Question

The brightest star in the night sky in the northern hemisphere is Sirius. Its distance from earth is estimated to be 8.7 light years. A light year is the distance light travels in one year. Light travels at a speed of \(3.00 \times 10^{8} \mathrm{~m} / \mathrm{s}\). Calculate the distance from earth to Sirius in miles. (1 \(\mathrm{mi}=\) \(5280 \mathrm{ft})\)

Short answer

The distance from Earth to Sirius in miles is calculated by multiplying the number of light years (8.7) by the speed of light in meters per second, then converting that to the number of seconds in a year, to feet, and finally to miles.

Step-by-step solution

Convert Light Years to Meters

Since a light year is the distance light travels in a year, we need to calculate the number of meters in 8.7 light years. To do this, we multiply the distance in light years by the speed of light in meters per second, and then by the number of seconds in a year. There are 31,536,000 seconds in a normal non-leap year (60 seconds/minute * 60 minutes/hour * 24 hours/day * 365 days/year).

Convert Meters to Feet

Having the distance in meters, we convert it to feet. There are approximately 3.28084 feet in a meter.

Convert Feet to Miles

With the distance now in feet, the final conversion is to miles. There are 5280 feet in a mile.

Calculate the Total Distance in Miles

By chaining these conversions together (light years to meters, meters to feet, feet to miles), we can find the total distance from Earth to Sirius in miles.

Key concepts

Speed of Light

The speed of light is a fundamental constant in physics, representing the fastest speed at which information or matter can travel through the universe. In a vacuum, light travels at a constant speed of approximately \(3.00 \times 10^{8} \text{m/s}\), which is about 186,282 miles per second.

Understanding the speed of light is crucial for calculating astronomical distances. When we say a star is 8.7 light years away, we are saying that it takes light 8.7 years to travel from that star to Earth, moving at the speed of light. By knowing this constant speed and the time light travels, we can calculate the distance in more familiar units like meters or miles, which helps us grasp the vastness of space.

Unit Conversion

Unit conversion is a critical skill in science for comparing and converting measurements between different units of measurement. To calculate the distance from Earth to Sirius in miles, we have to convert the astronomical distance of light years into a measure we use more commonly on Earth, like miles.

For this conversion, we follow a series of steps using conversion factors. A conversion factor relates one unit of measurement to another and is derived from the definition or relationship between those units. For example, knowing that 1 meter is equivalent to approximately 3.28084 feet, we can convert any length in meters to feet by multiplying by this conversion factor.

Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It's often used in science and engineering to handle extremely large or small values, like astronomical distances or the speed of light.

In scientific notation, a number is written as the product of a number between 1 and 10 and a power of 10. For instance, the speed of light, \(3.00 \times 10^{8} \text{m/s}\), is much easier to read and work with than writing out 300,000,000 meters per second. This notation helps simplify calculations by making it easier to multiply and divide numbers that span several orders of magnitude.

Astronomical Distance Measurement

Astronomical distance measurement involves quantifying the immeasurable expanse between celestial objects in the universe. A common unit of distance used in astronomy is the light year, which is the distance light travels in one year. Since light travels at a constant speed in a vacuum, a light year can be calculated by multiplying the speed of light by the number of seconds in a year.

In the context of our problem, the distance of the star Sirius from Earth is first identified in light years, a measure that conveys the time-component of travel at light-speed. To get a sense of this distance in terms of earthly measurements, we then translate light years into kilometers or miles through the process of unit conversion, which is vital for making such cosmic distances comprehensible in human terms.