# Lightyears and Parsecs **URL:** \ **Category:** Factual Questions\ **Created:** [June 19, 2002, 1:42am UTC](https://boards.straightdope.com/t/lightyears-and-parsecs/114778 "2002-06-19T01:42:53Z")\ **Posts on this page:** 7\ **Page:** 1
**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\ **Post date:** [June 19, 2002, 1:42am UTC](https://boards.straightdope.com/t/lightyears-and-parsecs/114778/1 "2002-06-19T01:42:53Z")
A lightyear is the distance light travels in one earth year. A parsec (parallax arc-second) is the distance of one half the mean orbit of earth in relation to one second of arc…approx. 3.2616 light years. Two units of distance defined by Earth’s orbit. Would the 3.2616 conversion factor remain constant even with a larger or smaller planetary orbit? If not, at what size orbit do the units coincide and equal each other? ---
**Author:** ![David\_Simmons](https://avatars.discourse-cdn.com/v4/letter/d/9de053/32.png) [@David\_Simmons](https://boards.straightdope.com/u/David_Simmons)\ **Post date:** [June 19, 2002, 4:30am UTC](https://boards.straightdope.com/t/lightyears-and-parsecs/114778/2 "2002-06-19T04:30:40Z")
> [@](#): > > \*Originally posted by Enola Straight \* > \*\*A lightyear is the distance light travels in one earth year. > > A parsec (parallax arc-second) is the distance of one half the mean orbit of earth in relation to one second of arc…approx. 3.2616 light years. > > Two units of distance defined by Earth’s orbit. > > Would the 3.2616 conversion factor remain constant even with a larger or smaller planetary orbit? > > If not, at what size orbit do the units coincide and equal each other? \*\* Based on similar triangles wouldn’t it be 92 million miles divided by 3.26? ---
**Author:** ![David\_Simmons](https://avatars.discourse-cdn.com/v4/letter/d/9de053/32.png) [@David\_Simmons](https://boards.straightdope.com/u/David_Simmons)\ **Post date:** [June 19, 2002, 4:52am UTC](https://boards.straightdope.com/t/lightyears-and-parsecs/114778/3 "2002-06-19T04:52:55Z")
Sorry about that. I answered the second question without answering the first. If the speed of light is constant then a light year is a light year no matter were you are. The parallax of an object depends on the length of the baseline, so an object that has a parallax of 1 second for us would have a different parallax for a different orbit. The ratio of parsec to light would be different in that case. ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [June 19, 2002, 8:26am UTC](https://boards.straightdope.com/t/lightyears-and-parsecs/114778/4 "2002-06-19T08:26:47Z")
“Would the 3.2616 conversion factor remain constant even with a larger or smaller planetary orbit?” No, because, the period of a planet’s orbit and the distance from the sun are not directly proportional. For any given planet, their equivalent parsec would be 206265a, where a is the distance from the Sun. Their light-year would be T/c, where T is the time it takes them to orbit the sun, and c is the speed of light. If we measure a in AUs and T in years, then Kepler’s Third law tells us: a[sup]3[/sup] = T[sup]2[/sup] Now, if we want their equivalent parsec and their equivalent light-year to be the same, we’d say 206265a = T/c The easiest way to solve this would be to cube both sides: 206265[sup]3[/sup]a[sup]3[/sup] = T[sup]3[/sup]/c[sup]3[/sup] 206265[sup]3[/sup]T[sup]2[/sup] = T[sup]3[/sup]/c[sup]3[/sup] 206265[sup]3[/sup] = T/c[sup]3[/sup] T = 206265[sup]3[/sup]/c[sup]3[/sup] = (206265/c)[sup]3[/sup] In the appropriate units (AUs/year), c = 63200, so T = 34.76 years, and a = 10.65 AUs. ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [June 19, 2002, 9:23am UTC](https://boards.straightdope.com/t/lightyears-and-parsecs/114778/5 "2002-06-19T09:23:56Z")
I forgot to point out two things. First, that 206265/c is, more exactly, that 3.2616 ratio that the OP mentioned. That might make the calculation a little easier: T = 3.2616[sup]3[/sup] year and a = 3.2616[sup]2[/sup] AU. Second, Kepler’s Third Law also has an implicit dependence on M, the mass of the Sun. So, this only holds true around our star. In a different system, you’d have different values. ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [June 19, 2002, 10:54am UTC](https://boards.straightdope.com/t/lightyears-and-parsecs/114778/6 "2002-06-19T10:54:35Z")
Whoops, I made a small typographical error up there. A light-year is not T/c, but T×c. Light times year, get it? Anyway, I realized what I meant, and the last line (the one starting with "T = ") is correct. It doesn’t change the final numbers any, though. Sorry about that! ---
**Author:** ![David\_Simmons](https://avatars.discourse-cdn.com/v4/letter/d/9de053/32.png) [@David\_Simmons](https://boards.straightdope.com/u/David_Simmons)\ **Post date:** [June 19, 2002, 7:26pm UTC](https://boards.straightdope.com/t/lightyears-and-parsecs/114778/7 "2002-06-19T19:26:22Z")
> [@](#): > > \*Originally posted by David Simmons \* > \*\*Sorry about that. I answered the second question without answering the first. > > If the speed of light is constant then a light year is a light year no matter were you are. > > The parallax of an object depends on the length of the baseline, so an object that has a parallax of 1 second for us would have a different parallax for a different orbit. The ratio of parsec to light would be different in that case. \*\* Oh my! In a long life of my now and then giving bad answers, the one quoted is one of the bad ones. Achernar’s answer is correct. The proper formula for the relationship between the distance ® of a satellite from its primary to the period (t) of the satellite orbit is k = r[sup]3[/sup]/t[sup]2[/sup], “k” being the constant of proportionality. So the period is proportional to r[sup]3/2[/sup]. The measured parallax is directly proportional to the radius of the orbit. So the ratio of the distance in light-years, measured in the years of the particular satellite in question, to the distance in parsecs measured from the same satellite is proportional to r[sup]3/2[/sup]/r = r[sup]1/2[/sup]