Why is Saturn invisible in this radar image of its rings?

Question

The image below is from Radar imaging of Saturn’s rings Nicholson, P. D. et al., Icarus 177 (2005) 32–62, doi:10.1016/j.icarus.2005.03.023 and discussed further in this answer to How did Arecibo detect methane lakes on Titan, and image Saturn's rings?

I believe I can see Saturn's radar "shadow" blocking the furthest parts of the ring near the top of the images, but I can not see any reflections from Saturn itself.

The paper addresses a possible signal from the planet itself in Section 3.2. Ring images:

Echoes at low Doppler shifts might be expected to arise either from near the subradar point on the planet itself or from ring material far outside the main rings. Our images show no evidence for any echo from the subradar point on Saturn, which would appear near ν = 0 and τ = −2RS/c = −402 ms.

The same answer describes radar measurements of Saturn's moon Titan as well.

It might be possible to make a hand-waving argument that Saturn itself is invisible because it's "just gas" but according to ESA's Saturn's atmosphere

The top visible cloud deck, made of ammonia clouds, is found at about 100 kilometres below the top of the troposhere (tropopause), where the temperature is about -250°C.

The second cloud deck, made of ammonium hydrosulphide clouds, is found at about 170 kilometres below the tropopause, where the temperature is -70°C.

The lowest cloud deck, made of water clouds, is found at about 130 kilometres below the tropopause, where the temperature is about 0°C (freezing point of water).

So I'm thinking that the various clouds of different colors and compositions will contain droplets or particles, rather than just be regions of different gas composition.

Why then does there seem to be nowhere near as much radar return from the planet as there is from the rings?

Answer 1

The main issue is that there is relatively little material in Saturn's atmosphere that can efficiently scatter radar waves, so the radar basically just gets absorbed.

The key point is that it's much harder to get a radar return from the very small objects (aerosol droplets or tiny ice particles) that would make up clouds in Saturn's upper atmosphere than it is to detect the meter-sized ice chunks in the rings.

The figure below (from here) shows $\sigma / (\pi r^{2})$, which is the "effective" scattering cross section relative to the geometric cross section of a scattering object, plotted versus the size of the object $r$ relative to the wavelength of light $\lambda$ (e.g., the radio waves used in radar). When the size of the object is roughly the same size as wavelength of light, or larger, it efficiently scatters them: its effective cross section is roughly the same as its geometric cross section (ignoring other effects such as chemical composition, surface roughness, etc.). Sometimes it can even be several times larger!

But when the objects become smaller than the wavelength, you transition to the Rayleigh scattering regime, where the scattering depends on the size of the object to the sixth power.

Imagine a sphere 10 cm in radius, roughly the same as typical radar wavelengths (so $2 \pi r / \lambda \sim 1$). It will have an effective cross section similar to its geometric cross section: about 300 cm$^2$. Now imagine subdividing that object into spheres one-tenth the radius. This would mean about 1000 objects 1 cm in radius (so that the total volume is the same), with a total geometric cross section of about 3000 cm$^2$ (realistically a little less, because of shadowing). But the plot shows that the effective cross section goes down by a factor of 1000, so the total radar cross section of all the small objects -- and thus the backscattered energy of the radar waves -- would be ten times smaller than it was for the single big object of the same total volume. And that's just for a reduction in size of ten; in reality, we're talking about going from centimeter-to-meter-sized objects (the ring) to sub-mm and micron-sized cloud particles in Saturn's atmosphere.

This is why weather radar on Earth can show you precipitation (rain, snow, hail) -- because the raindrops/etc. are large (a few mm in size), and can scatter radar waves with some efficiency -- but usually can't show you clouds -- because the water droplets making up clouds are so small. (And obviously radar wouldn't be nearly as useful to the military if it was easily scattered by clouds.)

So -- absent floating chunks of centimeter- or large-sized ice in the upper atmosphere of Saturn -- you're not going to get much of a radar return from Saturn itself. The original discovery of radar returns from the rings surprised people, because the assumption had been that the rings were made up of very small (e.g., micron-sized) bits of ice, which wouldn't efficiently reflect radar waves either.

Answer 2

The images you present aren't literal images of Saturn's rings. They're "Doppler-delay" plots: the vertical axis represents distance from Earth, while the horizontal axis represents speed towards or away from Earth. Since the ring particles are moving in circular paths around Saturn, this produces an elliptical chart.

Further evidence that this isn't a literal image is the fact that the rings are non-concentric. The B ring is closer to Saturn than the A ring (less variation on the vertical scale), but because it's closer, the particles move faster (more variation on the horizontal axis).

The absence of Saturn, as noted briefly in the paper you mention, is because it simply isn't very bright. Only a small portion of Saturn is moving at any given combination of speed and distance compared to the large areas of the rings, and atmospheric gasses are only weak radar reflectors where rock is a strong reflector. According to figures 3 and 4 in the paper, if it were visible, Saturn would be an arc in the middle of the plot, extending vertically from 0 down to about -200, and horizontally from -141 to +141.