8.4: Parallax and Naming Stars

Measuring Distance

The nearest star is hundreds of thousands of AU from Earth. Even so, we can, in principle, survey distances to the stars using the same technique that a civil engineer employs to survey the distance to an inaccessible mountain or tree, the method of triangulation.

Triangulation in Space

Depth perception requires two views of an object some distance away from each other. Humans and other animals that have eyes on the front of their head are able to perceive distance, while animals with eyes on opposite sides of their heads, such as cows or horses, have nearly 360 degrees of vision but decreased depth perception. For humans, depth perception fails for objects more than a few tens of meters away. In order to see the shift of an object a city block or more from you, your eyes would need to be spread apart a lot farther.

Let's see how surveyors take advantage of the same idea. In Figure \(\PageIndex{1}\), a surveyor has set up two observing stations some distance apart to measure the distance to a tree across a deep river. That distance, the distance between the two stations, line AB, is called the baseline. The direction to the tree at position C in relation to the baseline is observed from each station. The tree appears in different directions from the two stations. This apparent change in direction of the remote object due to a change in vantage point of the observer is called parallax.

Figure \(\PageIndex{1}\) : Triangulation. By measuring the angle to a distant tree from two points along a known baseline, the properties of the resulting triangle reveal the tree's distance. (CC BY 4.0; Fraknoi, et al. via Openstax Astronomy 2nd ed.) Accessible description of Figure \(\PageIndex{1}\).

The parallax is also the angle that lines AC and BC make. A knowledge of the angles at A and B and the length of the baseline, AB, allows the triangle ABC to be solved for any of its dimensions, such as the distance AC or BC. The solution could be reached by constructing a scale drawing or by using trigonometry to make a numerical calculation. If the tree were farther away, the whole triangle would be longer and skinnier, and the parallax angle would be smaller. The general rule is that the smaller the parallax, the more distant the object we are measuring must be.

The farther away an astronomical object lies, the longer the baseline has to be to give us a reasonable chance of making a measurement. Unfortunately, nearly all astronomical objects are very far away. To measure their distances requires a very large baseline and highly precise angular measurements. With the aid of telescopes, astronomers were able to measure the distances to the nearer planets and asteroids using Earth's diameter as a baseline. This is how the AU was first established. To reach for the stars, however, requires a much longer baseline for triangulation and extremely sensitive measurements. Such a baseline is provided by Earth's annual trip around the Sun.

Distances to Stars

As Earth travels from one side of its orbit to the other, it provides us with a baseline of 2 AU, or about 300 million kilometers. Although this is a much bigger baseline than the diameter of Earth, the stars are so far away that the resulting parallax shift is only detectable with telescopes. Figure \(\PageIndex{2}\) demonstrates how the orbit of Earth is used to measure the distance to stars. Two observations of the same star, the red dot, are collected 6 months apart, so that the orbit of Earth is the baseline. At these two positions, A and B, the star appears to be in two different positions relative to the background stars. That change in position can be measured and used to calculate the star's distance.

Even for the nearest stars, parallax angles are usually only a fraction of a second of arc. One second of arc (arcsec) is an angle of only 1/3600 of a degree. A coin the size of a US quarter would appear to have a diameter of 1 arcsecond if you were viewing it from a distance of about 5 kilometers. The first successful detections of stellar parallax were in the year 1838, when Friedrich Bessel in Germany, Thomas Henderson, a Scottish astronomer working at the Cape of Good Hope, and Friedrich Struve in Russia independently measured the parallaxes of the stars 61 Cygni, Alpha Centauri, and Vega, respectively. Even the closest star, Alpha Centauri, showed a total displacement of only about 1.5 arcseconds during the course of a year. Astronomers actually define parallax to be one-half the angle that a star shifts when seen from opposite sides of Earth's orbit, the angle labeled P in Figure \(\PageIndex{2}\). The reason for this definition is just that they prefer to deal with a baseline of 1 AU instead of 2 AU.

Figure \(\PageIndex{2}\) : Parallax. As Earth orbits the Sun, a nearby star appears to shift position against the background of more distant stars; parallax is defined as one half of this total angular shift, typically measured in arcseconds. (CC BY 4.0; Fraknoi, et al. via Openstax Astronomy 2nd ed.) Accessible description of Figure \(\PageIndex{2}\).

Units of Stellar Distance

With a baseline of one AU, how far away would a star have to be to have a parallax of 1 arcsecond? The answer turns out to be 206,265 AU, or 3.26 light-years. This is equal to 3.1 × 10¹³ kilometers. We give this unit a special name, the parsec (pc)—derived from “the distance at which we have a parallax of one second.” The distance (D) of a star in parsecs is just the reciprocal of its parallax (p) in arcseconds:

\[ D = \frac{1}{p} \nonumber\]

Thus, a star with a parallax of 0.1 arcsecond would be found at a distance of 10 parsecs, and one with a parallax of 0.05 arcsecond would be 20 parsecs away. A parsec was a useful unit of distance, but it is not as intuitive as the light-year. One advantage of the light-year as a unit is that it emphasizes the fact that, as we look out into space, we are also looking back into time. The light that we see from a star 100 light-years away left that star 100 years ago. What we study is not the star as it is now, but rather as it was in the past. The light that reaches our telescopes today from distant galaxies left them before Earth even existed.

In this text, we will use light-years as our unit of distance, but many astronomers still use parsecs when they write technical papers or talk with each other at meetings. To convert between the two distance units, just bear in mind: 1 parsec = 3.26 light-year, and 1 light-year = 0.31 parsec.

The Nearest Stars

Other than the Sun, no known star is within 1 light-year or even 1 parsec of Earth. The stellar neighbors nearest the Sun are three stars in the constellation of Centaurus. To the unaided eye, the brightest of these three stars is Alpha Centauri, which is only 30^○ from the south celestial pole and hence not visible from the mainland United States. Alpha Centauri itself is a binary star—two stars in mutual revolution—too close together to be distinguished without a telescope. These two stars are 4.4 light-years from us. Nearby is a third faint star, known as Proxima Centauri. Proxima, with a distance of 4.3 light-years, is slightly closer to us than the other two stars. If Proxima Centauri is part of a triple star system with the binary Alpha Centauri, as seems likely, then its orbital period may be longer than 500,000 years.

Proxima Centauri is an example of the most common type of star, and our most common type of stellar neighbor. Low-mass red M dwarfs make up about 70% of all stars and dominate the census of stars within 10 parsecs (33 light-years) of the Sun. For example, a recent survey of the solar neighborhood counted 357 stars and brown dwarfs within 10 parsecs, and 248 of these are red dwarfs. These stars only produce a fraction of the Sun's light, and nearly all of them require a telescope to be detected.

The nearest star visible without a telescope from most of the United States is the brightest appearing of all the stars, Sirius, which has a distance of a little more than 8 light-years. It too is a binary system, composed of a faint white dwarf orbiting a bluish-white, main-sequence star. It is an interesting coincidence of numbers that light reaches us from the Sun in about 8 minutes and from the next brightest star in the sky in about 8 years.

Measuring Parallaxes in Space

The measurements of stellar parallax were revolutionized by the launch of the spacecraft Hipparcos in 1989, which measured distances for thousands of stars out to about 300 light-years with an accuracy of 10 to 20%. In Figure \(\PageIndex{3}\), the data from Hipparcos show that there the stars within 300 light years have a large range of temperatures and luminosities. However, even 300 light-years are less than 1% the size of our Galaxy's main disk.

Figure \(\PageIndex{3}\) : H-R Diagram of Stars Measured by Gaia and Hipparcos. This H-R diagram of 16,631 stars with well-measured parallaxes shows most stars falling along a diagonal main sequence, with a separate clump of red giant stars above and to the right. (CC BY 4.0; Fraknoi, et al. via Openstax Astronomy 2nd ed.) Accessible description of Figure \(\PageIndex{3}\).

In December 2013, the successor to Hipparcos, named Gaia, was launched by the European Space Agency. It completed its observations in 2025. Gaia measured the position and distances to almost one billion stars with an accuracy of a few millionths of an arcsecond. Gaia's distance limit extended well beyond Hipparcos, studying stars out to 30,000 light-years, 100 times farther than Hipparcos, covering nearly 1/3 of the galactic disk. Gaia has measured proper motions for thousands of stars in the halo of the Milky Way.

Naming Stars

You may be wondering why stars have such a confusing assortment of names. Just look at the first three stars to have their parallaxes measured: 61 Cygni, Alpha Centauri, and Vega. Each of these names comes from a different tradition of designating stars.

The brightest stars have names that derive from the ancients. Some are from the Greek, such as Sirius, which means “the scorched one”—a reference to its brilliance. A few are from Latin, but many of the best-known names are from Arabic because much of Greek and Roman astronomy was “rediscovered” in Europe after the Dark Ages by means of Arabic translations. Vega, for example, means “swooping Eagle,” and Betelgeuse (pronounced “Beetle-juice”) means “right hand of the central one.”

In 1603, German astronomer Johann Bayer (1572-1625) introduced a more systematic approach to naming stars. For each constellation, he assigned a Greek letter to the brightest stars, roughly in order of brightness. In the constellation of Orion, for example, Betelgeuse is the brightest star, so it got the first letter in the Greek alphabet—alpha—and is known as Alpha Orionis. (“Orionis” is the possessive form of Orion, so Alpha Orionis means “the first of Orion.”) A star called Rigel, being the second brightest in that constellation, is called Beta Orionis. Figure \(\PageIndex{4}\) is a star map of the constellation Orion. The bright stars in Orion still have their original names, but the other stars are labeled with Greek letters. Since there are 24 letters in the Greek alphabet, this system allows the labeling of 24 stars in each constellation, but constellations have many more stars than that.

Figure \(\PageIndex{4}\) : Stars in Orion. Image (a) shows the constellation Orion as seen in the sky, while Diagram (b) labels its brightest stars with both their proper names and Bayer's Greek-letter designations, along with three nebulae cataloged by Charles Messier. (CC BY 4.0; Fraknoi, et al. via Openstax Astronomy 2nd ed.) Accessible description of Figure \(\PageIndex{4}\).

In 1725, the English Astronomer Royal John Flamsteed introduced yet another system, in which the brighter stars eventually got a number in each constellation in order of their location in the sky or, more precisely, their right ascension. In this system, Betelgeuse is called 58 Orionis and 61 Cygni is the 61st star in the constellation of Cygnus, the swan. There is also a completely different system for keeping track of stars whose luminosity varies, and another for stars that brighten explosively at unpredictable times. Today's astronomers often refer to stars by their precise locations in the sky rather than by their names or various catalog numbers.

Examples and Exercises

Example: How Far Is a Light-Year?

A light-year is the distance light travels in 1 year. Given that light travels at a speed of 300,000 km/s, how many kilometers are there in a light-year?

Solution

We learned earlier that speed = distance/time. We can rearrange this equation so that distance = velocity × time. Now, we need to determine the number of seconds in a year.

There are approximately 365 days in 1 year. To determine the number of seconds, we must estimate the number of seconds in 1 day.

We can change units as follows (notice how the units of time cancel out):

\[ 1 \text{ day} \times 24 \text{ hr/day} \times 60 \text{ min/hr} \times 60 \text{ s/min} = 86,400 \text{ s/day} \nonumber\]

Next, to get the number of seconds per year:

\[ 365 \text{ days/year} \times 86,400 \text{ s/day} = 31,536,000 \text{ s/year} \nonumber\]

Now we can multiply the speed of light by the number of seconds per year to get the distance traveled by light in 1 year:

\[ \text{distance} = \text{velocity} \times \text{time} = 300,000 \text{ km/s} \times 31,536,000 \text{ s} = 9.46 \times 10^{12} \text{ km} \nonumber\]

That's almost 10,000,000,000,000 km that light covers in a year. To help you imagine how long this distance is, we'll mention that a string 1 light-year long could fit around the circumference of Earth 236 million times.

Exercise: How Far Is a Light-Year?

The number above is really large. What happens if we put it in terms that might be a little more understandable, like the diameter of Earth? Earth's diameter is about 12,700 km.

Answer

 

\( 1 \text{ light-year} = 9.46 \times 10^{12} \text{ km} = 9.46 \times 10^{12} \text{ km} \times \frac{1 \text{ Earth diameter}}{12,700 \text{ km}} = 7.45 \times 10^8 \text{ Earth diameters} \) That means that 1 light-year is about 745 million times the diameter of Earth.

Example: Calculating the Diameter of the Sun

To estimate the diameter of the Sun we can use parallax. The Sun spans about half a degree on the sky. A full circle has 360^○. The circumference of the circle centered on Earth and passing through the Sun is given by:

\[ \text{circumference} = 2\pi \times 93,000,000 \text{ miles} \nonumber\]

Then, the following two ratios are equal:

\[ \frac{0.5°}{360°} = \frac{\text{diameter of Sun}}{2\pi \times 93,000,000} \nonumber\]

Calculate the diameter of the Sun. How does your answer compare to the actual diameter?

Solution

To solve for the diameter of the Sun, we can evaluate the expression above.

\[ \text{diameter of the sun} = \frac{0.5°}{360°} \times 2\pi \times 93,000,000 \text{ miles} = 811,577 \text{ miles} \nonumber\]

This is very close to the true value of about 848,000 miles.

Exercise: Distance to a Star

What is the distance to a star that has a parallax of 1 arcsec? (Hint: Remember that the parallax angle is defined by 1 AU, not 2 AU, and that 3600 arcseconds = 1 degree.)

Answer

 

206,265 AU

Further Exploration

• Use the Astronomical Parallax model to explore how the Earth's motion around the Sun causes nearby stars to appear to wobble back and forth compared to background stars.
• The European Space Agency (ESA) maintains a Gaia mission website where you can learn more about the Gaia mission.
• To learn more about Hipparcos, explore the Hipparcos mission website.