# Measuring the Stars: Calculating Astronomical Distances

FROM THE LECTURE SERIES: INTRODUCTION TO ASTROPHYSICS

## How do astronomers locate objects in the universe? How do we measure these coordinates of a star? In maths, a good way to specify the location of an object in three dimensions is with Cartesian coordinates. One chooses an origin, and then picks three perpendicular directions to be the X, Y, and Z axes. But in astronomy, we usually don’t do that, for the same reason we don’t label points on the globe according to X, Y, and Z. Then what is the solution?

### The Spherical Polar Coordinate System

In astrophysics, the location of a body is specified using latitude, longitude, and elevation. This is the spherical polar coordinate system. It makes sense for the Earth because it is a sphere. And it makes sense for astronomers, too, not because the universe is a sphere, but because we’re trapped on a sphere. That makes it natural to use an Earth-centered coordinate system and to extend our concepts of latitude, longitude, and elevation up into the heavens.

Our ancient ancestors imagined the sky to be a glass ‘celestial sphere’ upon which the stars and planets were painted. Of course, this is not physically correct; the stars that we see forming constellations are actually at very different distances from the Earth. But the celestial sphere is still a useful fiction.

So, let’s imagine a giant, transparent sphere, centered on the Earth, marked with grid lines of latitude and longitude. The latitude lines tell us how far we are from the celestial equator, that’s the projection of the Earth’s equator up into the sky. And the longitude lines tell us how far east or west we are from the celestial equivalent of the prime meridian. That way when we look at a distant star, we can read off the star’s angular coordinates by seeing where it appears relative to the grid. That leaves only the third dimension, the distance to the star, the celestial equivalent of elevation, which is much trickier to measure.

### Measuring the Coordinates

Let’s start with the angular coordinates and take an example of two stars that happen to be located along nearly the same line of sight, from the Earth. They’ll appear close together on the celestial sphere. However, if they’re too close, they will blend together and appear as a single point of light, rather than two. So, what determines whether we can perceive the double star? As one might guess, it depends on how good our telescope is. But we can be more specific than that, and it’s worth going into detail because the question gets right at the basic dilemma of astronomy, which is that all we have is light.

With few exceptions, our only source of knowledge is the electromagnetic radiation that happens to hit the spinning ball of rock we live on. So, we need to understand the physics of light.

### Diffraction

Imagine a telescope as a big lens pointed at star straight overhead, which focuses the starlight into a tight spot on our camera. If there’s another star in a slightly different direction, then ideally, the lens would focus its light onto a different spot in the image. So, our image shows two dots, star 1 and star 2. The problem, though, is we can’t focus light into as small a point as we might like. Moreover, the stars blend together when the angle between them, Delta-theta, is on the order of lambda over D, where lambda is the wavelength of light, and D is the diameter of our lens, or mirror, or whatever we’re using to collect and focus light.

The reason why the stars blend together and, inevitably, blurr is owing to the phenomenon called diffraction, a consequence of the wave nature of light. Light is an electromagnetic wave, a traveling pattern of oscillating electric and magnetic fields. So, we can imagine the light from star 1 as an ocean wave, a traveling pattern of crests and troughs of electromagnetic energy, with a wavelength—a separation between crests—of lambda.

This wave then passes through the diameter D of our telescope and then a lens or a mirror responds to that pattern of energy by redirecting it toward a camera.

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

### Delta-theta

The energy gets redirected to a different position on the camera, if the starlight comes in from a different angle, tilted by Delta-theta. But if Delta-theta is really tiny, the wave energy is smeared out with a spatial extent of lambda. So, the telescope still sees a crest filling the opening. The optical system responds by directing the energy to the same spot on the detector.

As we increase Delta-theta, at what point do the waves from star 2 start to look different from star 1? It’s when we no longer have a crest extending across the opening. The tilt is large enough that there’s a crest at one end, and a trough at the other end. For this minimum value of Delta-theta, it’s at least possible for the optics to distinguish between the two waves.

### Blurry Images

Since the diffraction limit is proportional to lambda over D, the way to improve our angular resolution is to increase D, build a bigger telescope. And that does work, to a point.

But in practice, there are lots of other reasons why our images might be blurry, besides diffraction. Maybe our lens isn’t perfectly polished, or our mirror has defects. And then there’s the constantly fluctuating atmosphere, which scrambles the directions of light rays at the level of about an arcsecond, even at our best mountaintop observatories. So, even with a large telescope we usually can’t achieve the ultimate diffraction limit. That’s one reason why we launch telescopes into space, above the atmosphere.

However, the problem remains that even with a perfect telescope and no atmosphere, we can’t resolve details on angular scales smaller than of order lambda over D radians, where lambda is the wavelength of light, and D is the diameter of the telescope. Moreover, turbulence in the Earth’s atmosphere usually limits us to about an arcsecond, regardless of the size of our telescope.

### Common Questions about Measuring the Stars

Q: How can we read off a star’s angular coordinates?

The latitude lines tell us how far we are from the celestial equator, that’s the projection of the Earth’s equator up into the sky. And the longitude lines tell us how far east or west we are from the celestial equivalent of the prime meridian. That way when we look at a distant star, we can read off the star’s angular coordinates.

Q: What is caused due to diffraction?

The reason why the stars blend together and, inevitably, blurr is owing to the phenomenon called diffraction, a consequence of the wave nature of light.

Q: What limits us, regardless of the size of our telescope?

Turbulence in the Earth’s atmosphere usually limits us to about an arcsecond, regardless of the size of our telescope.