Astroparticle Physics

13.2 Motivation for Dark Matter 267[44] had argued that clusters of galaxies would not be gravitationallystable without additional invisible dark matter.Recent observations of high-z supernovae and detailed measurementsof the cosmic microwave background radiationhave now clearly demonstrated that large quantities of darkmatter must exist, which fill up the universe. An argumentfor the existence of invisible dark matter can already be inferredfrom the Keplerian motion of stars in galaxies. Keplerhad formulated his famous laws, based on the precision measurementsof Tycho Brahe. The stability of orbits of planetsin our solar system is obtained from the balance of centrifugaland the attractive gravitational force:circumstantial evidencefor dark mattermv 2r= G mMr 2 (13.1)(m is the mass of the planet, M is the mass of the Sun, r isthe radius of the planet’s orbit assumed to be circular). Theresulting orbital velocity is calculated to bev = √ GM/r . (13.2)The radial dependence of the orbital velocity of v ∼ r −1/2is perfectly verified in our solar system (Fig. 13.2).The rotational curves of stars in galaxies, however, showa completely different pattern (Fig. 13.3). Since one assumesthat the majority of the mass is concentrated at the centerof a galaxy, one would at least expect for somewhat largerdistances a Keplerian-like orbital velocity of v ∼ r −1/2 .Instead,the rotational velocities of stars are almost constanteven for large distances from the galactic center.The flat rotational curves led to the conclusion that thegalactic halo must contain nearly 90% of the mass of thegalaxy. To obtain a constant orbital velocity, the mass ofthe galactic nucleus in (13.1) has to be replaced by the nowdominant mass of invisible matter in the halo. This requirementleads to a radial dependence of the density of this massofϱ ∼ r −2 , (13.3)becausemv 2= G m ∫ r0 ϱ dVrr 2 ∼ G mϱVr 2 ∼ G mr−2 r 3r 2⇒ v 2 = const . (13.4)Fig. 13.2Rotational curves of planets in oursolar system, 1 Astronomical Unit(AU) = distance Earth to Sunorbital velocity [km/s]150100500radius [kpc]NGC 6503halogalacticdiskgas0 10 20Fig. 13.3Rotational curves of the spiralgalaxy NGC 6503. Thecontributions of the galactic disk,the gas, and the halo are separatelyshown