For most astronomical measurements, the following approximate relation is true,
The proportionality constant depends upon the units we use for the angle.
The most familiar unit for angular measurement is a degree. We
know that there are
in one circle.
A more natural way to measure
angles is to use what is known as a
Radian.
There are 2
= 6.28 Radians in one circle.
Why are Radians a natural unit?
Because the formula for angular size is
In terms of degrees, the same formula is
In most astronomical measurements, the angles measured are very small; they are much smaller than 1 degree. Because of this, new units for the measurement of angles were invented.
The idea is similar to what we do for time. We have many units for time --1 year = 365 days; 1 day = 24 hours; 1 hour = 60 minutes; 1 minute = 60 seconds. It is silly to talk about how old you are in seconds; you use years. By the same token it is silly to talk about how long this class is in terms of years; you use minutes or hours. You usually try to choose the units to match the size of the thing you are measuring.
For angles, because most relevant astrophysical angles are tiny, smaller units for angles were invented;
The typical astronomical measurement is on the order of arc seconds (i.e., 1/3600 of a degree!!!). This is an impressively small number. The average human being can resolve an angle on the order of an arc minute.
Comment--what does resolution mean? Well, return to the railraod track. We know that up close the rails on a railroad track are parallel. However, as the track moves off into the distance, the rails appear to get closer and closer together and that somewhere near the horizon we cannot make out that there are two tracks; we can only see (resolve) one track. What resolution measures is the smallest angle that we can see with our eyes.
The parallax angle is defined as
Mathematically, it is given by
Here, A.U. is the astronomical unit and is defined as the average radius of the Earth's orbit and D is the distance to the star. Okay, what sort of an effect does this motion produce or more precisely, how large are typical parallax angles (that is, how large are the typical annual shifts in the positions of stars)? The astronomical unit is around 93,000,000 miles or 150,000,000 kilometers. Outside of the Sun, the closest star, Proxima Centauri, to the Earth is at a distance of 100,000,000,000,000 kilometers (4.3 light years). The parallax angle is then
This works out to be 9
degrees which
is around
one-third of one arc second which is << than the resolving
power of the human eye.
Because of the astronomical distances to even the closest stars, parallax is not measurable with the unaided human eye.