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My question concerns the following problem, published in Galactic Dynamics by Binney and Tremaine and based on a work by Schwarzschild.

Specifically, I am having difficulties in deriving the given result in part (c), which I suppose is due to the confusion surrounding the definition of strip brightness. Could someone please explain, with the aid of a diagram if possible, how $x$ is related to the projection, $R$, of the radius $r$ onto the plane of the sky? I am having trouble visualising what $x$ is measuring here. enter image description here

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It is the luminosity in a straight strip like this:

(Credit: West 29 / CC BY-SA from here)

where the center of the strip passes a distance $x$ from the center of the circle at closest approach. So starting from the center and moving out, each successive strip would have a progressively larger value of $x$, but $x$ is a constant for a given strip (with $x = 0$ for the center strip).

The only differences from the picture are that (a) the strips have an infinitesimal width $dx$, and (b) they are infinite in length rather than the finite circle shown here.

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  • $\begingroup$ Which part of the diagram is the strip? I'm still having trouble visualising. How does it relate to $R$? $\endgroup$
    – wrb98
    Aug 3, 2020 at 19:14
  • $\begingroup$ They are all strips, each black or white stripe in the picture. That formula is for the luminosity of one individual strip, but there are an infinite number of strips that cut across the cluster, at different values of x. The variable R is the radial distance of any point from the center of the cluster. $\endgroup$
    – Eric Jensen
    Aug 4, 2020 at 2:15

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