Understanding Elliptical Orbits

This activity is aimed at making middle school students understand the meaning of elliptical orbits. Many students have an idea of what the word ‘elliptical’ means. But when they are asked to draw the elliptical orbit of a planet around the Sun, they often place the sun at the ‘center’ of the ellipse, not at the focus. This simple experiment will help clarify the notion of an elliptical orbit.

What you don’t need to know

We will assume that the reader (of age 10-12) is not exposed to trigonometry or even the concept of a tangent to the circle. The emphasis is on observations with a handy tool to measure the angles, in particular, the altitude. The altitude of Venus (or Mercury) at the time of sunset or sunrise is (roughly) the angle between it and the Sun as viewed from Earth.

What you need for the experiment

What we need is a simple extension of a protractor. A hole is made at the centre and a thread inserted. A small weight is tied so that it serves as a pendulum, which is always perpendicular to the horizon. A straw serves as a sighting tube and can be glued/fixed with cellotape. When the device is held horizontally, the pendulum would be vertical. We will take this as the zero angle.

The instrument needed for altitude measurement.

When a planet is seen through the straw, you would need to measure the angle ‘A’ between the vertical pendulum and the zero of the protractor (see above). Then (90-A) will give the altitude.

Better methods of making such devices are available at the following websites:
Making a mini-clinometer

Measure for measure

Finding the altitude of the Pole Star

Orbits of Venus and Mercury

The idea is to measure the altitude of Venus every evening. As you may have noticed, when seen from the Earth, Venus reaches a maximum angular distance of roughly 47° from the Sun. This happens because Venus is an inner planet. The date of maximum elongation and the reading of the angle is recorded.

Detailed steps are described below. The dates correspond to the readings recorded by students during a workshop that was held in 2006 at the Jawaharlal Nehru Planetarium, Bangalore. Eg, the maximum (western, in the evening) elongation was recorded on 12 January.

Step 1: Draw a circle of radius 10cm to represent the orbit of earth; Mark the position of earth from different months; convention is to start with March21 as zero.

Step 2: Mark the positions of earth for the dates of observations as A (28/12) and B (9/1). The observed angles were 45 and 46; Mark them as SAA’ and SBB’. (Figure 1)

Figure 1

Step 3: Mark the angles for another two evenings in the same graph as in Figure 2. P (25/1) and Q (7/2) with angles 47 and 43 degrees. Among these values 47 deg is the maximum; this is called maximum elongation. Draw a perpendicular from S on to the line PP’. That fixes the position of Venus on the line PP’, which is the radius of the orbit.

Figure 2

Step 4:Draw the circle with this radius to get the orbit as in Figure 3.

Figure 3

Let us now draw the orbit of Mercury. In the same month, Mercury also reached maximum elongation at dawn. The measurements were done by students. The maximum occurred on 19th; the angle provided was 200°. They were asked to repeat the procedure as was done for Venus.

They got a circle, representing the orbit.
Step 1: On the orbit of Earth, mark positions A, B C and D corresponding to 7th, 10th, 12th and 15th of December. Draw the angles of elongation of Mercury as was done for Venus. Note that the angles are measured in the opposite sense as compared to Venus because Mercury was visible at dawn. (Figure 4)

Step 2: The maximum elongation occurred on the 10th and the angle was 20°.

Figure 4

Step 3: Draw a perpendicular from S to the orange line corresponding to maximum elongation. Use this as the radius and get the orbit as shown in Figure 5.

Figure 5

Step 4: Mercury reached maximum elongation at dusk on April 1st 2016. The angle as measured was 180°. Get the radius by the same method by drawing the elongation angles on the same figure. P, Q and R correspond to 28th March, 1st April and 3rd April. The maximum elongation line (blue) was used to get the radius of the orbit and a circle was drawn. The two points are obtained this way are M and N.

Figure 6

Step 5: Notice that this is a different circle. The points M and N are both on the orbit of Mercury. We can join the two to get a smooth ellipse as the orbit of Mercury. Thus emerges the idea of an elliptical orbit. (Figure 7)

Figure 7