So in Monday's "what's up in the solar system" post I mentioned that New Horizons will soon be observing Uranus and Neptune using its LORRI camera, but since the planets will be so far away -- roughly three and four billion kilometers, respectively -- LORRI wouldn't resolve them. Well, I was wrong. Three billion sounded like such a big number that it seemed obvious to me that the distant planets would look no different from stars. I should have done the math! Shame on me. (And shame on me also for a couple of earlier errors in the analysis below. Sigh. My elementary school teachers would tell you I've always been too rushed and too sloppy with math!)
Here's a back-of-the-envelope calculation. The LORRI camera has a field of view of about 0.3 degrees. That field of view is projected onto a detector about 1000 pixels wide. So each pixel covers about 0.3/1000 = 0.0003 degrees. It's easier to work with that number in radians, so we'll convert to radians by multiplying by (2π/360) or about 1/60 -- so each pixel covers about 0.000005 radians. Now, for a target 3 billion kilometers away, that means each pixel will cover (3 billion times 0.000005, which is 3 million times 0.005, which is 3000 times 5, or) 15,000 kilometers. Uranus is about 50,000 kilometers across, so at New Horizons' distance from Uranus, it should be roughly 3 pixels in diameter. It won't be the greatest picture ever, but it definitely won't look like a star. It'll look like a little disk.
It's even possible you'll even be able to tell that it's not a full disk; New Horizons will be looking at it from a phase angle of about 30 degrees, so Uranus will appear to be in a gibbous phase. In fact, that's the main reason for New Horizons to be looking at Uranus in the first place. From Earth's position, which is much, much closer to the Sun than Uranus is, Uranus (and all the other outer planets) always appear almost completely full. In more technical terms, we see them from very low phase angles, near zero degrees. Looking at solar system bodies from different phase angles, studying the way that the brightness of a body decreases with increasing illumination angle, helps scientists understand physical properties of its surface or atmosphere.
There won't really be any detail visible, though. Another important aspect of LORRI (and any other camera) is something called the point spread function. It's a description of how sharp the camera is -- how much details that should only appear in one pixel on the detector actually spread out across multiple pixels. LORRI is a very, very sharp imager, with a point spread function of only 1.5 pixels. Still, when you are looking at an object only 3 pixels across, a point spread function of 1.5 pixels means you won't be seeing much detail.
Doing the same math above for Neptune is pretty easy -- all that changes is the range to the target, since Neptune has virtually the same diameter as Uranus. It'll be a bit farther away at nearly 4 billion kilometers, so LORRI should get a resolution of about 20,000 kilometers per pixel, so Neptune will appear a little more than 2 pixels in diameter.
Hmm. I wonder if LORRI will be able to detect the larger moons of these planets? It seems very likely. None of them will be resolvable -- the largest moon of Uranus is Titania, only 1,600 kilometers in diameter, and the largest moon of Neptune is Triton, 2,700 kilometers in diameter. But at the same time, LORRI is going to be used to look at Pluto, which is only 2,300 kilometers in diameter. It seems quite plausible that LORRI will be able to see a family of moons around Uranus as little specks of light, and at least Triton at Neptune. We'll have to wait and see.