How to calculate a solid angle in steradians from arcminutes?

[Kayla Martin]

Homework Statement

I am given the angle (theta) as 4.3 arcminutes, and need to calculate the solid angle in steradians using the solid angle formula (non-differential). Could someone show me how to do this?

Relevant Equations

$$\Omega = \frac{A}{r^2}$$

I tried $$\Omega = \pi(\frac{\theta}{2})^{\frac{1}{2}}$$

 

[andrewkirk]

More information is needed for the question to have a unique answer. A solid angle is a measure that relates to a shape on the surface of a notional sphere. In order to calculate a solid angle, you first need to specify what shape you are talking about. Just giving an angle does not tell us what the shape is, or how to calculate the solid angle. Three completely different possible interpretations of your question are:

1. The shape is the smallest lune bounded by two great circles that intersect at angle theta (area between two lines of longitude)
2. The shape is the area bounded by the equator and the line of latitude theta.
3. The shape is a spherical cap and theta is the polar angle between the rays from the center of the sphere to the apex of the cap (the pole) and the edge of the disk forming the base of the cap. If it's this one, you can find a formula for the area of the cap in the table on this wiki page. If you set the radius r to 1, the area will equal the solid angle.

 

[Kayla Martin]

More information is needed for the question to have a unique answer. A solid angle is a measure that relates to a shape on the surface of a notional sphere. In order to calculate a solid angle, you first need to specify what shape you are talking about. Just giving an angle does not tell us what the shape is, or how to calculate the solid angle. Three completely different possible interpretations of your question are:

1. The shape is the smallest lune bounded by two great circles that intersect at angle theta (area between two lines of longitude)
2. The shape is the area bounded by the equator and the line of latitude theta.
3. The shape is a spherical cap and theta is the polar angle between the rays from the center of the sphere to the apex of the cap (the pole) and the edge of the disk forming the base of the cap. If it's this one, you can find a formula for the area of the cap in the table on this wiki page. If you set the radius r to 1, the area will equal the solid angle.

Okay, so the question that I am curious about this for, specifically says "A supernova remnant has an angular diameter θ = 4.3 arcminutes and a flux at 100 MHz of F100 = 1.6×10−22 Jm−2s−1Hz−1. Assume that the emission is thermal."

Does that help at all? I figured that to calculate the value of intensity (which the question later asks for), I needed to compute F/$\Omega$.

 

[etotheipi]

I tried $\Omega = \pi(\frac{\theta}{2})^{\frac{1}{2}}$

Suppose we consider a small, circular surface element that subtends an angular diameter $\theta$ at a distance $d$ from our coordinate origin. The angle between the centre and edge of this circular element is $\theta / 2$.

The arc length between the centre of this circular element and the edge of the element, which is approximately the radius of the circle in the small angle regime, is then $\frac{\theta}{2}d$. Finally the area of the element is $\pi (\frac{\theta}{2}d)^2$, and we divide this by $d^2$ to obtain the solid angle,
$$\Omega = \pi \left(\frac{\theta}{2} \right)^2$$So it would appear you just have the wrong exponent.

On another note, don't you think the given units of flux are slightly odd? Given that$$F_{100} = 1.6×10^{−22} \text{J} \text{m}^{−2} \text{s}^{-1}\text{Hz}^{−1} \equiv 1.6×10^{−22} \text{J} \text{m}^{−2}$$is probably better described as an intensity, and $\text{s}^{-1} \text{Hz}^{-1}\equiv s^{-1} s \equiv 1$? Although I'm not familiar at all with the conventions in this field