1The Virial Theorem [CO 2.4]• For gravitationally bound systems in equilibrium• Time-averaged kinetic energy = - ½ time-averaged potential energy.E = total energyU = potential energy.K = kinetic energy.E = K + U• Can show from Newton’s 3 laws + law of gravity:•½(d2I/dt2) -2K= U where I = Σmiri2= moment of inertia.• Time average < d2I/dt2> = 0, or at least ~ 0.• Virial theorem Î -2<K> = <U><K> = - ½ <U><E> = <K> + <U> Î<E> = ½ <U>Mass determinations fromabsorption line widths• Virial Theorem 2K = -U<v2> = 3 <vr2>Ε See pp. 959-962, + Sect. 2.4• Applied to nuclei of spirals Î presence of massive black holes• Also often applied to• E galaxies• Galaxy clustersRGMU253−=M32GRMrvirial25σ=[CO 25.14]2Mass determinations fromabsorption line widths• Virial Theorem 2K = -U<v2> = 3 <vr2>Ε See pp. 959-962, + Sect. 2.4• Applied to nuclei of spirals Î presence of massive black holes• Also often applied to• E galaxies• Galaxy clustersRGMU253−=Fourier TransformsE galaxy = K star convolved with Gaussian velocity distribution of stars.K starStarGalaxyRatioGaussian fit: • Convolution turns into multiplication in F.T. space.• F.T. of a Gaussian is a Gaussian.Observed SpectrumGRMrvirial25σ=Faber-Jackson relation: Le~ σ04(Absolute magnitude)−−=133.34/110)(eRReIRIIe= surface brightness at ReLe= luminosity within Re3Mass-Luminosity relationships−−=133.34/110)(eRReIRIFrom Binney & Merrifield, Galactic AstronomyCO give different coefficients???re∝σ01.24Ie-0.82L ∝σ02.65re0.65• Faber-Jackson relation: Le~ σ04• Dn-σ0correlation.• Dn= diameter within which <I> = 20.75µB• Fundamental plane in log Re, <I>e, log σ0space• Re= scale factor in R1/4law• <I>e= mean surface brightness within ReDifferent from Ie!• Intro. to Principle Component Analysis: astro-ph/9905079Distribution of galaxy types• Dense regions (cluster centers) predominantly ellipticals.• Field galaxies predominantly spirals.• On average, roughly even split between E and S.ESS0Log (Projected surface density of galaxies) ÎFraction of population ÎTotal number of galaxiesDressler 19804Schechter Luminosity Functionφ(L)dL = Lαe-L/L*dLφ(M)dM = 10 –0.4(α+1)M e –10dM• The Milky Way is an L*galaxy.0.4(M *-M)[CO 25.36][BM 4.12]moreluminous-22 -20 -18 -16MB-1-3-5-7Log
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