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Binary Stars: Weighing Stars with Orbits

A binary star is a pair of stars bound by gravity, circling their common centre of mass. The heavier star is closer to that point and moves in a smaller circle: r1 / r2 = m2 / m1. Kepler's third law gives the total mass: m1 + m2 = a³ / P² (a in AU, P in years, masses in Suns). This is how astronomers weigh stars.

🎬 Step-by-step story

  1. Two stars are bound together. They go round and round, like two dancers holding hands.
  2. They do not circle each other's centre. They circle a balance point in between: the centre of mass. It is the black dot.
  3. Make star A heavier. The balance point slides towards A, and A moves in a smaller circle. Light star, big circle.
  4. Move the stars farther apart. The orbit grows and one round trip takes much longer.
  5. Free play: the total mass is a³ / P². Change a, mA and mB and read the period. Measure a and P in the sky and you can weigh both stars.

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

🤔 Common doubts, cleared

Do the stars really circle each other?

Both circle the balance point. Neither star is fixed. See the black dot and the two rings.

Why is the heavy star's circle smaller?

The balance point is nearer the heavy star, like a seesaw. Slide A to heavier and watch the dot move towards it.

Why does the period get longer when the stars are far apart?

The pull is weaker and the path is longer. Both make one trip slower. Watch the speed when a grows.

Why do we need the period to weigh a star?

Gravity pulls the stars round. A stronger pull (more mass) makes a faster orbit, so the period tells the mass.

What is a binary star?

Many stars are not alone. A binary star is two stars held together by gravity, going round each other. Even the Sun is unusual: it has no partner. Binaries are common, so they are important for the study of stars.

Some binaries can be seen as two separate dots in a telescope (visual binaries). Others are too close to separate but show their motion in the changing light (spectroscopic binaries).

The balance point (centre of mass)

Each star pulls the other. Because of this, both stars circle a point between them, called the centre of mass. The point is on the line joining the stars.

If the separation is a and the stars have masses m1 and m2, their distances from the balance point are r1 and r2, with

r1 / r2 = m2 / m1 and r1 + r2 = a.

The heavier star is nearer the balance point and moves in the smaller orbit. Measuring the two orbit sizes gives the ratio of the masses.

Weighing the stars: Kepler's third law

The time for one round trip is the period (P). Kepler's third law for two stars says:

m1 + m2 = a³ / P²

Use these units: a in AU (1 AU = Earth to Sun distance), P in years, masses in Suns. Check with the Earth and Sun: a = 1, P = 1, so the total mass = 1 Sun. It works.

So you measure a and P, find the total mass, then split it with the ratio r1 / r2. That is how most star masses are known.

Try it: a spinning pair

  1. Tie two different balls to the ends of a stick and balance it on your finger. The balance point is closer to the heavy ball.
  2. Spin the stick slowly. The light ball travels farther.
  3. In the 3D, set A = 3 and B = 0.5. Predict: which star has the big circle? Then check.

Key formulas and definitions

Worked examples

1. Two stars are 4 AU apart and take 2 years for one orbit. Find the total mass.

m1 + m2 = a³ / P² = 4³ / 2² = 64 / 4 = 16 Suns.

2. Check the formula with Earth and Sun (a = 1 AU, P = 1 year).

a³ / P² = 1 / 1 = 1 Sun. Correct.

3. Stars of 3 and 1 Suns are 4 AU apart. How far is each from the balance point?

r1 (the 3-Sun star) = a × m2 / (m1 + m2) = 4 × 1 / 4 = 1 AU. r2 (the 1-Sun star) = 3 AU.

4. Two stars of total mass 8 Suns are 2 AU apart. Find the period.

P² = a³ / M = 8 / 8 = 1, so P = 1 year.

5. Star A is 2 AU and star B is 6 AU from the balance point. The period is 4 years. Find both masses.

a = 8 AU. Total mass = 8³ / 4² = 512 / 16 = 32 Suns. Ratio mA : mB = r2 : r1 = 6 : 2 = 3 : 1, so mA = 24 Suns and mB = 8 Suns.

Common mistakes

Practice quiz

1. Two stars in a binary go round:
2. The heavier star of a binary moves in the:
3. For a = 2 AU and P = 1 year, the total mass is:
4. If the stars are moved farther apart (same masses), the period:
5. The ratio r1 / r2 equals:

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

What is a binary star?

Two stars bound by gravity that go round their common centre of mass.

How do we find the mass of a star?

Using a binary. Measure the separation a and period P, then total mass = a³ / P² in Suns (with a in AU and P in years).

Is the Sun a binary star?

No. The Sun has no partner star, though many stars do.

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