Asteroid Size: Fixing Another Mistake in the Aerospace Engineers Handbook

Introduction

My attention was drawn yesterday (21st March 2023) to some more bizarre equations in the Aerospace Engineers Handbook. Not only were they wrong, but they were overly complex and not reduced to their simplest and most usable forms. This had the effect of bamboozling the reader.

I am fully aware some will be angry at me for pointing out yet another AEH error. Sadly, it's a very deep well to draw from.

The Weird Maths

Page 99 of the Aerospace Engineers Handbook has some calculations for asteroid size. The Math [sic, it's supposed to be written in British English] is given in this paragraph.

Firstly, let us note that the formula (1) for the volume of a sphere is incorrect. It reads as:

(1)

Which is obviously incorrect. The correct formula is:

(2)

This might be pedantic, but further examination gets interesting. In the rearrangement, tonnage (t) is used, and we'll assume t = V, and be charitable. I will also assume that R is supposed to be r; they're not the same. Remember, dividing by 4/3 is the same as multiplying by 3/4.

The attempted rearrangement (3) is:

(3)

The text then explicitly states this rearrangement, whilst forgetting the pi to yield another equation (4):

(4)

This is all patently nonsense. (3) is not the rearrangement of (1) or (2), and it is written in a bizarre way in its' original form. Ultimately, when you write it as (3) and compare to (6) below, it's clear that pi is the problem, and the final answer will be wrong by pi^(2/3). For a 1,000,000 dTon asteroid, the true value of r is just under 150 m, whereas (3) would give a value just over 320 m.

 

Fixing and Simplifying The Maths

So, what should the correct formula be, and how far can we simplify it? For maths to be used in a game, we want the equations to be as simple as possible, but correct.

Taking the correct equation (2), and rearranging for r yields (5-8):

(5-8)

Where (8) is rounded and is ca. 0.3% off the real value.

Rearranging to get volume (in dTons) from the radius:

 (9)
 

With the constant less than 0.3% from the true value by rounding. I think everyone can agree that (8) and (9) are elegant and usable. If I want the dtonnage of an asteroid, I cube the radius and multiply by 0.3. No messing about. The 1,000,000 dTon asteroid is just under 150 m in radius by (8).

I thus commend the corrected equations (8) and (9) to the Mong2k3 community.