Problem 44 - Astronomy: At Play in the Cosmos

Question

An object at a distance of 200 meters is 0.5 meter wide. What is its corresponding angular width in arcseconds? in arcminutes?

Step-by-Step Solution

Answer

The angular width is 515.4 arcseconds and 8.59 arcminutes.

Step 1: Understand the Problem

We need to find the angular width of an object that is 0.5 meter wide and located at a distance of 200 meters. We are asked to provide the angular width in both arcseconds and arcminutes.

Step 2: Use the Angular Width Formula

The formula to calculate the angular width \( \theta \) in radians is given by \( \theta = \frac{w}{d} \), where \( w \) is the width of the object and \( d \) is the distance. In this case, \( w = 0.5 \) meters and \( d = 200 \) meters. So, \( \theta = \frac{0.5}{200} \).

Step 3: Calculate Angular Width in Radians

Calculate \( \theta \): \[ \theta = \frac{0.5}{200} = 0.0025 \text{ radians} \].

Step 4: Convert Angular Width to Degrees

To convert from radians to degrees, use the conversion factor \( 180/\pi \). Therefore, \[ \theta_{degrees} = 0.0025 \times \frac{180}{\pi} \approx 0.1432 \text{ degrees} \].

Step 5: Convert Degrees to Arcminutes

There are 60 arcminutes in one degree. So, multiply the angular width in degrees by 60 to convert it to arcminutes: \( 0.1432 \times 60 \approx 8.59 \text{ arcminutes} \).

Step 6: Convert Arcminutes to Arcseconds

There are 60 arcseconds in one arcminute. So, multiply the angular width in arcminutes by 60 to convert it to arcseconds: \( 8.59 \times 60 \approx 515.4 \text{ arcseconds} \).

Key Concepts

Radian to Degree Conversion

Converting angles from radians to degrees is an essential task in many fields involving mathematics and physics. Radians are the standard unit of angular measure in mathematics, mainly because they lead to simpler and cleaner formulas in calculus. However, in many practical situations, degrees are more intuitive and easier to understand.To convert an angle from radians to degrees, you multiply the angle in radians by the conversion factor \( \frac{180}{\pi} \). This factor arises because there are \( 180 \) degrees in a half-circle, and \( \pi \) radians also represent a half-circle.Here's a handy refresher:

• Radian to degree: Multiply by \( \frac{180}{\pi} \)
• Full circle: \( 360 \) degrees = \( 2\pi \) radians

Using our example, converting an angle of \( 0.0025 \) radians to degrees gives: \( 0.0025 \times \frac{180}{\pi} \approx 0.1432 \) degrees.

Arcseconds

Arcseconds are a smaller subdivision of degrees, typically used in astronomy and other fields that require precision. When you've already converted an angular measure to degrees and possibly to arcminutes, the arcseconds provide a further breakdown.A single degree consists of 60 arcminutes, and each of these arcminutes further divides into 60 arcseconds.Breaking it down further:

• 1 degree = 60 arcminutes
• 1 arcminute = 60 arcseconds
• This means 1 degree = 3600 arcseconds (since \( 60 \times 60 = 3600 \))

For example, if you have an angle of \( 8.59 \) arcminutes, you multiply by 60 to convert to arcseconds, resulting in \( 8.59 \times 60 = 515.4 \) arcseconds. Arcseconds allow you to measure very small angles accurately.

Arcminutes

Arcminutes are a crucial intermediary step in the conversion process from degrees to more precise angular measurements like arcseconds. They offer a more precise breakdown of angles when degrees are too large of a unit for the level of detail needed.The main relationship for converting degrees to arcminutes is simple: one degree equals sixty arcminutes.Keep these handy notes:

• 1 degree = 60 arcminutes
• Used when more precision than degrees alone is necessary in the calculation

In practical terms, this allows you to take an angular measurement in degrees and further subdivide it into 60 arcminute segments; for instance, \( 0.1432 \) degrees becomes \( 0.1432 \times 60 \approx 8.592 \) arcminutes. Here, each of these arcminutes can be subdivided into even finer measurements, enabling very precise angular calculations.