*Gerald Lemay*Show bio

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Instructor:
*Gerald Lemay*
Show bio

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson, we define and relate two very important distance measures: the parsec and the light year. These measures are typically used for describing the immensely long distances between astronomical objects. We'll learn how to convert from parsecs to light years.

Both the parsec (pc) and the light year (LY) are measures of distance useful in astronomy where the distances are huge. For example, the nearest large galaxy cluster is the Virgo Cluster. From the earth, this celestial object is 59 million light years away while the Andromeda Galaxy is about 780 thousand parsecs away.

So let's begin working out the relationship between parsecs and light years!

For a right-triangle, the tangent of an angle is the opposite side over the adjacent side.

In the figure, the tangent of θ is the opposite side, *y*, divided by the adjacent side, *x*:

The **astronomical unit (AU)** is the average radius of the earth's orbit around the sun. You may know 93 million miles as this number. What if we convert 93 million miles to meters?

- One mile is 1609.3 meters.
- 93 million miles is 93 times 1609.3 million meters.
- This is approximately 1.4960 x 1011 meters.

We use 1.4960 x 1011 meters as the average distance from the sun to the earth. This is 1 AU. In the right-triangle figure, *y* is 1 AU. The *x* is the distance we will call 1 parsec (1 pc). Our triangle now looks like:

In terms of the tangent, just like tan θ is *y*/*x*, we now have

The word ''parsec'' is a contraction of ''parallax'' and ''second''. **Parallax** is the angle subtended at a celestial object by one AU. In our figure, it is the angle θ. And one second of arc is 1/3600 of one degree. This comes from 60 minutes/degree times 60 seconds/minute.

The **parsec** is the distance when the subtended angle θ is one second of arc.

This is easily computed on a calculator. You can verify the tangent function on your calculator is set to degrees by taking the tangent of 45. The answer should be 1.

Remember tan θ = 1 AU/1 pc? Then, 1 pc = 1 AU/tan θ (when θ = 1/3600 degrees). In the calculation, we also use 1.4960 x 1011 meters as the value of AU.

This is a fun conversion exercise. We use:

- 365 days
- 24 hours/1 day
- 60 minutes/1 hour
- 60 seconds/1 minute

When we multiply these four factors together, the remaining units are seconds and the number is 3.1536 x 107. From this, we get:

Thus, in one year, there are 3.1536 x 107 seconds.

A **light year (LY)** is the distance light travels in 1 year. We use the speed of light as 3 x 108 meters per second.

We now have the distances in meters for both the pc and the LY. To compare them, we make a ratio of 1 pc to 1 LY, substitute the distances in meters and simplify:

Comparing the distances in meters of 1 pc and 1 LY we get

Tau Ceti is a star in the constellation Cetus. It has sparked the interest of extra-terrestrial life searches because of the 5 planets circling this star. The distance of Tau Ceti from the earth is 3.65 pc. How many LY is this?

If we could travel at the speed of light, it would take us almost 12 years to reach the Tau Ceti star system.

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