6.6 Satellites and kepler’s laws: an argument for simplicity  (Page 3/5)

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${F}_{\text{net}}={\text{ma}}_{\text{c}}=m\frac{{v}^{2}}{r}\text{.}$

The net external force on mass $m$ is gravity, and so we substitute the force of gravity for ${F}_{\text{net}}$ :

$G\frac{\text{mM}}{{r}^{2}}=m\frac{{v}^{2}}{r}\text{.}$

The mass $m$ cancels, yielding

$G\frac{M}{r}={v}^{2}\text{.}$

The fact that $m$ cancels out is another aspect of the oft-noted fact that at a given location all masses fall with the same acceleration. Here we see that at a given orbital radius $r$ , all masses orbit at the same speed. (This was implied by the result of the preceding worked example.) Now, to get at Kepler’s third law, we must get the period $T$ into the equation. By definition, period $T$ is the time for one complete orbit. Now the average speed $v$ is the circumference divided by the period—that is,

$v=\frac{2\pi r}{T}\text{.}$

Substituting this into the previous equation gives

$G\frac{\text{M}}{r}=\frac{{\mathrm{4\pi }}^{2}{r}^{2}}{{T}^{2}}\text{.}$

Solving for ${T}^{2}$ yields

${T}^{2}=\frac{{4\pi }^{2}}{\text{GM}}{r}^{3}\text{.}$

Using subscripts 1 and 2 to denote two different satellites, and taking the ratio of the last equation for satellite 1 to satellite 2 yields

This is Kepler’s third law. Note that Kepler’s third law is valid only for comparing satellites of the same parent body, because only then does the mass of the parent body $M$ cancel.

Now consider what we get if we solve ${T}^{2}=\frac{{4\pi }^{2}}{\text{GM}}{r}^{3}$ for the ratio ${r}^{3}/{T}^{2}$ . We obtain a relationship that can be used to determine the mass $M$ of a parent body from the orbits of its satellites:

$\frac{{r}^{3}}{{T}^{2}}=\frac{G}{{4\pi }^{2}}M\text{.}$

If $r$ and $T$ are known for a satellite, then the mass $M$ of the parent can be calculated. This principle has been used extensively to find the masses of heavenly bodies that have satellites. Furthermore, the ratio ${r}^{3}/{T}^{2}$ should be a constant for all satellites of the same parent body (because ${r}^{3}/{T}^{2}=\text{GM}/{4\pi }^{2}$ ). (See [link] ).

It is clear from [link] that the ratio of ${r}^{3}/{T}^{2}$ is constant, at least to the third digit, for all listed satellites of the Sun, and for those of Jupiter. Small variations in that ratio have two causes—uncertainties in the $r$ and $T$ data, and perturbations of the orbits due to other bodies. Interestingly, those perturbations can be—and have been—used to predict the location of new planets and moons. This is another verification of Newton’s universal law of gravitation.

Making connections

Newton’s universal law of gravitation is modified by Einstein’s general theory of relativity, as we shall see in Particle Physics . Newton’s gravity is not seriously in error—it was and still is an extremely good approximation for most situations. Einstein’s modification is most noticeable in extremely large gravitational fields, such as near black holes. However, general relativity also explains such phenomena as small but long-known deviations of the orbit of the planet Mercury from classical predictions.

The case for simplicity

The development of the universal law of gravitation by Newton played a pivotal role in the history of ideas. While it is beyond the scope of this text to cover that history in any detail, we note some important points. The definition of planet set in 2006 by the International Astronomical Union (IAU) states that in the solar system, a planet is a celestial body that:

1. is in orbit around the Sun,
2. has sufficient mass to assume hydrostatic equilibrium and
3. has cleared the neighborhood around its orbit.

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