3.5 Interactive orbits
We couldn’t explore the way star and planet orbits work with our tactile example in the previous activity because the planet’s orbit is so much bigger than the star’s. But using interactive applications we can do this. Generally, mathematical treatments like that underlying the interactive application allow us to understand and predict the behaviour of things which are too big, too small, too hot, too dense or too complex to allow us to build physical models. The ball and stick model is a physical model, while the interactive application is a mathematical model, built with the equations which describe the motions of masses. Mathematical models are so useful and powerful that generally when scientists use the word model, they mean a mathematical model.
This interactive application lets you see the motion of a star and planet around a centre of mass. As before, you can set the value of the mass of a star and the mass of its planet. The orbital paths of the star and planet are indicated by black circles. This application has two additional sliders, the first of which allows you to choose the distance of the planet from the centre of mass.
Play with the top three sliders for the masses and distance of the planet from the centre of mass and watch how the orbital paths respond. As before, you can also use the arrow keys on the keyboard to amend the values. You may need to click anywhere on the application screen to ensure that it calculates and displays the correct orbits for your choice of values. Again you can work with the application in a separate page by clicking on the [ ] button. You may need to adjust the magnification in your browser to view the full application panel.
Set the mass of the star to 0.3 MSun, the mass of the planet to about 10 MJ and the distance of the planet from the centre of mass to about 4.5 AU.
- Take the fourth slider, which controls simulated time, to the value 0, and then move it slowly to the value 1. Watch the planet.
Question: Describe the motion of the planet.
The planet starts off at the right-hand side of the application, at 3 o’clock and moves anticlockwise all the way around the circle representing its orbital path, returning to 3 o’clock when t = 1.
Time on the slider is given in units of the orbital period. Based on what you observe when you vary the time from 0 to 1, explain what is meant by the orbital period.
The orbital period is the time taken for the planet to complete one loop around its orbit. The orbital period is the same as the planet’s year. So, the Earth takes one Earth year to complete its orbit. Astronomers talk about the orbital period of the planet, rather than the planet’s year because there is a danger of becoming confused between the planet’s year and our own Earth year.
- Take the fourth slider back to the value 0, and then move it slowly to the value 1 again. Watch the star this time.
Question: Describe the motion of the star.
The star starts off at the left hand side of its small orbit near the centre of the application, at 9 o’clock, and moves anticlockwise all the way around the small circle, returning to 9 o’clock when t = 1. Because the orbit is so small, the star’s motion looks like a small circular wobble.
- Move the fourth slider slowly from 0 to 1 again. Watch the relative positions of the planet and the star. You may find it helpful to adjust the zoom using the + and − buttons.
Question: Describe the how the positions of the planet and the star change relative to each other and the centre of mass.
The star and the planet are always on exactly opposite sides of the centre of mass. You could always connect the centre of the star and the centre of the planet with a straight line that would pass exactly through the centre of mass. The orbital periods of the planet and the star about the centre of mass are exactly the same.