Understanding Circumstellar Disks: Exploring Surface Density in 3D

[robotopia]

Hi everyone,

I'm trying to understand circumstellar disks. One term that keeps cropping up in this context is "surface density". The disks, though flattened, are still three-dimensional, so I'm confused about what "surface density" actually means. I can think of a couple of possibilities:

1. The hypothetical surface density of the projection of the disk onto its mean plane
2. The surface density of its cross section taken at some distance R from the star

Try as I might, I can't seem to find a clear-cut definition for it.

Cheers

 

[robotopia]

I'm tending to think that #1 is correct, and that it's synonymous with "column density". Can anyone confirm?

 

[Mordred]

I was hoping someone more familiar than my self would have responded to this question.

As far as the 2 definitions go they both mean approximately the same thing one good example is the surface density of say an asteroid belt any value to describe the surface density of one region to another region in the case of the first definition it would have to be hypothetical as indeed there is no surface, so in essence its a conveneient descriptive. number 2 is essentially the same thing as you are allowing for surface density of non flat shapes such as heliosphere or circumstellar habitable zone. Thats about the only differents I can see between the two, hope that helps. As I stated I was hoping someone more familiar with circumstellar disk terminology would have assisted you

 

[robotopia]

Thanks Mordred. Not a popular question to answer, it seems... :-)

 

[Mordred]

might help if you provide some references so myself or others can better grasp where you see surface density applied. The term is used in a variety of interstellar models.

 

[cepheid]

I had always thought is was definition 1, i.e. if you take all the mass of the disc, and divide it by the 2D surface area of one of its faces, then you get $\Sigma$. I have to admit to not being absolutely sure.

EDIT: Column density (of an object, e.g. molecular cloud in the ISM) is a little different. If you add up the mass of all the gas in a 1 m² cylindrical tube going along the line of sight from the observer to the object (presumably out to as far as you can see, so about optical depth of 1), that number (divided by the 1 m²) is the column density of that object in that area. So it is sort of a surface density of all the gas that lies along the line of sight of the observer as projected onto the plane of the sky.

 

[robotopia]

might help if you provide some references so myself or others can better grasp where you see surface density applied. The term is used in a variety of interstellar models.

Quite right, Mordred. I just stumbled on this paper which, though it is talking about surface density profiles, confirms what cepheid says:

It is then a trivial task to sum the masses of the particles in each [radial] bin and divide by the area of the annulus represented to obtain a surface density (Hall 1997).

Summing masses between r₁ and r₂ and dividing by annulus area is the same as integrating mass density over the z (axis of rotation) direction, right? Or is it subtly different because summing masses between r₁ and r₂ means summing the masses between two spherical shells whereas integrating ρ(r) over z is more like summing the masses between two cylindrical shells? Granted, for a thin disk the difference is negligible.

Thanks everyone, for your input.