**By **Joshua Winn**, **Princeton University

## In astrophysics, while calculating distance, the astronomical system of units are angular measurement. In ordinary life, we measure angles in degrees, with a right angle being 90°, and 360° going around the whole circle. In astronomy, an angle is measured in ‘natural units’ or radians. What other units are used while measuring distances to stars?

### AU, Arcseconds and Parsec

In astronomy, when we talk about units, as is the case with the diffraction limit, alpha equals 1 AU over d works when alpha is expressed in radians. If an object of size S is located a distance d from Earth, its angular size is S over d radians. But what if we want to use arcseconds? Well, one radian works out to be 206,265 arcseonds. So, if we’re expressing alpha in arcseconds, the right side of our parallax equation becomes 206,265 AU over d.

The tradition at this point is to also do a little sleight of hand and define a new unit of distance, the parallax-second, or parsec, equal to 206,265 AU. That way the numbers are easier: d equals one parsec divided by alpha in arcseconds. For example, the bright star Sirius shows a parallax of 0.38 arcseconds, so its distance in parsecs is 1 over 0.38, or 2.6.

Out of these, the parsec is a handy unit for measuring the distances between stars, and it happens to have the same order of magnitude as the light-year. One parsec is about 3.3 light years. Over the course of a year, nearby stars appear to move on the celestial sphere in little ellipses of with a maximum angular size, in arcseconds, equal to one divided by the distance in parsecs. So, a star 10 parsecs away has a parallax angle of a 10th of an arcsecond.

### Units for Angles

When we talk about units for angles: lambda over D is a dimensionless number, it’s a length divided by a length. That means the angle is measured in ‘natural units’ or radians. In ordinary life, we measure angles in degrees, with a right angle being 90°, and 360° going around the whole circle. However, it’s simpler to calculate in radians, the system in which a right angle is pi over 2, and the whole circle is 2pi. For example, if we’re observing visible light with a 10-centimeter telescope, then the diffraction limit comes out to be 6 times 10 to the minus-6 radians. This, according to the Babylonian system, would be about 1.2 arcseconds.

To put that into perspective, our eyes probably have an angular resolution of around 50 or 100 arcseconds. And the full Moon has an angular diameter of half a degree, or 1800 arcseconds.

### Best Parallax Measurements

However, when it comes to measuring distances to stars, parallax is by far the most reliable method. Yet, as the distance gets larger, eventually the parallax angle becomes too small to measure, if for no other reason than the diffraction limit. Right now, our best parallax measurements come from a space telescope, called *Gaia*, which was launched by the European Space Agency in 2013.

*Gaia* measured parallaxes as small as a ten-thousandth of an arcsecond; that’s good enough to make maps of the galaxy out to 10,000 parsecs, 10 kiloparsecs. That is impressive. But to go beyond our galaxy—and there’s a lot beyond our galaxy—we need to take another step in the quest to measure distances.

This quest is actually a long story, and it’s central to the history of astronomy. As of now, there are two best ways to measure the distances to very remote objects. They both rely on the flux-luminosity relation: F equals L over 4pi d-squared, wherein, L is the power that an object emits, and F is the power per unit area measured by Earthlings. So, we can measure F, but we can’t figure out L unless we also know the distance, d.

But suppose there were some light source out there for which we already knew L. In that case, one could calculate the distance by re-arranging the flux-luminosity equation: d equals square root L over 4pi F.

This article comes directly from content in the video seriesIntroduction to Astrophysics. Watch it now, on Wondrium.

### Cepheid Variables

Unfortunately, stars don’t come with manufacturer’s labels! Yet, in some special cases, we can figure out the wattage even without a label. Let’s take the example of a famous star called *RS Puppis*, surrounded by a cloud of material that was ejected by the star. It’s an example of a category of stars called Cepheid variables. They are called ‘Cepheid’ because the first known example was in the constellation of Cepheus, and ‘variable’ because these stars vary in brightness.

They pulse, they get brighter and fainter, in an endless cycle, with a period—the time for a full oscillation—that’s typically a few weeks. The important thing is that the average luminosity of a Cepheid can be predicted accurately from the period of the pulsations. This was first discovered by Henrietta Leavitt, in 1912. Stars that pulse more slowly are intrinsically more luminous.

One reason we know this to be true is that some Cepheids are close enough for parallax measurements, so we can determine their luminosities. And among that collection, one can easily observe that L is linked to the pulsation period, P. A schematic chart of luminosity versus period shows this increasing relationship.

So, if we spot a Cepheid a megaparsec away, in some other galaxy, we can’t measure its parallax, but we can measure its pulsation period. We just monitor the flux, and see how long it takes to rise and fall. We can then use the period-luminosity relationship to determine L, and boom, we’ve got both F and L, and we can calculate d, figuring out the distance to the galaxy where the Cepheid resides.

### Common Questions about the Astronomical System of Units

**Q: Why is the parsec a handy unit for measuring distances between stars?**

The parsec is a handy unit for measuring the distances between stars, and it happens to have the same order of magnitude as the light-year.

**Q: Were does our best parallax measurements come from?**

Our best parallax measurements come from a space telescope, called *Gaia*, which was launched by the European Space Agency in 2013.

**Q: Why are Cepheid variables are called ‘Cepheid’?**

Cepheid variables are called ‘Cepheid’ because the first known example was in the constellation of Cepheus, and ‘variable’ because these stars vary in brightness.