You may ask me after setting everything up...
You may also know by this point that Saturn is the furthest a human can see in the solar system with naked-eye observation, being at ~9,5 AU away from the Sun, 8,5AU from Earth at maximum approach, yet, is fairly logical to assume that a Saturn-sized planet can be seem up to ~10 Habitable Zone radii away, ie, assume the maximum you can see in a planetary system from a habitable planet is ~10x it's orbit radii...
Yep, we can go with that, even, comparing with other solar system objects works well, like if your planet is a Jupiter-sized gas giant and it is twice as close to it's sun-like star, is fairly accurate to state it may appear four-times as bright as Jupiter in the night sky, hence the similar disk-size available to reflect sunlight, given the Inverse Square Law.
As well, we can go through getting the difference of your planet to it's solar counterpart to get more accurate comparizons, a planet with a diameter 20% larger that of Jupiter, at Jupiter-distance from our Sun, will have 44% more surface area, and thus appear 44% as brighter if it has similar albedo of course.
Getting magnitudes for superior planets is pretty easy, considering that you only need to count it's magnitude at opposition (closest to the observer's planet) and conjunction (far side of the sun), the magnitude at conjunction must is roughly 1/4 as bright, a bit less than 1/4 because you have to add your observer planet's orbit radii to the distance count.
For inferior planets, we have the fact that they show phases like the Moon does, Venus may show a thin slit of it's day-time side when close to Earth, but it's disk is also nearly twice as wide in the sky than on superior conjunction.
For a roughly but fairly accurate measurement, let's assume we know how bright it should appear if it were a superior planet to our own, if it receives ~2x as much light than our planet from the Sun (similar to Venus), being 0,7 AU from the sun, at sup.conj. it's disk is 1/5th the full size when at superior conjunction (assuming the observer is at 1 AU), hence it's brightness must be as well 1/5 of what it should be if we were facing it from 0,3 AU inwards it's orbit, now, how much bright is that full disk when observed from an inward perspective?
Let's call the full illuminated disk at closest approach,1 Brightness-unit, is a value we don't know yet, but we know how wide the full-close disk is, let's say 1 unit, then the superior conjunction disk is 0,2 units wide, from the observer's perspective the maximum elongation of the planet is ~44,42º, which cosine is 0,7 it's orbit radius, let's re-do the math for the new distance from our observer planet, the disk is 0,38 units wide.
Now:
Planet at superior conjunction, 0,2 units wide, full disk, 1/5 as bright.
Planet at maximum enlongation, 0,38 units wide, half disk, 2,63x as bright than at superior conjunction.
Planet at inferior conjunction, ~1 unit wide, 10% disk.
If we measure it's brightness at sup. conj. to be -3,8 magnitude, even tho the planet has only 10% of it's disk illuminated during inferior conjunction, it should appear 5x as bright, then the magnitude of the 10% illuminated disk is magnitude -5,3, and when at half disk it is -4,3 mag.
As well, it's magnitude as seem from an inward perspective is -10,18.
What makes a lot of sense, if you're questioning why -5,3 mag is way out of what we see at Venus, is because I'm just assuming the brightest phase of the planet is when at 10% of the disk is lit, considering this is an 0,5 error margin, consider it to be ~50% less bright than calculated, ie, +0,5 mag dimmer...
A difference of y mag means one of two objects is ~2,512^yx fainter or brighter than the other, if two objects are 2 mag apart, then one is ~2,512^2x (6,3x) brighter than the other, and so on.
Again, if your planet is x-times it's solar counter part, it's magnitude should be the solar counterpart mag plus the difference in mag, if further, minus, if closer, negative value for bright objects, positive for dim objects.
You can use this calculator to find magnitudes given brightness differences:
Or calculate it yourself, given solar counterparts as shown before, using the table bellow:.
SOLAR MAGNITUDE TABLE
App. Mag. (V) Celestial object
–4.89 Maximum brightness of Venus when illuminated as a crescent
–3.82 Minimum brightness of Venus when it is on the far side of the Sun
–2.94 Maximum brightness of Jupiter
–2.91 Maximum brightness of Mars
–2.45 Maximum brightness of Mercury at superior conjunction (unlike Venus, Mercury is at its brightest when on the far side of the Sun, the reason being their different phase curves)
–1.61 Minimum brightness of Jupiter
–0.49 Maximum brightness of Saturn at opposition and when the rings are full open (2003, 2018)
1.47 Minimum brightness of Saturn
1.84 Minimum brightness of Mars
5.32 Maximum brightness of Uranus
5.73 Minimum brightness of Mercury
5.95 Minimum brightness of Uranus
7.78 Maximum brightness of Neptune
8.02 Minimum brightness of Neptune
13.65 Maximum brightness of Pluto (725 times fainter than magnitude 6.5 naked eye skies)
EXTRA
For a matter of curiosity, Saturn's disk as seem from Titan appears 10x larger than the Moon, having ~31x more area to reflect light from the Sun, yet receiving only 1,1% as much light as the Moon, it would appear 9,6x as bright as the Moon, but only 0,96x as bright per unit of area, with a magnitude of -14,3, not counting the ring system.
-M.O. Valent, 16/07/2019
