A right ascension is a date

Right ascension is written in hours, and that is not a historical accident. An object’s right ascension is the sidereal time at which it crosses the meridian, so a catalogue of positions is also a timetable: the Orion Nebula, at 5 hours 35 minutes, is due south whenever the local sidereal time reads 5:35. Where a star is depends on who is asking, and on this clock the answer to “when is it highest?” is simply its first coordinate.

That timetable is written in the wrong kind of time for anybody who wants to observe. What matters to an observer is not only whether the object is high but whether the sky is dark, and darkness runs on the solar clock. The two clocks drift apart by three minutes fifty-six seconds a day, one whole turn a year, so the sidereal hour at which Orion is due south falls at a different solar hour every night. For most of the year it falls in daylight. Seeing the object at all requires both clocks to agree at once, and the part of the year in which they do is the object’s season.

The date the object is opposite the Sun #

The diagonal line on that chart carries the whole idea. Each night the transit comes four minutes earlier by the solar clock; after a month it has moved two hours, and after a year it has gone once round the clock and returned. Somewhere on that journey it crosses local midnight, and on that night the object is due south at the moment the Sun is due north below the horizon — the object is opposite the Sun in right ascension. For the Orion Nebula that night is 15 December.

The date follows from the coordinate alone. The Sun’s right ascension advances through the year, reaching 0 hours at the March equinox, 6 at the June solstice, 12 at the September equinox and 18 in December. An object at right ascension $\alpha$ is opposite the Sun when the Sun is at $\alpha + 12$ hours, so Orion’s 5.6 hours puts its midnight transit where the Sun is at 17.6 hours — mid-December. The chart bears it out: the midnight crossing found from the computed transit times lands on a date when the Sun is within the equation of time of being exactly twelve hours away. A right ascension is a date: add twelve hours, and read the Sun’s calendar.

The season is the stretch of nights either side of that date on which the transit still falls inside the dark hours, and its width follows from two numbers already on the chart. A mid-winter night at 52 degrees north holds about fourteen hours of astronomical darkness, and the transit moves through the clock at two hours a month, so a transit that entered the dark hours at dawn would take about seven months to leave them at dusk. The season an observer actually gets is shorter than that, because at its edges the transit falls in the last or first hour of darkness and the object is only high enough for part of it — which is where the altitude limit, below, takes its share.

Darkness is not simply the hours between sunset and sunrise, and the dark band on the chart is narrower than the night. At 52 degrees on the equinox the Sun takes 34 minutes to go from its apparent setting to six degrees below the horizon, another 40 to twelve and another 42 to eighteen, so the band begins almost two hours after sunset and ends almost two hours before sunrise. Those two hours come off both ends of every night, and they grow towards midsummer, when the dusk and dawn curves close together and finally meet: at this latitude the Sun then never gets eighteen degrees down, and the band vanishes for two months whatever an object’s coordinates.

The chart shows the season’s shape as well as its width. Early in the season the object is best late in the night; late in the season, in the early evening. Its edges are where the transit line leaves the dark band — in autumn the object is up only before dawn, in spring only after dusk — and on either side of those edges it is above the horizon in daylight.

The altitude that is demanded decides as much as the date #

The shading on the first chart asks for twenty degrees of altitude. That is a generous requirement. Light from an object twenty degrees up passes through $1/\sin 20° \approx 2.9$ times as much air as light from the zenith, and every property of the image degrades with that path: the extinction that has to be corrected in every magnitude, the atmosphere’s dispersion of colours, and the turbulence that sets the sharpness of every ground-based image. A more common working limit is thirty degrees, where the path is exactly twice the vertical one.

Ten degrees of extra altitude cost more than half the hours in the year. The reason is that from 52 degrees north the nebula never climbs higher than 32.6 degrees, so a thirty-degree limit leaves only the top of its daily arc — a couple of hours around transit — and that short window has to fall inside the dark hours. The declination and the latitude have not changed the date of the season; they have decided how much of the season there is.

How much can be read off one more line of spherical trigonometry. An object of declination $\delta$, seen from latitude $\varphi$, stands above an altitude $a$ for hour angles within $\pm H_a$ of its transit, where

$$\cos H_a = \frac{\sin a - \sin\varphi\,\sin\delta}{\cos\varphi\,\cos\delta}.$$

For the Orion Nebula from 52 degrees north that gives $H_a = 20.5°$ for a thirty-degree limit and $47.3°$ for a twenty-degree one — windows of 2.7 and 6.3 hours around transit. Those are exactly the best nights on the two charts. The best night of a season is simply a night on which the whole altitude window fits inside the dark hours, and every other night of the season is one on which only part of it does. The declination and the latitude fix the width of the window, the darkness decides how much of it survives, and the right ascension decides on which nights of the year the window and the darkness line up.

The same object from the other hemisphere #

Move the observatory to thirty degrees south, the latitude of the large observatories in Chile and South Africa, and the same arithmetic gives a different sky.

From the south the Orion Nebula passes almost overhead, and the thirty-degree limit that cut its northern year to 375 hours allows 958. The Galactic centre, at declination −29 degrees, passes through the zenith of a station at −30 and gives the longest season of the four. Andromeda, well north of the equator, never reaches thirty degrees, and Polaris never clears the horizon at all.

The northern view of the same four objects makes a matching picture with every entry reversed.

Two things are visible in the northern curves that the southern ones hide. The first is a hole in every curve around June. At 52 degrees north the Sun never gets eighteen degrees below the horizon for sixty-four nights around midsummer, so no object of any declination can be observed in astronomical darkness then. The latitude at which that begins is 48.6 degrees, and a station north of it simply has no June. The second is Polaris’s curve, which is tallest in December and has no season at all in the usual sense — it is up all night, every night, and its hours are the hours of darkness.

An unequal pair of hemispheres #

The window figures invite a comparison that turns out to be subtle. The Galactic centre passes near the zenith of a station at −30 and never rises far above the horizon at 52 north, which is plainly a matter of declination. But declination is not the whole of it, and a figure that holds declination fixed shows what else is there.

Two stations the same angular distance either side of the object’s declination see it climb to exactly the same altitude, and one of them gets forty per cent more hours. The difference is the other clock. The Galactic centre is opposite the Sun on 17 June, so its season is June everywhere on the Earth. In June the southern hemisphere has its longest nights and the northern its shortest, so a southern station watches the centre’s season through long winter darkness while a northern station at the same distance watches the same season through short summer nights. The curve also peaks at −50 degrees rather than at the object’s own declination, because a station further south trades a little altitude for a good deal more night.

The size of the effect can be checked against a single date. On 17 June the Sun stands 23.4 degrees north of the equator, and the hour angle at which it reaches eighteen degrees below the horizon gives a night of astronomical darkness 9.1 hours long at latitude 5 degrees north and 13.0 hours long at 65 degrees south. The ratio of those two nights is 1.44. The ratio of the two stations’ annual totals in the figure is 1.43. Almost the whole of a forty per cent difference in a year’s observing is the length of one night — the night the object’s right ascension names.

The southern observatories’ advantage for the centre of the Galaxy is partly a matter of calendar. The object is in the southern sky, and its season also happens to fall in the southern winter. If the Galactic centre lay at the same declination but at six hours of right ascension instead of eighteen, its season would be December, and the asymmetry would run the other way.

That counterfactual can be checked on an object that actually has a December season.

It runs exactly the other way. The same angular offset gives the northern station a quarter more hours, because Orion’s season is the northern winter. Nothing about the nebula favours the north; the Sun’s declination in December does.

An object that never sets is best at the wrong time of year #

Every object so far rises and sets, so its season is bounded by the dates on which its transit enters and leaves the night. An object close enough to the pole never sets, and the logic of a season changes.

The nebula’s right ascension names June. Its best observing is in December, half a year away, because it is above thirty degrees for almost the whole of every night and the only question is how long the night is. The date its coordinate names is, from this latitude, the date on which it cannot be observed at all.

The split nights are the one trace of its right ascension that remains. Around December the nebula’s lower culmination, when it passes below the pole, falls near midnight, and its lowest altitude, 28.6 degrees, is just under the thirty-degree limit. So the long winter night is interrupted for a while in the middle and the good time comes in two pieces. For a circumpolar object the date of its midnight transit is when it is highest in the middle of the night; half a year later it is lowest in the middle of the night, and that is when the nights are long enough to use. The chart shows both at once.

A year of telescope time is cut into right ascensions #

The charts describe one observer’s year. Large observatories turn the same arithmetic into administration. Time on most of them is allocated by semester, usually two a year, and a proposal is written for a semester rather than for a night — because the semester, not the night, is what an object’s right ascension fixes. A proposal to observe the Orion Nebula belongs to the half of the year from October to March and one for the centre of the Galaxy to the half from April to September, at every observatory on the Earth, whatever its latitude and whatever its telescope.

That has a consequence nobody chose. The sky is not uniform in right ascension: the plane of the Milky Way, with its star-forming regions and crowded fields, crosses the meridian at midnight around the middle of the year, and the direction looking out of the Galaxy, towards the distant galaxies that deep surveys want, does so around the turn of the year. So the demand for a telescope’s time varies from one semester to the next with the structure of the Galaxy, and a southern observatory’s winter semester, with the Galactic centre overhead through long nights, is the one in which the two clocks and the Galaxy all line up at once.

The semester is also an admission. A given target has one good half-year and one bad half-year at every site, and a proposal that misses its half has not lost a night but a year. The only freedom a right ascension leaves an observer is the choice of which year.

What the charts leave out #

The Moon. A night with the Moon above the horizon is not dark in any sense that matters for faint work, and a large share of the dark hours on these charts falls in the half of each month when the Moon is bright. Observatories schedule faint targets for the fortnight around new moon, which cuts every season shown here into monthly pieces.

The weather and the site. Every hour counted assumes a clear sky and a natural night floor. Neither holds at most places, and the night that has stopped arriving under artificial skyglow removes the faint end of the chart altogether.

The object’s own motion. The coordinates are fixed stars and nebulae. A planet moves in right ascension, so its season drifts from year to year, and its opposition is also the date on which it loops backwards across the sky. Even a fixed star’s coordinates drift, as the equinox they are counted from moves — by about three seconds of right ascension a year for an object near the equator. Over a century that is about five minutes, and at four minutes a night it moves the object’s season by a little over a day.

Local mean time. The charts use the mean solar time of the station’s own meridian, the time in which the Sun is due south at noon on average. Civil clocks keep zone time and summer time, which shift every line on the chart by up to an hour or two without changing anything about the sky.

What the two clocks decide #

The argument fits in a sentence per clock. The sidereal clock turns a right ascension into a time of night and, through its one-turn-a-year drift against the Sun, into a date. The solar clock decides how long the night is on that date at a given latitude. An observing season is the overlap, and its length is set by the declination and the latitude, while its position in the year is set by the right ascension alone — except for objects that never set, whose season is the season of long nights wherever their right ascension says.

Still open: a telescope that has no night #

Every chart here depends on the Sun going below a horizon. A telescope in orbit has no horizon to put it behind, and its constraint is instead an angle it must keep between its line of sight and the Sun, the Earth and the Moon. The sidereal clock still turns each object’s direction into a date, but the season becomes a band of the sky on either side of the Sun’s direction that moves round once a year — and the one part of the sky that is never inside it lies near the poles of the telescope’s own orbit.