Motion Within our Solar System - Do Planets Really Behave as we Think?

    As children grow up, most are told that the Earth orbits the sun. In fact, it is said that all planets within our solar system orbit our very own star. What if I told you that every adult who ever told you this as a child was lying straight to your face? Not maliciously, of course! That being said, if you were to think long and hard about everything we have covered up until this point in Physics 111, does it really make sense that a bunch of objects would just choose to orbit another, more massive object? How do they know that they should follow this star and not the next pretty, shiny thing coming along? They don't, and that is why it is absurd to think that all of the objects in our solar system (including the sun) orbit anything other than the center of mass of our solar system. 

    This point is known as the "barycenter" of our solar system. As Jupiter has a mass much larger than any of the other planets, the barycenter of it and the sun is just outside the sun's surface. This would be the point to hold the sun and Jupiter at if you wanted to balance them! The barycenter of our solar system as a whole, however, constantly changes based on where the planets are in their orbits. The solar system's barycenter can range from being near the center of the sun to being just outside its surface! One direct cause of this fluctuation is the fact that the planets all have elliptical orbits with the barycenter as just one focus. What is an ellipse, you may ask?

Each dot on this ellipse represents one focus point of the ellipse. This is a very
stretched-out ellipse; in reality, the Earth's focus points are much closer together
making its orbit much more like a circle than an oval. 

  

    In simple terms, an ellipse can be coined as a stretched-out circle with two focus points that allow each planet in our solar system to be much closer to the barycenter at certain points than others. One very special consequence of this was discovered in 1609 by Johannes Kepler. He said that the line from a focus point of any planet's elliptical orbit will sweep out equal areas in equal times. What does this mean? If we were to make multiple triangles by saying that the same amount of time passes between each starting and stopping point, we would see that in some triangles our planet travels farther than others. Since time remains the same and distance changes, we must agree on the fact that our planet is traveling at different speeds at different points in its orbit. 

    This is all very astonishing, but you may ask, why does any of it matter? Well, let's consider the idea that our planet moved as fast throughout its whole orbit as it does when it is closest to the barycenter (a position known as perihelion). For this situation, let's model Earth's orbit as a perfect circle (since it almost is in real life). At perihelion, Earth is moving at about 30,280 m/s. Since v=rω, and the distance from Earth to the barycenter at perihelion is 1.475 x 10^11 meters, ω=2.05 x 10^-7 rad/s. Now, we can calculate how long a year would be if we were to travel at this speed throughout our entire orbit using ω=2π/T → T=2π/ω → T=2π/(2.05 x 10^-7 rad/s) = 30,606,665 seconds, or 354.24 days. Since our current orbit takes us about 365.25 days, our year would be eleven days shorter! To put this in perspective, over 4.2 million babies are born in that amount of time! Our current sense of how long a year takes shapes almost everything in our daily lives including seasons, holidays, pay dates, anniversaries, and so much more. If this time frame was suddenly compressed, everything about our lives and society as we know it would suffer momentous consequences. 

    To take this one step further, we can also use our knowledge of rotational motion to calculate the moment of inertia of our very own Earth! We will use the parallel axis theorem to determine this value since Earth is spinning on its own axis as well as orbiting the barycenter. The mass of the Earth is 5.97 x 10^24 kg, the radius of Earth is 6.378 x 10^6 meters, the Earth is usually ~ 1.5 x 10^11 meters from barycenter, and I=(2/5)mR^2 + mh^2. So, I = (2/5)(5.97 x 10^24 kg)(6.378 x 10^6 meters)^2 + (5.97 x 10^24 kg)(1.5 x 10^11 meters)^2 → I = 1.34 x 10^47 kgm^2. Luckily for us, our Earth is very, very resistant to changes in its motion meaning we won't start spinning crazily or go flying away into deep space anytime soon!

Sources:

    - Earth Fact Sheet (nasa.gov)

    - What Is a Barycenter? | NASA Space Place – NASA Science for Kids

    - Planetary Motion: The History of an Idea That Launched the Scientific Revolution (nasa.gov)