sky (left) and the chart (right), centred on the RA that is on your meridian now. The shaded part of the chart is below your horizon and the green curve is the horizon itself. Run the night and watch the chart slide under the fixed meridian. Click a star in either view: its hour angle (brass arrow) and declination give its altitude and azimuth.#

Run the night and notice how the two views agree. The dashed cyan meridian is fixed in your sky, and the chart slides under it from left to right as the night goes on. A star climbs until its RA reaches the meridian (hour angle zero), then sinks. Pick Vega at 40° N and watch it pass nearly overhead: its declination, +38.8°, is close to your latitude. Pick Antares and watch it skim the southern horizon.

Why the night sky changes with the seasons

If the stars stay put on their grid, why are Orion and Sirius winter stars in the north, and Vega and the Summer Triangle summer ones? Because the Sun isn’t fixed on the grid. As Earth orbits, the Sun appears to move eastward against the stars along a path called the ecliptic (the dashed brass curve), going all the way round in a year: about 2h of RA a month, or about 4 minutes a day.

Figure 3 Left: Earth’s orbit seen from above. The night side of Earth (cyan arrow) faces the stars opposite the Sun. Right: the whole 24-hour chart, with the Sun on the ecliptic and its glare hiding the sky around it. Run the year and watch the midnight sky march east through the stars.#

Whatever is near the Sun on the chart is up during the day and lost in its glare. Whatever is opposite, 12 hours of RA away, is on your meridian at midnight. In early November the Sun is near RA 14h 45m, so the midnight sky is centred near 2h 45m, with the Pleiades and Aldebaran just east of it, climbing toward the meridian. In May it’s the reverse.

The same 4 minutes a day explain the gap between two kinds of day. A sidereal day, one turn of Earth relative to the stars, lasts 23h 56m 4.09s[3]. A solar day, Sun to Sun, is about 4 minutes longer, because Earth has moved along its orbit and must turn a little further to bring the Sun back. So each star rises about 4 minutes earlier every night, and a year contains one more sidereal day than solar days.

A planisphere does all of this with cardboard

The rotating star wheel sold in every museum shop is a mechanical computer for equations (i.2) and (i.3). Its star disk is a map in right ascension and declination around the pole. Its cover has a window cut to the shape of your horizon, for one latitude. Turning the disk until today’s date lines up with the time on the cover sets the sidereal time, and the window then shows exactly the sky above you.

Figure 4 A working planisphere. Drag the star disk to turn it (or use the sliders). The month marks on the disk sit at the RA that is on the meridian at midnight on that date; the time scale is on the cover. The window’s shape is your horizon at the chosen latitude, and the cross marks your zenith.#

Try changing the latitude and watch the window change shape. Far north, it shrinks toward the pole: more stars never set and fewer ever rise. That’s why a planisphere bought in London is wrong in Sydney, and why good ones state the latitude they were cut for.

The grid drifts: epochs

The equinox, RA’s zero point, isn’t quite fixed. Earth’s axis precesses, tracing a circle among the stars every 26,000 years or so, and the celestial equator and equinox move with it by about 50″ a year[3]. As a result every star’s RA and declination creep slowly; for most stars the RA increases by about 3 seconds of time a year.

So a star catalogue has to say which year’s grid it uses, its epoch. The modern standard is J2000.0, the grid as it stood at noon on 1 January 2000. Software, including this site, starts from J2000 positions and precesses them to the date you’re looking at. Try the year slider in the first figure: drag back a few thousand years and watch the stars slide across the grid, and the pole wander away from Polaris.

Astronomers think in RA and LST. Navigators rearranged the same numbers so they could be added up quickly on a moving deck, in degrees and measured westward, the way the sky turns[4].

  • Sidereal hour angle replaces RA: SHA = 360° − RA, measured westward from ♈. Vega’s SHA in 2026 is about 80° 33′.
  • Greenwich hour angle of Aries, GHA♈, is how far west of Greenwich the equinox is right now. The Nautical Almanac tabulates it for every hour of the year[5].
  • A star’s own GHA is then a sum: GHA★ = GHA♈ + SHA★.

And here is the payoff. At any instant a star is directly overhead at exactly one point on Earth, its geographic position, at latitude equal to its declination and longitude equal to its GHA, counted west. The almanac is really a timetable of points racing west across the globe at 15° an hour.

Figure 5 Each dot is the point on Earth directly beneath a navigational star at this moment. Run time and they all sweep west together at 15° per hour, keeping their shape. The cyan meridian is where RA 0h is overhead (GHA of Aries); a star’s point lies its SHA further west.#

Your altitude above the horizon then tells you how far you are from that point: the zenith distance, 90° − altitude, in degrees of arc, at 60 nautical miles per degree. Measure two stars and you have two circles around two moving points, crossing where you stand. That is the whole of celestial navigation in one sentence, and it’s what Chapter 5 lets you do by hand. First, though, there’s the matter of knowing the time to the second, which is Chapter 3.