I need help figuring out both questions Consider the surface of a two-dimensiona

Question

I need help figuring out both questions

Consider the surface of a two-dimensional sphere of radius R, as illustrated. Circles are drawn on the sphere which have radius r as measured on the surface on the sphere, centred on the North Pole. Show that the general formula for the circumference c of such a circle, as a function of r, is given by c = 2pi sin theta / theta r = 2pi R sin r / R, where theta is the angle between a line drawn from the centre of the sphere to the North Pole and one drawn to the circle. [Remember that by definition of angles in radians we have r = thetaR.] Demonstrate that for small theta (i.e. tau R) this gives the normal flat geometry relation. Evaluate the relation for the case when the circle is at the equator. Consider the spherical geometry of the previous problem, staying with the two-dimensional analogy to the real Universe. Suppose that galaxies are distributed evenly in such a Universe, with a number density n per unit area. Show that the total number N of galaxies inside a radius r is given by N = 2pinR2 [1 - cos r / R]. Expand this for r R to show that the flat space result that the number is npir2 is recovered (remember we are working in only two dimensions). Do you see more or fewer galaxies out to the same radius, if the Universe is spherical rather than flat?

Explanation / Answer

The main problem you will face with this is that they have defined the radius r not as the radius of the circle but the radial length of the sphere.

To find out the circumference, simply compute the radius of the circle formed as a function of theta.

you will see that the radius of the 2d circle and the radius of the sphere form a right angled triangle, you can compute the radius of the 2d circle as R sin (theta)

then the circumference of that circle is 2* pi * R * sin (theta)

now theta is r/R so cicumference is 2 * pi * R * sin (r/R)

also r/R = theta so R = r/theta, substituting c = 2 * pi r * sin(theta) / theta

2. Previosly we had calculated the radius of the circle to be R sin (theta)

then the area is pi * R^2 * sin ^2(theta)

then the number of galaxies in this area is n * pi * R^2 * sin ^2(theta)

we know cos (2x) = 1 - 2 * sin^2(x)

or sin^2 (theta) = (1 - cos (2*theta))/2

using this we have number of galaxies is (1/2) * n * pi * R^2 * ( 1 - cos (2*theta))

= (1/2) * n * pi * R^2 * ( 1 - cos (2*r/R))

I think your book has got the second expression wrong

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