radiation-part2

radiation-part2 - NASA, NOAO, ESAand The Hubble Heritage...

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Unformatted text preview: NASA, NOAO, ESAand The Hubble Heritage Team (STScl/ AURA) Angles and Solid Angles Solid angles describe a fraction of a sphere. If an emitting source covers an area A on a spherical surface of radius 7“, then the solid angle is Q:A/r2 Solid angles are measured in steradians How many steradians in a hemisphere? AST 203 (Spring 2011) Intensity I(u)du = energy/unit time/unit surface area in the frequency range 1/ to V + dz/ being emitted into a cone of solid angle d9 Radiation moves through a small area dA into the cone described by d9 Energy moving through this area into the solid angle dB is dEzIV COSQdAdI/dfldt Intensity is measured in units of erg s"1 cm"2 Hz"1 ster'1 AST 203 (Spring 2011) Intensity (.059 {Actor [Vt lVl‘l‘eL/lsl‘ly ciefmd‘wh AST 203 (Spring 2011) Intensity In terms of wavelength, I(A)d)\ = energy/unit time/unit area in the wavelength range A to A + dA emitted into solid angle d9 Then dE : I,\ cos QdA dA d9 dt It is important to remember that intensity has a direction. Note—your book is a bit sloppy here. It leaves out the solid angle part. AsT 203 (spring 2011) Blackbody - We won't go deeply into what intensity means (AST 341 topic) — We'll use intensity to derive some important results. - Important concept: blackbody (thermal radiator) — Object in thermodynamic equilibrium — Emission = absorption — Emitted light has well-known intensity spectrum - Function ofT only - Isotropic, unpolarized. - Blackbody emission spectrum can be very different than that it absorbs — Only requirement is that the net energy gain is zero, is it is in equilibrium. AST 203 (Spring 2011) Blackbody A blackbody absorbs ALL incident radiation —> black. If emission ¢ absorption, then T would change —> no equilibrium. Stellar interior: atoms opaque to radiation—absorb and re-emit it. We see the stellar surface —> stars act as good blackbodies. (Fimoozwmema) AST 203 (Spring 2011) Thermal Radiation 1433.2? mm .123 .100 uflr Seeing in IR maritime 352-0 NA Elfin-"I P A C Ashttpwmnlopsmos.ipac.caltech .ed u/cosmic_kids/learn_ir/ind ex. html Blackbodies infrared visibleUV Ximy Gammuimy 1015 As T increases, the peak of the emission moves to higher frequencies (shorter wavelengths) 10m 105 Hotter objects appear bluer. 107m “3715 105 10‘“ 10‘5 102” Notice that the higher temperature blackbody emits more radiation than the lower temperature blackbodies at all frequencies. AST 203 (Spring 2011) Wien's Law Position of the peak emission moves to shorter A (higher u) with increasing T. Wien's law: AmaXT = 0.29 cm K = 2.9 X 106 nm K For a star: peak of its spectrum —> surface temperature Note, T of a star increases toward the center. AST 203 (Spl inc; 2011 i Wien's Law Consider the constellation Orion. Betelgeuse's spectrum peaks at 800 nm (what color does it appear?) What is its temperature? 2£3X UT3HH11{ T:—:3600K 800nH1 Rigel's spectrum peaks at 220 nm, what is its temperature? X 11m K (Mouservwniams) T = — = 13000 K 220nH1 AST 203 (Spring 2011) Wien's Law The average T of the earth is 300 K. What wavelength does it radiate in? Why don't we see it glowing? 2£lx UTSHH1I{ cr' Amax : W : nm (NASA/Apollo11) What part of the electromagnetic spectrum is this? AST 203 (Spring 2011) Rayleigh-Jeans Law What's a first principles explanation of the blackbody spectrum? Classical physics: 2kTV2 2 I(V,T) : C Works well at low V, but does not work at higher frequencies. As we see, as V —> 00 this predicts an infinite amount of energy. ultraviolet catastrophe AST 203 (Spl ing 2011 ) Blackbod ies 10‘5— I Rludiol I I I lnfrhredlvisibleuvl >l<emyI I Gangtmuilroy I NOtice the ' Rayleigh-Jeans law gets the slope right for low frequencies. AST 203 (Spl ing 2011 i Planck's Law Planck: emperical formula that fits the entire blackbody spectrum 2hV3/62 [(y’ : Bhu/kT _ 1 For low frequencies, hy << kT, and we can use expand: 3132 m3 (B— _ _ e —1+£L‘+2! +3! —l—... and write ehll/kZT _ 1 % h_y kT resulting in [(y, T) m ZkTZ/2/c2 —Rayleigh-Jeans expression. Development required quantization of energy of photon: E = nhv = nhc/A AST 203 (Spring 2011) Planck's Law Wavelength instead of frequency: ‘ I T _ 2hV3/02 (y’ )— Bhu/kT _ 1 Simply replace V with C/A—incorrect. Look at the definition of I so dE : L, cos 0dA d9 dt dz/ : I,\ cos QdA d9 dt dA Judy : IAdA AST 203 (Spring 2011) Planck's Law Since V = C/A , we have C dl/ : —Ed/\ Minus sign: frequency increases = wavelength decrease. Ignoring this sign, we have I(/\, T) : [(u, T)% or 2h02/A5 [(A’T) : ehc/AkT _ 1 Wien's law: define cc : hc/AkT and find maximum of 1(95) AsT 203 (Spring 2011) Planck's Law Note, to differentiate between a general intensity, [(y, T) , and a blackbody, most books write 2hz/3/c2 B04 : Bhv/kT _ 1 AsT 203 (Spring 2011) Photons Photons are particles which are the “units” of light Countable Described by a frequency and wavelength Carries momentum and energy hc E=h :— V /\ AST 203 (Spring 2011) Intensity and Flux (Zeilik and Smith) Intensity has a direction, i.e. it is the energy/time/area/frequency emitted per unit solid angle in a specific direction. Detectors measure the energy flux (erg s‘1 cm'2 Hz'1) hitting the detector. Records energy hitting the detector area from all directions. Frequency dependent—monochromatic flux. Integrate over all frequencies —> total flux (erg s‘1 cm'z). AST 203 (Spring 2011) Flux (Karttunen et al.) If we start with the definition of intensity, dE : IV cos 6dA d1/ (19 (175 then we can write the fluX as: dE f_/—dAdt _/IV cosddfldu Flux from a star: intensity = Planck function. We are interested in the outward emitted fluX from a patch dA, so f /dE fr /7r/2/OOB 0 ' 0d0d¢d : — : V cos sm V dAdt ¢=0 9:0 u=0 Note: we will not be tested on the details of this derivation, but we will need to understand the end result. AST 203 (Spring 2011) Flux The angular integration gives f—7r/ Budu =0 Integral over the frequency (we won't do this integral here) gives: f : 0T4 This is called the Stefan-Boltzmann law. Here, a = 5.67 X 10—5 erg 01111—2 K_4 53—1 (see Karttunen et al. for a detailed derivation). We will use this result often. AST 203 (Spring 2011) Luminosity and Flux Note the very strong T dependence. If we double T, the radiated flux increases by 16x! Integrate over the stellar surface —> luminosity (erg/s) L : 47rR20T4 We measure this in erg 3'1. ln SI units, this is measured in Watts. What is the luminosity of a common light bulb? How hot does the filament get? AST 203 (Spring 2011) Luminosity and Flux 100W=10"ergs‘1 The filament in the lightbulb is typically a thin wire coiled up, with an overall surface area of about 1 cm2 The temperature is then T Z A 1/4 2 109 erg 8—1 1/4 GA 5.67 X 10—5 erg s—1 cm—2 K—4- 1 cm2 T m 2000 K AST 203 (Spring 2011) Luminosity and Flux Sun's spectrum peaks at 500 nm, Using Wien's law, this corresponds to a temperature of 2.9 x 106 nm K T : — = 5800 K 500 nm The radius of the Sun is R9 : 7 x 1010 cm , giving a luminosity of L : 47r(7 x 1010 em)2 (5.67 x 105 erg s’1 errr2 K41) (5800 K)4 : 3.9 x 1033 erg 8’1 We call this the solar luminosity and denote it as LG We will use this as a reference luminosity when we compare to other stars. AST 2031;8pring 2011) Luminosity and Flux Luminosity is invariant with distance, but flux falls off as 1/r2 Measure the same star at two different distances, the flux will be different. The flux of the Sun at its surface is f = (5.67 x 105 erg s—1 errr—2 K—4) (5800 K)4 = 6.4 x 1010 erg s—1 em—2 Earth is 1.5 x 1013 cm (1 AU) away —> flux of the Sun at earth is Lo f 7 47rd2 —2 : 1.4 X 106 erg 8—1 cm This is called the solar constant. AST 203 (Spring 2011) Flux and 1/r2 Closer planets intercept more energy per second mmswplanet's area) Think about the brightness of a lightbulb as you move closer or farther from it. The power is not changing. Luminosity distributed over a sphere. Move further from the Sun, the energy flux (W/m2) decreases as 1/r2 The same total luminosity (4 x 1033 erg/s) is distributed over the surface of both spheres. Flux is L/(4Trr2) A Note About Fluxes - Flux just means energy / unit time / unit area - We've now talked about flux in 2 different contexts - Flux at the surface of a star: — Blackbody, f = 0T4 - Flux received from some distant star: — f = L/(47rr2) ,where 7’ is the distance to the star. - This is the flux that enters into the magnitude equation. AST 203 (Spring 2011) Parallax Apparent magnitudes: measure of how bright the star appears to be from earth. To compare different stars, we need to know the luminosity of the star. This requires that we know the distance to the star. If a star has a luminosity Land is at a distance d, then its observed flux is L* fl : 47rd2 (This assumes that there is nothing in between the star and us that absorbs some of its radiation). AST 203 (Spring 2011) Parallax We can measure the distance to nearby star using simple trigonometric parallax. Where do we observe this in everyday life? Over 6 months, the baseline for the measurements will be 2 AU We define the parallax angle to be 1/2 of the total shift. 1AU tanpzpzT 1AU AST 203 (Spring 2011) Parallax Parallaxes are measured in arcseconds (1/3600 of a degree). We denote the parallax angle in arcseconds as p” Recalling 7r rad : 180°, ,, 180° 60’ 60” 6( ) _ 6(1raid)7T rad 10 T _ 206265 9(rad) so the distance to the star is Note: this is written ,7 different than your d 2 206265 < book to make the 1 AU p units clearer. Distance at which a star has a parallax angle of 1” is the parsec. 1 pC : 206265 AU : 3.09 x 1018 cm : 3.261y We use: 1” d 1190 p AST 203 (Spl ing 2011i Parallax Parallax is the only direct way to measure the distances to the stars. There are other distance indicators built upon the distances that we know from parallax measurements. The Hipparcos satellite measured parallaxes down to 0.001”, finding the distances to over 120,000 stars. AST 203 (Spring 2011) Proper Motion Stars orbit around the galactic center —> causes shift in the position of a star Motion of “j” overtime: proper motion. 3333;“ - Must subtract the proper motion of the _ 33m“???- _ star from the parallax effect. ., , Barnard's star has a proper motion of 10.3” per year. l — AST203(Spring2011) a H m: m Absolute Magnitudes Absolute magnitude: magnitude if object were 10 pc away. A star whose luminosity is Land is a distance d away will have a measured flux of f* and apparent magnitude m L* fl : 47rd2 if instead it were 10 pc away, its flux would be L* _ f0=47rdg d0—10pc and its magnitude would be its absolute magnitude, M AST 203 (Spring 2011) (uonmdmo 5U!|lS!lqnd Ibis) Absolute Magnitudes Putting this together, we have: f0 d2 d — M _ 2. 1 — _ 2. 1 — _— 1 m 5 0g (J: 5 0g d8 5 0g 10 pc We call m — M the distance modulus. Given any two of m , M, or d, we can compute the third. We can estimate M for a star given its observed properties. homework: you will derive a relationship between L and M. AsT 203 (Spring 2011) Absolute Magnitudes Consider a star that has an apparent magnitude of -0.4 and a parallax of 0.3”—what is its absolute magnitude? First, we compute the distance in parsecs, d 177 177 : — —> d : 3.33 pc 1 pc p 0.3” Next we can find the absolute magnitude, d m—M:5log<10pc> d M = m — 5 log : —0.4 — 5 log(0.333) : 2.0 10 pc AST 203 (Spring 2011) Stellar Colors We've been talking about the flux and magnitude over the entire EM spectrum—all wavelengths or frequencies. Usually called the bolometric flux or magntiude. Instruments usually measure the flux over a small wavelength band. A standard set of filters is defined, that have a central wavelength and an associated width. This gives us information about the color of a star. AST 203 (Spring 2011) Stellar Colors U B V 1.0 (emso one "01:20) 0.8 0.6 0.4 Sensitivity function S(/\) 0.2 0.0 l . l . 300 400 500 600 700 Wavelength (nm) Note that these filters have a rather complex shape. AST 203 (Spring 2011) Stellar Colors fB =f|ux through the B filter fv = fluX through the V filter magnitude difference: mB — my 2 2.5log B (usually written as B — V) This is a measure of the color of the star. Stars are good blackbodies, so this determines of the temperature of the star. As T increases, fB/fv increases, so B — V decreases (because the magnitude scale runs backwards). AST 203 (Spring 2011 :1 Stellar Colors Table 9.2. Spectral type, color, and effective temperature.3 Main sequence Giants Spectral type B — V T‘, (K) B — V Te (K) 05 70.45 35,000 i — BO 7031 21,000 — 7 BS *0.17 13,500 — — A0 0.00 9,700 — —- A5 0.16 8,100 — — F0 0.30 7,200 — — F5 0.45 6,500 — — GO 0.57 6,000 0.65 5,400 G5 0.70 5,400 0.84 4,700 KO 084 4,700 1.06 4,100 K5 1.11 4,000 1.40 3.500 M0 1.24 3,300 1,65 2,900 M5 1.61 2,500 — i ‘ Adapted from C. W . Allen. Astrophysical Quantities. h“...— (Shu) AST 203 (Spring 2011) ...
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This note was uploaded on 05/04/2011 for the course AST 203 taught by Professor Simon,m during the Spring '08 term at SUNY Stony Brook.

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