I'm designing a habitable planet orbiting an M-dwarf, which is orbiting a G star at 5.5 AU. The orbit around the G star keeps the "night-side" (not technically an accurate term in this context) warm enough that it doesn't trap too much water. I don't want to put the M dwarf-planet system any closer to the G star, because I don't want to mess with the stability of the orbit (I don't want the gravity of the G star to disturb the planet's orbit around the M star). So I'm wondering if any photosynthesis could occur on the back side of the planet. Is the amount of sunlight available at 5.5 AU enough to power any photosynthesis?
At that distance, during the (week or so?) half orbit when the planet's "dark" side is facing the G star, the peak illumination will approach 1/(5.5^2) that of Earth's noonday tropics (basic inverse square law).
Many house plants thrive in only artificial room light that seldom exceeds 1/100 of temperate midday sun (which itself is less than half that in the tropics). This value is based on light metering for photography; temperate noonday sun in summer will give exposure of f/11 to f/16 at 1/500 on ISO 400 film; the same film in a well lit office will require f/2 at 1/30, give or take one stop. That's a difference of 9 stops, which is a ratio of 2^9 = 512x.
I'd confidently say that the (weeklong?) "nights" when facing away from the G star are a bigger problem than the light level when facing toward it, at least for the sort of plants that would, on Earth, natively grown on forest floors or beneath shrubs and brush. Given some of those same plants will survive sub-Arctic winters with weeks of nearly no sunlight at all, I'm sure some plants could survive the "G nights" as well.
Whether this life could evolve in those conditions is an open question, potentially subject to huge amounts of hand-waving -- but survive there? Very much so.