Parallax

###### Extra stellar systems are so far away from our own that we couldn’t even hope of developing a tape measure long enough to determine how far away they really are. On the other hand, astronomers can still determine how far away a stellar system is from our own. Have you ever wondered how this is possible?

Astronomers have developed several techniques to indirectly measure the vast distances between Earth and the stars and galaxies. In many cases, these methods are mathematically complex and involve extensive computer modelling. It turns out that measuring the distance to a star is an interesting problem! Astronomers have come up with two different techniques to estimate how far away any given star is.

The first technique uses triangulation (a.k.a. parallax). The Earth’s orbit around the sun has a diameter of about 186 million miles (300 million kilometres). By looking at a star one day and then looking at it again 6 months later, an astronomer can see a difference in the viewing angle for the star. With a little trigonometry, the different angles yield a distance. This technique works for stars within about 400 light-years of earth.

Parallax is “the best way to get the distance in astronomy,” said Mark Reid, an astronomer at the Harvard Smithsonian Center for Astrophysics. He described parallax as the “gold standard” for measuring stellar distances because it does not involve physics; rather, it relies solely on geometry.

It works like this: hold out your hand, close your right eye, and place your extended thumb over a distant object. Now, switch eyes, so that your left is closed and your right is open. Your thumb will appear to shift slightly against the background. By measuring this small change and knowing the distance between your eyes, you can calculate the distance to your thumb.

To measure the distance of a star, astronomers use a baseline of 1 astronomical unit (AU), which is the average distance between Earth and the sun, about 93 million miles (150 million kilometres). They also measure small angles in arcseconds, which are tiny fractions of a degree on the night sky.

If we divide the baseline of one AU by the tangent of one arcsecond, it comes out to about 19.2 trillion miles (30.9 trillion kilometres), or about 3.26 light-years. This unit of distance is called a parallax second, or parsec (pc). However, even the closest star is more than 1 parsec from our sun. So astronomers have to measure stellar shifts by less than 1 arcsecond, which was impossible before modern technology, in order to determine the distance to a star.

Measuring this distance is no small feat. The parallax angle by which even the closest stars shift is very small. For Proxima Centauri, it’s 0.77 arc second. An arcsecond is 1/3,600 of a degree. If you hold one of your hairs about 10 meters (or 33 feet) away, the hair covers an angle of 1 arc second. It wasn’t until 1838 that astronomers were able to measure such small angles. In that year, Friedrich Bessel measured the parallax of 61 Cygni as 0.314 arc second or 11.4 light-years.

Fun fact: A star with a parallax of 1 arc second would be 3.26 light-years away. This distance became known as the “parallactic second,” or parsec for short.

There is no direct method currently available to measure the distance to stars farther than 400 light-years from Earth, so astronomers instead use brightness measurements. It turns out that a star’s colour spectrum is a good indication of its actual brightness. The relationship between colour and brightness was proven using the several thousand stars close enough to earth to have their distances measured directly.

Astronomers can, therefore, look at a distant star and determine its colour spectrum. From the colour, they can determine the star’s actual brightness. By knowing the actual brightness and comparing it to the apparent brightness seen from Earth (that is, by looking at how dim the star has become once its light reaches Earth), they can determine the distance to the star.

For example, they use a class of variable known as Cepheids, which pulsate in and out like beating hearts. There is a direct relationship between the length of a Cepheid’s pulsation and its true brightness. Measuring a Cepheid’s apparent brightness – how bright it looks from Earth – allows astronomers to calculate its true brightness, which in turn reveals its distance. For this technique to work correctly, though, astronomers must first use the parallax method to get the distances to some of the closer Cepheids. This allows them to calibrate a Cepheid’s true brightness, which then can be used to calculate its distance. Cepheids are especially bright stars, so they are visible in galaxies that are tens of millions of light-years away.

For more distant galaxies, astronomers rely on the exploding stars known as supernovae. Like Cepheids, the rate at which a certain class of supernovae brighten and fade reveals their true brightness, which then can be used to calculate their distance. But this technique also requires good calibration using parallax and Cepheids. Without knowing the precise distances to a few supernovae, there is no way to determine their absolute brightness, so the technique would not work.

Unfortunately for objects that are billions of light-years away, astronomers’ predictions become significantly less accurate. We’re compelled to consider factors such as redshift and the expansion of the universe, which result in largely theoretical data. While we can determine an estimated distance based on redshift, the expansion of the universe is always throwing a wrench in our data.

While these methods work for most observations, some things just aren’t bright enough to see at all… let alone measure the distance to.

### Limitations of Distance Measurement Using Stellar Parallax

Parallax angles of less than 0.01 arcsec are very difficult to measure from Earth because of the effects of the Earth’s atmosphere. This limits Earth-based telescopes to measure the distances to stars about 1/0.01 or 100 parsecs away. Space-based telescopes can get accuracy to 0.001, which has increased the number of stars whose distance could be measured with this method. However, most stars even in our own galaxy are much further away than 1000 parsecs, since the Milky Way is about 30,000 parsecs across. The next section describes how astronomers measure distances to more distant objects.

#### Some examples to try

- A star has a parallax angle p of 0.723 arcseconds. What is the distance to the star?
- Sirius, a binary star in our galaxy, is a distance of 2.64 parsecs away from us. What would the parallax angle in arcseconds be for this binary star?
- Star A has a parallax angle of 0.82 arcseconds, and Star B has a parallax angle of 0.45 arcseconds. Which star is closest to Earth, and by how much?

###### Answer

1/0.723 = 1.38 parsecs

1/2.64 = 0.34 arcseconds

Star A is closest to Earth. It is 1 parsec closer than Star B.