In astronomy, parallax is the apparent motion of nearby stars against their background caused by the Earth's travel around the sun. Because closer stars seem to move more than distant ones, the amount of apparent motion allows astronomers to determine their distances by measuring the change in the angle of observation as it appears from Earth.
The apparent motion and the change in angle are so small that they are imperceptible to the naked eye. In fact, the first stellar parallax was only measured in 1838 by German astronomer Friedrich Bessel. Applying the tangent trigonometric function to the measured parallax angle and the distance traveled by the Earth around the sun gives the distance to the star in question.
TL;DR (Too Long; Didn't Read)
The motion of the Earth around the sun produces an apparent motion in nearby stars, resulting in a small change in the angle of observation of the star from Earth. Astronomers can measure this angle and calculate the distance to the corresponding star using the tangent trigonometric function.
How Parallax Works
The Earth moves around the sun on a yearly cycle with the distance from the Earth to the sun being one astronomical unit (AU). This means that two observations of a star six months apart take place from two points that are two AU apart as the Earth travels from one end of its orbit to the other.
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The angle of observation of a star changes slightly during the six months as the star seems to move against its background. The smaller the angle, the less the star seems to move and the further away it is. Measuring the angle and applying the tangent to the triangle formed by the Earth, the sun and the star gives the distance to the star.
An astronomer might measure an angle of 2 arc seconds for the star he is observing, and he wants to calculate the distance to the star. Parallax is so tiny, it is measured in seconds of arc, equal to one-sixtieth of one minute of arc, which in turn is one-sixtieth of a degree of rotation.
The astronomer also knows that the Earth has moved 2 AU between observations. In other words, the right-angled triangle formed by the Earth, the sun and the star has a length of 1 AU for the side between the Earth and the sun, while the angle at the star, inside the right-angled triangle, is half the measured angle or 1 arc second. Then, the distance to the star equals 1 AU divided by the tangent of 1 arc second or 206,265 AU.
To make it easier to handle the units of parallax measurement, the parsec is defined as the distance to a star that has a parallax angle of 1 arc second, or 206,265 AU. To give some idea of the distances involved, one AU is about 93 million miles, one parsec is about 3.3 light years, and a light year is about 6 trillion miles. The closest stars are several light years away.
How to Measure the Parallax Angle
The increasing accuracy of telescopes allows astronomers to measure smaller and smaller parallax angles and to accurately calculate the distances to stars farther and farther away. To measure a parallax angle, an astronomer has to record the angles of observation of a star six months apart.
The astronomer chooses a stationary target close to the star in question, usually a distant galaxy that does not move. He focuses on the galaxy and then the star, measuring the angle of observation between them. Six months later he repeats the process and records the new angle. The difference in the angles of observation is the parallax angle. The astronomer can now calculate the distance to the star.