# Is delay-doppler radar imaging of NEO asteroids possible only if it spins fast enough?

Emily Lakdawalla's blog on the Planetary Society's page explains some of the basic principles in the delay-doppler radar imaging of asteroids, and other astronomical bodies by extension.

Radar from a single dish does not (usually!!) have anywhere near enough resolution to image an asteroid. However, the miracle of math allows one to reconstruct an image of the object anyway. A complicatedly modulated (coded) long radar pulse is broadcast toward the asteroid, and each point on it reflects a little of the pulse back to Earth. However each reflection is slightly modified in a unique way.

The parts of the asteroid that are farther return reflections that are delayed by the extra distance travelled, and each reflection is doppler shifted by the line-of-sight projection of the rotation velocity of that point around the asteroid's center of mass. The doppler shift is in frequency as well as in phase, and therefore compresses/stretches the reflection in time as well.

By correlating the original coding pattern with the received signal, one can build up information about the 3D structure of the asteroid and generate a kind of image. Because you have both absolute time delay differences as well as rotation, you now have a fairly precise physical measurement of the absolute size (and of course shape) of the asteroid as well as its precise distance.

## But, What if it wasn't rotating?

This leads me to the possibly inescapable conclusion that an asteroid that wasn't rotating, or even worse, was rotating very very slowly such that it presented the same face towards Earth as it passed by would be impossible to generate an image or transverse size with radar using the delay-doppler technique. You could still get some kind of size information along the line of sight, and of course precise distance, but the transverse shape and size would be un-image-able.

It could be 100x100m or 10x10km (extreme example) in the transverse plane, and except for assumptions on radar albedo, you'd never know which.

Am I wrong? Is there something I've missed?

note: In order to not get bogged down too deeply in math within the set-up of the question, I've avoided the complications of 3D rotation (tumbling) or that rotation along the line of sight direction is almost as bad as no rotation at all, or the potential use of polarization or advanced modeling. However that doesn't mean these need to be excluded from answers as well!

Asteroid 2014 JO25 is passing relatively close to Earth now, and the radar images are already starting to be published! See for example NPR's An Asteroid Is Swinging By Earth Today For Its Closest Visit In 400 Years and the image and video linked there.

below: Cropped image from NPR's article, with the caption:

This composite of 30 images of asteroid 2014 JO25 was generated with radar data collected using NASA's Goldstone Solar System Radar in California's Mojave Desert. NASA/JPL-Caltech/GSSR

• Well, if you were to assume that the face it is showing you is stationary, then I see no reason why you wouldn't be able to get a topographic map of that face, providing you had high enough resolution. – Phiteros Apr 20 '17 at 3:44
• @Phiteros Nope! Not with delay-doppler. Not on a 1km asteroid 1,000,000km away. If you have a method - propose it, and don't forget the math! Read through Lackdawalla's explanation first. This is not traditional imaging. – uhoh Apr 20 '17 at 3:45
• I understand the concept of DDM. But if you have a high enough resolution, you could simply bounce a signal off a small part of the asteroid and calculate the distance to that little area. Repeat until you've built up a map of the whole surface. Essentially you'd be doing radar range-finding on every small part of the surface. Of course, you'd have to be extremely precise and have spectacular resolution. – Phiteros Apr 20 '17 at 4:15
• @Phiteros but you don't have that. If you had a 100m dish and a 1cm wavelength, you'd have a resolution of $10^{-4}$. At a distance of 1,000,000km each pixel would be 100km. In this image, each pixel is only few meters! – uhoh Apr 20 '17 at 4:16
• You didn't put any specifications about our instrumentation or the distance to this object in your question ;) And if you used an instrument like the VLBA you can get an angular resolution of 0.17 milliarcseconds. That's actually enough to resolve your asteroid. Now, if you throw something like RadioAstron in there, you'd get an even higher angular resolution. – Phiteros Apr 20 '17 at 4:20