You don´t even need big power for longrange HF contacts (human or natural interference excluded). I talked to a guy in Baltimore, Maryland using only 100 Watts and a mediocre antenna. Obviously around the poles the earth magnetic field distorts the normal conductive layers, but, since the charged particles from the solar wind get pulled towards the poles and concentrated there (see polar lights), the effect must even be stronger.
For us hams, who are limited in transmission power by law (e.g. for HF 2 KW in the US, 750 W in Germany or even lower in many other places) the biggest problem is human interference. E.g. I can hear hamsd from the US or Japan very well, but my 100 Watts simply get shouted over by the big guns with their power amplifiers (though I bought a 500W amplifier two weeks ago and am just fixing up my antenna so that I can use it. I will also get a commercial East German made HF transmitter as used for ATC or merchant marine ground and coastal stations, as well as HF broadcasters, within the next few weeks, which delivers a maximum of 2 kW, but output power can be lowered by turning a knob, so that I will stay legal. Together with a matching receiver, which I already own, I´ll then have the equipment of an equivalent of Shanwick Oceanic at home and should be able to talk to radio hams all over the world).
AM as a means of modulation is not very efficient. Most transmitter power (about 75% ) is being used to transmit the leftover carrier wave, which doesn´t carry any information. Since the signal also contains two side bands (which each contain the full information), the signal used a huge bandwidth. Single sideband transmission is much more efficient. Basically you´ll start with an AM signal, but in the transmitter, before you´ll get to the final power amplifier stages, you´ll filter out the carrier frequency and one sideband (historically hams use the lower sideband below 10MHz, and the upper one above 10MHz, while commercial operators almost exclusively use the upper sideband). Then only the remaining sideband gets amplified and transmitted.
On the receiver side the missing carrier gets added again and the signal then demodulated to get the audio signal.
This means that while AM equipment is cheap and easy to produce (to receive an AM signal a simple detector receiver consisting of an antenna, a coil, a variable capacitor a diode and an earphone) is enough, SSB equipment is much more expensive and complicated.
FM uses quite a bit of bandwidth, but has the advantage that it isn´t affected by interference like lightning (which change the signal amplitude). Due to the bandwidth requirements, it is more used in higher frequencies, like VHF and UHF, where more stations at given bandwidth, can fit into a frequency band. It gives better quality of sound.
Other modulation modes, like CW (Morse code through switching the carrier wave on and off), RTTY (Radioteletype, using a two tone audio signal transmitted via SSB) or various other digital modes like PSK (Phase shift keying) etc.
use less bandwidth and can be better received even if the received signal is weak.
Je Suis Charlie et je suis Ahmet aussi