Parallax and Parsecs
Published on Jun 18, 2007 at 11:53 pm.
9 Comments.
Filed under astronomy.
Every now and then, I refer to astronomical distances such as Astronomical Units or Parsecs, but I don’t think that I’ve yet actually had a posting about the meaning of those terms. So, I thought that I’d write about these distance measurements today.
Basically, an Astronomical Unit is defined to be the semi-major axis of Earth’s orbit. Planets orbit the Sun in elliptical orbits. The semi-major axis of an ellipse is determined by taking half of the long diameter of an ellipse. For a planet, that means adding the perihelion (the closest distance to the Sun) and the aphelion (the farthest distance from the Sun) and dividing by two. Mathematically, that, of course, is the average of those two numbers. So, it would seem that the semi-major axis of a planet’s orbit is its average distance from the Sun. Well, that is not exactly true. Yes, it is the average of the extrema of the orbit, but there is more of the orbit farther from the Sun than near the Sun (because of how ellipses work). Furthermore, planets speed up and slow down as they orbit the Sun, and they move slowest when they are farthest from the Sun. So, that means that a planet (or asteroid, comet, etc) spends more time in its orbit farther from the Sun than the semi-major axis distance than it does closer than the semi-major axis. So, in that sense, it isn’t really the “average” distance. But, that’s probably close enough of an understanding for a non-astronomer.
Frequently we measure distances within the Solar System in Astronomical Units (AU). Aesthetically, we find numbers ranging from about 0.1 to 100 to be easiest to look at and grasp. Within the Solar System, all planets, and almost all other bodies (other than a few of the Kuiper Belt Objects) fall within this range. By convention, we also typically measure exoplanets’ distances from their stars using the same units: AU. And, there are historical reasons that we use Astronomical Units. Long before we were able to actually measure the distances to the planets, Johannes Kepler showed that there was a relationship between a planet’s semi-major axis and the time that it takes to orbit the Sun. So, we could find the relative spacing of the planets from determining how long it took them to orbit the Sun. But, that was relative to Earth. Unfortunately, not knowing how far Earth actually was from the Sun left astronomers in a quandary. Well, by defining the Earth to be 1 AU from the Sun meant that we could then determine planetary distances in terms of astronomical units. Only later was an astronomical unit finally measured in terms of terrestrial units (like kilometers).
And this leads into stellar distances. Stars are so very far away that measuring distances to them is very difficult. The most reliable way to measure distances is to use parallax. You are already familiar with parallax. Hold a pen or pencil out in front of you at arms length. Then, close one eye and look at the pencil. It appears to be in front of some of the more distant background objects. Then, without moving the pencil, look at it with your other eye. Now, the pencil seems to be in front of different background objects. This is parallax!
For stellar parallax, though, you don’t use your two eyes. They are far too close together. Instead, you look at a star from Earth. Then, you can wait six months and look at the same star from the other side of Earth’s orbit. If you are looking at the star against a background of much more distant stars, then the star appears to shift. If the star is near enough, you can measure the angular shift of the star against the background stars. The more distant the star, the smaller this shift, so the more difficult it is to measure. Half of that angular shift is the parallax angle. Stars are so far away that the nearest star has a shift that is so small as to yield a parallax angle of only about 1/4000 of a degree! It is no wonder that it took so long for astronomers to be able to measure that small of an angle. The diagram above shows the geometry of this sort of measurement. If you know the base of the triangle, the distance between the Sun and the Earth, then you can compute the distance.
However, as I said earlier, the distance between the Sun and Earth is about 1 AU. In point of fact, 1 AU is the semi-major axis of the Earth’s orbit, but it is only fairly recently that measurements have been accurate enough to account for the fact that Earth’s orbit is not really circular. The distance at which a star would have a parallax angle of one arcsecond was defined to be a parallax-second, or simply parsec. This makes finding distances in parsecs easy. You simply take the reciprocal of the parallax angle (measured in arcseconds) and that is the distance in parsecs. So, a star with a parallax of 0.25 arcseconds would be a distance of 4 parsecs away. A more modern definition of a parsec, which gets past the whole eccentricity of Earth’s orbit issue, is that a parsec is the distance at which one astronomical unit subtends one arcsecond of parallax. This turns out to be about 3.26 light years in distance.
So, that is what is meant by astronomical units and parsecs.
-Astroprof
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JT on December 2, 2007 at 12:39 pm: 4
Exactly how is the parallax angle found, how is it measured?
Astroprof on December 2, 2007 at 3:42 pm: 5
The parallax angle is found by measuring the position of a star against the background sky. Then, it is measured again six months later.
Michael on February 12, 2008 at 10:16 am: 6
What is the distance between the earth and the sun in parsecs and also in light years?
Astroprof on February 12, 2008 at 12:44 pm: 7
You can figure that out by dividing the Earth-Sun distance (1 Astronomical Unit) by the length of a light year.
So,
1 AU = 1.58e-5 light years
1 AU = 4.85e-6 parsecs
Daniela on June 12, 2008 at 11:51 pm: 8
how far in parsecs, is an object that has a parallax of 1 arc-seconds? anda howfar is it, in light years?
Astroprof on June 13, 2008 at 11:19 am: 9
As I said, by definition, one parsec is the distance that creates a parallax of 1 arc-second. That is about 3.26 lyrs.