One day you accidentally fall into a wormhole and by some freak of nature emerge alive and unscathed. You find yourself marooned on a tiny star (a brown dwarf). Rather than being scared of the unknown and worrying about the how you will ever return to earth you jump at the chance to study the star and its surroundings.

After initial observations of the star you determine that its radius is 1.00 m and that the acceleration due to gravity at its surface is 1.00 m/s². You quickly realise that on this star you should be very careful when jumping!

• Find the mass and density of this star. ("What is it made of?" you ask yourself. You are encouraged to ask someone else this question!)
• What is the acceleration due to gravity at 1.00 m above the surface of the star?

• Calculate the escape velocity at the surface of the star.

As you gaze with awe at the beautiful vista a low orbit planet passes by 1 m above the ground travelling at 0.90 m/s in a direction tangential to the surface. Thankfully it misses you and continues to orbit the star. You realise this is the perfect opportunity to study orbital mechanics and decide to study the motion of this planet. First you wonder if the orbit is circular (Believe me you do!).

• What would be the speed of a planet in circular orbit 1.00 m above the surface of this star?

You realise immediately that it is not a circular orbit and so expect according to Kepler's 1^st law that the motion is elliptical. In order to make accurate predictions about the behaviour of the planet you model the motion using the above information and your computer.

1. Plot an x-y graph showing the orbit of the planet. Is the orbit elliptical? (make sure you have plotted on a graph where the axes are scaled equally)
2. Plot a radius vs. time graph and determine the period of the orbit of the planet? (Make sure you duck at the right moments.)
3. Plot a speed vs. radius graph and determine the speed and radius of the planet at its apogee and perigee. Do these results satisfy conservation of angular momentum?
4. Use the above plots to verify Kepler's 2^nd Law. (i.e. determine the area swept out by the planet during equal time intervals near the apogee and near the perigee -- choose one suitable time interval and draw the area swept out on your xy plot.)

In order to investigate Kepler's third law you search for other planets, but sadly there are none. Then you think of a wonderfully horrible idea and in a callous act of environmental insensitivity you catch the planet. (You think to yourself, "I can always put it back when I am finished".) You decide to throw the planet starting at a radius of 2 m above the surface of the star thrown tangential to the surface with a variety of different initial speeds.

Please plan this experiment carefully to avoid damaging the planet unnecessarily. Before throwing the planet establish the range of allowable initial speeds.

• What is the maximum speed above which the planet would disappear forever into outer space?
• What is the minimum speed below which the planet would smash into smithereens on the star's surface? (This one you may have to determine experimentally – but please catch the planet before it hits the star)

Choose about 5 different initial speeds and for each resulting orbit find the period T and the length of the semi-major axis R (i.e. find the mean of the apogee and perigee radius).

Make a plot of T vs. R and verify Kepler's third law.

After you have finished this experiment gently place the planet onto the surface of the star and try to kick it back into its original orbit. Can you do it? Give a plot of your attempts.