## parallax [try it]

Vinay

How do we know how far anything is? For nearby objects, we can pace off the distance, of course, but what about places we can’t walk to? How does your brain know that that door is 10 feet away without marking it off on a ruler? Suppose you are standing at point A (see figure) and looking at object O, and you want to know how far it is from you. Behind the object is some background scenery, and it will appear projected onto the background around the point P. If you now move sideways by some distance to the point B and look at the object again, it will now appear to have moved against the backdrop, and will now be near Q. If you can measure the angle θ by which the object appears to have moved when you move by a distance b, then it is easy to work out that the distance AO ≡ d = b/tanθ. When the angle θ is small, we can use the approximation tanθ ≈ θ, i.e., d = b/θ. (Careful with the units here; θ is in radians, not degrees.)

This method of measuring distances is called parallax.

Note 3/21: In astronomical situations, say when you are observing from the Earth and taking measurements at half-year intervals, the astronomer is located at B, not A, and the distance of interest is OB. The parallax displacements measured are . In such a case, OB = b/sinθ ≈ b/θ. (Thanks to Jagadeesh for pointing out this correction.) For θ = 1 arcsecond, and b = 1 AU (the distance of the Earth from the Sun), the corresponding distance is a popular distance unit in astronomy called the parsec. 1 parsec is 3.1 1018 cm, or 3.26 light-years.

Your eyes are about 3 inches apart, and typical human visual acuity is about 1 arcminute. How far out can you reliably estimate distances then? What do you have to do to extend the range beyond this limit?

Can you build a simple instrument that uses rulers and protractors to measure distances to objects? For a given baseline separation, how precisely should you be able to measure angles to match its accuracy to that of our eyes?

How large can the baseline b be, in practice? If you wanted to measure astronomical distances, can you think of a natural length that we on Earth have easy access to?