What are ‘Orbits’? by Gordon Dennis

We often say that the planets in the solar system ‘orbit the Sun’; or that the Moon ‘orbits the Earth’.  But what exactly is an ‘orbit’?

Geocentric theories
For many centuries it was believed that the Earth was at the centre of the universe and that all the stars and planets revolved in circular motion around the Earth.  It is a curiosity that this theory, usually attributed to Ptolemy, persisted so long. It could only explain the observed pattern of movements of the planets by explaining that they moved in epicycles – effectively circles within circles, but how this movement came about was unexplained. 
  
Copernicus and heliocentric models
The Prussian Nicolaus Copernicus  proposed a theory of a heliocentric system, with the Sun at the centre of the universe, in which Earth was one of the six planets known at the time.  Copernicus was not the first to espouse this theory, but was the first to extensively document it in his book published shortly before his death in 1543.  As in the Ptolemaic system, Copernicus’ orbits were circular, so epicycles still had to be invoked to explain some of the observed motions of the planets. In fact, more epicycles were needed in the Copernican system than the Ptolemaic system, and yet there were still unexplained gaps in explaining astronomical observations.

Kepler’s breakthrough
Johannes Kepler, a prolific German astronomer and mathematician made the key breakthrough in the study of orbital motion by a combination of observation and deductive reasoning empirically deriving three laws which stand in good stead to this day.
Kepler’s key discovery was that planetary orbits are NOT circular, but elliptical.  An ellipse is an example of a conic section.
In case you’re not familiar with conics, and with terms such as ‘semi-major axis’, ‘eccentricity’ and ‘focus’, the next section contains a quick primer on the subject.  If you are familiar with conics, you could skip this and go to the following section.

Newton’s Laws of Gravitation: formalizing Kepler’s laws
Due to the operation of gravity, two massive objects will move in space-time around their common centre of mass, known as the barycentre.  This was established by Sir Isaac Newton who defined his laws of gravitation, our first formal description of gravity. This provides our first definition of what an orbit is. 
Newton’s Laws added formal mathematical reasoning to Kepler’s empirically-derived Laws.  In his words:
”I deduced that the forces which keep the planets in their orbs must be reciprocally as the squares of their distances from the centres about which they revolve: and thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the Earth; and found them answer pretty nearly.”

Kepler’s First Law
Planets orbit on elliptical paths, with the Sun at one focus of the ellipse. 
As we’ve seen, an ellipse is an example of a conic section.
More precisely, Newton defined the focus as being at the barycentre of the Sun–planet pair. 
So, using Newton’s Laws, how can we determine where the barycentre is?  Let’s consider a very theoretical solar system consisting of just the Sun and Jupiter. 
With only two objects in a system, it's simple to locate the barycentre of the system around which the two bodies orbit.   The Sun’s mass, M⊙=1.99*1030kg.  Jupiter’s mass, MJ=1.898*1027kg.  Using Newton’s Laws, we have:

Which means the ratio of Jupiter’s mass to the Sun’s mass is about 1/1,000.
In turn that means that the distance from Sun to the system's barycentre approximately 1/1,000 times the distance from Jupiter’s distance to the barycentre.
The mean distance from the Sun to Jupiter is 7.785*1011m; we shall see how to work that out in a moment, using Kepler’s Third Law; please accept it for now.  One thousandth of that distance is 7.785*108m. The radius of the Sun is slightly less:  6.96*108m.  This means the barycentre of our very simplified two-body solar system is just above the Sun’s surface.
In the real solar system, consisting of the eight major planets, several known minor planets, asteroids etc., the position of the barycentre of the whole system changes all the time.  Jupiter is by far the most massive planet and so has the greatest influence on the position of the barycentre.  However, the alignment of the other solar system bodies is constantly changing as the planets orbit at different speeds (see Kepler’s Second Law below).  So the relative influence of their respective masses changes the position of the barycentre constantly.  Technically, this is known as an “n-body problem”, the mathematics of which is very complicated, so we’ll conveniently declare outside the scope of this Blog.

Kepler’s Second Law
Planets sweep out equal areas in equal times.

What this means is that the angular momentum is conserved throughout the course of the orbit.

If the orbit was circular, then the distance of the orbiting body from the primary, r, would be constant.  However, since the orbit is elliptical, then r changes.  The mass, m is constant, so the velocity, v must change to keep the angular momentum constant.
In practice,

• Near perihelion: planet is closer to the Sun > r is smaller >  v is proportionally greater.
• Near Aphelion: planet is farther from the Sun > r is larger > v is proportionally less.

 
 
Kepler’s Third Law
The orbital period squared is directly proportional to the size of the semi-major axis cubed.

Where k is a constant. In his gravitational theory, Newton formalized this constant as:

Where M1 and M2 are the masses of the two bodies. As we have seen, in the solar system even the largest planet, Jupiter, has a mass of only about 1/1,000 solar masses, and this simplifies the equation to become:

However, this can be simplified even further if we pick our units carefully.  If the orbital period, P is in Earth years (yr); and the semi-major axis, a is in AU, then by definition k=1. So we don’t even need to know the mass of either the star or the planet to calculate the furthest orbital distance between them.

Let’s again take the case of Jupiter.  From observations, we know that the orbital period of Jupiter, PJ=11.86 yr.  So:

So the mean distance of Jupiter from the Sun is 5.20AU is approximately 7.78*1011m, just as we assumed when describing Kepler’s First Law above.

Putting it all together

1. In a gravitationally bound system, as in the cases of the major and minor planets in the solar system, orbits are elliptical.
   

The eccentricities of the orbits of the major solar system planets are shown here:

This plot clearly demonstrates the huge effect Jupiter has on the solar system.  Jupiter’s mass is greater than the sum of the masses of all the other major planets.  Notice that Mercury has by far the greatest orbital eccentricity, despite being closest major planet to the Sun. The huge mass ratio between the Sun and Mercury would mean a near circular orbit were it not for the influence of other large bodies – most notably Jupiter. 
The high orbital eccentricity of Mars is due to the high mass ratio between Jupiter and Mars, and Mars mean orbital distance being relatively close to the mean orbital distance of Jupiter.
Saturn, despite being further from the Sun, has a higher orbital eccentricity than Jupiter, due to the more massive planet’s gravitational influence.
Uranus and Neptune, each orbiting further from the Sun than the mean distance at which Jupiter orbits the Sun, are less gravitationally influenced by Jupiter, and so have lower orbital eccentricities.
Less easy to explain is the very low orbital eccentricity of Venus – the lowest of any planetary body in the solar system. This may have something to do with Venus’ retrograde rotation affecting its angular momentum.  I’d be very interested in comments on this (NOT a trick question: I don’t have the answer!)

1. Our current best description of gravity is general relativity (GR), announced by Einstein in 1915.  However for many purposes (e.g. getting a spacecraft to the Moon or any of the major planets), Newton’s Laws of Gravity are sufficiently accurate.

The greatest success of Newton's gravitational theory was when it was successfully used by others to predict the existence of Neptune based on observed perturbations in the orbit of Uranus.
Newton’s Laws were unable to account for the orbital precession of Mercury, yielding values well below the observed movements.  GR predicts this to very close accuracy (see my August Blog).

1. If the bodies in the system are gravitationally unbound, the orbits will still be conics, but in this case, they will both follow either parabolic or hyperbolic paths.   

Examples of this behaviour are ‘visitors from outside the solar system’ such as Oumuamua, the first known interstellar object detected passing through the Solar System, discovered on 19 October 2017.