This tutorial will guide you through using the Newton Tool to simulate a simple gravitational orbit. Also, in this tutorial, we'll introduce the idea of 'scaling'. Scaling is a technique that will help keep the numbers used during the computation to a reasonable size and will prevent problems that may occur when dealing with very large or very small numbers. If you have not already done so, launch the Newton Tool.
Before Scaling
Accelerations
First, we'll briefly go over the equations required to simulate a planetary orbit in 2 dimensions. There are two equations that will govern the motion of this problem. They are:
The variable, G, above is just a constant that refers to the strength of the gravitational attraction. In SI units, G = 6.673*10-11 m3kg-1s-2. Now we need the value of M. To model a planet around our sun, M = mass of the sun. In SI units, M = 1.98892*1030 kg. The very small G, and the incredibly large M foreshadows why we'll need to scale this problem. For now, enter the equations in the Newton Tool as they appear below.
Initial Conditions
Since we've entered G in SI units, our other initial conditions must be in SI as well. For the 'Initial x:' enter 150e9. This is the average distance the earth is from the sun as it orbits. 150e9 is the same as 150,000,000,000 meters. Enter 29805 (29.805 km/s) into 'vy'. This is the average velocity the earth travels around the sun. For the 'Final Time', enter 3.2e7 (32,000,000 seconds). Or roughly one year.
Plots
Because we used the average value of the earth's orbit as the initial x, and the average value of the earth's speed as the initial vy, the graph should be almost a perfect circle. However, since our solution was not scaled, we lost too much precision and get a graph like this.
This error could also be caused by an inappropriate 'Adaptive Step Factor'. However, trying to find the right size for the 'Adaptive Step Factor' will require lots of trial and error, can be time consuming, and due to the size of the units involved may not work at all. A more effective method to propagate this motion is to scale the units to something more reasonable.
Scaled Solution
In order to correctly model this system, we should not use SI units. One convenient way to scale is to replace the constants or initial conditions in the problem with values of 1. For example, replace 1.98892*1030 kg with 1 solar mass. And 150*109 m with 1 orbit radius. In addition, using years instead of seconds would also be more appropriate. This means we've scaled the length, time and the mass of the problem. Changing these will affect the value of the initial velocity and the constant G used earlier.
After converting units, we find that G=-39.1627 (orbit radii)3 (solar masses)-1(years)-2, vy becomes 6.2705 orbit radii per year, and the final time is just 1 year.
Accelerations
After scaling all the numbers in the problem, enter the new acceleration equations as shown below.
Initial Conditions
Enter the initials conditions as shown below.
Plots
Making a plot with 'x position' on the x-axis and 'y position' on the y-axis produces a circular graph as we would expect.
Table
In order to get the numbers in SI units again, use the table tool to import the data to any data manipulation software to scale the numbers back to SI. Below is an example of some of the numbers generated in the Newton Tool being scaled back to SI units.
Challenge
Use the output from the Newton Tool to verify the conservation of:
- Energy
- Angular Momentum
- Laplace-Runge-Lenz Vector
