Parallax: measuring a star with Earth’s two eyes
Nobody has gone out to measure the distance to a star. It is deduced, and the method is the one you use without thinking every time you close one eye and then the other.
What you are looking at
- On the left, what happens: Earth travelling round its orbit. On the right, the only thing you actually see through a telescope.
- The background stars do not move a single pixel. The one dancing is the near one, and only because you changed position.
- The two dashed circles mark the ends of the travel. That separation is twice the parallax, and it is all that gets measured.
01Do it right now with your thumb
Hold your arm out with your thumb up and close one eye. Now switch eyes: your thumb jumps against whatever is behind it.
The background did not move. Neither did your thumb. The only thing that changed is where you are looking from, about six centimetres. And the closer you bring your thumb, the bigger the jump.
That jump is measurable, and if you knew the spacing between your eyes you could work out how far away your thumb is without touching it. That is all parallax is.
02Earth’s two eyes are January and July
For a star, six centimetres is useless. But Earth moves: in six months it crosses its orbit from one side to the other, giving a separation of two astronomical units, about 300 million kilometres.
You photograph the same star in January and in July, and measure how far it shifted against the background of distant stars, which are so far away they do not move at all.
03The distance comes out of a division
Half that shift is the angle we call the parallax. And there is a unit built to fit the problem: the parsec is, by definition, the distance at which a star would show a parallax of one arcsecond.
With that definition the arithmetic becomes trivial: distance in parsecs is one divided by the parallax in arcseconds. That is it.
04Why this is so hard in practice
Proxima Centauri, the closest star to the Sun, has a parallax of 0.77 arcseconds: less than a thousandth of a degree. That is the width of a coin seen from five kilometres away.
The animation exaggerates the angle about ten thousand times, because at real scale the two lines of sight would be the same line and the right-hand panel would sit completely still.
This is why parallax was not successfully measured until 1838, two hundred years after Galileo. And why the Gaia satellite, which measured almost two billion stars this way, is one of the most precise instruments ever built.
Where this is used
Your own eyes
Your brain does this all day: it compares the two images and extracts depth from the difference. It is why pouring water into a glass is hard with one eye covered.
The GPS in your phone
The same idea with times instead of angles: the phone measures its distance to several satellites and crosses the lines. Triangulation is triangulation, here or in space.
The scale of the whole universe
Parallax is the first rung of the cosmic distance ladder. It calibrates Cepheids, which calibrate supernovae, which give the expansion of the universe. Move the rung and everything above it moves.
Maps made with drones
Photogrammetry builds a 3D model of terrain from photos taken at different points. The same object seen from two places gives you the height of a hill.
Explaining it to a child
The idea, in plain wordsWhen you close one eye and then the other, nearby things seem to jump while far-away things stay still. Astronomers use that jump to work out how far away a star is, without going to measure it.
- Have them hold an arm out with the thumb up and close the left eye. Note what the thumb lines up with across the room.
- Now switch eyes without moving the arm. The thumb "jumps" to a different spot on the background.
- Have them bend the elbow to bring the thumb closer to their face and repeat. The jump is much bigger.
- Try it with something across the street: it barely jumps at all.
The question to leave them withIf near things jump a lot and far things barely jump, how far away would something have to be to never jump at all?
For the classroom
Ten minutes with the animation, ready to take into a room. Free under CC BY 4.0, like everything else here.
- Have them hold an arm out with the thumb up and close one eye, then the other, alternating. The thumb jumps against the background.
- Now with the arm bent, thumb close to the face. It jumps much more. Ask: "what changed?" The eyes are the same distance apart; what changed is the distance to the thumb.
- Play the animation. The Earth's orbit is "the gap between the eyes", and the star is the thumb.
- Close with the question that matters: if a star is so far that you can no longer see it jump, what do you do?
Why wait six months between the two measurements?
Answer: Because in six months the Earth moves from one end of its orbit to the other, giving the largest possible gap between the two "eyes": about 300 million kilometres. Wait less and the angle is smaller, so the error weighs more.
The farther the star, is the angle larger or smaller?
Answer: Smaller. That is why the method has a limit: eventually the angle is smaller than the instrument's error and the measurement stops saying anything.
If this method cannot reach galaxies, what is it good for?
Answer: It is the first rung of the ladder. Parallax measures nearby stars, and those calibrate the methods that reach farther. If the first rung is wrong, everything above it is shifted.
Many will say the thumb "moves". It did not: what moved is the point they are looking from. That distinction —between what changes and where you look from— is the whole concept, and it is worth dwelling on even though it sounds like a point about wording.
FísicaEterna classroom guide · CC BY 4.0 · fisicaeterna.cl
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The code behind it
It is written in Remotion, which renders video by writing React. The bar and the number are not drawn next to the figure: they come out of the calculation on the same geometry you are watching. If the drawing lies, the number lies with it.
Licence
This animation is under CC BY 4.0: you may copy, redistribute, adapt and use it commercially, as long as you credit the source.
Suggested attributionParallax: measuring a star with Earth’s two eyes — FísicaEterna (fisicaeterna.cl), CC BY 4.0