Apparent magnitude of my moon as seen from the planet's surface
Setting: we have a fantasy world which is a planet with much of the same conditions that can be seen on Earth, but with the difference that it is orbited by a moon with a mean diameter of 3 500 km. The moon has an albedo of 0.12 and its distance from our planet is 20 000 km at perigee.
The moon/planet system orbits a star that has an apparent magnitude of -27, when observed from the moon/planet system. Assume that we have the same atmospheric conditions as on Earth, and - if other parameters have to be taken into account besides the atmosphere - assume those are equal to Earth's as well.
I would like to learn two things here:
- What would the apparent magnitude of the moon in question be to an observer looking up at the night sky from the planet it orbits, when the moon is at perigee and in its full phase?
- What does the equation that gives you the solution to this problem look like?
This post was sourced from https://worldbuilding.stackexchange.com/q/19494. It is licensed under CC BY-SA 3.0.
1 answer
There's a relatively easy way to calculate this number. Compare your moon's parameters to ours and multiple those differences
Size
Start with out Moon (diameter 2,156 km) and compare to your moon (diameter 3,500). This means that the size difference in moons makes yours $ \left( \frac{3500}{2156} \right)^2 = 2.6 \times $ brighter due to the size difference.
Distance
Your moon is ~ $19 \times$ closer than ours, which would make it ~$ 19^2 = 361 \times$ brighter due to its distance.
Albedo
Luna (our Moon)'s albedo runs around 7%, while yours comes in around 12%. Your moon comes in around $ \frac{12}{7} = 1.7 \times $ brighter due to its reflectivity.
Sun's brightness
Your world's star has an apparent magnitude of -27, while Sol's (our Sun's) apparent magnitude runs around -26.7. The difference in brightness from this is negligible.
Combined
When you combine these factors, you get the following:
$$ 2.6 * 361 * 1.7 = 1631 \times \text{brighter than our Moon, Luna}$$
$ m_{Luna} = -12.6$ (Moon's magnitude)
$ m_{Moon2} = \text{Unknown} $
$ \frac{F_{Moon2}}{F_{Luna}} = 1631 $ - Ratio of the two moon's brightness
The equation looks like this:
$$ m_{Moon2} - m_{Luna} = -2.5 log_{10} \left( \frac{F_{Moon2}}{F_{Luna}} \right) $$
Plug in the numbers:
$$ m_{Moon2} - -12.6 = -2.5 log_{10}\left(1631\right) $$
Simplify
$$ m_{Moon2} = -2.5 \times 3.21 - 12.6 = -20.6 $$
So, the visual apparent magnitude of your Moon would have is about -20.6.
NOTE:
My above numbers were performed using our Moon as a comparison and factoring those differences. I did NOT derive the value directly from your Moon's values. You can do that as an alternative method of getting the same answer.
This post was sourced from https://worldbuilding.stackexchange.com/a/19495. It is licensed under CC BY-SA 3.0.
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