Problem 36 - Astronomy: At Play in the Cosmos

Question

The average orbital radius of a star around a galactic black hole has an angular size of 0.25 arcsecond when observed from a distance of 6.2 kpc. What is the orbital radius in kilometers? in astronomical units?

Step-by-Step Solution

Answer

The orbital radius is approximately 2.32 billion kilometers or 155 AU.

Step 1: Convert Angular Size to Radians

First, we convert the angular size from arcseconds to radians, because astronomical calculations typically use radians. 1 arcsecond is equal to \( \frac{1}{3600} \) of a degree and \( 1 \) degree is \( \frac{\pi}{180} \) radians. Thus, the conversion formula is: \(\theta_{radians} = 0.25 \times \frac{1}{3600} \times \frac{\pi}{180} \)Calculating this gives:\[ \theta_{radians} = 0.25 \times \frac{\pi}{648000} \approx 1.212 \times 10^{-6} \text{ radians}\]

Step 2: Use Small Angle Formula to Find Distance

The small angle formula relates the angular size \( \theta \) in radians to the physical size \( D \) and the distance \( d \) by \( \theta = \frac{D}{d} \). Given \( d = 6.2 \text{ kpc} \) (which we convert to kilometers using the conversion 1 kpc = 3.086 \times 10^{16} \text{ km}), we solve for \( D \):\[ D = \theta \times d = 1.212 \times 10^{-6} \times 6.2 \times 3.086 \times 10^{16} \]Calculating this gives:\[ D \approx 2.32 \times 10^{10} \text{ km}\]

Step 3: Convert Kilometers to Astronomical Units

Next, we need to convert the orbital radius from kilometers to astronomical units (AU). 1 AU is approximately equal to 1.496 \times 10^{8} kilometers. So, we use the conversion:\[ D_{AU} = \frac{2.32 \times 10^{10}}{1.496 \times 10^{8}} \]Calculating this gives:\[ D_{AU} \approx 155 AU\]

Key Concepts

Angular Size Conversion

When observing objects in the sky, astronomers often need to convert the object's apparent size from angles measured in arcseconds to radians. This conversion is crucial because radians are the standard unit used in many astronomical calculations.

• 1 arcsecond is defined as \( \frac{1}{3600} \) of a degree because there are 60 arcseconds in a minute of arc and 60 minutes in a degree.
• 1 degree is \( \frac{\pi}{180} \) radians since there are \( 2\pi \) radians in a circle (360 degrees).

To convert an angular measurement from arcseconds to radians, multiply by \( \frac{\pi}{180 \times 3600} \). For instance, for an angular size of 0.25 arcseconds, the conversion to radians is:\[ \theta_{radians} = 0.25 \times \frac{\pi}{648000} \approx 1.212 \times 10^{-6} \text{ radians} \]Making these conversions helps simplify the handling of astronomical distances.

Small Angle Formula

The small angle formula is a handy tool in astronomy connecting three parameters: angular size \( \theta \), physical size \( D \), and distance \( d \). This formula is particularly helpful because it allows us to find one of these parameters if the other two are known, using:\[ \theta = \frac{D}{d} \]

• \( \theta \) is the angular size in radians.
• \( D \) is the actual size of the object.
• \( d \) is the distance to the object.

Let's consider the exercise example:

• Given \( \theta = 1.212 \times 10^{-6} \) radians and \( d = 6.2 \text{ kpc} \).
• Substitute these into the formula to solve for physical size \( D \).

In our case:\[ D = \theta \times d = 1.212 \times 10^{-6} \times 6.2 \times 3.086 \times 10^{16} \approx 2.32 \times 10^{10} \text{ km} \]This result represents the star's orbital radius in kilometers.

Kiloparsec to Kilometers Conversion

In astronomical terms, a kiloparsec (kpc) is often used to describe large distances between cosmic objects. To perform calculations involving kiloparsecs, converting them into kilometers (km) is necessary because kilometers are more prevalent in detailed calculations.

• 1 parsec is equivalent to approximately 3.086 \( \times 10^{13} \) kilometers.
• Therefore, 1 kiloparsec equals 3.086 \( \times 10^{16} \) kilometers.

Converting these units makes detailed distance calculations possible.For example, the given distance of 6.2 kpc equals:\[ 6.2 \times 3.086 \times 10^{16} \approx 1.91 \times 10^{17} \text{ km} \]Understanding these conversions is fundamental for solving problems in orbital mechanics and other astronomical applications.

Kilometers to Astronomical Units Conversion

Astronomical units (AU) are often more convenient for expressing large distances within the solar system or nearby stars.One AU is equivalent to the average distance from Earth to the Sun, about 1.496 \( \times 10^{8} \) kilometers.

• To convert from kilometers to astronomical units, divide the distance in kilometers by 1.496 \( \times 10^{8} \).

Continuing with the exercise, where the star's orbital radius is calculated to be approximately \( 2.32 \times 10^{10} \) km, we find its value in AU:\[ D_{AU} = \frac{2.32 \times 10^{10}}{1.496 \times 10^{8}} \approx 155 \text{ AU} \]This conversion provides a more comprehensible measurement in astronomical terms, suitable for discussing orbits and other large scale distances.