StudySmarter - The all-in-one study app.
4.8 • +11k Ratings
More than 3 Million Downloads
Free
In astrophysics, orbits are described as the movement of astronomical objects due to the gravitational attraction of other bodies.
To understand the appearance of such a movement, we first need to remember some concepts of general motion and gravitation.
Let’s recall some of Newton's laws!
Newton’s law of gravitation
Newton’s law of gravitation is the first rigorous description of the force of gravity. It quantifies the gravitational strength between two bodies with mass as a function of the radial distance between them. Newton also developed a formula for the energy of a body under the influence of a gravitational field, namely the gravitational potential:
Here, G is the gravitational constant (with an approximate value of 6.67 ⋅ 10-11m3/kg⋅s2), M and m are the masses being attracted to each other, r is the radial distance between them (whose presence indicates spherical symmetry), and the vector er is the vector joining both masses. The force F is measured in Newtons in the international measurement system, and the U is the potential in joules.
Newton’s second law of motion
Newton’s second law of motion states that
with the vector F the net force acting on a body, m the mass of this body, and the vector a the acceleration applied to the body by the force F. (Note that the mass of a body is constant in time.)
Check out our explanation of Newton’s Laws!
Circular motion
We use the concept of circular motion as a starting point until we study more complex shapes of orbits in the following section. Circular motion is a movement around a fixed point (called the centre) due to a force pulling the body towards it.
Circular motion is maintained thanks to a velocity perpendicular to the line joining the centre and the body. In this kind of setting, we are usually concerned with the modulus of the acceleration (a) causing the movement, which we can calculate with this equation:
Here, v is the object’s velocity in a circular motion, and r is the radius of the circular path. The symbol | | indicates the modulus of a certain vector.
With all these tools, we can now analyse a general astronomical setting: a system formed by two masses. Usually, these systems feature an object much heavier than the other.
Newton's law of gravitation states that an attractive force will appear between these two objects. Let’s take the frame of reference of one of the objects (usually the heavier one). If the other one is still from this frame of reference, the movement will be an attraction in a straight line until they both collide. However, a circular motion will appear if it has a velocity perpendicular to the segment joining them.
This circular motion occurs because the attractive force is constantly changing the direction of the orbiting body but not its magnitude. A way to see a circular motion is to equate the force of gravitational attraction and the force appearing in Newton’s second law of motion using the equation of acceleration in a circular motion. This gives us:
This shows us that if the speed of the orbiting body is constant, its radius will be constant too (or vice versa). We know that the force of gravitation does not change the speed of a perpendicularly moving object and that the radial distance will remain constant, which is the definition of circular motion.
We also know (due to the formula for the gravitational potential) that the potential energy of an orbiting body remains constant since its radius remains constant. In addition, the relationship between speed and radius implies that for several orbits at different radial distances, their speed is fixed and different. It also means that objects in the same orbit have the same speed.
For determining the orbit of a body, we need to consider the gravitational potential energy AND kinetic energy.
Let’s explore other possibilities of orbits that occur in our world and how they deviate from the ideal description we explored above.
A complex and general study of Newton's potential energy gives us three kinds of orbits, each depending on the body's energy. So, what would happen if the body had an excessively high speed compared to a scale provided to its radial distance?
In the end, it all boils down to the total energy a body has, which is the sum of gravitational potential energy and kinetic energy. Depending on whether the energy is positive, negative, or zero, a different motion will appear. Below is the total energy equation:
U is the gravitational potential energy and Ek is the kinetic energy.
You may also see V or U_g used for the gravitational potential energy.
Here are the three types of orbits generated by Newton’s law of gravitational attraction.
Ellipses have two points that determine their shape and the proximity of their points to the source of the gravitational interaction. If these two points are the same, we obtain a circle. Thus, circles are a special case of ellipses.
Although circular orbits are only a special case of elliptical orbits, we find that many of the aspects studied, such as the dependence of the speed on the radial distance, remain as a general feature (although a more complex analysis is needed).
The possible orbits of a body under gravitational influence depending on its total energy, Oğulcan Tezcan - StudySmarter Originals
These orbits are the result of small objects close to planets (satellites) and planets close to stars. Usually, these are elliptical orbits.
Planets in the Solar System describe elliptical orbits around the Sun, although they are almost circles. For the Earth, the farthest point from the sun is reached in July, while the closest point is reached in January. We can also find other objects that describe extreme elliptical orbits, such as Comet Halley. It can be seen from Earth every 76 years since this is the moment when it comes closer to the Sun in its elliptical path.
Nevertheless, most of the orbits for planetary satellites orbiting planets can be taken as circular since the initial conditions of the movement, which determine the elliptical character of the orbit, are close to the ones yielding circular orbits.
Some facts about satellites in our Solar System:
Mercury and Venus have none (they are too close to the Sun and it probably attracted them), Earth has one (the moon), Mars has two, Jupiter has 79, Saturn has 82, Uranus has 27, and Neptune has 14.
Mercury and Venus have none (they are too close to the Sun and it probably attracted them), Earth has one (the moon), Mars has two, Jupiter has 79, Saturn has 82, Uranus has 27, and Neptune has 14.
Orbits of planets are the movement described by planets due to the gravitational attraction of other bodies, usually stars.
The three types of orbits generated by Newton’s law of gravitational attraction are hyperbolic orbits, parabolic orbits, and elliptical orbits.
There are hyperbolic orbits, parabolic orbits, elliptical orbits, and circular orbits as a particular case of elliptical.
Be perfectly prepared on time with an individual plan.
Test your knowledge with gamified quizzes.
Create and find flashcards in record time.
Create beautiful notes faster than ever before.
Have all your study materials in one place.
Upload unlimited documents and save them online.
Identify your study strength and weaknesses.
Set individual study goals and earn points reaching them.
Stop procrastinating with our study reminders.
Earn points, unlock badges and level up while studying.
Create flashcards in notes completely automatically.
Create the most beautiful study materials using our templates.
Sign up to highlight and take notes. It’s 100% free.