Thread by @markmccaughrean

As we wait for the #JWST sunshield tensioning to begin after a day of well-deserved down time for the mission team, let’s talk in a bit more detail about how having a big, cold telescope helps us detect faint things.

Yes, folks, it’s Signal-To-Noise Sunday 😬

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Now, a health & safety warning – this could get a bit technical, maybe mathematical, & possibly even quantum mechanical 😱

I don’t know – I haven’t planned this thread at all, so it’ll just come out as a stream of consciousness. But hopefully a semi-intelligible one 🤷‍♂️🙂

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In the optical & infrared parts of the electromagnetic spectrum (i.e. light), as well as at higher energies like the ultraviolet, X-rays, & gamma rays, we typically thing of it comprising photons, individual packets or quanta, like little bullets of energy.

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Conversely, at longer wavelengths, in the far-infrared, millimetre, & radio, we usually think about it in terms of waves (hence the words wavelength & frequency, for example). Truth of it is that light is both: a wave-particle duality. But that’s too complicated for a Sunday.

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Fortunately for our relaxing Sunday thread, JWST is an infrared telescope & thinks about light as photons.

Its detectors capture individual photons & turn them into individual electrons (well, with an average efficiency of about 80% across the 1-5 micron wavelength range).

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Those electrons are then counted: the more electrons in a given pixel on the detector, the more photons that arrived there, & the brighter the source at that position in the image (or spectrum).

So in essence, we’re in the business of counting things.

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Now, the arrival times of those photons are uncorrelated, independent; sometimes you get a bunch, sometimes few. The way they’re emitted at the source (a star, say) is “stochastic”.

And when you count things which arrive stochastically, you turn to Siméon Denis Poisson.

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And what Monsieur Poisson tells us is that if you count things arriving in a stochastic way, the uncertainty in your measurement is equal to the square root of the number of things you counted, at least once the number exceeds 10 or so.

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en.m.wikipedia.org/wiki/Poisson_d…

The maths is complicated, but the outcome is fairly simple.

If I count 100 photons in a second, say, then the uncertainty is the square root of 100, so 10.

Then I can say the average rate is 100 ± 10 photons per second. In turn, you can say that’s a 10% error.

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To be technical, when I say “uncertainty” here, I mean the standard deviation or “sigma” of the distribution. I won’t go into more detail about the statistics of normal distributions here, but there’s plenty of reading material out there, e.g. 👇

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en.m.wikipedia.org/wiki/Standard_…

The other way astronomers refer to this error or uncertainty is to talk about “signal-to-noise”.

In this example, the signal is the number of photons we detected (100) & the noise is the Poisson error (10).

So the signal-to-noise or S/N is 100/10 = 10.

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Now, to be sure of a good detection & to get enable some good scientific analysis, we want to have good signal-to-noise & we will go to great lengths to improve it.

To do that, we need to collect more photons.

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We can do that by making a longer exposure, by building a bigger telescope, or both.

Let’s say we do that & collect 10,000 photons rather than 100. M Poisson now tells us that the uncertainty on that number is the square root of 10,000, so 100.

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So now the signal-to-noise is 10,000/100 = 100.

That’s good, but note we had to collect 100 times as many photons as in the first experiment, either with a 100 x longer exposure or with 100 x more collecting area, to improve the S/N by a factor of 10.

It’s tough going.

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So far we’ve only talked about the source of light itself, the star we’re interested in.

But in the optical & infrared, we also have to deal with that source being seen against a background, a glow of extra photons coming from the sky and/or the telescope.

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And because that background glow is made of photons too, it also follows Monsieur Poisson’s rules & thus also introduces more noise into the experiment.

And while you can subtract the background, you can’t remove the noise 😱

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A brief aside on the sky backgrounds that affect optical & infrared astronomy, starting on the ground.

During daytime, the obvious one is sunlight scattering off molecules in the atmosphere, making the sky blue. It's so bright, you can't see any stars with the naked eye.

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At night, things get much better, although if the Moon is up, then it acts in the same way, making the sky (relatively) bright through molecular scattering. For optical astronomers at least, being scheduled during "bright time" or "dark time" can make a big difference.

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Because the sunlit sky is blue (& even more so when the Moon is responsible) though, it's not very bright in the red or infrared.

But we infrared astronomers get clobbered by other problems instead.

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Earth's atmosphere not only scatters Sun & Moon light (& is brighter if you're close to humans & their blazing roads, gardens, & buildings), it also glows of its own accord.

Various atoms & molecules, often pumped up in daytime, give out light, including NO, Na, O, & Li.

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In the near-infrared, the main culprit is the hydroxyl radical, OH, at about 80km altitude. It gets pumped by daylight & is also affected by distant thunderstorms, leading to gravity waves & a rapidly-changing bright glow across the sky.

scielo.br/j/rbg/a/v86rQ3…

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The glow in the near-infrared is much brighter than the glow at visible wavelengths & makes ground-based astronomy in the near-IR quite challenging.

But at wavelengths beyond 2.5 microns (mid-visible is ~0.5 microns), it gets even worse.

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Then the atmosphere & even the telescope start to glow because they're (relatively) warm.

At ~0ºC on a cold mountaintop, their peak "blackbody emission" comes out around 10 microns, but there are plenty of photons already at 2.5 microns & they dominate the background.

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The colder the telescope, the better, hence IR telescopes are best on very high sites like Maunakea or even Antarctica, where it's cold & there's less water in the atmosphere the precious few IR photons you're seeing in the first place.

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Going into space is better. It gets rid of the atmospheric airglow &, if you can make your telescope cold (as #JWST will be doing behind its sunshield), you can reduce the thermal background emission to negligible levels too.

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Indeed, a key goal of #JWST is to reduce the background emission as far as possible, to be limited by one that's much harder to get rid of, i.e. the zodiacal light, sunlight scattered off small dust particles in the solar system.

en.wikipedia.org/wiki/Zodiacal_…

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Pic: ESO/Y Beletsky

Most of the dust that makes the zodiacal light comes from comets (thanks Rosetta, for that result) & is widely distributed, mostly beyond Earth. That means that #JWST at its L2 location "only" 1.5 million km away can't get rid of that background: it's the baseline.

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Of course, if you could build a space telescope that could get out of the solar system & away from the zodiacal light, it'd be even more sensitive ... but the data rates wouldn't be great 😬

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So let's go back to those signal-to-noise calculations for a bit before finishing for today.

Imagine we have a star that yields 100 photons per second all into a single pixel, but it sits on a bright background giving 10,000 photons per second per pixel.

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Thanks to M Poisson, we know that the noise on the star signal is the square root of 100 = 10. But the noise on the background is sqrt(10,000) = 100.

So the signal-to-noise of the star above the background is 100/100 = 1.

That's hopeless – the star's lost in the noise.

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Let's say I want a signal-to-noise of 100 instead, a clear, strong detection of the star. How long will that take?

If we expose for 100 seconds, we get 100 x 100 photons from the star = 10,000 & 100 x 10,000 from the background = 1 million.

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The noise on the background is sqrt(1 million) = 1,000.

So now the S/N is the 10,000 signal from the star divided by the 1,000 noise from the background (the sqrt(10,000) = 100 noise from the star is negligible).

The S/N is now only 10, even after 100 x more exposure.

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You can easily work out that if I want to increase the S/N by another factor of 10 to 100, I need to go 100 x longer again, for a cumulative 10,000 seconds.

When you're looking at faint sources & the background is the dominant source of emission & noise, life is hard.

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So reducing the background is vital. With #JWST, that means a cold telescope high above Earth's atmosphere.

And it also means a superclean telescope, to avoid the scattering of light from bright stars off dust on the optics into the field, raising the background.

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It also means a big telescope to collect more light from your faint sources in the Universe.

And a big telescope in space can mean sharper images, another way of effectively beating the background & getting better S/N ... but that's for another #JWST megathread.

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There's a lot more to these S/N calculations, of course, including cases where detector read-noise & dark current become dominant, e.g. in high-res spectroscopy.

But hopefully this gave some idea why we've worked so hard to build a cold telescope & kill the background.

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As a metaphor, we're trying to detect tiny boats bobbing on a wild ocean with high waves.

While we don't care how deep the ocean is, the deeper it is, the bigger the waves.

So the more we can do to bring those little boats to shallower waters, the easier we'll see them. 37/37

Coda: there will be many more weeks & months before #JWST deployment, alignment, & commissioning is complete 😬

So if you're lucky, there will be more #SignalToNoiseSundays 🎉

Otherwise, you can always mute or unfollow me 🤷‍♂️🙂