All righty...
First...The range of several thousands km is nothing spectacular. But there is a caveat...
Radar Realities from AOPA Magazine
To illustrate the problems of small antennas on small airplanes, let's use AOPA's Beechcraft A36 Bonanza as an example. Pilots new to this airplane get all misty-eyed when they see the wing-mounted radar pod and the King KWX-56 display screen on the instrument panel. They think this gear can get them through or around a line of thunderstorms, no sweat.
Hah! Behind that radome is a 10-inch antenna that throws out a beam 10 degrees wide. By the time that beam reaches 80 miles ahead of the airplane, it's 80,000 feet high and 80,000 feet wide. That beam goes above the tropopause at that range, and if there are any storm cells out there, their features have been distorted and merged by a process called beam smearing. "It's like trying to paint a nail head with a mop," says Archie Trammel, president of AJT, Inc. Individual cells of storm activity are blended together by beam smearing at this range, and the storm will appear to be much wider than it really is. Needless to say, the imagery you may see on the display screen will probably be useless for locating easy passages through storm systems.
Set the display for the 20- or 10-nautical-mile range, however, and the beam is narrower -- a 20,000-foot or 10,000-foot disc, respectively. More useful for seeing storm and cell shapes, but still subject to some beam smearing. As for using the tilt control to scan up and down the height of a storm, forget it. At the 80-mile range, the Bonanza's beam tops the highest of thunderstorms -- even if the tilt control is set at 0. Leave vertical scanning to the airliners, whose 30-inch-diameter antennas put out 3-degree beams that really can do the kind of vertical profiling that lesser general aviation units can promise only on a good day.
Another antenna-related problem comes into play when a small-diameter antenna's beam misses precipitation echoes. For example, aim a small antenna at a storm that is 40 miles away and 4 miles high by 4 miles wide. About 75 percent of the radar energy in that 8-mile-high by 8-mile-wide beam at that range will sail over, under, or around the storm. The precipitation echoes in this situation are said to be less than "beam filling," and this is not a good thing. The radar averages the radar energy returned to the antenna and therefore shows a storm that's much weaker than it really is -- all because of a non-beam-filling return. Couple beam-smearing effects with an absence of beam-filling returns and you have a deceptive picture in the cockpit. The radar display can show precipitation echoes both wider and weaker than they really are.
"The bigger the antenna, the better," says radar expert Dave Gwinn, who likens the focus of small-diameter antenna beams to that of a patio light. Large 30-inch-diameter airliner-sized antennas, on the other hand, "have the focus of a spotlight" and are much more accurate in defining storm shapes. There is less beam smearing and more beam filling.
The caveat here is that the larger the antenna, classical dish or modern phase array, the greater the distance. Pave Paws array dimensions and the above general aviation radar source illustrate that point nicely. So never mind the math, this is informational, not instructional. If anyone want to play with numbers, then get himself to a university. The basic information here is the larger the antenna, the greater the range.
Next is freq and beamwidth...
AN/FPS-115 PAVE PAWS Radar
The active portion of the array resides in a circle 22.1 meters (72.5 feet) wide in the center of the array. Each radiating element is connected to a solid-state transmit/receive module that provides 325 watts of power and a low-noise receiver to amplify the returning radar signals. The RF signals transmitted from each array face form one narrow main beam with a width of 2.2 degrees.
Frequency 420 megahertz (MHz) to 450 MHz
The mhz range is the VHF/UHF bands with meters length wavelength. So for Pave Paws, using meters length freqs and to have as narrow a beam as possible required a meters wide diameter antenna. The relationship and problem between range and antenna dimension is obvious.
This begs the question of why would Pave Paws uses such low freqs when the higher freqs are already in operation? The answer is that Pave Paws' mission is volume search. The lower freqs will have a wider dispersion pattern than the ghz freqs. For volume search, a wide beamwidth is good, but at the expense of target resolutions, such as fine speed or altitude resolutions. Remember, this is basically a cone. So for volume search, we want to cover the volume as soon as possible. The narrower the beam the longer it will take to cover that volume.
There is something else called 'resolution cell'...
Definition: radar resolution cell
radar resolution cell: The volume of space that is occupied by a radar pulse and that is determined by the pulse duration and the horizontal and vertical beamwidths of the transmitting radar. Note: The radar cannot distinguish between two separate objects that lie within the same resolution cell.
The height of the cell and the width of the cell do increase with range.
The narrower the beam, the longer it will take us to cover a volume, but if there are multiple targets we want to be able to tell them apart. Another way of looking at 'resolution cell' is that for X-distance, a B-52 can span across hundreds or thousands of cells at the ghz bands but only dozens at the mhz bands. So if we have two F-16s inside a cell, the radar will tell us there is only one target. But if we use a higher freq which produces smaller cells, then we can see two F-16s because they will occupy, or span across, multiple cells. If this is about two F-35s loaded with bombs, then the defender is 's-o-l', as the Americans would say, no matter what ghz freq use until it is too late. This is only a raw example as this is informational, not instructional.
So these main items are very important:
- Antenna size
- Freq use
- Beamwidth
- Resolution cell
The relationship between them and the type of target, which set our mission, determine the type of radar we finally design. For aircrafts, we have limited real estate or volume. For land we have no such limitations. That is why Pave Paws is the way it is and why fighter aircraft radars the way they are.
Clearly...Depending on the mission, range may not be as important as volume search and vice versa. The reason why the X-band, or ghz freq or centimetric wavelength, is used here is because of the nature of the target, which is far smaller than a fighter aircraft and is moving, or descending, at double digit Mach. Remember, the type of target set our mission. The target is a descending warhead, possibly nuclear, moving at double digit Mach. Unlike Pave Paws, we would have a better idea of the source direction of this target, therefore we can afford to use higher freqs, which give us tight beams, at the expense of longer coverage time for a given volume, or sector of the sky. This six-inch target size is nothing more than sales brochure blurb. It is the freq that should be the focus of our attention because the ghz bands produces tighter beams than the mhz bands at any distance, producing smaller resolution cells, and this give us better odds at distinguishing decoys, which is a very real possibility, from the genuine warhead. The larger the antenna, the longer the range and the greater the response time we have.
and as explained by a senior (Thomas) in previous post that US just shares the data with israel, when israel could not get it, how could you 
