Lecture 2

ExaminationThe course is assessed in three parts:

•Problem sets. The problem sets and the peer reviews. One of the sets will randomly be corrected by me

•Observational labs. The laborative exercises + a lab report.

•Written Exam. It consists of two parts. Part I is needed for grade E: at least 75% needs to be correct. It is also needed to be able to go on to Part II, which is used to assess for grades A-D. The grade is set according to the score.

To pass the course, all three of the above assessments must be ready.

Objectives for grade E: For passing the course with grade E you need to know the fundamentals of the course.

Objectives for grades A-D: Be able to solve problems on topics covered in the course, that need a well developped and mature understanding its content.

Pedagogical background for peer review:

- See other solutions, comparing solutions.

- The understanding gets deeper, when applying criteria for good/bad solutions; ”Is it really the correct solution for this problem?

- Select good evidence to be able to convince others.

- Learn to judge a good performance. You are included in the critique process.

- All this leads to increased self-supervision capabilities.

Other positive effects of this is that the feedback is prompt, and is relevant. In addition, learning from errors is very powerful. The possibility of erroneous thinking without the risk of being negatively judged by the teacher is important.

Repetition

Vernal equinox”Vårdagsjämningspunkten”

Right ascension

Declination

• The Sun appears to trace out a circular path called the ecliptic on the celestial sphere tilted at 23 ½ degrees to the equator

• The ecliptic and the celestial equator intersect at only two points

• Each point is called an equinox

• The point on the ecliptic farthest north of the celestial equator that marks the location of the Sun at the beginning of summer in the northern hemisphere is called the summer solstice

• At the beginning of the northern hemisphere’s winter the Sun is farthest south of the celestial equator at a point called the winter solstice

June 21

March 31

Dec 21

Sept 21

Stellar Motions: Proper motion

r

Nature of Light - Cnt’d

Spectral Lines

Kirchhoff’s Laws

Bohr’s formula for hydrogen wavelengths

1/ = R x [ 1/N2 – 1/n2 ]

N = number of inner orbit

n = number of outer orbit

R = Rydberg constant

(1.097 X 107 m-1)

= wavelength of emitted or absorbed photon

Ly Ly

H H

Balmer Lines in Star Spectrum

Photons’ interaction with matter:Extinction

Absorption:

Scattering:

Interstellar grains and gas

Rayleigh scattering is responsible for the scattering of visible light in the daytime sky

* Scattering on electrons

* Proportional to 1/4

Scattering

Gravitation

Ancient astronomers invented geocentric modelsto explain planetary motions

• Like the Sun and Moon, the planets move on the celestial sphere with respect to the background of stars

• Most of the time a planet moves eastward in direct motion, in the same direction as the Sun and the Moon, but from time to time it moves westward in retrograde motion

A planet undergoes retrograde motion as seen from Earth when the Earth and the planet pass

each other

Johannes Kepler proposed elliptical pathsfor the planets about the Sun

• Using data collected by Tycho Brahe, Kepler deduced three laws of planetary motion:

1. the orbits are ellipses2. a planet’s speed

varies as it moves around its elliptical orbit

3. the orbital period of a planet is related to the size of its orbit

Kepler’s First Law:A planet orbits the Sun in an ellipse, with the Sun at one focus of the ellipse

r + r0= 2a

b2 = a2(1à e2)

Eccentricity

Semiminor axis

Focal pointSemimajor axis

Actually, the center-of-mass is at the focus

Kepler’s Second Law:A line connecting a planet to the Sun sweeps out equal areas in equal time intervals

Orbital speed dependson location

dtdL = 0

dtdA = 2

1öL

ö = m1+m2

m1m2

Angular momentum of a system is a constant for a central force law

The time rate of change of the area swept out by a line connectinga planet to the focus of an ellipse is a constant = one-half of the orbital angular momentum per unit mass

Kepler’s Third Law (harmonic law): Relates the average orbital distance of a planet from the Sun to its period

P2 = a3

P = planet’s sidereal period, in yearsa = planet’s semimajor axis, in AU

Kepler’s Third Law

P2 = G(m1+m2)4ù2

a3

Simple derivation using Newtonian mechanics gives that Kepler’s third law is

Galileo’s discoveries with a telescope stronglysupported a heliocentric model

• The invention of the telescope led Galileo to new discoveries that supported a heliocentric model

• These included his observations of the phases of Venus and of the motions of four moons around Jupiter

• One of Galileo’s most important discoveries with the telescope was that Venus exhibits phases like those of the Moon

• Galileo also noticed that the apparent size of Venus as seen through his telescope was related to the planet’s phase

• Venus appears small at gibbous phase and largest at crescent phase

Geocentric

• To explain why Venus is never seen very far from the Sun, the Ptolemaic model had to assume that the deferents of Venus and of the Sun move together in lockstep, with the epicycle of Venus centered on a straight line between the Earth and the Sun

• In this model, Venus was never on the opposite side of the Sun from the Earth, and so it could never have shown the gibbous phases that Galileo observed

• In 1610 Galileo discovered four moons, now called the Galilean satellites, orbiting Jupiter

Isaac Newton formulated three laws that describefundamental properties of physical reality

• Isaac Newton developed three principles, called the laws of motion, that apply to the motions of objects on Earth as well as in space

• These are1. the law of inertia: a body

remains at rest, or moves in a straight line at a constant speed, unless acted upon by a net outside force

2. F = m x a (the force on an object is directly proportional to its mass and acceleration)

3. the principle of action and reaction: whenever one body exerts a force on a second body, the second body exerts an equal and opposite force on the first body

Newton’s Law of Universal Gravitation

F = gravitational force between two objectsm1 = mass of first object

m2 = mass of second objectr = distance between objects

G = universal constant of gravitation

• If the masses are measured in kilograms and the distance between them in meters, then the force is measured in newtons

• Laboratory experiments have yielded a value for G of G = 6.67 × 10–11 newton • m2/kg2

Kepler’s First Law:A planet orbits the Sun in an ellipse, with the Sun at one focus of the ellipse

r + r0= 2a

b2 = a2(1à e2)

Eccentricity

Semiminor axis

Focal pointSemimajor axis

Actually, the center-of-mass is at the focus

Elliptical orbits result from an attractive r-2 central force law such as gravity, when the total energy of the system is less than zero (a bound system).

Parabolic path is obtained when E=0.

Hyperbolic path in an unbound system E>0

Orbits may be any of a family of curves called conic sections

Orbits

• The law of universal gravitation accounts for planets not falling into the Sun nor the Moon crashing into the Earth

• Paths A, B, and C do not have enough horizontal velocity to escape Earth’s surface whereas Paths D, E, and F do.

• Path E is where the horizontal velocity is exactly what is needed so its orbit matches the circular curve of the Earth

FIN

Tidal force

Gravitational forces between two objectsproduce tides

The Origin of Tidal Forces

Light has properties of both waves and particles

• Newton thought light was in the form of little packets of energy called photons and subsequent experiments with blackbody radiation indicate it has particle-like properties

• Young’s Double-Slit Experiment indicated light behaved as a wave

• Light has a dual personality; it behaves as a stream of particle like photons, but each photon has wavelike properties

Determining the Speed of Light

• Galileo tried unsuccessfully to determine the speed of light using an assistant with a lantern on a distant hilltop

Light travels through empty space at a speedof 300,000 km/s

• In 1676, Danish astronomer Olaus Rømer discovered that the exact time of eclipses of Jupiter’s moons depended on the distance of Jupiter to Earth

• This happens because it takes varying times for light to travel the varying distance between Earth and Jupiter

• Using d=rt with a known distance and a measured time gave the speed (rate) of the light

• In 1850 Fizeau and Foucalt also experimented with light by bouncing it off a rotating mirror and measuring time

• The light returned to its source at a slightly different position because the mirror has moved during the time light was traveling

• d=rt again gave c

• The number of protons in an atom’s nucleus is the atomic number for that particular element

• The same element may have different numbers of neutrons in its nucleus

• These three slightly different kinds of elements are called isotopes

Spectral lines are produced when an electron jumps from one energy level to another within an atom

• The nucleus of an atom is surrounded by electrons that occupy only certain orbits or energy levels

• When an electron jumps from one energy level to another, it emits or absorbs a photon of appropriate energy (and hence of a specific wavelength).

• The spectral lines of a particular element correspond to the various electron transitions between energy levels in atoms of that element.

• Bohr’s model of the atom correctly predicts the wavelengths of hydrogen’s spectral lines.

A planet’s synodic period is measured with respect to the Earth and the Sun (for example, from one

opposition to the next)

Tycho Brahe’s astronomical observations disproved ancient ideas about the heavens

Parallax Shift

There is a correlation between the phases of Venus and the planet’s angular distance from the Sun

Newton’s description of gravity accounts for Kepler’slaws and explains the motions of the planets and

other orbiting bodies

Each chemical element produces its own unique set of spectral lines

Bohr’s Model of the Hydrogen atom

Spectral line series of the hydrogen atom

Doppler Shifts

• Red Shift: The object is moving away from the observer• Blue Shift: The object is moving towards the observer

= wavelength shift

rest = wavelength if source is not movingv = velocity of source

c = speed of light

õrest

õobsà õrest = õrest

4 õ = cvr

The wavelength of a spectral line is affected by therelative motion between the source and the observer