Orbiting Objects

Orbiting Objects Types

Orbiting objects are frequently called "satellites".

Satellites can be categorized as natural or man-made. Examples of natural satellites are the moon, planets, and comets. Some examples of man-made satellites are a host of satellites that are launched from the earth for the purposes of communication, weather forecasting, or intelligence.

There are three major types of orbits: galactocentric, heliocentric, and geocentric orbits:

• A galactocentric orbit is one that orbits around the center of a galaxy. Our solar system is an example of this because it follows this type of orbit around the Milky Way.

• A heliocentric orbit is one that orbits around the sun. All the planets in our solar system, along with all the asteroids in the asteroid belt and all comets, are an example of this kind of orbit.

• A geocentric orbit is one that orbits around the Earth. Our moon follows a geocentric orbit, and so do most man-made satellites.

The highest orbiting object on record is called Vela 1A, and it is geosynchronous, meaning that its orbital period is the same as the rotational period of the Earth. The satellite orbits Earth at an altitude greater than 22,223 miles.

Orbiting Objects Facts

Whether orbiting objects are considered natural or man-made, every satellite's motion is governed by the same physics principles and use the same mathematical equations. The motion of an orbiting object can be described using the same characteristics we use for any other object in circular motion: velocity, acceleration, and force vectors. You can even calculate the force and acceleration due to the gravity of objects while they’re in orbit. Or you can apply your knowledge of gravity and circular motion to find the velocity of any circular orbit.

A system that involves orbital motion will exchange kinetic energy and gravitational potential energy. In basic terms, kinetic energy is the energy of motion, and potential energy is stored energy due to the relative position of objects in a system. An example of kinetic energy and potential energy working together is a yo-yo. When the yo-yo is in your hand before it starts to fall towards the ground, it is all potential energy. When the yo-yo begins to fall towards the ground, the energy turns to kinetic energy because it is moving. Similarly, as planets orbit further away from the Sun, there is an increase in gravitational potential energy which is balanced by the decrease in kinetic energy.

Orbiting objects are made up of a wide range of things like satellites, planets, asteroids, and even what we call "space junk". A satellite is considered to be a "projectile" because it is an object upon which the only force is gravity. This is because, once the satellite is launched into orbit, the only force governing the motion of a satellite is the force of gravity. Today, there are thousands of objects or satellites that are orbiting the Earth. Satellites have many uses but are most commonly known for uses associated with the government or military, internet access, television, GPS, weather forecasting, and more. SpaceX has the largest number of satellites in orbit. According to NASA, there are more than 27,000 pieces of "space junk" or space debris floating around in space. Space debris can be both natural meteoroid and human-made. Meteoroids are in orbit about the sun, while most artificial debris is in orbit about the Earth. This is why we use the term "orbital" debris to describe them.

Here are some important expressions for orbiting objects in circular motion:

Fig. 4 - System of two orbiting bodies. StudySmarter Originals

• The gravitational force,

$$\begin{align*}\vec{F_{\text{g}}}&=-\frac{GMm}{|\vec{r}|^2}\hat{r},\\F_{\text{g}}&=\frac{GMm}{r^2}.\end{align*}$$

• The gravitational force provides a centripetal force,

$$F_{\text{c}}=\frac{mv^2}{r}.$$

• If we equal the forces and solve for \(v\) we can determine the orbital speed of the object,

$$v=\sqrt{\frac{GM}{r}}.$$

• Because gravity is a conservative force, we just need to know the object's initial and final position to determine the work done by the force The work done by the gravitational force to bring \(m\) and \(M\) from being infinitely apart to just a distance \(r\):

$$\begin{align*}W&=\int_{\infty}^{r}\vec{F_{\text{g}}}\cdot\vec{\text{d}r},\\W&=\int_{\infty}^{r}-\frac{GMm}{r^2}{\text{d}r},\\W&=\int_{\infty}^{r}-{GMm}{r^{-2}}{\text{d}r},\\W&=-\left(-\frac{GMm}{r}\right)\Big|_{\infty}^{r},\\W&={GMm}\left(\frac{1}{r}-\cancelto0{\frac{1}{\infty}}\right),\\W&=\frac{GMm}{r}.\end{align*}$$

• The change in potential energy of a system of two approximately spherical distributions of mass,

$$\begin{align*}\Delta{U_{\text{g}}}&=-W,\\\Delta{U_{\text{g}}}&=-\frac{GMm}{r}.\end{align*}$$

• Now we know the change in potential energy of two orbiting objects. The gravitational potential energy of a system consisting of two orbiting objects is defined to be zero when they are infinitely apart. We can determine the potential energy of the system when the bodies are separated by a distance \(r\):

$$\begin{align*}\Delta{U_{\text{g}}}&=U_{\text{gf}}-U_{\text{gi}},\\\Delta{U_{\text{g}}}&=U_{\text{gf}}-\cancelto0{\frac{1}{\infty}},\\U_{\text{gf}}&=-\frac{GMm}{r}.\end{align*}$$

• The gravitational potential energy of a system of mass \(m\) in a gravitational field of magnitude \(g\) at a certain height \(y\) with respect to the surface,

$$\begin{align*}\Delta{U_{\text{g}}}&=-W,\\\Delta{U_{\text{g}}}&=\int_{y_1}^{y_2}-\vec{F_{\text{g}}}\cdot\vec{\text{d}y},\\\Delta{U_{\text{g}}}&=\int_{y_1}^{y_2}-\left(-mg\text{d}y\right),\\\Delta{U_{\text{g}}}&=mg\left(y_2-y_1\right),\\\Delta{U_{\text{g}}}&=mg\Delta{y}.\end{align*}$$

• The kinetic energy of the system made up by two orbiting objects can also be expressed in terms of the gravitational potential energy,

$$\begin{align*}K&=\frac{1}{2} mv^2,\\K&=\frac{1}{2} m\left(\frac{G M}{r}\right),\\K&=\frac12\frac{GMm}{r},\\K&=-\frac12U.\end{align*}$$

• The total energy of the two orbiting objects can be expressed in terms of the gravitational potential energy and the kinetic energy,

$$\begin{align*}E&=K+U,\\E&=\frac12\frac{GMm}{r}-\frac{GMm}{r},\\E&=-\frac{GMm}{2r},\\E&=\frac12U.\end{align*}$$

Orbiting Objects Examples

Let's do a few example problems for orbiting objects.