Parallax & Parsecs…

One down, two to go. This time, we’re going into the bigger deals. Parsecs, which is (as far as I know) the largest unit for distance. Before getting into this boring topic, I wanna say that H-Twins have amazed me once again by their wonderful flash animation where they’ve scaled the whole universe. From the strings & branes (at Planck scale) to the galactic clusters, all that you might wonder watching – have been scaled. And, I can assure you that it’d not only be interesting, but also productive. Alright, don’t spend much time with them. You have a post to read…

“How much” relative to the background…

Well, parsec is not a very well-known or a great way of measuring distance. Instead, it’s an easy way to measure the stars that seem to show the Stellar parallax. and it’s the fundamental step of the cosmic distance ladder (it’s just a list of the different ways to “see” & count stars in the night sky – we’ll discuss about that soon). Hey, parallax is an everyday phenomena. Say, you have an object at some distance from your laptop. This is the configuration. Laptop, Object and you. Now, you move a little bit sidewards and you can see some movement in the object relative to the background (which is hopefully, your laptop). But, since the laptop is quite closer to you, you may also notice a relative motion of your laptop with respect to the background (say, your door, window, or wall).

There are other examples like, your eyes which experience a parallax quite often. It’s just your fantastic brain which has lot of wires, and enormously complicated subtlety of clockworks going on inside, that compiles everything, and provides a beautiful artwork of a nice girl standing elsewhere, disturbing your indulgence in my post.

Your eyes have a variable focal length. For instance, bring something closer, and you can feel your eyes adjusting themselves to bring the object into their focus. But, we’re not dealing with what the muscles and fibers do inside the eye. Let’s do a stupid experiment to experience a parallax now. Point your finger upwards, and keep in mind that you do these actions relative to some background object (it’s rather worthless to point it over a blank wall).

Close one eye and see the finger. Then, close the other eye, and see it again. That apparent change in the field of view is called the Parallax. Pretty boring. Now, I can now safely assume that I don’t have to get my hands dirtier than that…

Now, this effect becomes more pronounced (magnified) when the background is at relatively very large distances. As an example, say you’re traveling in a bus (train, bike, whatever) and you’re looking out of the window. Objects that are closer to you (such as trees, posts, windmills, angels & fairies, etc.) may appear to move backwards at some rate, based on their distance from you – (that’s incomplete), relative to the farther objects such as hills, mountains, demons & minotaurs, etc.

Those things, being at somewhat larger distance, appear to move very slowly (because they’re farther). Try extending this basic example to clouds, moon, sun, stars (??). They’re at unimaginable distances, while your objects being very near to you, appear to show a clean parallax. But, that’s the fate of astronomers…

As shown, they’ll try to trace out how a star moves relative to some distant star (or say, a giant galaxy). Now, how would you tell whether a specific star is relatively closer or farther?

Observations, of course. A titanic heck of them. You’ve to keep track of a certain patch of stars at say, south east at some 20 degrees elevation. Now, Earth rotates. After few hours, when you observe the same patch, you’ve to plug a lot of factors into your equations like say, rotation of Earth, it’s inclination (i.e) how the direction has changed, etc. and observe it again.

I’ve to stress this point. Stellar parallax relies wholly on the revolution of Earth. Because, you have to move – to make the closer star appear as if it were moving, for your small steps. For our convenience, we choose two positions. The positions are on either end of the elliptic orbit. When Earth arrives at one end, we capture the star patch. And, after six months of long waiting, we shoot our next image of the same patch and notice something terrible. A star has moved from right to left, showing an effect to our motion from left to right. So, that’s a parallax.

Here’s where parsec is of great importance. What we do is, just apply the basic trigonometry to estimate the distance to the star. Firstly, we know Earth-Sun distance (which is just the mean of distance to Sun from perihelion & aphelion) to be 1 AU (150 million kms) and by playing with the telescope for sometime, we can find the angle between the sun’s axis (we’ve taken sun’s axis as reference, because we’re moving around, ours is variable) and our view of the star.

If you’ve taken a basic trigonometry lesson or two, you would’ve figured out what I’m trying to spew here. Yep, the thing forms a right-angled triangle and the ratio of the opposite side length to that of the adjacent side, gives the tangent to its angle to the hypotenuse.

Now, we’re forced to do a trick. One-degree may seem to be a small measure in your protractor. But think a while, as the hypotenuse extends, extends, and it becomes so extremely gigantic. So, we’ve assumed the parsec to a minimal value. A degree can be sliced into 3600 arc seconds. And based on our stupid definition, one parsec is distance of the adjacent side of the triangle (sun-star) when the angle is 1 arc-second. Simply,

\mathrm{tan \theta=} \mathrm{\frac{Earth-Sun\ (opposite)}{Sun-Star\ (adjacent)}}

\mathrm{Sun-Star=} \mathrm{\frac{1\ AU}{tan\ 1''}}

\mathrm{\implies d=} \mathrm{\frac{1\ AU}{arctan(\frac{1}{3600})\times \frac{\pi}{180}}}

Going along with this math, we’ll get a value of 206264.8 AU, which is equivalent to 3.26 light years. Now, why did I use “arc-tangent” up there? Because, it’s the only function that’s been defined to measure the angular distance (arc on a circle) in radians, corresponding to a given tangent. In other words, the Earth is sweeping an elliptical orbit. In such a case, we’d also see the star sweeping an elliptical arc with respect to us (well, see the figure).

If we use a tangent, it’s gonna result a value for a straight line (the opposite side). On the other hand, an “arc-tangent” gives the swept angle (which is what we want now).

It may appear to you, as if parsec doesn’t need to be preferred at all, being something that’s only three times a light-year. Why do we use it then? As I said above, it’s very easy, and a convenient way for measuring distances when a parallax is encountered. Although light years are mostly preferred in schools (as they follow SI units), and popular studies, astrophysicists & cosmologists mostly prefer parsecs (even kiloparsecs & megaparsecs) to light years. Don’t ask me why…

Okay then, is it very accurate? Bad luck..!!! Accuracy is something based on our instruments. You take any experimental data and you’re highly likely to find a \pm sign sticking to the end.

Duh, we can’t necessarily blame Technology for that. But instead, it’s the inability of humans, with their classical instruments, which is what quantum mechanics is all about


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