Problem Set 4 is due Thursday. Problem Set 5 will be out today or tomorrow. Launch Latest from MASCOT

Transcription

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2 Problem Set 4 is due Thursday. Problem Set 5 will be out today or tomorrow. Launch Latest from MASCOT

3 3 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-Boltzmann Law

4 8 Thermal Radiation and Circumstellar Disks

5 9 Thermal Radiation and Circumstellar Disks

6 10 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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8 14 Luminosity vs. Flux 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 4x106 Watts. The Inverse Square Law says that Solar flux drops off with distance as R. Luminosity Flux= 4π R

9 15 Spherical Blackbodies (and Cows) The emergent flux from each square meter (watts/m ) of a blackbody is st4 (Stefan-Boltzmann Law). The surface area of a sphere is 4pR so it is easy to take the unit area flux and turn it into a luminosity. 4 L=(4 π R )(σ T )

10 17 Spherical Blackbodies (and Cows) The emergent flux from each square meter (watts/m ) of a blackbody is st4 (Stefan-Boltzmann Law). The surface area of a sphere is 4pR 4 L=(4 π R )(σ T ) Stars are spherical. Stars approximate blackbodies A star s color gives away it s temperature via Wien s Law. Given you know T. Measure the distance and flux and thus luminosity determine R Have a means of measuring the stellar radius determines L and thus the distance (because you can measure the flux and apply the inverse square law)

11 18 Equilibrium Temperature of Planets The Sun's Luminosity is 4x106 Watts. The Inverse Square Law says that Solar flux drops off with distance as R. At Earth s distance (1.5 x 1011 meters) the solar flux is 1400 W/m Luminosity Flux= 4π R

12 19 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 R 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)

13 0 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 Fraction absorbed Flux= Luminosity 4 π R cross section L sun 4πR solar flux at R r is the radius of the object R is the distance to the sun A is the albedo of the object (its reflectivity)

14 1 Equilibrium Temperature of Planets The planet/asteroid radiates with emission efficiency relative to a perfect blackbody (i.e. emissivity), e 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 4πR input = (1 A) π r L sun 4π R r is the radius of the object R is the distance to the sun A is the albedo of the object (its reflectivity)

15 Equilibrium Temperature of Planets Setting the input equal to the output yields: T= ( L star (1 A) 16 π σ ϵ R ) 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 4πR input = (1 A) π r L sun 4π R r is the radius of the object R is the distance to the sun or star A is the albedo of the object (its reflectivity) e is the emissivity

16 3 Equilibrium Temperature of Planets For our Solar System, and assuming (1-A) = e (so equal absorption and emission efficiency) Calculate out everything for the Earth (R=1AU, Lstar = Lsun ) and the result is 77K. T= ( L star (1 A) 16 π σ ϵ R 1 T =77 K R AU Flux= Luminosity 4πR input = (1 A) π r L sun 4π R 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 ) 1/ 4

17 4 Equilibrium Temperature of Planets Generalizing to any stellar luminosity (where Lstar is in units of solar luminosities) T= ( L star (1 A) 16 π σ ϵ R 1/ 4 star L T =77 K R AU Flux= Luminosity 4πR input = (1 A) π r L sun 4π R 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 ) 1/ 4

18 5 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. A e~1

19 6 Spectral Line Emission/Absorption Individual atoms produce/absorb light only at precise discrete wavelengths/colors (or specifically at certain exact energies).

20 7 Spectral Line Emission/Absorption Emission at specific wavelengths 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.

21 8 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.

22 9 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. 1 Ze F elec = 4 π ϵ0 r me v F centrip = r

23 30 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. 1 e F elec = 4 π ϵ0 r me v F centrip = r Add a quantum mechanical twist that the angular momentum is quantized in integer multiples (n) of h/p. nh me v r = π

24 31 A Classical Approach to Atomic Energy Levels Equating the force laws and substituting in the quantized angular momentum equation to get rid of v, solving for r we get: ϵ0 h x 10 meters r (n) = n = n Z π Z e me Never forgetting that n is an integer. F centrip me v = r me v r = F elec 1 e = 4 π ϵ0 r nh π v = nh π me r

25 3 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 for K.E. Ze K.E. = 8 π ϵ0 r Ze P.E. = 4 π ϵ0 r Ze Total = 8 π ϵ0 r (n) F centrip me v = r me v r = nh π F elec 1 e = 4 π ϵ0 r ϵ0 h 5.9 x meters r (n) = n = n Z π Z e me

26 33 A Classical Approach to Atomic Energy Levels The absolute energy of an energy level is not nearly as interesting as the energy difference between two levels (corresponding to the energy of an emitted or absorbed photon). Converting the physical constants to a quantitative energy (in electron volts): Δ E = 13.6 ev F centrip me v 1 e F elec = = 4 π ϵ0 r r me v r = nh π [ 1 n lower ϵ0 h 1 n upper ] 5.9 x meters r (n) = n = n Z π Z e me

27 34 A Classical Approach to Atomic Energy Levels Converting this energy to the equivalent wavelength via E=hc/l yields the wavelength of a photon emitted or absorbed in an electron transition between two energy levels λ = nm Z nlower nupper [ F centrip me v = r me v r = nh π 1 ] F elec 1 e = 4 π ϵ0 r ϵ0 h 5.9 x meters r (n) = n = n Z π Z e me

28 35 A Classical Approach to Atomic Energy Levels If nupper = then the wavelength/energy calculated is the energy required to ionize the last electron from an element (or the only electron from hydrogen λ = nm Z nlower n upper [ 1 ]

29 36 A Classical Approach to Atomic Energy Levels A series of hydrogen lines has a common lower state Dn = nlower nupper = 1 is alpha, Dn = is beta... 1 = Lyman (ultraviolet): Lyman a is 11.6 nm 1 (a) 3 1 (b) 4 1 (g).. = Balmer (visible): Balmer a, known as Ha, is nm Red Teal Blue 3 = Paschen (near infrared) 4 = Brackett (infrared), Brackett g is 165 nm 5 = Pfund (infrared) 6 = Humphries (infrared) λ = nm Z nlower n upper [ 1 ]

30 37 H-alpha Solar Images

31 38 The Highway to Apache Point / Sunspot, NM

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33 40 A Classical Approach to Atomic Energy Levels A series of hydrogen lines has a common lower state Dn = nlower nupper = 1 is alpha, Dn = is beta... Energy spacing between transitions decrease with increasing n upper, becoming vanishingly small as nupper approaches infinity. These closely packed transitions create a band edge at a wavelength corresponding to ionization from the nlower energy level. The Balmer Jump in stellar spectra results from this clumping of highorder lines λ = nm Z nlower n upper [ 1 ]

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36 43 Radiative Lifetime Accelerated charges radiate. Consider an electron in the nd 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.

37 44 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.

38 45 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.

39 46 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.

40 47 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.

41 48 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

42 49 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

43 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 50

44 51 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.

45 5 Spectral Line Emission/Absorption Collisions between atoms can shorten lifetimes and broaden lines. ℏ Δ E ΔT >

46 53 Giant vs. Dwarf Star Spectra The difference is surface gravity / photospheric pressure. The collision time between gas atoms for stars of the same surface temperature will be pressure dependent with a longer time between collisions in the low pressure (giant star) configuration.

47 54 Giant vs. Dwarf Star Spectra The difference is surface gravity / photospheric pressure. The collision time between gas atoms for stars of the same surface temperature will be pressure dependent with a longer time between collisions in the low pressure (giant star) configuration.