Question

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 \((\mathrm{AU})\) is equal to the average distance from Earth to the Sun, \(1.50 \times 10^{8} \mathrm{~km}\). 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 \(3.00 \times 10^{5} \mathrm{~km} / \mathrm{s}\), would cover in 1 year. ( \(a\) ) Express the distance from Earth to the Sun in parsecs and in light-years. (b) Express a light-year and a parsec in kilometers. Although the light-year is much used in popular writing, the parsec is the unit preferred by astronomers.

Short answer

The distance from Earth to the Sun can be expressed as \(\frac{1}{206265}\) Parsecs or \(0.0000158\) light-years. A light-year and a parsec can be expressed as \(9.461 \times 10^{12}\) kilometers and \(3.086 \times 10^{13}\) kilometers, respectively.

Step-by-step solution

Express Distance from Earth to the Sun in Parsecs

Understand that 1 parsec is the distance at which an astronomical unit (AU) subtends an angle of 1 arcsecond. Since there are \(206,265\) arcseconds in a radian, we can find that \(1 AU = \frac {1}{206265} Parsec\). Thus, the distance from Earth to the Sun, which is \(1 AU\), can be expressed as \(\frac{1}{206265} Parsec\).

Express Distance from Earth to the Sun in Light-years

To express this in light-years, understand that 1 light-year is the distance light would cover in 1 year traveling at a speed of \(3.00 \times 10^{5} km/s\). There are \(60 \times 60 \times 24 \times 365.25 = 31557600\) seconds in a year, and therefore, light travels \(3.00 \times 10^{5} km/s \times 31557600 s = 9.461 \times 10^{12}\) kilometers in a year. Since we know that \(1 AU = 1.50 \times 10^{8} km\), it can be expressed in light years as \(1 AU = \frac{1.50 \times 10^{8} km}{9.461 \times 10^{12}} = 0.0000158 Light-years\).

Express a light-year and a Parsec in Kilometers

As established in the previous steps, 1 light-year presents a distance of \(9.461 \times 10^{12}\) kilometers. Now, since \(1 AU = 1.50 \times 10^{8} km\) and \(1 AU = \frac {1}{206265} Parsec\), we can calculate that \(1 Parsec = 206265 \times 1.50 \times 10^{8} km = 3.086 \times 10^{13}\) kilometers.

Key concepts

Astronomical Unit (AU)

The astronomical unit (AU) is a vital measurement in astronomy. It serves as the baseline for measuring vast distances in space. Specifically, 1 AU represents the average distance between the Earth and the Sun. This distance measures approximately \(1.50 \times 10^{8}\) kilometers.

Using AU allows astronomers to easily express and communicate the distances between planets in our solar system. For example, when you hear that Mars is roughly 1.52 AU from the Sun, it becomes more comprehensible than quoting an enormous number of kilometers.

• 1 AU = \(1.50 \times 10^{8}\) km

Parsec

A parsec is another crucial unit used by astronomers. Unlike the AU, the parsec is often employed to measure interstellar distances.

One parsec is equivalent to the distance at which one astronomical unit would subtend an angle of one arcsecond. This can be calculated using the small angle approximation in radians. Since there are 206,265 arcseconds in a radian, 1 parsec equals \(206,265\) times the distance of 1 AU, which is approximately \(3.086 \times 10^{13}\) kilometers.

• 1 parsec = \(3.086 \times 10^{13}\) km
• 1 AU = \(\frac{1}{206265}\) parsec

Light-Year

The light-year is a popular unit in both science fiction and scientific circles. It measures the distance that light travels in one year. Since light moves at a speed of \(3.00 \times 10^5\) kilometers per second, we can calculate the total distance light covers in a year.

There are roughly \(31,556,926\) seconds in a year, which means that the distance light travels in that time is \(9.461 \times 10^{12}\) kilometers. Thus, when distances in space are described in light-years, they illustrate vast expanses that even the speed of light takes time to cross.

• 1 light-year = \(9.461 \times 10^{12}\) km

Distance Calculation

Calculating astronomical distances involves converting between units like AU, parsecs, and light-years.

To express the distance from the Earth to the Sun in parsecs, remember that 1 AU is approximately \(\frac{1}{206265}\) parsec. For light-years, use the fact that \(1 AU = 1.50 \times 10^{8}\) km and that a light-year is \(9.461 \times 10^{12}\) km, leading to \(1 AU \approx 0.0000158\) light-years.

• Transform AU to parsec using: \(1 AU = \frac{1}{206265}\) parsec
• Transform AU to light-year using: \(1 AU = 0.0000158\) light-years

Speed of Light

The speed of light is a fundamental constant, integral to numerous physical theories including relativity.

In vacuum, light travels at \(3.00 \times 10^5\) kilometers per second. This speed allows us to calculate the vast distances light can cover in space over time, such as the span it covers in a year, forming our unit of a light-year.

• Speed of light = \(3.00 \times 10^5\) km/s

Understanding the speed of light helps in comprehending astronomical distances, underpinning the calculations of both light-years and related space phenomena.