Gerald has taught engineering, math and science and has a doctorate in electrical engineering.
Converting 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!
Step 1: Review the definition of the tangent.
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:
Step 2: Relate x and y to astronomy.
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
Step 3: Define the parsec.
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.
Step 4: Compute tan θ when θ = 1 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.
Step 5: Calculate the number of meters in 1 pc.
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.
Step 6: Calculate how many seconds of time in 1 year.
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.
Step 6: Define and compute the distance of 1 light year.
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.
Step 7: Compare the length of 1 parsec to the length of 1 light year.
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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