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agnimusayoti

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- Homework Statement
- Great Nebula in Andromeda called M-31. The nearest of the large regular galaxies it is still 2 500 000 lightyears from solar system. Its diameter is about 125 000 light-years and it contains more than ##10^{11}## stars. a) Determine the angle subtended by the diameter of the Great Nebula M-31 when observed from the earth. Express it in radians and in degree of arc. (b) Find the solid angle subtended by the nebula. (FUndamental University Physics I, Chapter 2, Prob 2.18; Finn and Alonso)

- Relevant Equations
- Plane angle ##\theta ## is defined by:

$$\theta = \frac{l}{R}$$

where l is arc of circle with radius R.

Solid angle ##\Omega## is defined by

$$\Omega = \frac{S}{R^2}$$

where S is the area of spherical cap intercepted by the solid angle.

Or,

$$d\Omega = \frac{dS}{R^2}$$

If I assume the nebula is a circle, than the length of arc viewed from Earth is a half of the circumference. So, here

$$l = \frac{1}{2} \pi D$$

From the problem, ##D = 125 000 ly##.

Because the distance of nebula is much larger than the diameter; I try to approximate R with the distance of nebula from earth. Therefore, ##R = 2 500 000 ly##

a) From the equation, I get

$$\theta = \frac{\pi}{20} radian$$

$$$\theta = 9^o$$

b) Actually, I don't really understand what solid angle is, so I try to find the problem to get solid angle. Here is what I try.

First I use the infinitesimal form of the equation. Using the spherical coordinate,

$$dS = R^2 \sin {\theta} d\theta d\phi$$

So,

$$d\Omega = \int \int \sin {\theta} {d\theta} {d\phi}$$

My domains of integration are:

$$0 <= \theta <= \frac{\pi}{20}$$

$$0 <= \phi <= 2\pi$$

Is it right? Thankss

$$l = \frac{1}{2} \pi D$$

From the problem, ##D = 125 000 ly##.

Because the distance of nebula is much larger than the diameter; I try to approximate R with the distance of nebula from earth. Therefore, ##R = 2 500 000 ly##

a) From the equation, I get

$$\theta = \frac{\pi}{20} radian$$

$$$\theta = 9^o$$

b) Actually, I don't really understand what solid angle is, so I try to find the problem to get solid angle. Here is what I try.

First I use the infinitesimal form of the equation. Using the spherical coordinate,

$$dS = R^2 \sin {\theta} d\theta d\phi$$

So,

$$d\Omega = \int \int \sin {\theta} {d\theta} {d\phi}$$

My domains of integration are:

$$0 <= \theta <= \frac{\pi}{20}$$

$$0 <= \phi <= 2\pi$$

Is it right? Thankss

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