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When Piti arises and you stay equanimously watching the arising and passing knowing its impermanence (with no attachment or craving to this feeling) then Piti will fade away and Passadhi will arise. When Passadhi initially arises you may think your meditation is regressing as you loose Piti and also board after a few hours of Passadhi.
Piti is a stage which is conducive to tame Sankara resulting from craving.
Passadhi is a stage which is conducive to tame Sankara from Ignorance or Sanna.
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There is no correlation between sukha (the 4th component of the jhanangas) and passadha (the 5th component of the sambojjhanga) because there is no correlation between passadha and jhana.
The rapture of sambojjhanga is not the rapture of jhana. This is quite obvious because the rapture of sambojjhanga arises before the concentration of sambojjhanga, as follows:
For one whose mind is uplifted by rapture the body becomes tranquil and the mind becomes tranquil.... For one whose body is tranquil and who is happy the mind becomes concentrated.
SN 46.3
The rapture of sambojjhanga is merely the rapture that occurs when the 1st and particularly the 2nd sambojjhanga are being practised successfully.
This type of rapture is similar to the rapture below that occurs due to faith in the teachings about ending suffering, which is not jhana:
"Tranquillity, monks, also has a supporting condition, I say, it does not lack a supporting condition. And what is the supporting condition for tranquillity? 'Rapture' should be the reply.
"Rapture, monks, also has a supporting condition, I say, it does not lack a supporting condition. And what is the supporting condition for rapture? 'Joy' should be the reply.
"Joy, monks, also has a supporting condition, I say, it does not lack a supporting condition. And what is the supporting condition for joy? 'Faith' should be the reply.
"Faith, monks, also has a supporting condition, I say, it does not lack a supporting condition. And what is the supporting condition for faith? 'Suffering' should be the reply.
SN 12.23
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Sukha isn't one of the 7 Bojjhangas, although Piti (rapture) is the fourth of them so I think that's what you meant. Yes, there is a connection. When Piti is developed correctly it will lead to Passadha (which simply means tranquility or peace.) For example, in the section on the Bojjhangas in the Anapanasati Sutta it says:
"[4] In one whose persistence is aroused, a rapture not-of-the-flesh arises. When a rapture not-of-the-flesh arises in one whose persistence is aroused, then rapture as a factor for awakening becomes aroused. He develops it, and for him it goes to the culmination of its development.
"[5] For one enraptured at heart, the body grows calm and the mind grows calm. When the body & mind of a monk enraptured at heart grow calm, then serenity as a factor for awakening becomes aroused. He develops it, and for him it goes to the culmination of its development. http://www.accesstoinsight.org/tipitaka/mn/mn.118.than.html
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It's not just a strong correlation, it's the direct cause. http://lucid24.org/sn/sn46/sn46-003/toc-addon/index.html
It's not just that sutta, SN 46.3.
Doing a search for passaddha-kāyo sukhaṃ returns (60 hits). At least 60 sutta references to that same 7sb awakening factor sequence. passaddha-kāyo sukhaṃ: (with) pacified-body, [physical] pleasure (results)