6. Gravity, Dark Matter, and Cosmic Expansion

Transcription

1 A1143: Histoy of the Univese, Autumn Gavity, Dak Matte, and Cosmic Expansion Reading: The eading is sections 4.2 and 4.3, which you should have aleady ead but would be woth evisiting. If you want to get ahead, you could also stat eading chapte 6. Density Density is mass divided by volume. It is usually designated with the Geek lette ρ (ho). The metic unit of density is kg/m 3. The volume of a sphee of adius R is 4 3 πr3. The mass of a sphee of adius R and constant density ρ is theefoe M = 4 3 πr3 ρ. If the density isn t constant (e.g., if the mass is concentated in the cente), what mattes is the aveage density inside the adius R. Measuing mass inside a galaxy In the pevious section, we equated acceleation in a cicula obit (a = v 2 /) to acceleation poduced by gavity (a = GM/ 2 ) to obtain the equation M = v2 G, fom which we can detemine the mass M of an object if we know the speed v and adius of an object that is obiting it in a cicle. What do we do in a galaxy, whee the mass is not all at the cente (like it is in the sola system) but spead out? Newton demonstated the emakable fact that inside a spheical shell of matte thee is no net gavitational foce the pull fom diffeent sides of the shell cancels exactly. If you ae at adius in an extended spheical mass distibution, theefoe, you gavitational acceleation is detemined entiely by the mass inteio to adius ; the gavitational effects of all the moe distant matte cancels out. We can theefoe genealize ou equation to M int () = v2 G, v = speed of obiting object [m/s] = adius of (cicula) obit [m] G = Newton s gavitational constant [m 3 kg 1 s 2 ] M = mass inteio to adius [kg] Stictly speaking, this equation only woks fo a spheical galaxy, but it is not fa off even fo a flattened (e.g., disk-shaped) galaxy. We will theefoe continue to use it, without woying about the small inaccuacies fo flattened galaxies. 1

2 Galaxy otation cuve, simple expectation It is also useful to conside a eaanged fom of this equation: v cic () = A1143: Histoy of the Univese, Autumn 2012 GMint () whee v cic () stands fo the cicula velocity, i.e., the velocity that an object (such as a sta) obiting in a cicle would have at a distance in a mass distibution M int (). By measuing Dopple shifts, we can measue the velocities of stas at diffeent locations inside galaxies. A plot of the cicula velocity in a galaxy against adius (distance fom the galaxy cente) is called a otation cuve. Suppose we have a spheical galaxy of total mass M tot, made up of stas on andomly oiented cicula obits, held togethe by the gavity of those stas. Fo the simplest case, suppose that the aveage density of stas within the galaxy is constant, with a shap edge at adius R gal. What do we expect to measue fo v cic ()? Inside adius R gal, the mass inteio to is M int () = 4 3 πρ3 = ( ) 4 3 πρr3 /R 3 3 = (M tot /R 3 ) 3. i.e., the mass is inceasing in popotion to the enclosed volume, hence to 3. The cicula speed is theefoe v cic () = G (Mtot /R 3 ) 3 = R 2 R 2 = R R. Thus, within the galaxy, we expect the cicula velocity to ise as we go futhe out, v cic (). Suppose we find a few stas o gas clouds at > R gal to measue the cicula velocity. Beyond R gal, the inteio mass is just M tot, so we expect v cic () = 1. Outside the edge of the galaxy, the cicula velocity should dop in popotion to 1/, just like in the sola system. Galaxy otation cuve, moe ealistic expectation In eal galaxies, the density of stas is highest in the middle, and dops off as you go futhe fom the cente. Just as in the simplest case, we expect the otation cuve to ise in the cental egions whee the density is high, then dop in the oute egions once we ae outside most of the mass. The tunove fom ising to falling should be gadual athe than shap, since the density dops of smoothly instead of instantly. A ealistic otation cuve should theefoe look like a smoothed out vesion of the one we figued out fo the constant density, spheical galaxy. 2

3 A1143: Histoy of the Univese, Autumn 2012 Galaxy otation cuve, obsevations In disk galaxies, the stas and gas clouds in the disk ae moving on nealy cicula obits (like planets in the sola system). We can measue Dopple shifts at diffeent locations in the disk to detemine v cic (). Gas clouds can often be detected fa out in the disk, enabling measuements at lage adii whee thee ae vey few stas left. As expected, obseved otation cuves ise in the middle, then begin to tun ove. But instead of falling at lage, obseved galaxy otation cuves stay flat, with constant v cic (). The discovey of flat otation cuves was made by seveal astonomes using diffeent methods, with paticulaly impotant contibutions by the Ameican astonome Vea Rubin. Dak Matte Halos A flat otation cuve implies M int () = v2 cic G. Thus, at lage adii whee the light fom the stas has faded out, the mass of the galaxy is continuing to gow, with M int (). The visible, stella potion of a galaxy sits in a much lage, and moe massive, halo of dak matte, which povides gavity but does not poduce light. A vaiety of obsevations (e.g., motions of satellite galaxies) indicate that the extent of a galaxy s dak matte halos is typically the visible extent of the galaxy, and that the mass of the halo is typically 10 times the mass of the stas. Fo example, the oute adius of Milky Way s stella disk is about 20 kpc, and its dak matte halo extends to about 200 kpc. The mass of the Milky Way s halo is about M, compaed to about M of stas. Dak Matte Dak matte is non-luminous matte that is detected via its gavitational effect on visible matte. (Thee ae attempts to seach fo it in othe ways, but they have so fa poven unsuccessful.) Dak matte was fist discoveed by the Swiss astonome Fitz Zwicky in 1933, based on the motions of galaxies in the Coma galaxy cluste. He agued (coectly) that they wee moving too fast to be held togethe by the gavity of stas alone. Without exta, unseen matte, the cluste would fly apat. Evidence fo dak matte became stonge in the 1970s, when technological advances enabled measuements of galaxy otation cuves at lage adius. These technological impovements included moe sensitive detectos fo visible light and adio telescopes that could measue motions of hydogen gas clouds. In pinciple, the obsevations of flat otation cuves and apid galaxy motions in clustes could be explained by changing the theoy of gavity itself instead of invoking a new fom of matte e.g., by saying that gavitational foces fall slowe than 1/ 2 at lage distances. But it is vey had to constuct any theoy of gavity that explains all of the obsevational evidence fo dak matte. Escape Speed Thow a ball up in the ai. It falls back down. Fie it up with a cannon. It goes highe. Then falls back down. Fie it up faste than 11 km/s (and ignoe ai esistance). It goes foeve, and doesn t fall back. 11 km/s is the escape speed fom the suface of the eath. 3

4 A1143: Histoy of the Univese, Autumn 2012 The geneal fomula fo the escape speed at a distance fom a mass M is v esc() = 2GM = 2v cic (). It is not supising that it is lage than but simila to v cic (). Gavity and expansion Conside a spheical egion of the expanding univese, with adius, lage enough to contain an aveage amount of matte (both luminous and dak). The shell at the edge of this sphee is expanding outwad with speed v = H. Note that I have witten H instead of H 0, because the agument we ae going to make could also be applied at othe times when the Hubble constant is diffeent fom today s value (which is what the subscipt-0 denotes). Gavity is pulling back on the shell, causing it to deceleate at a ate a = GM int() 2. (Ou spheical symmety agument allows us to ignoe the effects of all of the matte outside the shell. This is not totally obvious, but it tuns out to be tue.) Thus, the expansion of the shell should slow down ove time. The ate of deceleation will depend on M int (), and thus on the aveage density of matte in the univese. Citical Density It is natual to ask whethe the shell is moving above o below the escape speed. Let s calculate what would be equied fo it to move exactly at the escape speed: implying v = H = v esc () = H 2 2 = 2GM int() 2GMint () = 2G 4 3 π3 ρ = 8πG 3 ρ2, whee ρ is the aveage density of matte in the univese. We can divide out 2 and eaange this into an equation fo the citical density: ρ cit = citical density H = Hubble constant G = Newton s gavitational constant ρ cit = 3 8πG H2. If the aveage density is ρ > ρ cit, then the shell is moving slowe than v esc (), so it will eventually stop and ecollapse. If the aveage density is ρ < ρ cit, then the shell will expand foeve. If the aveage density is ρ = ρ cit, then the shell will always slow down, but it will neve come to a stop. 4

5 A1143: Histoy of the Univese, Autumn 2012 Note that dopped out of ou equation. If one shell is above the escape speed, then they all ae, and vice vesa. Thus, the aveage density of the univese detemines its fate: expand foeve, o ecollapse to a big cunch. As the univese expands, ρ dops, but H also dops. If the density is above the citical density at one time, it stays above it at all late times, and vice vesa. Afte plugging in H 0 = 70kms 1 Mpc 1 and doing lots of unit convesions one gets ρ cit = kg/m 3. This is equivalent to about 1 hydogen atom pe cubic mete, oughly a tillion-tillion (10 24 ) times less dense than ai. Waning! The aguments above ae coect given the assumptions that went into them. Howeve, we will lean late in the couse about an impotant and supising complication. Rewinding the Cosmic Movie, Again Peviously: Running Hubble flow backwad, assuming galaxies go at constant speed, implies a big bang at a time t 0 = 1/H 0 in the past. Gavity slows expansion = galaxies wee moving faste in the past, big bang occued moe ecently. Fo a univese filled with matte at the citical density, a full calculation gives t 0 = H 0, 33% younge. Fo a nealy empty univese, with ρ ρ cit, gavity has little effect on expansion, so t 0 1/H 0. The waning above will also apply to these conclusions. An impotant geneal conclusion: if gavity is always attactive, and the expanding univese is appoximately homogeneous on lage scales, thee must have been a time in the past when eveything was tightly packed. Hubble flow plus gavity makes a stong case fo a big bang, but we still want diect empiical evidence that it happened. 5