How Far the Moon

Copyright © Karl Dahlke, 2023

How far away are the stars? That's a great question, don't you think? After all, we can't reach out and measure them.

Take a step back and ask the same question about the planets, or even our own moon. These distances we can measure. We've sent probes to all the planets, and many moons and comets. Since it takes years to get there, we know where the planets will be years from now, so that our probes reach their destinations. We know where everything is, and has been, and will be, in our own solar system, but what about beyond? How can we possibly know that?

Rewind to 1690, when we were finally able to ascertain, with some accuracy, the distance to the moon. It wasn't by going there, or bouncing radar signals off the surface. How did we do it?

The best technology in 1790, still a century off, was the Brothers Montgolfier and their hot air balloon. They could get perhaps a thousand feet in the air, and once they didn't die doing it, King Louis XVI took the credit, a scientific triumph for France. That was the height of science and technology in 1790. Surely nobody was measuring the distance to the moon in 1690. But we were, and with high precision.

We had sailed the world by then, and actually, mostly, didn't get lost. We knew how large the earth was, to at least a few kilometers. And we had a stopwatch that could time how fast an object fell. This in turn gives us the strength of gravity at the earth's surface. From there, Isaac Newton put it all together in his Principia Mathematica, the most important book in the history of science. It included an explanation of gravity, just another chapter, just something we'd been speculating on for 3000 years. Then there's all the math that follows from an understanding of gravity, the orbits of moons around planets, the orbits of planets around the sun, and why they all have to be ellipses. The calculus that he invented, just another chapter in his book, the calculus to do this, is the most beautiful thing you've ever seen in your life. Once it's done, you don't need the high math any more. The high math derives the formula, but if you're willing to assume a circular orbit, to keep things simple, a high school student can understand the formula. Gravity on earth, combined with its circumference, implies an at-the-surface orbit, if such were possible, of 85 minutes. Thus our close satellites, and the space station, just above the atmosphere, orbit the earth every 90 minutes, more or less. Now apply Kepler's third law, as verified and proved by Newton. This relates the height of an orbit to the time it takes to go around. We know how long it takes the moon to go round the earth. People have been tracking that for 6000 years. Lots of calendars are based on it. It's 27.32 days to go around the earth. That's a lot more than 90 minutes, so the moon is a lot farther away. do the math and boom, 239 thousand miles away. So yes, they knew the mean distance to the moon, probably to a precision of 4 digits, 238,800 miles, centuries before we could get there. And when we went there in 1969, it was exactly where it was suppose to be.

We left a reflector on the moon, a fancy mirror. Send a laser pulse to the moon and it bounces back to us. Time the pulse, divide by the speed of light, and we can measure distance to the moon to a fraction of an inch. In fact we now know it is moving away from us at 3 cm, a little more than an inch, each year. I could explain why, but that will have to wait for another day. Newton explained it, why the moon must be moving, very slowly, away from the earth. Just another chapter in his book. It's connected to the tides, something else we didn't understand for thousands of years, yet another chapter in his book.


Here's an experiment you can do at home. Hold your hand in front of you at arms length, and point your index finger straight up. Close your left eye and note the position of your finger against the back wall, or the trees if you are outside. Now close your right eye. Your finger seems to move. Close one eye, and then the other, and repeat, and note how your finger jumps back and forth. (The effect is more pronounced if the background is far away.) Each eye sees the finger from a slightly different angle, although the background looks the same, since it is farther away. This is called parallax.

With both eyes open, your brain blends the two images into one. It notes the different positions of the finger, and after some complex calculations, your brain tells you that your finger is in front of your face, closer than the background. Thus your visual cortex creates a 3 dimensional image of the world. You might not realize it, but your brain is capable of performing trigonometry.

Move your finger even closer to your face and repeat the experiment.

This phenomenon has been employed to create 3D movies. Two cameras take the place of two eyes, recording the scene from slightly different angles. These images are then put onto one screen, using different colors, or polarizations. Special glasses then allow the left image to pass through to your left eye, and the right image to pass through to your right eye. Your eyes see what the two cameras recorded when the movie was made. It looks 3 dimensional to you, just as it did to the director. It's a fun little trick.

How Far the Star

Parallax can be used to determine the distances to our nearest stars. These are 100 million times as far away as the moon, so we're going to need two eyes far apart. To achieve this, we take advantage of the earth's orbit. Take a picture of the stars, then wait six months, when the earth has gone halfway around the sun, and take another picture of the same stars. The closest stars should shift. This is like having two eyes 186 million miles apart. Still, the technology would not support these measurements until 1832-1838, for the stars alpha Centauri, Vega, and 61 Cygni. These are just a few light years away, which seems small relative to the size of the cosmos, but was unimaginably large in its day. (remember that a light year is about 6 trillion miles, or 10 trillion kilometers.) People had no idea their universe was so vast. In fact these results were met with some skepticism, but they have been replicated again and again over the ensuing decades.

In 1989, the Hipparcos satellite was launched, primarily for obtaining parallaxes of nearby stars. It measured distances out to fifteen hundred light years, a little more than one percent of the diameter of our Milky Way Galaxy. The Hubble telescope can establish the distance of certain bright stars up to ten thousand light years away. Therefore, we know, with high certainty, the distances to the stars in our corner of the galaxy.

Bright objects in other galaxies, or the galaxies themselves, cannot be measured by parallax. They are millions or billions of light years away. Their distances are established by red shift, which is beyond the scope of this book. However, the use of red shift assumes a calibration, which is made possible, in part, by determining the distances to the near by stars by parallax, that being an independent method of measurement.