Question

Sirius is the name of a star that is approximately 86 trillion kilometers (8.6 light-years) away from the Earth. How long does the light from Sirius take to reach the Earth? (A) 86 trillion seconds (B) 300,000 seconds (C) \(8.6\) seconds (D) \(8.6\) years

Short answer

The light from Sirius takes approximately \(8.6\) years to reach Earth. The correct answer is (D) \(8.6\) years.

Step-by-step solution

Convert light-years to kilometers

First, let's convert the distance from light-years to kilometers. We are given that 1 light-year is approximately equal to 9.461 trillion kilometers. Therefore, \[8.6 \textrm{ light-years} \times \frac{9.461 \textrm{ trillion km}}{1 \textrm{ light-year}} = 81.3646 \textrm{ trillion km}\]

Calculate the time it takes for light to travel from Sirius to Earth

Now, we can use the formula of distance, speed, and time: \[distance = speed \times time\] We know the distance and the speed of light, so we can solve for time: \[time = \frac{distance}{speed}\] Plugging in the values: \[time = \frac{81.3646 \textrm{ trillion km}}{3 \times 10^5 \textrm{ km/s}}\]

Convert the time to years

To find the time in years, we need to convert the time from seconds to years. There are approximately 31,557,600 seconds in a year. Thus, \[time \approx \frac{81.3646 \textrm{ trillion km}}{3 \times 10^5 \textrm{ km/s}} \times \frac{1 \textrm{ year}}{3.15576 \times 10^7 \textrm{ seconds}}\] \[time \approx 8.6 \textrm{ years}\] So, the light from Sirius takes approximately 8.6 years to reach Earth. The correct answer is (D) \(8.6\) years.

Key concepts

Light-Year Conversion

The concept of a light-year is essential to understanding astronomical distances. A light-year is the distance that light travels in one year, which is approximately 9.461 trillion kilometers (\textasciitilde9.461\times10^{12} km). Conversion between light-years and kilometers can be daunting, but it's quite straightforward with a simple multiplication.
When you're given a distance in light-years and you need to find out how many kilometers it translates to, you multiply the number of light-years by the number of kilometers in one light-year.
For example, to convert 8.6 light-years to kilometers, the equation would be:
\[8.6 \times 9.461 \times 10^{12} \text{ km} = 81.3646 \times 10^{12}\text{ km}\]
Picturing this immense distance can be challenging, but this conversion provides a tangible number that we can use in further calculations, just as presented in the exercise.

Speed of Light

The universe's speed limit, the speed of light, is an integral part of physics. It is approximately 299,792 kilometers per second (or roughly 300,000 km/s for estimation). This constant speed is what enables us to calculate how long it takes for light to travel across the vast distances of space.
Understanding this speed helps not only in scientific computation but also in grasping the sheer scale of the cosmos. Light zipping around the Earth almost 7.5 times in just one second is a testament to its incredible speed!
In our Sirius example, knowing the distance in kilometers and that light travels at a speed of 300,000 km/s allows the calculation of the time taken for light from Sirius to reach Earth to be a simple exercise in division, underscored by the awe-inspiring fact that this journey spans several years.

Distance-Time Calculation

At the heart of many astronomical problems is the ability to calculate the time it takes for light to travel a given distance—this is a fundamental distance-time calculation. The relationship between distance, speed, and time is expressed in the formula:
\[time = \frac{distance}{speed}\]
This equation tells us that time is equal to the distance traveled divided by the speed. To apply this to real-world problems, such as determining how long it takes for light from Sirius to reach Earth, you substitute the actual numbers for distance and speed. After getting the time in seconds, converting it to more manageable units like years often makes the result more comprehensible:
\[time_{years} = time_{seconds} \times \frac{1 year}{3.15576 \times 10^7 seconds}\]

• If the distance from Sirius to Earth is 81.3646 trillion km,
• and light travels at 300,000 km/s,
• then the time taken in seconds is the quotient of the distance by speed.
• Converting that result into years gives us a reasonable understanding of the time span involved in interstellar communication or observation.

Such calculations emphasize the vastness of space and provide profound insights into how we understand the universe's dimensions and scale.