# Number density/average distance relationship question

1. ### QuantumX

31
I am having trouble with this number density astrophysics question. Any help is greatly appreciated:

Consider a galactic disk with radius much larger than its thickness. Let R be the radius and the thickness be 2H where H is the ‘scale height’ of the disk. For a population of objects with large n3, the mean distance is small and a 3-dimensional approach can be taken. For a sparse population with large mean distance, a 2-dimensional (area) approach is appropriate where n2 is the integral of n3 through the disk vertically. At what mean distance does the transition from a 2D to 3D approach occur?

This question has to do with the relationship between number density and average distance, which is (I think) average distance = 1/cube root(density) for 3D space and 1/square root(density) for 2D space.

It involves calculus which I'm not too comfortable with and I'm not sure where to start... Please help!

2. ### abitslow

140
If this is a homework problem, then you need to post it in the homework forum. An integral is just the sum of a variable quantity over the path or distance (or volume, area, set, space, ...). The average over any of these things is just this sum divided by the whole (length, area, path, distance, set, space...). In other words the average is proportional to the integral, only differing by the divisor (which often is constant).
I find the question a bit confusing:"At what mean distance does the transition from a 2D to 3D approach occur?"
""approach"" ??? approach TO WHAT?? Based on your comments I guess it is using area to compute the average(?) number density compared to using volume. Well, volume (assuming we are talking about 3D space (where we live) and assuming that time evolution is not involved) is always correct, area will only be an approximation. Its generally used because it simplifies the calculations (although sometimes it is used because one or more of the assumptions in the 3D calculation are inappropriate when the 3 dimensions aren't roughly equal). Given this, the question becomes: "When does the 2D approach fail to provide accurate n (number density) ?" You need to construct a 3D model of the system. What is appropriate? A disc? A flying saucer type shape (ellipsoidal)? A disc is a cylinder, right?