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In comments below this answer to the question Which satellites can hear emergency signals from Scott Kelley's watch? we are (or at least I am) trying to figure out definitively if a wristwatch can communicate with satellites in geostationary orbit. The protocol comes from the international Cospas-Sarsat programme.

These are ~60 times farther away than those in LEO, so $1/r^2$ is about 3,600 times smaller.

Rather than litigate it there, I thought it would be better to focus on these few satellites in a separate question and see what the link budget turns out to be.

In order to estimate this link budget, one needs some details; specifically, the gains of the antennas and the transmit power at 406 MHz are needed to estimate the receive power, and the bandwidth used for the signal is needed to estimate the signal to noise ratio (similarly to this).

Presumably the bandwidth is quite small, but the beacon does transmit a bit of self-identification information. It's not CW. Still, when received in GEO, there will be no Doppler shift and so receive bandwidth could be narrower than that used in spacecraft in LEO, which would have to be roughly 20 kHz no matter how narrow the transmission bandwidth was. Lack of Doppler also means there is no information obtainable about position.

Question: What are reasonable estimates or known values for these and the resulting received power and SNR, and how realistic is it for emergency beacons from a wristwatch (Scott Kelley's or otherwise) to be heard by the few of the receiving satellites that reside in geostationary orbits?

There seems to be some information about a particular watch linked here and here but I can't vouch for it.

From this answer from this page found in this answer, the current spacecraft in GEO capable of receiving these 406 MHz emergency beacons are:

GEOSAR:

GOES:      two spacecraft       (15, 16) 
MSG:       three spacecraft     (1, 3, 4) 
Misc:      three spacecraft     (Louch-5A, INSAT-3D, INSAT-3DR)

It is possible that information about the receiving antenna gain of one or more of these will be available.

Link Budget

From this answer which is from this answer:

$$ P_{RX} = P_{TX} + G_{TX} - L_{FS} + G_{RX} $$

  • $P_{RX}$: received power by spacecraft
  • $P_{TX}$: transmitted power by wristwatch
  • $G_{TX}$: Gain of wristwatch's transmitting antenna (compared to isotropic)
  • $L_{FS}$: Free space Loss, what we usually call $1/r^2$
  • $G_{RX}$: Gain of spacecraft's receiving antenna (compared to isotropic)

$$G \sim 20 \times \log_{10}\left( \frac{\pi d}{\lambda} \right)$$

$$L_{FS} = 20 \times \log_{10}\left( 4 \pi \frac{R}{\lambda} \right).$$

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    $\begingroup$ Ptx seems to be 3.2 W, pretty much for a wristwatch. $\endgroup$ – Uwe Apr 30 at 8:08
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    $\begingroup$ @uhoh the data sent does not need to contain gps location. The current time is enough as that can also be used to get the location. I think those watches work kind of like opposite gps satellites. $\endgroup$ – GittingGud Apr 30 at 10:00
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    $\begingroup$ @GittingGud if you want to use the current time of the watch to get the location, you need a very precise time. At the equator one scond error is about 460 m. But how do you get the individual local sun-time? A time zone time does not help. For "watches work kind of like opposite gps satellites" an extreamly precise atomic clock time is needed, remember the speed of light. $\endgroup$ – Uwe Apr 30 at 10:24
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    $\begingroup$ @uhoh Only some beacons transmit GPS coordinates, but the GEOSARs are still useful if the beacon doesn't do that. I've added information about the role of the GEOSARs to my answer on the other question. $\endgroup$ – Polygnome Apr 30 at 11:03
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    $\begingroup$ I don't think high gain narrow beam antennas may be used for the wristwatch and the GEOSAR satellite. The watch should have a 180 ° antenna. The satellite should receive emergency signals from all over the world, at least the visible part as seen from GEO orbit. $\endgroup$ – Uwe May 5 at 9:42

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