A Cosmic Focus Knob

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There's a general problem in astronomy, which is that we almost totally lack depth perception. This is not to say that we don't know the distances to things, for we frequently do. That's what the cosmic distance ladder is for. However, except for quite nearby things, for which parallax measurements can be done, you're quite some ways up the ladder and you often have only approximate distances only for certain types of objects.

Case in point, at some journal club talk last year a number of interesting conclusions hinged on whether a chunk of radio emission was coming from a particular galaxy however many Megaparsecs distant, or else was far in front of or behind it. For the galaxy, the distance is reasonably well known (certainly via the cosmological redshift, possibly via other means as well). For the radio emission, not so much. It was a continuum source, which means there were no spectral lines to give a redshift, and it wasn't a galaxy, which rules out most of the other higher rungs of the distance ladder.

With present techniques, the Magellanic Clouds are the only galaxies for which one could really conceive of getting parallax measurements. This would have to be done using long-baseline interferometry of radio point sources, of course, using something like the VLBA. The limit is set by the angular resolution of your instrument, since the parallax is nothing more than measuring a (tiny) angle on the sky. For the VLBA, observing at 10 cm from stations around 10,000 km apart, you can get about a milli-arcsecond. Using the Earth's orbit as your separation, that gets you out to a few kiloparsecs.

What could you conceive of building, anyway? If you want a super-long baseline, you need to stick to radio techniques, where you record the waveforms and feed them into a correlator elsewhere. Electronics are getting better, so I can imagine that working up to several hundred GHz, so millimeter waves. We're pretty good at chucking things into solar orbits, and at powering things off solar energy at Earth-like distances from the sun, too. So I can conceive of building a millimeter wavelength interferometer array with a baseline of a couple of AU. And that gets you to about a nanoarcsecond. With this kind of resolution, you could just resolve a penny held up to the sun by an astronaut at Saturn. (Or, someone else with this telescope could see the city lights of Earth from halfway across the Galaxy.)

If you consider that the Solar System moves at about 220 km/s around the galactic center, if you're willing to wait a year as with traditional parallaxes, you get a baseline of 50 AU or so. In principle, you can then measure a parallax out to basically the edge of the universe, 50 Gigaparsecs or so, although you'll have trouble defining a fixed background if you do that. However, this probably wouldn't help with the sort of diffuse source that I started out discussing.

I often wonder if this would work. Measure distance by defocusing an interferometer. By this I mean, interferometric correlators work on the assumption that the incoming wavefronts are flat. The constant-phase surface of a radio signal leaving a point source is actually a sphere (usually), but at a distance of light years, you don't especially care. But I guarantee that you have seen this effect before.

Turn the focus knob of a pair of binoculars, or one of those old cameras that actually made you focus it yourself. Objects at one distance will appear crisply, while objects in the foreground and background become fuzzy. The optics of the focus mechanism are compensating for a specific amount of this wavefront curvature. You would be correct in imagining that this could be used to measure distance, but only out to a certain maximum.

If you push the focus knob all the way to one end, generally marked as "infinity", then everything beyond some distant point will be in focus. Past that is the far field of your device, beyond which the wavefront curvature doesn't matter, and because of which nobody worries about depth of field when photographing landscapes or nebulae. The far field distance is roughly the square of your aperture size divided by wavelength. For your binoculars (3 cm, 550 nm) it's a kilometer or so. For the VLBA (10,000 km, 10 cm) this is about a tenth of a light year, but for our really ambitious yet conceivable telescope (2 AU, 1 mm) this becomes a few Gigaparsecs.

Now, I think an algorithm based on this technique would probably work, even on the diffuse cloud discussed above. You just have to adjust the depth of field until the cloud is at its smallest. You also wouldn't need a fixed background against which to compare (usually distant quasars today, but they probably wouldn't be distant enough for this kind of work). Now, it's rather complex to make a good interferometric image of a spread-out thing, but my understanding is that it's possible. Maybe some of the radio astronomers reading this will set me straight if not.

3 Comments

Don't forget radar ranging. Everyone always forgets the radar ranging rung on the bottom of the distance ladder. Without it our scale for the Astronomical Unit would not be fundamentally set.

Another thought on this. It would work optimally for a long-focus instrument. The longer the focal length, the more "resolution" you've got on your focus axis. So you'd want the longest focal length instrument you could design to do this.

The drawback on this is the at longer focal lengths your plate scale at the focal plane gets huge. Effectively your magnification factor increases and the number of photos falling per unit area on your detector drops off precipitously.

Most extra galactic work requires long exposures on short focus instruments. We're talking F-ratios of 2 to 4. You'd want an F-ratio so high that your exposure times would get really long to see the same features you make out with the shorter F's.

In short you've got the physics of ray-tracing optics causing you to make a trade-off that screws you if you try to do the distance from focus thing.

Hm, I can honestly say I didn't think about the plate scale argument, but I'm also not entirely sure it applies here. After all, I'm not proposing a real AU-wide Cassegrain dish, but a network of telescopes doing synthetic aperture interferometry. In which case you still have a microscopically small beam, and if you don't fill in the short baselines, no ability to map structures, which you'd definitely need for this trick to work.

But I think the analog to a long focal length as you discuss above, would be the amount of phase resolution at each of your receivers. That's a different kind of engineering problem, although certainly not trivial.

I did think a little about the problem of integration time, though. Since synthetic apertures work strictly in the wave domain, the right way to think about it is to note that when you do the integrations in the correlator, virtually all of your incident signal power will cancel out. So that adds up to the same thing; for any reasonable source, the power coming from any given square-nanoarcsecond solid angle is going to be infinitesimal.

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This page contains a single entry by Milligan published on January 19, 2008 2:00 PM.

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