Planets

I started trying to calculate the movements inside our solar system, but it is a lot more complicated than I ever thought it would be.

First of all, you have the Earth rotation. The easiest way to describe the Earth rotation is to do it relative to the sun. The rotation in a day is not 360 degrees. Earth is orbiting around the sun, so since the position of the Earth is different the angle to the sun changes over the course of a day. One day is of course the time it takes for the sun to appear in exactly the same position as before, for example zenith. If you look at the time of zenith and the rotation of the Earth between two such times, there is 360 degrees rotation of the Earth plus the orbit movement so that the Earth has to rotate a little bit more before the sun is back at zenith again. This turns out to be 360.9856 degrees.

Earth’s orbit is what we call a year. It takes 365.2422 days (Earth rotations) for one year (orbit) to complete. The orbit is actually an ellipse, it is not a circle. The amount of “ovality” that an ellipse has is called eccentricity. Earth’s orbit does not have a constant eccentricity, it changes over time from 0.000055 to 0.0679. When the orbit is more eccentric, there are larger differences in the amount of sunlight during summer/winter on the northern/southern hemispheres because the sun is not in the center of the orbit. It takes 413’000 years for the orbit eccentricity to complete one cycle. The current eccentricity is 0.017 and we are going towards a more circular orbit. Jupiter and Saturn are the influences that cause the eccentricity to change, because they pull the Earth away from the sun during part of their orbits. As the eccentricity changes, the speed of orbit also changes, so that the time of one year is always the same. Earth has a slower orbit when it is close to the sun and a faster orbit when it is far away from the sun. When the eccentricity of orbit is close to zero then the changes in orbital velocity is also close to zero. The eccentricity itself is also circulating, so that the point of the egg-shape is rotating around the sun. This is called apsidal precession. Apsidal precession takes 112’000 years to complete a full cycle. Finally, there is some wobble to the orbit, like if you spin a plate before it lies flat on a table there is a wobble, but the orbital inclination has a rather small effect with a 100’000 year cycle.

We also have something called axial precession. The rotation of the Earth has a center from the north pole to the south pole that we call the axis. This is where the rotation speed is slowest. Compared to Earth’s orbit, the axis is tilting. This makes the northern hemisphere be closer to the sun for part of the year and farther from the sun for another part of the year. It also means that if we look up on the stars in the night, one star will appear to stand still whereas the others move in circles due to the Earth’s rotation. Axial precession means that the axis is moving in a circle, so that the star that appears to be standing still constantly changes. Right now, the star Polaris in the Ursa minor constellation is our northern star, but eventually the north star will be in the constellation Draco instead. The axial precession has a cycle of 25’771.5 years. Additionally, the angle of the axle against the orbit changes over time. Right now, the axial tilt is 23.44 degrees, but it changes between 22.1 degrees to 24.5 degrees. The next minimum occurs in 11’800 CE.

That’s it for Earth. The sun is fairly simple, it has a rotation time around 25 days that doesn’t really impact much at all. The moon has an orbit around 29.5 days and a rotation that keeps it facing the Earth with the same side despite the orbit.

When you get into other planets, it gets a whole lot more complicated. Mercury, Venus, and Mars don’t have moons so they are not too complicated by themselves, but for Saturn and Jupiter there are lots of moons to consider. All large objects will modify gravity wells and the movement of heavenly bodies. Making an accurate model of the whole solar system is just way more complicated than I ever imagined. Not sure if I will ever finish this project.