Why Are Planetary Orbits Elliptical?

I hope everyone enjoys my Extremely Fancy images on this post.  I built most of this myself and I a) don't have the best technical art skills and b) lack any actual graphics software.  Someday hopefully we can afford a graphics department.  :)

It may seem like planetary orbits and orbits of other objects should be circular.  Circles are simple and many patterns in nature adopt a roughly circular shape.  However there are two factors that dictate a planet's orbit that will help us understand why orbits are elliptical and not circular. 

Just be sure we're on the same page, an ellipse is what we usually call an oval.  This is an ellipse:

The two dots are the ellipse's foci (I just sort of ballparked it here).  The exact math and concept of what foci are is a bit outside the scope of this post but you can think of it this way: a circle is a special case ellipse in the same way that a square is a special case rectangle.  In a circle the two foci overlap and are at the center.  The more stretched an ellipse is, the further away from the center of figure they will be (yes, this is a simplification).  You can also check out this neat little demo to get a sense of how the placement of the foci dictate what the ellipse will look like.

Let's also get a little bit of possible confusion out of the way.  The way you often see Earth's orbit drawn in text books or diagrams looks a bit like this:

There was a time when I was hearing people talk about the idea that Earth's orbit was actually circular and that the way it's drawn in textbooks is highly exaggerated.  This is sort of half true.  The way it is drawn is often exaggerated but the orbit is elliptical.  It's just not all that elliptical.  How elliptical an orbit is can be described with a measurement called eccentricity which goes form 0 to 1 with 0 being a perfect circle.  Earth's orbit has an eccentricity of about 0.0167. So rather than the very noticeably elliptical path in the image above it's closer to something like this (again, ballparked):

To see why planets have elliptical orbits we need to take a look at what happens when two massive objects interact in space.  We're going to keep the systems to two objects for simplicity and we're also going to imagine one object is stationary and the other is moving towards it.  Neither of these things is really what you see in space as objects are all hurtling around towards and away from one another constantly but it gives us a nice clean system that should hopefully make things relatively straightforward.

When one object approaches another in space there are two forces acting on the object: its own momentum and the force of gravity attracting the two objects.  There are a few different things that can happen depending on how strong these two forces are.  If an object has a very high momentum and the gravitational attraction between the two objects is fairly low, the path of the object will be bent but it will not be "caught" in an orbit.  The red arrow represents the object's momentum and the blue arrow, gravity, resulting in the purple arrow.  These arrows will change as the object proceeds on its path but hopefully this illustrates how this combination of forces will result in the path shown:

The second possible outcome is that the force of gravity is quite high and the momentum quite low.  This results in the smaller object crashing into the larger.

When we have a situation in between these two extremes, when the momentum and force of gravity are relatively in balance, this is when the smaller object is "caught" and falls into orbit around the larger.  We're going to visualize this to see why the orbit will be an ellipse.  We're going to look at a relatively extreme (high eccentricity) shape compared to Earth's orbit but this situation still applies to our planet.  I'm also going to change the orientation so we can visualize the formation of an orbit as the smaller object *falling* into the gravitation field of the larger.  This is not really a metaphor, as we'll see.  Orbits are a type of falling...they just don't end up in a crash.  At least for a great many billions of years.  I'm also going relabel the larger object "object A" and the smaller object "object B."  OK here we go!

I put numbers on a few points in object B's new orbital path to try to break down what's going on.  Imagine at point 1 that object B is hurtling along in space but it has begun to get close enough to object A that their gravitational attraction is going to start to change its course.  As it moves toward point 2 its initial momentum has it wanting to continue on its original path but gravity now has it on a path falling towards object A.  As B moves towards point 3 gravity is getting stronger, accelerating it, but because it maintains momentum, it does not crash here but instead moves on a curved path along the "bottom" of object A.  As it moves "up," away from the "bottom" of the orbit here gravity is now going to decelerate object B.  I removed the other force arrows at point 4 because the important point here is that its momentum is carrying it "up," away from object A but gravity is still pulling towards object A.  Object B decelerates until it reaches the "zenith" of the orbit at point 5 where it then begins to fall again back towards object A. 

You can think of this a bit like tossing a ball into the air in an arc from one hand to the other.  As you put a force on the ball it rises but is decelerated by Earth's gravity.  As it reaches a zenith it's upward velocity momentarily becomes 0, then it is accelerated by gravity, falling back down to your other hand. 

So why can't this system form a circle?  Well...actually it can.  It's just extremely unlikely.  In order for the orbital path to be a perfect circle the balance between the momentum and gravity need to be exactly matched and at exact right angles as it is caught in orbit.  This is what that would look like:

Imagine the smaller object moving towards the larger and it's position and momentum enter the orbit at the top of this diagram.  Right at this moment, the force of the momentum and the force of gravity would have to be exactly the same and exactly at right angles to one another to create the perfect 90 degree arc, bringing it to the point to the right of the larger object, at which point momentum and gravity are still exactly matched and exactly at right angles, continuing neat 90 degree arcs all the way around. 

Sorry for holding out on you: circular orbits can happen and do happen.  They're just very, very rare because the conditions you need for them to form are exceedingly unlikely.  Think back to the analogy of throwing the ball in an arc from one hand to the other.  Imagine how unlikely it would be for you to get that ball to go in an exact semicircle.  The force and direction you apply to the ball with one hand would need to be super precise for this to happen.  It's not hat you couldn't ever get a ball to fall into a perfect semicircle, it's just that you would have to be extremely lucky or try over and over again.  And remember, while Earth's orbit is an ellipse, it is actually pretty close to a circle. 

Except!  There is one little caveat.  I've simplified a lot here but one thing I've glossed over a bit is that gravity is a force that attracts two objects together.  It doesn't actually pull object B towards object A, it pulls them towards one another.  Now obviously with objects that differ in mass by extremes like the Sun and Earth or the Earth and a ball, the acceleration on the smaller object is going to be much, much more noticeable.  But there is a non-zero acceleration on the more massive object. 

One of the things that this means for orbits is that smaller objects don't actually orbit larger objects.  They orbit each other.  The point where they orbit one another is their center of mass.  You can think of this as the point at which the combined mass of both objects is equal on both sides.  The center of mass also represents one focus (remember our ellipse's foci at the very beginning?) of an elliptical orbit.  While the Earth is a large object, the Sun is much much more massive so for our orbit the center of mass is actually pretty close to the center of the Sun.  Again, I'm totally ballparking this and completely simplifying but just to illustrate:

Warning!  Not remotely to scale!

Even if Earth's orbit wanted to be a perfect circle (err...conditions were lined up exactly perfectly so the momentum and gravitational forces were balanced and at exact right angles), that tiny shift in where the objects orbit one another would mean it would still end up as an ellipse.  Just by being two massive objects orbiting one another and having a center of mass they introduce eccentricity into the orbit.  

Now, for an object the size of Earth, the eccentricity introduced due to the center of mass not being at the very center of the Sun is pretty small.  For something like an asteroid that is 100th or 1,000th the mass of Earth that eccentricity is going to be...not even measurable.  We're talking ten, thirty, fifty decimal places out.  It's not zero but it's tiny.  But it's always going to be there.  Even in a system where the momentum and gravitational forces are lined up just right, the center of mass of the system is going to introduce, at the very least, a tiny amount of eccentricity into the system.  So even in the most unlikely, perfect situations, we're never going to get an exactly perfectly circular orbit. 

This post was super fun and I did a ton of research.  I hope you enjoyed it!  As with other topics, I'm not an expert but this definitely took me out of my comfort zone.  If you have comments, corrections, questions etc. please drop them in the comments or tweet at me @paulsfenton. 

Thanks for reading!

Sources:

email correspondence: Richard Binzel, Professor of Planetary Science, Joint Professor of Aerospace Engineering and MacVicar Faculty Fellow, Massachusetts Institute of Technology

https://www.scienceabc.com/nature/universe/planetary-orbits-elliptical-not-circular.html

https://www.youtube.com/watch?v=DurLVHPc1Iw

https://www.youtube.com/watch?v=xaCyQvwJ_Vs

https://www.youtube.com/watch?v=59qniggFpFQ

https://www.quora.com/Why-are-planets-orbits-elliptical-Why-not-circular

https://www.windows2universe.org/physical_science/physics/mechanics/orbit/eccentricity.html&edu=high

https://www.education.com/science-fair/article/orbital-eccentricity/

https://gravity.wikia.org/wiki/Orbital_eccentricity

https://everything.explained.today/Circular_orbit/

https://everything.explained.today/barycenter/

https://physicsabout.com/keplers-laws/

https://www.quora.com/What-causes-comets-on-an-elliptical-orbit-to-return-to-the-sun-Theres-no-friction-in-space-and-the-comet-should-keep-going-out-of-the-solar-system