By Peter Higgins, PhD

Isaac Newton lived from 1642 (the year Galileo died) until 1727. He was considered both the first of the age of reason, joining the likes of Laplace, and the last of the alchemists-astrologers who bedazzled Europe’s royalty.¹

He was a lonely figure, and a mystic, envied by many. Especially Robert Hooke, who tried to claim credit for much of Newton’s work. Newton’s greatest contribution however, and one that was his own work, was his proof describing the orbit of Mars that won the Wren prize² enabling astronomy to be viewed as a secular science.

Science in Newton’s time favored theory not observations, as is common today. Johann Kepler’s reduction of Tycho Brahe’s observations of the daily celestial position of Mars establishing its trajectory to be an ellipse having the sun as a focal point, was not considered then as a fact. Instead, Kepler’s findings were taken as only an explanation of the data, not a reality. In the same way, Nicolas Copernicus’s conclusion upon updating the tables of planetary position in common use then, that a heliocentric planetary model made more sense, was discounted as a mere mathematical convenience. It was believed that the Earth was stationary and the center of the Universe, and that the Sun, and other celestial objects, revolved around the Earth. Such stellar motion, based on the perfection of circular trajectories on divine spheres, was held as irrefutable. A heliocentric model would only be accepted if planetary motion could be explained using secular mathematics. But such a proof was unknown for at least two reasons and one logical criticism. The two reasons were (1) the mathematics of objects in motion didn’t exist, and (2) gravitational force was not qualified.  Of course, the pull of gravity was known, and Galileo was exploring it, but its role in determining a planet’s motion was not. The universally held criticism was the incongruity that the earth could be hurtling through space at 33 thousand miles/hour around the Sun. At such a speed, it was reasoned, why wasn’t everything blown away?

The Model of the Universe before Newton

When an observer records the position of Mars each night. It is seen that about every 26 months, Mars appears to make a loop in the sky as illustrated in Figure 2.

 

Before Newton this looping was assumed to actually occur, and to account for this, around 140 BC, Hipparchus (the greatest of the Greek astronomers), proposed that planetary motion³ followed circular trajectories called epicycles, centered on a larger orbit known as its deferent as sketched in Figure 3. Planetary movement on this path was explained as being divine.

 

As was necessary to make horoscopes for royal courts, elaborate tables were published for astrologers  giving planetary coordinates. These were first formalized by Claudius Ptolemy. Planetary motions described by these tables are very accurate (this system is still used today in planetarium software), and it’s accuracy helps explain the reluctance to accept a heliocentric view then..

The apparent looping being observed is now understood as an illusion brought about by the relative positions of Earth and Mars in their orbits when Earth passes by Mars. Earth is orbiting the Sun about twice as fast as Mars orbits the Sun. About every 26 months, Earth comes up from behind and overtakes Mars. Normally, Mars appears to move from West to East, but in this 26 month overtaking Mars appears to move from East to West as Earth overtakes Mars as diagrammed in Figure 4:

 

Everything Changed when Christopher Wren Offered a Prize

Shortly after 1666, Christopher Wren, made wealthy and famous as the architect of London following the great London fire, offered a prize to anyone proving that conventional forces (not angels) were responsible for planetary motion explained by Johann Kepler in 1609. To win the prize, the proof had to use secular ideas and mathematics (no magic, no divine intervention).  Both Hooke and Newton claimed to have done such  a proof, but only Newton produced it to Wren’s satisfaction.⁴

Newton’s Proof

According to laws of inertia, a force is required to alter the course of a body in motion.  A force diagram (below) aids in understanding how Mars changes direction to follow a path in space. Newton proposed this force to be the pull of gravity from the Sun. Newton asserted that the gravitational force between objects is proportional to the product of their masses divided by the square of the distance between them. To mathematically associate gravitational pull with geometry, Newton made use of similar triangles first recognized by the Greek mathematician Thales of Miletus. Using this the force components  (F_x,F_y)  are proportional to the distance components (x, y) because their angles are the same as seen in Figure 5

 

In figure 6, Mars is drawn at the starting position at the beginning of the analysis (1), and after one step in the analysis (2). Each step is an interval of time advancing the planet around the Sun until it has made a complete orbit. It is the shape of this trajectory that Newton was challenged to predict (in particular proving the orbital shape matched Kepler’s ellipse). The planet is subject to two accelerations, a_y and a_x, determined from the gravitational force and directed towards the Sun aligned in cartesian coordinates. The planet will then move from these accelerations assuming it has an initial inertia. Each step is done recursively. recursively:

 

Three calculations are needed in the solution to advance from (1) to (2).:

1. Compute the components of acceleration from the x, y position at (1)
2. Compute a new velocity average from this acceleration which is v = old v + a dt
3. Compute new position coordinates at (2) by adding the vector change v dt

Once a new position is computed, the same process is repeated (recursion) until the planet has travelled completely around the sun.

 

Students can program these calculations by inserting instructions for steps 1-3 in cells in Excel, letting Excel repeat the calculation through enough steps to complete a path around the sun, then plotting all the positions.  Richard Feynman presents the details for this in a lecture². The resulting plot, Figure 7, is given below showing just what we were looking for: The trajectory of Mars is an ellipse with the Sun at a focal point!

 

The Importance to us of this event

Newton’s proof described above was a truly remarkable achievement. It leveraged his understanding of the force of gravity, and his new mathematics, calculus. Recalling the legend of the falling apple, watching it fall, Newton made the association of a planet falling towards the Sun, but due to its inertia it orbits the Sun instead of falling into it.

Since about 500 AD, the papacy held firm to the concept of heaven and earth. In this view, everything in the heavens, such as the celestial spheres, was the realm of the divine, and were perfection. Apparent stellar movements were caused by angels. Conversely, everything on Earth was considered flawed, being corrupted by sin. The universe described by Ptolemy adhered to the papal view. Copernicus’s proposal of a heliocentric model of the universe written in De revolutionbus ran afoul of Pope Paul III prompting the addition of a forward to his book recanting its conclusion⁴. Later, Galileo was subject to an inquisition⁶ resulting in house arrest when his views were contrary to the church.

In opposing the heliocentric view of Copernicus, the pope knew that planetary motion could not be predicted by current secular mathematics that would validate Kepler’s elliptical Mars trajectory about the Sun. It took the intervention of an influential, and rich man (Wren), and the brilliance of Newton, to bring about papal acceptance of the heliocentric model.  Eventually this acceptance would allow intellectual freedom to pursue such strange concepts as entropy, dark matter and dark energy, the Big Bang, etc. without threatening proponents with inquisitions.

 

Sources

1. John Maynard Keynes: Newton, the Man. https://mathshistory.st-andrews.ac.uk/Extras/Keynes_Newton/
2. The Feynman Lectures on Physics, July, 1966. Reading, Massachusetts, Addison Wesley, V2, 9-3
3. Evans, J; 1998. The History & Practice of Ancient Astronomy, New York. Oxford University Press
4. Asimov’s Biographical Encyclopedia of Science & Technology. 1982. Garden City, N.Y. Second Revised Edition. Doubleday & Company, Inc.
5. Gingerich, O. The Book Nobody Read.  2004. Penguin Books.
6. The Galileo Affair. https://en.wikipedia.org/wiki/Galileo_affair
7. Baker, R.H. Astronomy. Seventh Edition; Princeton, New Jersey, D. Van Nostrand Company, Inc.