7 Gravitation and Orbits

Universal gravitation

Every pair of objects with mass attracts every other pair. For two spherical bodies, or objects modeled as point masses, the magnitude of the force is

where $m_1$ and $m_2$ are the masses, $r$ is the distance between their centers, and $G$ is the gravitational constant, approximately $6.67\times10^{-11}\ \mathrm{N\,m^2/kg^2}$. The force points along the line joining the centers and is attractive. Each body exerts an equal and opposite force on the other.

The inverse-square dependence means that doubling the separation makes the force one-fourth as large. For an extended, spherically symmetric body, the external gravitational effect is the same as if its mass were concentrated at its center. When multiple bodies contribute, add their gravitational forces as vectors.

Takeaway: Gravitational attraction depends on both masses and on the square of their separation.

Fields and potential energy

A Gravitational field describes the gravitational force per unit mass at each point. For a spherical body of mass $M$, the field at distance $r$ from its center is

Here, $\hat r$ points outward, and the minus sign shows that the field points toward the body. A test object of mass $m$ in the field experiences force $\vec F=m\vec g$. Near Earth’s surface, the field magnitude is approximately $9.8\ \mathrm{m/s^2}$, but it decreases with distance from Earth’s center.

The gravitational potential energy of a mass $m$ at distance $r$ from an isolated mass $M$, taking potential energy to be zero infinitely far away, is

The negative sign indicates a bound system: energy must be added to separate the masses to infinity. For a small height change $h$ near Earth’s surface, the change can be approximated by $\Delta U\approx mgh$. This approximation works when the height change is small compared with Earth’s radius.

For example, at twice Earth’s radius from its center, the field is one-fourth its surface value because it varies as $1/r^2$. The relevant distance is measured from Earth’s center, not from its surface.

Takeaway: The field describes gravitational force locally, while potential energy tracks the energy associated with the configuration of masses.

Circular orbits and orbital energy

A Circular orbit occurs when an object’s sideways motion continually carries it forward while gravity accelerates it inward. Gravity provides the centripetal acceleration; it is not an additional force separate from gravity.

For an object of mass $m$ orbiting a much more massive body of mass $M$ at radius $r$, set gravitational force equal to the required centripetal force:

Solving gives the circular orbital speed and period:

The orbiting object’s mass cancels. Therefore, objects at the same radius around the same central body have the same ideal circular-orbit speed, regardless of their own masses. A higher circular orbit has a lower speed but a longer period.

For a circular orbit, the kinetic and potential energies are

Their sum is the total mechanical energy:

Negative total energy corresponds to a bound orbit. In the ideal two-body Newtonian model, orbits are ellipses, with a circle as the special case of constant radius. An object with enough energy to escape follows an open, unbound trajectory.

Takeaway: Orbital speed and period depend on the central body and orbital radius; negative total energy indicates a bound orbit.

Kepler’s laws

Kepler’s laws describe the shapes of orbits, how orbital speed changes, and how orbital period relates to orbit size. They apply to planets and other objects moving under gravity.

1. Law of orbits: A planet follows an elliptical path, with the central body at one focus. A circle is a special ellipse.

2. Law of areas: The line from the central body to the orbiting object sweeps out equal areas in equal times. The object moves faster when closer to the central body and slower when farther away. This reflects conservation of angular momentum when gravity is the only force.

3. Law of periods: The square of an orbital period is proportional to the cube of the ellipse’s semimajor axis. For an object orbiting a much larger mass $M$,

   $$T^2=\frac{4\pi^2}{GM}a^3,$$

   where $a$ is the semimajor axis. For two bodies with comparable masses $M$ and $m$, use

   $$T^2=\frac{4\pi^2a^3}{G(M+m)}.$$

Together, these laws connect orbit shape, speed, and period. Newton’s law of gravitation combined with Newton’s laws of motion explains why these patterns hold.

Takeaway: Kepler’s laws summarize regularities in orbital motion, while gravitation explains them.