Solar series 142 and 147 started same year - series 147 and 152 will be ending same year. Why? Coincidence? This question was left over from the last question I had, so I am asking it separately here:


I have been looking into two solar eclipses (Saros series 147 and 152) that will both be occurring in 2021. Each of these eclipses will be happening near the opposite poles of the earth.  Both are part of young saros series. Saros 152 is comprised of 70 eclipses, while saros 147 contains a total of 80 eclipses. These series are 10 eclipses difference in length and began 180 years apart. So, I happened to check out their ending dates and found that both series are projected to end during the exact same year - 177 day apart – together in the year 3049. I thought that both of these series ending together one semester apart in the very same year was a pretty amazing coincidence.

So, checking further – it seems new saros series start from as few as 7 years apart to as many as 100 years apart (this is extremely variable spacing, but it averages to about 31 years apart over the usual 1260+ year active series lifespan). I wanted to know why series 147 and 152 would both be ENDING at the same time.  I went back to the NASA website, and checked the 5000 year listing of all solar saros series numbering 0 through 180. I only found one other such instance of two solar saros series ENDING in the same year – series 49 and 54 both ENDED together in 32 CE – so it seems like a pretty rare event.

But then, I checked to see if there were any occasions of two saros series BEGINNING together in the same year. Going back to the NASA website again with this question, I checked the 5000 year listing of all solar saros series numbering 0 through 180. I only found one single example of two saros series both BEGINNING in the same year. This time series 142 and 147 both BEGAN together in the year 1624 CE.

Not only is solar saros 147 involved in BOTH of these "same year" cases, but all three series eclipses will be presenting themselves during a single upcoming semester series in 2021 – solar series 142 (Dec 14, 2020), 147 (June 10, 2021),
and 152 (Dec 4, 2021). 


  Can anyone explain to me the actual mechanics behind what causes saros series to begin and end? I am familiar with Saros/Inex
  Panorama – but unfortunately they do not answer this question, and I
  can’t even read them online anyway.
  
  Does anyone know why solar saros 147 may have STARTED in the same year as saros 142 or why saros 147 will be ENDING together in the same
  year as saros 152? Are these two occasions both very rare
  coincidences, while at the same time, both involve the same series 147?
The reason Saros Cycles begin and end is down to the positions of the orbits of the Sun, the Moon and the Earth. From the main Wikipedia page on Saros:
For an eclipse you need the 3 bodies to line up at a full or new moon,

and repeat occurrences of these lunar phases result from solar and lunar orbits producing the Moon's synodic period of 29.53059 days...however for the majority of these, the shadow of the Earth or Moon fall to the north or south of the other body.

It can only happen when a full or new moon is around the ecliptic.

The period of time for two successive lunar passes through the ecliptic plane (returning to the same node) is termed the draconic month, a 27.21222 day period. The three-dimensional geometry of an eclipse, when the new or full moon is near one of the nodes, occurs every 5 or 6 months when the Sun is in conjunction or opposition to the Moon and coincidentally also near a node of the Moon's orbit at that time, or twice per eclipse year.

Also:

Two Saros eclipses have the same appearance and duration due to the distance between the Earth and Moon being nearly the same for each event because the Saros eclipse cycle is also an integer of the anomalistic month, the period of the eccentricity of lunar orbit, 27.5545 days.
After one saros, the Moon will have completed roughly an integer number of synodic, draconic, and anomalistic periods (223, 242, and 239) and the Earth-Sun-Moon geometry will be nearly identical

The reason we get varying start and ends is because of this. Those roughly integer periods vary.
This diagram from that Wikipedia page illustrates perfectly why they can be different lengths, stop and start at changing intervals etc.