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EDIT: With the third Starlink launch happening about right now, it's time to update this answer! We now know that each satellite has a weight of 260kg, and that they launch 60 satellites per mission. They are flat packed, stacked 30 up side by side in the Falcon 9 Fairing, and far smaller than what I thought would be possible in my first answer.

Elon Musk stated that they need at least 6 launches of 60 satellites to cover the US and Canada, and about 24 launches to provide global coverage (Phase 1 calls for 1584 satellites in the constellation). The full 12000 satellites constellation would be complete after 200 Falcon 9 launches.

So yeah, that about answer really the question.

With their new in-development Starship system, the payload goal is 100T to orbit. That would be about 384 satellites per launch. This would be about 32 launches from Starship to deploy the full 12000 satellites constellation. However, I personally believe the Starlink constellation will be operational before Starship is available to fly at full capabilities.

The dimensions of the test satellites are indeed 400kg each, in a box measuring 1.1 m × 0.7 m × 0.7 m .

Something I did not take into account in these calculations is the likely reduction in weight of the satellite during the mass production. For example, the Orbcomm OG2 fleet weighs in at 172 kg per satellite. This is a likely goal for SpaceX's satellites (and this would allow twice as much satellites per launch).

Also, the satellites are already fitted with antennas. Phased array antennas can be really tiny!

Falcon 9

Payload Mass

As per the Falcon 9 User Guide, the payload to LEO when flying fully expendable is 22 800kg. However, it's unlikely SpaceX will orbit their satellite fleet (all 12 000 of them, from this link) flying expendable. The Falcon 9 Wikipedia page cites a 9 600kg payload capability in the reusable configuration.

This would amount to a staggering 24 satellites per launch.

We know SpaceX has already launched thirty new satellites of the IridiumNEXT constellations in three launches. Their injection orbit is not far off the one of SpaceX Starlink, and the satellites weighs in at about 860kg (as per this article).

This is consistent with our findings too.

However, as you mentioned, all of this did not account for the payload attach fitting (the thing to hold and release the satellites).

If we use again the Iridium flights as reference, the PAF configured with 10 satellites was about 1 000kg max. This would drive the available payload mass down to 21.5 satellites.

I will round that down to 20 satellites, arranged in 5 rows of 4, with an alternate pattern (i.e. first row North/South/East/West, next row NE/NW/SE/SW, etc...) since they will want to have them detach from the PAF two by two in a coordinated fashion.

Moreover in this configuration, the PAF can weigh up to 1 600kg, which should be plenty enough for 20 satellites smaller than Iridium's.

Physical

However, we also have to check those would fit in the fairings. The Falcon 9 User Guide also gives us the internal dimensions of the fairing. Fitting 5 rows of 4 satellites should not be a problem here.

View from top:

Also, this would amount to a total of 600 launches on the Falcon 9.

Some people suggest that volumetrically, 40 could fit as follows:

Falcon Heavy

Payload Mass

The Wikipedia page for Falcon Heavy, and available info on SpaceX's own website, does not reference a payload mass when flying fully reusable. I will assume it's the same ratio as with the Falcon 9. When flying reusable vs fully expendable, payload mass is reduced by 2.375 (22 800 / 9 600).

The maximum payload mass is 63 800 kg when flying expendable, so this would gives us a payload mass of roughly 26 800kg when flying reusable.

This would allow us to send 64 satellites in orbit (exact number is 67, but I want to keep the *4 or *8 symmetry.).

64 satellites would gives us a margin of 1 600kg for the PAF. Seems a bit light to me for 64 satellites.

Physical

Regarding the physical space, we can assume the Falcon Heavy fairings are about the same size as the Falcon 9 ones (same core diameter).

If we fit 8 satellites per row, that would gives us 8 rows of satellites. It's starting to get a bit cramped, especially near the top. However, with some special arrangement, I believe this could still be done.

This configuration and the launches on Falcon Heavy would amount to 188 launches.

BFR

Payload Mass

This is going to be a bit easier but less precise. The Wikipedia page for BFR gives a maximum payload mass of 150 000kg when flying reusable. (No, that is not a typo. It is 150 metric tons to LEO.)

This would allow us to send 368 satellites in orbit (exact number is 375, but again here, I want to keep the *4 or *8 symmetry).

368 satellites would gives us a margin of 2 800kg for the PAF. Definitely, that is not enough for that number of sats.

I reckon the number of satellites per launch on the BFR will be in the ballpark of 304 satellites (this would make the PAF weigh up to 100kg/satellite, same as the IridiumNEXT launches).

Physical

Core diameter is 9m. The rest of the available payload space is not known.

However, fitting 304 satellites in 19 rows of 16 satellites should be possible I believe.

With this configuration, there would be 40 launches on the BFR.