How Do We Measure Distance in Space Using Parallax and Parsecs

A look at the ways which astronomers measure distance in space covering parallax, distance modulus, variable stars, supernova and redshifts.

When we talk about distance in astronomy, we are usually talking very, very large numbers. Far too many to describe them in terms of miles or kilometres. When we realised just how big space was, we needed some new units. In modern astronomy, we often use the Astronomical Unit or the Lightyear.

Space is big. Really big. You just won't believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it's a long way down the road to the chemist, but that's just peanuts to space.Douglas Adams, The Hitchhiker's Guide to the Galaxy

The Astronomical Unit is the average distance from Earth to the Sun. We say average because the Earth's orbit is elliptical, varying from a maximum (aphelion) to a minimum (perihelion) and back again once a year.

Due to this variation, the Astronomical Unit is now defined as exactly 149,597,870,700 metres (about 150 million kilometres, or 93 million miles). You can see why we don't express this as kilometres! For objects in the solar system, their orbits are typically given in terms of the Astronomical Unit (AU). Earth is 1AU, Venus at 0.72AU, and Jupiter at 5.2AU. These values are much easier to work with. If you want to convert AU to KM or Miles, simply multiply the Earth's orbital radius by the AU value.

For distances outside of the solar system, the light-year distance is often used. A light year is defined as the distance light travels in a year. Since the speed of light is constant, the distance is also constant. Light travels at around 300,000 kilometres per second, so these numbers can get very big, very fast. In one year, light travels about 10 trillion km. One light-year is equal to 9,500,000,000,000 kilometres, or 63,241 AU.

For large numbers like this, we often use scientific notions. We write a light year as 9.5x10¹² km. This is called scientific notation. We simply move the decimal place to the left until we get to the smallest significant figure and count the number of times we moved the decimal place. Even using light years as a measure of distance we still deal with very large numbers. The Andromeda galaxy is the nearest galaxy to the Milky Way, and at a distance of 2.5 million light years, it's quite a bit further than walking down the road to the chemist. The furthest observed galaxy is EGS8p7 which is more than 13.2 billion light years away. Because we know how far away it is, and that the speed of light is constant, we know that the light from that galaxy has travelled for 13.2 billion years to arrive here. We are effectively looking back in time, to a point only a few hundred million years after the big bang. How cool is that?

Comparing Distances in Space

How Do We Measure Distance in Space?

There are various methods for measuring distance in space and a different method is used depending on how far away the target is.

• For stars within our galaxy we can use distance modulus or parallax.
• For stars outside our galaxy we can use cepheid variables, supernovae and redshift.

For Stars within our Galaxy

Using Parallax to Measure Distance in Space

Astronomers use an effect called parallax shift to measure distances to nearby stars. The principle of Parallax can easily be demonstrated by holding your finger up at arm's length. Close one eye, then the other and notice how your finger appears to move in relation to the background. This occurs because each eye sees a slightly different view because they are separated by a few inches.

If you measure the distance between your eyes and the distance your finger appears to move, then you can calculate the length of your arm.

This same principle can be used on a larger scale to calculate the distance to an object in the sky, only we use different points on the Earth's orbit instead of looking through alternate eyes. This is a fantastic way of measuring distance as it relies solely on geometry. Parallax calculations are based on measuring two angles and the included side of a triangle formed by the star, Earth on one side of its orbit and Earth six months later on the other side of its orbit.

Calculating parallax requires that the objects Right Ascension and Declination be recorded accurately so that we know the object's precise location on the celestial sphere.

We measure the position of an object relative to the other background stars during the winter months, and then again 6 months later, in the summer when the Earth has moved 180° around its orbit around the Sun to give maximum separation distance.

In this diagram (not to scale) during the summer the position of the object appears to be at point A in the sky. Six months later, during the winter, it appears to be at point B. The imaginary line between the two opposite positions in the Earth's orbit is called the baseline. The half baseline is the Earth's orbit radius.

We know the radius of the Earth's orbit radius (r), and we can calculate the angle, θ from the observed apparent motion, measured in radians. Finally, we just need a little trigonometry to calculate the distance, d.

Equation 8 - Pythagoras Triangle Trig

Since the value of theta measured is going to be very small, we can approximate tan θ = θ. Rearranging to solve for d gives us:

Equation 9 - Pythagoras Triangle Trig

This equation forms the basis for a new unit of length called the parsec (PC). A parsec is defined as the distance at which 1 AU subtends 1 arcsecond. So an object located at 1pc would, by definition, have a parallax of 1 arcsecond.

The parallax measured for α Centauri is 0.74 arcseconds Calculate the distance in light years to α Centauri.

Equation 10 - Distance Calculation using Parallax

Equation 11 - Distance Parallax Calculation

1 AU is equal to 1.4960x10¹¹ meters and 1 parsec is equal to 3.26 light years, which makes α Centauri 4.405 light years away.

Parallax can be used for distances of up to 10,000 lightyears / 3 kpc.

Using Distance Modulus to Calculate Distance

Beyond 100 light years, but within our galaxy, we can use a technique called distance modulus. Using the distance modulus it is possible to establish a relationship between the absolute magnitude of a star, its apparent magnitude, and its distance. Distance modulus can be obtained by combining the definition of absolute magnitude with an expression for the inverse square law and Pogson's relation.

Equation 25 - Distance Modulus

Distance Modulus is the difference between the apparent and absolute magnitudes of a star. If you know both values through observations we can derive distance. Distance modulus can be used on any stars within our galaxy where we can obtain absolute and apparent magnitude, or variable stars at further distances when we can observe the star's period.

Barnard's Star observed an (apparent) magnitude of 9.54 and an absolute magnitude of 13.24. Using these we can derive the distance.

Equation 63 - Distance Modulus solved for d

d = 100.2(m - M + 5)
d = 100.2(9.54-13.24+5)
d = 100.26
d = 1.82 parsecs

For stars Outside our Galaxy

Larger distances mean measuring things like magnitude and parallax becomes difficult due to very small numbers and differences. Instead, there are some other techniques we can use to measure distance.

Using Cepheid Variables to Calculate Distance

For objects outside our galaxy, we can use the unique properties of a Cepheid variable star. These stars vary in brightness over time, in a frequency that is exactly in ratio to their apparent brightness, thus we can measure their frequency and brightness and compute how far away it is using distance modulus. Every galaxy has a bunch of Cepheid variables, so it's quite easy to map fairly accurate distances of all the galaxies we can see.

Cepheid variable stars pulsate predictably and a star's period (how often it pulsates) is directly related to its luminosity or brightness. Once the period of a distant Cepheid has been measured, its luminosity can be determined from the known behaviour of Cepheid variables. Then its absolute magnitude and apparent magnitude can be related by the distance modulus equation, and its distance can be determined. Cepheid variables can be used to measure distances from about 1kpc to 50 Mpc.

Cepheid variables can be used to measure distances from about 1kpc to 50 Mpc.

Equation 63 - Distance Modulus solved for d

A Cepheid variable star has a period of 3.7 days, and from this, we know its absolute magnitude is -3.1. Its apparent magnitude is 5.5.

Equation 65 - Worked example for Cepheid distance modulus

Using Supernovae to Calculate Distance

Supernovae is a supermassive explosion caused by a star imploding. There are different types of supernovas (see linked article for details. The one type we are interested in is a type 1a supernova. Type Ia supernovae are not common, instead, they are quite rare events with one supernova occurring in a galaxy every 100 years or so. Having said that, we do know that all type Ia supernovae always reach the same brightness at their peak, they are very close to a standard candle. This peak brightness corresponds to an absolute magnitude of -19.3.

The spectrum of a supernova and its brightness in different light colours (photometry) and how these evolve to determine the peak apparent magnitude. Knowing the absolute magnitude and apparent magnitude we can again use distance modulus to determine the distance to a supernova, and thus the distance to the galaxy it is within.

Using Redshift to Calculate Distance

The light from galaxies stretches out as the Universe expands, shifting it towards the colour spectrum's red end. Edwin Hubble discovered that redshift increases with distance. To work out how far away the furthest galaxies are we just analyse their light to determine redshift.