# Attraction to barycenter

If an object was a distance from a planet away that equaled the distance from the object to a moon of the said planet, would it be drawn into the direction of the barycenter as it is the common center of mass?

• In general, no. – David Hammen May 20 '15 at 15:43
• why? where to if not the barycenter? – Astrony May 20 '15 at 16:05
• This would only happen if the moon and planet had the same mass. Imagine a very light moon and a very heavy planet, but with a barycenter about 100 miles above the planet's surface. An object between the moon and planet would be attracted to the planet and eventually hit the planet's surface. It wouldn't hover 100 miles above the surface. Instead of the halfway point, did you perhaps mean the en.wikipedia.org/wiki/Lagrangian_point – barrycarter May 20 '15 at 16:38
• Even that's not true, @barrycarter. – David Hammen May 20 '15 at 17:13
• Which Lagrangian point? There are five of them, and in general, none of them is at the barycenter. – David Hammen May 20 '15 at 18:07

## 1 Answer

The Newtonian gravitational acceleration of a test object (a small object with negligible mass) toward a point mass is given by

$$\vec g =\frac {GM}{||\vec r||^3} \vec r \tag{1}$$

where $\vec g$ is the acceleration vector, $G$ is the Newtonian gravitational constant, $M$ is the mass of the point mass, and $\vec r$ is the displacement vector directed from the test object toward the point mass.

Equation (1) cannot be used to compute the net gravitation acceleration of a test object toward a pair of point masses by pretending that the pair of point masses is equivalent to a single point mass located at the barycenter of the two point masses. Instead, what needs to be done is to apply equation (1) to the point masses individually and then form the vector sum of these individual accelerations. This in general will not point toward the barycenter.

There are two classes of points where the net gravitational acceleration vector to a pair of point masses is directed toward the barycenter of the point masses:

• Points along the line connecting the point masses, less for two regions where the net gravitational acceleration vector points away from the point mass, and
• Points away from this line and whose projection onto this line is exactly midway between the point masses.

For a test object located anywhere else, the net gravitational acceleration vector does not point toward the barycenter.