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 + dinto a unit solid angle (d= dsindin spherical coordinates) 1exp1223kThchBW/m2/Hz/srFlux and MagnitudesA2290-13 3A2290-13 Flux and Magnitudes 5Blackbody Spectra6000 K5000 K7000 KIntensityUV VISIBLEIR462The peak shifts with T(Wien’s law).Tpeak/1The area under the curve increase rapidly with T(Stefan-Boltzmann law).4TF = 5.7x10-8Wm-2 K-4R = 12,000 mRsun= 6.96108mL is net radiationNotes1.010265.7101629 A106Neutron Star3.91026~100L (W)6.51070.5 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/2900in 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 )4TFWatts424 TRLA2290-13 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/m2Observed 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 rLfInverse 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/(41m)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.fdLsun242211 W/m1000m105.114.34 sunLW10326sunLfLdsun4/21026W/m104.214.34W/103dpc10m10317dstarsunstarstarsunsunsunstarfffLLfddstarsunsunstarffdd 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 pdpdstan1 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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