r i r j 3. (2) Gm j m i r i (r i r j ) r i r j 3. (3)

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1 N-body problem There s a lot of nterestng physcs related to the nteractons of a few bodes, but there are also many systems that can be well-approxmated by a large number of bodes that nteract exclusvely by gravty (meanng that, e.g., physcal collsons are rare). Ask class: can they thnk of some examples? Stellar systems, e.g., open or globular clusters, or galaxes, are a good example because the dstance between stars s much greater than the sze of the stars, usually. Dark matter partcles n a halo are another example. It s completely hopeless to attempt to solve such a many-body problem exactly. However, we can get a remarkable amount of understandng of some systems f we go to the extreme of havng lots and lots of partcles. That s because one can then make statements about the statstcal average of varous nterestng propertes. That s, although predcton of the movement of an ndvdual star n a cluster s mpossble n the long run, we can say how the cluster as a whole wll evolve. Once agan we see echoes of thermodynamcs: you re not gong to predct the moton of an ndvdual molecule, but you can say how the pressure, temperature, etc. of a system wll evolve wth hgh accuracy f there are enough molecules. To start, let s notce somethng curous about a partcle n a crcular orbt, where we back up to our one-body moton. Ths partcle has an orbtal speed of GM/r, and therefore a knetc energy of K = 1 (GMm/r), where m s the mass of the partcle, M s the mass of 2 the object n the center, and r s the radus of the orbt. The gravtatonal potental energy s W = GMm/r, so W = 2K or W +2K = 0. Ths may seem a specal statement about crcular orbts, but amazngly we can prove a much more general statement, called the vral theorem, that apples to self-gravtatng systems n general. To start, note that the equaton of moton for some partcle out of a collecton of partcles s m r = Gm j m (r r j ). (1) r r j 3 j Now take the dot product wth r. m r r = j Gm j m r (r r j ) r r j 3. (2) Summng over, we get m r r = j Gm j m r (r r j ) r r j 3. (3) Note that ths sum s essentally a sum over a square matrx, by j, that s mssng the dagonal. It s therefore the sum of two trangles n ths matrx, whch may be reexpressed

2 as m r r = 1 r (r r j ) Gm j m r r j 3 j j=1 j 1 r (r r j ) Gm j m. (4) r r j 3 Snce and j are dummy ndces, we can swtch ther names n the second term and combne them to get m r r = 1 j=1 Gm m j (r r j ) (r r j )/ r r j 3 = 1 j=1 Gm m j / r r j (5) = W. Therefore, m r r s the potental energy. Havng establshed that, now consder C p r = where I m r 2 s the moment of nerta. Then =1 m ṙ r = 1 d m r 2 = 1 d 2 dt 2 dt I, (6) dc/dt = 1 2 d2 I/dt 2 = m r r + m ṙ ṙ = m r r + 2 m ṙ 2 /2 = W + 2K, (7) where K m ṙ 2 /2 s the knetc energy. Therefore, we have derved the vral theorem: W + 2K = 1 d 2 I 2 dt. (8) 2 Whew! But what does ths mean, physcally? Frst of all, Ask class: what s the statement of ths theorem for a tme-ndependent system? Tme ndependence means that all tme dervatves are zero, so we would smply have W + 2K = 0. In fact, that s true as long as the moment of nerta I = m r 2 s constant or varyng at a constant rate (so that the second dervatve s zero). For example, for a partcle n a crcular orbt, the dstance from the center of force s a constant, so I s constant and therefore W + 2K = 0, as we found. But what f the system s varyng? At frst glance we re hosed, because we d need to determne d 2 I/dt 2 n an extremely complcated stuaton. However, the key here s to take a tme average. If a system s roughly n statstcal equlbrum, then the tme average I s constant, so that over tme the second dervatve s small and can be neglected. In fact, we can go farther than that. Suppose that the moton of a system s bounded n physcal space and momentum space. Then, over a very long tme compared to oscllaton perods, I goes through maxma and mnma, but ts tme average s some constant. Therefore, for any bounded system, d 2 I /dt 2 tends to zero over a suffcently long perod. Then agan we have W + 2 K = 0, tme-averaged. Ask class: can they thnk of a counterexample that

3 doesn t follow ths? An exploson s an example. If all the partcles are flyng off to nfnty, then the system s not bounded and the tme-average of W + 2 K s not zero. For most systems we care about though, t s. For example, a partcle n an ellptcal orbt has W and K that vary over the orbt, but the tme-average follows W + 2 K = 0. Ths s a profoundly useful theorem. Agan, t doesn t say anythng about a partcular partcle. However, t does relate the physcal sze of a system to ts velocty dsperson and mass, or ts physcal sze to ts mass and total energy. Let s look at some examples. Frst, consder a star cluster. Suppose t has mass M and radus R. Ask class: not botherng wth factors of order unty, what s the approxmate potental energy of the cluster? It s roughly fgm 2 /R, where the factor f depends on the exact mass dstrbuton. Therefore, Ask class: what s the approxmate typcal speed? From the vral theorem, the speed must be v f 1/2 GM/R. We d lke to be able to use ths to estmate the mass of the cluster. Ask class: suppose we are observng a globular cluster (nce and sphercal!). What quanttes can we measure wth bearng on the mass, usng the vral theorem? We can estmate the radus of the cluster, based on ts angular radus and some measure of the dstance to the cluster. We can also, star by star or as a whole, measure the radal component of the speed because of Doppler shfts. However, the transverse components are a mystery because usually a cluster s too far away to see proper motons. Ask class: for a sphercal cluster, what s a reasonable approxmaton that may allow us to move from the lne of sght speed to the total speed? We can assume that there has been enough scramblng of velocty drectons that, on average, the speeds are dstrbuted sotropcally. Suppose we measure the lne of sght speed for a number of stars, and come up wth a velocty dsperson. Snce we measure one component (call t σ v ), we can assume that the total squared three-dmensonal velocty dsperson s 3σ 2 v for an sotropc dstrbuton. But how does ths get us a mass? For that we have to get some estmate of how the stars are dstrbuted. Ths can come from observatons of the lght: we see a projecton of the total lght, from whch we make educated guesses about the three-dmensonal dstrbuton. For example, f the stars are dstrbuted unformly then The vral theorem then gves us W = 3 GM 2 5 R. (9) 2K = W 2( 1 2 M3σ2 v) = 3GM 2 /(5R) M = 5Rσ 2 v/g. The exact coeffcent wll be dfferent from 5 f the mass s dstrbuted non-unformly. However, the pont s that one can wegh a cluster usng the vral theorem. If the stars are movng n a dsk (as n a spral galaxy) then t s more useful to use Kepler s laws to fgure (10)

4 out how much mass s nteror to the orbt of a partcular star, but t s the same dea. The net result of all ths s that all galaxes wth good data show evdence of greater gravtatonal attracton than s accounted for by the vsble stars and gas. In fact, the bgger the collecton of mass, the larger the correcton factor; clusters of galaxes show ths, too. Ths s known as dark matter, and s thought to add up to somethng lke sx tmes as much mass as all the baryons n the unverse. Oddly, globular clusters show no evdence for dark matter. The nature of dark matter s a mystery. It has to be somethng other than baryons, based on constrants from bg bang nucleosynthess. The leadng canddate s that dark matter s some form of elementary partcle that nteracts only by gravty (.e., t doesn t collde or radate). Stll, no partcular canddate has been verfed yet. The vral theorem also allows nsght nto cosmologcal structure formaton. The basc dea behnd structure formaton s that n the early unverse varous processes mprnted fluctuatons n the densty. That s, some regons of the unverse were a lttle denser than others (and some a lttle less dense). The slghtly denser regons contracted gradually under ther self-gravty, and eventually settled nto equlbrum. But how much dd they eventually contract? Consder a cloud of partcles of mass M and ntal radus R. Let the partcles nteract only by gravty. Assume that at the begnnng the partcles are movng very slowly, so that ther knetc energes can be neglected. Ask class: how can we use the vral theorem to estmate the fnal radus of the cloud n equlbrum? Wth purely gravtatonal nteractons, the energy of the system s conserved (e.g., no energy escapes to nfnty as radaton). The orgnal energy of the system s purely potental energy, of magntude GM 2 /R. The total energy, E tot = W + K, s conserved. From the vral theorem, K = W/2, so E tot = W W/2 = W/2. Therefore, W/2 = GM 2 /R. If the equlbrum radus s R eq then W = GM 2 /R eq, so R eq R/2! The exact amount of contracton agan depends on how the stars are dstrbuted, but the radus contracts by approxmately a factor of two, so that the densty goes up by about a factor of 8. In cosmologcal structure formaton theory, now beautfully confrmed by results from the Wlknson Mcrowave Ansotropy Probe (among other experments), the ntal fluctuatons n densty are largest at small scales. That means that, all else beng equal, structure wll form at small scales frst. Snce the unverse s expandng, the average densty of the unverse s gong down wth tme. Therefore, consder a densty enhancement at small spatal scales. It collapses out and forms a system n vral equlbrum at, say, 8 tmes the surroundng densty. Snce ths happens at an early tme, the surroundng densty was relatvely large and thus the system has hgh densty tself. Now consder a densty enhancement at large spatal scales. It also collapses out and forms a system wth 8 tmes the surroundng densty, but by ths later epoch the surroundng densty s less, so the system tself has a lower densty n vral equlbrum. What we therefore expect (and see!) s that lower-mass systems have

5 hgher average densty than hgher-mass systems. For example, a globular cluster mght have 10 5 M n a radus of 10 pc, for an average densty of ρ 20M pc 3. The Mlky Way has a mass of about M (ncludng ts dark matter halo) n a radus of about 10 5 pc, for an average densty of about M pc 3. Galaxy clusters are even less dense. It s thought that larger systems were formed by the herarchcal assembly of smaller systems. That s, the lttle stuff formed clumps, but then those clumps often assembled nto bgger super-clumps, and so on. We now see structure on sze scales out to 100 Mpc, the scale of superclusters. There s one more comment to make, whch wll lead us nto the next class. When we start wth a low knetc energy cloud and let t collapse nto vral equlbrum, t s a volent process. The partcles essentally free fall to the center, where the rapdly changng potental scrambles ther veloctes. Ths s a process called volent relaxaton (I love that term!). Effectvely, t means that a cluster can settle nto somethng close to vral equlbrum n the short tme that t takes to fall. After a couple of bounces, t s vralzed (meanng that ts moment of nerta s changng slowly f at all). You mght thnk that would be the end of t, and that a cluster could reman n ths equlbrum ndefntely. But t s not true, so n the next class we need to thnk about how somethng n vral equlbrum evolves.

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