Download Notes - SFA Physics and Astronomy

Gravity
The first quantitative model for gravity was given by Newton. He described the force
acting between two objects with masses M and m separated by the distance d and showed
Fg 
GMm
d2
It’s important to note that the separation is measured between the centers of mass of
the two objects. Although we will not use the equation for problem solving, we can learn
some important points about the way gravity operates by studying the form of the
equation.

Gravity is an example of an inverse square force, i.e., the force
depends on the inverse square of the separation. By doubling the separation, the
force falls by a factor of four. If the separation shrinks by a factor of three, the
force increases nine times.

Gravity can never become zero. We can increase the separation and
make the force as small as we like, but we can’t make it go away. Every piece of
matter in the universe has a gravitational attraction to every other piece of matter.

Mass causes gravity.

Unlike the electrical force which exists as both an attractive and
repulsive force, gravity is only attractive. To account for the two types of
electrical force, we require two types of charge (positive and negative). Only one
type of mass is required.

The constant G must be very small so that objects of ordinary mass
have negligible gravitational attraction for each other (I didn’t say zero!). You
must accumulate a lot of mass before the gravitational force is appreciable.
Gravity is the weakest force in nature.
Normal forces give us the sensation of having weight. If we remove the supporting
(normal) force, we have the sensation of “apparent weightlessness.” So a skydiver feels
weightless before opening the parachute. In the tragic example of an elevator breaking
loose, you would experience weightlessness. Certain amusement park rides are designed
to remove the supporting force and induce a feeling of weightlessness.
Galileo correctly concluded that in free fall, all objects must fall at the same rate (g is
a constant). While in orbit our astronauts are constantly falling. Since everything around
them is also falling and everything falls at the same rate, they have a continuing
experience of weightlessness.
An interesting consequence of the inverse square nature of the gravitational force is
tides. The major influence on the ocean tides of Earth is our Moon. Various parts of the
Earth are different distances away from the Moon. Those parts closer to the Moon
experience more force than the parts farther away. If we take the gravity acting on the
center of the Earth as the average and form force differences, we see that the gravity
difference for the side nearest the Moon is directed toward the Moon, but the gravity
difference for the side farthest from the Moon is directed away from the Moon. The
waters of the oceans respond to this gravity difference and move in the direction of the
gravity difference. Thus the sides of the Earth in line with the Moon experience tidal
bulges (high tides) while the sides of Earth perpendicular to the Earth-Moon line
experience tidal dips (low tides). These tidal variations stay in place with respect to the
Earth-Moon line. As the Earth rotates any given position moves into two high tides and
two low tides per day (since the Moon orbits the Earth, the actual period is 24h50m for the
complete cycle of tides).
The Sun contributes a force here as well, but only about half as great as the Moon’s
contribution due the distance of the Sun. Greater tides (Spring tides) are experienced
when the Sun joins the Earth-Moon line (New Moon or Full Moon). Lesser tides (Neap
tides) occur when the Sun is perpendicular to the Earth-Moon line (First or Last Quarter
Moon).
Other forces can affect the size and exact timing of the tides. Most notable here are
friction of the water with the ocean bottom and the shape of the continents.
Since ocean tides are obviously present, Earth must also experience atmospheric and
crustal tides. The tides of the atmosphere are larger than ocean tides and the tides of the
crust very much smaller. Crustal tides have been measured for the Moon and Io, a moon
of Jupiter.
An alternate view of gravity is to give each body that has mass a gravitational “field.”
In the construct we use arrows to visualize the field. For an object like the Earth the field
lines are straight and radial, directed toward the center of the Earth. Objects coming near
the Earth will experience a force directed in the same way as the field lines. The size of
the force depends on how close together the field lines are. Clearly, for the field
described for Earth, the arrows are closer together nearer the surface.
The concept of a gravitational field allows us to compute the gravity inside the planet.
The size of the gravitational field depends only on the mass between you and the center
of the Earth. As you move toward the center, the amount of inward mass decreases and
so does the gravitational field. At the center of the Earth, the field would have to be zero.
The understanding of black holes requires general relativity, a new theory of gravity
due to Einstein. Here the universe is viewed as four-dimensional, the three spatial
dimensions and time. A universe with no mass has no distortions in space-time. Mass
distorts space-time. For example the space-time around the Sun is distorted by the
presence of the Sun. Planets are moving in the straightest possible paths given the
distorted space-time in which they move. The predicted orbital shapes are ellipses, just
like in Newtonian physics. New theories must be able to predict all of the known facts as
well as make new predictions to be tested. General relativity does each.

The Advance of Mercury's Perihelion - this is a long-standing problem in which
the orbit of Mercury does not exactly close onto itself. The point of closest
approach to the Sun is slowly moving (advancing) in space. Newtonian physics
has no explanation for this effect, but General Relativity predicted it exactly.

The Bending of Starlight - gravity affects light just as it affects objects having
mass. When light from a star passes close to the Sun, its trajectory is bent. We
see the star in a slightly different position than we would if the Sun were not
there. This prediction was confirmed in the solar eclipse of 1919.
If we could view the vicinity of a collapsing star, as it becomes a black hole, we
would notice that the gravity from the star is increasing as the collapse proceeds. At first
the gravity is weak and the light is bent very little. As the collapse proceeds the bending
becomes greater and greater until light is bent so much that it orbits the star. The orbital
radius is called the photon sphere. An exit cone may be defined such that light
projected inside the cone is bent, but escapes. Light moving just along the edges of the
cone orbits the star, and light projected outside the exit cone is bent so much that it falls
back onto the star. As the collapse continues gravity gets stronger and the exit cone
narrows. General relativity predicts that the exit cone disappears when the escape
velocity becomes equal to the speed of light. Nothing can move faster than the speed of
light, so when the gravity has increased to make the escape velocity equal to the speed of
light, nothing (not even light) can escape. The surface of the collapsing object at this
moment is called the event horizon and its radius is the Schwarzschild radius. The
object is truly black. The collapse continues inside the event horizon, but observers on
the outside can never see it. The object collapses into a mathematical point called the
singularity. The Schwarzschild radius is three times the mass of the collapsed object
measured in solar masses. For example a three solar mass black hole has an event
horizon of radius 9 km.
Near a black hole space-time is severely distorted and time itself is affected. As
measured by an outside observer, time stops on the event horizon. Because the original
object is no longer accessible from our universe, we cannot know very much about a
black hole. In fact, theoretically only three things are knowable about a black hole:



Mass - gravity can be felt at a distance
Charge - electric forces act at a distance
Rotation - material may corotate with the black hole
Black holes are frustrating to astronomers - they don't emit any light. Perhaps the
only way of finding a black hole is if one formed in a binary star system. If the black
hole and the normal star are close enough, material will be pulled off of the normal star
and an accretion disk will form. As the material in the disk moves closer to the event
horizon of the black hole, it heats up. Just before dipping within the event horizon and
disappearing from the universe, the disk gas is hot enough (several million K) to emit Xrays. We may be able to find black holes by looking for X-ray sources. So far several
promising candidates have been found, Cygnus X-1 being the best known.
Edwin Hubble made a major contribution to the understanding of the structure of the
whole universe. He measured the spectra of distance galaxies. He used independent
means to determine the distances to the galaxies. Two methods he used were to look at
supergiant elliptical galaxies that tend to dominate clusters of galaxies. These galaxies,
he found, were about the same brightness and size regardless of where they appeared in
the universe. When they appear fainter or smaller, it was because they were farther away.
Upon taking the spectra of these galaxies, Hubble found that the each one was
redshifted - the galaxies were moving away from us. Even more he found that the more
distant galaxies were moving away faster. The information was presented in the form of
a graph of recessional velocity vs. distance which we call Hubble's Law. The
conclusion we draw from Hubble's Law is that the universe is expanding. Using the
graph we can get the distance of anything in the universe as follows:
The use of Hubble's Law does rely on the assumption that the redshifts are caused by
the universal expansion. The relationship, however, would not be nearly so good if this
conclusion were not the fact. Hubble's Law stands today as one of the pillars of Modern
Cosmology. The question still remains, however, as to how the universe began.
Two cosmologies developed from Hubble's work:
 The Big Bang Cosmology assumes that the expansion can be drawn back in time
to the time when the universe occupied a point. Some process began the
expansion (the Big Bang), which continues to this day. Based on the presently
observed rate of expansion, we can judge that the universe is 12- 15 billion years
old.
 The Steady State Cosmology assumes that the universe always looks more or less
the same. This idea accounts for the expansion of the universe by assuming that
matter is continually created to fill in the voids left by the expansion. The amount
of material required would be entirely too small to be observed.
Prior to the mid-1960's observations could not distinguish between these two
cosmologies. Arno Penzias and Robert Wilson at Bell Labs changed all that with their
discovery that the universe is emitting blackbody radiation indicative of a blackbody at 3
K. At the same time the Big Bang Cosmologists had determined that the temperature of
the Big Bang was sufficient to make Hydrogen and Helium only and, therefore, was
limited. Starting with that temperature and expanding the universe for 12 - 15 billion
years at the observed rate yielded a universe that was, by now, much cooler. Recall that
ideal gases cool when expanded. The present temperature of the Big Bang is 3 K, in
exact agreement with the observed microwave background radiation discovered by
Wilson and Penzias. Everyone now agrees that the universe began as a Big Bang. But
many variations of the basic theory still exist to be sorted out. The 3 K background
radiation is another of the pillars of cosmology today.