Astronomy/Motion and Gravity

Gravity is a type of force that causes bodies with mass to be drawn toward each other. Any object that has mass also radiates a gravitational force. The strength of the force is directly proportional to the amount of mass possessed by the body.

Relative to other fundamental physical forces, gravity is extremely weak. For example, the electrical force between a proton and an electron is about 10³⁹ times as strong as their mutual gravitational attraction. The only reason that gravity dominates our day-to-day world is that the positive and negative charges tend to balance each other out, while the force of gravity is cumulative with mass. I.e. there is no known comparable force of anti-gravity working to cancel out the gravitational attraction, as there is attractive and repulsive forces in electrical forces.

Newton's laws of motion

Isaac Newton first derived the law of universal gravitation in 1666. This law holds that the force of gravity between any two particles with masses m₁ and m₂ that are separated by a distance r is a mutual attraction that acts along the line between the particles, and which has the value:

$F=G\frac{m_1{m}_2}{r^2}$

The parameter G is the universal constant of gravitation, and it is the same for all particles. The value of G has been determined by experiment to be equal to 6.67×10^-11 N·m²/kg².

Note that there is no limit to the size of the nominator. The force of gravity can be greater than any other; it can even collapse a star!

The reason why the force is proportional to the distance squared can be understood by surrounding the particle by an imaginary sphere. Now imagine a very small postage stamp on the surface of the sphere. The force of gravity radiated by the particle acts the same in all directions, and it must pass through the outer surface of this sphere. So regardless of where the stamp is placed on the surface, it experiences the same force.

The surface area of any sphere is proportional to its radius squared. Thus if the same sphere were to be expanded in radius by a factor of r, the surface area of the sphere would likewise increase by r². Now if the postage stamp is placed on the surface of this sphere, it covers a smaller portion of the total area. Thus the portion of the force acting on the stamp has decreased by exactly the same amount by which the sphere's surface has increased in area.

The formula for gravitational potential energy ($U$) is

$U=\frac{3G{M}^2}{5r}$

where $G$ is the Gravitational Constant, $M$ is the object's mass, and $r$ is the object's radius.

Attraction of a spherical body

Newton's law above applies to very tiny particles. In the case of a planet, however, the attractive force is created by an immense number of particles occupying a volume in space. It is not immediately obvious whether the entire planet will create a force on a particle that follows the law of universal attraction. Some parts of the planet will be closer, and so exert a stronger force, while other parts will be further away and the force will be weaker.

It was exactly this problem that drove Newton to formulate the calculus. This is a branch of mathematics that allows huge quantities of very tiny amounts to be added up. When he applied the calculus to a spherical body, Newton added up the attractions of the many tiny particles. By this means was able to demonstrate that the forces of the particles precisely balance each other out. The resulting cumulative force acts exactly as if all of the mass were concentrated at a single point at the center. I.e. to determine the force of gravitational attraction from a spherical planet upon a particle, it is only necessary to know the total mass of the planet.

As we know the force of the Earth's gravity on a body standing on its surface, and the radius of the planet, we can use this to determine the total mass of the planet. The gravity at the surface of the Earth is commonly represented by the symbol g, which has a value of 9.80665 meters/seconds². By convention, the surface gravity of a planet is given in multiples of g. Thus the surface gravity on the planet Mars is 0.37 g.

Orbital motion

Based on the observations of Galileo Galilei, Newton also devised a series of three laws of motion. The first of these laws states that, every body persists in its state of rest of uniform motion in a straight line unless it is compelled to change that state by forces impressed upon it.

In the absence of gravity or other forces, if a bottle rocket was fired into the air it would continue to travel in a straight line. The gravity of the Earth acts upon the rocket, however, and bends its trajectory back toward the surface in a curving arc. Suppose, however, that you continue to improve upon the rocket and give it greater and greater thrust. The rocket would still continue to follow an arc, and the distance to where it eventually lands steadily increases.

As the Earth is a sphere rather than a flat surface, such very long arcs would begin to benefit from the curving surface and the rocket would land further away than they might on a flat plane. (The landing spot is, in effect, "lower", allowing the rocket to travel further before it lands.) If this cycle is continued with ever more powerful rockets, eventually the curvature of the surface would fall away from the rocket as quickly as it descended. Thus it would continue to fall forever around the Earth. This type of trajectory is called an orbit.

Orbits occur whenever an object in space is moving along a trajectory that will carry it past another object, rather than directly toward it. During the course of this movement, the gravitational attraction between the two objects draws them toward each other and deflects the otherwise straight line path of the first object.

When the velocity of the first object is too low, the gravitational force will draw them together and they will collide. If the velocity is sufficiently high, however, the first object will whip past the second and continue onward along its new path. (The second object also gains a new trajectory in the process.) It is only when the velicity difference of the two objects is within a certain range relative to their masses and separation that the two objects will orbit each other in a stable configuration.

Tides

Tides occur when two large objects, such as planets, are in relatively close proximity to each other. In this case Newton's tidy summation of the gravitational attraction from each of the planets still applies. However the attraction of a planet differs between the near and far sides of its companion. That is, because the back of a planet is further away from the companion than the front side, the attraction is weaker. The difference between these two forces creates a net tidal "force" on the planet.

A tide acts along a line joining the two objects. It acts to elongate the bodies along this line, while there is no net tidal force acting perpendicular to this line. As a result, the body tends to become elongated in this direction, forming an oval shape.

Example

As an example, suppose there is a small moon in a perfectly circular orbit around a large planet. The radius of the moon is indicated by R. Let the separation of their mid-points by indicated by S (which must be at larger than R). On the side of the moon nearest to the other planet, the gravitational force from the planet is proportional to:

$\frac{1}{(S-R{)}^2}$

irrespective of their mutual masses. On the opposite side of the moon, the gravitational force from the planet is proportional to:

$\frac{1}{(S+R{)}^2}$

The difference between these two forces is the net tidal force on the moon. It is left as a algebraic exercise for the reader to show that the difference between these two forces can be reduced to the following expression:

$\frac{4SR}{(S^2-R^2{)}^2}$

When the separation of the two bodies is reasonably large compared to the diameter of the moon, the bottom term becomes approximately equal to S⁴. Thus the force is nearly proportional to:

$\frac{4SR}{(S^2{)}^2}$

Since the S in the numerator cancels out one of the exponents of the S⁴ in the denominator, you can see that the tidal force becomes roughly proportional to the inverse cube of the separation, or S³.

This approximately inverse-cube relation of the tidal force turns out to be fairly typical. Thus since the gravitational attraction between the bodies only decreases as the inverse square, the tidal force decreases much more rapidly with distance than does gravity.

Roche limit

The inverse cube relation of tidal forces begins to have an interesting effect when a small moon is orbiting very close to a large planet. The tidal force elongates the moon, bringing the near side closer to the planet and the far side further away. This increased distance along the direction of the tidal force only serves to accentuate the force.

When a sufficiently strong stretching force is exerted on any physical body, eventually this tug will overcome the internal forces holding the body together and it will be ripped apart. A similar effect can occur with tidal forces.

The distance at which a tidal force is strong enough to tear a moon apart is called the Roche limit. If a moon were to slowly spiral inward, moving closer and closer to a large planet, eventually it would reach this limiting distance and be ripped apart. The resulting debris would still continue to orbit around the planet, but would spread out to form a ring. An example of such a ring of debris can be seen around the planet Saturn.

The actual formula for the roche limit depends on the ratio of densities for the two bodies, as well as wether they behave like rigid, inflexible spheres or bodies of liquid. Here are some sample roche limits derived for rigid bodies within our solar system.

Body	Satellite	Roche limit
		Distance (km)	Radii
Earth	Moon	9,495	1.49
Earth	Comet	17,883	2.80
Sun	Earth	554,441	0.80
Sun	Jupiter	890,745	1.28
Sun	Moon	655,322	0.94
Sun	Comet	1,234,186	1.78