nobody wrote:
100 m W can 7 Km

according http://uk.youtube.com/watch?v=xNFg8l5tjDI

I draw conclusion from the movie at 0.22 second from the above movie


1000 m W = 1 W

1 W can reach 7KM * 10

5 W = 7KM * 10 * 5

= 350 KM the signal still cannot reach the AO 51 satellite which is
850 Km away from the earth.


to reach AO 51 the power required at least 12.5 W

7KM * 10 * 12.5
=875 KM


is my calculation correct?

No.Bear in mind that signal strength goes as inverse square of
distance.. Doubling the distance requires 4 times the power. Or
conversely, 10 times the power only gives you about 3x the range (square
root of 10)).

However, the initial assumptions for required power (100mW = 700m) is
invalid.


For example Pioneer 10 radiated about 8 Watts at 2.2 GHz from well past
the orbit of Saturn, and we detected it here on Earth (but we used a big
70m dish with a cryogenic receiver to do it). And it's not like it
carried voice.. more like telemetry at 8 bits/second.

So, let's get back in the realm of possibility.

Since you're looking at AO51, that's an FM receiver on the bird. It's
spec'd to require -125dBm on 2meters. The range to the satellite varies
from 800km (satellite overhead) to 3000km (satellite on horizon).

On 2M uplink, the path loss due to distance is about -145 to -134 dB
plus another 3 or 4 dB for ionosphere and polarization losses. Working
backwards: -125dBm +140dB = about +25dBm (300mW) into a omnidirectional
antenna should do it. In reality, you might want to use a 4-5 Watt
transmitter and a 6 dB gain antenna to really get a good signal in there.



http://www.qsl.net/kb7ino/2_sat_calc.html has more info, as does
http://www.amsat.org/amsat-new/echo/Echo_Linkbudget.php