Problem 14 - Physics

Question

When we look at the star Polaris (the North Star), we are seeing it as it was 680 years ago. How far away from us (in meters) is Polaris?

Step-by-Step Solution

Answer

Polaris is approximately \(6.439 \times 10^{18}\) meters away from Earth.

Step 1: Understand Light Travel Time

When we look at a star, the light we see has traveled from the star to Earth. The time it takes for the light to travel from the star to us is called the light travel time. If we see the star as it was 680 years ago, it means the light we see now left the star 680 years ago.

Step 2: Convert Light Travel Time to Distance

The distance to the star can be found by multiplying the light travel time by the speed of light. The speed of light is approximately \(3.00 \times 10^8\) meters per second. First, we need to convert 680 years into seconds.

Step 3: Convert Years to Seconds

To convert 680 years to seconds, follow these conversions:- 1 year = 365.25 days (including leap years)- 1 day = 24 hours- 1 hour = 3600 secondsThus, \[680 \text{ years} = 680 \times 365.25 \times 24 \times 3600 \text{ seconds} \] = 21,462,816,000 seconds.

Step 4: Calculate Distance in Meters

Using the converted time, calculate the distance using the formula:\[\text{Distance} = \text{Time} \times \text{Speed of Light}\]Substitute the values:\[\text{Distance} = 21,462,816,000 \times 3.00 \times 10^8 \]= 6.439 \times 10^{18} \text{ meters}.

Key Concepts

Speed of Light

The speed of light is a standard metric often used in various scientific calculations. It is denoted by the symbol \(c\) and valued at approximately \(3.00 \times 10^8\) meters per second (m/s). This is a huge speed, allowing light to travel the entire circumference of the Earth over seven times in just one second! The speed of light is consistent in a vacuum—meaning it's reliably unchanging. This consistency makes it a foundational constant in physics and essential for ensuring precision in calculations.
The fantastic speed of light means it can transfer information from distant objects in space to our eyes almost instantaneously on our time scale. However, on an astronomical scale, it takes considerable time for light from the outer universe to reach us. For example, when we observe objects like stars, we're seeing them as they existed in the past due to this light travel time.
Understanding this speed helps us discern how we perceive the universe and allows for calculations that determine distances based on how long it takes light to reach us.

Distance Calculation

Calculating astronomical distances often relies on the light travel time, which is the time light takes to travel from an object, like a star, to an observer on Earth. The primary formula used is the distance formula:

• \[\text{Distance} = \text{Time} \times \text{Speed of Light}\]

To find the distance, we need to know how long the light has been traveling (time) and the speed of light. In Polaris's case, since it takes 680 years for its light to reach us, we first need to convert those years into seconds. This conversion step is crucial because the speed of light is measured in meters per second.
Begin with the conversion factors:

• 1 year \(= 365.25\) days (including leap years)
• 1 day \(= 24\) hours
• 1 hour \(= 3600\) seconds

Calculating the total number of seconds for 680 years gives us \(21,462,816,000\) seconds. Then, multiplying by the speed of light gives us the distance
Finally, this calculation reveals the distance in meters: \(6.439 \times 10^{18}\) meters from Earth to Polaris.

Astronomical Measurement

Astronomical measurements take into account vast distances, like those between Earth and stars. Such measurements are typically made in light-years or parsecs, with a light-year being the distance light travels in one year. These widespread units help astronomers express and comprehend the universe's immense scale.
Measuring such distances sheds light on cosmic events occurring eons ago, revealing how stars and galaxies were structured and how they've evolved. While today's distance from Earth to Polaris is expressed in meters for precision, it's more commonly expressed in light-years—a more practical unit for astronomical purposes.
By understanding light travel time and distance calculation, we can map out the universe and learn about its ever-expanding nature. These measurements are crucial not just for comprehending space but also for making predictions of stellar movements and phenomena. Moreover, they assist astronomers in triangulating positions of celestial objects and understanding their dynamics over time.
This comprehension of astronomical measurements is key to expanding our grasp of the cosmos and unraveling mysteries that have puzzled humanity for centuries.