A null filled in by the star itself

Every method of imaging a planet beside its star has to do something about the star, and the methods that subtract a model of the star’s halo from the image are all limited by how well the halo can be predicted. The alternative is to cancel the starlight before it forms an image at all. Combine the light from two telescopes with a half-wave phase shift in one arm, and a source exactly on the axis — the star — arrives at both telescopes in step, is shifted, and cancels. A source slightly off the axis arrives with a path difference that undoes the shift, and at the right angle it is transmitted fully. The pattern of transmission on the sky is a set of fringes, dark on the star and bright where the planet is.

It is the only technique that reaches inside the diffraction limit of the individual telescopes, because the fringe spacing is set by the distance between them rather than by either one’s diameter. And it works best in the mid-infrared, around ten microns, where a temperate planet emits its own heat and its contrast against a Sun-like star is about $10^{-7}$ rather than the $10^{-10}$ of reflected visible light. It has been demonstrated on the ground and designed for space for thirty years. What limits it turns out not to be the planet, the telescopes or the atmosphere, but the star, and specifically the fact that the star has a size.

Why ten microns, and why two telescopes #

The choice of wavelength is forced by the planet. In visible light a planet like the Earth shines by reflected sunlight, and its brightness relative to the Sun is its albedo times the fraction of the Sun’s light it intercepts — about $10^{-10}$. In the mid-infrared it shines by its own heat. A 255-kelvin body emits most strongly near eleven microns, while a 5,772-kelvin star is far down the tail of its spectrum there, and the ratio of their blackbody surface brightnesses at ten microns is a thousandth. Times the ratio of their areas, $8\times10^{-5}$, the contrast is $8.4\times10^{-8}$ — a thousand times more favourable than in reflected light. And the mid-infrared carries the molecules that would say what the planet is: the carbon dioxide band at fifteen microns, the ozone band at 9.6, the broad water continuum and the window between them through which a temperate surface radiates.

The same wavelength forces the interferometer. The diffraction limit of a telescope of diameter $D$ is about $\lambda/D$, and at ten microns a 6.5-metre mirror resolves 0.3 arcseconds. An Earth at the distance where it receives the Earth’s insolation around a Sun-like star 10 parsecs away is 0.1 arcseconds from it — inside the diffraction limit of any single telescope that could be launched. Resolution has to come from the separation of apertures rather than their size, which is what an interferometer is for. A coronagraph on a single large mirror works in the visible, where the diffraction limit is twenty times finer but the contrast a thousand times worse; a nuller works in the infrared, where the contrast is kinder and only a baseline gives the resolution. They are not competitors doing the same thing better and worse but two different trades between contrast and angle.

A null that is only quadratic #

The transmission of a two-telescope nuller at angle $\theta$ from the axis, measured along the baseline $B$, is $\sin^2(\pi B\theta/\lambda)$. It is zero on the axis, rises to one at $\theta = \lambda/2B$, and falls back to zero at $\lambda/B$. To put the first bright fringe on a planet at angular separation $\theta_p$, the baseline has to be $\lambda/2\theta_p$ — for an Earth at 1 AU from a star 10 parsecs away, 100 milliarcseconds out at 10 microns, that is 10.3 metres.

The star at that distance is 0.93 milliarcseconds across, a hundredth of the planet’s separation, and on the figure’s scale it is a sliver at the origin. The null looks perfect across it. It is not, because near the axis the transmission rises as the square of the angle: $\sin^2 x \approx x^2$. The star’s centre is fully cancelled, but its edge, half a milliarcsecond from the axis, is transmitted at about $(\pi B\theta/\lambda)^2$ — a small number, but a finite one, and the star is ten million times brighter than the planet. Averaged over the disc, the fraction of the star that leaks through is

$$N_2 = \left(\frac{\pi B \theta_*}{4\lambda}\right)^2,$$

where $\theta_*$ is the star’s angular diameter.

A leak that does not depend on distance #

Put in the baseline that finds the planet, $B = \lambda/2\theta_p$, and the leak becomes $(\pi\theta_*/8\theta_p)^2$. The star’s angular diameter is $2R_*/d$ and the planet’s separation is $a/d$, where $d$ is the distance, and the distance cancels:

$$N_2 = \left(\frac{\pi R_*}{4a}\right)^2.$$

So does the wavelength. At the working baseline, the stellar leak through a two-telescope null depends only on the star’s radius divided by the planet’s orbital radius — a property of the planetary system, not of the observation. Moving the system farther away shrinks the planet’s separation and the star’s disc in proportion, the baseline grows to keep the planet on its fringe, and the leak stays exactly where it was. Observing at a longer wavelength needs a longer baseline for the same reason, and the leak stays where it was again.

For the Sun and the Earth, $R_*/a$ is 0.00465, and the leak is $1.3\times10^{-5}$. The Earth’s contrast against the Sun at ten microns — the ratio of an Earth-sized blackbody at 255 kelvin to a Sun-sized one at 5,772 kelvin, both at that wavelength — is $8.4\times10^{-8}$. The star leaks through the null at 158 times the planet’s brightness, and that ratio is the same for a twin of the solar system at 3 parsecs or at 30.

A steady leak can in principle be subtracted, since it is starlight of a known spectrum at a known position. But subtraction removes its mean and not its noise. The leaked light brings photon noise with it, and at 158 times the planet’s signal that noise alone would take many times longer to average down than the planet’s own photon noise; and any fluctuation in the leak — from the star’s own variability, from the telescope pointing wandering by a fraction of the stellar disc — does not subtract at all. The two-telescope design that is simplest to build is the one that cannot find an Earth by the light of the Sun, however well it is built.

Around smaller stars #

The ratio $R_*/a$ is a property of the kind of star and of where its temperate planets orbit, so the comparison can be carried along the main sequence.

A smaller, cooler star has a habitable zone much closer in, because its luminosity falls far faster than its radius. The star’s radius is then a larger fraction of the planet’s orbit, and the leak rises: around a star of 0.3 solar masses the two-telescope leak is $1.4\times10^{-4}$, ten times the Sun’s. But the planet’s contrast rises faster, to $1.2\times10^{-6}$, because the star is both smaller and cooler and emits much less at ten microns relative to a 255-kelvin planet. The ratio of leak to planet improves from 158 to about 120 — not enough to rescue a two-telescope design, and bought at a cost the figure does not show: the planet’s separation from a small star is tiny, a few milliarcseconds at 10 parsecs, and the baselines needed to reach it are hundreds of metres.

The kink in the curves near 0.43 solar masses is the break in the mass–luminosity relation used to place the habitable zone, where the stars become fully convective; it is a feature of the stars, not of the null.

How equal two paths have to be #

The null has a second leak that has nothing to do with the star’s size: the two arms of the interferometer have to be matched in length and in amplitude, and any mismatch fills the null in on its own.

A path difference $\delta$ between the arms is a phase error $2\pi\delta/\lambda$, and a phase error $\delta\phi$ leaves a fraction $(\delta\phi/2)^2$ of the light uncancelled. At ten microns a path error of 12 nanometres leaks as much as the Sun’s disc already does; to hold the path error’s leak below the Earth’s contrast requires the two paths to be equal to 0.9 nanometres. An amplitude mismatch leaks in the same way, with the fractional mismatch in place of the phase, so the two beams must also be balanced in intensity to a few hundredths of a per cent.

As with the stellar leak, the requirement is really on the fluctuations rather than the mean: a constant path error could be calibrated, and the demanding number is how steadily the paths are held over the hours of an observation. Through the atmosphere that stability is out of reach, which is why the ground-based nullers, which worked, were used to measure the dust around stars — the exozodiacal light, a thousand times brighter than a planet — and not planets. In space, formation-flying telescopes or a single structure holding the apertures would have to keep their optical paths equal to a nanometre while separated by tens of metres. That has been demonstrated in laboratories and not yet in orbit.

A deeper null, and what it costs #

The escape from the stellar leak is to make the null deeper than quadratic. With four or more apertures, arranged and phased so that the contributions cancel in pairs and the pairs cancel again, the transmission near the axis can be made to rise as the fourth power of the angle — $\sin^4$ instead of $\sin^2$ in the simplest arrangement. Averaged over the star’s disc, the leak becomes $(\pi R_*/2a)^4/8$, still independent of distance, and for the Sun and the Earth it is $3.6\times10^{-10}$ — 237 times fainter than the planet. The quartic null solves the problem the quadratic one cannot.

Its price is in the other direction. The same fourth power that suppresses the star’s edge suppresses anything else near the axis, including planets on small orbits.

A real nuller rotates its baseline, so that a planet at a given separation passes through bright and dark fringes and its signal is modulated — the modulation is how the planet is distinguished from the leak, which does not vary with rotation. The figure averages the transmission over that rotation. Far from the star the two designs transmit comparable fractions, a half and three-eighths. Close to it the quartic null is much darker, and its disadvantage grows as the inverse square of the separation. To reach the same inner orbit, a quartic array needs a longer baseline, and a longer baseline brings the stellar leak up again as the fourth power of its length.

The modulation deserves a sentence of its own, because it is how a nuller turns a leak it cannot remove into a background it can reject. As the array turns, the transmission at the planet’s position follows a known pattern — a function of the rotation angle fixed by the planet’s separation and position angle — while the leak from a centred, symmetric star and the emission of a symmetric dust disc do not vary at all. The detection is then a matched filter: the recorded signal against rotation angle is compared with the pattern each trial planet position would produce, much as a periodogram tests trial periods against a time series. Arrays with four or more telescopes can also be combined in two different ways at once and their outputs subtracted — phase chopping — which cancels every symmetric contribution and keeps only the asymmetric one a planet makes. What no chopping cancels is the photon noise of the leak, so the requirement on the null’s depth is a requirement on counting statistics rather than on subtraction.

The designs that emerged from two decades of study — a four- or five-telescope array flying in formation, working between four and eighteen microns — are the compromise this implies: a quartic or higher null for the star, a baseline tuned target by target so that the habitable zone sits near the first bright fringe, and a rotation that turns the planet into a modulated signal. The figures here are the arithmetic behind that compromise, and none of it depends on the details of the hardware.

What the drawing leaves out #

The star is drawn as a uniformly bright disc. Real stars are darker at the limb than at the centre, and since the leak comes from the edge, limb darkening reduces it — but only by a few per cent for the Sun in the mid-infrared, where the darkening is weak, so the uniform disc is a fair stand-in. The planet is drawn as a blackbody at 255 kelvin; the real Earth’s spectrum at ten microns has a deep ozone band and a broad window, and its brightness depends on which part of the planet faces the observer. The zodiacal dust around the target star — the exozodiacal light — is an extended source that fills the fringes just as the planet does, and for a system with ten times the solar system’s dust it is brighter than the planet; the modulation that separates a planet from a stellar leak does not separate it from a clumpy disc of dust as cleanly. And the photon noise of the local zodiacal light, from the solar system’s own dust seen from a telescope near the Earth, sets a background floor that no null affects.

None of these changes the central result, because the central result is geometric. A null that rises as the square of the angle leaks the star’s edge in proportion to the square of the star’s size relative to the orbit being searched, and for the Sun and the Earth that is a hundred and fifty times the planet.

A number the star chooses #

It is unusual for the limiting noise of an instrument to be independent of both the distance of the target and the wavelength of the observation. For a coronagraph the limit is set by how well the optics are made; for an interferometer used to measure a star’s angular diameter, the stellar size is the signal. For a nuller searching for a planet it is the noise, and it is set entirely by the ratio of two lengths in the system being observed — the star’s radius and the planet’s orbit. Every design choice follows from that ratio. It fixes how deep the null must be, and so how many telescopes are needed; the depth fixes how the planets closest in are suppressed, and so how long the baselines must be; and the baselines fix how steadily the paths must be held. A mission to find another Earth by its heat is, in this sense, designed by the Sun’s radius divided by an astronomical unit.

Still open: whether a nanometre can be held for a day #

The geometry is settled; the engineering is not. The quartic null deals with the star, the rotation with the separation of planet from leak, and the choice of target baseline with the habitable zone’s position. What remains is whether four or five spacecraft tens of metres apart can hold their optical paths equal to a nanometre, and their beam intensities equal to a part in ten thousand, stably enough and for long enough to integrate down the photon noise of what does leak — and whether the dust around the nearest stars is faint enough to let an Earth through at all. The first question is being answered in laboratories. The second has been answered, in part, by the ground-based nullers themselves: a survey of several dozen nearby Sun-like stars with a large binocular telescope, nulling at eleven microns, found that the typical star carries warm dust at a few times the level of the solar system’s, and that a minority carry tens to hundreds of times more. The quadratic null was precisely the right instrument for that measurement, because dust is an extended source far brighter than a planet, and the stellar leak that rules out an Earth is small beside it. The same survey sets the list of stars around which an Earth could be seen through its own system’s dust at all. The answer to both decides whether the first spectrum of a temperate rocky planet’s heat, with its carbon dioxide, water and ozone bands, is taken with this technique or with a coronagraph in reflected light — the other way of dealing with a star, which is limited by different physics entirely.