You may have learned that the Moon goes around the Earth and the Earth goes around the Sun. But that’s not quite right. Strictly speaking, the Moon and Earth orbit *each other*.

The **barycenter **is the point in space around which two objects orbit. For the Moon and Earth, that point is about 1000 miles (1700 km) beneath your feet, or about three-quarters of the way from the Earth’s center to its surface. That means the Earth actually wobbles around a point deep in its interior, pulled around by the Moon.

The same thing happens for the Sun and all the planets. Usually we think of the Sun sitting still in the center of the solar system while the planets whiz around. But, in reality, the Sun is wobbling too, orbiting the barycenter of the solar system. Right now, that point is about halfway between the Sun’s center and its surface. But because all eight planets are constantly in motion, the solar system’s barycenter wanders over time. In 2023, it will be way above the Sun’s surface! Later in 2030, it will return to a point closer to the sun's center.

In other systems, the barycenter is permanently located in the space between the two objects. Pluto and its largest moon, Charon, orbit around a point nearly 600 miles (960 km) *above* the surface of Pluto! This has led some astronomers to think of Pluto and Charon not as a dwarf planet and moon, but rather as a **binary planet**, a term sometimes used to describe planets that orbit a point located beyond the surface of the larger planet.

In this project, we’re going to make some models of planets and moons and see if we can figure out what determines the location of their barycenters.

### Problem

Where are the barycenters of orbiting bodies (moons, planets, stars)?

### Materials

- Modeling clay
- Food scale
- Long wooden dowel
- String
- Index card
- Hole puncher
- Ruler

### Procedure

- Tie one end of the string to a hook in the ceiling, the edge of a table, or any other point suspended above the ground.
- Cut a rectangular strip of paper from the index card about ½ an inch wide and use the hole puncher to punch one hole in each end. Fold the paper strip in half and tie the loose end of your string through the holes to create a sling for the dowel.
- Measure out two 9-ounce (255-gram) balls of clay using the food scale.
- Stick one ball on one end of the dowel.
- Put the dowel through the sling and put the other ball of clay on the other end of the dowel.
- Slide the dowel back and forth in the sling until it balances.
- Use the ruler to measure how far the center of each ball of clay is from the center of the sling.
*How do the two distances compare to each other?* - Take one of the balls of clay off the dowel and add more clay until you’ve doubled its mass to 18 ounces (510 grams).
- Put the new clay ball on the dowel. Slide the dowel around until it’s balanced again.
- Use the ruler to measure the new distances from the clay balls to the center of the sling.
*How do these distances compare to each other? How have they changed from the first time you measured them?* - Repeat steps 5 and 6, this time tripling the original clay ball mass to 27 ounces (765 grams).
*How do the distances from the balls to sling compare now?*

### Results

In step 7, you should have found that the balls balance when each is the same distance from the sling. After doubling the mass of one of the balls, the lighter ball must be twice as far from the sling as the heavier ball to balance. After tripling the mass, the lighter ball must be three times as far from the sling as the heavier one.

### Why?

Two objects balance when the ratio of the distances from sling is equal to the inverse of their mass ratio. Put another way, if one ball is twice as heavy as another, the heavier ball has to be twice as far away from the balancing point. If one ball is five times as heavy, it must be five times as far away from the sling. This can be summed up in a simple equation: * m_{1}d_{1} = m_{2}d_{2}*, where

**and**

*m*_{1}**are the masses and**

*m*_{2}**and**

*d*_{1}**are the distances from the sling.**

*d*_{2}The balls can be thought of as a planet and a moon, or a planet and a sun. The dowel is like the gravitational force that holds the two near each other. Because more mass means more gravitational force, when a moon and planet orbit each other, they orbit around their **center of mass**—the place where their masses balance each other. The barycenter is the same as the center of mass!

### Going Further

So if a planet is three times heavier than its moon, where is the barycenter? Based on your ball-and-dowel model, it would be three times closer to the planet than the moon. The Earth has nearly 80 times more mass than the Moon does. Can you guess how much closer the Earth-Moon barycenter is to the Earth’s center? What about the barycenter of the Earth and Sun, where the Sun is 333,000 times more massive than Earth?

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