Thursday January 24
Kepler discovered how the planets orbit the Sun. But not why. Did they orbit the Sun because of magnetism? Magic? Wind from the beating of angels wings? He didn't know.
The modern understanding of gravity dates the the work of Galileo. (You may know of the famous and possibly apocryphal story of Galileo dropping two cannonballs of the leaning tower of Pisa.)
In Galileo's time, mechanics was still based on the philosophy of Aristotle. And, while Aristotelian physics has a certain aesthetic appeal, it doesn't make any sense when confronted with the actual world. Rocks keep moving through the air after you throw them, even though you are no longer exerting any force on them.
Galileo was the first modern physicist in that he was the first person to conduct his own experiments. This was truely revolutionary.
Among the things he discovered was that objects do not fall at a constant rate. Instead, they accelerate as they fall to the ground.
He also discovered (and this is where the Pisa story comes in) that the rate at which an object fall DOES NOT DEPEND ON ITS MASS!!!!!. Heavy objects fall no more quickly than light objects.
Everyone can think of an obvious counter-example to this statement: If you drop a book and a sheet of paper, the book falls faster. But that's not because the book is heavier. It's because the paper is held up by air resistance. This was the jumping off point for a famous camera-op during the Apollo 15 Moon landing. Astronaut David Scott dropped a hammer and a feather for the TV audience. Because the moon has no atmosphere, the hammer and feather dropped together to the lunar surface.
One other crucial discovery he made was that moving objects continue to move unless some force acts to stop them.
This is as far as Galileo got with the subject. It occupied the last ten years of his life, while he was under house arrest. The matter was taken up, and put in its modern form by Isaac Newton.
Isaac Newton first examined inertial or unaccelerated motion. This led him to what are now called Newton's Laws of Motion:
And the second law can be written algebraically as
From here, Newton went on to consider the behavior of falling objects. As falling objects accelerate downward, they MUST be acted on by a force. But, as noted by Galileo, the rate of acceleration does not depend on the mass of the falling object. This is inconsistent with Newton's second law UNLESS the force of gravity depends on the mass of the falling object.
Newton realized that his third law of motion (the reaction law) meant that any two massive bodies must exert equal gravitational forces on each other. So the force of gravity between any two objects has to depend on the masses of both objects.
Newton asked the question "Is this force on the Moon the same as the force of gravity we feel on the surface of the Earth?"
Tuesday January 29
Now, the Moon is in a roughly circular orbit around the Earth. Circular motion also requires a force. This is because any change in velocity requires an acceleration. And velocity isn't the same as speed. Two objects that are moving at the same speed (say two cars on the highway, both moving 50 mph), but in opposite directions (one car moving north, and the other south) do NOT have the same velocity (in this example the velocities differ by 100 mph). To change the direction that an object is moving in requires a force, even if the object's SPEED remains constant.
Newton reasoned that the force should follow an Inverse- Square Law. This is just a reflection of the geometry of a sphere. The surface area of a sphere increases as the square of the radius. Take the example of a light source of some fixed luminosity. If you get twice as far away from it, it appears only a quarter as bright.
Newton reasoned that gravity should behave in this way also. Taken together, this led Newton to the following formulation for the law of gravity:
The m1, M2 are the two masses involved, the R1,2 is the seperation between the masses. The G is a constant (the Gravitational Constant).
Newton knew this law was consistent with his Laws of Motion, and with the observed behavior of falling objects near the Earth's surface (local gravity). Now we return to his question: "Is the force that keeps the moon in a constantly accelerating (nearly circular) orbit around the Earth the same as the force of gravity we feel on the surface of the Earth?"
Among the consequences of Newton's Laws of Motion are rules that govern objects in circular motion (and you'll need to take a physics class if you want to see the derivation of this). So he knew that any object in circular motion must have an acceleration that depends on its speed and the radius of the circle involved:
Newton knew the distance to the Moon (this was known from parallax measurements), and he knew the moon's orbital speed (from its distance and its orbital period). So he knew the acceleration the Moon HAS to be experiencing to be in the orbit it is in:
Newton knew the size of the Earth, and he knew that the distance between the Earth and the Moon was about 60 Earth radii. Given this, he calculated
Or about 1/3600 of the acceleration due to gravity at the Earth's surface. The acceleration of gravity on the Earth's surface is about 9.8 m/sec/sec. This means the acceleration on the Moon due to the Earths gravity should be about 2.7x10^-3 m/sec/sec. This is exactly the acceleration that the Moon must be experiencing in order to stay in a circular orbit around the Earth.
I can't stress enough how important this was. Kepler had figured out the motions of the planets, but not why they move as they do. Newton figured out why. His law of gravity is one of the most important tools we have for understanding the behavior of objects in the Universe.
This is worth playing around with some, as we will be using this relationship for the rest of the semester. First of all, it is consistent with the 3rd law of motion (action-reaction). Notice that this means you exert as strong a force on the Earth as the Earth does on you. But the Earth is much bigger than you.
But consider the result of two bodies with masses that are fairly close to the same. Or, by analogy, consider the problem of riding a see-saw with someone who is only 3/4 of your weight. How does that work?
The two masses orbit a common center of mass.
One can derive a few more useful relations from the gravitational force law. The first is an equation for a circular orbit:
The second is an equation for the escape velocity of an object:
Don't get caught up the algebra. Just think about this for a bit. If there are two satellites in circular orbits around the Earth, the one in a higher orbit will move more slowly than the one in the lower orbit (because the radius is larger for the one in the higher orbit). And if you have two satellites in circular orbits, at the same altitudes, but one is orbiting the Earth, and the other is orbiting the Moon, the one orbiting the Earth will move more rapidly than the one orbiting the Moon (because the Earth is more massive than the Moon).
Now, recall Kepler's first law. The planets orbit the Sun in ellipses. Not circles. Why? It isn't because circular orbits are not allowed. It's because a circular orbit is a special case. For any circular orbit, there are an infinite number of elliptical orbits with the same starting point. So if you pick one, at random, it will be elliptical.
Newton's discussion of the properties of orbits involved a classic thought experiment involving shooting cannonballs off of a cliff. The first, and pretty obvious, point is that the faster the muzzle velocity, the farther the cannonball will go. And, because the world is round, the farther the cannonball travels, the longer it will have to drop before it strikes the surface of the Earth. If one can get the cannonball moving fast enough, it will end up in an orbit.
Now I want to go back to a point I made when I was discussing the observed properties of the Moon: We always see the same face of the moon. This means that its period of rotation has to be the same as its orbital period around the Earth (a sidereal month). I noted that this behavior is an example of Tidal Phase Locking. Now it's time to look into the matter.
The Moon orbits the Earth due to gravity. This causes tides. The essential point about gravity is that it obeys an inverse-square law:
This means that the oceans on the side of the Earth closest to the Moon feel a larger force of gravity due to the Moon than does the center of the Earth. And the center of the Earth feels a larger force than does the ocean on the far side of the Earth from the Moon. The result of this is that two tidal bulges are raised on the Earth. One on the side facing the Moon, and one on the side facing away from the Moon.
The Sun also contributes to the tides raised on the surface of the Earth. But the Sun is much further away than the Moon is, so it has a relatively minor effect on the Earth's tides. Still the strongest tides (Spring Tides) are those that occur near New and Full Moon, when the Moon and Sun are aligned with the Earth. And the weakest tides (Neap Tides) are those that occur near First and Third Quarter Moon, when the Moon is perpendicular to the Earth--Sun line.
Thursday January 31
Now it's time to talk about the "phase locking" part of Tidal Phase Locking. Tides also occur in solid rock, but they are much less dramatic than those occuring in oceans. The solid body tides on the Earth only produce tidal bulges with amplitudes of about 1cm. Still, this has a dramatic long term effect. The Earth spins much more rapidly than the orbital period of the moon (one day versus 27.3 days), and thus the actual tidal bulge on the Earth is always ahead of the Earth--Moon line. This causes a tidal torque: The Moon is dragging the Earth, slowing its rotation rate down. This is born out by the fossil record: The day was only 22 hours long 400 Myr ago. Eventually, this will cause the Earth's rotation rate to slow down to match the orbital period of the moon.
The Earth is much more massive than the Moon, and gravity is a symmetric force, so the Earth is much more effective at slowing down the Moon than the Moon is at slowing down the Earth. In fact, the Earth has already slowed down the rotation period of the Moon to match its orbital period. This is why we always see the same face of the moon. Tidal Phase Locking is a common phenomenon, both for moons around planets, and for stars in binary star systems.
