15-06 Mass of Galaxies and Galaxy Clusters


  • Estimating the mass of a galaxy ...
    • Rotation curve method  (for spirals; use with care; only inner mass)
       
    • Velocity dispersion method  (for ellipticals)

      Assuming a spherical distribution with radius R of N stars, each with mass m, the mass of the galaxy is M=Nm.
      From the virial theorem
      - 2 ∑i=1,N (1/2 mivi2 ) = U
      - m/N ∑i=1,N  vi2 = U/N
      and assuming random motions
      <v2> = <vr2> + <vθ2> + <vφ2> = 3<vr2>
      1/N ∑i=1,N  vi2 = <v2> = 3<vr2> = 3 σr2 (<vr>=0)

      σr2 = - U/3mN = GM2/(5RmN)   (U = - 3GM2/5R , for constant density)

      =>   M = 5Rσr2/G
      This is the virial mass.
      The virial mass of an elliptical galaxy is usually found to be much larger than its luminous mass.
       

    • Double-galaxy method
      quite difficult...
       
  • Estimating the mass of a galaxy cluster ...
    • Cluster method
      Applying the velocity dispersion method to a galaxy cluster.
      The cluster must be gravitationally bound, otherwise over-estimated.
      The virial mass of a galaxy cluster is usually found to be much larger than its luminous mass.
       
    • Gravitational lensing


      [Two HST images of gravitational lensing.]

      The inferred mass is usually larger than luminous mass.
       
  • X-ray emissions from hot intracluster gas also require a mass similar to the virial one to confine the high-temperature gas; the mass of the gas itself is not sufficient.


    [The Virgo cluster]


    [The Coma cluster]


    [The Abell 2029 cluster in X-rays and optical light; 300 Mpc away from the earth.]