There is always a Part of Fortune even when there is not any meaningful aspect between the Sun and Moon. Furthermore the Part of Fortune doesn't care what Houses the Sun and Moon are in - just their signs.
The Sun and Moon have been married for a long time. Their Monthly union and parting dominates calendars and behavior regardless of astrology. The Full Moon is a Meaning-point all its own (and the Part of Fortune is always on the Descendant). How to deal with the ceremony / power / earth-magic of that cycle in one's own person? When the gathering at a New Moon begins, where am I? [The Part of Fortune is at the Ascendant at that moment.]
The Ascendant is the only thing (in my opinion) that dies. I can 'meet' a recently deceased person's life-energy (Sun); his women (Venus), his habits and life station, etc. But I can't meet the person Himself. His Here Self, not his There Self. That's gone. Occasionally I may meet My own Here Self (when my own Ascendant is stimulated); or see it reflected in others (through their Descendant, not mine) and through that mirror see it a bit more clearly.
Example. My Ascendant is 13+ Sagittarius; the Sabian symbol for that degree is: "A vast panorama of sand and time is unfolding; the pyramids and the sphinx in their glory rise before the eye." When I went to work for PB, he took one look at me and hung a tapestry in his work area - of the Sphinx and Pyramids. He took the Monets and Renoirs down in the office area and put up a couple of Turners. These unexpected unsolicited actions strengthened those particular qualities in my Here Self - I love Turner's work, and the Sphinx has symbolized my trust / bond with such Ancient Ways as the traditions of India, Greece, China, and lately, even early Egypt.
So how / where / can I find a mirror to see / check in upon my Here Self? How can I differentiate those elements of personality and preference through which light (sunrise) is most likely to shine from those that bind me to the endless repetitions of Nature's compulsions?
In Egypt (and in the movies like the Labyrinth, the 5th Element, the Mummy) mirrors were used to reflect the Sun's light deep into underground rooms, solving the problem of cooling vs. light.
The Pars Fortuna is a mirror in which the sorcery of the Sun / Moon Spanda can be housed and placed so we can see the Ascendant. And vice-versa - the PF is a mirror which uses the light of my Here-Self to personalize the Space-Between Sun and Moon (the awareness of the forces of Destiny/Sun and Necessity/Moon).
In my own chart the Moon is pretty tired out - it's a day short of a New Moon - so it's on its last few hours of the lunar cycle. The Moon is Dark, the Sun is dominant. The natural life is dim compared to the purposive life. The dark Moon also moves in Darkness, so that's okay. But the darkness of the Moon, the natural moves are not fused with the light source - nor is the Moon contradicting the Sun, it's just putting in those last few moonths before retirement, as it were.
My PF is in Scorpio in the 11th house. Its position in the 11th is fixed for the day - anyone born on that day anywhere in the lower 48 is bound to have their Part of Fortune in the 11th, no matter what time of day they were born (and hence no matter what sign is on their Ascendant). On that day both the Sun and Moon were in Virgo; with the Sun near the end of the sign, and the Moon just at the beginning. So for everyone born with the double Virgo Sun Moon, their Part of Fortune is placed in the house of friendship, in the mansion of student-teacher relations. In fact, anyone born on the day before any New Moon in any sign will most probably have their Part of Fortune in the 11th.
In my own case, each teacher-student experience I have had, starting with my 4th grade teacher who gave me Pilgrim's Progress & Dante, has allowed me to deepen (Scorpio) or ritualize my Sun-Moon dilemma in terms of one being the teacher to the other's student. What kind of teacher/student? A Sagittarius one - the relationship is to bring about spiritual journeys; to bring about fighting cleanly; to bring about pioneering work. This makes the relationship a very private one, with all the attendant advantages and dangers. Furthermore, each of my teachers has disappeared or died = Scorpio; leaving me to find myself through my own self-reliance = Sagittarius.
Conversely, through my own explorations of the ancient spiritual pathways, I've been able to use my relationship, friendship even, with my dead teachers to see more clearly into my own destiny, and to understand, if not accept my own Virgo nature.
So, even when we don't know the birth-time of an individual, we can still know the house of the Part of Fortune, even though its sign, and the rising sign remain unknown. This little piece of information gives that individual a promise - a promise of where to look for light, a light which will illuminate their own inner state, as well as fill the frightening, mysterious space between our daily selves (the Moon) and our larger identity (the Sun).