Line BroadeningInitial questions: What processes can change the profile of an atomic line? Whatabout in the practical case, where we have a finite signal to noise?Spectral lines are not arbitrarily sharp. There are a variety of mechanisms that givethem finite width, and some of those mechanisms contain significant information. We’llconsider a few of these in turn, then have a detailed discussion about how line shapes andprofiles have given information about rapidly rotating accretion disks around black holes.First, Ask class: why can’t a spectral line be arbitrarily sharp? Ultimately, it comesfrom the uncertainty principle in the form ∆E∆t > ¯h/2. If a line were arbitrarily sharp,this would imply perfect knowledge of E, which can’t happen unless the atom spends aninfinite amount of time before decaying into a lower state. If instead the decay time is finite,say τdecay, then the approximate width of the line is ∆E ∼ ¯h/τdecay. This is called naturalbroadening, and represents the limit on how sharp a line can be. If one has an atom in staten, and the spontaneous decay rate to a lower energy state n′is Ann′, then the spontaneousdecay proceeds at a rateγ =Xn′Ann′, (1)Ask class: Is this the only contribution to the decay? No, there are also induced decayprocesses (stimulated emission). These should be added to the spontaneous rates. Theenergy decays at a rate exp(−γt). The energy is proportional to the square of the coefficientof the wave function, so that coefficient decays at a rate exp(−γt/2). The decaying sinusoidthat is obtained for the electric field then gives a line profile of the Lorentz form, as we sawin the semiclassical picture:φ(ν) =γ/4π2(ν − ν0)2+ (γ/4π)2. (2)Ask class: from the above discussion, can they name a state that will have zero breadthbecause it can persist indefinitely? The ground state is stable, so its energy can be definedwith (in principle) arbitrary sharpness. If instead the level n′is itself an excited level, thatenergy level has breadth as well. Then, approximately, the effective width of the transitionis γ = γu+ γl, where γuand γlare respectively the widths of the upper and lower states.Ask class: what’s a way to broaden this line further for a single atom? Collisions willdo it. Effectively, a collision produces an abrupt change in the phase of the wave function.Suppose that collisions occur at random times with an average frequency νcol. Then theresulting profile still looks like a Lorentzian:φ(ν) =Γ/4π2(ν − ν0)2+ (Γ/4π)2, (3)where now Γ = γ + 2νcolincludes contributions from both natural broadening and collisions.Now suppose we have a collection of many atoms, and we are measuring the line profilefrom all of them combined. Ask class: what is another mechanism that will broaden theobserved line? Doppler shifts are one way. Each atom, individually, will emit a line that hasthe natural width plus a collisional width, but its motion towards us or away from us willproduce blueshifts or redshifts, so its line center will be displaced. Many atoms, moving indifferent directions with different speeds, will produce a line blend with significant width.For example, suppose the atoms are thermalized and thus have a Maxwellian distribution ofvelocities with some temperature T . Then if the line center frequency is ν0, the line profileis Gaussian:φ(ν) =1∆νD√πe−(ν−ν0)2/(∆νD)2(4)where ∆νD, the Doppler width, is∆νD=ν0cs2kTma. (5)A similar profile is obtained if one has microturbulence. However, if the Doppler shifts arefrom ordered motion (e.g., orbits), the profile will be different. Fundamentally, one calculatesthe Doppler profile by adding up the Doppler shifts from all the atoms individually. Onecan imagine a situation in which collisions and Doppler shifts are both important. If theDoppler shifts are due to isotropic thermal motion, the resulting line profile is called theVoigt profile, and is a convolution of a Lorentzian and a Gaussian. Note that because aLorentzian dies off like a power law, whereas a Gaussian dies off exponentially, the linewings sufficiently far from the center will always be dominated by the Lorentzian.Let’s examine a couple of examples in which the line profile gives us physical information.Suppose you are observing stars moving in the center of a distant galaxy. Ask class:If there is a supermassive black hole in the center of the galaxy then what, qualitatively,do you expect to see when you focus on a particular spectral line? It depends on whetherthe motion near the black hole is ordered or random. If the motion is ordered, then asone scanned across the central regions one would expect the net velocity (as measured bythe redshift or blueshift of the line) to increase quickly towards the center, then abruptlychange sign when the center was crossed. If the motion is random, then the line wouldhave a width that increased towards the center. Either way, one can define a velocity orvelocity dispersion that indicates the mass of the black hole. In more distant galaxies, othermethods are used to estimate or constrain the mass of the black hole, because one can’tobserve the optical lines of stars with enough spatial resolution.Now consider another example. The inner regions of accretion disks around black holesare hot places, and various processes mean that there are photons of energies reachingup to many keV to tens or even hundreds of keV. When a photon with an energy of 6-7keV or more hits the accretion disk, it can photoionize the inner K shell electrons of iron,which is relatively abundant for a metal and has a high cross section for this effect. Whenan electron drops down into the K shell from the next shell up, it emits a line that, inthe rest frame of the atom, is relatively sharp and has an energy of 6.4 keV. Motion ofthe atoms in an accretion disk can change this sharp rest-frame line into a broader line.Detailed interpretation of this line has given a tremendous amount of information aboutthe properties of accretion disks and strong gravity. Let’s try our hand at it. Suppose thatthe observed line looks like:Ask class: What effects might account for this profile? We’ll need to identify importantparts and interpret them separately to put together the picture. We see that the line is(1) broad, (2) asymmetric, (3) sharply peaked. We also note that the line goes a little bitabove the rest-frame energy, but a lot below. All this can be