A schedule written by where the Sun is

A telescope in space has no horizon and no night, and it still cannot look at most of the sky for most of the year. Its sunshield, its solar panels and the heat of its instruments confine it to a band of directions at a fixed range of angles from the Sun. As the Earth goes round the Sun, the band sweeps around the ecliptic, and each target comes into view in windows — typically two a year, each lasting some weeks, for a target near the ecliptic; one continuous window for a target near the ecliptic poles.

The seasons of a single target are fixed by its position. The problem a telescope’s operators actually face is about thousands of targets at once, chosen by a peer-review process that cares about science and not about where the Sun is, and fitted into one year that has the same number of days in every season.

Windows fixed by the date #

The constraint is simple to state. A target at ecliptic longitude $\lambda$ and latitude $\beta$ is at an angle $E$ from the Sun given by

$$\cos E = \cos\beta\,\cos(\lambda - \lambda_\odot),$$

with $\lambda_\odot$ the Sun’s longitude, which advances a degree a day. The James Webb Space Telescope may observe only between solar elongations of 85 and 135 degrees. For a target on the ecliptic that allows two windows a year, each about fifty days long, centred about 110 days before and after the date when the target is opposite the Sun. A target at higher ecliptic latitude has longer windows; within five degrees of the ecliptic poles the target is never outside the band, and it can be observed every day of the year.

A right ascension is a date for a ground-based observer, who sees a target best at the time of year when it is opposite the Sun at midnight. For a space telescope the same is true with the dates shifted: a target is best placed not when it is opposite the Sun but when it is about a hundred and ten degrees from it, on either side. The calendar of each target is fixed by its coordinates, and it is the same every year.

Demand does not follow the Sun #

The targets astronomers want are not scattered evenly. Some regions are uniquely interesting and every year’s programme returns to them: the centre of the Galaxy, with its black hole, weighed by the stars orbiting it, its nuclear star cluster and its crowded star-forming clouds; the deep extragalactic fields, where years of earlier observations make each new one more valuable and the faintest galaxies crowd the image; the nearest large star-forming region in Orion; the Magellanic Clouds.

The background, spread over the whole sky, has some target visible on almost every day of the year. The popular regions do not: each is visible only in its own windows, and the windows are the same for every proposal that asks for it. A programme that gives a fifth of the year to the Galactic centre asks for seventy days of observing inside two windows that together last about a hundred.

Filling around a crowd #

A schedule is built day by day, and on each day the question is which of the visible targets to observe. The rule that copes best with fixed windows is to serve first the targets whose remaining windows are shortest: a target that can be seen on only twenty more days this year goes before one that can be seen on two hundred. This earliest-deadline rule is close to what real schedulers do, with many refinements for the cost of slewing between targets, the duration of each observation and the constraints of particular instruments.

The rule has a striking consequence. When a popular region’s window opens, its requests have the shortest deadlines and take the telescope’s time; the background, which could be observed on many other days, is deferred and catches up during the quiet seasons. The pile-up in the first figure is absorbed — not by any cleverness about the crowded region, but by the flexibility of everything else. A schedule can carry a crowd in a window as long as the rest of the sky is willing to wait.

That flexibility has a limit, and it is not the total.

What binds is each region’s demand against its own windows. The Galactic centre can be given up to about a third of the telescope’s year and every hour of it delivered, provided the rest of the programme does not also need those days. Past that share, the excess cannot be observed at all: there are not enough days on which the target can be seen, and no deferral of other targets makes more. The rest of the programme continues to be delivered in full, because it never needed those days.

Who loses the time #

The loss falls entirely on the crowded region and is shared among the proposals that asked for it. That has a consequence for how time is allocated, because the peer-review process that selects proposals does not see windows. A review panel that ranks a dozen excellent Galactic-centre proposals above everything else, and awards them time accordingly, may be awarding time that cannot be delivered, while the next-ranked proposals elsewhere on the sky, which could have been observed, are rejected.

Observatories handle this in several ways. Proposals are checked for schedulability before they are approved, and some regions carry an explicit cap on the time that can be awarded in a cycle. Programmes that need a crowded window are sometimes split across two cycles, so that each year’s demand fits. And targets in the continuous viewing zone near the ecliptic poles — which for this telescope includes the Large Magellanic Cloud, about five degrees from the south ecliptic pole — are invaluable as fillers, because they can be observed on any day the crowded windows leave free.

The surprising part: the oversubscription that does not matter #

Large telescopes are said to be oversubscribed — proposals asking for many times the available time — and the number is quoted as a measure of demand. It is the wrong number for the schedule. A telescope oversubscribed nine to one overall can still deliver every hour it accepts, if what it accepts is spread across the sky; a telescope accepting only ninety per cent of its year can still fail to deliver, if too much of that ninety per cent is in one window.

The binding constraint is local in ecliptic longitude. A year is not one pool of time but a sequence of overlapping windows, and each region of the sky draws only on its own. The total oversubscription measures how much the community wants the telescope; the shape of the accepted demand on the sky measures whether the telescope can do what was accepted.

The same structure, at a different scale, governs ground-based telescopes, where each target’s window is set by the time of year when it is above the horizon at night. A ground telescope at a southern site sees the Galactic centre in the southern winter, when the nights are long, and the demand for those months is notoriously high; the months when the Galactic centre is in the daytime sky are when the extragalactic fields away from the Galactic plane get their time. The constraint has always been there. A space telescope makes it cleaner, because there is no weather and no Moon to add noise to it.

A year that turns once #

The window structure is an annual clock, and it is worth seeing why it is annual and not something else. The telescope’s orientation is tied to the Sun, and the Sun’s direction, seen from a telescope at the Sun–Earth Lagrange point, turns once a year against the stars — the same turning that makes the sidereal day four minutes shorter than the solar one, accumulated over a year into a full circle. A target’s windows are the times when this turning brings the Sun to the right angle from it.

So the schedule has a period of exactly one year, and every target’s windows recur on the same dates. That regularity is useful — a programme that needs to follow a target over several years knows its dates in advance — and it is also why the crowding cannot be relieved by waiting. A region whose windows are full this year will have the same windows next year, and whatever demand was not met cannot be absorbed into a later season of the same year. It can only be deferred to the next year, where it adds to that year’s demand for the same weeks.

The clock is also slow to shift. The telescope’s orbit around the Lagrange point is a large loop, and the field of regard is defined relative to the Sun’s direction as seen from the telescope, not from the Earth, so the windows move by a day or two from year to year as the loop’s phase changes. That is negligible against windows several weeks long, and the scheduling problem can be planned years ahead on the assumption that the windows are fixed.

What “visible” hides #

Being inside the field of regard is necessary for an observation, not sufficient. Within a window, a target can be observed only at orientations of the telescope that keep its sunshield toward the Sun, and those orientations rotate through the window. An instrument whose slit or coronagraphic mask must be aligned with something on the sky — the axis of a disc, the line joining a star to a faint companion — can make its observation only on the days when that orientation is allowed, which may be a few days out of a window of fifty.

Some observations must also be made at a particular moment: a transit of a planet across its star, an eclipse in a binary, a predicted occultation, a scheduled observation coordinated with another telescope. Those are fixed in time and must be placed first; everything flexible is fitted around them. A crowded window that also contains several time-critical observations has less room left for its ordinary requests than the window’s length suggests.

Each such constraint shortens the effective window of a target without changing its coordinates, and in the language of the figures above, it moves the threshold to the left. The simple model, in which every hour of a window is usable for every target visible in it, is the optimistic bound; real schedules reach the threshold sooner, for reasons no change in the allocation can remove.

The best days inside a window #

Even within a window the days are not equal, and the inequality pushes demand into still narrower slots.

At infrared wavelengths a space telescope’s background is dominated by the zodiacal light — sunlight scattered and heat radiated by the dust in the plane of the solar system — which is a hundred times brighter than the faint light beyond it. Its brightness in any direction depends on the angle from the Sun and on the ecliptic latitude: it is brightest looking along the ecliptic toward the Sun and faintest looking out of the ecliptic away from it. For a target on or near the ecliptic, the background at the near-Sun edge of each window can be several times higher than at the far edge.

A deep exposure on a faint target is limited by the background, so its signal-to-noise for a given exposure time depends on which part of the window it is taken in. A scheduler that wants to minimise exposure time places faint-target observations on the low-background days, and every faint-target programme on the same region wants the same days. The effective window for a background-limited deep observation is therefore a fraction of the visibility window, and the threshold above which time cannot be delivered is correspondingly lower.

A window that moves a day a lifetime #

The windows are tied to ecliptic longitude, and ecliptic longitude is measured from an equinox that is not fixed. The equinox slides westward along the ecliptic by about fifty arcseconds a year, so a fixed point on the sky gains fifty arcseconds of longitude a year, and its windows arrive a little later in the calendar each year — about a day every seventy years.

Over the twenty-year life of a space telescope that is a few hours, negligible against windows weeks long. Over the longer history of an observing programme it is not entirely negligible: a target’s best season for ground-based observation moves through the calendar by a month in two thousand years, which is why the seasons of the constellations in ancient texts no longer match the calendar’s months. A space telescope’s schedule inherits the same slow drift, and nobody planning one needs to care.

Why a third #

The figure’s threshold, about a third of the year for the Galactic centre, is specific to the telescope’s range of elongations and the target’s latitude. It is worth seeing where it comes from, because it tells how other telescopes compare.

A target on the ecliptic is visible while its angle from the Sun lies between 85 and 135 degrees, which happens on two arcs of the Sun’s annual path, each fifty degrees long — a hundred days of the year. A target at latitude $\beta$ off the ecliptic has a smaller range of elongations over the year, and its windows lengthen; at the Galactic centre’s latitude of about six degrees south, the total is a little over a hundred days. A telescope with a wider range of permitted elongations would have longer windows and a higher threshold; one confined more tightly would have shorter windows and a lower one.

The Hubble Space Telescope is in low Earth orbit, where the Earth blocks each target for part of every ninety-minute orbit, and its solar constraint is weaker — it can point to within about fifty degrees of the Sun. Its windows are longer, its threshold higher, and its scheduling problem is dominated by the Earth’s occultations rather than by the Sun. A telescope at the Sun–Earth Lagrange point, a million and a half kilometres away, has no occultations and a stricter solar constraint, which is why the window structure drawn here describes it so cleanly.

The deep fields that sit in a window #

The extragalactic deep fields are a revealing case. They were chosen, for earlier telescopes, to be far from the Galactic plane and dust, in regions of low foreground, and several of the most famous lie in the same part of the sky. A new telescope inherits them because their existing data make new observations more valuable, and so its demand concentrates where the earlier telescope’s choices put it.

The second bump in the demand figure is such a field. At its modest share of a tenth of the year it fits inside its windows. If several large programmes on the same field are accepted in one cycle, it can approach its threshold, and the field that was chosen for being the easiest place in the sky to look deep becomes one of the hardest places to schedule. The constraint is not on the science; it is on how many of the best programmes can share the same few weeks.

What the model leaves out #

One day of time per day is an idealisation. Real telescopes lose time to slewing between targets, to calibrations, to momentum management and to instrument changes, and an efficient schedule groups targets that are close together on the sky. Grouping favours crowded regions — many targets close together cost little slewing — which partly offsets their window problem.

Requests are divisible. The model lets a request be split into any number of pieces across any number of days. Many real observations must be made in a single block, or at a particular orientation of the telescope, which within a window may be available only on certain days; others must be repeated at particular intervals to follow something that varies. Each such constraint shortens the effective window, and the threshold falls.

The scheduler is greedy. The earliest-deadline rule is good but not optimal, and real schedules are built with optimisation methods over the whole year. They can do better in the crowded-but-not-overfull regime; they cannot do better past the threshold, where the days simply do not exist.

The demand is invented. The shares assigned to each popular region are illustrative. A real cycle’s distribution of demand is published by each observatory and changes from year to year with the questions astronomers are pursuing.

Still open: allocating time that knows about windows #

The schedules of the major space telescopes are now good enough that the losses described here are rare, because time-allocation processes check schedulability before approving proposals. What is not settled is the fairness of the check. A proposal for the Galactic centre competes against other Galactic-centre proposals for a limited window, while a proposal for a target elsewhere competes against the whole sky; the bar for acceptance is higher in a crowded window, by an amount that depends on how many good proposals happen to target the same region in the same year.

Whether that is the right outcome — whether a region’s scientific value should be weighed against the scarcity of its windows, or whether the scarcity should be spread over several years by deferring some programmes — is a question about how a community allocates a resource. The geometry says only that a telescope’s year is not a single quantity of time but a ring of windows indexed by the Sun’s position, and that the ring is the same every year.