[18.1] G is the "gravitational coupling
constant", which sets the size of the force between two massive objects
separated by a given distance. Because gravity is the weakest of the 4
fundamental forces of nature, G is hard to measure experimentally with
any precision. Newton did not know the value of G, but he was able to
pose his problems in ways that G drops out mathematically, thus to him
it was just a constant of proportionality.
The first experimental measurment of G was done by British physicist
Henry Cavendish in experiments performed between 1797 and 1798, using a
torsion balance to measure the force of gravity between two weights in
the laboratory. However, Cavendish's explicit goal for this experiment
was to accurately measure the density - and hence the Mass - of the
Earth, and he never once mentions G in his work or explicitly derives a
value for it. Like Newton, Cavendish posed his problems so that G
canceled mathematically. We'll do much the same in this class, which
is why you'll never need to know G operationally for exams or homework
It was not until much later (almost 75 years later) that his
experimental data was used by others to derive a value for G. It was
not until the later part of the 19th century that astronomers needed to
know G so they could, among other things, compute the densities of
celestial bodies like the Moon and Sun.
[19.1] This formula is only strictly
valid in the case that the orbiting object is much less massive than the
central massive body. For example, it is correct for a satellite or
astronaut, but not for the Moon.
[19.2] Like with circular speed, this
formula is only strictly valid in the case that the moving object is
much less massive than the central massive body it is "escaping from".
Also, strictly speaking an object never reall "escapes" or "breaks free"
from another object's gravity, it just becomes unbound and is on a
trajectory that will never turn back on the bigger object like a closed
or "bound" orbit.
[19.3] This formula is a simplification
for the case of a circular orbit, but it illustrates the basic effect.
In general, the angular momentum is a vector, equal to the mass times
the vector cross-product of the radial vector and the velocity vector in
[20.1] The value of 0.0023 seconds
per century quoted is the present-day ("instantaneous") value. The
process of tidal braking is made complicated by the fact that the
interior of the Earth is semi-molten, and the actual rate of change of
the rotation rate is very complicated. While tidal braking is one of
the primary components of the Length-of-Day change, it is not the only
process at work, and the rate at which it has changed is itself different
at different times (see also Note 20.3).
[20.2] The rate of lunar recession
quoted is the present-day measured rate. However, please use caution if
using this number to extrapolate back into the past or forward into the
future, as the rate depends on a number of factor that make the actual
recession rate over long periods of time non-linear.
[20.3] This value for the length of the
day, 1.7 ms/century, is the average change over the past 2700
years. Note that this is different than the instantaneous
present-day value of 2.3 ms/century (see Note
20.1). The length of the day, and the rate of change of the
length of the day, has changed throughout Earth's history. The best
data come from the study of tidal rhythmites - layered sediments laid
down in tidal estuaries that record the cycle of tides, and hence a
combination of the lunar month and the rotation rate of the Earth
("length of the day"). The study of paleorotation and paleotides
helps us see these effects over scales of billions of years. At
present, data goes back reliably to 620Myr ago, when the day was
21.9+/-0.4 hr long, 18.2+/-0.2 hours about 900Myr ago, 18.9+/-0.7
hours 2.5Gyr ago (see Williams, G.E. 1997, Geophysical Research
Letters, 24, 421).
[20.4] The time required for the
Moon and Earth to get into perfect tidal synchronization is actually
longer than the time remaining in the Sun's life. Thus other changes
will occur in the Earth-Moon system that somewhat obviate these effects,
so they are shown for illustration only. The main point to carry away
is that, all other things being equal, it takes a very long time for a
well-separated system like the Earth and Moon to tidally lock.
[21.1] In reality, because the
Earth/Moon system has a mutual elliptical orbit, the locations and the
properties of the Lagrange points differ in detail from the simple
circular-orbit approximation used by Lagrange to make the problem
soluble by solving Newton's equations of motion algebraically. In
general, the real problem is solved numerically using computers.
[21.2] Galileo's notebooks showed that
he observed Neptune twice during observations of Jupiter on 1612 Dec 28
and 1613 Jan 27. At this time, Neptune and Jupiter were in conjunction
in the sky. Because this conjunction also occurred near when Neptune
was in opposition, its motion was very small and undetectable to
Galileo's small telescope, and he thought it was a fixed star.
[22.1] The speed of light, c, is
the only constant of nature we can state "exactly", only because our
metric system of measuring lengths is defined in terms of the speed of
light. This seems a somewhat circular argument, but what it amounts to
is saying that our measurement system is tied directly to the speed of
[22.2] The wavelengths assigned to the
colors in this table are meant only to be illustrative. Color is a
physiological response of our brains to visible light of different
energies. Our division of the visible spectrum into named colors is very
subjective, and certainly not a matter of uniform divisions at every 50
nm of wavelength! An oft-cited example of the subjectivity of color
names is the color "orange". It did not enter the language until the
Middle Ages, when the fruit of the same name reached Europe from the
Middle East. Before that, the color would have been called a reddish
shade of yellow.
[22.3] Most traffic radar guns are of
the portable microwave doppler radar type in this example. A problem of
microwaves is that their excess emission can be detected by small
receivers mounted in cars and used to warn drivers that a radar gun is
in use on the road. Some types of police "radar" guns don't use
microwaves but instead use laser ranging techniques called LIDAR. This
works by time-of-flight calculation rather than the Doppler effect.
[23.1] The process of neutron decay is
an example of beta decay. The products are 3 particles: a
proton, and electron, and a neutrino. The neutrino is specifically a
electron anti-neutrino, but I have omitted that detail from the notes to
avoid the problem of introducing the concept of antimatter (or, more
correctly, antiparticles) at this time.
Most subatomic particles have associated antiparticles that
have the same mass but opposite charge. For example, the electron's
associated antiparticle is the positron, a particle with the same
mass as the electron but a positive charge.
Neutral particles also have antiparticles, but since they are
electrically neutral the "opposite charge" characteristic is more
subtle. For example, the neutron has an associated antineutron that has
the same mass but it is also electrically neutral. The crucial
differences lies in the internal details of the neutron. Neutrons are
composed of 3 quarks: 1 up quark and 2 down quarks. Up quarks have
charge +2/3, down quarks have charge -1/3, so 1 up + 2 down adds up to
give a net zero charge. The antineutron is composed of 3 anti-quarks: 1
anti-up and 2 anti-down quarks. These anti-quarks have the same masses
but opposite charges (-2/3 for anti-up and +1/3 for anti-down) as their
associated quarks. If you combined 1 anti-up and 2 anti-down quarks
into an antineutron, it also has zero net charge. Thus the same mass
but opposite charge definition still holds for the neutron/antineutron
pair because it is vested in the quarks that make them up.
Electrons are leptons or light particles that are not made up of
quarks on a lower level, and are in fact truly elementary particles in
that they have no further internal structure. There are three flavors
of leptons: electrons, muons, and tau particles. Each lepton has a
partner neutrino: electron, muon, and tau neutrinos, respectively.
Similarly, the antileptons have partner antineutrinos. Thus the
relationship between neutrinos and their associated antineutrinos is a
lot more subtle, since they are related through their lepton partners,
unlike the case of heavy particles (baryons) like protons and neutrons.
The only possible wrinkle to this is that the above is true only if
neutrinos are "Dirac fermions" (a fermion is the generic name for spin
1/2 particles). There is a proposed class of "Majorana fermions" that
are fermions that are their own antiparticle. If neutrinos are in fact
Majorana fermions, then a radioactive decay mode called "neutrinoless
double beta decay" is possible. Experiments to try to detect this
radioactive decay are currently in the planning stages. Demonstrating this
would be about as close to an instant Nobel Physics prize as you can get.
You see why I wanted to avoid specifically talking about antimatter
unless I need the concept, which I don't for Astronomy 161.