1 Chapter 1 Astronomical distances are so large we typically measure distances in lightyears: the distance light can travel in one year, or 9.46 10 12 km or 9, 600, 000, 000, 000 km. Looking into the sky is looking into the past. The light we see may have been traveling a long time. Light from the moon takes about 1.3 s to reach us, from the sun 8 minutes, from the nearest star (α Centauri) about 4.4 years, from the brightest star (α Canis Major, or Sirius) about 8 years, and from the most distant naked eye object (Andromeda galaxy) about 2.5 million years. The light from the most distant object ever observed has taken about 13 billion years to reach us. This is very close to the limit of the observable universe, since the age of the universe is about 14 billion years. On a grapefruit scale, where we represent the sun as a grapefruit, the earth would be about 1 mm in diameter and about 10 m away from the grapefruit. The moon, about a quarter mm, would orbit 3 cm around the earth. Pluto would be 320 m away, and the nearest star would be 2,800 km away! The Earth moves around the Sun in an (slightly) elliptical orbit, with a mean radius of about 150 million (150 10 6 ) km. This distance is also known as an astronomical unit or AU, and the earth completes this orbit in one year. The axis rotation of the earth is tilted 23.5 with respect to its orbital plane, and points very close to polaris, the north star. The perihelion, or closest approach to the sun, of the earth is around January 2nd. The direction of rotation of the earth is the same as its orbital motion (see conservation of angular momentum in chapter 4). The solar system itself moves around the galactic center of the Milky Way galaxy every 230 million years, although the motion relative to nearby stars is essentially random. This is somewhat analogous to a whirlpool of gas: There is collective motion visible on a large scale around the center, but the gas molecules still have random thermal motion. The Milky Way galaxy seems to again be moving fairly randomly with respect to nearby galaxies, but on a larger scale we see distant galaxies are moving away from us; the more distant the galaxy is, the faster it is receding. This puzzling situation is evidence of the
2 expansion of the universe, and was the basis for the Big Bang theory or the origin of the universe. Chapter 2 With our naked eye we cannot tell the distance to objects in the sky; they all appear to be at the same distance. It looks as if they are all tied to a large sphere in the sky called the celestial sphere. We can measure positions of objects in the sky using several different coordinate systems. An altitude and azimuth coordinate system seems easiest at first, but celestial objects do not move simply in this system. A system using declination and right ascension are much more useful, being latitude and longitude on the celestial sphere. The north (south) celestial pole is directly over the earths north (south) pole, the celestial equator is a projection of the earths equator onto the celestial sphere, and the ecliptic is the path the sun follows around the celestial sphere, or in other words the plane of the earths orbit around the sun projected onto the sphere. It crosses the celestial equator at an angle of 23.5 due to the tilt of the earths axis. The stars were grouped into images known as constellations by the ancients, and we use many of their names today, although the names today correspond to abstract areas of the celestial sphere. The stars appear to rise and set similar to the sun and moon due to the rotation of the earth. As the earth rotates under these stars, the north star, polaris, is very close to the celestial north pole and does not appear to move, while the other stars move in circles around it. A star whose entire circle is above the horizon is known as circumpolar. This depends on the observers latitude: at the north pole, polaris is directly overhead and every visible star is circumpolar; at the equator, polaris is on the horizon, and no visible star is circumpolar. The stars and constellations which are visible depend on both the latitude of the observer and the time of year. Stars and constellations that are near the suns position on the celestial sphere are not visible, since they are above the horizon during daylight hours. Latitude also restricts the visible stars. An observer in the souther hemisphere would never be able to see polaris, for instance. The seasons on earth are due to the tilt of the earths axis with respect to its orbital plane. This tilt has two effects which combine to cause the seasons: the change of the length of the day, and the angle of incidence of the suns light. For instance, the day is longer in
3 the northern hemisphere in the summer when the tilt of the axis is towards the sun. In addition, the sun hits the surface more squarely, increasing the flux of light on a given area. These effects combine to create warmer days. The reverse happens in the winter, when the axis tilts away from the sun. The summer (winter) solstice is when the earths axis points toward (away) from the sun, and are the longest (shortest) days of the year. The vernal and autumnal equinoxes are when the earths axis points perpendicular to line connecting the earth and the sun, and have equal length days and nights. The word equinox in fact comes from Latin equal night, while solstice comes from stationary sun, since the suns motion (along the ecliptic) away from the celestial equator stops and it begins to move back toward it. The length of the day can be measured against the sun or the stars. The time it takes for a star to return to the same point in the sky is called a sidereal day. This day is very constant in length and is 23 hours 56 minutes long. The solar day, however, is slightly longer due to a small extra rotation that the earth must undergo to account for the movement of the earth relative to the sun during that day. Since the motion of the earth relative to the sun is not constant (the earth moves faster during our winter, when we are actually closer to the sun) the length of the solar day varies throughout the year. Our 24 hour solar day is technically a mean solar day, or the average length of a solar day. The precession of the Earths axis is a slow wobbling of the Earths axis due to a torque on the (non-spherical or oblate spheroid) Earth by the Sun. This precession causes the north celestial pole to sweep out a circle in the sky, returning to polaris in 26,000 years. This cycle combines with other slow celestial cycles to form the Milankovitch cycles, a suspected cause of the earths ice ages. The moon is a natural satellite of the earth. Its diameter is slightly more than 1/4 the earths, 3400 km, and its orbit has a mean radius of about 380,000 km. The plane of this orbit is inclined about 5 relative to the ecliptic. The moons orbital sidereal period (relative to the stars) is 27.3 days, but the more noticeable synodic period (full moon to full moon) is 29.5 days. The moons always has the same face pointing at the Earth, a condition known as being tidally locked. See chapter 3 for an explanation of tidal forces, but friction due to flexing of the moon in response to these tidal forces of the earth have converted rotational energy into heat and slowed the rotation of the moon until there was no more flexing.
4 The phases of the moon become easily explained by considering the moon as a sphere lit by the distant sun. One half of the moon (pointing toward the sun) is always lit, and the other half (pointing away from the sun) is always dark. As the moon moves around the earth in its orbit, we see this half lit moon from varying vantage points, sometimes seeing more or less of the lit half as we (the earth) moves from the same side of the moon as the sun, when we see a full moon, to the opposite side of the moon as the sun, when we see a new moon. The relative sense of rotation is the same for the orbit of the Earth around the sun, the orbit of the moon around the earth, the rotation of the earth on it s axis, and the rotation of the moon on its axis. The fact that these rotations are all in the same direction is related to the conservation of angular momentum during the collapse of the cloud of gas that formed the solar system. See angular momentum in chapter 4 (and more in later chapters). Eclipses happen when the shadow of one celestial body falls on another. The shadow is divided into a region where the sun is totally blocked, called the umbra, and a region where it is partially blocked, called the penumbra. Two important types of eclipses we observe on earth are solar and lunar eclipses. A solar eclipse is when the moons shadow falls on the earth, and a lunar eclipse is when the earths shadow falls on the moon. For either type of eclipse the three bodies: the sun, the earth, and the moon must lie on a line. This can only happen when the line of nodes (the intersection of the moons orbital plane and the earths orbital plane) points at the sun. This happens twice a year, and these times are known as eclipse seasons Lunar eclipses can be total lunar, when the moon is completely in the earths umbra, or partial lunar if only part of the moon enters the umbra, or penumbral lunar if the moon enters the earths penumbra without every touching the umbra. Solar eclipses may appear to be total solar if the viewer is under the umbra of the moon, or partial solar if the viewer is under the penumbra. If the eclipse occurs when the moon is relatively far from the earth, the umbra will not reach the earth and an annular solar eclipse will result.
5 Chapter 3 The scientific method is a technique for understanding nature which tests models against nature itself. The usual (although perhaps overly idealized) enumeration of steps is: 1. Look at nature (observation) 2. Make a guess explanation or model (hypothesis) 3. See what happens in a new situation with your model (prediction) 4. Try out new situation in nature (test) If your test matches model, good! Try another test. If it does not match, good! Try another hypothesis. Occam s razor is a philosophical bias which allows us to choose from two models that make equivalent predictions: the simpler model is better. Eratosthenes measured the earth around 240 BC. Looking at the angle shadows made with respect to vertical in two towns separated by a known distance, he could figure out what distance was required to make the angle a full circle, i.e. the circumference of the earth. (see book, page 64) The ancients adopted a geocentric model of the solar system, with the earth at center and the sun, moon, and planets revolving around it. This seemed natural since the sun and moon move in circles around the celestial sphere. The planets also almost moved in circles, but there was a problem with apparent retrograde motion of the planets, when they appeared to change direction in the sky. The geocentric model explained retrograde motion with epicycles, small circular orbits whose centers moved around large circular orbits around the earth. Finer corrections resulted in the final Ptolemaic system which had epicycles around epicycles. A sun centered model or heliocentric was proposed around 200 BC by Aristarchus, but lost until 1543 when Copernicus proposed it again. This model naturally explains retrograde motion. Tycho Brahe made careful observations of the positions of the sun, moon, and planets. These data allowed his assistant Johannes Kepler to deduce his three law s which, combined with the Copernican model predicted the positions of the planets extremely well.
6 Kepler s three laws of planetary motion are: 1. Planets move around the sun in ellipses with the sun at one focus of the ellipse. 2. As a planet moves around it s orbit, it sweeps out equal areas in equal times. 3. More distant planets from the sun move with slower average speeds, obeying the relation p 2 = a 3. Chapter 4 The description of the motion of a body in space requires the definition of some terms: 1. position: where an object is 2. velocity: how quickly something s position is changing (direction also!) 3. speed: just the magnitude of velocity, regardless of direction. 4. acceleration: how quickly something s velocity is changing (direction also!) 5. momentum: Newtons quantity of motion. p = m v 6. force: causes a change to an objects momentum. The mass of an object is how much matter is in it. We will see two kinds of mass: inertial mass, defined by Newton s laws of motion, and gravitational mass defined by Newton s universal law of gravitation. These turn out to be the same, and we refer to the mass of the object as the quantity used in either sense. This is different from the weight of an object, which is a force (for instance, the force a scale measures when you stand on it). A body is weightless in freefall. Newton s laws of motion describe how the momentum of objects are affected by the forces acting upon them. 1. An object that is at rest will stay at rest unless an unbalanced force acts upon it. An object that is in motion will not change its velocity unless an unbalanced force acts upon it. 2. Force equals mass times acceleration, or F = m a 3. To every action there is always an equal and opposite reaction
7 Physical quantities that do not change over time are called conserved quantities. Angular momentum is conserved. Angular momentum of an orbiting body is the radius (r) of the orbit times the momentum (mv) or, angular momentum = mvr. Kepler s second law is equivalent to conservation of angular momentum. Energy is conserved. Energy of a system can be thought of as the ability to do work. It can appear in various forms which can be grouped into three broad categories: kinetic, potential, and radiative. Energy can be transformed from one type into another, but the total amount of energy is conserved. The force of gravity between two masses is described by Newton s universal law of gravitation. The force is proportional to the product of the two masses and inversely proportional to the distance between them squared, or F = G m 1 m 2 d 2 so the force gets stronger as the masses get bigger, and weaker as the distance between them grows. Newton s version of Kepler s third law allows (with some assumptions) the calculation of the masses of orbiting bodies. It can be written p 2 = 4π 2 G(m 1 + m 2 ) a3 where p and r are the period and radius of the orbit. The sum of Kinetic and Potential energies for an orbiting body is called the Orbital Energy of the object. To change it s orbit it must change it s orbital energy. If an object has enough kinetic energy to climb out of the potential well it is in and leave the body it was bound to, it is said to have escape velocity. The tides are caused by the differential pull of the moon (and sun, to a lesser extent) on the near and far side of the earth, due to the different distances to the moon. The moon pulls more on the near side of the earth because the gravitational force decreases at larger distances.