Isaac Newton invented physics with his theory of universal gravitation: Every pair of bodies attract each another by a force directly proportional to both masses, and inversely proportional to the square of the distance between them, F = GMm/r². As you will see, he arrived at this rule by applying his ideas about force and motion to the orbits of planets and moons in the solar system. This combination of abstract theory (three mathematical laws of motion) with experimental observations (Kepler's description of planetary motion) has served as the archetype for physical science ever since.
The impact of Newton's ideas on the way we think about and work with nature cannot be overstated. The excitement over understanding celestial mechanics very quickly spread throughout the scientific and engineering world as people began to apply Newton's mathematical modeling approach to countless different problems. Newtonian mechanics ignited, then fueled the industrial revolution.
The first law is, in a sense, a special case of the second - namely the case of zero force. However, the first law really defines what "straight line" means in space-time according to Einstein, as you'll see later. The second law is the iconic F = ma, which defines both force and mass in a deliberately circular manner. The third law is what physicists now call conservation of momentum; it limits how bodies can interact.
Newton applied these laws to the case of the planets: At a given instant of time, you know from Kepler's laws that the Earth is accelerating directly toward the Sun at at a rate inversely proportional to the square of the distance between them, say a = K/r². Furthermore, things accelerate directly toward the Earth according to a different inverse square law, say g = k/r², where the Earth's constant of proportionality k is smaller than the Sun's K by a factor of about 330,000. The second law of motion asserts that the force of the Sun pulling on Earth is proportional to the mass m of the Earth, F = ma, and also that the force of the Earth pulling on the Sun is proportional to the mass M of the Sun, F = Mg.
Enter the third law: The force of the Sun pulling on the Earth must equal the force of the Earth pulling on the Sun, F = ma = Mg. (They act in opposite directions along the line connecting them.) Therefore the acceleration constants K and k are related by mK = Mk, since r refers to the same Earth-Sun distance. Newton's law of equal action and reaction plus the experimental inverse square acceleration laws for the Sun and Earth require that the observed ratio of acceleration constants K/k must equal the ratio of masses M/m (and the ratio of accelerations a/g). Thus, the Sun must be 330,000 times more massive that the Earth.
Furthermore, the ratio K/M must equal k/m. Call this ratio G. You could have made this argument with Jupiter or any other planet instead of Earth, so the ratio G = K/M must be the same for every body. In other words, any two bodies attract each other with a force proportional to both masses, F = GMm/r², where G is a universal constant of nature - the first such constant ever discovered. Since the acceleration constants K = GM and k = Gm, this universal law of gravitational force identifies the mass of any object as the source of the gravitational acceleration of all other objects toward it.
You have learned something completely new by applying the second and third laws of motion to the planets: Mass is the source of gravity! Newton's reasoning is quite simple, but turns out to be very deep. (The technically difficult part of universal gravitation is Newton's demonstration that Kepler's laws are equivalent to an inverse square law of acceleration - see Touching the Solar System.) The next step is to try to visualize how the inverse square law of gravitational attraction operates in a geometrical way.
Gravity is an acceleration vector at every point in space - the gravitational acceleration a small mass would experience at that point. According to universal gravitation, to compute this, you have to add the acceleration vectors GM/r² directed toward every other particle of mass M and distance r in the universe. Start by considering the acceleration produced by just a single idealized point mass - a particle so much smaller than its distance r that you can ignore its dimensions. The planets are good examples of point masses, since they appear as mere pinpoints of light from anywhere else in the solar system.
A vector associated with every point in space is called a vector field. Michael Faraday invented a way to visualize vector fields about a century after Newton. Faraday pictured lines of force filling space - a force field. He was looking at the lines traced by iron filings on a sheet of paper with a magnet beneath, but the picture works for gravity as well. The idea is to connect the vectors into (possibly curved) lines - the path you follow if you keep moving in the direction the vector points. So you picture the gravity of a point mass as myriad radial rays converging to the point, like long pins in a tiny pincushion. Each ray is called a field line. Notice that a field line is a directed line - like a one-way street. This field picture captures the direction of the acceleration vector everywhere, but at first glance it seems to lose any indication of magnitude - the length of the arrow.
However, the spacing between the lines decreases as they converge to the point mass. You can define their density at a given position along some field line by imagining a small loop around the field line. The loop will encircle other nearby lines, and you can define the density of field lines at that location to be the number of field lines threading through this tiny loop, divided by the area of the loop. If you move farther away from the point mass, you need to scale up the diameter of the loop in proportion to the distance for it to encircle a fixed set of field lines, and the loop area scales as the square of the diameter - hence distance from the point mass. Thus, the number of field lines per unit area scales inversely as the square of the distance! If you make the total number of lines emanating from the point proportional to its mass, the magnitude of the gravitational acceleration will be proportional to the number of field lines per unit area!
So far, you assumed the plane of the loop was perpendicular to the field lines. What happens if the plane of the loop is tilted relative to the field lines? The number of lines threading the loop will be proportional to the area of the projection of the loop into the plane perpendicular to the field line. The ratio of that projected area to the loop area when it was oriented normally is exactly the same as the ratio of the component of the acceleration normal to the tilted loop to the full acceleration, which is the component along the field line. Therefore, for any small loop, the component of acceleration normal to the plane of the loop is the number of field lines threading the loop per unit area. More subtly, you need to choose the sign of the component to be negative if the directed field lines thread the loop in the direction opposite to its normal.
Imagine a closed surface of any size (the surface of the Earth will soon become an example). A point mass at some arbitrary position produces an acceleration which varies as you move around to different points on the surface. The component of that acceleration normal to the surface (here the inward normal) thus also varies around the surface. The field picture makes it easy to compute the area-weighted average of this normal component of acceleration over the whole surface:
Since the normal acceleration component is the number of lines per unit area threading any small loop on the surface, its area-weighted average is just the total number of field lines piercing the surface divided by the total surface area. The one caveat is that you have to count each inward piercing line as plus one, and each outward piercing line as minus one, in order to be consistent about the sign of the acceleration relative to the surface normal direction you have selected.
There are two cases: First, if the point mass is outside the surface, then each field line pierces the surface twice - once going inward and a second time going outward, so the net number of piercings is zero. Second, if the point mass is inside the surface, then each field line pierces just once, on its way inward. (If the surface is not convex, there may be additional pairs of inward and outward piercings for some rays, but such pairs cancel each other.) Thus, if the point mass is outside the surface, the mean value of the normal component of gravitational acceleration over the whole surface is always zero. When the point mass is inside the surface, on the other hand, the mean value of the normal acceleration is always the total number of field lines converging on the point mass, divided by the total surface area.
A second point mass adds its acceleration vector to the acceleration from the first at every point in space. Because the acceleration vectors simply add, the averages of the inward component of the acceleration also simply add. Thus, if the new point is inside the surface, the average increases by the new mass divided by the surface area, otherwise the new mass contributes nothing. Accumulating more point masses in this way, you find that the inverse square law of universal gravitation has this remarkable property:
The average value of the inward acceleration component over any closed surface is proportional the the total mass inside that surface divided by the surface area.
The constant of proportionality is 4𝝅G, as you can work out from a spherical mass distribution (see below). This is a major result of vector calculus, called Gauss's law, discovered about a century after Newton. Faraday's powerful field line picture has made it almost obvious, showing you its geometrical significance - total number of field line piercings over total surface area. It is equivalent to the inverse square law, with the advantage that instead of "the square of the distance" you can think about the area of a surface.
Newton faced an immediate technical problem with his theory of universal gravitation: In the argument that the Earth's gravity diminishes as the square of distance between your downward acceleration on its surface and the acceleration of the Moon in its orbit, you assumed that your "distance from the Earth" was the radius of the Earth, that is, your distance from the center of the Earth. But with universal gravitation, the idea is that each little piece of the Earth - every rock and grain of sand and drop of magma in the whole ball - is individually pulling on you. When you add up all those individual forces in different directions, why should you get the same answer as if all that mass were concentrated at the center of the Earth? Unlike Newton, you can use Gauss's law to answer this question.
For any spherically symmetric object, the acceleration at its surface must point directly inward toward the center, and have the same value at every point of the surface - which is therefore the average inward component of the acceleration. Thus, the gravitational acceleration at the surface depends only on the total mass of the object divided by the surface area. How the mass is distributed does not matter, as long as it is spherically symmetric (otherwise the acceleration would differ from place to place around the surface). In particular, the acceleration at the surface is the same as if all the mass were condensed to a single point at its center. You have resolved Newton's technical problem, and universal gravitation has passed an important consistency test. (And you can work out the 4𝝅G constant mentioned above!)
Kepler's laws are equivalent to Newton's inverse square gravitation only when there are just two bodies, a smaller one orbiting a very much larger one. The motion of the planets is dominated by the Sun; the motion of the moons by their planet. However, according to universal gravitation, all the planets and moons are attracting one another.
In fact, our Moon is accelerating toward the Earth at about 1/3600 of the "one gee" surface gravity you feel, while it is accelerating toward the Sun over twice as fast, about 1/1650 gee. Although the Earth accelerates toward the Sun at nearly the same rate, their distances from the Sun differ by up to one part in 400, so according to the inverse square law, the accelerations differ by up one part in 200. Newton himself spent years calculating the details of how this time varying half percent difference perturbs the orbit of the Moon, showing that the gravity of the Sun accounts for all the irregularities of the Moon's orbit known since ancient times.
The idea that every mass attracts every other according to the inverse square law similarly explains a great many small deviations from Kepler's laws in the solar system as the result of attractions between multiple bodies. In fact, small unexplained perturbations in the orbit of Uranus led astronomers to predict and then discover Neptune.
Universal gravitation has immense built-in complexity because every pair of bodies attract each other, while at the same time a great simplicity in the rule of attraction between each pair. Complexity is a crucial feature, inasmuch as the motions in the solar system really do become more and more intricate the closer you look. Far from being completely deterministic, as people long assumed Newtonian mechanics to be, the fate of the solar system is provably unknowable on timescales of a hundred million years (called its Lyapunov time).