Astronomy 161: An Introduction to Solar System Astronomy Prof. Richard Pogge

# Notes to Unit 4 The Physics of Astronomy: Gravity, Matter, & Light

These notes give references to sources for information provided in the individual lecture notes, sometimes giving more detail, qualifiers, or more advanced information for those interested. Material in these footnotes is supplementary to the lectures, and is formally outside the scope of the course (in other words, details here will not appear in homework assignments or exams).

## Notes for Individual Lectures

Lecture 18: Newtonian Gravity

[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 problems.

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.

Lecture 19: Orbits

[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 3 dimensions.

Lecture 20: Tides

[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.

Lecture 21: Dance of the Planets

[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.

Lecture 22: Light the Messenger

[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 light.

[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.

Lecture 23: Worlds Within: Atoms

[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.