How many Earths a mirror can reach

The largest question in the planning of the next great space telescope has been stated as a single number. The American decadal survey of 2020 recommended a mission able to take spectra of about twenty-five potentially habitable planets, and it attached that number to a telescope roughly six metres across. Elaborate codes stand behind the pairing. They optimise a schedule of visits to thousands of possible targets against exposure times, a model of the coronagraph, the dust around each star and the keep-out zones of the sky, and they return a yield with an error bar. The outcome is a graph that can be read but not easily reasoned about.

Most of what that graph says follows from two limits, and it can be derived on one page. A star can be used only if its habitable zone lies outside the angle the instrument can see, and only if the time needed to find a planet there is affordable. Those conditions behave very differently as a telescope grows. One is a wall set by diffraction and the other a wall set by photon arithmetic. The second, in particular, contains a surprise: an exponent of twelve-sevenths, which is why doubling a mirror’s diameter gains more than doubling a mission’s lifetime can by a large margin.

This does not replace the codes. The reasons they are elaborate are real, and the last part of this account lists them. What the two-wall version supplies is what no table of simulated yields can: a reason. Reaching nine orders of magnitude half an arcsecond from a star is the instrument’s problem, and it has been treated at length. Here that problem is taken as solved, and the question is what a solved instrument can then get.

Two walls #

Take the first wall. A coronagraph suppresses a star by blocking or cancelling its light, and it cannot work arbitrarily close to the star’s position. The closest separation at which it delivers its contrast, the inner working angle, is a few times the diffraction angle $\lambda/D$. In visible light at a wavelength of 0.55 microns and with three times that angle, a six-metre mirror reaches in to 57 thousandths of an arcsecond.

The planet to be seen is an Earth twin, one that gets the same sunlight as the Earth, and so it sits at a distance from its star of $\sqrt{L}$ astronomical units for a star of luminosity $L$ in solar units. That scaling is the first thing known about any habitable zone. A separation of $\sqrt{L}$ astronomical units at a distance of $d$ parsecs subtends $\sqrt{L}/d$ arcseconds, so the planet clears the inner working angle only out to a distance proportional to the mirror’s diameter and to the square root of the star’s luminosity:

$$d_{\rm IWA} = \frac{\sqrt{L}\,D}{n\lambda}.$$

Luminous stars can be used out to great distances, and faint ones hardly at all. A red dwarf’s temperate planet hides in the star’s glare beyond three parsecs, and a Sun’s stays visible out to more than seventeen.

The second wall treats all stars alike, and that is the less obvious of the two. An Earth twin receives the same light from its star whichever star it is, so it reflects the same flux, and what reaches the telescope depends only on the distance, falling as $d^{-2}$. A dim star’s planet is closer in and has the same brightness as a bright star’s planet further out. So the time to detect one depends on distance and aperture, not on the type of star.

It is a coincidence, though a telling one, that the two walls meet near the luminosity of the Sun for a six-metre telescope given a year of exposure. Its survey is as limited by what it can resolve as by how long it can stare, at exactly the kind of star it most wants to look at. That balance is much of the reason six metres keeps being called the minimum for this purpose.

The arithmetic of a fixed budget #

The exposure wall is where the exponent comes from. In the arithmetic of counted photons, a faint source on a brighter background needs a time proportional to the background rate over the square of the signal rate. The signal grows with the collecting area, as $D^2$, and falls with distance as $d^{-2}$. The background is dominated by dust, and it is a surface brightness, spread uniformly across the sky near the star. What matters for it is the patch of sky the planet’s image covers, and that area shrinks as the image sharpens, as $(\lambda/D)^2$. Multiply by the collecting area, and the background rate per image comes out independent of the mirror’s size. So

$$t \propto \frac{b}{D^4 d^{-4}} = b\,\frac{d^4}{D^4}.$$

The power of four in $D$ is the headline. A larger mirror increases the signal, and it concentrates the planet’s light against less of the background, and those effects add up to a much faster exposure. The same dependence on distance has already been seen for dust-limited exposure, and it is what makes a survey different from a pointed observation. The stars at the edge of the sample cost far more than the nearest, so a limited budget has to be spent from the inside outward.

Now spend a total time $T$ on all stars out to a distance $d_t$ in a uniform neighbourhood. The number of stars in a shell of thickness $dd$ grows as $d^2$, and each costs $d^4/D^4$, so the total cost is the integral of $d^6/D^4$, which is $d_t^7/(7D^4)$. Setting this equal to $T$ gives

$$d_t \propto T^{1/7}D^{4/7}, \qquad N \propto d_t^3 \propto T^{3/7}D^{12/7}.$$

This is the argument in one line. Seven appears because a uniform density, a distance-to-the-fourth cost and a volume integral give a seventh power of the survey’s reach. The yield rises with the aperture at a power near two, and with time at a power below a half.

The computed curve is not a pure power, and the reason is worth reading off. For small mirrors the angular wall binds, and a larger mirror brings in more stars of every type in proportion to the cube of the diameter. Its slope is then steeper than the time-limited one. As the mirror grows, the exposure wall takes over for the luminous stars, and the slope comes down. It never quite settles at 12/7 for the fleet of stars actually available, since the angular wall is still retreating past the red dwarfs, which are by far the commonest stars. At large aperture the slope sits a little above 12/7. Full simulations commonly report an exponent a little below two, and the calculation here shows why the number comes out where it does, even though it does not include their refinements.

Time as the expensive currency #

The other side of the arithmetic is unwelcome to anyone planning a mission’s duration. Time appears at the power 3/7.

There are two reasons the time slope is so poor, and each has its own remedy. The first reason is the seventh power: an extra stretch of mission buys a thin shell at the edge of the sample, where every star is expensive. The second is the wall. At six metres a longer mission soon runs out of stars the coronagraph can reach at all, and it would then be spending its extra time looking again at what it has already seen. A mission extended from one year of exposure to ten yields only about half again as many candidates.

Repeat visits have a value of their own that this calculation ignores, since a planet not found on the first visit might be at an unfavourable phase of its orbit or hidden inside the inner working angle at that moment. The yield here has a fixed completeness of sixty per cent standing in for all that, and it is where the codes do much of their work. Orbits have to be confirmed with several visits, spectra take longer than detections by a large factor, and overhead per target penalises long lists of faint stars. Each of these lowers the height of the curve. They change the slope less, because they are mostly multiplicative.

The comparison remains as the essential guidance. If an instrument designer is offered either more aperture or more exposure, the exchange rate is about four to one. That is not always a free choice, since the mirror size is set by the launch vehicle and the money, and the throughput of the coronagraph and its working angle can substitute in part for aperture. Those improvements behave like an increase in the effective area and a shift in the angular wall, so a telescope of a given size can trade design effort for aperture. Stark and his collaborators reached this conclusion through extensive simulation in 2024: to give a real margin on the goal, a combination of design improvements and some additional diameter is required, as extra years alone will not do it.

Whose planets are found #

The two walls also determine what kind of star a survey ends up studying, and that changes as the mirror grows.

A small telescope has an easy choice. It can reach an Earth twin only around relatively luminous stars at distances where few stars exist, and those F stars, whose planets stand far enough from them to clear the inner working angle, dominate its list. In a sense it looks where it can, not where it would like to. As the mirror grows the Sun-like stars take over, since their abundance beats that of the F stars once both are within reach.

At very large mirrors a new population comes in. Red dwarfs are by far the commonest stars, and until the angular wall has been pushed back to the scale of their tight habitable zones, they are invisible to the survey. Earth twins around them are then the same brightness as any others, and they would be the bulk of a very large telescope’s targets. The contrast, too, turns favourable: an Earth twin next to a star a hundred times fainter than the Sun has the same brightness as ever, so it needs a hundred times less suppression, which is much less demanding of the coronagraph. A ground-based extremely large telescope, with its much smaller $\lambda/D$, is designed around that fact. It cannot reach $10^{-10}$ through the atmosphere, and it does not have to reach so far if the star is a red dwarf.

It is worth remembering that this changes the question asked, not only the answer. Planets around red dwarfs face their own difficulties: they are tidally influenced, their stars were bright and active when they were young, and they may have lost their water. A survey dominated by Sun-like stars and one dominated by red dwarfs would be samples of different things.

What the dust costs #

The background so far has been treated as a fixed quantity, and that is the term a census of warm dust around nearby stars was built to supply. Each star’s habitable zone is filled with some amount of dust, and the survey’s yield is sensitive to it through the same exponent. The exposure is proportional to the background, so a background $b$ enters the budget as a multiplier on $T$, and the yield depends on it at the power $-3/7$.

That is a gentler penalty than one might fear. A thirtyfold increase in the dust around every star costs a factor of a little over two in yield, and the reason is the same one that makes extra time so unrewarding. Going less far out is not much of a loss, when so few stars were reached in the last stretch. This is reassuring for planners, and it also shows the limits of a uniform assumption. In reality some stars will be far dustier than others, and a good scheduler would drop those from the list, so the real loss is smaller still. The trouble that remains is that dust around each star has been measured in heat and is needed in light, so the scheduler does not know which star is which before it looks.

The factor that is in nobody’s hands #

Every number so far has been multiplied by one fraction that no instrument can change: the share of stars with a planet of the right kind, usually written $\eta_\oplus$. The calculation has used 0.24. Measuring that fraction from the planets a transit survey did find means extrapolating from larger planets and shorter periods, because the survey had hardly any at the size and distance in question, and its radii are biased. Estimates span about a factor of three or more, and the same published study said that this uncertainty dominates the yield.

The figure separates two kinds of uncertainty. Moving between curves is ignorance of the sky: it is what the occurrence rate might be. Each curve’s slope is the plain chance of a counted sample, which remains even if the rate were known exactly. A survey aiming at a goal should be sized for the pessimistic curve, because a null result has to be informative. Finding no Earth twin among a hundred well-observed stars would be one of the more interesting results in the history of the subject, though it would be a statement about planets only if the survey’s sensitivity were known to have been adequate for them. Otherwise it would be a statement about the survey.

There is also a sense in which aperture beats the occurrence rate as well: a low rate multiplies the yield by the same factor at any aperture, so the telescope that is marginal in the optimistic case is badly short in the pessimistic one, and doubling the diameter recovers a factor of three at a stroke. This asymmetry is what a large part of the argument for margin rests on.

The surprise in an instrument’s budget #

The connection worth carrying away is between the exponent and what it says about a design. Most large astronomical programmes reward persistence: a longer exposure goes deeper, and longer surveys cover more. An exposure of the deepest fields rewards time at the plain square root of a photon count, and an astronomer’s instinct is to count time and aperture as near substitutes. In this survey they are not, because the volume is limited by geometry and the reward of depth is a seventh power of reach. A blind survey of nearby stars is an odd animal in that sense: it is not hungry for more of the sky, since there is only so much nearby sky to have. It is hungry for a sharper image and a larger area.

The same arithmetic has a cruel consequence for one planned use of a large telescope. Comparing the planets of stars at different distances is only possible for those stars the survey has reached, and because the sample is bounded by a seventh-power cost, the number of stars within reach changes slowly. The neighbourhood of the Sun is finite, and a few tens of parsecs of it are all that exist for this purpose, however much time and however many telescopes are spent. The nearest Sun-like stars will be looked at first and most often, and any future survey will revisit that same small list.

What the picture cannot show #

This account takes the instrument’s inner working angle as a sharp wall, and real coronagraphs have a throughput that rises gradually across a range of angles, and one that depends on the size of the star’s disc. For the nearest large stars, the star itself is partly resolved at this wavelength, and that resolved light fills in the null in a way this model ignores. It takes detection at one wavelength as the whole cost, though identifying water requires a spectrum near a micron, where the inner working angle is about twice as large and exposure is many times longer. So the numbers for spectra are lower than these, and for that reason the shape of the first figure is more forgiving than the real requirement. The stellar densities come from a census within ten parsecs and are assumed to continue unchanged, and binaries, which a target list has to treat with care, have not been removed.

The height of the yield curve in the second figure is put in, not calculated: an exposure scale was chosen so that a 6 metre telescope and one year of exposure give 25 candidates. So nothing here independently confirms that a six-metre telescope can reach the goal. What is calculated is the shape: how the yield changes with diameter, time, dust and rate, relative to that point.

The figures also assume that the contrast needed can be delivered. A 6 metre telescope has to hold its optics stable to a few tens of picometres for the length of an exposure, to reach $10^{-10}$. No space telescope has yet come close. If that is not achieved, the survey is limited by residual starlight rather than by dust, and the exposure no longer falls as the fourth power of the diameter. Then the exponents derived here do not apply at all.

Still open: how many stars have the planet at all #

The size of the survey is limited by a fraction nobody can yet measure well. An improved measure of it cannot come from the transit surveys already flown, as they have used up what they can do at these periods. Radial-velocity programmes on nearby stars are approaching the precision needed but are fighting the motions of the star’s own surface. Astrometric detection of temperate planets needs precision well beyond what the catalogues offer. So the mission may well have to be designed without knowing the answer it is built to find, and it will measure the number itself in the course of its first years. Whether its target list should be fixed in advance from the stars’ properties, or adapted as the first detections come in, is a question for the schedulers that bears directly on what the yield means when it is finally reported as a fraction and not as a count.