Determining the Age, Distance & Relative Motion of Stars

Instructor: Damien Howard

Damien has a master's degree in physics and has taught physics lab to college students.

Even though we could never travel to a star in the sky to study it, astronomers still find ways to learn a lot about stars. In this lesson, you'll learn how they can find a star's age, distance, and relative motion to us.

Exploring Our Galaxy

If you look up at the sky on a clear night you can see thousands of stars. Even the closest of these stars are so far away that our fastest current spacecraft would take thousands of years to reach it. Yet still, astronomers can gather a lot of information on these stars just based on the specs of light we see in the sky.

You can see more stars than you can count in the sky at night
stars at night

Some examples of the data astronomers gather include a star's age, distance, and motion. By the end of this lesson, you'll understand how they are able to get this data on stars that seem so impossibly far away to us.

Age of Stars

One of the main methods for determining the age of stars is to look at star clusters instead of individual stars. As stars age, they go through different stages of life. What we think of as a normal star, like our Sun, is one in the main sequence of its life. Once a star uses up most of its fuel it leaves the main sequence, and becomes either a red giant, white dwarf, black hole, or goes supernova.

An example of a star cluster that astronomers might analyze
star cluster

In a star cluster, all the stars were born at the same time. We can use this to determine the age of the stars in the cluster. Their age will be equal to the lifetime of the stars that are just about to leave the main sequence, which is the most massive main sequence stars. This is called the main sequence turnoff method for determining the age of stars.

The lifetime (T) of a main sequence star is related to its mass (M) through the following formula.


lifetime formula


Astronomers can't measure a star's mass directly, but luckily there is a direct relationship between mass and luminosity (L) which they can measure.


luminosity formula


To see how these formulas let us determine the age of a star, let's imagine a star cluster with a star about to leave its main sequence that has a luminosity 10^2 solar units. We can first use this to find its mass:


age part1


From its mass that star's lifetime can be found.


age part2


Since this is the star about to leave its main sequence, it means the stars in the cluster are all around 317 million years old.

Distance to Stars

Measuring the distance (d) to stars is more straightforward than finding their age. To do this we use stellar parallax (p), which is the phenomenon where the Earth's rotation around the sun causes a star to appear to shift positions in the sky compared to a star further away in the background.


Visual representation of stellar parallax
stellar parallax


The parallax of a star is inversely related to the distance the star is from us.


distance formula


For example, one of our closest stars, Bernard's Star, has a parallax of approximately 0.55 arc seconds. This gives us the following distance from us.


distance example


This method of judging a star's distance works best for close stars. If a star is too far away and there are no background stars to compare it to, this method can't be used.

Motion of Stars

Parallax makes stars appear to move in the sky, but this isn't the actual movement of the star. The actual motion of a star relative to us can be broken down into two parts, tangential and radial velocity. To the naked eye stars at night seem stationary, but astronomers can track the slight movement of stars in the sky. We call this the proper motion of stars. From the proper motion (u) of a star and its distance from us, astronomers can find it's tangential velocity (vtan), the perpendicular velocity component of the star compared to us.


tangential velocity formula


Tracking radial velocity is more complicated than tangential velocity. The radial velocity of a star is its velocity component directly towards or away from us. This can't be tracked directly, as a star won't appear to move in the sky due to radial velocity. Astronomers can instead find a star's radial velocity by looking at the wavelength of light given off by it. If the star is moving away from us the wavelength is stretched a.k.a. redshifted, and if it's moving away toward us its wavelength is compressed a.k.a. blueshifted. This changing of wavelength is called a Doppler shift. The star's radial velocity (vrad) can be compared to its wavelength through the following formula.


radial velocity formula


Here Δλ is the shifted wavelength of light (λshift) minus the unshifted wavelength (λ), and c is the speed of light, 3x10^5 km/s.

Once you know a star's tangential and radial velocities, you can find it's total velocity (vtot) using the Pythagorean theorem.


This diagram shows why we can use the Pythagorean theorem to find total velocity
total velocity diagram



total velocity formula


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