How to calculate the size of a planet using a moon?

[Edutyn]

Basically, as the title says, I want to calculate the size of a planet using the moon that orbits it. Is there a formula for doing that, or is it simply not possible?

 

[JPTheGamer]

Basically, as the title says, I want to calculate the size of a planet using the moon that orbits it. Is there a formula for doing that, or is it simply not possible?

Usually, you'd have to pixel-scale the moon's given size and use it to find the planet's diameter, but if only part of the planet is shown, you need to find the panel's size, properly fit it around an ellipse, and then scale it using either the moon's size or something else. Here's an example of a calculation I made.

 

[Edutyn]

Usually, you'd have to pixel-scale the moon's given size and use it to find the planet's diameter, but if only part of the planet is shown, you need to find the panel's size, properly fit it around an ellipse, and then scale it using either the moon's size or something else. Here's an example of a calculation I made.

In this case, the character is on the surface of the planet and has a clear view of the moon

 

[JPTheGamer]

In this case, the character is on the surface of the planet and has a clear view of the moon

Can I see a picture of it?

 

[Edutyn]

 

[JPTheGamer]

You'll need an image with the planet in it as well; this one's a bit obscure.

 

[Edutyn]

You'll need an image with the planet in it as well; this one's a bit obscure.

Unfortunately, there are no shots from space of the planet next to the moon. Anyways thanks you!

 

[JPTheGamer]

Unfortunately, there are no shots from space of the planet next to the moon. Anyways thanks you!

Then I'm afraid you can't calculate the planet's size, especially with images like this; sorry about that. You're welcome.

 

[CodeCCLL]

Unfortunately, there are no shots from space of the planet next to the moon. Anyways thanks you!

So, if you somehow have the information/images to calculate the moon's orbit (so, depending on the verse, why not?) and period (I think you can use the game time), you can find the planet's mass using this formula.

A Newtonian form of Kepler’s third law:

Planet mass (+ Moon mass [for moons large enough to significantly affect the result]) = (4 × pi squared × orbital radius cubed) / (G × orbital period squared)

Spoiler

Centripetal force = moon mass * v^2 / r

Set gravitational force equal to centripetal force: G * (planet mass + moon mass) * moon mass / r^2 = moon mass * v^2 / r

Cancel moon mass from both sides and multiply both sides by r

Orbital velocity v = 2 * pi * r / T
So: v^2 = 4 * pi^2 * r^2 / T^2

Substitute v squared into the equation: G * (planet mass + moon mass) / r = 4 * pi^2 * r^2 / T^2

Solve for planet mass + moon mass:
Planet mass + Moon mass = (4 * pi^2 * r^3) / (G * T^2)

Next, using the Earth's density as an average value, calculate the volume of sphere (assuming that the mass is homogeneously distributed), and using the volume, calculate the diameter and radius of sphere, and finally, using all the values, calculate the planet's GBE.