Blog #10 - Trigonometric Parallax

Trigonometric Parallax 

One of the most common ways to measure the distance to a star is using the "trigonometric parallax." This works by measuring the angular distance a star appears to move, with respect to a background field of much more distant stars, as the Earth moves one quarter of an orbit (i.e. as the Earth translates "sideways" by a distance of b = 1 AU). 

There are 60 arcminutes in a degree and 60 arcseconds in an arcminute. What is the distance, measured in cm and light years, of a star that moves by 1 arcsecond when the Earth moves by 1 AU? Give a general formula relating the distance D to the angle, \(\theta\), moved by the star. 

We begin with a sketch of the situation described in the problem seen below:

In the drawing we see an exaggerated angle, \(\theta\) created when between when viewing a star from Earth's original position around the sun and then once the Earth makes a quarter orbit around the sun and is now directly vertical when original it was 1 AU away from the sun. 

First we must convert the distance of 1 arcsecond to degrees and then to radians in order to find a distance from the geometry of the situation. We convert with dimensional analysis \[\frac{1 \space degree}{60 \space arcminutes} * \frac{1 \space arcminute}{60 \space arcseconds} * 1 \space arcsecond = \frac{1}{3600} \space degrees \]Then converting from degrees to radians \[\frac{1}{3600} \space degrees * \frac{2 \pi \space radians}{360 \space degrees} \approx \frac{1}{200,000} \space radians \]With the radian found, we can use the geometry of the problem to find the distance to the star. \[tan(\theta) \approx \theta = \frac{1}{200,000} \space radians = \frac{1 AU}{d} \rightarrow d \approx 200,000 AU \]Converting from AU to cm and then to lightyears we get \[200,000 AU \approx 3*10^18 cm \approx 3 lightyear \approx 1 parsec \] This unit is known as a parsec because it is the trigonometric parallax (or geometric angle) passed through after a star moves 1 arcsecond when the Earth moves 1 AU. 

Moving this relationship to a general equation for trigonometric parallax, we get \[d_{Earth Translation} = \theta_{Star Translated}*d_{star} \]