Towards the Physical Universe

Although Kepler did succeed in providing a "correct" model of the Solar System, there was no physics in it. All was empirical. While empirical models clearly have some value, they are ultimately unsatisfying as we have a strong desire to understand the reasons that the empirical model works. Without the underlying foundation there is no physical description and the Universe can appear to be ad hoc and capricious (i.e. what guarantee is there that Kepler's empirical laws apply to other solar systems?). Fortunately, Isaac Newton (1642-1727) was able to provide the foundation. Newton postulated that there was an attractive force between the Sun and the planets that had to account for Kepler's third law. Knowing that the Kepler's second and third laws meant that the orbital velocity of a planet must be a function of distance from the Sun, Newton deduced the functional form of the gravitational force law that would be required to reproduce these laws. As Newton was able to show, Kepler's third law requires that the gravitational force between two objects decrease as the inverse square of their distances.

We can derive Kepler's Third Law from Newton's equations as follows:

Step 1: Assume that M₁ is a small mass in a circular orbit about a much larger mass M₂ . We can write down the Force law on M₁ using Newton's formulations:

Step 2: Combining terms yields:

Step 3: In an orbit governed by a central force, the centripetal acceleration, a is given by:

Step 4: For a circular orbit, the circular velocity, V_c is the total distance traveled (the circumference of the circle) divided by the orbital period, P or V_c = 2p R/P which then yields:

in which the harmonic law of Kepler is now apparent as the R³/P² term.

Measurements of R and P for any lesser body ( M₁ ) in orbit about a larger body ( M₂ ) now gives directly a physical quantity, mass.

The genius of Newton was his ability to use calculus and coordinate transformations to reveal the underlying physics. That is beyond the scope of this book. The point of this derivation was to show that only the R^-2 force law can yield Kepler's Third Law. No other force law will work. If this gravitional force law is universal then Kepler's laws must also be universal.

The Distances to the Stars

Newton's formulation of gravity allows Kepler's laws to become natural, which signifies the arrival of the first physical cosmology. In this cosmology, all observations can be explained by some physical model. In turn, observations provide information on how the models can be refined and improved. However, even though the mechanics of the Solar System was now understood, there was certainly no understanding of the nature of the stars. For the next two centuries, advances in our cosmology were linked to efforts to determine the overall size of the Universe (note that this effort continues today). During the 18th century, accurate positions of many stars were recorded. This was really the first era of observational astronomy - cataloging the positions and brightnesses of stars based on visual observations through telescopes. This also was likely an age of great anticipation because the first accurate measurement of the stellar parallax of even just one star would provide the first observational determination of the size of the Universe.

The most substantial catalog was that of William Herschel (1738-1822) who used a massive 72-inch telescope to make his naked eye observations. Herschel produced extensive maps of the positions and brightnesses of stars. In fact, Herschel's maps suggested that the Sun was actually at the center of the distribution of stars so, once again, we appear to occupy a special location in the Cosmos. We now know that this is an illusion caused by interstellar dust. As the Sun is located in the plane of the Milky Way galaxy, our ability to see distant stars in that plane is severally limited because interstellar dust strongly absorbs the light from distant stars, making them much fainter than they otherwise would be. Had there been no dust, Herschel would have easily noticed that one direction of the sky (the direction towards the Galactic center) contained many more stars than the opposite direction and consequently would have been able to make a fairly accurate map of the structure of our Galaxy.

From Herschel's catalog, the first accurate stellar parallax measurement of 0.32 seconds of arc for the star 61 Cygni was published by the F.W. Bessel in 1838. This was followed in 1839 by Thomas Henderson's measurement of about 1 arc-second for Alpha Centauri and the 1840 measurement of Alpha Lyrae (Vega) at 0.26 arcseconds. Recall that 1 parsec is the distance a star would have to be at to have a parallactic angle of 1 arcsecond. This distance is equivalent to 3.26 light years. By 1840, it became clear that even the closest stars were more than one million times farther away than our Sun. The Universe was now a large place and the intrinsic energy output of the stars had to be huge in order that their light could reach us. The source of that energy would remain mysterious until Einstein develops relativity approximately 70 years later.

Summary

Kepler used Tycho's observations of the position of Mars to develop his three empirical laws of planetary orbits. Embodied in those laws was the notion that a planet had to move faster when it was closer to the sun. Newton established the dynamical basis for Kepler's Laws by postulating the existence of an attractive force, gravity, whose magnitude depended on the inverse square of the distance between two objects with mass.

Herschel cataloged the positions of stars to help map out the structure of the Galaxy. In 1839/40 the first credible stellar parallax measurements were made of some stars in this catalog. These measurements showed that the stars had to be many millions of times farther away from the Earth as our Sun was. The source of the energy that could sustain such incredible brightnesses was unknown until the early 20th century.