Expert: Courtney Seligman - 12/14/2010

Question
Hello, Mr Seligman

I enjoy your answers so much I wish I could have you as a teacher. Your classes must be fascinating!

Now, I've been discussing some alternative scenarios for Earth (this will eventually go into developping a fantasy story but for now it's just world-building time, and we're going through some interesting, unexplored hypothesis)...

If the object that impacted Earth hadn't given rise to the Moon, but if everything had simply remained glued together, how much mass and gravity woult it have? I'm basically joining Moon and Earth together. I came up with a diametre of 16.000km (12,742 + 3,474); I rounded the number as they usually do for Earth. This part was very easy.

I intended to do the same for mass, but I can't seem to understand the figures (I've studied literature and languages and my grasp on Maths is very basic as I never even got around to work with the fancy-full-of-strange-functions calculators). Could you help me with how to go about summing figures with x10 (so next time I won't have to bug anyone to do the maths for me), please?

As for gravity, if the Moon has 1/5 of our gravity, then joining the two would give us a planet which has an extra 1/5 of gravity (in my simplistic logic). However, I've got the feeling I should calculate gravity from the size and mass, shouldn't why? Is there a formula?

Thank you so much for your help,
Sara

Answer
Thanks for the kind comments. As for your question, I'm afraid you're going at things backward, starting with a combined radius. You should start with the mass and volume of the combined object, which are just the sum of the corresponding values for the Earth and Moon.

The Moon has about 1.2% the mass and 1.5% the volume of the Earth, so if it and the Earth were a single object, that object would have 1.012 the mass and 1.015 the volume of the Earth.

Presuming the object were as round as the Earth, to have 1.015 the volume of the Earth, the radius would have to be about 1.005 the radius of the Earth (about 20 miles more than now).

The gravity would be proportional to the mass divided by the square of the radius, which in percentage terms is 1.012 divided by 1.010, or about 1.002. So someone who weighs 150 pounds now would weigh about 5 ounces more on the "new" Earth.

In other words, we'd hardly notice the difference, if the Moon could be (quietly) amalgamated into the Earth. The Moon is a big object, but insignificant in comparison to the Earth, so adding it to the Earth wouldn't make much of a difference.