Astronomical distances are so large compared to terrestrial ones that much larger units of length are used for easy comprehension of the relative distances of astronomical objects. An astronomical unit (AU) is equal to the average distance from Earth to the Sun, `1.50xx10^(8)km`. A parsec (pc) is the distance at which 1 AU would subtend an angle of 1 second of arc. A light year (ly) is the distance that light, traveling through a vacuum with a speed of `3.00xx10^(5)` km/s, would cover in 1 year.
Express a light - year and a parsec in kilometers.

Options

A. `9.48xx10^(11)km,4.08xx10^(13)km`
B. `9.48xx10^(12)km,3.08xx10^(13)km`
C. `9.48xx10^(12)km,3.08xx10^(14)km`
D. `8.48xx10^(12)km,4.08xx10^(13)km`

Answer

To express a light-year and a parsec in kilometers, we will follow these steps:

### Step 1: Calculate the distance of a light-year in kilometers.

1. **Understand the definition of a light-year**: A light-year is the distance that light travels in one year.
2. **Use the speed of light**: The speed of light is given as \(3.00 \times 10^5\) km/s.
3. **Convert one year into seconds**:
- 1 year = 365 days
- 1 day = 24 hours
- 1 hour = 3600 seconds
- Therefore, \(1 \text{ year} = 365 \times 24 \times 3600 \text{ seconds}\).

\[ 1 \text{ year} = 365 \times 24 \times 3600 = 31,536,000 \text{ seconds} \]

4. **Calculate the distance**:
- Distance = Speed × Time
- Distance in one light-year = \(3.00 \times 10^5 \text{ km/s} \times 31,536,000 \text{ s}\).

\[ \text{Distance} = 3.00 \times 10^5 \times 31,536,000 \approx 9.46 \times 10^{12} \text{ km} \]

### Step 2: Calculate the distance of a parsec in kilometers.

1. **Understand the definition of a parsec**: A parsec is the distance at which 1 AU subtends an angle of 1 arcsecond.
2. **Use the given value for 1 AU**: \(1 \text{ AU} = 1.5 \times 10^8 \text{ km}\).
3. **Convert 1 arcsecond to radians**:
- \(1 \text{ radian} = \frac{180}{\pi} \text{ degrees}\)
- \(1 \text{ degree} = 3600 \text{ arcseconds}\)
- Therefore, \(1 \text{ arcsecond} = \frac{1}{3600} \text{ degrees} = \frac{\pi}{180 \times 3600} \text{ radians}\).

\[ 1 \text{ arcsecond} \approx 4.848 \times 10^{-6} \text{ radians} \]

4. **Calculate the distance of 1 parsec**:
- Using the small angle approximation, \( \text{tan}(\theta) \approx \theta \):
- \(1 \text{ parsec} = \frac{1 \text{ AU}}{\theta}\)
- Thus, \(1 \text{ parsec} = \frac{1.5 \times 10^8 \text{ km}}{4.848 \times 10^{-6}}\).

\[ 1 \text{ parsec} \approx \frac{1.5 \times 10^8}{4.848 \times 10^{-6}} \approx 3.09 \times 10^{13} \text{ km} \]

### Final Results:
- **1 light-year** = \(9.46 \times 10^{12} \text{ km}\)
- **1 parsec** = \(3.09 \times 10^{13} \text{ km}\)