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

## Historically, the hardest problem in astronomy —is measuring the third dimension, the distance from Earth. Imagine we discover some new galaxy and measure its angular extent on the celestial sphere. But with only that information, we can’t tell if the galaxy is relatively small and nearby, or if it’s huge and billions of light years away. Thus, measuring the distance to an astronomical object, in order to make maps of the solar system, remains a crucial problem in astronomy.

### The Flux-Luminosity Relation

Imagine a situation in which we’re on the Earth, and an object is located a distance d from the Earth. The object has a true size of S, and an angular size alpha, that is, the rays arriving from opposite sides of the object have an angle alpha between them. Using the small angle approximation, alpha approximates to S divided by d, or equivalently, S equals alpha times d. The problem is that if all that we know is alpha, we can’t figure out S without knowing d.

A similar situation arises when we measure the brightness of a source. For a given brightness, we can’t tell if the source is intrinsically luminous, and very far away, or if it’s actually intrinsically faint and happens to be nearby. Suppose the luminosity is L. That’s the power, the energy per unit time, that our luminous object is pouring out into space. We can measure L in units of watts, for example. And if all that power spreads out as the radiation goes farther from the source.

The Earth is far away, at a distance of d. So, by the time the light reaches the Earth, it’s been spread out over a huge sphere of radius d. We can’t measure L directly. Instead, we’ve got a telescope with a certain collecting area, and we measure the power received by the telescope. We can then calculate the power per unit area, which we call the Flux, whereby F equals L divided by the surface area of the sphere, 4pi d-squared. That’s an important equation, the flux-luminosity relation: F equals L divided by 4pi d-squared. It’s another example of an inverse square law: The flux goes down as the inverse square of the distance.

### Using a Radar

If we measure F, we need to know d in order to deduce L. And we really want to know L, the true luminosity, if we want to figure out what’s physically going on to produce the radiation. The message again is that measuring the distance to an astronomical object is a crucial problem that we need to solve. And like any tough problem, it’s useful to break it down into manageable parts.

The first step is using a radar. Build a giant radio transmitter, aim it at a nearby planet, and fire. If we hit the target, and our receiver is sensitive enough, we can detect the echo, the reflected radio waves. The echo will be delayed by a time interval Delta-t equals 2d over c, where 2d is the round-trip distance, and c is the speed of radio waves, that is, it’s the speed of light. And since we know the speed of light, we can calculate d.

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

### Making Maps of the Solar System

With the world’s biggest radio transmitters, we can measure the distances to Mercury, Venus, Mars, and even some asteroids. That allows us to make maps of the solar system with a precision of a few parts in 10 billion. Even though it’s an astonishing level of detail, unfortunately, this method is limited to relatively nearby objects. We can show that the amplitude of the echo falls off like 1 over d to the 4th power. So, to go beyond the solar system, we need other techniques.

This brings us to the next step in our solution: parallax. Parallax is based on simple geometry. It can be understood by simply holding out our arm and raising a finger. Now, if we close our left eye and look at our finger and the scene in the background and then repeat the same after switching eyes and closing the right one, it will look like our finger just jumped! That’s because our right eye views from a slightly different angle, so it sees your finger projected against a different part of the background scene. That’s parallax. And if we measure that shift in angle, as well as the distance between our eyes, we could use trigonometry to calculate the distance to our finger.

### Parallax Equation

The parallax equation in astronomy takes advantage of the Earth’s motion around the Sun. In order to make the calculations, we take a picture of a nearby star, and then we wait six months for the Earth to go halfway around, and then we take another picture. That’s like closing our left eye and opening our right eye. The nearby star, like our finger, will appear to have shifted in position relative to the much more numerous background stars.

Following the the parallax equation, over the course of a year, nearby stars appear to move on the celestial sphere in little ellipses. These elipses are of 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.

If the star is directly above the plane of Earth’s orbit or off to the side somewhere, that doesn’t change the basic idea; it just means that the star will appear to move in an ellipse, rather than a circle, and the parallax angle is the semimajor axis of the ellipse. And if the star is right on the ecliptic—the projection of the Earth’s orbit onto the celestial sphere—it’ll go back and forth along a straight line.

### Common Questions about How the Solar System Is Mapped

**Q: How do we calculate power per unit area?**

The power per unit area is what we call the Flux, whereby F equals L divided by the surface area of the sphere, 4pi d-squared.

**Q: How can we measure the distances to Mercury and Venus?**

With the world’s biggest radio transmitters, we can measure the distances to Mercury, Venus, Mars, and even some asteroids.

**Q: What does parallax equation in astronomy take advantage of?**

The parallax equation in astronomy takes advantage of the Earth’s motion around the Sun.