## Experiment II-8

Predict a Comet's Orbit Using Gravity With others plotting to kill the Holy Roman Emperor and overthrow the government, Johann Kepler (1571-1630) fled from his job as royal astrologer, astronomer, and mathematician in Prague.  His understanding of the sky allowed him to sell horoscopes to provide subsistence for his family.  While teaching school in Linz, Kepler attempted to improve his explanation of the planets' spacing by considering regular polygons that could be easily constructed.  Guided by the Pythagorean penchant for harmonies, Kepler wrote music for each planet and considered the cause of their motion.  Kepler suspected the rotating Sun equally swept the planets around the sky against a resistance proportional to the square root of the orbit radius, R.  With a particular planet's period of orbit proportional to both the size of the orbit (T ∝ R) and the resistance, (T ∝ resistance ∝ √R), Kepler proposed that the period, T ∝ R3/2.  This fits the actual periods of the planets!

• the spacings of the planets orbits: determined by the geometry of the 5 perfect solids
• the shape of their orbits: ellipses
• the changing speeds of each planet: each radius sweeps out equal areas in equal times
• and the relative speeds of the planets: T ∝ R3/2
These ideas revolutionized astronomy!  By finding the motions of the planets did not fit the Greek idea of perfect circular perpetual motion, Kepler suggested the motions needed CAUSES that matched rules he had discovered.

Isaac Newton (1642-1727) an English introvert explained the needed cause.  While his teaching job at Cambridge University was suspended due to the plague, he avoided infection by living on the family farm.  Here he pondered the motion of the Moon and considered a possible relationship with falling bodies on Earth.  Newton realized the orbit of the Moon could also be viewed as perpetually falling towards Earth but at a lesser rate perhaps due to reduced gravity further from Earth.  Newton found his hunch that gravity faded with the square of the distance was supported by Kepler's laws describing the planets as they sampled gravity at different distances from the Sun.  But the reclusive Newton kept his thoughts to himself.

Later, back teaching at Cambridge, Newton did experiments with prisms and wrote a controversial Theory of Light and Colors.  Its publication in 1672, brought him recognition as a scientist and member of the Royal Society.  In 1684 another member of the Royal Society, Edmond Halley, asked for Newton's opinion about a controversy with Christopher Wren and Robert Hooke about the force needed to cause heavenly bodies to move in ellipses in accord with Kepler's laws.  Halley was surprise to hear that Newton had already solved this problem.  Halley persuaded Newton to write down the solution which Halley arranged to publish as the Principia in 1687.  The synthesis of astronomy and physics in this work quickly established Newton as one of the great thinkers of all time.

### Experiment

The force of gravity, Fg, according to Newton depends on the respective masses of two bodies, m1 and m2, and the distance between their centers dG is a constant:
Fg = G•m1•m2 / d2.
This equation apparently applies to all objects in the universe!  This was a revolutionary break with the ancient belief that the heavens followed different rules than earthly occurrences.

In this experiment, we shall attempt to predict the path of a comet using Newton's law of universal gravity.

Until Newton's time, observers of comets believed they were earthly phenomena, below the heavens because they moved rapidly past the stars and were not permanent.  But Halley and others identified that comets are heavenly objects, often with periodic reappearances.

According to Newton, any force on an object not countered by an opposing force will cause a proportional acceleration, a, inversely related to that object's mass:
a = F / m.
Gravity attracts objects such as a comet and the Sun depending on their distance of separation.  The force of gravity causes the comet to accelerate.  The amount of acceleration can be determined by combining Newton's two equations above:
a = [G•ms•mc / d2] / mc
a = G•ms / d2.
So gravity continuously accelerates a comet, changing its speed and direction of motion, depending on its distance to the Sun.

### Procedure

But it would be difficult for us to continuously calculate the constantly changing acceleration and simultaneously plot the orbit.  So we will elect to approximate the path of the comet by applying 60 days worth of acceleration all at once, then letting the comet coast ahead while we calculate the next acceleration.

The only variable controlling the change in the comet's velocity (direction and speed) is distance to the Sun.  So to predict the path of a comet, we need to construct a graph of acceleration compared to the distance.  Use the following information based on Newton's equations.

1. Construct a graph of change of velocity verses distance from the Sun real size using either inches or centimeters so that the specified 1.87 inches = 4.75 cm really is that length on your graph paper.
 Distance from Sun Change in Velocity AU inches cm AU inches cm 0.75 1.87 4.75 1.76 4.44 11.3 0.8 2.00 5.08 1.57 3.92 9.97 0.9 2.25 5.72 1.23 3.07 7.80 1.0 2.50 6.35 1.00 2.50 6.35 1.2 3.0 7.62 0.69 1.74 4.42 1.5 3.75 9.52 0.44 1.11 2.82 2.0 5.0 12.7 0.25 0.62 1.57 2.5 6.25 15.9 0.16 0.40 1.02 3.0 7.50 19.1 0.11 0.28 0.71 3.5 8.75 22.2 0.08 0.20 0.51 4.0 10.00 25.4 0.06 0.16 0.41 4.5 11.25 28.6 0.05 0.13 0.38
1. Using a large piece of paper (perhaps 4 letter papers fastened together), place a dot in the middle representing the Sun
2. Using a scale of 1 Astronomical Unit (AU) = 2.5 inches = 6.3 cm, locate the comet 4 AU from the Sun (10 inches ≅ 25 cm).
3. We shall assume the comet is initially traveling 2 AU per year which on our scale is 0.83 inches = 2.14 cm in 60 days.  At a right angle to the direction towards the sun, accurately construct ray this long for the 60 day path of the comet.
4. If no gravity applies, the comet would continue to coast through space in this direction at this speed.  For construction purposes, lightly draw a second equal ray showing the continuing coasting path of the comet at the same direction and speed for another 60 day period without gravity. 1. But we wish gravity to apply.  Place the origin of you graph at the dot for the Sun.
2. Align distance to Sun axis of the graph with the direction to the comet's location at the start of this 60 day period.
3. Note (measure?) the change in velocity at this distance the comet is from the Sun
4. Rotate this change in velocity 90° towards the sun (because gravity pulls towards the Sun) and draw a ray that length from the comet towards the sun.
5. Recalling that vectors such as velocities need to be add sequentially, carefully duplicate the length and direction of this change in velocity to the other end of the coasting path. 6. Combine the coasting path and the change of velocity by drawing the comet's path from its location at the beginning of the 60 days to the tip of the change of velocity ray, finding the location of the comet at the end of the 60 days.
7. Duplicate this new 60 day path to determine where the comet would next coast without gravity.
5. Repeat steps a to g predicting the comet's path.
Is the path nearly an ellipse as Kepler suggests for all heavenly bodies?  How does the comet's speed change with its distance from the Sun? Is kepler's equal areas confirmed?  Find the center of the orbit and calculate the eccentricity.  How well does the technique of applying gravity in bursts of attraction work?

### References

• Based on an experiment developed for teaching physics by Gerald Holton, F. James Rutherford, Fletcher Watson, directors, Project Physics, at Harvard University, funded by U.S. Dept. of Education and N.S.F., 1964-1975.
• David Cassidy, Gerald Holton, F. James Rutherford, Understanding Physics, Springer
• Johann Kelper, Harmonices Mundi, (The Harmonies of the World), Linz, 1619.
• Johann Kelper, Epitome Astronomiae, Linz, 1621.
• Isaac Newton, Theory of Light and Colors, London, 1672.
• Isaac Newton, Philosophiæ Naturalis Principia Mathematica, London, 1687. created 4/24/2003
revised 5/6/2003
by D Trapp 