Derivation of the distance equation used in the Hyades lab

Transverse motion of a star

Let's say that we observe a distant object, far away at a distance D, that shows a transverse velocity V_t. At a given time, the observer notes the object at location A at a time t_A. After a given amount of time, the observer notes the same object at location B. The object has traveled an actual distance of X, which at the distance D is measured as the small angle . How can we determine the actual distance, D?

The time the object has taken to travel the distance X is , and thus we use the relationship between distance traveled, velocity, and time:

Since the distance D is extremely large, we may use the small angle formula to obtain the relationship between D and X:

Note that since we always measure

in arc seconds, we must do some converting between arc seconds and radians (there are approximately 206,265 arc seconds in a radian. Also, D and X must be measured in the same units. If we substitute for X, this equation becomes

The proper motion of a star, is defined by . To get the equations into the form where we are using the common units for proper motion--arc seconds per year--and distance--parsecs--we must do some unit conversions. In final form, the equation we use is

where the number 4.74 comes from

Once again, the small-angle approximation makes our calculations much easier, although it may not look like it at first glance!