# Illustration of an elliptical orbit

##### Demonstration

Three methods of visualising elliptical orbits.

#### Apparatus and materials

Either a

Retort stand with rod and clamp, or tripod

Small steel ball approximately 5 mm diameter, or a marble

Large glass funnel at least 20 cm diameter

Or b

Steel ball approximately 3 mm diameter

Rubber sheeting

Toy hoop or embroidery frame

Retort stand, rods, clamps, boss head

Or c

Alpha-particle scattering analogue apparatus

#### Health & Safety and Technical notes

For method b, stretch the rubber sheeting over the rigid horizontal circular frame and secure it tightly with tape. It should be stretched a little, equally in all directions.

Fix a vertical metal rod over the sheet so that it is pushing the sheet into a curved well, thus imitating, roughly, an inverse-square-force potential.

For method c, the apparatus is available from a number of suppliers (Philip Harris, Griffin Education, ASCOL), see Alpha particles scattering analogue (plastic hill and ramp).

#### Procedure

a Firmly hold the glass funnel vertically and let the ball fall into it. Friction will affect the orbit and make it precess, but the motion around the funnel will be elliptic. Select a ball which will fall right through the funnel.

b Project the small steel ball across the sheet. By choosing suitable initial conditions the ball can be made to describe an oval like an ellipse with one focus on the axis of the well.

c Balance the plastic or aluminium hill upside down. Wood blocks can be used to hold it in position. Project the ball across the inverted hill so that it will orbit the centre. The elliptical path will be visible.

#### Teaching notes

1 Method a is not a true representation of Kepler's Second Law. The ball starts in a circular orbit, but it gradually moves lower in the funnel because of friction and so the time for a circular orbit gets smaller. The orbit also precesses as a result of friction.

2 Method c uses a valley with a 1/r profile. This corresponds to a 1/r2 force of gravitational attraction and so will closely model the movement of a satellite around a larger mass.