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Distance to Stars: Parallax and Brightness

Nearby stars are measured by parallax: the star seems to shift as Earth moves round the Sun, and distance in parsecs d = 1 / p (p in arcseconds). Farther stars are measured by brightness: the distance modulus m āˆ’ M = 5 log10(d / 10) links how bright a star looks (m) with how bright it really is (M). 1 parsec = 3.26 light-years.

šŸŽ¬ Step-by-step story

  1. Earth goes round the Sun. This gives us two viewing places, six months apart. A star close to us can be seen from both.
  2. Look at the star in January and in July. It seems to shift against the far stars. Half of that shift is the angle p, called parallax.
  3. Now bring the star closer. The angle p gets bigger. A far star gives a tiny p. So the distance is d = 1 / p.
  4. Brightness is a second ruler. The same star looks fainter when it is far. At 2 times the distance it looks 4 times fainter.
  5. Free play: move the slider. Read p, the distance in parsecs and light-years, and the number m āˆ’ M.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

šŸ¤” Common doubts, cleared

Why do we use six months and not one day?

In one day Earth moves too little. Six months put Earth on opposite sides of its orbit, the biggest possible baseline, so the shift is largest.

Why is p called half the shift?

The total shift is between two places on the opposite sides of the Sun. The angle at the Sun-Earth distance (one side) is half of it. Look at the two lines to the star.

Why can't parallax measure very far stars?

For a far star p becomes smaller than our telescope can measure. Slide the star far away and see p shrink.

If a star looks faint, is it always far?

No. It may be a weak star that is close. You need to know its real power too.

Why does a smaller magnitude mean brighter?

It is an old Greek scale: the brightest stars were called first magnitude. The habit stayed.

Why we need a ruler for stars

Stars are very, very far. Even the nearest one, after the Sun, is about 4 light-years away. A light-year is the distance light travels in one year, about 9.5 million million km. We cannot use a tape or a radar. So astronomers use the star's own light and the movement of Earth.

A bigger unit is the parsec (pc). 1 pc = 3.26 light-years. It is made for the parallax method, as you will see.

Method 1: Parallax (nearby stars)

Take a picture of a star in January. Take another in July. Earth has moved to the other side of its orbit. The near star seems to move a little against the far, fixed stars. Half of the total shift is the parallax angle p.

The angle is tiny, so we measure it in arcseconds (ā€³). 1 degree = 3600 arcseconds.

d (in parsecs) = 1 / p (in arcseconds)

Small p means big distance. Telescopes in space can measure p down to about 0.00001ā€³, so parallax works for stars thousands of parsecs away. From the ground it works only to a few hundred parsecs.

Method 2: Brightness (apparent and absolute magnitude)

Light spreads out. At twice the distance, the same light covers 4 times the area, so a star looks 4 times fainter. This is the inverse square law: brightness āˆ 1 / dĀ².

Astronomers use magnitude. A smaller number means a brighter star. Apparent magnitude (m) is how bright it looks from Earth. Absolute magnitude (M) is how bright it would look from 10 pc, a standard distance. M shows the star's real power.

The difference tells the distance:

m āˆ’ M = 5 log10(d / 10) (d in pc). This is the distance modulus.

You need M. It comes from the star's type, for example a star of a known kind has a known power.

Try it: parallax at home

  1. Stretch your arm and raise a thumb. Close your left eye and look at a far tree or building. Now close the right eye instead. Your thumb jumps.
  2. Bring the thumb halfway to your nose and repeat. The jump is bigger.
  3. Predict first, then check: will a near pencil or a far pencil show the bigger jump?

Then use the 3D above: slide the star closer and see p grow.

Key formulas and definitions

Worked examples

1. A star has parallax p = 0.25ā€³. Find its distance in parsecs and light-years.

d = 1 / p = 1 / 0.25 = 4 pc. In light-years: 4 Ɨ 3.26 = 13.0 light-years.

2. A star is 50 pc away. What is its parallax?

p = 1 / d = 1 / 50 = 0.02ā€³.

3. A star has m = 7 and M = 2. How far is it?

m āˆ’ M = 5. So 5 = 5 log10(d / 10), log10(d / 10) = 1, d / 10 = 10, d = 100 pc.

4. A star is 3 times farther than before. How many times fainter does it look?

Brightness āˆ 1 / dĀ², so it is 3Ā² = 9 times fainter.

5. A star with M = 0 is 1000 pc away. What is its apparent magnitude m?

m āˆ’ M = 5 log10(1000 / 10) = 5 log10(100) = 5 Ɨ 2 = 10. So m = 10 (too faint for the naked eye).

Common mistakes

Practice quiz

1. Parallax is measured by looking at a star from:
2. A star with a bigger parallax angle is:
3. The distance of a star with p = 0.2ā€³ is:
4. If m āˆ’ M = 5, the star is at:
5. 1 parsec is about:

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

How do astronomers measure the distance to a star?

For near stars they use parallax. For farther ones they compare how bright the star looks with its true power (the distance modulus).

What is a parsec?

The distance of a star whose parallax is 1 arcsecond. It equals 3.26 light-years.

What is the distance modulus?

It is m āˆ’ M, the difference between apparent and absolute magnitude. It equals 5 log10(d / 10), so it gives the distance d in parsecs.

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