What this calculator does
The centre of mass of two objects is the point they balance about, and it always sits closer to the heavier one. For a pair of very unequal masses it can lie inside the larger body entirely.
That is exactly the situation for the Earth and Moon. Their common centre of mass, the barycentre, is about 4,668 km from Earth's centre, which is well inside a planet with a radius of 6,371 km. Earth does not sit still while the Moon circles; both orbit that internal point.
The formula
The distance from each mass to the centre of mass is the total separation multiplied by the other mass, divided by the sum of the two masses. The heavier body is therefore closer to the balance point.
| Term | Meaning |
|---|---|
| Centre of mass | The mass-weighted average position of a system, which moves as though all the mass were concentrated there. |
| Barycentre | The same idea applied to orbiting bodies: the point both objects actually orbit. |
| Mass ratio | The ratio between the two masses, which determines how the separation is divided. |
The inputs explained
| Field | What to enter |
|---|---|
| Mass 1 (kg) | The first mass, in kilograms. |
| Mass 2 (kg) | The second mass, in kilograms. |
| Separation between the masses (m) | The separation between the two centres, in metres. |
When to use it
Finding an orbital barycentre
Two bodies orbit their common centre of mass, and its position determines how much the larger one wobbles.
Balancing a two-mass system
A beam with weights at each end balances at the centre of mass, which this locates directly.
Detecting exoplanets
A star wobbles about the barycentre it shares with its planets, and measuring that wobble is one of the main detection methods.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does the balance point shift with the second mass?
Earth paired with second masses of increasing size.
| Second mass | Distance from mass 1 | Distance from mass 2 |
|---|---|---|
| 7.34e+22 kg | 4,668,435 m | 379,731,565 m |
| 5.97e+23 kg | 34,945,455 m | 349,454,545 m |
| 5.97e+24 kg | 192,200,000 m | 192,200,000 m |
Questions
Why is the Earth-Moon barycentre inside the Earth?
Because Earth is about 81 times heavier than the Moon, so the balance point sits 81 times closer to Earth's centre. That works out to roughly 4,668 km, which is comfortably within Earth's 6,371 km radius.
Does the Earth orbit the Moon?
Both orbit their shared barycentre. Since that point lies inside Earth, the effect is that Earth wobbles about a point beneath its own surface once a month rather than circling the Moon in any obvious sense.
What if the two masses are equal?
The centre of mass sits exactly halfway between them, and both bodies trace identical orbits about that midpoint. Some binary star systems do exactly this.
Does the centre of mass move?
Not from internal forces. However the two bodies move relative to each other, their common centre of mass continues at constant velocity unless something external acts, which is a direct consequence of momentum conservation.
For the gravitational attraction between the two, see the Newton's law of gravitation calculator. For orbital periods about that point, see the Kepler's third law calculator.