Have you ever taken a moment to think about how some objects orbit each other? This may be especially thought-provoking if you think about how celestial bodies orbit each other in space. Perhaps you have thought about how the Moon orbits the Earth, or how the Earth orbits the Sun. It can also be interesting to contemplate how astronauts are considered to be weightless when floating in space inside of their space shuttle. It should be fascinating to know that astronauts are actually not weightless in their space shuttle at all. Instead, the Earth's gravity is acting on them and their space shuttle and pulling them toward the center of the Earth with force. As gravity plays a major role in this scenario, it also plays a very big role in every orbiting object. Let's read on to learn more about the physics of orbiting objects.
Orbiting Objects Definition
An orbit is a type of trajectory, but a trajectory is not an orbit.
A trajectory is a path followed by a moving body.
An orbit is a trajectory that repeats periodically.
The word trajectory is used in relation to projectiles, while the word orbit is usually used in connection with celestial bodies or artificial satellites. For example, the path followed by a satellite around a planet is an orbit as it occurs repeatedly. The path followed by a launching rocket is a trajectory, as it only occurs once.
Kepler's First Law states that planets move in elliptical orbits with the sun at its focus.
Fig. 1 - Elipses have two focus points as well as a semi-major axis and a semi-minor axis. StudySmarter Originals
The shape of the orbit between objects will be defined by its eccentricity. This quantity describes how an orbit deviates from a perfect circle. The larger the eccentricity, the more elongated the orbit. In the other hand, an eccentricity of zero refers to a circular orbit.
Fig. 2 - Different types of orbital trajectories depend on eccentricity. The circular orbit is blue, the elliptical orbit is green, the parabolic orbit is red, and the hyperbolic orbit is purple. StudySmarter Originals
Kepler's Second Law states that the distance vector between the orbiting objects sweeps equal areas of the ellipse at equal times. In a circular orbit, the orbiting object will move at the same speed throughout the orbit. Consequently, in a circular orbit, the angular momentum, total energy, kinetic and potential energies will be constant. However, according to Kepler's Second Law, in an elliptical orbit, the orbiting object travels faster when it is nearer the central mass and moves slower when farthest away from the central mass. In elliptical orbits, only the angular momentum and the total energy of the system are conserved.
Fig. 3 - A visual representation of Kepler's Second Law. Notice how equal areas of space are swept across equal periods of time. StudySmarter Originals
Kepler's Third Law states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of the ellipse.
Kepler's Third Law explains that the period for a satellite to orbit Earth increases rapidly with the radius of its orbit. For an elliptical orbit, Kepler's Third Law can be expressed as follows,
$$T^2=\frac{4\pi^2}{GM}a^3,$$
where \(T\) is the orbit's period in seconds \(\mathrm s\), \(G\) is the gravitational constant \(6.67\times10^{-11}\;\frac{\mathrm N\;\mathrm m^2}{\mathrm{kg}^2}\), \(M\) is the planet's mass in \(\text{kg}\), and \(a\) is the semi-major axis in \(\text{m}\). The semi-major axis is equal in length to half of the longest diameter of an ellipse. In the case of a circular orbit, the semi-major axis can be replaced with the radius of the orbit.
An orbit is a repeating path that an object takes around another object or center of gravity. It can also be studied using Newton's second law. We will assume circular motion for simplicity. An object orbiting a celestial body like a planet in a circular orbit is held in its circular path by a centripetal force. The orbital speed of an object in circular orbit under the influence of gravitational forces does not depend on the orbiting object’s mass, it only depends on the central mass. If an object moves fast enough, it can orbit the Earth, Sun, or another planet. An orbit can be thought of as a projectile motion where the ground curves away equally as fast as the object falls. Keep in mind that for all orbiting satellites, the total angular momentum of the satellite is constant.
Objects in orbit, including satellites, move around the planet in a circular/elliptical motion.
Orbiting Objects Types
Orbiting objects are frequently called "satellites".
A satellite is any object that is orbiting the Earth, Sun, or other massive body.
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.
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
$$\begin{align*}\vec{F_{\text{g}}}&=-\frac{GMm}{|\vec{r}|^2}\hat{r},\\F_{\text{g}}&=\frac{GMm}{r^2}.\end{align*}$$
$$F_{\text{c}}=\frac{mv^2}{r}.$$
$$v=\sqrt{\frac{GM}{r}}.$$
$$\begin{align*}W&=\int_{\infty}^{r}\vec{F_{\text{g}}}\cdot\vec{\text{d}r},\\W&=\int_{\infty}^{r}-\frac{GMm}{r^2}{\text{d}r},\\W&=\int_{\infty}^{r}-{GMm}{r^{-2}}{\text{d}r},\\W&=-\left(-\frac{GMm}{r}\right)\Big|_{\infty}^{r},\\W&={GMm}\left(\frac{1}{r}-\cancelto0{\frac{1}{\infty}}\right),\\W&=\frac{GMm}{r}.\end{align*}$$
$$\begin{align*}\Delta{U_{\text{g}}}&=-W,\\\Delta{U_{\text{g}}}&=-\frac{GMm}{r}.\end{align*}$$
$$\begin{align*}\Delta{U_{\text{g}}}&=U_{\text{gf}}-U_{\text{gi}},\\\Delta{U_{\text{g}}}&=U_{\text{gf}}-\cancelto0{\frac{1}{\infty}},\\U_{\text{gf}}&=-\frac{GMm}{r}.\end{align*}$$
$$\begin{align*}\Delta{U_{\text{g}}}&=-W,\\\Delta{U_{\text{g}}}&=\int_{y_1}^{y_2}-\vec{F_{\text{g}}}\cdot\vec{\text{d}y},\\\Delta{U_{\text{g}}}&=\int_{y_1}^{y_2}-\left(-mg\text{d}y\right),\\\Delta{U_{\text{g}}}&=mg\left(y_2-y_1\right),\\\Delta{U_{\text{g}}}&=mg\Delta{y}.\end{align*}$$
$$\begin{align*}K&=\frac{1}{2} mv^2,\\K&=\frac{1}{2} m\left(\frac{G M}{r}\right),\\K&=\frac12\frac{GMm}{r},\\K&=-\frac12U.\end{align*}$$
$$\begin{align*}E&=K+U,\\E&=\frac12\frac{GMm}{r}-\frac{GMm}{r},\\E&=-\frac{GMm}{2r},\\E&=\frac12U.\end{align*}$$
Orbiting Objects Examples
Let's do a few example problems for orbiting objects.
Calculate force and acceleration due to gravity while in orbit. If the space shuttle orbits the Earth at an altitude of \( 380 \mathrm{~km}\), find an approximation to the gravitational field strength due to the Earth at that altitude. The mass of the Earth is \( 5.98 \times 10^{24} \mathrm{~kg}\), and the radius of the Earth is approximately \(6.37 \times 10^6 \mathrm{~m}\).
$$\begin{align*}F_{\text{g}}&=m_{\text{shuttle}} g,\\F_{\text{g}}&=\frac{G m_{\text{shuttle}} m_{\text{Earth}}}{r^2},\\g&=\frac{G m_{\text { Earth }}}{r^2},\\g&=\frac{\left(6.67 \times 10^{-11}\,\frac{\text{N}\text{m}^2}{\text{kg}^2}\right)(5.98 \times 10^{24}\,\text{kg})}{\left(6.37 \times 10^6\,\text{m}+380 \times 10^3\,\text{m}\right)^2},\\g&=8.75 \mathrm{~N} / \mathrm{kg},\\g&=8.75 \mathrm{~m} / \mathrm{s}^2.\end{align*}$$
What is the period of a satellite orbiting \( 200\, \text{km}\) above the Earth?
Assume a circular orbit.
$$\begin{align*}V&=\frac{2 \pi \mathrm{r}}{\mathrm{T}},\\T&=\frac{2 \pi \mathrm{r}}{V},\\T&=\frac{2 \pi\left(6.58 \times 10^6 \mathrm{~m}\right)}{7.78 \times 10^3 \mathrm{~m} / \mathrm{s}},\\T&=5314 \mathrm{~s},\\T&=88 \mathrm{~min}.\end{align*}$$
Determine the total mechanical energy for a satellite of mass \(m_2\) in orbit around a much larger object of mass \(m_1\) in terms of the two masses and the distance between their centers of mass.
When solving this problem, we should recognize first that the total mechanical energy is the sum of the kinetic and potential energies. Next, we substitute in expressions for the kinetic and gravitational potential energies. Finally, we utilize the relationship for the velocity of a satellite in orbit to remove the equation’s dependence on \(v\) and solve for the total energy:
$$\begin{align*}E&=K+U,\\E&=\frac{1}{2} m_2 v^2-G \frac{m_1 m_2}{r},\end{align*}$$
where the orbital speed of the satellite is \(v=\sqrt{\frac{Gm_1}{r}}\), so we can rewrite the equation:
$$\begin{align*}E&=\frac{1}{2} m_2 \frac{G m_1}{r}-G \frac{m_1 m_2}{r},\\E&=-G \frac{m_1 m_2}{2 r}.\end{align*}$$
Orbiting Objects - Key takeaways
- An orbit is a trajectory that repeats periodically.
In a circular orbit, the orbiting object will move at the same speed throughout the orbit. Consequently, the angular momentum, total energy, kinetic and potential energies will be constant in a circular orbit.
According to Kepler's Second Law, in an elliptical orbit, the orbiting object travels faster when it is nearer the central mass and moves slower when farthest away from the central mass. In elliptical orbits, only the angular momentum and the total energy of the system are conserved.
The potential energy of a system with two orbiting objects that are separated by a distance \(r\) is \(U_{\text{g}}=-\frac{GMm}{r}\). 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 of the spherical body is \(\Delta{U_{\text{g}}}=mg\Delta{y}\).
The kinetic energy of a system made up of two orbiting objects can be expressed in terms of the gravitational potential energy, \(K=-\frac12U\). The total energy of the two orbiting objects can also be expressed in terms of the gravitational potential energy, \(E=\frac12U\).
References
- Fig. 1 - Elipses have two focus points as well as a semi-major axis and a semi-minor axis, StudySmarter Originals
- Fig. 2 - Different types of orbital trajectories depend on eccentricity. The circular orbit is blue, the elliptical orbit is green, the parabolic orbit is red, and the hyperbolic orbit is purple, StudySmarter Originals
- Fig. 3 - A visual representation of Kepler's Second Law. Notice how equal areas of space are swept across equal periods of time, StudySmarter Originals
- Fig. 4 - System of two orbiting bodies, StudySmarter Originals
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