Let me discuss here the underlying math of the following basic task in the context of spherically symmetric surface brightness profiles of globular clusters
and recently also of galactic halos
The basic surface brightness measurements represent 3D distributions of light (stars) as projected onto the 2D skyplane! Correspondingly, the important question arises: Can we infer from the measured surface brightness profiles across the 2D skyplane the corresponding 3D distribution of light (stars!) assuming spherical symmetry of the objects under study?
That is crucial information, to allow for a full 3D rendering of globular clusters and galactic halos in agreement with observation!
Answer: Yes, we can!
Have a look at this illustration of the geometrical aspects of the problem that I prepared with Maple and GIMP:[Click on image for a larger display]
sphere3d.jpg [ 59.93 KiB | Viewed 2330 times ]
The sphere stands for a spherical shell of constant light intensity at a distance r
from the object's center O
. The straight red line at distance R
from O is the line of sight along the z-direction, with the 2D skyplane shown across the sphere's equator. The latter is placed in the x-y plane.
Note the connection between r, R and z according to the familiar Pythagorean Theorem:
As evident from the illustration above, the projection I(R)
of the spherical 3D distribution Φ(r)
of light onto the 2D skyplane amounts to integrating over Φ(r (x,y,z)) along the z coordinate that is transverse to the skyplane in the x-y plane:
where in the second integral, I made a change of integration variables like so
What we want to do next is to invert
this integral relation such as to express Φ(r) in terms of an integral over the measured 2D skyplane distribution I(R)!
Actually, this is relatively straightforward, since the relation of I(R) in terms of Φ(r) is nothing but an Abel integral equation for Φ(r)
, the analytical solution of which is known. Here it is:
Soon we'll use this important result explicitly for implementing the full 3D intensity distribution as resulting from observation into the celestia.Sci code. Stay tuned...