Planetary Orbits

To understand the appearance of such a movement, we first need to remember some concepts of general motion and gravitation.

What is the mathematical description of orbits?

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 e_r 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.)

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.

What types of orbits are there?

Let’s explore other possibilities of orbits that occur in our world and how they deviate from the ideal description we explored above.

Mathematical implications of Newton’s potential energy

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 E_k is the kinetic energy.

Types of orbits

Here are the three types of orbits generated by Newton’s law of gravitational attraction.

• Hyperbolic orbit. If the total energy of a body under the gravitational influence is positive, it will describe a hyperbole. This means that it will escape from the source of gravitational interaction and move towards regions where the strength is weaker. The positive energy means that for an infinite time, the body would reach an infinite radial distance with a certain speed (only kinetic).
• Parabolic orbit. If the total energy of a body under the gravitational influence is exactly zero, it will describe a parabola. This means that it will escape from the source of gravitational interaction and move towards regions where the strength is weaker. However, for an infinite time, the body would reach an infinite radial distance with exactly zero speed (only kinetic, but zero in this case).
• Elliptical orbit. If the total energy of a body under the gravitational influence is negative, it will describe an ellipse. This means that it will not escape from the source of gravitational interaction and will orbit it. It would not make sense for the body to reach an infinite radial distance with negative kinetic energy since that cannot be (squared velocity is always positive), so we conclude that the body never reaches an infinite radial distance.

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

Examples of planetary orbits and satellite orbits

These orbits are the result of small objects close to planets (satellites) and planets close to stars. Usually, these are elliptical orbits.

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.