Appearance of double planets with tilted orbit

Question

Consider a pair of double planets. They are roughly the same size and also roughly earth-sized, but their orbital plane around each other is tilted with respect to the ecliptic (see sketch below). The planets are tidally locked to each other. I'm having some trouble figuring out the appearance of the planets in each other's sky depending on the tilt.

Distances obviously not to scale

It seems to me that with a large tilt (upper image) there would be a situation where the shadow of planet 1 (P1) does not fall onto planet 2 at all. So P2 would appear full in the sky of P1 at inner-side night, and I guess P1 would be fully in darkness during inner-side day of P2. But given i.e. the appearance of the new moon on Earth, would it be visible at all?

Then if the tilt is lesser (lower part) such that the shadow of P1 actually does fall onto P2, what would they look like respectively? I guess P2 would be a crescent with the lower part in darkness, the size of which depends on exactly how much of it is in P1's shadow. But what does P1 look like to P2?

Sorry if I'm missing something obvious.

Sizes and distances in case that is helpful:

• Radius P1 ~ radius P2 ~ 6300 km (Earth radius)
• Semi-major axis of orbit around each other: 53000 km (assume roughly circular orbit since they have similar size and mass)
• Orbital period of planets around each other: 24h
• Semi-major axis of orbit around the star: 26.9 million km (it's a red dwarf, that's why they are so close)
• Orbital period around star: 50 days

Answer 1

You're forgetting an important thing... the axial tilt stays pointing in the same direction during the whole orbit, not always tilted towards the parent star. This is why Earth gets seasons! This means that even if at day 0 your binary planets cannot eclipse each other, the same is not necessarily true at day 12. Eclipsing will therefore be seasonal with high-axial-tilt binary planets, but it will happen.

(you will probably get axial precession, but on Earth that cycle takes tens of thousands of years... there's no way it can happen during a single orbit)

I guess P2 would be a crescent with the lower part in darkness

Not necessarily! Consider the real world example of the partial lunar eclipse. Here's a nice photo:

(image credit Farhan Perdana (Blek))

There's a bright crescent where the sun is still directly illuminating the moon, sure... but there's also a wide area lit by light that's been refracted through the Earth's atmosphere (something that doesn't happen for Solar eclipses, because the Moon lacks an atmosphere). This is dominated by long wavelengths of light, hence the reddish tone. It illuminates quite a large area of the moon, but how much of your planet would be illuminated in this way is hard to tell without reaching for a simulator. Earth is a less neutral color than the moon, but at the very least visible cloud systems (of which there are probably quite a few) should dim and redden during the partial eclipse.

Simulation though is probably what you really need. Setting such a thing up is a bit of a hassle to start with, but it will massively help visualising stuff.