Calculate Satellite Constellation Launch Interval

I am trying to deploy three relay satellite constellations.

One between the orbit of Moho and Eve with four satellites.

One between Kerbin and Duna with six satellites.

One between Dres and Jool with eight satellites.

I would like the satellites in each constellation to be equally spaced.

For the first, I settled on an orbital altitude of 7.5Gm. That gives an orbital period of roughly 184 days. So, for four satellites, they should be spaced about 46 days apart.

I tried launching one satellite every 46 days, but that didn't work. By the time 46 days passed after launching the first satellite, the periapsis moved ahead a bit and the second satellite was ~26 days behind the first.

I figure I could add 20 days to the launch window (launch every 66 days). That still didn't work. It brought me ~30-40 days separation.

I found the Resonant Orbit Calculator

I suppose I could launch all of my satellites at (about) the same time (they're too big to practically launch as a single vessel – especially eight of them) and use the calculator to space them out. That would probably be close enough.

But I would like something better.

Is there a way to calculate my launch interval (knowing how many satellites in the constellation, orbital period, and apoapsis/periapsis) so I wind up with an equally spaced constellation?

Launch later (as long as you know your target orbital period)

Considering circular orbits only (or close enough), the thing is that when you launch 46 days apart, Kerbin goes 46 days forward, reducing the actual interval between each satellite (if your target orbit is smaller, otherwise it increases it).

Quick back of the envelop calculation :

Let's say p is the period of your target orbit and k the period of Kerbin orbit. We consider a smaller target orbit, so p<k.

After t days have passed : Kerbin has done a portion t/k of its orbit, which corresponds, for a satellite on target orbit, of a advance of t2 = p*t/k.

Hence, by launching a second satellite at t, this one is not only t days behind the first, but also t2 forward.

So the actual separation, in time, is actually t - t2 = t - t*p/k = t*(1 - p/k).

In order to get a separation of p/4 (4 satellites evenly spaced on the orbit of period p), you need to have p/4 = t*(1 - p/k) with t the time between launches.
Which gives t = p/(4(1 - p/k))
Or, to put it more simply, you have to divide your initial 46 days by the factor 1 - p/k.
For an inner orbit (p<k), this means your have to launch later.

But what about a larger orbit ? That factor goes negative, so you should launch back in time ? Well... actually kinf of yes. Since Kerbin is faster than, say, Jool, the second satellite you launch will end up forward the first anyway. So there, the separation will be of p*t/k - t, but "forward".

You then need to have (for 8 satellites) p/8 = t*(p/k - 1) with t the time between launches.
Which gives t = p/(8(p/k - 1)). Basically, for a larger target orbit, you have to divide your initial assumption (of splitting the target period in 8 equal chunks) by p/k - 1 (which is then positive).

In fact, all you have to do is divide the period of the target orbit by the number of satellites times the absolute value of the factor 1 - p/k

Let's try it for your first constellation :
Period of target orbit p = 184 days
Period of Kerbin orbit k = 426 days
Factor of division = 1 - 184 / 426 = 0.568
184 / (4*0.568) = 81 days

So you see that you should further increase the delay between each launch to 81 days.

A quick final note : this answer supposes a series of identical Hohmann transfers for each satellite, and it does not consider elliptical orbits, nor modulos of orbital period. That is, if this calculation shows you should launch every 550 days for a 100 days period target orbit, of course you should modulo that to a 50 days delay.

Also, it's interesting to see that in the case of p = k, the calculations divides by 0 ! Of course if you want to place your satellites evenly on the same orbit as they already are, without fiddling with temporary elliptical orbits, then they'll just stay right where they are for eternity...

And since I did not test all of this and could have made a huge and humiliating mistake, please tell me if your tests prove me right ! :D