## Universal Gravitation

Satisfied that he's on to something, he analyzed the motions of the moons of Jupiter and Saturn and eventually comets and showed that they obeyed Kepler’s Law, like Kepler’s planets and now, like the Moon. Suddenly, the whole solar system, including the moons of all planets hung together in a single mathematical system. One set of rules for the all of the celestial bodies in our visible universe *and apparently fruit on our terrestrial home*. But apples, planets, and moons…what's special about them? Not much, but one quality is common to all of them: **they have mass**.

### Breathtaking Leap

In *Principia* he made a breathtaking leap. He assumed that a gravitational attraction exists between *any two objects with mass*! Not just moons and planets. Right now you are being attracted by the Earth, but also by the sun, and the Moon, and Jupiter, but also...by the banana on your desk that you’re saving for lunch . All objects in the Universe attract one another according to his original planetary universal rule. For example:

<img align="center" src="./../_images/fruitforce.png">

<BR CLEAR="all">

<figcaption> In the top, two objects, 1 and 2, with masses $M_1$ and $m_2$ , are a distance $R_{12}$ from one another. In the bottom, they each exert a gravitational force on the other of equal magnitude, but opposite direction (according to Newton's First law)
</figcaption>

```{aside}
An interesting fact about this equation is that it can only be solved exactly for two objects. Add a third object—or a fourth, or fifth, etc—and the equation cannot be solved. Rather it is necessary to solve it *approximately* and after Newton people became very skilled at doing very complicated approximation calculations called “perturbations.

```

The force $F_{1,2}$ that some object #1 with a mass $M_1$ attracts some other object \#2, with a mass $m_2$…and *visa versa*. This attraction is along a line connecting their centers which are $R_{1,2}$ apart. The relation that describes this is just an abstraction of his original Moon rule:

$$
F_{1,2} = G\cfrac{M_1m_2}{R_{12}^2},(\#eq:universal)
$$

the **Universal law of Gravitation**. The constant of proportionality, $G$ is called **Newton’s Constant** or the **Gravitational Constant** and is very tiny and not known well. The uncertainty on that number is 0.021 out of 6.67, or about 0.3%. For a fundamental constant of nature, that’s not very precise.