### Measurement by Triangulation

Let’s try to understand what exactly do we mean by the parallax method? How can triangulation help us measure the distances of faraway stars. The parallax method uses the fact that a triangle can be described completely with only three elements. This method of finding the values of the triangle to yield the location of an object is termed as Triangulation.

Such methods are regularly used by surveyors and architects. Triangulation the process of determining the location of a point by measuring the angles to it from two known points rather than measuring distances directly.

Let’s try to apply this in a real life example. How might we measure the distance of a ship from the shore without actually measuring it? Using triangulation, let’s create a fixed baseline with two different points AB. Now let’s name the points A and B. The angle extended by point A to the ship can be denoted as α (BAC) and the angle extended by B to the ship is denoted by a β. Now the baseline AB is known and so are the angles and hence we can deduce the rest of the properties of a triangle such as the position of the third point which is the ship.

Figure: Triangulation method of distance measurement

### Parallax

Parallax is the displacement or the change in the apparent position of the object when viewed from two different point of views. The two points of view have their own line of sights and parallax is measured as half of the angle between the two line of sights.

When you are travelling in a vehicle and you look around as you are moving, you will see that objects in the distances appear to move more slowly than the objects closer to you. This is the effect of parallax. Nearby objects have a larger parallax than the distant objects, so the parallax can be used to determine distances.

The phenomenon of parallax when combined with triangulation will yield the location of the object with considerable accuracy. Astronomers regularly use the parallax method to measure the distances of the closer stars.

### Distance Measurement by Parallax

Distance measurement by parallax is a special application of the principle of triangulation. From triangulation we known that a triangle can be described completely if two angles and side are known. In the image below, the distance of a distant star is being calculated.

The star being closer to earth than the far away starts exhibits a finite parallax value. This way, we can obtain the value of the parallax angle by viewing the star from two known points on Earth forming the baseline of the triangle. Let’s define the parallax half angle from two distinct points on Earth as ‘p’. Here the maximum value of‘d’ is the radius of Earth and the distance of the star can be assumed to be only slightly farther than the sun.

Here the value of the distance from the sun is many magnitudes smaller than the radius of the earth due to which the value of the parallax angle we get is extremely small. Too small to be even detected, which is why the great Astronomer Cassini and his colleagues were unsuccessful in finding the parallax for even a single star. This was considered by his detractors to be the proof that heliocentricity was wrong. Cassini’s theory was correct but the least count of the equipment used then wasn’t small enough.

### Application

Assuming the angle ‘p’ is small, the distance to the object measured in parsecs (in terms of speed of light) is equal to the reciprocal of the parallax angle measured in arcseconds.

$$D (parsec)= \frac{1}{p}(arcsec)$$

In order to overcome the problem of the small ratios, star parallax is most often measured using annual parallax which is defined as the difference in position of a star as seen from the earth and sun. Instead of taking the fixed baseline as the Earth’s radius, the fixed baseline is taken as the Radius of Earth’s revolution around Earth which increases the size of the baseline therefore the top angle making it easier to measure.

Unfortunately, the ground-based telescopes can only measure parallax reliably for stars that are within a few hundred light years from us. Telescopes above the atmosphere such as the Hubble Telescope can measure smaller parallax shifts and thus larger distance, but even in that case the most distant objects for which distance can be determined by parallax of a few thousand light years away.

Figure: Measurement of stellar distances and the parallax angle in terms of distances