How bright would these four stars look on the planet you're standing on right now?

Question

Most of the numbers here have been plugged to me by someone you would find on both Reddit and the Speculative Evolution Forum.

Imagine yourself standing on a planet in a quaternary solar system. The first two stars are a binary of G0 yellow dwarves (105% as wide, 110% as massive and 126% as bright as our sun). Separating the two are 0.46 AUs of space.

Where you are standing right now is 56 billion miles, or more than 602 AUs, from the yellow dwarves. The planet itself orbits the other two, a binary of K0 orange dwarves (85% as wide, 78% as massive and only 40% as bright as our sun). Separating the two are 0.26 AUs of space. The planet you are standing on orbits the two orange dwarves from a distance of 1.34 AUs, so you're pretty much standing on a Mars analogue.

This planet rotates once every 24 hours, just like Earth, so that simplifies the math a bit.

So with the information provided above, what would the 24 hours of day and night look like on this planet?

Answer 1

The two yellow dwarves are 602 AU away, so they will look like stars only.

Two orange dwarves will be seen as Sun from Mars but with lesser intensity of light.

Part A will see both dwarves, so its brightness will be more.

Part B will see one of the dwarves, so its brightness will be less.

Part C will be dark.

Answer 2

The stars that actually matter are the parts of the K-binary, because the G-binary is too far away to meaningfully affect daylight levels.

The orbital period of the K-binary will be ~930 hours, (or ~38 Earth-days and 18.5 hours). This period needs a good name, like "month", but I can't think of one right now. Depending on the relationship of the orbital plane of the stars, and the orbital plane of the planet, there will be changes in the irradiance of the planet during the period of the star's orbit.

I'm going to assume that they are coplanar (like most things in a more conventional planetary system) and as such the stars will eclipse each other periodically. There will be a steady up-and-down change in stellar irradiance, with a sharp dip during eclipses. I'm also going to assume that the stars and the planet orbit in the same direction.

An eclipse will occur on the synodic period of each star and the planet. You can easily compute this using $\frac{1}{T_{syn}} = \left \lvert \frac{1}{T_{star}} - \frac{1}{T_{planet}} \right \rvert$, and remembering that there are two stars you'll see that you get eclipses at a ~21.2 day interval.

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At their brightest, the pair of G0 stars will have 2 x 1.26 or 2.52 times the luminosity of the sun, but because of the distance from your planet their apparent brightness (relative to the apparent brightness of the Sun as seen from Earth) will be reduced by a factor of 602². This means that at their brightest (when both stars are fully visible from your planet), they'll be ~145000 times dimmer. That's equivalent to an apparent magnitude of roughly -13.8 which is a little over twice as bright as a full moon on Earth.

The G-binary has an orbital period of ~76 of your Earth days. If the orbital plane of the G-binary is sufficiently inclined relative to the observers on the planet, then twice per its orbital period its apparent luminosity will drop as one star occludes the other. If the plane of the binary's orbit is more-or-less edge-on to the planet, then it will be an eclipsing binary. When one half of the pair occludes the other the apparent magnitude will be more like -13.1, or more like 20% brighter than the full moon on Earth.

Their contribution to daylight brightness will be negligible, but they'll have a big influence on night-time light levels, assuming they were visible in the sky. If the planet orbits the K-binary in the same plane as the K-binary orbits the G-binary, then for half the year the G-binary will be obviously visible in a clear sky at daytime, and for the other half of the year they'll be visible at night, probably brightly enough to cast shadows if the sky is clear.

At maximum separation, to an observer on the planet they'll have an apparent separation of ~2.75 seconds of arc. This is some way below the angular resolution of a human eye (commonly 60 arc seconds) so they'll appear as a single point-source without the aid of a telescope or binoculars. They can be safely observed through these devices, which will clearly separate them.

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what would the 24 hours of day and night look like on this planet?

Most days will end up with Mars-like illumination. For half the year, the nights will have a very bright point-like star that can cast shadows. For the other half of the year, that bright star will be visible throughout the day in the sky, but will not otherwise contribute to light levels. Assuming suitable orbital inclinations, then every 38 days or so that "star" will have a dip in light levels as its components eclipse. I don't think this will be obvious to the naked eye, but on a suitably clear night it might be possible to notice.

Assuming that the K-binary and the planet orbit in the same direction and in the same plane, then the solar irradiance received at the top of the atmosphere will vary like this:

(for comparison, the solar irradiance at the top of Earth's atmosphere is ~1367W/m²)

You get a slight variation as the stars rotate, being brightest and warmest when one star is as close as possible without occluding the other, coolest and dimmest when the stars are equidistant (and so most widely separated). Eclipses will last a few hours, and involve a sharp dip in irradiance:

The small ticks are at 5 minute intervals, the big ones at 1 hour intervals. "Totality" will last a bit less than an hour. I modelled the stars as simple circular emitters so limb darkening has not been considered. The pre-eclipse irradiance and irradiance at totality is correct, only the slope of the "attack" and "decay" phases will be different if the model were more accurate.

The nature and frequency of eclipses changes when the orbital planes of the stars and the planet are not aligned. I'm not going to consider them here.

The precise irradiance and light levels experienced on the surface of the planet depend on so many factors that I'm going to leave them as an exercise for the reader.