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It's my understanding of observational astronomy that the size of a telescope limits its effective angular resolution, which is why scientists needed to use radio telescopes all over the globe to look at the M87 black hole and Sagittarius A*.

For the sake of convenience, let's define a "good picture" as at least 100-by-100 pixels, and assume that all neutron stars have a diameter of 20 km. With this assumption, the nearest known neutron star is PSR J0108−1431, at 424 light years away. This gives our neutron star an angular diameter of $2.9\times10^{-13}$ degrees, so the image would need a resolution of $2.9\times10^{-15}$ degrees per pixel. How big would a telescope need to be to see this accurately?

zucculent
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2 Answers2

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The angular resolution is just $\sim \lambda/D$ (in radians), where $\lambda$ is the wavelength and $D$ is the telescope diameter (or the size of an interferometer). So plug in the numbers you like.

To resolve the optical emission (say $\lambda = 500$ nm) at the angular power you specify would require $ D =10^{10}$ m.

Details

A 20000 m diameter neutron star at a distance of 424 light years (= $4.01\times 10^{18}$ m), subtends an angle of $5\times 10^{-15}$ radians.

To resolve this into 100 pixels requires a resolution of $5\times 10^{-17}$ radians.

Then $D = \lambda/5\times 10^{-17} = 10^{10}$ m.

Edit:

As blademan9999 points out, one can show that the nearest (undiscovered) neutron star to Earth is likely an order of magnetude closer than this - see How far away is the nearest compact star remnant likely to be? If so, then a telescope of diameter $\sim 10^9$ m might resolve it. However, such a neutron star is likely to be old and very cold. Its surface brightness might be very low in which case it may not be a good target for telescopic observation.

ProfRob
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    Would interferometry at a 500 nm wavelength be possible with our current technology? – zucculent Nov 01 '21 at 22:42
  • @zucculent Has optical interferometry been done at radio frequency using heterodyning with a laser in a nonlinear material? and Will the Magdalena Ridge Optical Interferometer be able to image extended objects like the surface of the Moon? and Can the interferometer called "Gravity" measure "a few centimeters on the Moon"? there are ways to extend baselines for optical interferometry, but ~10 AU is still quite a challenge. – uhoh Nov 02 '21 at 00:08
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    That is one big dish. Carving out the insides of a supergiant star, perhaps? – Stian Nov 02 '21 at 10:30
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    @StianYttervik it wouldn't be a dish (for optical work), it would be two telescopes with an interferometric connection. – ProfRob Nov 02 '21 at 16:03
  • @ProfRob That's still an equivalent dish size of ~67 AU. 9 and a quarter hours at speed of light from edge to edge. A reflector that large would basically be a solar death ray. – David S Nov 02 '21 at 16:38
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    @DavidS you misunderstand. An interferometer can be made from two small telescopes (well big enough to actually detect the neutron star though), separated by $10^{12}$ m. Some day, such things will exist. Probably requires some technology to record phase and amplitude of the received light. – ProfRob Nov 02 '21 at 16:45
  • @Zucculent interferometry at 500 nm is possible now. Just not with big baselines. – ProfRob Nov 02 '21 at 16:46
  • @ProfRob I understand you can have two 1 meter telescopes with a baseline that long and have your telescope. Separately if you had a parabolic reflector that large it would probably produce very interesting effects at its focal point. – David S Nov 02 '21 at 16:56
  • @zucculent interferometry is just a cheaty way of pretending you have a disk as wide as the two dishes in the interferometry are apart. Putting 2 dishes 1e12m apart would be problematic, That's two dishes a bit further apart than Earth and Jupiter. and you need to synchronize their reception to a tiny fraction of a wavelength of your observing frequency, so accurate in distance and timing down to 500 nanometer and microsecond synchronization. Microsecond accuracy, over a lightspeed gap of about an hour. Impossible? no. Doable? likely also no. And that's to resolve the whole target as ONE PIXEL – PcMan Nov 02 '21 at 16:58
  • @PcMan that's why (in a comment) I suggested this might only be possible when the technology exists to record the phase and amplitude at optical frequencies. Then off-line correlation can be done, just as it is for radio VLBI. We will also have to see how the free-flyer interferometer used by eLISA performs, which will have $>10^9$ m arms. – ProfRob Nov 02 '21 at 17:21
  • @ProfRob Hanbury Brown and Twiss effect can be done as off-line correlation which would give a measure the angular size of the neutron star. – D Duck Nov 02 '21 at 19:27
  • So... one dish on Pluto, another hanging out at its L3... – J... Nov 02 '21 at 19:52
  • I redid the math myself, and it looks like I gave the wrong angular resolution, and it should be $10^{10}\space\rm m$, not $10^{12}\space\rm m$. I've fixed it now. – zucculent Nov 04 '21 at 16:34
  • Good news though is that there probably is an undiscovered neutron adleast an order of mangitude nearer https://astronomy.stackexchange.com/questions/16678/how-far-away-is-the-nearest-compact-star-remnant-likely-to-be?noredirect=1&lq=1 So you'll "only" need $10^{9}$m, instead of $10^{10}$ m – blademan9999 Jul 02 '23 at 11:24
  • Interferometry would be a poor choice, because you are limited not just by angular resolution but by how many photons from the source you can capture. An interferometer may have the angular resolution of a much bigger scope, but its light collecting power is just that of the scopes in the array itself, and almost certainly not enough to actually image such a small, dark target. Better to use a solar gravitational lens scope. – Dan Hanson Jul 07 '23 at 21:58
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Closest neutron star is RX J1856.5−3754, which is about 400 light-years from us. HOWEVER the Hubble did visibly spot a 200 lightyear distant otherwise known moving Bare neutron star streaking space. They flash the sky, producing that twinkle stars are known for. Here is a photograph of a neutron star taken by Hubble. Never the less any neutron star close enough to obtain detailed resolution images would be lethal.... Neutron stars produce so much radiation and magnetic fields they're potentially lethal at distances of a few lightyears.

LazyReader
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