Astrophysics of Krynn, part 3

 In the previous and previous posts, we figured:

• Earth: 6,371.00 km (mean radius)
• Krynn: 2,376.45 km (radius)
• Luna (Earth's moon): 1,737.4 km (mean radius)

... and:

• Solinari (36 days): 460,540 km radius orbit around Krynn
• Lunitari (28 days): 389,498 km radius orbit around Krynn
• Nuitari (8 days): 168,963 km radius orbit around Krynn

In this post, we'll try to deduce the radius of each of the moons of Krynn.

What's the data

There is no data. But we can use the following hints:

• Let's assume that Krynn's sun is of the same size, and same distance as Earth's sun.
• Let's assume that Lunitari is of the same size as Luna

This has interesting consequences:

• Lunitari is able to make a beautiful eclipse, hiding the sun (as Luna hides the sun)
• Solinari will hide more than the sun during an eclipse (i.e. probably no visible corona)
• Nuitari is unable to hide the sun, but will put a black point within the sun during an "eclipse"

Also, I want the following data to be true:

• When Solinari, Lunitari and Nuitari are in conjunction, I want a beautiful "eye" to form in the sky, as below:

In other words, the apparent radius of each moon need to have a specific size in relation to the others. For example, using the drawings I did:

• Solinari: 64 px
• Lunitari: 40 px
• Nuitari: 20 px

Two-moons conjunctions will result the following image in the sky:

For reference, you'll find below the image of each individual moon, with a green circle representing the size of Luna, for comparison's sake:

Also, all the moons are tidally locked with Krynn.

So? Lunitari

Disclaimer: Again, I'll be rounding a lot. In particular, the difference at sea altitude, between Krynn and Earth is negligible, and thus, neglected. Also, the difference between the radius of Krynn, and the radius of any moon's orbit can also be considered as negligible, so distances will be taken from the center of Krynn, not the surface. Last but not least, we'll assume angles are small enough to approximate A, tan(A) and sin(A) as the same values, in radians.

The good point about the data is that it simplifies some calculations: Luna and Lunitari have the same orbital period, and are at the same distance from their planet, and have the same apparent size, so, they have the same radius: 1,737.4 km

The angular diameter of Lunitari (and Luna) is thus (with rMoon being the radius of the moon, and D being the distance between the center of Krynn and the center of the moon):

δ = 2 arcsin(rMoon / D)

δ = 2 * rMoon / D

For Lunitari, this would be:

δ = 2 * 1,737.4 km / 389,498 km

δ = 0.00892122

Et voilà...

How to calculate the other moons?

The apparent diameter of the other moons is a factor of Lunitari's:

• Solinari = Lunitari * 1.6
• Nuitari = Lunitari / 2

This means the angular diameter of the moons are thus:

• δSolinari = 0.00892122 * 1.6 = 0.01427395
• δLunitari = 0.00892122
• δNuitari = 0.00892122 / 2 = 0.00446061

We can then, knowing their real distance, find their real radius:

δ = 2 * rMoon / D

rMoon = δ * D / 2

... which resolves into:

• rSolinari = 0.01427395 * 460,540 km / 2 = 3,287 km
• rLunitari = 1,737 km
• rNuitari = 0.00446061 * 168,963 km / 2 = 377 km

Et voilà...

Conclusion

We know have the following data:

• Earth: 6,371.00 km (mean radius)
• Krynn: 2,376.45 km (radius)
• Luna (Earth's moon): 1,737.4 km (mean radius)

... and:

• Solinari (36 days): 460,540 km radius orbit around Krynn
• Lunitari (28 days): 389,498 km radius orbit around Krynn
• Nuitari (8 days): 168,963 km radius orbit around Krynn

 ... and:

• Solinari (radius) = 3,287 km
• Lunitari (radius) = 1,737 km
• Nuitari (radius) = 377 km

So... yeah, Solinari is larger than Krynn... That is awkward, but unavoidable because our calculations our correct.

(Remember the cause of all of this is the small size of Krynn)

😁

The good thing is:

• No one will go there to verify (even if this can be deduced, the same way the ancient Greeks deduced the size of Luna when it passed through Earth's shadow)
• Even if one did, it's a fantasy world
• Also, Solinari (and the other moons) might have a very low density to counteract their size
• If we assume all the moons have the same albedo as Luna, then nights on Krynn can be much more luminous than on Earth