Densities of the Planets |
In Chapter 7 you learned about planetary densities (Section 7.1). If we can determine the density of a planet, we can make intelligent guesses as to what it might in fact be made of. However, density cannot be observed directly, but must be derived from other quantities (specifically, mass and volume). When we calculate the density of a planet, what we have is a bulk density or average density of the planet as a whole. Physics dictates the likelihood that the actual densities of materials at the surface of the planet will be less than those at its core.
Having determined the distance to the Sun, and having a few laws under our belts, we are ready to drop in a few numbers, but we will hold that for the Activity. As you have learned, density is the product of Mass divided by volume. The accuracy of our knowledge of a planet's volume depends on the accuracy of our knowledge of its distance from the Earth. Before the the invention of radio and radar, depended on our data on the distance of other planets depended on the accuracy of determination of the distance to the Sun. (We discussed this on the Further Explorations for Chapter 7). Today we know the planetary distances with high accuracy. By observing a planet an noting its angular size, we can easy compute the true size (and volume) if we know the distance.
What remains, though, is the mass of the planet. Sir Isaac Newton studied Kepler's Laws, and realized that the inverse square relation between distance and period implied that there was a force whose strength varied with this same relationship. Kepler's Law represented a proportionality, defining how one quantity varies in relation to another. Newton introduced the concept of the Gravitational Constant, G , into the proportionality, which turned it into a genuine equation:
where a is the distance of the planet to the Sun in Astronomical Units, and P is the period of the planet's orbit expressed in years. (See figure 7.5 in your text.)
(Don' t let the equation frighten you. The calculation is really no more complicated than a series of multiplication and divisions!) The problem for Newton was simply that G was a quantity that had to be determined by measurement -- and he couldn't figure out how. He could have done it if he had known the mass of the planet to begin with, but of course that was what he was trying to find. Newton incorporated the Gravitational Constant into in Universal Law of Gravitation as well, but never successfully determined the value of G .
Some 48 years after Newton's death, Nevel Maskelyn substituted a mountain for the Sun and a small pendulum bob for the Earth, and successfully measured the force between them (by how much the pendulum bob moved toward the mountain). Unfortunately, the precision of his result depended on his estimate of the mass of the mountain, which wasn't very good. Another 23 years later, in 1798, Henry Cavendish performed an experiment in which he measured the gravitational force between two lead balls. Oddly, although his experiment eventually led others to a good value for G , that was not what Cavendish was trying to find. He was looking for the density of the Earth!
Today's value for G is 6.67259 * 10 -11 m 3 kg -1 s -2 . The periods of the planets were easy to observe, the distances and planetary sizes are equally easy once the distance to the Sun was known, so mass and density can be calculated.
Ready? Click here for the Activity .
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