CORNELL ASTRO 290 - Luminosity, Flux and Magnitudes

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Flux and MagnitudesA2290-13 1Luminosity, Flux and MagnitudesRelativity and AstrophysicsLecture 13Terry HerterA2290-13 Flux and Magnitudes 2Outline Blackbody Radiation Flux and Luminosity Inverse square law MagnitudesFlux and MagnitudesA2290-13 2A2290-13 Flux and Magnitudes 3Radiation from objects All objects have internal energy which is manifested by the microscopic motions of particles. There is a continuum of energy levels associated with this motion. If the object is in thermal equilibrium then it can be characterized by a single quantity, it’s temperature. An object in thermal equilibrium emits energy at all wavelengths. resulting in a continuous spectrum We call this thermal radiation.A2290-13 Flux and Magnitudes 4Blackbody Radiation A black object or blackbody absorbs all light which hits it. This blackbody also emits thermal radiation, e.g. photons! Like a glowing poker just out of the fire. The amount of energy emitted (per unit area) depends only on the temperature of the blackbody. In 1900 Max Planck characterized the light coming from a blackbody. The equation that predicts the radiation of a blackbody at different temperatures is known as Planck’s Law. h = Planck’s constant, k = Boltzmann’s constant, c = speed of light This is the power radiated per unit area in the frequency range to + dinto a unit solid angle (d= dsindin spherical coordinates) 1exp1223kThchBW/m2/Hz/srFlux and MagnitudesA2290-13 3A2290-13 Flux and Magnitudes 5Blackbody Spectra6000 K5000 K7000 KIntensityUV VISIBLEIR462The peak shifts with T(Wien’s law).Tpeak/1The area under the curve increase rapidly with T(Stefan-Boltzmann law).4TF = 5.7x10-8Wm-2 K-4R = 12,000 mRsun= 6.96108mL is net radiationNotes1.010265.7101629 A106Neutron Star3.91026~100L (W)6.51070.5 m5800Sun5269.4 m310PersonF (W/m2)peakT (K)ObjectA2290-13 Flux and Magnitudes 6Properties of Blackbodies The peak emission from the blackbody moves to shorter wavelengths as the temperature increases (Wien’s law). Hot objects look blue, cool ones look red The hotter the blackbody the more energy emitted per unit areaat all wavelengths (Stefan-Boltzmann law). Note - bigger objects emit more radiation Except for their surfaces, stars behaves as a blackbodiesTpeak/2900in m and T in K4TF= 5.7x10-8Wm-2 K-4Flux and MagnitudesA2290-13 4A2290-13 Flux and Magnitudes 7Energy Flux and Luminosity The Energy Flux, F, is the power per unit area radiated from an object. The units are energy, area and time. Luminosity is the total energy radiated from star of radius R is given by: So the luminosity, L, is: If stars behave like blackbodies, stars with large luminosities must be very hot and/or very big. W/m2(at all )4TFWatts424 TRLA2290-13 Flux and Magnitudes 8Luminosity and Flux Luminosity, L The total energy radiated from an object per second. Measured in Watts Emitted Flux, F The flow of energy out of a surface.  Measured in Watts/m2Observed flux, f The power per unit area we receive from an object Depends on the distance to the object. Measured in W/m2e.g. fsun= 1 kW/m2 Also called flux or apparent brightness Meaning of Observed Flux Make a sphere of radius, r, around an object (such as the Sun or a light bulb) which is radiating power. All energy radiated from the object must pass through this sphere The size of the sphere does not matter!Flux and MagnitudesA2290-13 5A2290-13 Flux and Magnitudes 9Flux falls off with distancerr rThe flow of energy per square meter passing through a give sphere decreases as the size of the sphere increases. f  1/r2.All energy radiated from the object must pass through each sphere --The size of the sphere does not matter!A2290-13 Flux and Magnitudes 10ABIf bucket B is twice as far away as bucket A, it collects 1/4 as much water.The sprinkler is the star, one of the buckets is the telescope (or your eye), and the water jets are the photons.Flux: Sprinkler and Bucket AnalogyFlux and MagnitudesA2290-13 6A2290-13 Flux and Magnitudes 11Inverse square law The flux, f, of energy through a sphere of radius, r, is given bywhere L is the luminosity of the object Why do we care about flux? The flux is what we measure. We use a telescope (or our eye) and measure a small fraction of the light passing through this sphere.( W/m2 )24 rLfInverse square lawA2290-13 Flux and Magnitudes 12An illuminating example? A 100 W light bulb about 1/5 of power goes into light It’s total power output is always 100 W. It’s apparent brightness to us depends upon how far away it is. For instance at 1 m the flux is:  Flux = 0.08 W/m2[ f = 100 W/(41m)2) W/m2] If we double the distance away from the light bulb, the flux drops by a factor of 4. At 2 m, the flux is 0.02 W/m2Flux and MagnitudesA2290-13 7A2290-13 Flux and Magnitudes 13Observed Flux – Distance Example: A star like the sun has an observed flux of 2.4x10-10W/m2. If the flux of the sun at the Earth is 1 kW/m2, how far away is the star? Now We could also have used ratios rather than compute Lsunfirst. Where we know everything but dstar.fdLsun242211 W/m1000m105.114.34 sunLW10326sunLfLdsun4/21026W/m104.214.34W/103dpc10m10317dstarsunstarstarsunsunsunstarfffLLfddstarsunsunstarffdd A2290-13 Flux and Magnitudes 14Distances to stars: Stellar Parallax As stars get further away, their parallax becomes smaller. Parallax can not be measured to better than ~0.02” from the ground (d< 50 pc). Interferometry is improving on this for selected applications Parallax is measured in arcseconds. Equations are for distances in AU and parsecs (pc), respectivelypsd pdpdstan1 AUdEarthNow6 MonthsLaterp = parallax (angle)d = distancepSuns)arcsec(206265)AU(pd pd1)pc( 1.0 arcsec => 1 pc,0.5 arcsec => 2 pcorFlux and MagnitudesA2290-13 8A2290-13 Flux and Magnitudes 15The closest starsStar Parallax Distance Luminosity(arcsec) (pc) (Lsun=1)Proxima Centauri 0.763 1.31 5x10-5 Centauri A 0.741 1.35 1.45 Centauri B 0.741 1.35 0.4Barnard’s Star 0.522 1.81 4x10-4Wolf 359 0.426 2.35 2x10-5Lalande 21185 0.397 2.52 5x10-3Sirius A 0.377 2.65 23Sirius B 0.377 2.65 2x10-3Parallax


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