   Chapter 7.8, Problem 66E

Chapter
Section
Textbook Problem

# Astronomers use a technique called stellar stereography to determine the density of stars in a star cluster from the observed (two-dimensional) density that can be analyzed from a photograph. Suppose that in a spherical cluster of radius R the density of stars depends only on the distance r from the center of the cluster. If the perceived star density is given by y ( s ) , where s is the observed planar distance from the center of the cluster, and x ( r ) is the actual density, it can be shown that y ( s ) = ∫ s R 2 r r 2 − s 2 x ( r ) d r If the actual density of stars in a cluster is x ( r ) = 1 2 ( R − r ) 2 , find the perceived density y ( s ) .

To determine

To find:

The perceived density of stars y(s)

Explanation

Given:

The actual density of stars in a cluster is x(r)=12(Rr)2,

Perceived density y(s)=sR2rr2s2x(r)dr

Formulae used:

Integration by parts

The actual density of stars in a cluster is:

x(r)=12(Rr)2

Note that r is the variable, s and R are constants.

y(s)=sR2rr2s2x(r)dr=sR2rr2s2(12(Rr)2)dr=sRr(Rr)2r2s2dr

When r=s, the integrand is undefined so that integral is improper.

Use integration by parts.

u=(Rr)2dv=rr2s2drdu=2(Rr)dr=(2r2R)drv=r2s2

On solving further:

udu=uvvdusRr(Rr)2r2s2dr=(Rr)2r2s2r2s2(2r2R)dr=(Rr)2r2s22rr2s2dr+2Rr2s2dr=(Rr)2r2s223(r2s2)32+2Rr2s2dr

Notice that for second integral we used the fact that derivative of r2- s2 is 2r.

Now, u2a2du=u2u2a2a22ln|u+u2a2|+C

Use above property for the integral at the end.

s R r ( Rr ) 2 r 2 s 2 dr= [ ( Rr ) 2 r 2 s 2 2 3 ( r 2 s 2 ) 3 2 +2R( r 2 r 2 s 2 s 2 2 ln| r+ r 2 s 2 | ) ] s R s R r ( Rr ) 2 r 2 s 2 dr= [ ( Rr ) 2 r 2 s 2 2 3 ( r 2 s 2 ) 3 2 +Rr r 2 s 2 R s 2 ln| r+ r 2 s 2 | ] s R s R r ( Rr ) 2

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### Find more solutions based on key concepts 