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?
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.
“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.
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.
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!
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]