Gravity

FAS Astronomers Blog, Volume 32, Number 4.

Objects fall to the ground. That’s the nature of things on the surface of the Earth, and everywhere else.

Gravity is something that humankind has been familiar with since ancient times. Drop something and it falls to the floor. Jump up and you end up back on the ground. Throw a ball into the air and it arcs, eventually falling back to the ground. Sit under an apple tree, and eventually, a few apples will fall off the tree landing on the ground next to you. This might have happened to someone, some time ago.

Aristotle

Aristotle was one of, or possibly, the greatest philosopher of the ancient world. But he got it wrong. He, and the other philosophers of the day, made general assumptions about nature and then developed conclusions from them. In their thinking, objects have a natural state of rest and require a force to keep them moving. 

Galileo

Galileo took a different approach. He experimented with objects and based on his experiments, drew conclusions about how things moved. He developed three laws of motion.

1. Law of Inertia: Once moving, objects continue at a constant speed unless acted on by a force.
2. A falling object’s velocity is proportional to the time taken to fall.
3. Objects fall at the same rate regardless of their mass.

With this, Galileo discovered how things work here on the Earth. 

Kepler

Kepler looked up and developed the first understanding of how objects moved in the heavens. Kepler understood the motion of the planets. Although he didn’t discover the full laws of gravity, he did identify three laws of planetary motion.

1. Planets move around the Sun in ellipses, with the Sun at one focus.
2. The line connecting the Sun to a planet sweeps out equal areas in equal time.
3. The square of the orbital period of a planet is proportional to the cube of the mean distance from the sun.

Newton

The full understanding of gravity didn’t come until 1687 when Isaac Newton published his The Mathematical Principles of Natural Philosophy (Principia). He did more than just understand how gravity worked, he showed that the motion of the Earth, Moon, and planets is controlled by the same force of gravity that we experience here on the Earth. In other words, he combined the work of Galileo with Kepler, basing them on a firm set of mathematical laws.

In Principia, Newton introduced his three laws of motion.

1. Law of Inertia: Objects stay at rest or continue in uniform motion unless acted on by a force.
2. A force is required to accelerate an object (that is, change an object’s momentum) and the resulting acceleration is equal to the force divided by the object’s mass, a = F / m. 
3. For every force, there is an equal and opposite force.

Newton also developed his law of gravity, which says that the gravitational force between two objects is equal to a constant multiplied times the product of the objects’ masses divided by the square of the distance between the objects (F = Gm₁m₂ / d²). G is the gravitational constant (6.67259 x 10^-11 N m²/kg²), which is our measure of the strength of gravity.

We all have gravity. If any two of us were cast out into space, we would, over time, slowly move toward each other. 

Newton’s ideas were so successful that Alexander Pope said: “Nature and Nature’s Laws lay hid in night: God said, Let Newton be – And all was light.”

Falling Objects

If you drop two objects of different masses at the same time from the same height, they will hit the ground and the same time. This is because objects that have mass also have inertia (the resistance to acceleration). The mass and inertia of falling objects cancel each other out. Gravity pulls harder on a larger object, but it is harder to move a larger object. Therefore, objects, no matter their mass, fall to the Earth at the same rate, and are accelerated at: a = G m_e / r² = 32 ft/sec² = 9.8 m/sec². 

When objects fall toward the ground, they do encounter air resistance (aka drag). This is a frictional force that slows an object down. Air resistance depends on the speed of an object (it increases with speed) and the object’s cross section (area). It is roughly, F_air = – ½ CrAv², where C is a constant for air, v is the velocity of the object, r the density of air (1.2 kg/m³), and A the cross-sectional area of the object. Because of this, falling objects eventually reach a terminal velocity when they are no longer accelerating, that is, when the force of gravity equals the air resistance. 

Einstein

Newton looked at gravity as a force that reaches out over space instantaneously between objects with mass. Albert Einstein took a different view of gravity. His General Theory of Relativity, published in 1915, treated gravity as a warping or curvature of space-time. Gravity is not a force that attracts object, but a curved geometry that objects follow. Einstein’s equations are much more complex than Newton’s and look something like this: Rmn – ½ gmn R = -k Tmn. In a quote attributed to John Wheeler, Einstein’s equations are summarized as: “matter (with mass) tells space-time how to curve, space-time tells matter (with mass) how to move.”  

Einstein later modified his equation to include a “cosmological constant:” R_mn – ½ g_mnR – Lg_mn = -kT_mn. He supposedly called this his greatest mistake – maybe. Today, Lambda (the cosmological constant) is a center piece to the Lambda Cold Dark Matter (LCDM) model of the universe.

One interesting aspect of Einstein’s view of gravity is his “equivalence principle.” If you are placed in a closet with no windows and you feel a force, you cannot tell if the force is due to gravity or due to acceleration. In other words, gravity and acceleration are the same. Marcus Chown (The One Thing You Need to Know) explains how this principle shows that light bends in a gravitational field. If you are accelerating fast enough in a closed box and a light beam shines horizontally through the box, the beam will exit the box at a lower point than where it entered. The beam appears to curve downward. Because gravity has exactly the same effect, the beam would appear to curve downward in a gravitational field. In other words, per Marcus Chown, gravity is acceleration and we feel gravity because we are accelerating in curved space-time.

Another effect of Einstein’s view of gravity is that time is affected by a gravitational field. The stronger a gravitational field, the slower time runs. Therefore, time runs just a little faster for a satellite orbiting the Earth than it does for us here on the ground. If you were to find yourself at the event horizon of a black hole, the gravitational field is so great that time would stop.

Quantum Gravity

Despite all the success of Einstein’s theory of gravity, it doesn’t really work when we dive down into the constitutes of matter. There are some theories that attempt to describe gravity at the quantum level (e.g. Quantum Loop Gravity and String Theory), but for now, gravity at the macro level and at the quantum level just doesn’t mesh together. 

Four Fundamental Forces

Today, we consider gravity to be one of the four fundamental forces of nature (see The Standard Model of Particle Physics). The four forces are governed by four sets of particles – maybe. The graviton is still hypothetical and there is something called the Higgs. 

One would think that gravity is the strongest of the forces because it is all around us and it affects everything. It is not. Gravity is, by far, the weakest of the four forces. The four forces have the following relative strengths.

• Strong nuclear force: 1
• Electromagnetism: 1/137 (around 10^-2)
• Weak nuclear force: 10^-6
• Gravity: 6 x 10^-39 (around 10^-38)

Density

Still, gravity can have a significant effect on objects in the cosmos. We see this as the mass of an object is squeezed into smaller and smaller volumes. In this case, the distance over which gravity acts becomes smaller and smaller. In other words, the object’s density (mass per unit volume) increases, and the effect of gravity becomes “stronger.”

A rocky object around 500 to 600 miles in diameter (or possibly more) is large enough for gravity to overcome its structure and crush it into a sphere. For an icy object it is closer to 250 to 300 miles in diameter. This is why planets and dwarf planets are round and smaller asteroids are not. 

Gravity is how a star is formed. Gravity pulls dust and gas together over a long period of time. The star becomes denser as gravity squeezes the material into a smaller volume. Temperature rises, and eventually, it becomes hot enough that hydrogen atoms are fused into helium and a star is born. The pressure from this dense core pushes out and just balances the inward force of gravity. This balance is maintained over the billion or even trillion year lifetime of a star. 

The remnant material left over from a supernova can have a density so great that strange objects result. If the mass and density is high enough, gravity will overcome the electromagnetic force that keeps negative electrons and positive protons apart. The particles combine creating neutrons and a neutron star is born. If the mass is even higher, the density is so great that light cannot escape, and a black hole is the result.

Planetary Orbits and Radial Velocity (Exoplanets)

As we all know, the Sun’s gravity keeps planets in their orbits. Planets are moving just fast enough so that their orbital velocity just balances the gravitational force of the Sun, and planets stay in their orbits.

As a star’s gravity pulls on a planet, the planet’s gravity also pulls back on its star. As a result, the star wobbles due to gravitational force of the planet. Because the planet’s orbit takes it around the star, the star’s wobble moves in different directions. When viewed from the Earth, it is sometimes toward us and sometimes away from us. We can determine this “radial velocity” by measuring the Doppler shift of the light coming from the star. Today, this is one of the primary ways astronomers detect exoplanets orbiting distant stars.

Tides

Just like the planets, gravity keeps the Moon in its orbit around the Earth. However, there is another feature to gravity called tides (aka tidal forces). It is a little hard to explain, but tidal forces are due to the difference in the gravitational force (e.g., between the Earth’s surface and the Earth’s center). The Moon’s gravity pulls on the Earth’s oceans more on the side of the Earth facing the Moon and less on the side facing away from the Moon. The result is that the oceans bulge out toward the Moon on the near side of the Earth and away from the Moon on the far side of the Earth. 

Hypothetically, this would happen to someone falling into a black hole. The tidal forces between his/her head and feet would be so great that the individual would be stretched out. We call this spaghettification.

While the force of gravity varies with the square of the distance (see Newton’s law of gravity above), tides vary with the cube of the distance. This means that the tidal force of the Moon on the Earth is greater than that of the Sun because the cube of the Moon’s distance is much larger than the cube of the Sun’s distance. The equation for this is: dF = (2GMm/r³)dr, where dF is the difference in the gravitational force of M on m across a difference in distance of dr.

Gravitational Waves

Just like light, gravity propagates through the universe in waves and at the speed of light. But gravity is so weak that even Einstein thought we’d never be able to detect these so-called gravitational waves. That is, until the LIGO observatories came online. In 2015, LIGO was able to detect gravitational waves from the collision of two black holes in the distant universe. Eventually, they detected the collision of two neutron stars as well.