RemoteTech antenna ranges in RO

Eventually you will notice that the effective ranges in the antenna descriptions don't seem to match the stats shown in the informational window or the RemoteTech Player Guide. This is because RO uses different RemoteTech settings than the RemoteTech defaults. Specifically, the range model and the ground station ranges differ.

TL;DR: The effective ranges in antenna descriptions use the root range model (see below) with a ground station range of 1.14E+14 m.

Root range model

The default RemoteTech range model is "Standard". A pair of antennae can link if the distance between them is smaller than the minimum of their ranges:

linkRange_Standard(antennaRange1, antennaRange2) = min(antennaRange1, antennaRange2)

RO instead uses the "Root" range model. In this model, a weak antenna can link with a stronger one even if it's out of the range listed in the part info:

linkRange_Root(antennaRange1, antennaRange2) = min(antennaRange1, antennaRange2) + sqrt(antennaRange1 * antennaRange2)

However, the link range is limited to 100 times the omni range or 1000 times the dish range, whichever is smallest:

linkRange_Limited(range1, range2) = min(linkRange(range1, range2), ceiling(range1), ceiling(range2))

ceiling(range) =  100 * range   for omni antennas
                 1000 * range   for dish antennas

For example, the link range between a Communotron 16 (4 Mm) and a Reflectron KR-7 (75 Mm) is 4 Mm + sqrt(4 Mm * 75 Mm) = 21 Mm.

Quoting ABZB from the KSP forums:

This is more realistic, in reality, probes have relatively low-power antenna, and we rely on having very powerful earth-based antennas to send & receive.

Realism Overhaul changes the ranges of the various antennas, as well as those settings, to create a reasonable approximation of that RL setup. If you change it back to Standard, it would be harder than RL - which is quite insane enough as it is!

Ground station ranges

The antenna ranges of ground stations is also different from the 75 Mm listed in the RemoteTech Player Guide. By inspecting the RemoteTech_Settings.cfg shipped with RealSolarSystem v10.6.2, we find in particular three stations whose names start with "DSS". These are stations in the Deep Space Network, which have antenna ranges of 1.14E+14 m. These are the stations used to calculate the effective ranges in the antenna descriptions. Take for example the HG-55 High Gain Antenna with its listed range of 1 Gm:

linkRange_Root(1E+09 m, 1.14E+14 m) = 1E+09 m + sqrt(1E+09 m * 1.14E+14 m) = 339E+09 m

These three DSS stations are located in Australia, Spain and the USA, so they should be able to keep you covered most of the time.

A useful tool for network planning is Visual RemoteTech Planner for KSP
Enter this data in the settings tab page. Stock Data: Real Solar System Range Multiplier: 1 Range Model Type: Root Multiple Antenna Multiplier: 1