ASTR271 Angles, Powers-of-Ten, Units. Chapter 1

Transcription

1 ASTR271 Angles, Powers-of-Ten, Units Chapter 1

2 Announcements Research Experience for Undergrads (REU) for summer 2018 applications due now. ASTRON/JIVE due Feb 1 First Homework is due January 25

3 The process of astronomy research What is it that we see? (observing) How does it work? (analysis) How was it formed, and how will it evolve? (theory)

4 What we need in order to explore To describe: angular measure coordinate system phases, apparent motion etc. To explain: orbital motion atoms and molecules EM radiation

5 Angles We lack depth perception, therefore we use the angular measure to describe: The apparent size of a celestial object The separation between objects The movement of an object across the sky To estimate angles, extend your arm and use your hands and fingers like this =>

6 Example of angular distance: the pointer stars in the big dipper The Moon subtends about one-half a degree

7 How do we express smaller angles? One circle has 2π radians = 360 We subdivide the degree into 60 arcminutes (a.k.a. minutes of arc): 1 = 60 arcmin = 60 ' An arcminute is split into 60 arcseconds (a.k.a. seconds of arc): 1' = 60 arcsec = 60"

8 Angular size - linear size - distance Use the small-angle formula: D = αd where D = linear size of an object (any unit of length), d = distance to the object (same unit as D) α = angular size of the object (in arcsec), The 206,265 is required in the formula - it's the number of arcseconds in a circle divided by 2π. Where does this formula come from?

9 The same idea in pictures: the angular size depends on the linear (true) size AND on the distance to the object. See Box 1-1. Moving an object farther away reduces its angular size. Angular size Distance Physical size As long as the angular size is small, we can think of the object's physical size as a small piece of a circle.

10 Examples 1. The Moon is at a distance of about 384,000 km and has an angular diameter 30. From the small-angle formula, its physical size is 3350 km. 2. M87 (a big galaxy) has angular size of 7', corresponding to diameter 120,000 light years at its large distance (1ly = about 100 trillion km).

11 More Basic Trig r Circle/Disk Circumference = 2πr Area = πr 2 Diameter = 2r Sphere Circumference = 2πr Area = 4πr 2 Volume = (4πr 3 )/3

12 Powers-of-ten notation Astronomy deals with very big and very small numbers we talk about galaxies AND atoms. Example: distance to the Sun is about 150,000,000,000 meters. Hard to handle! Use powers-of-ten, or exponential notation. All the zeros are consolidated into one term consisting of 10 followed by an exponent, written as a superscript.

13 Note some familiar numbers in powers-of-ten notation (ch. 1.6): One hundred = 100 = 10 2 hecto One thousand = 1000 = 10 3 kilo One million = 1,000,000 = 10 6 mega One billion = 1,000,000,000 = 10 9 giga One one-hundredth = 0.01 = 10-2 centi One one-thousandth = = 10-3 milli One one-millionth = = 10-6 micro One one-billionth = = 10-9 nano

14 Examples power-of-ten notation 150 = 1.5 x ,500,000 = 8.45 x = 3.2 x = 4.5 x 10-6 The exponent (power of ten) is just the number of places past the decimal point.

15 We can write the size of just about anything on this chart! (Sizes given in meters): An atom Cell is about 10-4 m Taj Mahal is about 60 meters high Earth diameter is about 10 7 m

16 Units in astronomy Astronomers use the normal metric system and powers-of-ten notation, plus a few special units. Example: Average distance from Earth to Sun is about 1.5 x m = 1 Astronomical Unit = 1 AU Used for distances in the Solar system. This spring we are working on much larger scales, in which case we normally use the unit of parsec.

17 The parsec unit Relation between angular size and linear distance => basic unit of distance in astronomy: the parsec Short for "parallax of one second of arc" The distance between Earth and a star at which the radius of the Earth's orbit around the Sun (1AU) subtends an angle of 1" The nearest star is 1.3 pc away (1 pc = 3.09 x m)

18 Coordinate systems (Box 2.1) Purpose: to locate astronomical objects To locate an object in space, we need three coordinates: x, y, z. Direction (two coordinates) and distance. At Earth we use coordinates of longitude and latitude to describe a point (ignoring altitude)

19 0º Position in degrees: Longitude: connecting the poles, 360º, or 180º East and 180º West Latitude: parallel to the equator, 0-90º N and 0-90º S 90º N A location is the intersect of a longitude and latitude line (virtual) 0º Albuquerque: 35º05' N, 106º39' W 90º S

20 Positions of celestial objects Same idea when we describe the position of a celestial object The Sun, the Moon and the stars are so far away that we cannot perceive their distances - no depth perception. Instead, the objects appear to be projected onto a giant, imaginary sphere centered on the Earth. All objects seem to be on the surface of this imaginary sphere

21 The celestial sphere Stars are really distributed in 3-dimensional space, but they look like they re on a 2-dimensional sphere centered on Earth. To locate an object, two numbers (angular measures), like longitude and latitude are sufficient. Useful if we want to decide where to point our telescopes.

22 Celestial coordinate systems The Horizon coordinate system (used for telescopes): Meridian Altitude Angle above the horizon 0-90º The altitude of the north celestial pole equals the observer's latitude on Earth. Azimuth Angle measured eastward along horizon, starting from the north 0-360º

23 Pros and cons of the Horizon system Pros Easy to tell and understand Cons At different position on the Earth, the same object has different coordinates At different times, the same object has different coordinates The coordinates of an object change in this system!

24 The Equatorial system A system in which the coordinates of an object does NOT change. The coordinates are called Right Ascension and Declination, and are analogous to longitude and latitude on Earth. The equatorial coordinate system rotates with stars and galaxies.

25 RA and Dec Declination (Dec) is measured in degrees, arcminutes, and arcseconds. Right ascension (RA) is measured in units of time: hours, minutes, and seconds. Example 1: The star Regulus has coordinates RA = 10 h 08 m 22.2 s Dec = 11 58' 02" Zero point of RA: The vernal equinox, which is the point on the celestial equator the Sun crosses on its march north - the start of spring in the northern hemisphere (cf. Greenwhich 0 longitude).

26 Properties of Light Chapter 5

27 Light carries information to astronomers How hot is the Sun? How does it compare to other stars? What is its composition? Is there anything between the stars? Between galaxies? How do we know that the Universe is expanding? All of this comes from the study of light! What do you see if you look around? Astronomers use telescopes to collect light of distant objects, and then extract information from this light.

28 What is light? Light is electromagnetic (EM) radiation Light can be treated either as EM waves Photons (particles of light) Both natures have to be considered to describe all essential properties of light

29 Light as electromagnetic waves EM waves: self propagating electric and magnetic fields (changes in strengths of E- and B-fields). Traveling (in vacuum) at the constant speed of light c = 3 x 10 8 m/s. ν = c/λ, where ν is the frequency [Hz] and λ is the wavelength [m]. EM waves are different from other waves, since it doesn t need a medium to propagate in!

30 Properties of light It heats up material (e.g. your skin), thus it is a carrier of energy. This is radiative energy, one out of three basic categories of energy (kinetic, potential and radiative) It has a "color", especially in the visible regime you are used to the colors of the rainbow:

31 The human eye is sensitive to light with colors from violet to red, which corresponds to a wavelength range: 4,000 Å < λ < 7,000 Å where an Å is m (Å=Ångström). = 400 nm < λ < 700 nm where 1 nm =10-9 m. White light: all colors of the rainbow Black: no light, hence no colors

32 The electromagnetic spectrum Light extends beyond the visible regime, and the continuous and infinite distribution of wavelengths is called the electromagnetic spectrum Light of different wavelengths interacts in different ways with matter

33 Different objects in the Universe give off EM radiation in different ways, depending on their physical condition.

34 The dual nature of light Light often acts like a wave but also can act like a particle in interactions with atoms and molecules. The particles are called photons. A photon is a mass-less particle that carries energy E at the speed of light. E = hc/λ = hν where h = 6.6 x J s (Planck's constant) Question: which has more energy, a photon of blue light, or of red light? UV or radio?

35 Young's double slit experiment Demonstrates the wave-like nature of light: Like water waves (right), EM waves can interfere.

36 The photoelectric effect Illustrates the particle nature of light: Photons hitting a piece of metal will knock out single electrons, but only if they have enough energy. Above this energy level the kinetic energy of the photon does NOT depend on the intensity of the incident radiation, but on the frequency. IF light had only wave-nature, the kinetic energy of ejected electrons should depend on the amplitude (intensity) of the wave, but instead it depends on the frequency (E=hν)

37 How do light and matter interact? Emission - light bulb, star Absorption - your skin can absorb light, in turn the absorbed energy heats your skin Transmission - glass and air lets the light pass through Reflection and scattering - light can bounce off matter leading to reflection (same direction of reflected light) or scattering (random direction of reflected light) Materials that transmit light are transparent. Materials that absorb light are opaque.

38 Three basic types of spectra Kirchoff's laws of spectroscopy: 1. A hot, opaque body, or a hot, dense gas produces a continuous spectrum. 2. A hot, transparent gas produces an emission line spectrum. 3. A cool, transparent gas in front of a source of a continuous spectrum produces an absorption line spectrum.

39 Illustration Kirchoff's laws

40 Astrophysical examples: - Continuous: asteroids, planets, etc. - Emission line: hot interstellar gas -- HII regions, planetary nebulae, supernova remnants. - Absorption line: stars (relatively cool atmospheres overlying hot interiors), cool interstellar gas. Important concepts: temperature and blackbody.

41 Temperature A measurement of the internal energy content of an object. Solids: Higher temperature means higher average vibrational energy per atom or molecule. Gases: Higher temperature means more average kinetic energy (faster speeds) per atom or molecule.

42 At high temperatures atoms and molecules move quickly, and slower at lower temperatures. If it gets cold enough, all motion will stop. How cold is that? Corresponds to a temperature of -273 C (-459 F) - absolute zero.

43 Kelvin temperature scale An absolute temperature system in which the temperature is directly proportional to the internal energy. Uses the Celsius degree, but a different zero point. 0 K: absolute zero 273 K: freezing point of water 373 K: when water boils

44 Temperature conversions Fahrenheit, Celsius, Kelvin (absolute) 0 K = -273 C (T K = T C +273) Room temp about 300 K T F = 9 5 T C + 32 T C = 5 9 (T F 32) T F = temperature in degrees Fahrenheit T C = temperature in degrees Celsius

45 Thermal radiation Let's consider the concept of a "blackbody" Tenuous gas: most light can pass through relatively unaffected Denser matter (like a rock) will not allow light to pass as easily A dense body will absorb light photons over a wide range of wavelengths This also means the absorbed photon cannot easily escape - once reemitted it immediately interacts with another molecule The photons bounce around inside the body for a long time, and when they finally escapes it has randomized wavelength => results in a smooth, continuous spectrum A perfect absorber/emitter is called a blackbody

46 Blackbody (thermal) radiation A blackbody is an object that absorbs all light, at all wavelengths: perfect absorber No reflected light As it absorbs the light, it will heat up Characterized by its temperature A black body will emit light at all wavelengths (continuous spectrum): perfect emitter Energy emitted depends on temperature

47 The radiation the blackbody emits is entirely due to its temperature. Intensity, or brightness, as a function of frequency (or wavelength) is given by Planck s Law: I ν = 2hν 3 1 c 2 e hν / kt 1 where k is Boltzmann constant = 1.38 x J/K Units: J s -1 m -2 ster -1 Hz -1 I λ = 2hc 2 1 λ 5 e hc / λkt 1

48 Example: 3 blackbody (Planck curves) for 3 different temperatures.

49 Wien's law for a blackbody λ max = /T where λ max is the wavelength of maximum emission of the object (in m), and T is the temperature of the object (in K). => The hotter the blackbody, the shorter the wavelength of maximum emission Hotter objects are bluer, cooler objects are redder.

50 The spectrum of the Sun is almost a blackbody curve.

51 Example 1: How hot is the photosphere of the Sun? Measure λ max to be about 500 nm, so T sun = /λ max = /5.0x10-7 = 5800 K Example 2: At what wavelength would the spectrum peak for a star which is 5800/2 = 2900 K? For a star with T= 5800 x 2 = 11,600 K? What colors would these stars be?

52 Which stars are hotter, and which are cooler?

53 Stefan-Boltzmann law for a blackbody F = σt 4 F is the energy flux, in joules per square meter of surface per second, σ is a constant = 5.67 x 10-8 W m -2 K -4, T is the object s temperature (in K). The hotter the blackbody, the more radiation it gives off at all wavelengths.

54 At any wavelength, a hotter body radiates more intense.

55 Example: If the temperature of the Sun were twice what it is now, how much more energy would the Sun produce every second? See box 5-2 for more examples.

56 Kirchhoff s laws illustrated

57 Spectrum of the Sun what kind of spectrum is this?

58

59 Spectrum of a quasar what kind of spectrum is this?

60 Atomic structure Nucleus: heavy, subatomic particles: Protons (positively charged) Neutrons (uncharged) Cloud of electrons in a cloud orbiting the nucleus (negatively charged) Old-fashioned model of the atom we now understand that electrons exist in "clouds", but this picture is useful for some calculations.

61 Chemical elements Atoms are distinguished into elements by the total number of protons in the nucleus. This is called the atomic number: 1 proton: Hydrogen 2 protons: Helium 3 protons: Lithium and so on.

62 Each element can have isotopes Same number of protons, but different number of neutrons. 12 C has 6 protons and 6 neutrons 13 C has 6 protons and 7 neutrons 14 C has 6 protons and 8 neutrons Isotopes of an element are chemically identical, but have different masses.

63 Inside the atom Quantum mechanics determines the details: Electrons can only orbit the nucleus in discrete orbitals Each orbital corresponds to a particular energy of the orbiting electron The electron must have exactly the right energy - otherwise it cannot be in that orbital

64 The simple Bohr model for the hydrogen atom: a single electron orbiting a single proton (the nucleus). The different energy levels represents allowed orbitals for electrons (not to scale, the n=2,3,4 orbits should be 4, 9 and 16 times larger than the n=1 orbit). First orbital = ground state (n=1). Lowest energy orbital the electron can reside in Higher orbitals: excited states (n=2,3,4 )

65 Energy level diagram of H Orbitals come in specific and exact energies

66 Emission and Absorption Electrons can get into excited states by either Colliding (other atoms or free electrons) Absorbing photons (of specific energy) Absorption: only photons with exact excitation energy are absorbed, ALL others pass through unaffected. Electrons get out of excited states by emitting photons in random directions.

67 An atom can absorb a photon, causing electron to jump up to a higher energy level. An atom can emit a photon, as an electron falls down to a lower energy level.

68 Emission and absorption line spectra Hot, low density gas where atoms are isolated will cause emission lines Lines at specific wavelengths, dark between lines Light from continuous spectrum through cooler gas cause absorption lines Dark regions corresponding to wavelengths of emission lines seen when the gas is hot, bright between lines

69

70 Energy level diagram for hydrogen:

71 Excited, low-density hydrogen gas. Red due to H-alpha emission line, n = 3 to n = 2.

72 Key points Different elements have different sets of energy levels => electron transitions produce or absorb photons of different and unique energies. Different energies = different wavelengths. The wavelengths of light tell you what elements you re dealing with.

73 Example: This is part of the Sun s spectrum, along with emission lines of vaporized iron.

74 Fingerprints of matter H has an easy orbital structure, but more complex atoms (several electrons) will have more difficult structure This cause more energy levels and a more complex line spectra Each element has a unique spectrum, reflecting the unique electron orbital structure. Isotopes has same lines, but slightly shifted in wavelength.

75 Molecules Even more complex, compounds of two or more atoms of same or different elements Share some electrons in common orbitals Have vibrational and rotational energy levels as well (IR, microwave, radio) => Very complex spectra!

76 Ionization If an atom or a molecule absorbs enough energy from a photon or a collision, an electron can escape the force field => positive ion By adding electrons, you can get a negative ion Ions differ from the neutral atoms/molecules: different spectral line signatures, and different chemical properties

77 Spectroscopy The spectrum of an objects tells us: Which atoms and molecules are present, and in which proportions Which atoms are ionized, and in which proportions How excited the atoms are, which tells us about the physical state (cold, hot) => Spectroscopy is a very important tool of the astronomer.

78 Doppler shifts Happens for all wave phenomena: sound => change of pitch light => change of wavelength (or color) V = λ observed λ emitted λ emitted c = Δλ λ emitted c where V is the velocity of the emitting source (m/s), c is the speed of light (m/s).

79 V = λ observed λ emitted λ emitted c = Δλ λ emitted c Redshift if receding, blueshift (negative sign) if approaching. Spectral lines are used to measure Doppler shift => gives us information about the motion of an object.

80 Example Doppler shift A spectral line normally seen at 400nm is shifted to 401nm due to relative motion of the source. What is the velocity of the source? Is it approaching or receding?