MASCOT Lander to deploy on Wednesday

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4 4 MASCOT Lander to deploy on Wednesday More visual imaging, hopping, and infrared spectroscopy = mineralogy

5 5 Synodic vs. Sidereal Month As with the sidereal vs. solar day, the Moon moves in its orbit (significantly) during the course of a lunar month. The next Full Moon does not occur for some time (about 1/12 of a sidereal lunar period of 27.3 days) after the Moon complets a sidereal rotation. Since somebody on the Earth could be considered to orbit the center of the Earth once a day and the Moon orbits this person like a superior planet the superior planet relation for sidereal vs. synodic periods applies to the Lunar Synodic Month (29.5 days) = P syn P Earth P sid

6 6 Time Between Lunar Meridian Transits The Moon moves in its orbit (significantly) during the course of a solar day. The Moon s motion is about one 27.3th of 360 degrees (the lunar sidereal orbital period is 27.3 days) a little more than 10 degrees. Since the Earth turns 15 degrees per hour this must add about an hour. Since somebody on the Earth could be considered to orbit the center of the Earth once a day and the Moon orbits this person like a superior planet, the superior planet relation for sidereal vs. synodic periods applies to the time between meridian crossings. 1 P moon at culmination = 1 P Earth day 1 P sid Moon orbit ``Culmination is when an astronomical object reaches its highest altitude. For objects tied to the celestial sphere this happens at meridian crossing.

7 7 Time Between Lunar Meridian Transits The calculation yields a lunar transit period of 24h 50m Implying an average time between high tides of 12h 25m You can go to the beach and infer the existence of and the orbital period of the Moon 1 P moon at culmination = 1 P Earth day 1 P sid Moon orbit

8 8 Synodic Lunar Month The Synodic Lunar Month is the time it takes the Moon to execute a cycle of phases - Full to Full or New to New. Since the phases are tied to the Sun and the Earth orbits the Sun about 1/12 the way around in the course of a lunar sidereal month. The synodic month is about 1/12th a lunar sidereal period (couple of days) longer than a sidereal month.

9 9 Synodic Lunar Month Mathematically the Moon orbits the Earth at an angular rate wsid_moon (so units of radians per second, degrees per day.) Relative to the Sun the Moon appears to go around the Earth more slowly because the Earth is orbiting the Sun at an angular rate, wsid_earth_orbit The synodic orbital rate of the Moon is the difference of these two rates. ω syn_moon = ω sid_moon_orbit ω sid_earth_orbit 2π P syn_moon = 2π Psid_moon_orbit 2π P sid_earth_orbit

10 10 Synodic Lunar Month Mathematically the Moon orbits the Earth at an angular rate wsid_moon (so units of radians per second, degrees per day.) Relative to the Sun the Moon appears to go around the Earth more slowly because the Earth is orbiting the Sun at an angular rate, wsid_earth_orbit The synodic orbital rate of the Moon is the difference of these two rates. ω syn_moon = ω sid_moon_orbit ω sid_earth_orbit = 29.5 days

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14 14 Note the nearly identical eclipses 18+ years apart. The Moon's exact position repeats relative to the Earth and Sun every 18 years creating a family of eclipses the Saros cycle.

15 15 Saros Cycles The coincidence of three periodic lunar phenomena after 18 years (and 11 days) leads to repeats of identical eclipse circumstances at 18 year intervals. Except the Earth is turned by 8 hours for each successive one.

16 16 Saros Cycles The coincidence of these three periodic phenomena after 18 years (and just 11 days) leads to repeats of identical eclipse circumstances at 18 year intervals. Except the Earth is turned by 8 hours for each successive one.

17 17 Solar Eclipses from Charlottesville? You have to wait several hundred years on average for a total solar eclipse to happen at your location. Partial eclipses, which cover more area, are common however.

18 18 Eclipses in the 21st Century

19 19 Solar Eclipses on Other Worlds The outer planets have lots of moons...

20 20 Lunar Eclipses The Earth casts a shadow on the Moon To be specific the Moon moves through the Earth's shadow

21 21 Lunar Eclipses The Moon takes a couple of hours to cross the Earth's shadow. The Earth's umbral shadow is large enough to consume the whole Moon.

22 22 The Shape of Earth s Shadow Round Substantially bigger than the Moon.

23 23 Lunar Eclipses Everybody on the night side of the Earth (and then some) can see the eclipse.

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25 25 Lunar Eclipses The Moon takes on a reddish hue during the total eclipse because of light refracted through the Earth's atmosphere. This is the combined light of all of the world's sunrises and sunsets! The View from the Moon

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27 27 Works for Pluto, too!

28 28 Annular vs. Total Eclipses The Moon follows an elliptical orbit. When it is close to the Earth it easily covers the Sun (total eclipse) when far away its angular size is smaller than the Sun's (annular eclipse).

29 29 Annular vs. Total Eclipses If any solar photosphere is visible during an annular eclipse then none of the spectacular phenomena is visible.

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31 31 MASCOT Descent

32 32 Light and Wavelength Wavelength alone distinguishes types of light At visible wavelengths short wavelengths are blue; long are red Wavelength, color, and energy of a photon are all the same thing λ ν=c hc E=hν= λ

33 33 Wavelength vs. Frequency The product of the wavelength and frequency of a photon (the separation between wavecrests times the number of wavecrests passing per second) naturally equals the speed at which the wave moves. λ ν=c c ν = λ

34 34 Wavelength vs. Energy hc E=hν= λ

35 35 Photon Production Accelerated charges (typically electrons) produce photons. Synchrotron radiation (from electrons spiraling in a magnetic field) may be conceptually the most easily motivated. Consider whirling an electron around at the end of a string. An observer at a distance sees a varying electromagnetic field.

36 36 Photon Production Accelerated charges (typically electrons) produce photons. Synchrotron radiation (from electrons spiraling in a magnetic field) may be conceptually the most easily motivated. Consider whirling an electron around at the end of a string. An observer at a distance sees a varying electromagnetic field.

37 37 Sorting Light Filters and Spectra Light can be sorted and/or restricted by wavelength.

38 38 Spectra Light can be sorted and binned by wavelength. The resulting spectrum can be projected on a screen or plotted on a graph.

39 39 Two Fundamental Types of Spectra Spectra can be from one of two classes Continuous a smoothly varying distribution of all colors Discrete emission (or absorption) at precise wavelengths Often a spectrum is a combination of both

40 40 The Solar Spectrum

41 41 Pluto s Infrared Spectrum

42 42 Continuous Spectra: Thermal Radiation Any hot object glows The hotter the object the brighter and bluer the glow

43 43 The Nature of Temperature Temperature is a measure of the energy of motion of particles in a gas or in a solid. In a gas the particles (atoms or molecules) are independently flying about colliding with one another or with the walls of the chamber. At high temperature the particles move quickly. At low temperatures they are sluggish. In a solid the particles are vibrating in place. The lowest possible temperature is the point at which all thermal energy has been removed absolute zero.

44 44 The Nature of Temperature Temperature is a measure of the energy of motion of particles in a gas or in a solid. In a gas the particles (atoms or molecules) are independently flying about colliding with one another or with the walls of the chamber. At high temperature the particles move quickly. At low temperatures they are sluggish. In a solid the particles are vibrating in place. The lowest possible temperature is the point at which all thermal energy has been removed absolute zero.

45 45 Continuous Spectra: Thermal Radiation Any hot object glows The hotter the object the brighter and bluer the glow

46 46 Continuous Spectra: Thermal Radiation Dense spheres of gas (stars) are good approximations to blackbodies as well. The hot stars below are blue. Cooler ones are yellow and red.

47 47 The Planck Equation The Blackbody/Planck equation defines, for a given temperature, the spectrum of emergent energy per unit time into a unit solid angle (i.e. the specific intensity) from a unit area of a blackbody per unit frequency. 2hν B ν (T ) = 2 c 3 1 ( ) Watts / m2 Hz sr hν kt e 1 In general, we care about the amount of energy launched into a given solid angle from a unit area of a blackbody Watts = B ν (T ) Δ ν Δ Ω Δ Area

48 48 The Planck Equation The Blackbody/Planck equation defines, for a given temperature, the spectrum of emergent energy per unit time into a unit solid angle (i.e. the specific intensity) from a unit area of a blackbody per unit frequency. 2hν B ν (T ) = 2 c 3 1 ( ) Watts / m2 Hz sr hν kt e 1 In general, we care about the amount of energy launched into a given solid angle from a unit area of a blackbody Watts = B ν (T ) Δ ν Δ Ω Δ Area

49 49 Statistical Mechanics kt Energy injected into a coupled/interacting system (imagine a network of springs or a gas of colliding atoms) tends to distribute itself evenly amongst the degrees of freedom of the system. A typical degree of freedom has energy, ½ kt. A free particle has a typical energy of 3/2kT (three degrees of translational freedom) Bulk system properties e.g. the equilibrium temperature, the distribution of velocity of particles in a gas - are dictated by statistics/probabilities. States which require higher energy are less probably populated by the factor ni α e Ei kt However, each energy may have many identical configurations corresponding to that energy ni = (density of states) x e Ei kt

50 50 Solid Angle A solid angle is the three-dimensional equivalent of a two dimensional angle basically a cone defined by its apex angle. Solid angle is measured in units of steradians, where there are exactly 4p steradians on a full sphere (41,253 square degrees) For small cone apex angles the solid angle, W, is given by: 2 θ Ω=π 4 In spherical coordinates a differential unit of solid angle is d Ω = sin θ d θ d ϕ

51 51 Visualizing Solid Angle

52 52 Filter Profiles Delta(l) vs. Delta(n)

53 53 Maybe the Most Important Thing You'll Ever Learn in Astronomy c c ν= so d ν = 2 d λ λ λ ( ) 2hν B ν (T ) = 2 c 3 2 hc B λ (T ) = 5 λ 1 ( ) ( ) 2 1 watts m Hz sr hν kt 1 e e hc λ kt watts m m sr 1

54 54 The Planck Function

55 55 The Planck Function

56 56 Continuous Spectra: Thermal Radiation The equations below quantitatively summarize the light-emitting properties of solid objects. The hotter the object the bluer the glow. The Sun (6000K) peaks in the middle of the visible spectrum (0.5 micrometers / 500 nanometers) Room temperature objects (300K) peak deep in the infrared (10 um). Wien's Law The hotter the object the brighter the glow. The power emitted from each square centimeter of the surface of a hot object increases as the fourth power of the temperature. Double the temperature and the emission goes up 16 times! Stefan-Boltzman Law

57 57 Motivations for Derivation Stefan-Boltzman Law Integrate the Planck Law over all wavelengths and 2p solid angle to get emergent total flux. - Solution Wien's Law Take the derivative vs. l and set equal to zero. Must be solved iteratively, not analytically. Solution 2 hc B λ (T ) = 5 λ 2 ( 1 e hc λ kt 1 )

58 58 The Planck Function Two Extremes

59 Two Extremes 2 hc λ5 ( 1 e hc λ kt 1 59 ) Rayleigh Jeans Long wavelengths for a given temperature Longward of the Wien's Law peak 2 ckt B λ (T ) = 4 λ B λ (T ) = 2 hν = The exponential dominates 2 2 hc B λ (T ) = e 5 λ hc k T λ because e x = 1+ x for x 1 Wien tail hν = hc λ kt hc k T λ

60 61 Sunspots and Thermal Radiation Sunspots are relatively cooler regions of the Sun's 6000K surface. Being only about 1000K cooler than their surroundings, they do glow brightly, but due to the strong, T4, dependence of a hot solid object's brightness on its temperature they appear dark.

61 62 Thermal Radiation and Circumstellar Disks

62 63 Thermal Radiation and Circumstellar Disks

63 64 Submillimeter Galaxies The study of the first galaxies in the distant universe benefits from the fact that much of the stellar radiation gets reprocessed by dust via absorption and re-emission at a temperature of around 30K, thus a peak wavelength around 100um. Cosmological redshift moves this peak into the radio/submillimeter part of the spectrum. Galaxies actually become brighter as they become more distant in a given radio band.

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65 70 Flux and the Inverse Square Law Flux is the amount of energy passing through a unit area per unit time. Luminosity is the total amount of energy leaving a source. The Sun's Luminosity is 4x1027 Watts. The Inverse Square Law says that Solar flux drops off with distance as R2. Luminosity Flux= 2 4π R

66 71 Spherical Blackbodies (and Cows) The emergent flux from each square meter (watts/m 2) of a blackbody is st4 The surface area of a sphere is 4pR2 2 4 L=(4 π R )(σ T ) Stars are good approximations to blackbodies. To get the distance measure th flux of a star and its distance (not so easy). Alternatively determine its temperature from the Wien Law and you can estimate it's size knowing the easily measured flux.

67 72 Equilibrium Temperature of Planets The Sun's Luminosity is 4x1027 Watts. The Inverse Square Law says that Solar flux drops off with distance as R2. Luminosity Flux= 2 4π R

68 73 Equilibrium Temperature of Planets A spherical planet or asteroid presents a circular cross section to the Sun's light. The intercepted energy per unit time is: input = (1 A)π r Flux= Luminosity 4 π R2 2 L sun 4πR r is the radius of the object R is the distance to the sun A is the albedo of the object (its reflectivity) 2

69 74 Equilibrium Temperature of Planets The planet/asteroid radiates with emissivity, e 2 4 output = 4 π r σ T ϵ where e is the emissivity (think of it as the radiative efficiency) and would in general be equal to (1-A), however both are wavelength dependent and if you are absorbing visible light but emitting infrared the two terms can be quite different. Flux= Luminosity 2 4πR input = (1 A) π r 2 L sun 4π R 2 r is the radius of the object R is the distance to the sun A is the albedo of the object (its reflectivity)

70 75 Equilibrium Temperature of Planets The planet/asteroid radiates with emissivity, e T= ( L star (1 A) 16 π σ ϵ R 2 ) 1/ 4 the temperature of an object falls off as the square root of its distance from its star and depends weakly on the luminosity of the star (one-quarter power) Flux= Luminosity 2 4πR input = (1 A) π r 2 L sun 4π R 2 r is the radius of the object R is the distance to the sun A is the albedo of the object (its reflectivity) e is the emissivity

71 76 Equilibrium Temperature of Planets T= ( L star (1 A) 16 π σ ϵ R 2 ) 1/ 4 1 T =280 K R AU Flux= Luminosity 2 4πR input = (1 A) π r 2 L sun 4π R 2 r is the radius of the object R is the distance to the sun A is the albedo of the object (its reflectivity) e is the emissivity

72 77 Asteroid Radiometry Asteroids can have quite different visual reflectivity, but their emissivities are similar, typically close to e=1. Infrared flux measurements are used to pin down asteroid sizes.

73 78 Spectral Line Emission/Absorption Individual atoms produce/absorb light only at precise discrete wavelengths/colors (or specifically at certain exact energies).

74 79 Spectral Line Emission/Absorption This property arises from the discrete nature of electronic orbits in atoms. Electrons can only be in configurations that have a specific energy. Jumping between these configurations (higher to lower energy) emits light. A photon of exactly the right energy can kick an electron from a lower to higher energy.

75 80 Spectral Line Emission/Absorption This property arises from the discrete nature of electronic orbits in atoms. Electrons can only be in configurations that have a specific energy. Jumping between these configurations (higher to lower energy) emits light. Conversely, a photon of exactly the right energy can kick an electron from a lower to higher energy.

76 81 A Classical Approach to Atomic Energy Levels Consider the electron in orbit around the atomic nucleus (in this case a proton) held in orbit by the electrostatic attraction between two opposite charges. 2 1 Ze F elec = 2 4 π ϵ0 r me v F centrip = r 2

77 82 A Classical Approach to Atomic Energy Levels Consider the electron in orbit around the atomic nucleus (in this case a proton) held in orbit by the electrostatic attraction between two opposite charges. 2 1 e F elec = 4 π ϵ0 r 2 me v F centrip = r 2 Add a quantum mechanical twist that the angular momentum is quantized in integer multiples (n) of h/2p. nh me v r = 2π

78 83 A Classical Approach to Atomic Energy Levels Equating the force laws and substituting in the quantized angular momentum equation we get: ϵ0 h x 10 meters 2 r (n) = n = n 2 Z π Z e me F centrip me v 2 = r me v r = 2 F elec nh 2π 1 e2 = 4 π ϵ0 r 2

79 84 A Classical Approach to Atomic Energy Levels What is the energy of an orbit as a function of r(n)? Equate the force equations and rearrange to get ½ mv 2 for K.E. 2 Ze K.E. = 8 π ϵ0 r 2 Ze P.E. = 4 π ϵ0 r 2 Ze Total = 8 π ϵ0 r (n) F centrip me v 2 = r me v r = nh 2π F elec 1 e2 = 4 π ϵ0 r 2 ϵ0 h x meters 2 r (n) = n = n 2 Z π Z e me 2

80 85 A Classical Approach to Atomic Energy Levels The energy of each quantized level depends inversely on the level number squared. The difference in energy between levels (and thus the energy of a radiated or absorbed photon is: Δ E = 13.6 ev F centrip me v 2 1 e2 F elec = = 4 π ϵ0 r 2 r me v r = nh 2π [ 1 n 2 lower ϵ0 h 2 1 n 2 upper ] 5.29 x meters 2 r (n) = n = n 2 Z π Z e me 2

81 86 A Classical Approach to Atomic Energy Levels The energy of each quantized level depends inversely on the level number squared. The difference in energy between levels (and thus the energy of a radiated or absorbed photon is: λ = 2 nm 2 2 Z nlower n upper [ F centrip me v 2 1 e2 F elec = = 4 π ϵ0 r 2 r me v r = nh 2π ϵ0 h 2 1 ] 5.29 x meters 2 r (n) = n = n 2 Z π Z e me 2

82 87 A Classical Approach to Atomic Energy Levels A series of hydrogen lines has a common lower state Dn = 1 is alpha, Dn = 2 is beta... 1 = Lyman (ultraviolet), Lyman a is nm 2 = Balmer (visible), Balmer a, known as Ha, is nm 3 = Paschen (near infrared) 4 = Brackett (infrared), Brackett g is 2165 nm 5 = Pfund (infrared) 6 = Humphries (infrared) λ = 2 nm 2 2 Z nlower n upper [ 1 ]

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84 89 Radiative Lifetime Accelerated charges radiate. Consider an electron in the 2nd hydrogen energy level. How long does it take, classically, to radiate enough energy to reach the energy of the 1st level? Answer = 3x10-9 s (you get half that for a formal quantum mechanical solution). Bottom line an excited state decays quite rapidly emitting a photon equal in energy (wavelength) to the energy difference between levels. Selection rules driven by quantum mechanical principles (exclusion law, angular momentum restrictions) limit allowed transitions and can make lifetimes substantially longer that this simple classical result.

85 90 Spectral Line Emission/Absorption This property arises from the discrete nature of electronic orbits in atoms. Electrons can only be in configurations that have a specific energy. Jumping between these configurations (higher to lower energy) emits light. A photon of exactly the right energy can kick an electron from a lower to higher energy.

86 91 Spectral Line Emission/Absorption This property arises from the discrete nature of electronic orbits in atoms. Electrons can only be in configurations that have a specific energy. Jumping between these configurations (higher to lower energy) emits light. A photon of exactly the right energy can kick an electron from a lower to higher energy.

87 92 Spectral Line Emission/Absorption Spectral lines can reveal the elemental content of a planet or star's atmosphere. Line intensity reveals both the quantity of the element as well as the temperature.

88 93 Spectral Line Emission/Absorption Spectral line absorption arises when light from a continuous source passes through a cold gas. The gas atoms selectively remove (actually scatter) specific colors/energies.

89 94 The Doppler Shift The observed wavelength of a spectral line depends on the velocity of the source toward or away from the observer. The amount of the shift is proportional to the object's velocity relative to the speed of light (so typically the shift is tiny but measurable). λ shifted λ rest Δλ v = = λ rest λ rest c

90 95 The Doppler Shift Objects approaching an observer have wavelengths artificially shifted toward shorter wavelengths a blueshift. Objects moving away toward longer wavelengths a redshift Note that these are directions in the electromagnetic spectrum, not absolute colors. λ shifted λ rest Δλ v = = λ rest λ rest c

91 The Doppler Shift λ shifted λ rest Δλ v = = λ rest λ rest c Using the Doppler Shift we can measure the subtle motions (towards or away from us) of stars, galaxies and interstellar gas without ever seeing actual movement 96

92 97 Spectral Line Emission/Absorption Doppler velocities of individual atoms in a gas broaden spectral lines in a way that is characteristic of the temperature and density.

93 98 Spectral Line Emission/Absorption Collisions between atoms can shorten lifetimes and broaden lines. ℏ Δ E ΔT > 2