# Parsec confusion

parsec is defined as the distance at which average radius of earth's orbit around sun would subtend an angle of 1''(second of arc). But suppose im looking at this star when earth is on one side of the sun, now six months later earth is on the other side of the sun and when im looking at this star theres some sort of parallax error right. Now the angle between these two observations say the angle between the line connecting the earth when it is on one side of the sun to the star and the line connected the earth when it is on the other side of the star to the sun. That's parallax right? So my baseline is the whole of the diameter of earths orbit not only the radius explain please

Basically... you're right. But I'll just say exactly what's going on because your question has some imprecise phrasing.

The parsec is defined as a distance such that if you change your observation position by $$1\text{ AU}$$ perpendicular to the direction toward a distant object, that distant object will move by 1 arc second with respect to much more distant stars. The wikipedia explains this clearly with this image:

As you point out, the earth moves by $$2\text{ AU}$$ in 6 months, not $$1\text{ AU}$$. So if a star was 1 parsec away from the earth in the direction perpendicularly up from the plane of the earth's orbit, it wouldn't move by 1 arc second in six months, it would move by two.

They could have defined the parsec either way - but they chose to define it this way. Which I support because radii are in general more convenient to work with than diameters in physics. Equations are just more often nicely written in terms of radius. Like the centripetal acceleration formula $$a=v^2/r$$

• wait okay oh yes. we are concerned with the distance if we take 2au 2 arcsecs its gonna come the same as 1 arcsecs and 1 au oh yes. god thankyou this simple thing was not clicking. Commented May 2, 2023 at 18:49
• I apologize if im disturbing you but this formula 1 parsec = 1 AU / 1 ". This was derived by the tangent of 1'=1Au/ 1 parsec right? Now im aware tangent of arcsecond isn't possible so if we converted it into radians we can work it out ? my book ignored the tangent tho I dont understand why. Commented May 2, 2023 at 19:14
• @sanya sounds right to me. Note that 1 arc second is $4.8\times 10^{-6}$ radians, and if the angle is small, then $\tan\theta\approx\theta$, so $\tan(4.8\times 10^{-6})\approx 4.8\times 10^{-6}$, and 1 parsec is indeed $1\text{ AU}/4.8\times 10^{-6}$ Commented May 2, 2023 at 19:20

Parallax is a strictly visual phenomenon caused by the fact that we have two eyes which see faraway objects from slightly different angles. Wikipedia defines parallax as "a difference in the apparent position" (emphasis mine) of an object, which is not the same as a difference in the actual position. The distance between Earth and the Sun at any given time is defined as the difference between their centers of mass, which can be computed exactly with modern techniques without worrying about the effects of parallax. Thus, the average radius of the Earth's orbit is a perfectly good "measuring stick" that can be used to define units such as the parsec.

• I don't understand then how do we measure the angle and necessary for calculating its distance? Commented May 2, 2023 at 15:14
• replace and with that is Commented May 2, 2023 at 15:26