Homework #2

1. Homework #2 Homework #2 is due Thursday, September 20. Be sure to always show the unit in your final answer (every physical quantity has a unit). Remember to only write the appropriate number of significant digits for your final answer.

2. Spectra and Spectral Lines • Need to “break apart” light into its constituent wavelengths • Infer composition • Temperature/Density • Velocity • Mass • One of the primary tools in astronomy

3. 2-dimensional vs. 1-dimensional spectra 2-d 1-d 2-d: dispersed spectrum straight from the dispersing slit 1-d: slice through the 2-d spectrum

4. What are the three basic types of spectral features? Emission line spectrum Continuous spectrum (black body emission) Absorption line spectrum Spectra of astrophysical objects are usually combinations of these three basic types.

5. Continuous Spectrum • The spectrum of a common (incandescent) light bulb spans all visible wavelengths, without interruption (blackbody radiation). Continuous light source: Incandescent bulb, star, any blackbody (including you)

6. Emission Line Spectrum • A low-densitycloud of hot gas emits light only at specific wavelengths that depend on its composition and temperature, producing a spectrum with bright emission lines - lines correspond to electron transitions Gaseous nebula - gas is too low density to emit like a blackbody

7. Absorption Line Spectrum A cloud of cold gas between us and a light bulb can absorb/re-direct light of specific wavelengths, leaving dark absorption lines in the spectrum. Note: If gas cloud is hotter than light source, we get emission lines on top of continuum.

8. Reasons Lines Occur: The Bohr Atom • In atoms, electrons can only have certain allowed orbits (energy levels). Orbital energy is quantized (quantum mechanics). Electrons can change orbits only by absorbing/emitting energy (a photon), and can only move from one discrete orbit to another.

9. Electrons in Atoms • In quantum mechanics, an electron in an atom does not orbit in the usual sense. • We can know only the probability of finding an electron at a particular spot. • But for simplicity, we will adopt a visual picture like in the last slide Different energy levels of hydrogen atom

10. Electron Energy Level Transitions Hydrogen • In a crude sense, electron energy levels can be thought of as orbits around the nucleus of the atom. • A larger orbit implies a higher (excited) energy level. Note: 1 eV = 1 electron volt = 1.6 x 10-12 ergs = 1.6 x 10-19 J

11. Electron Energy Level Transitions Increasing energy Allowed upward energy level transitions occur when a photon of the correct wavelength (from another source) is absorbed by the atom. Allowed downward electron energy level transitions release a photon at a specific wavelength. Not Allowed Allowed Energy of photon released/absorbed is equal to difference in energy levels.

12. Electron Transitions Energy of photon= Energy(level n) - Energy(level m) = h *  = h * c /  For example, an electron moving from the n=2 to n=1 level will release a photon of energy 10.2 eV (or equivalently a wavelength of 121.6 nm).

13. Electron Transitions • Each type of atom has a unique set of electron energy levels • Each electron transition corresponds to a unique photon energy, frequency, and wavelength • Level 1 = ground state Energy levels of hydrogen (oxygen, neon are much more complicated)

14. Electron Transitions • Downward electron transitions produce a unique pattern of emission lines Wavelength in nanometers (nm) = 10-7 cm = 10 Å

15. Electron Transitions • Because those atoms can absorb photons with those same energies, upward transitions produce a pattern of absorption linesat the same wavelengths. Wavelength in nanometers (nm) = 10-7 cm = 10 Å

16. Balmer Transitions Balmertransitions are important hydrogen transitions in the optical part of the spectrum. Transition down to/up from the n=2 level. n=3 to n=2 transition is more simply called “H”. Same for H (4 to 2 transition), etc. Wavelength in nanometers (nm) = 10-7 cm = 10 Å

19. Three Types of Spectra

21. Chemical Fingerprints • Each type of atom has a unique spectral fingerprint • Other elements are MUCH more complicated than hydrogen

22. Chemical Fingerprints Observing the fingerprints in a spectrum tells us which kind(s) of atom(s) are present. Sometimes a non-trivial task identifying emission or absorption lines.

23. Energy Levels of Molecules Molecules are composed of more than one atom, such as H2, H2O, or CO2 • Molecules have additional energy levels because they can vibrate and rotate

24. Energy Levels of Molecules • The large numbers of vibrational and rotational energy levels can make the spectra of molecules very complicated • Many of these molecular transitions are in the infrared part of the spectrum spectrum of molecular hydrogen

25. Identifying Elements in Stars Composite stellar spectrum Problem: Which elements are seen in the spectrum of this star? Answer: hydrogen and calcium are strong

26. Ionization State Carbon has 6 electrons, when neutral An electron can be knocked off by a photon or collision with another particle (electron, atom, proton…) Common way of showing how many electrons an ion has: C I Roman numeral showing ionization; I = neutral; II = missing one electron, etc. Symbol for Element Problem: What is the symbol for oxygen (8 electrons) that has only one electron left? Answer: O VIII

27. Keeping Terminology Straight 2H – hydrogen with 1 proton and 1 neutron (deuterium, a rarer isotope of normal hydrogen 1H) H2 – hydrogen molecule (2 hydrogen atoms chemically bonded) H II – hydrogen atom missing its electron

28. How do we interpret an actual spectrum? • By carefully studying the features in a spectrum, we can learn a great deal about the object that created it.

29. What is this object? Reflected sunlight: Continuous spectrum of visible light is like the Sun’s except that some of the blue light has been absorbed—object must look red.

30. What is this object? Thermal radiation: Infrared spectrum peaks at a wavelength corresponding to a temperature of 225 K.

31. What is this object? Carbon dioxide: Absorption lines are the fingerprint of CO2 in the atmosphere.

32. What is this object? Ultraviolet emission lines: indicate a hot, low-density upper atmosphere

33. What is this object? Mars!

34. How does light tell us the speed of an object? The Doppler Effect – a change in frequency (and therefore a change in wavelength and energy) of a wave caused by motion of the object emitting the wave – applies to both sound and light waves.

35. The Doppler Effect • Light emitted from an object coming toward us has a higher (frequency, energy) and shorter wavelength than light from a stationary object. The light is said to be blue-shifted. • Light emitted from an object moving away from us has a lower (frequency, energy) and longer wavelength than light from a stationary object. The light is said to be red-shifted.

36. Emitter is approaching waves compressed (blueshift) Emitter is receding waves lengthen (redshift) Relative Velocity = 0 Motion between Observer & Emitter

37. Relationship between wavelength shift and radial velocity Order matters! Velocity is negative if it moves toward us For blueshift, vobj is negative. Only valid for v << c Very Important Tool: Measure wavelength shift  learn velocity

38. Measuring the Shift Stationary • We generally measure the Doppler effect from shifts in the wavelengths of spectral lines. Moving away Away faster Moving toward Toward faster

39. Doppler shift tells us ONLY about the part of an object’s motion toward or away from us: True velocity of object measured. No redshift or blueshift measured! True velocity of object underestimated.

40. Doppler shift tells us ONLY about the part of an object’s motion toward or away from us: Angle  Velocity component along our line of sight = v * cos  = vradial v  If  = 90°, cos  = 0 --> no radial velocity component, no Doppler shift measured  Doppler effect measures v cos  NOTv. If  is unknown the measured velocity is a lower limit.

41. Example A particular spectral line is measured to be located at 5007 Å in the laboratory (actually, it is O III). This same spectral line from an object is measured to be at 5017 Å. We know the object is moving away from us at a 60° angle from our line of sight. What is the object’s velocity? v cos  / c = (observed - rest )/ rest v = (c / cos ) * (observed - rest )/ rest = (3 x 1010 cm s-1/ 0.5) * (5017 Å - 5007 Å) / 5007 Å = 1.2 x 108 cm s-1 (or 0.004c)

42. How does light tell us the rotation rate of an object? • Different Doppler shifts from different sides of a rotating object spread out its spectral lines • Each point on the surface of the star has a different cos  with respect to us • At edge of star,  = 0° or 180° (maximum velocity), at center of star =90° (no measured red/blue shift)

43. Spectrum of a Rotating Object • Rotation causes spectral lines to broaden • Spectral lines are wider when an object rotates faster

44. Chapter 6Telescopes: Portals of Discovery

45. Eyes and Cameras: Everyday Light Sensors To manipulate light, we need to know: • How can we control light paths? • How do we record images? • How does your eye form an image?

46. Refraction:Light path changes direction when entering a new material of different density Speed of light in water is only 75% of its speed in a vacuum. Index of refraction = n = (speed of light in vacuum)/(speed of light in medium). Dependent on wavelength  shorter wavelengths are bent more

47. Another Example: Refraction at Sunset Refraction by dense air near the horizon distorts Sun's appearance.

48. Extreme Example: The “Green Flash” Look for this at sunset if you ever have a very flat horizon (i.e, in Hawaii)!

49. Focusing Light by Refraction • Refraction can focus parallel light rays at a point called the focus (obviously). • Distance between lens and focus is called the focal length.

50. Your Eye Versus a Camera • Light enters through the pupil • Lens refracts light • Focused image lands on the retina • Exposure time ~1/20 seconds • Light enters through the opening in the camera • Lens refracts light • Focused image lands on the detector (film or CCD) • Exposure time set by shutter