# Solid Angle of the Sun Derivation

• springBreeze
In summary, the solid angle is derived by dividing the subtended area (pi*r^2) by the square of the radial distance to that area (R^2), giving us pi*r^2/R^2. In the given example, the small angle approximation is used, where the radius (r) is approximated by half of the subtended angle (d), giving us pi*d^2/4. However, this approximation may not be accurate for larger angles. The solid angle is based on a section of the surface of a sphere and can be visualized as the area of the spherical cap inside the drawn circle on a transparent sphere. The maximum solid angle that can be subtended is 4 pi.

#### springBreeze

I read in a paper the following passage:

"We take the sun to subtend a linear angle of 32 arc-minutes. The solid angle is derived as $$\Omega=\pi sin^{2}16'=6.8x10^{-5} sr$$"

I don't understand how the formula to go from linear angle to solid angle is just found by taking the area treating sin(16') as a radius. Can someone explain?

Use diameter of 900,000 miles and distance of 94 million miles. So linear subtended angle is d = 9.57 milliradians = 32.9 arc-minutes. So radius is 16' and area is pi R2, so solid angle is pi x sin2(16') = 7.2 x 10-5 sterads.

Bob S said:
Use diameter of 900,000 miles and distance of 94 million miles. So linear subtended angle is d = 9.57 milliradians = 32.9 arc-minutes. So radius is 16' and area is pi R2, so solid angle is pi x sin2(16') = 7.2 x 10-5 sterads.
This derivation is not clear to me. You say that "radius is 16' ", so you're equating a length to an angle, then you insert sine of that angle without explanation.

Here's what I would say instead: The solid angle is defined as the ratio of the subtended area (pi*r^2 for the "disk" of the Sun) to the square of the radial distance to that area (R, the distance to the Sun). This would give us pi*r^2/R^2.

In this case, we are given the subtended angle rather than either radius value, so for convenience we use the small angle approximation:

r/R = tan(theta) ~ sin(theta), where theta is half the angle subtended by the disk. Plug that in the expression above and you'll get the desired expression.

belliott4488 said:
This derivation is not clear to me. You say that "radius is 16' ", so you're equating a length to an angle, then you insert sine of that angle without explanation..

You are correct. Actually, to get the correct value, we have to go back to my post:
Use diameter of 900,000 miles and distance of 94 million miles. So linear subtended angle is d = 9.57 milliradians.
Solid angle = pi d2/4 = pi (.00957)2/4 = 7.193 x 10-5 sr

The previous posts give a perfectly good approximation for the case in question, but I don't want anyone to go away thinking it's exactly right - it would break down at larger angles.
The solid angle is not based on the area of the disc (or other 2-d shape) at a given distance. It is based on a section of the surface of a sphere.
E.g. for the sun, imagine a transparent sphere of unit radius centred on the observer. Draw the outline of the sun on the sphere, as perceived by the observer. The solid angle is the area of the spherical cap inside that drawn circle. This is slightly greater than the area of the flat 2-d circle.
In the extreme, the solid angle subtended by the entire enclosing sphere is 4 pi.

## 1. What is the definition of solid angle?

The solid angle is a measure of the amount of space an object or light source takes up in three-dimensional space, as viewed from a specific point.

## 2. How is the solid angle of the sun calculated?

The solid angle of the sun can be calculated using the formula: Ω = A/r^2, where Ω is the solid angle, A is the surface area of the sun, and r is the distance from the sun to the observer.

## 3. What is the unit of measurement for solid angle?

The unit of measurement for solid angle is the steradian (sr), which is equal to the area of a sphere with a radius of one meter on the surface of a sphere with the same center.

## 4. How does the solid angle of the sun change throughout the day?

The solid angle of the sun changes throughout the day as the sun moves across the sky. It is largest at noon, when the sun is directly overhead, and decreases as the sun moves closer to the horizon.

## 5. Why is the solid angle of the sun important in astronomy?

The solid angle of the sun is important in astronomy because it helps us understand the amount of energy the sun emits and how it affects the Earth and other celestial bodies. It also helps us calculate the amount of solar radiation that reaches the Earth's surface, which is crucial for studying climate and weather patterns.